Geometrically Nonlinear Analysis

Contents

1 Introduction

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1Structural Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2

Geometrical Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2Material Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2Contact (Boundary) Nonlinearities . . . . . . . . . . . . . . . . . . . . . 1-4

Solution Procedures of Nonlinear Problems . . . . . . . . . . . . . . . . 1-4Solution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4

Concept of Time Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6 NSTAR: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6

2 Geometrically Nonlinear Analysis

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1Large Displacement Nonlinear Analyses . . . . . . . . . . . . . . . . . . 2-2

Finite Strain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2Large Deflection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

3 Material Models and Constitutive Relations

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1Linear Elastic Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

Isotropic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1Orthotropic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

COSMOSM Advanced Modules 1

Contents

2

Laminated Composite and Failure Criterion for Laminated Compos-ites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3

Laminated Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3Failure Criterion for Laminated Composite Materials . . . . . . . 3-3

Nonlinear Elastic Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5Hyperelastic Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8

Mooney-Rivlin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9Ogden Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12Blatz-Ko Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15

Plasticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-17Huber-von Mises Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-17Drucker-Prager Elastic-Perfectly Plastic Model . . . . . . . . . . 3-20Tresca-Saint Venant Yield Criterion(or the constant maximum shearing stress condition) . . . . . . 3-21Comparison of Tresca and von Mises Criteria for Plasticity . 3-23

Superelastic Models: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-24Nitinol Model (Shape-Memory-Alloy) . . . . . . . . . . . . . . . . . 3-24The Nitinol Model Formulation: . . . . . . . . . . . . . . . . . . . . . . 3-25The Yield Criterion: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-27The Flow Rule: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-28

Creep and Viscoelastic Models . . . . . . . . . . . . . . . . . . . . . . . . 3-30Creep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-30Linear Isotropic Viscoelastic Model . . . . . . . . . . . . . . . . . . . 3-32

Wrinkling Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-35A Bounding Surface Model for Concrete . . . . . . . . . . . . . . . . . 3-37

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-37Damage Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-38The Model Parameters and Feature . . . . . . . . . . . . . . . . . . . . 3-39

User-defined Material Models . . . . . . . . . . . . . . . . . . . . . . . . . 3-42Preparing the NSTAR Executable File . . . . . . . . . . . . . . . . . 3-43Requirements for Windows NT/2000 . . . . . . . . . . . . . . . . . . 3-43Model Definition Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 3-44

COSMOSM Advanced Modules

Part 1 NSTAR / Advanced Dynamics Analysis

Useful FUNCTION Statements to Access Information from Data Base3-48Useful COMMON Statements to Access Information From Data Base3-50 Element Nodal connectivity Common . . . . . . . . . . . . . . . . . 3-54Useful Subroutines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-59

User-Defined Creep Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-60Model Definition Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 3-61Modifying the CREPUM Subroutine . . . . . . . . . . . . . . . . . . . 3-61

Strain Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-64Automatic Determination of Material Properties from Test Data3-67

MPCTYPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-67MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-69Evaluation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-69Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-70

Birth and Death of Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 3-82

4 Gap/Contact Problems

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1Hybrid Technique for Gap/Contact Problems: General Description4-2

Hybrid Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2Gap Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3Contact Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5

GAP Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6Two-Node Gap Element (Node-to-Node Gap). . . . . . . . . . . . . 4-6One-Node Gap Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9

Automatic Generation of Gap Elements . . . . . . . . . . . . . . . . . . 4-12Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12

Contact/Gaps Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15Triangular Sub-Surfaces for Target Surface . . . . . . . . . . . . . 4-15Automatic Soft Springs for Contact Source or Target . . . . . . 4-15


Contents

4

A New Solution Strategy for Initial Interference . . . . . . . . . . 4-15Troubleshooting for Gap/Contact Problems . . . . . . . . . . . . . . . 4-16

5 Numerical Procedures

Static Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1Incremental Control Techniques . . . . . . . . . . . . . . . . . . . . . . . 5-1 Thermal Loading for Displacement/Arc Length Controls . . . . 5-4Iterative Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4Line Search Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8Termination Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9

Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-11Rayleigh Damping Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 5-13Concentrated Dampers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-13Base Motion Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-13Inclusion of Dead Loads in Dynamic Analysis . . . . . . . . . . . 5-14

Adaptive Automatic Stepping Technique . . . . . . . . . . . . . . . . . 5-15Step Size Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15Safe-guard Against Equilibrium Iteration Failures . . . . . . . . 5-15Safe-guard Against Converging to Incorrect Solutions . . . . . 5-16

J-Integral Evaluation for Nonlinear Fracture Mechanics NLFM 5-17Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-18Modification for Temperature . . . . . . . . . . . . . . . . . . . . . . . . 5-19Axisymmetric Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19The Requirements in Selection of the Path . . . . . . . . . . . . . . 5-20Requirements for JI and JII Evaluation . . . . . . . . . . . . . . . . . 5-20Symmetric Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21J-Integral Path Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21

Frequencies and Mode Shapes in a Nonlinear Environment . . . 5-22Buckling Analysis in a Nonlinear Environment . . . . . . . . . . . . 5-23Release of Global Prescribed Displacements . . . . . . . . . . . . . . 5-24



Defining Temperatures Versus Time Relative to a Reference Tempera-ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-25Modified Central Difference Technique for Dynamic Time Integration5-26

Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28

Combination of Force Control and Displacement/Arc-Length Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-30Artificial Bulk Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-31

6 Element Library

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

7 Commands and Examples

Command Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1Elastoplastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2Geometrically Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . 7-5Elastoplastic Large Displacement Analysis . . . . . . . . . . . . . . . . 7-6Nonlinear Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7Linear Dynamic Analysis (Time-History) . . . . . . . . . . . . . . . . . 7-9Analysis Including Temperature Loading . . . . . . . . . . . . . . . . . 7-10Structural Analysis with Temperature-Dependent Material Properties7-11Elastic Creep Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-13Static Analysis Using Displacement Control Technique . . . . . . 7-14Static Analysis Using Arc-Length Control Technique . . . . . . . 7-15Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16

Elastoplastic Nonlinear Analysis Example . . . . . . . . . . . . . . 7-16Large Displacement Nonlinear Analysis Example . . . . . . . . . 7-24


Contents

6

8 Verification Problems

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1


1 Introduction

Introduction

Developing a reliable model capable of predicting the behavior of structural systems represents one of the most difficult tasks to face the analyst. The finite element method provides a convenient vehicle for performing this task due to its versatility and the great advancement in its adaptation to computer use.

However, the success of a finite element analysis depends largely on how accurately the geometry, the material behavior, and the boundary conditions of the actual problem are idealized.

While elements with their geometric characteristics and boundary conditions are used to describe the geometric domain of the problem, material models (constitutive relations) are introduced to capture the material behavior.

All real structures behave nonlinearly in one way or another. Still, in some cases, due to the particular nature of the problem a linear analysis may be adequate. However, in many other situations a linear solution has proven to be catastrophic and a nonlinear analysis becomes a must.

COSMOSM Advanced Modules 1-1

Chapter 1 Introduction

1-2

Structural Nonlinearities

In this section, major sources of structural nonlinearities encountered in practical applications will be presented.

Geometrical Nonlinearities

In nonlinear finite element analysis, a major source of nonlinearities is due to the effect of large displacements on the overall geometric configuration of structures. Structures undergoing large displacements can have significant changes in their geometry due to load-induced deformations which can cause the structure to respond nonlinearly in a stiffening and/or a softening manner. For example, cable-like structures (Figure 1-1a) generally display a stiffening behavior on increasing the applied loads while arches may first experience softening followed by stiffen-ing, a behavior widely-known as the snap-through buckling (Figure 1-1b).

Material Nonlinearities

Another important source of nonlinearities stems from the nonlinear relationship between the stress and strain which has been recognized in several structural behaviors. Several factors can cause the material behavior to be nonlinear. The dependency of the material stress-strain relation on the load history (as in plasticity problems), load duration (as in creep analysis), and temperature (as in thermo-plasticity) are some of these factors. This class of nonlinearities, known as material nonlinearities, can be idealized to simulate such effects which are pertinent to different applications through the use of constitutive relations. Yielding of beam-column connections during earthquakes (Figure 1-2) is one of the applications in which material nonlinearities are plausible.


Part 1 NSTAR / Nonlinear Analysis

Figure 1-1a. Cable-like Structure

Figure 1-1b. Pressure (p) Versus Center Deflection (D) for Shallow Spherical Cap

BRIDGEMODELING

Cable Nodes

Suspension Bridge

Generalized Displacement

Linear

Nonlinear

Gen

eral

ized

For

ce Qualitative Force-Displacement Curve for

Suspension Bridge [10]

Linear

Nonlinear

P

P

D

D



1-4

Figure 1-2. Loading and Unloading of Beam-Column Connection Under Dynamic Loading

Contact (Boundary) Nonlinearities

A special class of nonlinear problems is concerned with the changing nature of the boundary conditions of the structures involved in the analysis during motion. This situation is encountered in the analysis of contact problems. Pounding of structures, gear-tooth contacts, fitting problems, threaded connections, and impact bodies are several examples requiring the evaluation of the contact boundaries. The evaluation of contact boundaries (nodes, lines, or surfaces) can be achieved by using gap (contact) elements between nodes on the adjacent boundaries.

Solution Procedures of Nonlinear Problems

Solution Strategies

For nonlinear problems, the stiffness of the structure, the applied loads, and/or boundary conditions can be affected by the induced displacements. The equilibrium of the structure must be established in the current configuration which is unknown

ε

σ

Time

Force

Beam-ColumnConnection

Applied Loading S tress-S train Curve

Figure 1-3. Pounding of Structures Due to Seismic Motion



a priori. At each equilibrium state along the equilibrium path, the resulting set of simultaneous equations will be nonlinear. Therefore, a direct solution will not be possible and an iterative method will be required.

Several strategies have been devised to perform nonlinear analysis. As opposed to linear problems, it is extremely difficult, if not impossible, to implement one single strategy of general validity for all problems. Very often, the particular problem at hand will force the analyst to try different solutions procedures or to select a certain procedure to succeed in obtaining the correct solution (for example, “Snap-through” buckling problems of frames and shells (Figure 1-4) require deformation-controlled loading strategies such as Displacement and Arc-length based controls rather than Force-controlled loading).

Figure 1-4a. Load-Deflection Curve of William-Toggle Frame where Force Control Strategy Fails

Figure 1-4b. Load-Deflection Curve for Hinged Cylindrical Shell where Only Arc-Length Control Strategy Succeeds

D

PP

D

H

L L

P

w

h P

w

R

L

θ θ

L



1-6

For these reasons, it is imperative that a computer program used for nonlinear analyses should possess several alternative algorithms for tackling wide spectrum of nonlinear applications. Such techniques would lead to increased flexibility and the analyst would have the ability to obtain improved reliability and efficiency for the solution of a particular problem.

Concept of Time Curve

For nonlinear static analysis, the loads are applied in incremental steps through the use of “time” curves. The “time” value represents a pseudo-variable which denotes the intensity of the applied loads at a certain step. While, for nonlinear dynamic analysis and nonlinear static analysis with time-dependent material properties (e.g., creep), “time” represents the real time associated with the loads' application.

The choice of “time” step size depends on several factors such as the level of nonlinearities of the problems and the solution procedure. A computer program should be equipped with an adaptive automatic stepping algorithm to facilitate the analysis and to reduce the solution cost.

NSTAR: An Overview

This brief section is intended to introduce the COSMOSM nonlinear structural analysis module NSTAR and outline its major features for performing nonlinear structural static and dynamic analyses.

The following present some of NSTAR capabilities:

Extensive 1D, 2D and 3D Element Library (Chapter 6)

Geometric Nonlinearities Large displacements (total and updated Lagrangian formulations)

Large strain formulation for rubber-like materials (total Lagrangian formulation)

Large strain formulation for von Mises elastoplastic materials (updated Lagrangian formulation)



Material Models and Constitutive Relations Linear elasticity

Nonlinear elasticity

• Arbitrarily user-defined stress-strain curve

Hyperelasticity

• Mooney-Rivlin model

• Ogden model

• Blatz-Ko

Plasticity

• Huber-von Mises yield criterion with isotropic or kinematic hardening rules

• Tresca-Saint Venant yield criterion with isotropic or kinematic hardening rules

• Drucker-Prager elastic-perfectly plastic model

• Concrete model

Creep and viscoelasticity

• Classical power law for creep

• Exponential creep law

• Linear isotropic viscoelastic model

Failure criterion for laminated composite materials

Wrinkling membrane for fabric materials

Temperature-dependent material properties for thermo-elastic-plastic analysis

User-defined material models

Contact ProblemsGaps, contact lines, and contact surfaces with generalized friction

Numerical ProceduresSolution control techniques

• Force-controlled loading strategy

• Deformation-controlled loading strategies

- Displacement control technique

- Riks arc-length control technique

Equilibrium iterations schemes



1-8

• Regular Newton-Raphson (tangent method)

• Modified Newton-Raphson

• Quasi-Newton BFGS (Broyden-Fletcher-Goldfarb-Shanno) (secant method)

Termination schemes

• Convergence criteria

• Divergence criteria

Line search option to improve convergence

User-controlled solution tolerances and equilibrium iterations interval

Direct time implicit integration techniques

• Newmark-Beta method

• Wilson-Theta method

Damping effects

• Rayleigh damping

• Concentrated dampers

Base motion effects

Restart option

Adaptive automatic stepping algorithm

LoadingsConcentrated loads (forces and moments)

Pressure (with displacement-dependency option)

Thermal

Centrifugal

Gravity

Time curves to scale loading and control the rate of application

Other FeaturesBuckling analysis

• Limit load analysis

• Post-buckling analysis (snap-through, snap-through/snap-back, and multiple snap-through/snap back problems)

Coupled degrees of freedom

Prescribed non-zero displacements associated with time curves.



Initial conditions for dynamic analysis

Reaction force calculations

PostprocessingListing of displacements, strains, stresses, and gap forces.

Listing of extreme values of displacement, strain and stress components

Deformed shape plots at the user-specified steps

Color-filled, colored lines, colored vectorial contour plots on undeformed and deformed shapes for:

• Displacements

• Strains

• Stresses

Animation for:

• Displacements

• Strains

• Stresses

X-Y plots for the response of user-specified nodes during the analysis as a function of time. Responses include:

• Acceleration

• Velocity

• Displacement

• Reaction force

• Stress

X-Y plots for the displacement response of user-specified nodes versus load factor multiplier for post-buckling analysis.

References

1. Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, 1982.

2. Belytschko, T., and Hughes, T. (eds.), Computational Methods for Transient Analysis, North-Holland, Amsterdam, 1983.



3. Bergan, P. G., “Solution Algorithms for Nonlinear Structural Problems,” Comput. Struct., Vol. 12, pp. 497-509, 1980.

4. Chen, W. F., and Saleeb, A. F., Constitutive Equations for Engineering Materials, Vol. 1, Elasticity and Modeling, John Wiley, 1981.

5. Cook, D. R., Malkus, D. S., and Plesha, M. E., “Concept sand Applications of Finite Element Analysis, “Third edition, Wiley, 1989.

6. Grisfield, M. A., Finite Elements and Solution Procedures for Structural Analysis, Vol. I: Linear Analysis, Pineridge Press Limited, U.K., 1986.

7. Hill, R., The Mathematical Theory of Plasticity, Oxford University Press, London, 1950.

8. Kulak, R. F., “Adaptive Contact Elements for Three-Dimensional Explicit Transient Analysis,” Comp. Meth. Appl. Mech. Eng., 72, pp. 125-151, 1989.

9. Kardestuncer, H., “Finite Element Handbook,” McGraw-Hill, 1987.

10. Niazy, A-S.M., “Seismic Performance Evaluation of Suspension Bridges,” Ph.D. dissertation, Civil Eng. Dept., USC, 1991.

11. Owen, D. R. J., and Hinton, E., Finite Elements in Plasticity: Theory and Practice, Pineridge Press, Swansea, U.K., 1980.

12. Parisch, H., “A Consistent Tangent Stiffness Matrix for Three-Dimensional Non-linear Contact Analysis,” Int. J. Num. Meth. Eng., Vol. 28, pp. 1803-1812, 1989.

13. Ramm, E., “Strategies for Tracing the Nonlinear Response near Limit Points,” in Nonlinear Finite Element Analysis in Structural Mechanics, edited by W. Wunderlich, E. Stein, and K. Bathe, Springer-Verlag, Berlin, 1981.

14. Riks, E., “An Incremental Approach to the Solution of Snapping and Buckling Problems,” Int. J. Numer. Meth. Eng., 15:529-551 (1979).

15. Zienkiewicz, O. C., and, Taylor, R. L., The Finite Element Method, Fourth edition, Vol. 2, 1991.


2 Geometrically Nonlinear Analysis

Introduction

In finite element analysis, the overall stiffness of a structure depends on the stiffness contribution of each of its finite elements. Throughout the history of the load application, the structure is displaced from its original position and the nodal coordinates will change causing the elements to deform and to change their spatial orientations.

In small displacement analysis, the displacement-induced deformations of the element and the change in its spatial orientation (rotation) are assumed small enough that the change in its stiffness contribution to the overall structural stiffness can be ignored. On the other hand, in large displacement analysis the displacement-induced deformations can be finite and the local element stiffness will change due to the change in the element geometrical shape (length, area, thickness, or volume). Also, the spatial-orientation change can no longer be infinitesimal so that the transformation of the element local stiffness into global-stiffness contribution will also change. In order to consider the finite change in the geometry of the structures in the analysis, auxiliary strain measures (e.g., Green-Lagrange and Almansi strain tensors) with their conjugate stress tensors (e.g., Second Piola-Kirchhoff and Cauchy stress tensors) must be introduced.


Chapter 2 Geometrically Nonlinear Analysis

2-2

Large Displacement Nonlinear Analyses

The current literature on geometrically nonlinear analysis contain several large displacement formulations for finite element applications. The main differences arise from the simplifications and assumptions imposed on the kinematic relations and the form of stress rate.

The use of the most general large displacement formulation will render “correct” solutions, however, in many cases the use of a more restrictive formulation could be attractive because of its computational efficiency.

Two main approaches have been introduced, namely; the Total Lagrangian (T.L.) formulation and the Updated Lagrangian (U.L.) formulation. These two approaches are generally used with continuum elements (PLANE2D, TRIANG, SOLID, and TETRA4/10). Another approach, used with skeletal elements (TRUSS2D/3D, BEAM2D/3D, and IMPIPE) and shell elements (SHELL3/4, SHELL3T/4T, and SHELL3L/4L), incorporates a co-rotational system of axes attached to the element during motion and use strain measures of infinitesimal displacement analysis.

Finite Strain Analysis

In this analysis, as the structure deflects, the local-ized deformations are large such that the strains are no longer infinitesi-mal. The local ele-ment stiffness will change as a result of the element shape change. Large strain effect is considered by adjusting the ele-ment shape. No assumptions are made on the magnitude of the strains.

The deformation of rubber-like materials is a typical analysis in which finite strains are experienced.

Deformed

Undeformed

Symmetry

p

Circular Rubber Plate Under Displacement- Dependent

Pressure Loading

y

x

y'

L ∆∆∆∆∆

ψ

θ

Deformed Element

x'

Figure 2-1. Large Strain Analysis



Large Deflection Analysis

In this category, the change in the spatial orientation (rotation) of the elements can be finite but the induced strains must remain small. The overall stiffness of the structure will change as a result of the change of the global stiffness contribution of the element due to the change in its spatial orientation. Large deflection effect is taken into consideration by updating the element orientation during the analysis.

Figure 2-2. Large Deflection (Finite Rotation but Small Strains) Analysis

Snap-through buckling analysis is an example where large deflection analysis is required. This type of buckling is characterized by a loss of the stiffness of the structure at a certain loading condition, known as the limit load, at which the structure becomes unstable. To trace the structural behavior beyond the limit load in the postbuckling range, a deformation-controlled loading strategy (Arc-length or Displacement controls) is used (refer to NL_CONTROL (Analysis > NONLINEAR > Solution Control) command). It has to be noted that if a snap-back behavior (see Figure 2-3), in the postbuckling range is experienced, the Arc-length must be used.

θψ

θ ∆

∆∆∆∆∆Cantilever Beam with

End Moment

M

L< 0.4

y

x

y'

L

Deformed Element

x'

Figure 2-3. Multiple Snap-Through/Snap-Back Buckling of Cylindrical Shell

P

w

h P

w

R

L

θ θ

L


Chapter 2 Geometrically Nonlinear Analysis

2-4

References

1. Allen, H. G., and Al-Qarra, H. H., “Geometrically Nonlinear Analysis of Structural Membranes,” Comput. Struct., Vol. 25, No. 6, pp. 871-876, 1987.


3. Biot, M. A., Mechanics of Incremental Deformations, Wiley, N.Y., 1965.

4. Belytschko, T., and Hughes, T. (eds.), Computational Methods for Transient Analysis, North-Holland, Amsterdam, 1983.

5. Chen, W. F., and, Mizunu, E., Nonlinear Analysis in Soil Mechanics, Elsevier, 1990.

6. Cook, D. R., Malkus, D. S., and Plesha, M. E., “Concepts and Applications of Finite Element Analysis,” Third edition, Wiley, 1989.

7. Fujikake, M., Kojima, O., and Fukushima, S., “Analysis of Fabric Tension Structures,” Comput. Struct., Vol. 32, No. 3/4, pp. 537-547, 1989.

8. Nughes, T. J. R., “Nonlinear Finite Element Shell Formulation Accounting for Large Membrane Strains,” Computer Methods in Applied Mechanics and Engineering, Vol. 39, 1983.

9. Kardestuncer, H., “Finite Element Handbook,'' McGraw-Hill, 1987.

10. Malvern, L. E., “Introduction to the Mechanics of a Continuous Medium,” Prentice-Hall, 1969.

11. Oden, J. T., “Finite Element of Nonlinear Continua,” McGraw-Hill, 1972.

12. Timoshenko, S., and Woinowskey, S., “Theory of Plates and Shells,” McGraw-Hill, 1959.

13. Weaver, W., and Gere, J. M., “Matrix Analysis of Framed Structures,” third edition, Van Nostrand Reinhold, 1990.

14. Wempner, G. A., “Mechanics of Solids with Applications to Thin Bodies,” McGraw-Hill, 1973.

15. Vunderlich, W., “Incremental Formulations for Geometrically Nonlinear Problems,” in Formulations and Computational Algorithms in Finite Element Analysis: U.S.- Germany Symposium on Finite Element Method, edited by Bathe, K., Oden, J., and Wunderlich, W., MIT, 1976.



3 Material Models and Constitutive Relations

Introduction

In this chapter, a brief discussion will be presented on the different material models and constitutive relations implemented in NSTAR to simulate the complex behavior of engineering materials. The description is not intended to be detailed but rather illustrative to give the user some guide lines to select the material model best suited for the analysis. For the users interested in further theoretical details, a list of references is provided at the end of the chapter.

Linear Elastic Models

Isotropic

For definitions and usage, refer to Basic System User Guide and COSMOSM Command Reference Manuals.

The linear elastic isotropic material model can be used with the following element groups:

• TRUSS2D & TRUSS3D• BEAM2D & BEAM3D


Chapter 3 Material Models and Constitutive Relations

3-2

• PLANE2D 4/8 nodes (plane stress, plane strain, and axisymmetric)• TRIANG 3/6 nodes (plane stress, plane strain, and axisymmetric)• SOLID 8/20 nodes • TETRA4 and TETRA10• SHELL3, SHELL3T, and SHELL3L • SHELL4, SHELL4T, and SHELL4L • SHELL6 and SHELL6T• SPRING • IMPIPE

The parameters required to describe this model can be associated with temperature curves to perform thermo-elastic analysis.

Orthotropic

For definitions and usage, refer to the Basic System User Guide and COSMOSM Command Reference Manuals.

The linear elastic orthotropic material model can be used with the following element groups:

• PLANE2D 4/8 nodes (plane stress, plane strain, and axisymmetric)• TRIANG 3/6 nodes (plane stress, plane strain, and axisymmetric)• SOLID 8/20 nodes • TETRA4 & TETRA10• SHELL3, SHELL3T, and SHELL3L• SHELL4, SHELL4T, and SHELL4L• SHELL6 and SHELL6T




Laminated Composite and Failure Criterion for Laminated Composites

Laminated Composite

For definitions and usage, refer to Basic System User Guide and COSMOSM Command Reference Manuals.

The linear elastic laminated composite material model can be used with SHELL3L and SHELL4L elements.


Failure Criterion for Laminated Composite Materials

A failure criterion provides the mathematical relation for strength under combined stresses. Unlike conventional isotropic materials where one constant will suffice for failure stress level and location, laminated composite materials require more elaborate methods to establish failure stresses. The strength of a laminated composite can be based on the strength of individual plies within the laminate. In addition, the failure of plies can be successive as the applied load increases. There may be a first ply failure followed by other ply failures until the last ply fails, denoting the ultimate failure of the laminate. Progressive failure description is therefore quite complex for laminated composite structures.

A simpler approach for establishing failure consists of determining the structural integrity which depends on the definition of an allowable stress field. This stress field is usually characterized by a set of allowable stresses in the material principal directions. The table below lists the components of allowable stresses referenced in the failure theories and their material names in COSMOSM.

A failure criterion, the Tsai-Wu failure criteria, which makes use of the allowable stresses input by the user is currently available. This failure criterion is used to calculate a failure index (F.I.) from the computed stresses and user-supplied material strengths. A failure index of 1 denotes the onset of failure, and a value less than 1 denotes no failure.



3-4

Table 3-1. Required Material Strength Components for Failure Criteria of Composites

The failure criterion of Tsai-Wu is applicable to SHELL3L and SHELL4L elements only. All components of material strengths for all layers must be input in order to compute the failure indices.

The failure indices are computed for all layers in each element of the model.

The Tsai-Wu failure criterion (also known as the Tsai-Wu tensor polynomial theory) is commonly used for orthotropic materials with unequal tensile and compressive strengths. The failure index using this theory is computed using the following equation:

where,

The application of Tsai-Wu failure criterion is requested by specifying a value of 7 in option 5 of the EGROUP (Propsets > Element Group) command for SHELL3L and SHELL4L elements.

Symbol DescriptionMaterial

Property Name

X1T Tensile strength in the material

longitudinal directionSIGXT

X1C Compressive strength in the

material longitudinal directionSIGXC

X2T Tensile strength in the material

transverse directionSIGYT

X2C Compressive strength in the

material transverse directionSIGYC

S12In-plane shear strength in the material x-y plane

SIGXY



The program computes the failure indices for all layers in each element and prints them along with the element stresses in the output file.

Three types of failure status are shown:

where σ1 is the normal stress in the 1st material direction.

Nonlinear Elastic Model

For the particular case of stress history as related to proportional loading, where components of stress tensor vary monotonically in constant ratio to each other, the strains can be expressed in terms of the final state of stress in the following form:

where, is the secant material matrix, Es is the secant modulus, and ν is the Poisson's ratio.

To incorporate this model in the analysis, Poisson's ratio NUXY (not needed for TRUSS2D/3D or BEAM2D/3D) should be defined, the MPCTYPE (LoadsBC > FUNCTION CURVE > Material Curve Type) command with the elastic option should be activated, and the stress-strain curve should be defined using the MPC (LoadsBC > FUNCTION CURVE > Material Curve) command.

1. OK failure index < 1

2. FAIL1 failure index ≥ 1and σ1 < X1

T if σ1 > 0

or σ1 > – X1C if σ1 < 0

3. FAIL2 failure index ≥ 1and σ1 ≥ X1

T if σ1 > 0

or σ1 ≤ – X1C if σ1 < 0



3-6

The total strain vector is used to compute the effective strain to get the secant modulus from the user-defined stress-strain curve (using the MPC (LoadsBC > FUNCTION CURVE > Material Curve) command). For the three dimensional case,

The stress-strain curve from the third (compressive) to the first (tensile) quadrants are applicable to this model for two and three dimensional elements with some modifications. A method of interpolation is used to get the secant and tangent material moduli. Defining a ratio R which is a function of the volumetric strain Φ, effective strain, and the Poisson's ratio, R has the following expression:

Es and Et are then computed by using the equation

It is noted that R = 1 represents the uniaxial tensile case and R = -1 is for the compressive case. These two cases are set to be the upper and lower bound such that when R exceeds these two values, the program will push it back to the limit.

✍ Cable (no-compression) type behavior. For the element to behave as a CABLE (non-compression) element, the element has to be associated with a stress-strain curve as shown in Figure 3-2. If the stress-strain curve in the third quadrant is not input, the program assumes that a symmetric behavior in tension and compression exists. It should be mentioned that users may have to specify initial force (r3) and/or initial strain (r4) for the cable-type behavior of the TRUSS2D 3D elements.



✍ Nonlinear SPRING. For the nonlinear SPRING element (Element Group Op. 5 = 1), the MPC represents force-displacement curve for the axial spring (Element Group Op. 1 = 0) and torque-rotation curve for the torsional spring (Element Group OP. 1 = 1).

Figure 3-1. Curved Description Nonlinear Elastic Model

The nonlinear elastic material model can be used with the following element groups:

Figure 3-2. Stress-Strain Curve for Cable-Type Behavior

• TRUSS2D & TRUSS3D

σ 7

σ

σ

σ

σ

ε6 ε7 ε8 ε 9ε

Tension

Compression

ε1 ε2 ε3 ε4

Input Stress-Strain Curve

σ

σ

4

3

2

1

5

(R=-1)

(R=1)

(t)Es (0)Es

ε

σ

σ 9σ 8

6

ε5

Tension

Strain= (0, 0)

Stress

Compression

(ε , σ )1 1 (ε , σ )2 2



3-8

• BEAM2D & BEAM3D• PLANE2D 4/8 nodes (plane stress, plane strain, and axisymmetric)• TRIANG 3/6 nodes (plane stress, plane strain, and axisymmetric)• SOLID 8/20 nodes• TETRA4 and TETRA10• SHELL3T and SHELL4T• SHELL6 and SHELL6T• SPRING

Hyperelastic Models

Incompressible Rubber-Like Materials

Hyperelastic material models can be used for modeling rubber-like materials where solutions involve large deformations. The material is assumed nonlinear elastic, isotropic and incompressible.

The finite element formulation for such materials has numerical difficulties due to incompressibility. Two approaches have been devised to treat the incompressible constraint, namely; the mixed finite element formulation and the penalty finite element formulation. Mixed formulation uses a separate interpolation of a stress variable that is related to the hydrostatic pressure. The penalty function method assembles the additional degrees of freedom into the global stiffness matrix. This method introduces an artificial compressibility and has a formulation in which the displacement degrees of freedom are the only unknowns. Compared to the mixed approach, the penalty finite element has fewer independent variables.

In COSMOSM, the displacement formulation (full or reduced integration) is based on the introduction of compressibility to the strain energy density function. This treatment is identical to the finite element penalty approach in principle. The introduction of the penalty function modifies the strain energy function from incompressible type to the nearly incompressible one. The stability, the convergence, and the numerical results of nonlinear analysis depend on the type of penalty function employed.

A mixed or displacement-pressure (u/p) formulation is also available in COSMOSM. This technique explicitly replaces the pressure computed from the displacement field by a separately interpolated pressure using a general procedure.



A consistent penalty method is used for eliminating the separate pressure degrees of freedom by introducing an artificial compressibility and then for each element, using full integration to form the element stiffness matrix. Static condensation is used to eliminate the pressure degrees of freedom at the element level and therefore the global stiffness matrix contains only displacement degrees of freedom.

All the skills needed for nonlinear analysis apply to the hyperelastic models. The load step, the mesh size and its distribution, the integration rule (e.g., reduced or full),.. etc. require careful considerations. However, in some cases especially when lack of good understanding of the problem exists, nothing but trial and error will prove successful. Higher order elements provide more numerical stability than lower order elements.

Mooney-Rivlin Model

The Mooney-Rivlin strain energy density function is expressed as:

1. Displacement formulation:

(3-1)

(3-2)

(3-3)

where I, II, and III are invariants of the right Cauchy-Green deformation tensor and can be expressed in terms of principal stretch ratios; A, B, C, D, E, and F are Mooney material constants, and

(3-4)

(3-5)

2. Displacement - pressure formulation:

w = + Q



3-10

where

= Strain energy density function due to displacement field

= +

= A(J1 - 3) + B(J2 - 3) + 1/2 K(J3 - 1)2

= C(J1 - 3)(J2 - 3) + D(J1 - 3)2 + E(J2 - 3)2 + F(J1 - 3)3

J1,J2,J3 = Reduced invariants

J1 = I1 I3-1/3

J2 = I2 I3-2/3

J3 = J = I31/2

I1,I2,I3 = Invariants of the right Cauchy-Green deformation tensor.

Q = The additional strain energy density function due to displacement and pressure fields.

=

= The hydrostatic pressure as computed directly from the displacement.

= The hydrostatic pressure as computed from the separately interpolated pressure variables

=

For constant field,

For linear field,

K = bulk modulus = E / [3(1 - 2ν)], E = 6(A + B)

It has to be noted that as the material approach incompressibility, the third invariant, III, approaches unity, while constants Y and K approach infinity. Thus, for values of Poisson's ratio close to 0.5, the last term in w1 remains bounded, and a solution can be obtained. For problems with significant localized hydrostatic stresses, the displacement-pressure formulation is recommended so that “locking” of displacements does not occur.



The material properties for Mooney-Rivlin model are input through the use of the MPROP (Propsets > Material Property) command. Up to six Mooney-Rivlin constants can be input. The input quantities can be:

MOONEY_A, MOONEY_B, MOONEY_C, MOONEY_D, MOONEY_E, and MOONEY_F.

The Mooney-Rivlin material model can be used with the following element groups:

• PLANE2D 4/8 nodes (plane stress, plane strain, and axisymmetric)• TRIANG 3/6 nodes (plane stress, plane strain, and axisymmetric)• SOLID 8/20 nodes • TETRA4 and TETRA10• SHELL3T and SHELL4T• SHELL6 and SHELL6T

The displacement-pressure formulation is available for element groups: PLANE2D 4- to 8-node (plane strain and axisymmetric) and SOLID 8-node.

✍ Notes:

1. Always use the Total Lagrangian Formulation for element groups PLANE2D, TRIANG, SOLID, and TETRA4/10. For SHELL3T/4T elements the formulation is automatically adjusted inside the program when the large displacement option is used.

2. Use the NR (Newton-Raphson) iterative method.

3. If the structure is subjected to pressure loading, use the displacement-dependent loading option.

4. For PLANE2D and SOLID elements, if significant localized hydro-static stresses develop in the structure, use the option for displacement-pressure formulation.

5. Values of Poisson's ratio greater or equal to 0.48 but less than 0.5 are acceptable. When the displacement-pressure formulation is used, Poisson's ratio is recommended in the range from 0.499 to 0.4999.

Element Type No. of Pressure DOF

4-node PLANE2D 1

5- to 8-node PLANE2D 3

8-node SOLID 1



3-12

6. Rubber-like materials usually deform rapidly at low magnitudes of loads thus requiring a slow initial loading.

7. When dealing with rubber-like materials, due to the highly nonlinear behavior of the problem, rapid increase in loading will often result in either negative diagonal terms in the stiffness or divergence during equilibrium iterations. In either case, the restart option can be used to restart the solution from the last converged step after the loading is modified. A more convenient way is to use the option for the automatic-adaptive stepping algorithm.

8. The displacement or the arc-length control may prove to be more effective than force control when negative diagonal terms repeatedly occur under various loading rates.

9. For cases of PLANE2D plane stress option and SHELL3T/4T elements, the analysis is simplified since incompressibility does not result in unbounded terms. The formulation is derived assuming perfect incompressibility (Poisson's ratio of 0.5), and therefore NUXY is neglected. Displacement-pressure formulation is not considered.

10. Constants A and B must be defined such that (A+B) > 0. For more information about how to determine the values of the A and B constants, refer to the work by Kao and Razgunas cited at the end of this chapter.

Ogden ModelThe Ogden strain energy density function, defined as,

(3-6)

where

λi = Principal stretches

αi, µi = Material constants

N = Number of terms in the function

is considered one of the most successful functions in describing the large deformation range of rubber-like materials.

The penalty function used in COSMOSM Ogden model take the form of the one used in Mooney-Rivlin model. The strain energy function actually used is a modified type of the Ogden function:



1. Displacement formulation:

(3-7)

where

J = Ratio of the deformed volume to the undeformed volume

N = Number of terms in the function

G(J) = J2 - 1

4/α = , ν = Poisson ratio

2. Displacement-pressure formulation:

w = + Q

where

L1,L2,L3 = Reduced principal values of right Cauchy-Green deformation tensor.

LI = λi2 I3

-1/3

K =

The 3-term (modified Ogden) models are widely used. Up to 4-term models (N = 4) are available in the COSMOSM nonlinear module.

Besides the material constants mentioned above, Poisson ratio is another input to be defined by the user. For most cases, satisfactory results can be obtained by assign-ing Poisson's ratio from 0.49 to 0.499 for displacement formulation and 0.499 to 0.4999 for u/p (displacement-pressure) formulation. Further, for displacement



3-14

formulation, increasing Poisson's ratio will not have significant effect on numerical results unless considerable volumetric strain is involved. When Poisson's ratio is extremely close to 0.5, it may cause solution termination due to negative diagonal terms in the stiffness matrix or lack of convergence.

Like Mooney-Rivlin Hyperelastic model, the Total Lagrangian Formulation is used for the modified Ogden model.

The material properties for Ogden model are input through the use of the MPROP (Propsets > Material Property) command. The required quantities are:

• MU1, MU2, MU3, MU4• ALPH1, ALPH2, ALPH3, ALPH4• NUXY (not required for plane stress element)

The Ogden model can be used with the following element groups:

• PLANE2D 4/8 nodes (plane stress, plane strain, and axisymmetric)• TRIANG 3/6 nodes (plane stress, plane strain, and axisymmetric)• SOLID 8/20 nodes• TETRA4 & TETRA10• SHELL3T & SHELL4T

The displacement-pressure formulation is available for element groups: PLANE2D 4-to 8-node (plane strain and axisymmetric) and SOLID 8-node.

At present, COSMOSM provides two different incompressible hyperelastic models namely; the Mooney-Rivlin (M-R) model and the Ogden model. The following considerations may be of use to predict which model is more appropriate to incorporate for a particular problem.

• A 2-term M-R model is a special case of the Ogden model. Two-term Ogden function can be transformed to M-R strain function by taking:

α1 = 2

α2 = -2


4-node PLANE2D 1


8-node SOLID 1



µ1 = 2 *MOONEY_A

µ2 = -2 *MOONEY_B

• M-R model compares favorably with experimental data of Rubber-like materials undergoing large deformations. As in the cases of simple tension, pure shear, and equibiaxial extension, the two-term M-R model can fit the experimental data for a finite strain range up to 150% (principal stretch ratio 2.5). Ogden model, however, can describe these deformations for very large range of strain up to 500% or 600% (principal stretches 6 or 7).

• M-R model constants are easier to obtain from experimental tests than Ogden's constants. M-R strain energy function is considered the most widely used constitutive law in the stress analysis of elastomers.

• M-R model may have higher computational efficiency than the Ogden model does since for Ogden model extra calculations are needed for transformation between the global or the user-defined coordinates and the principal directions.

✍ Notes (1) through (9) under the Mooney-Rivlin model are applicable.

Blatz-Ko Model

Compressible Foam-Like Materials

The Blatz-Ko strain energy density function is useful for modeling compressible polyurethane foam type rubbers and can be expressed as:

(3-8)

where

G = Shear modulus under infinitesimal deformations = E/2(1+ν)

E = Young's modulus of elasticity

ν = Poisson's ratio = 0.25

Ik = Invariants of

= Cauchy-Green deformation tensor =

= Lagrangian strain tensor



3-16

= Identify matrix

The above expression, contains only one material constant G. Since ν = 0.25 for the Blatz-Ko model, the only material property which is considered is the Young's modulus. thus,

Similar to other hyperelastic models, this model works with total Lagrangian formulation only.

The Blatz-Ko model is currently supported by the following element groups:

• PLANE2D• SOLID• TRIANG• TETRA4• TETRA10

✍ The selected Blatz-Ko model is a simplified form of the expression obtained by Blatz and Ko (1962) to model the deformation of a highly compressible polyurethane foam rubber. The strain energy was approximated by the following expression:

(3-9)

where

A specific form of this three-parameter family of elastic potential was later proposed in which the following values of the constants α, β, and ν were assumed:

β = 0, ν = 0.25, and α = 0.5



Plasticity Models

Huber-von Mises Model

The yield criterion can be written in the form:

(3-10)

where is the effective stress and σY is the yield stress from uniaxial tests. Von Mises model can be used to describe the behavior of metals.

In using this material model, the following considerations should be noted:

• Small strain plasticity is assumed when small displacement or large displacement with total Lagrangian formulation is used.

• Large strain plasticity is assumed when large displacement with updated Lagrangian formulation is used for certain element groups.

• An associated flow rule assumption is made.

• A linear combination of isotropic and kinematic hardening is assumed, where both the radius and the center of yield surface in deviatoric space can vary with respect to the loading history.

• Pure isotropic hardening. The radius of the yield surface expnads but its center remains fixed in deviatoric space.

• Pure kinematic hardening. The radius of the yield surface remains constant while its center can move in deviatoric space.

The parameter RK defines the proportion of kinematic and isotropic hardenings. The default value of RK is 0.85 (mixed hardening). For pure isotropic and pure kinematic hardening, RK takes the values 0 and 1 respectively.

Figure 3-3. Bilinear Stress-Strain Curve

E

Et

1

1

Strain, ε

Stress, σ

σ y



3-18

Figure 3-4. Curved Description Plastic Model

• A bilinear or multi-linear uniaxial stress-strain curve for plasticity can be input. For bilinear stress-strain curve definition, material parameters SIGYLD and ETAN are input through the use of MPROP (Propsets > Material Property) command. For multi-linear stress-strain curve definition, the MPCTYP (LoadsBC > FUNCTION CURVE > Material Curve Type) (with plasticity option) and MPC (LoadsBC > FUNCTION CURVE > Material Curve) commands are used.

• The SIGYLD and ETAN parameters for bilinear stress-strain curve description can be associated with temperature curves to perform thermoplastic analysis.

• The use of NR (Newton-Raphson) iterative method is recommended.

The Huber-von Mises model can be used with the following element groups:• TRUSS2D & TRUSS3D• BEAM2D & BEAM3D (thermo-plasticity not available)• PLANE2D 4/8 nodes (plane stress, plane strain, and

axisymmetric)*• TRIANG 3/6 nodes (plane stress, plane strain, and

axisymmetric)*• SOLID8/20 nodes *• TETRA4 & TETRA10 *• SHELL3T & SHELL4T (thermo-plasticity not available)*

*Large Strain Plasticity

σ

ε1 ε2ε3 ε4

Strain, ε

y

Stress, σ

ε6ε5



U/P Formulation:

The displacement-pressure formulation is available for element groups: PLANE2D 4- to 8-node (plane strain and axisymmetric) and SOLID 8-node, for large strain plasticity.

Large Strain Analysis:

In the theory of large strain plasticity, a logarithmic strain measure is defined as:

where is the right stretch tensor usually obtained from the right polar

decomposition of the deformation gradient is the rotation tensor). The incremental logarithmic strain is estimated as:

where is the strain-displacement matrix estimated at n+1/2 and is the incremental displacements vector. It is noted that the above form is a second-order approximation to the exact formula.

The stress rate is taken as the Green-Naghdi rate so as to make the constitutive model properly frame-invariant or objective. By transforming the stress rate from the global system to the R-system

the entire constitutive model will be form-identical to the small strain theory.

The large strain plasticity theory in COSMOSM is applied to the von Mises yield criterion, associative flow rule and isotropic or kinematic hardening (bilinear or multilinear). Temperature-dependency of material property is supported by bilinear


4-node PLANE2D 1


8-node SOLID 1



3-20

hardening. The radial-return algorithm is used in the current case. The basic idea is to approximate the normal vector by:

The illustration of the process is shown in Figure 3-5.

The element force vector and stiffness matrices are computed based on the updated Lagrangian formulation. The Cauchy stresses, logarithmic strains and current thickness (shell elements only) are recorded in the output file.

The elasticity in the current case is modeled in hyperelastic form that assumes small elastic strains but allows for arbitrarily large plastic strains. For large strain elasticity problems (rubber-like), you can use hyperelastic material models such as Mooney-Rivlin.

✍ Cauchy (true) stress and logarithmic strain should be used in defining the multi-linear stress-strain curve.

Drucker-Prager Elastic-Perfectly Plastic Model

The yield criterion can be defined as:

(3-11)

where α and k are material constants which are assumed unchanged during the analysis, σm is the mean stress and is the effective stress. α and k are functions of two material parameters φ and c obtained from experiments where φ is the angle of internal friction and c is the material cohesion strength.

Drucker-Prager model can be used to simulate the behavior of granular soil materials such as sand and gravel.

Figure 3-5

σ3σ tr

n+1

σ2

σ1

σ n~

~

σ n+1~



In using this material model, the following considerations should be noted:

• Small strains assumption is made.

• Problems with large displacements can be handled provided that small strains assumption is still valid.

• The use of NR (Newton-Raphson) iterative method is recommended.

• Material parameters φ and c must be bounded in the following ranges:

• For most soil mechanics problems, gravitational acceleration can have significant effect, therefore the ACEL (LoadsBC > STRUCTURAL > GRAVITY > Define Acceleration) command is usually used along with the A_NONLINEAR (Analysis > NONLINEAR > NonL Analysis Options) command in which the loading option G is activated.

The material parameters for the Drucker-Prager model are input through the MPROP (Propsets > Material Property) command. The required inputs include the following:

COHESN = Material cohesion strength

FRCANG = Friction angle

The Drucker-Prager model can be used with the following element groups:

• PLANE2D 4/8 nodes (plane strain, and axisymmetric)• TRIANG 3/6 nodes (plane strain, and axisymmetric)• SOLID 8/20 nodes • TETRA4 & TETRA10

Tresca-Saint Venant Yield Criterion(or the constant maximum shearing stress condition)

This criterion is based on the assumption that in the state of yielding, the maximum shearing stress at all points of a medium is the same, and is equal to half of the yield stress that is obtained from a uniaxial tension test for the given material.

90 ≥ φ ≥ 0 (in degrees)c ≥ 0 (in force/unit area)



3-22

In the three-dimensional case, this is expressed as:

2 | τ1 | = | σ2 - σ3 | ≥ σy

2 | τ2 | = | σ3 - σ1 | ≥ σy

2 | τ3 | = | σ1 - σ2 | ≥ σy

The elastic state is represented by the inequality signs. In the state of yielding there must be equality in one or two of these conditions. In other words, yielding is based on the maximum shearing stress which is equal to half the difference between the maximum and minimum principal stresses. Thus, based on this criterion, the intermediate principal stress does not influence the state of yielding.

Shearing Stress Intensity

The shearing stress intensity is defined by the square root of the second invariant of the stress deviator and can be expressed as:

State of Pure Shear

The state of pure shear is defined as:

σ1 = τ , σ2 = -τ , σ3 = 0 . τmax = τ

For this state, the shearing stress intensity and the maximum shearing stress are equivalent:

Τ = τmax = τ

Using the Tresca conditions the shearing stress at the yield point is obtained to be half of the tensile yield stress:

τy = σ / 2 = 0.5 σy

Based on the von Mises yield criterion the shearing yield stress is equivalent to:



Comparison of Tresca and von Mises Criteria for Plasticity

It has been observed that for polycrystalline materials (ductile metals), von Mises condition of constant shearing stress intensity in the state of yielding agrees somewhat better, in general, with experimental data. There are other cases, however, that the Tresca-Saint Venant conditions appear to be in better agreement with experimental data. Thus, the two methods may be regarded as equally possible formulations of the yield condition.

✍ For the states of uniaxial or equibiaxial stress, the two criteria are equivalent.

✍ At other stress states yielding occurs at lower stress values according to the Tresca conditions; under equal loading conditions, the Tresca criterion predicts larger plastic deformation than von Mises.

✍ Maximum deviation between the two techniques occurs for the state of pure shear. At this stress state, based on the Tresca conditions, yielding occurs at 87% of von Mises stress.



3-24

Superelastic Models:

The term Superelastic is used for a material with the ability to undergo large deformations in loading-unloading cycles without showing permanent deformations.

Nitinol Model (Shape-Memory-Alloy) Shape-memory-alloys (SMA) such as Nitinol present the Superelastic effect. In fact, under loading-unloading cycles, even up to 10-15% strains, the material shows a hysteretic response, a stiff-soft-stiff path for both loading and unloading, and no permanent deformation.

Figure A typical stress-strain response for a Nitinol bar under uniaxial loading conditions.

✍ The material behaves differently in tension and compression)

As it is shown by the response curve, the shape-memory-alloys show a distinctive macroscopic behavior, not present in most traditional materials, which finds its justification in the underlying macro-mechanics.



These alloys present reversible martensitic phase transformations, that is, a solid-solid diffusion-less transformations between a crystallographically more-ordered phase, “austenite”, and a crystallographically less-ordered phase, “martensite”.

The soft portions of the response curve represent the areas where a phase transformation: a conversion of austenite into martensite (loading), and martensite into austenite (unloading) occurs.

For the sake of simplicity, however, we will refer to the soft behavior on the response curve as “plastic”, and to the stiff portions as “elastic”.

According to this definition, the material first behaves elastically until a certain stress level is reached (the initial yield stress in loading). If the loading continues, the material shows an elastoplastic behavior until the plastic strain reaches its ultimate value. From this point onward, the material behaves elastically again for any increases in loading.

For unloading, again the material always starts to unload elastically until the stress is reduced to the initial yield stress in unloading. The material will then unload in an elastoplastic manner until all the accumulated plastic strain (from the loading phase) is lost. And from that point onward, the material will unload elastically until it returns to its original shape (no permanent deformation) and zero stress under zero loads.

The Nitinol Model Formulation:Since Nitinol material is usually used for its ability to undergo finite strains, the large strain theory utilizing logarithmic strains along with the updated Lagrangian formulation is employed for this model.

The constitutive model is, thus, constructed to relate the logarithmic strains & the Kirchhoff stress components. However, ultimately the constitutive matrix and the stress vector are both transformed to present the Cauchy (true) stresses.



3-26

σst1, σf

t1 = Initial & Final yield stress for tensile loading, [SIGT_S1, SIGT_F1]

σst2, σf

t2 = Initial & Final yield stress for tensile unloading, [SIGT_S2, SIGT_F2 ]

σsc1, σf

c1 = Initial & Final yield stress for compressive loading, [SIGC_S1, SIGC_F1]

σsc2, σf

c2 = Initial & Final yield stress for compressive unloading, [SIGC_S2, SIGC_F2]



The exponential flow rule, utilizes additional input constants, βt1, βt2, βc1, βc2 :βt1 = material parameter, measuring the speed of transformation for tensile loading, [BETAT_1]

βt2 = material parameter, measuring the speed of transformation for tensile unloading, [BETAT_2]

βc1 = material parameter, measuring the speed of transformation for compressive loading, [BETAC_1]

βc2 = material parameter, measuring the speed of transformation for compressive unloading, [BETAC_2]

The Yield Criterion:To model the possibility of pressure-dependency of the phase-transformation, a Drucker-Prager-type loading function is used for the yield criterion:

F(τ) = σ + 3α p – Rfi

= 0Where:

σ = effective Stress



3-28

p = mean Stress (or hydrostatic pressure)α = (e2/3) (σs

c1 − σst1) / (σs

c1 + σst1)

Rfi

= [σfi (e2/3 + α)] : i = 1 Loading

= 2 Unloading

The Flow Rule:Through adoption of the logarithmic strain definition, the deviatoric and volumetric components of the strain and stress tensors and their relations can be correctly expressed in a decoupled form.

First, we consider the total plastic & elastic strain vectors to be presented by:

εp = εul ξs (n + α m)

εe = ε − εp

As a result, the Kirchhoff stress vector can be evaluated from:

τ = p m + t

p = K (θ − 3α εul ξs )

t = 2G (e − εul ξs n )

In the above formulations:

εul = scalar parameter representing the maximum material deformation [EUL]

(obtainable by detwinning of the multiple-variant martensite)

ξs = parameter between zero & one, as a measure of the plastic straining

θ = volumetric strain = ε11 + ε22 + ε33

e = deviatoric strain vector

t = deviatoric stress vector

n = norm of the deviatoric stress: t /σ



m = the identity matrix in vector form: {1,1,1,0,0,0}T

K & G = the bulk & Shear elastic moduli: { K = E/[3(1-2ν)], G =E/[2(1+ν)] }

The linear flow rule in the incremental form can be expressed, accordingly:

∆ξs = (1. − ξs) ∆F / (F – Rf1 ) : Loading

∆ξs = ξs ∆F / (F – Rf2) : Unloading

And the exponential flow rule, used when a nonzero βι is defined:

∆ξs = β1 (1. − ξs) ∆F / (F – Rf1) : Loading

∆ξs = β2 ξs ∆F / (F – Rf2) : Unloading

✍ In general, shape-memory-alloys are found to be insensitive to rate-effects. Thus, in the above formulation “time” represents a pseudo variable, and its length does not affect the solution.

✍ All the equations are presented here for tensile loading-unloading, since similar expressions (with compressive property parameters) can be used for the compressive loading-unloading conditions.

✍ The incremental solution algorithm here uses a return-map procedure in evaluation of stresses and the constitutive equations, for a solution step. Accordingly, the solution consists of two parts. Initially, a trial state is computed; then if the trial state violates the flow criterion, an adjustment is made to return the stresses to the flow surface.



3-30

Creep and Viscoelastic Models

Creep

Creep is a time dependent strain produced under a state of constant stress. Creep is observed in most engineering materials especially metals at elevated temperatures, high polymer plastics, concrete, and solid propellant in rocket motors. Since creep involves larger time scale than structural dynamics, its effect can be neglected in dynamic analysis.

Creep curve is a graph between strain versus time. Three different regimes can be distinguished in a creep curve; primary, secondary, and tertiary (see the following figure). Usually primary and secondary regimes are of interest.

Figure 3-6. Creep Curve

An elastic creep analysis is available in NSTAR. Two creep laws based on an “Equation of State” approach are implemented. Each law defines an expression for the uniaxial creep strain in terms of the uniaxial stress and time.

Classical Power Law for Creep (Bailey-Norton law)

(3-12)

ε

t

ε 0

xxxxx

Rt

σ and T are constants

PrimaryRange

Secondary Range

TertiaryRange



where:

T = Temperature (°Kelvin) (= inputting temperature + reference temperature + offset temperature)

CT = A material constant defining the creep temperature-dependency [command MPROP, CREEPTC, value (default = 0) (Propsets > Material Property)]

Exponential Creep Law

(3-13)

Both laws represent primary and secondary creep regimes in one formulae. Tertiary creep regime is not considered. In the above formula, the constants C0 to C6 are creep constants that must be defined by CREEPC and CREEPX through the use of MPROP (Propsets > Material Property) command based on the material creep properties. “t” is the current real (not pseudo) time and σ is the total uniaxial stress at time t.

To extend these laws to multiaxial creep behavior, the following assumptions are made:

• The uniaxial creep law remains valid if the uniaxial creep strain and the uniaxial stress are replaced by their effective values.

• Material is isotropic

• The creep strains are incompressible

For a numerical creep analysis, where cyclic loading may be applied, based on the strain hardening rule, the current creep strain rates are expressed as a function of the current stress and the total creep strain:

(3-14)

where:

= Effective stress at time t

= Total effective creep strain at time t



3-32

= Components of the deviatoric stress tensor at time t

✍ Automatic Selection of step-size based on solution accuracy when Auto-Stepping is used.

Non proportional loadings, in particular, lead to considerable stress variations from one step to next. The check on the creep strain increment is not enough to ensure accuracy of creep strains. Additional Checks are added to limit the creep strain rates using the total effective creep strain and ratio of the effective creep strain increment to the total effective creep strain, based on the creep tolerance that is input.

ORNL (Oak Ridge National Laboratory) auxiliary strain hardening rules are used to extend creep behavior to cyclic loading conditions.

The creep models can be used with the following element groups:

• TRUSS2D & TRUSS3D• PLANE2D 4/8 nodes (plane stress, plane strain, and axisymmetric)• TRIANG 3/6 nodes (plane stress, plane strain, and axisymmetric)• SOLID 8/20 nodes • TETRA4 & TETRA10

✍ Modification for Kelvin temperatures.

The absolute temperature is evaluated from:

T(absolute) = T + Tref + T(offset)

✍ Creep effects are included in evaluation of J-integral.

Linear Isotropic Viscoelastic Model

Elastic materials having the capacity to dissipate the mechanical energy due to viscous effects are characterized as viscoelastic materials. In COSMOSM, a linear isotropic viscoelastic material model is available in the time domain analysis. For multiaxial stress state, the constitutive relation may be defined as:



(3-15)

where and φ are the deviatoric and volumetric strains; G(t-τ) and K(t-τ) are shear and bulk relaxation functions. The relaxation functions can then be represented by the mechanical model, (shown in Figure 3-7) which is usually referred to as a Generalized Maxwell Model having the expressions as the following:

Figure 3-7. Generalized Maxwell Model

(3-16)

(3-17)

where G0 and K0 are instantaneous shear and bulk moduli; gi, ki, τiG, and τi

K are the i-th shear and bulk moduli and corresponding times.

The effect of temperature on the material behavior is introduced through the time-temperature correspondence principle. The mathematical form of the principle is:

(3-18)

εηiη1 η2 η3 ηN

GiG1 G2 G3 GN

G0

Gi= G0 gi

ηi = Gi τ iG



3-34

Linpar

Repar

WLpar

where γ t is the reduced time and γ is the shift function. In COSMOSM, the WLF (Williams-Landel-Ferry) equation is used to approximate the function:

(3-19)

where T0 is the reference temperature which is usually picked as the Glass transition temperature; C1 and C2 are material dependent constants.

The material properties for the viscoelastic model are input through the MPROP (Propsets > Material Property) command. The required parameters include the following:

The viscoelastic model can be used with the following element groups:

• TRUSS2D & TRUSS3D• BEAM2D & BEAM3D• PLANE2D 4/8 nodes (plane stress, plane strain, and axisymmetric)• TRIANG 3/6 nodes (plane stress, plane strain, and axisymmetric)• SOLID 8/20 nodes • TETRA4 & TETRA10• SHELL3T & SHELL4T

Parameter GEOSTAR Symbol Description

ear elastic ameters

EX Elastic modulus

NUXY Poisson’s ratio

GXY (optional) Shear modulus

laxation function ameters

G1, G2, G3, ......, G8 represent g1, g2, ..., g8 in EQ. (3-16)

TAUG1, TAUG2, TAUG3, ......, TAUG8 represent τ1

G, τ2G, ..., τ8

G in EQ. (3-16)

K1, K2, K3, ......, K8 represent k1, k2, ..., k8 in EQ. (3-17)

TAUK1, TAUK2, TAUK3, ......, TAUK8 represent τ1

K, τ2K, ..., τ8

K in EQ. (3-17)

F equation ameters

REFTEMP represents T0 in EQ. (3-19)

VC1 represents C1 in EQ. (3-19)

VC2 represents C2 in EQ. (3-19)



✍ For TRUSS2D/3D elements, only uniaxial stress state is considered. The relaxation functions are reduced to the extension part. The extension relaxation moduli are input through K1, TAUK1,.... . The parameters G1, TAUG2, are not required.

✍ For BEAM2D/3D elements, (1) is applied to the extension direction; G1, TAUG1, ..... are needed only when the torsional degree of freedom is considered.

✍ Creep strain is printed if requested (see the STRAIN_OUT (Analysis > OUTPUT OPTIONS > Set Strain Output) command) which is defined as the difference between the total mechanical strain and the linear strain.

Wrinkling Membrane

The Wrinkling Membrane material is usually used to model fabric tension structures such as covers of indoor tennis courts and swimming pools. The wrinkling membrane used in such structures is very thin and flexible. It has in-plane stiffness but does not have any flexural stiffness. For most of the cases, membrane prestresses are mechanically introduced in the fabric tension structures. The prestress and the geometry give the membrane out-of-plane stiffness.

The shape of the structure is known in most structural analysis applications. However, in the case of fabric tension structures, we first have to determine the shape of the membrane surface. The equilibrium shape depends on the boundary conditions and the prestress in the membrane. The above process is called shape-finding analysis and is usually an iterative process in which the user should try several shapes.

Significant changes in the geometries and stresses might occur when membrane structures are subjected to loads (wind or snow). Moreover, the membrane cannot resist any compressive stresses, therefore, wrinkling occurs.

During the analysis, the strains and stresses in the principal directions are calculated.

1. If ε1 > 0 andε1 > 0 and

ε2 > 0 or ε2 ≤ 0 and -ε2 / ε1 ≤ 0



3-36

The material properties required for this material model are the same as in the Linear Elastic-Isotropic material model.

It is noted that the Wrinkling Membrane model is supported by the plane stress option of the PLANE2D and TRIANG elements and the membrane option of the SHELL3/4T elements only.

Figure 3-8a. Fabric Tension Structure

Figure 3-8b. Principal Stresses in Membrane

where ε1, ε2, and σ1, σ2 are strains and stresses in the principal directions 1 and 2 and ν is the Poisson's ratio, then the wrinkling does not occur. A linear elastic isotropic material matrix is used for the membrane.

2. If ε1 > 0 and ε2 ≤ 0 and -ε2 / ε1 > ν

then wrinkling can occur. The membrane yields the stress state ε1 = E ε1; and ε2 = 0.

3. If ε1 ≤ 0 and ε2 ≤ 0

then biaxial wrinkling occurs and the element is inactive.

σσ

X

Y

Z

12



A Bounding Surface Model for Concrete

Introduction

The concrete model is a three-dimensional, rate-independent model with a bounding surface. The model adopts a scalar representation of the damage related to the strain and stress states of the material. The bounding surface in the stress space shrinks uniformly as the damage due to strain softening and/or tension cracks accumulates. The material parameters depend on the damage level, the hydrostatic pressure, and the distance between the current stress point and the bounding surface.

The bounding surface function is:

where

σij is the normalized stress tensor (with respect to the ultimate compression strength fc'), I1 and j2 are the first stress and the second deviatoric normalized stress invariants respectively, θ is the loading angle, and kmax is the maximum damage coefficient.



3-38

Figure 3-9. Bounding Surface

Damage Coefficient

The damage coefficient represents the damage due to strain hardening or softening. The damage coefficient value is always positive and its magnitude in conjunction with the hydrostatic pressure represents the damage due to compression and tension cracking. For instance, the damage in a uniaxial compression test at the ultimate strength is normalized to be 1.0 and its value is approximately 0.20 for uniaxial tension test. The damage is obtained by integrating the incremental damage coefficient that depends on the plastic strain and the distance from the current stress state and the bounding surface.

where

HP = Plastic shear modulus

FI(I1,θ) is function of I1 and θ and loading conditions

Stress Point

Hydroaxis

σ ΙΙ

σ ΙΙΙ

σ Ι

Projection of Tensile Semiaxis

Bounding Surface of Lowe r Damage Level

Bounding Surface of Highe r Damage Level

rR θ



D = Normalized distance r/R

r = Distance from the projection of the current stress point on the deviatoric plane to the hydroaxis

R = Distance of the bounding surface from the hydroaxis along the deviatoric stress direction

The Model Parameters and Feature

The model is defined by two material parameters which are:

1. FPC = the concrete ultimate strength (fc')

2. EPSU = the ultimate strain (the strain at stress of fc' in the uniaxial compression test, εo)

The low strain elasticity modulus (E), bulk modulus (Kt), and shear modulus

(HP) are set.



3-40

The parameters are temperature independent. Moreover, the model should be used in conjunction with small strain formulation.

The model can be used with the following element groups.

• TRUSS2D & TRUSS3D• PLANE2D & TRIANG (plane stress, plane strain, and axisymmetric)• SOLID (8/20 nodes),

TETRA4 & TETRA10

Due to the strain softening, it is preferable to use the Displacement Control or the Arc-Length Control technique with Newton-Raphson or Modified Newton-Raphson in the solution.

Figure 3-10. Uniaxial Compression Test

Tension Behavior

Under tension stresses, the model behaves as a nonlinear strain hardening material until it reaches the tension strength and starts to behave as a perfectly plastic material. The maximum tensile strength for uniaxial test is considered as:



Strain Softening

In concrete structure under loads, some zones may reach failure before the overall failure of the structure. Therefore, a realistic modeling of the post-failure behavior is of primary importance.



3-42

Stress-Strain Relationship

References

1. Moussa, R. A. and Buykozturk, O., “A Bounding Surface Model for Concrete,” Nuclear Engineering and Design, Vol. 121, pp 113-125, 1990.

2. Chen, E. S., and Buyukozturk, O., “Damage Model for Concrete in Multiaxial Cyclic Stress,” J. Engineering Mechanics, ASCE, 111(6) 1985.

User-defined Material Models

Material behavior is often complex, and the analysts are sometimes faced with problems that require more sophisticated material models than those commonly used. COSMOSM allows the users to define their own material models.

This option can be used with the following element groups:

• TRUSS2D & TRUSS3D• PLANE2D 4/8 nodes (plane stress, plane strain, and axisymmetric)• TRIANG 3/6 nodes (plane stress, plane strain, and axisymmetric)• SOLID 8/20 nodes • TETRA4 & TETRA10• SHELL 3/4/6/3T/4T/6T• BEAM2D & BEAM3D



The user will be required to modify a FORTRAN subroutine named UMODEL in a FORTRAN file named “NSUM.F” to be able to incorporate the new model(s). The following sections describe the procedure for defining these models and linking them to NSTAR. In addition, some useful Function statements, COMMON statements, and subroutines are presented to provide the user with the information needed to define a user's model.

Preparing the NSTAR Executable File

For Windows user’s the required files are stored in a subdirectory called NL User Material Model off the COSMOSM installation directory. These files must be used as explained in the following sections.

Requirements for Windows NT/2000

Microsoft Visual Studio Basic 6.0

Contents of the Program

The NSTAR material model utility files stored in the NL User Material Model subdirectory consist of two sample files, the object library, and the make file which creates the special NSTAR module. These files are as noted below:

1. FORTRAN sample files “Nsum.f” and Nsumc.F” for the user material model and the user creep law, respectively.

2. The Nstar.lib object library for Windows NT/2000 is required to create the special NSTAR executable file.

3. The make file “makex” which should be used to compile, link and generate the NSTAR executable file for Windows NT/2000 platforms.



3-44

Procedures

1. Modify subroutine UMODEL in the FORTRAN file named “Nsum.f” (and / or subroutine CREPUM in file “nsumc.f” for the user creep laws).

2. Modify the file “makex” to specify the proper COSMOSM directory where Nstar.exe is to be generated (rename the current Nstar.exe to a different name to save).

3. For compilation and linking type the command: “nmake makex” in a DOS window (nmake.exe nmakx.err are properties of Microsoft Linker).

The Nstar.exe file should then be copied into the COSMOSM directory. It is recommended that you save (or rename) the original execution file for future use.

Model Definition Procedure

1. Prepare the modified Nstar.exe file after modifying the UMODEL sub-routine and appending user subroutines to the “NSUM.F” file (an example to show the procedure for modifying the UMODEL will be presented in the next section).

2. When using the EGROUP (Propsets > Element Group) command to define an element group, assign a negative integer number in the range [-20, -1] to define the type of material model (i.e., MODEL = [-20, -1].

3. When using MPROP (Propsets > Material Property) command to define a material property set, the user can input up to 20 additional properties (MCij, i = 1, 6 and j = i, 6 and MC66 not included) to be used in the definition of the user model.

Modifying the UMODEL Subroutine

Subroutine UMODEL calls the proper user subroutine(s) based on the material model (MODEL) and the element type (IGTYP). Upon entrance, the STRAIN vector is known. The user must define the STRESS vector and the stress-strain matrix [SS], before returning to the main routine.

In addition, if there are state variables to be saved, they must be placed into common VSTORE, as explained in the Useful COMMON statements section.

In the following, the UMODEL subroutine and its arguments are explained.



SUBROUTINE UMODEL (STRAIN, STRESS, SS, NDIMS, IGTYP, NEL, IPT, NPT, IOUT, MODEL)

IMPLICIT REAL*8 (A-H,O-Z)DIMENSION STRAIN(NDIMS), STRESS(NDIMS), SS(NDIMS,NDIMS)

Variable Description

STRAIN Mechanical strain vector in global Cartesian coordinates (thermal and creep strains are removed)

Green-Lagrange strains (for Total Lagrangian) or Almansi strains (for Updated Lagrangian)

TRUSS2D, TRUSS3D,

BEAM2D or BEAM3D{E11} (Updated Lagrangian only)

PLANE2D or TRIANG {E11,E22,E12,E33}

SOLID, or TETRA4/10 {E11,E22,E33,E12,E23,E13}

SHELL 3/4/6/3T/4T/6T* {E11,E22,E123}

STRESS2nd Piola-Kirchhoff stresses (for Total Lagrangian) or Cauchy stresses (for Updated Lagrangian) in the global Cartesian directions

TRUSS2D/3D {S11}(Updated Lagrangian only)

PLANE2D or TRIANG {S11,S22,S12,S33}

SHELL 3/4/6/3T/4T/6T {E11,E22,E12}

SOLID, or TETRA4/10 {S11,S22,S33,S12,S23,S13}

SHELL 3/4/6/3T/4T/6T* {S11,S22,S12}

SS Strain-Stress Matrix: {STRESS} = [SS] {STRAIN}

NDIMS Number of Strain or Stress Components

TRUSS2D, TRUSS3D

BEAM2D or BEAM3D = 1

PLANE2D or TRIANG = 4

SOLID, TETRA4, or TETRA10 = 6

SHELL 3/4/6/3T/4T/6T = 3IGTYP Element Group Type

PLANE2D or TRIANG (Axisymmetric option) = 0

PLANE2D or TRIANG (Plane-Strain option) = 1

PLANE2D or TRIANG (Plane-Stress option) = 2

SOLID, TETRA4, or TETRA10 = 3

TRUSS2D or TRUSS3D = 4



3-46

*In the case of shells, the above stresses and strains refer to in-plane terms only. To define out-of-plane (transverse) effects, common SHLCUR need to be used (refer to the Useful Common statements section).

ExampleSUBROUTINE UMODEL (STRAIN,STRESS,SS,NDIMS,

IGTYP,NEL,IPT,NPT,IOUT,MODEL)IMPLICIT REAL*8 (A-H,O-Z)DIMENSION STRAIN(NDIMS), STRESS(NDIMS), SS(NDIMS,NDIMS)

MODL = MODEL - 100IGTP = IGTYP + 1GO TO (10,20), MODL

10 GO TO (11,11,11,13,14,15,14), IGTP

C--PLANE2D /TRIANG 11 CALL MOD2D1 (STRAIN,STRESS,SS,NDIMS,

IGTYP,NEL,IPT,NPT,IOUT)RETURN

C--SOLID /TETRA4 /TETRA1013 CALL MOD3D1 (STRAIN,STRESS,SS,NDIMS,


C--TRUSS2D / TRUSS3D / BEAM2D / BEAM3D14 CALL MODTR1 (STRAIN,STRESS,SS,NDIMS,

IGTYP,NEL,IPT,NPT,IOUT)

C--SHELL3 / SHELL415 CALL MODSH1 (STRAIN,STRESS,SS,NDIMS,


20 GO TO (21,21,21,23,24,25,24), IGTP

SHELL 3/4/6/3T/4T/6T = 5BEAM2D or BEAM3D = 6

NEL Element number

IPT Location (Gauss point) number in the element

NPT Total number of Locations (Gauss points) for this element

IOUT Output unit number for printing

MODEL User material model number + 100 [101-120]




C--PLANE2D /TRIANG 21 CALL MOD2D2 (STRAIN,STRESS,SS,NDIMS,


C--SOLID /TETRA4 /TETRA1023 CALL MOD3D2 (STRAIN,STRESS,SS,NDIMS,


C--TRUSS2D / TRUSS3D24 CALL MODTR2 (STRAIN,STRESS,SS,NDIMS,


END

Subroutine to request special treatments and/or additional information for the user model formulation.

This subroutine is provided in the Fortran file NSUM.f, and is called internally by the program per element group.

Based on the element type (IGTYP) and model number (MODEL>101), this subroutine can be altered to activate (or deactivate) flags, to request:

1. Additional storage for the state variables

2. Nodal Coordinate per element

3. Evaluation (Gaussian Point) Location & Jacobian

4. SHELL formulation based on initial-configuration (Lagrangian Strains, 2nd Piola-Kirchhoff stresses)

5. Orthotropic transformation matrix

6. Deformation Gradient Tensor

The activation of each feature, allows for the common statement that is associated with it to become available, during the model formulation.

Optional Common Statements:

IMPLICIT REAL*8 (A-H,O-Z)

COMMON /VSMORE/ MDROW, MOREX, VSMOR(63*10)

COMMON /ELXYZS/ EXYZ0(3,21), EXYZN(3,21), IELXYZ

SUBROUTINE UMINIT (IGTYP,MODEL)



3-48

COMMON RRR,SSS,TTT, WEIGHT, JACOB, XXX,YYY,ZZZ,IPOSIT

COMMON /FMUTYP/ ICONF

COMMON /DIRECS/ ORTC(3,3), T(6,6), NORTH

COMMON /DGRADS/ FMTX(3,3), IDFGRD

Default Setting:

MOREX = 0 -> Additional storage is not required.

IELXYZ = 0 -> Nodal coordinates are not required.

IPOSIT = 0 -> Location information is not required.

ICONF = 1 -> Shell formulation is based on the deformed geometry.

NORTH = 1 -> Orthotropic formulation is activated.

IDFGRD = 1 -> Deformation Gradient Tensor is required.

✍ For more information regarding the above common statements, see the Useful COMMON Statements section.

Useful FUNCTION Statements to Access Information from Data Base

To Get Any Real Constant, Use Function

Example

To assign the 5th real constant in the set for the current element to RX, set:

REAL*8 RX, RCNSTRX = RCNST (5)

REAL*8 FUNCTION RCNST (i)


i real constant number)



To Get Any Material Property, Use Function

Example

To get modulus of elasticity, Poisson's ratio, and density for the current element, set:

REAL*8 EX,NUXY,DENSITY, PROPRT,ETEMP, US1

EX = PROPRT (1,ETEMP)NUXY = PROPRT (3,ETEMP)DENSITY= PROPRT (16,ETEMP)US1 = PROPRT (-36,ETEMP)

Table 3.1. Property ID Table

REAL*8 FUNCTION PROPRT (id, temp)


id property id number (see the table below)

temp temperature (used only if property is temperature-dependent)

id =

1-EX2-ALPX3-NUXY4-SIGYLD5-ETAN6-EY

7-EZ8-NUXZ9-NUYZ10-ALPY11-ALPZ12-GXY

13-GXZ14-GYZ15-VISCM16-DENS17-FRCANG18-COHESN

id = 19-26 -> Hyperelastic Constants

id =

-ij -> User Defined Properties

where i and j are the indices associated with the user's material properties constants MCij (i=1, 6 and j=i, 6 & MC66 not included). For example, id=-36 refer to MC36.



3-50

Useful COMMON Statements to Access Information From Data Base

General Information Common

State Variables Common

Variables to be saved and recovered from one previous step, if any.

REAL*8 UTEMPO, UTEMPN, TIMCUR, TIMINCCOMMON /UVARBL/ UTEMPO, UTEMPN, TIMCUR, TIMINC, ISTEP, IEQT, IREF, ISPR, LGDFM


UTEMPN Average element temperature at the current step

UTEMPO Average element temperature at the last step

TIMCUR Time value at the current step (t + ∆t)

TIMINC Time increment for the current step (∆t)

ISTEP Current step number

IEQT Equilibrium iteration number

IREF = 0 -> Stiffness is to be reformed (otherwise IREF>0)

ISPR > 0 -> During stress printings (otherwise ISPR = 0)

LGDFM = 0 -> Small deflection theory

= 1 -> Large displacement analysis (Updated Lagrangian formulation, U.L.)

= 2 -> Large displacement analysis (Total Lagrangian formulation, T.L.)

REAL*4 VSTORCOMMON /VSTORE/ NEXIS, NDMV, VSTOR (126)


NEXIS > 0 -> previous values are restored.

= 0 -> parameters are initialized to zero (occurs only at the start of solution) (not to be changed).



Example

1. Bring in A1(n1), A2(n2), A3(n3) from last step.

2. Save these variables (updated to the current step by the user) to be used in the next step.

Define the common as:

REAL*8 A1REAL*4 A2INTEGER A3COMMON /VSTORE/ NEXIS,NDMV, A1(n1),A2(n2),A3(n3)

NDMV = (n1*2 + n2 + n3) < or = 126

Upon entrance, A1, A2, A3 represent values from last step. Therefore, They can be used to define material law and stresses for the current step. Before leaving, these parameters must be updated to represent values at the current step and will be saved for use in the next step.

Deformation Gradient Tensor Common

NDMV Length of the variables to be stored (126 Real*4 or 63 Real*8) (not to be changed).

VSTOR Vector of state variables to be saved or recovered (to be changed).

REAL*8 FMTXCOMMON /DGRADS/ FMTX(3,3), IDFGRD


IDFGRD > 0 -> Deformation gradient matrix is required

= 0 -> Deformation gradient matrix is not used (default = 1)

FMTX Deformation Gradient Tensor (only when IDFGRD>0)

T.L. -> Right deformation tensor

U.L. -> Left deformation tensor




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✍ If the deformation gradient tensor is not required, the user can set IDFGRD = 0 to save time.

Orthotropic Directions Transfo rmation Matrices

✍ If the user model is not orthotropic, user can set NORTH = 0 to avoid unnecessary calculations.

REAL*8 ORTC,ORTTCOMMON /DIRECS/ ORTC(3,3), ORTT(6,6), NORTH


NORTH > 0 -> Calculate the orthotropic transformation matrices= 0 -> do not calculate orthotropic matrices (default =1)

ORTC Matrix of the orthotropic direction cosines with respect to the global coordinate system

T.L. -> Global undeformed coordinatesU.L. -> Global deformed coordinates

[Global Stress Tensor] = [ORTC] * [Local Stress Tensor] * [ORTC(transpose)]

[ORTC] = [Tij], i=1,3 & j=1,3

Tij = Cosine angle between jth orthotropic direction and the ith global coordinate

ORTT Transformation matrix between the orthotropic directions and the global coordinate system (when stress or strain are represented by vectors)

{Global Stress Vector} =[ORTT(transpose)] * {Local Stress Vector}

[SS(global)] = Constitutive law in global system =[ORTT(transpose)] * [SS(local)] * [ORTT]

[ORTT] = [TTij], i=1,nc & j=1,nc

nc = number of stress (or strain) components



Additional State Variables Common

Provides additional storage for the state variables in excess of the storage capacity of common VSTORE.

✍ Parameter MOREX must be reset (if non-zero) in subroutine UMINIT, according to the element type and model.

✍ Select m such that: (m-1)*126 < no. of additional variables < m*126

Element Nodal Coordinates Common

Provides information regarding the initial & current positions of the nodes associated with the element

REAL*4 VSMOR

COMMON /VSMORE/ MDROW, MOREX, VSMOR(126*10)


MDROW > 0 -> Previous values are restored.= 0 -> Variables are initialized to zero. (not to be altered)

MOREX = 0 -> Additional storage is not required.= m: 1<m<10 -> m*126 additional variables are needed to be saved &

VSMOR Vector of additional state variables to be saved and recovered.

REAL*8 EXYZ0,EXYZN

COMMON /ELXYZS/ EXYZ0(3,21), EXYZN(3,21),IELXYZ


IELXYZ =1 -> Provide nodal coordinates=0 -> Do not provide nodal coordinates

EXYZ0(i,N) Coordinate i of the Nth node of the element (Initial position)

EXYZN(i,N) Coordinate i of the Nth node of the element (Current position)



3-54

✍ 1< i < 3, represents one of the 3 global directions.

✍ 1< N < Number of nodes per element

✍ Parameter IELXYZ must be reset in subroutine UMINIT, according to the element type and model.

Element Nodal connectivity Common

Evaluation Point Information Common

Provides information regarding the location of the point, inside the element, for which the U-Model subroutine is called.

Element Type Exceptions

COMMON /ELNODS/ NENOD(21), NNODMX, NELCUR


NELCUR Element id numberNNODMX Number of nodes for this elementNENOD Vector of Node numbers

REAL*8 RRR,SSS,TTT,WEIGHT,JACOB,XXX,YYY,ZZZCOMMON /POSITN/ RRR, SSS, TTT, WEIGHT, JACOB, XXX, YYY, ZZZ, IPOSIT


IPOSIT =1 -> Provide information about location=0 -> Do not provide location information

RRR,SSS,TTT Gaussian coordinates for this point [-1 to 1]WEIGHT Multiplier for integrationJACOB Jacobian evaluated at this pointXXX,YYY,ZZZ Global coordinates for this point



PLANE2D and TRIANG

TTT: not used.

ZZZ: Thickness at this point

Associated Volume = Weight *|JACOB| * Thickness

SOLID and Tetrahedral elements

Associated Volume = Weight *|JACOB|

All Shell elements

RRR,SSS: not used.

TTT: Gaussian coordinate in the thickness direction [-.5 to .5]

JACOB: Initial AREA

Associated Volume = Weight * Thickness *AREA

Beam elements

RRR: Gaussian coordinate in the Length direction [0 to 1]

SSS,TTT: Gaussian coordinates on the Cross Section [-.5 to .5]

JACOB: AREA of the Cross Section

XXX,YYY, ZZZ: Coordinates with respect to the element system

Associated Volume = Weight * Length * AREA

Solution Termination Common

To Stop/Restart solution when errors are detected.

Previous-Step Information Common for Beam Elements

COMMON /AUTOFLG/ IAUTSTP, JCHECK, KSTOP


JCHECK = 9 -> Set this parameter =9 when errors are detected. It prompts to Stop the solution, or restart the step with a smaller time increment in case of auto-stepping.



3-56

Additional information that is provided in regards to the previous solution step: Time=t, and can be used in the current solution step: Time=t+dt)

Formulation Type common for Shell Elements

Defines whether the material model formulation is based on the element’s initial state, or the deformed state.

✍ Parameter ICONF must be set in subroutine UMINIT, based on the model formulation for shell elements (IGTYP=5).

✍ Among the available material models for SHELL in NSTAR, only the Hyper-elastic models are formulated based on the initial configuration of the element.

REAL*8 EPST,SIGT,GTAN

COMMON/ BMLAST/ EPST, SIGT, GTAN


EPST Axial Strain at time=tSIGT Axial Stress at time=tGTAN Current Shear Strength for Beam with torsion

= Gxy (GTAN can be changed in the U-MODEL routines for BEAM3D with torsion)

COMMON/FMUTYP/ ICONF


ICONF = 1 -> Formulation is based on the current state of deformation [Eulerian Strains, Cauchy Stresses]= 0 -> Formulation is based on initial configuration at time 0 [Green-Lagrange Strains, 2nd PK stresses]



Right Cauchy-Green Strains Common for Shell Elements

This strain tensor is evaluated from Lagrangian strains, based on the element’s initial configuration.

Previous-step Information Common for Shell Elements

Additional information that is provided in regards to the previous solution step: (Time=t), which can be used in the solution of the current step (Time=t+dt).

REAL*8 C11,C22,C33,C12,C23,C13

COMMON /CGDEFO/ C11, C22, C33, C12, C23, C13


C11,…,...C13

Components of the right Cauchy-Green Tensor

REAL*8 STRNT,STNTH,THICKT,AREAT,AREAN

COMMON /SHLAST/ STRNT(6), STNTH(4), THICKT, AREAT, AREAN


STRNT Mechanical Strains at time t:{E11t, E22t, E12t, E33t, E13t, E23t}

STNTH Thermal Strains at time t:{ETh11, Eth22, Eth12, Eth33}

THICKT Thickness at time tAREAT Area at time tAREAN Area at time t+dt



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Transverse (Out-of-Plane) Strains/Stresses Common for Shell Ele-ments

Additional information that is required to be defined in the U-MODEL routine for the current solution step: time=t+dt.

✍ Here the shearing strains, {GAMA}, are provided by the program. The transverse shear stresses, {TAU}, and the GAMA-TAU matrix, [TAG], must be defined by the user.

✍ d{TAU} and d{GAMA} are used to represent variations of the shearing stresses & strains, at time t+dt.

✍ The normal-to-plane strain component, EPSZ, is used in evaluation of the current thickness. It is initially evaluated based on the assumption that the material is incompressible. However, it can be corrected in the U-Model routine, for compressible materials.

✍ The last parameter, ENRGY, is used only for printout.

REAL*8 GAMA,TAU,TAG,EPSZ,ENRGY

COMMON/SHLCUR/ GAMA(2), TAU(2), TAG(2,2), EPSZ, ENRGY


GAMA Transverse Shearing Strains at time t+dt (known){E13, E23}

TAU Transverse Shearing Stresses at time t+dt (to be defined){S13, S23}

TAG Current Transverse Shearing strain-stress matrix.d{TAU}=[TAG].d{GAMA} (to be defined)

EPSZ Strain normal to the plane at time t+dt (optional).{E33}

ENRGY Energy at time t+dt (optional).{E33}



Useful Subroutines

Subroutine to Form Transformation Matrix Between Vectors of Stress or Strain in Different Coordinate Systems

If [T] relates stress (or strain) tensors in two systems a and b:

[Tensor(a)] = [T] * [Tensor(b)] * [T(transpose)]

Then, [TT] is defined such that

{Vector(a)} = [TT(transpose)] *{Vector(b)}

[TT] is evaluated according to:

ij = refer to stress components in system (a)

mn = refer to stress components in system (b)

✍ Regardless of the size (LNG), the rows and columns of [TT] to be defined depend on the size of the stress vector and the order in which it is defined according to the element type. For example, in the case of PLANE2D elements, only the first four rows and columns of [TT] are evaluated. Rows 1,2, and 4 correspond to the normal components in the x, y, and z directions, and row 3 represents the shear term in the xy-plane.

REAL*8 T(3,3), TT(LNG,LNG)CALL SYSTRF (TT,T,IGTYP,LNG)



3-60

Subroutine to Get [TC] = [TA] (transpose) * [TB]

NA1, NA2 = number of rows, columns in [TA]

NB1, NB2 = number of rows, columns in [TB]

NC1, NC2 = number of rows, columns in [TC]

✍ Since the rows and columns may not be equal for the three matrices, minimums of (NA2, NC1), (NA1, NB1), and (NB2,NC2) are used in matrix multiplication.

User-Defined Creep Laws

This feature allows the user to create his/her own creep law and implement it into NSTAR in a procedure similar to the one used for user material models. Accordingly, a library of NSTAR object files will be provided to the customer plus directions as to how to code his/her own creep law and link with the NSTAR library. The user creep law(s) can be used to model creep behavior in conjunction with most available or user defined material models in NSTAR.

A creep law gives an expression for the creep behavior in an uniaxial environment; In the multi-axial case, the uniaxial stresses or strains are replaced by their effective (or equivalent) values in the creep formulation. Here, the user is only required to provide the uniaxial expression that yields the uniaxial creep strain rate; the extension of the model to the multi-axial case (based on the element type), the considerations for stress reversals, and the accumulation and storage of pertinent data is done internally by NSTAR.

User creep laws can be used with the following element groups:

• TRUSS2D & TRUSS3D

• PLANE2D 4/8 nodes (plane-stress, plane-strain, axisymmetric)

• TRIANG 3/6 nodes (plane-stress, plane-strain, axisymmetric)

• SOLID 8/20 nodes

• TETRA4 & TETRA10

REAL*8 TA(NA1,NA2), TB(NB1,NB2), TC(NC1,NC2)CALL MULATB (TA, TB, TC, NA1, NA2, NB1, NB2, NC1, NC2)



The user will be required to modify a FORTRAN subroutine named CREPUM in a FORTRAN file named 'NSUMC.F' to be able to incorporate the new creep law(s). The linking procedure is already explained in the section regarding the user-defined material models (Replace 'NSUM.F' by 'NSUMC.F' or just add 'NSUMC.F' to the list of files to be used). The following sections define the procedure for defining user creep laws.

Model Definition Procedure

1. Prepare the modified NSTAR.EXE file after the CREPUM subroutine and appending user subroutines to the 'NSUMC.F' file are completed (an example to show the procedure for modifying the CREPUM routine will be presented in the next section).

2. When using the EGROUP command to define an element group, assign a negative integer number (-1,-2,...,-n) for the creep option to define the type of user creep law.

3. When using MPROP command to define a material property set, the user can input up to 21 creep properties (Mcij, i=1,6,j=i,6) to be used in the definition of the user creep law. If the user creep law is defined in conjunction with a user material model, the above constants can be used to specify properties for both the material model and the creep law (for example, the user can assign MC11, MC12, and MC13 to represent properties for the user model, while MC22, MC23, MC24, MC25 are to serve as creep properties.

Modifying the CREPUM Subroutine

Subroutine CREPUM calls the proper user subroutine(s) based on the creep law (independent of element type). Upon entrance, current effective creep strain, current effective stress, and current temperature is known. The user must define the current effective creep strain rate before returning to the main routine.

✍ Creep behavior under stress reversals is based on the ORNL auxiliary hardening rules for all creep laws.

✍ The user does not need to be concerned with storage of data, since at each time step, the current creep strains and the current creep origins (for stress reversals) are stored and recovered internally by NSTAR.



3-62

✍ Function statements RCNST & PROPRT, as explained in the user material model section, can also be used for the user creep model to access information from data base.

✍ Among the common blocks defined for user material model, only common block UVARBL can be used in the user creep subroutine to access information such as current time, current time increment, element temperature, and etc.

✍ The customer is responsible to have the proper Compiler and linker (See section on user-defined material models).

In the following, the CREPUM subroutine and it's arguments are explained. In addition, an ex-ample of a power creep law is given.

SUBROUTINE CREPUM(EDOT,EHBAR,SIGBAR,TEMP,TREF,TOFSET,NCTYPU,NEL,IPT,IOUT)



EDOT Effective creep strain rate (to be defined)

EHBAR Effective Creep Strain

SIGBAR Effective Stress

TEMP Current Temperature = Temp(t+dt/2)

TREF Reference temperature

TOFSET Offset units for evaluation of absolute temperature

NCTYPU id number for the user creep law = -1,-2,...

NEL Element number

IPT Location (Gauss point) number in the element

IOUT Output unit number for printing



Example

NCTYPE=-NCTYPUGO TO (100,200,900,900,900) NCTYPEC C** NCTYPE=1 -> User creep law type 1C100 CONTINUECALL CREEP1 (EDOT,EHBAR,SIGBAR,TEMP,TREF,TOF-SET,NEL,IPT,IOUT)RETURNCCC** NCTYPE=2 -> User creep law type 2C200 CONTINUECALL CREEP2 (EDOT,EHBAR,SIGBAR,TEMP,TREF,TOF-SET,NEL,IPT,IOUT)RETURNCCC** Non-existent Creep Models:C900 CONTINUERETURNCEND

Example

Define Effective Creep Strain rate for a power law given by:

(1/sec)

Where effective stress (SIGBAR) is in ksi, and T is Temperature in Fahrenheit. Lets assume that the creep constants are input by User constants:

MPROP, , MC11, 4.64E-8MPROP, , MC12, 12.5MPROP, , MC13, 53712

And,



3-64

TOFSET, 460.TREF, 70.

(Note that the above law is equivalent to the NSTAR classical power law with C2 = 1.0)

Coding

SUBROUTINE CREEP1(EDOT,EHBAR,SIGBAR,TEMP,TREF,TOFSET,NEL,IPT,IOUT)


C0 = PROPRT(-11,TEMP)C1 = PROPRT(-12,TEMP)CT = PROPRT(-13,TEMP)

TKLVN = TEMP + TREF + TOFSET

EDOT = C0 * (SIGBAR)**C1 * DEXP(-CT/TKLVN)

RETURNEND

Strain Output

Five types of strain output are available in NSTAR namely,

1. Total strain

2. Thermal strain

3. Creep strain

4. Plastic strain

5. Principal strain



For Small Displacement formulation (Element Group Op. 6 = 0) for all structural elements (trusses, beams, pipes, and shells) which are not associated with hyperelastic or large strain plastic material models, the infinitesimal strains are output. For large strain plasticity model, the logarithmic strains are output. Otherwise, the Unit Extensions are adapted, which are defined as follows:

1. eR: Extension ratio with respect to the original length in the direction XR (R = 1, 2, or 3 to denote the direction).

where

λR = Stretch ratio in the direction XR

= Lagrangian strain tensor

2. g12, g23, g13: Change of angle between two axes originally in the XR and XS directions

θRs = θR + θs

= π/2 - ϑRs

= sin-1 (cos ϑRs)

Figure 3-11

dL1

dl1

X 2

X 1

Figure 3-12

2

1

1 2

dL1

dL 2

dl 2

θ

θ

dl 1ϑ

X 2

X 1



3-66

3. Logarithmic strain :

Principal strains are defined as follows:

1. Small displacement or large strain plasticity formulation:

where

= infinitesimal strain tensor or logarithmic strain tensor

(large strain plasticity)

e = principal strain

δij = 1 if i = j and δij = 0 otherwise

2. Large displacement formulation (excluding large strain plasticity):

e = λ -1

where

= Left and Right Cauchy-Green strain tensors

λ = Principal stretch ratio



Automatic Determination of Material Properties from Test Data

NSTAR offers automated facilities (curve-fitting) for the analyst to determine material constants for various models, namely, Mooney-Rivlin, Ogden, and viscoelasticity. Two commands, MPCTYPE (LoadsBC > FUNCTION CURVE > Material Curve Type) and MPC (LoadsBC > FUNCTION CURVE > Material Curve), are required in the data entry level. The commands and the corresponding options are explained below:

MPCTYPE

Geo Panel: LoadsBC > FUNCTION CURVE > Material Curve Type

For Mooney-Rivlin and Ogden Models


type type of material property curve

= 2 Mooney-Rivlin

= 3 Ogden

icode Decimal code for experiment type, IJK

I = 1 Uniaxial test is present

J = 1 Plane strain test is present

K = 1 Equibiaxial test is present

number1 Number of terms for approximation

= 2(linear), or 5(quadratic), or 6(cubic) for Mooney-Rivlin model

= 1 to 4 for Ogden model

value1, value2, etc.:

Minimum and maximum values of ALPH(1),...



3-68

✍ Notes:

1. An experiment can be used alone or combined with the other experimnts. For example, IJK = 101 means both uniaxial and equibiaxial test data are present.

2. To obtain a set of accurate material constants for Mooney-Rivlin model, both uniaxial and plane strain tests should be employed. Equibiaxial test data can also be collected.

3. For Mooney-Rivlin model, the singular use of plane strain data is not acceptable because of the linear dependency of the material constants in the plane strain solution.

4. Number of terms might be increased inside the program. The preset criteria are listed below for Ogden model:

with uniaxial and/or plane strain only: - strain > 125% ---> 2-term approximation

with equibiaxial added:- strain > 125% ---> 3-term approximation - strain > 700% ---> 4-term approximation

5. It is suggested that for the Mooney-Rivlin model the analyst use 5-term approximation when strain > 125% and 6-term approximation when strain > 600%.

6. Value1 and the rest are only for Ogden model. For 1-term approximation, value1 and value2 are required. Likewise are 2-term, etc. Default values will be used if they are not input.

For Viscoelastic Model


type type of material property curve

= 4

icode Decimal code for relaxation type, IJK

I = 1 Shear relaxation function is present

J = 1 Bulk relaxation function is present

K is not used (= 0)

number1 Number of terms for shear relaxation approximation

= 1 to 8

number2 Number of terms for bulk relaxation approximation

= 1 to 8



✍ Nonzero value of number1 is input only when shear relaxation is present. Likewise is for number2.

MPC

Geo Panel: LoadsBC > FUNCTION CURVE > Material Curve

For Mooney-Rivlin and Ogden models, stress is defined as the nominal stress, i.e., force/original area (> 0) and strain is defined as the stretch ratio, i.e., final length/original length (> 0). For viscoelastic model, the definition of stress versus strain is replaced by relaxation function versus time.

✍ Multiple stress-strain curves can be input under the same material set number. The input sequence is: uniaxial, plane strain, and equibiaxial.

✍ The relaxation function must be in a descending order starting with a non-zero time.

✍ The instantaneous shear and bulk moduli are calculated in the program by using properties EX and NUXY.

Evaluation Process

After the experimental data are input, the nonlinear program mathematically fits the data. There are a few criteria preset inside the program for the evaluation process. They are listed below:

1. Mooney-Rivlin constants A+B must be greater than zero.

2. Summation of Ogden constants ALPH(I) * MU(I) must be greater than zero.

3. Within certain range of ALPH(I), the program should be able to find a set of ALPH(I) and MU(I) which minimize the error of stresses between the experimental and estimated values.



3-70

4. The ith relaxation modulus G(I) or K(I) should contain a positive sign.

5. Summation of shear relaxation modulus G(I) and summation of bulk relaxation modulus K(I) should be less than the instantaneous moduli.

In the output file the analyst will see the comparison of the experimental data versus estimated ones as well as the error of stresses or moduli. The error is defined as below:

where S(i) is the measured stress or modulus at the ith given stress ratio or time and Se(i) is the estimated one from the formula. The program also creates an ASCII file named problem-name.PLT with the same format as the user-created file. The analyst can use commands ACTXYUSR (Display > XY PLOTS > Activate User Plot), etc. to plot the input data versus the estimated one for verification. Finally the program creates a set of necessary material constants for the material model chosen by the analyst.

It is important that the estimated material parameters fit the test data within the range of strain appearing in the actual nonlinear analysis. Therefore, it is suggested that the analyst always review the comparison either from the output file or the user-created file before the actual nonlinear analysis is performed.

Examples

Some examples are shown below to illustrate the curve-fitting procedure:

Mooney-Rivlin Model

1. Input commands:

Geo Panel: Propsets > Element Group (EGROUP)

EGROUP,1,PLANE2D,0,2,0,0,3,2,0, C* uniaxial, plane strain, and equibiaxial tests presentC* 6-term approximation



Geo Panel: LoadsBC > FUNCTION CURVE > Material Curve Type (MPCTYPE)

MPCTYPE,1,2,111,6, C* uniaxial test data

Geo Panel: LoadsBC > FUNCTION CURVE > Material Curve (MPC)

MPC,1,0,1,....C* plane strain test dataMPC,1,0,23,....C* equibiaxial test dataMPC,1,0,36,....

2. Results in output file:

Curve Fitting for Mooney-Rivlin Material Constants:

Material Property Set 1

6-term approximation

Term Number Mooney-Rivlin Constant

1 0.168951

2 0.007546

3 -0.001034

4 -0.000094

5 -0.000001

6 0.000038

PrincipalStretch Ratio

Stress (Data in Uniaxial)

Stress (Theory)

0.1031E+01 0.1640E-01 0.3181E-01

..... ..... .....

0.7375E+01 0.5303E+01 0.5235E+01



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Stress Error = 0.2949E+00

3. Graphical display from user file:

Figure 3-13. Comparison of Results for Uniaxial Test


Stress (Data in Plane Strain)

Stress(Theory)

0.1063E+01 0.4760E-01 0.8135E-01

..... ..... .....

0.4952E+01 0.1856E+01 0.1793E+01


Stress (Data in Equibiaxial)

Stress(Theory)

0.1033E+01 0.8060E-01 0.6473E-01

..... ..... .....

0.4429E+01 0.2465E+01 0.2465E+01



Figure 3-14. Comparison of Results for Plane Strain Test

Figure 3-15. Comparison of Results for Equibiaxial Test



3-74

Ogden Model

1. Input commands:

Geo Panel: Propsets > Element Group (EGROUP)EGROUP,1,PLANE2D,0,2,0,0,6,2,0,C* uniaxial, plane strain, and equibiaxial tests presentC* 3-term approximationC* Initial trial range: ALPH(1): (1.,2.), ALPH(2): (4.5,5.5), C* ALPH(3): (-2.5,-1.5)


MPCTYPE,1,3,111,3,1.,2.,4.5,5.5,-2.5,-1.5,C* Using the same stress-strain curves as Mooney-Rivlin model......


Curve Fitting for Ogden Material Constants:



Term Number ALPHI MUI

1 1.100000 0.738110

2 4.800000 0.002134

3 -2.100000 -0.008790


Stress (Data in Uniaxial)

Stress(Theory)

0.1031E+01 0.1640E-01 0.3765E-01

..... ..... .....

0.7375E+01 0.5303E+01 0.5111E+01



Stress Error = 0.4977E+00


Figure 3-16. Comparison of Results for Uniaxial Test


Stress (Data in Plane Strain)

Stress(Theory)

0.1063E+01 0.4760E-01 0.9672E-01

..... ..... .....

0.4952E+01 0.1856E+01 0.1823E+01


Stress (Data in Equibiaxial)

Stress(Theory)

0.1033E+01 0.8060E-01 0.7794E-01

..... ..... .....

0.4429E+01 0.2465E+01 0.2488E+01



3-76

Figure 3-17. Comparison of Results for Plane Strain Test

Figure 3-18. Comparison of Results for Equibiaxial Test



Viscoelastic Model

1. Input commands:

Geo Panel: Propsets > Element Group (EGROUP)EGROUP,1,PLANE2D,0,2,1,0,8,0,0,C* Instantaneous elastic moduli

Geo Panel: Propsets > Material Property (MPROP)MPROP,1,EX,9152,MPROP,1,NUXY,0.3,C* shear and bulk relaxation functions presentC* 8- and 2-term approximation for shear and bulk relaxationC* functions respectively


MPCTYP,1,4,110,8,2,C* shear relaxation test data

Geo Panel: LoadsBC > FUNCTION CURVE > Material Curve (MPC)

MPC,1,0,1,....C* bulk relaxation test dataMPC,1,0,81,....


Curve Fitting for Viscoelastic Material Constants





3-78

Relaxation Function Error = 0.2829E+03


Relaxation Function Error = 0.1545E+03

Term Number GI TAUGI

1 0.34245 0.00017

2 0.12260 0.00500

3 0.58632 0.0500

4 0.44963 0.39043

5 0.29849 0

6 0.53631 2.15719

7 0.019014 50.000000

8 0.002623 95.000000

TimeShear Relaxation Function

Data Theory

0.5000E-05 0.3221E+04 0.3217E+04

..... ..... .....

0.5000E+03 0.5112E+03 0.5097E+03

Term Number KI TAUKI

1 0.508247 0.050720

2 0.252678 5.759163

TimeBulk Relaxation Function

Data Theory

0.1500E-02 0.7513E+04 0.7513E+04

..... ..... .....

0.1500E+02 0.2002E+04 0.1966E+04




Figure 3-19. Comparison of Results for Shear Relaxation Function

Figure 3-20. Comparison of Results for Bulk Relaxation Function

TIM E

TIM E



3-80

References

1. Bathe, K. J., Dvorkin, E., and Ho, L. W., “Our Discrete-Kirchhoff and Iso-parametric Shell Elements for Nonlinear Analysis- An Assessment,” Computers & Structures, Vol. 16, pp. 89-98, 1983.


3. Blatz, P. J. and Ko, W. L., “Application of Finite Elastic Theory to the Deformation of Rubbery Materials,” Transactions of the Society of Rheology, Vol. 6, 1962, pp. 223-251.

4. Chen, W. F., and, Mizunu, E., Nonlinear Analysis in Soil Mechanics, Elsevier, 1990.

5. Chen, W. F., Plasticity for Structural Engineers, Springer-Verlag, 1988.

6. Chen, W. F., Plasticity in Reinforced Concrete, McGraw-Hill, 1982.

7. Chen, W. F., and Saleeb, A. F., Constitutive Equations for Engineering Materials, Vol. 1, Elasticity and Modeling, John Wiley, 1981.

8. Christensen, R. M., Theory of Viscoelasticity, Second edition, 1982.

9. Drucker, D. C., and Prager, W., “Soil Mechanics and Plastic Analysis or Limit Design,” Quarterly of Applied Mathematics, Vol. 10, pp. 157 165, 1952.

10. Hill, R., The Mathematical Theory of Plasticity, Oxford University Press, London, 1950.

11. T. J. R. Hughes, “Numerical Implementation of Constitutive Models: Rate Independent Deviatoric Plasticity,” Theoretical Foundation for Large-Scale Computations for Non-linear Material Behavior (eds. S. Nemat-Nasser, etc.), Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1984.

12. Kao, B. G., and, Razgunas, L., “On the Determination of Strain Energy Functions of Rubbers,” Proc. VI International Conference on Vehicle Structural Mechanics, Detroit, pp. 124-154, 1986.



13. Knowles, J. K., and Eli Sternberg, “On the Ellipticity of the Equations of Nonlinear Elastostatics for a Special Material,” Journal of Elasticity, Vol. 5, Nos. 3-4, 1975, pp. 341-361.

14. Kraus, H., Creep Analysis, Wiley-Interscience, New York, 1980.

15. Ogden, R. W., “Large Deformation Isotropic Elasticity - on the Correlation of Theory and Experiment for Incompressible Rubberlike Solids,” Proc. R. Soc. Lond. A. 326, 565-584 (1972).

16. Owen, D. R. J., and Hinton, E., Finite Elements in Plasticity: Theory and Practice, Pineridge Press, Swansea, U.K., 1980.

17. Peeken, H., Dopper, R., and Orschall, B., “A 3D Rubber Material Model Verified in a User-Supplied Subroutine,” Computers & Structures, Vol. 26, pp. 181-189, 1987.

18. Snyder, M. D., and, Bathe, K. J., “Formulation and Numerical Solution of Thermo-Elastic-Plastic and Creep Problems,'' Report 82448-3, Department of Mechanical Eng., MIT, 1977.

19. T. Sussman and K. J, Bathe, “A Finite Element Formulation for Nonlinear Incompressible Elastic and Inelastic Analysis”, Computers & Structures, Vol. 26, No. 1/2, pp. 357-409, 1987.

20. G. G. Weber, A. M. Lush, A. Zavaliangos, and L. Anand, “An Objective Time-Integration Procedure for Isotropic Rate-Independent and Rate-Dependent Elastic-Plastic Constitutive Equations”, International Journal of Plasticity, Vol. 6, pp. 701-744, 1990.


22. S. W. Key, C. M. Stone, and R. D. Krieg (1981),” Dynamic Relaxation Applied to the Quasi-Static Large Deformation, Inelastic Response of Axisymmetric Solids,” pp. 585-620 in Nonlinear Finite Element Analysis in Structural Mechanics, W. Wunderlich et al. (eds.), Springer-Verlag, Berlin.



3-82

Birth and Death of Elements

The element birth and death feature allows you to include elements in some stages of the solution and exclude them in other stages. Ele-ments are excluded from participating in the solution using the EKILL

(Analysis, Nonlinear, Element_Birth/Death, Kill elements) command. Killed elements can be brought to life to participate in the solution in subse-quent runs using the ELIVE (Analysis, Nonlinear, Element_Birth/Death,

Bring elements back to life) command. Killed elements can be listed using the EKILLLIST (Analysis, Nonlinear, Element_Birth/Death, List killed

elements) command.All elements are considered alive by default, unless declared dead by the EKILL command. All elements, dead or alive, must be defined prior to running. Each solution stage requires a separate run. The restart flag should be active for all stages other than the first stage. The time for a solution stage starts from the end-time of the previous stage and ends at the end-time for the current stage. The EKILL and/or ELIVE commands can be used prior to running each solution stage.

✍ Elements are assumed stress free at the time of their first birth. The un-deformed, un-stressed shapes of elements brought to life using the ELIVE command are defined by the deformed nodal positions obtained in the last solution stage.

Various stages of the solution must be properly associated with the time variable and time curves.


4 Gap/Contact Problems

Introduction

One of the major areas of nonlinear analysis is the solution of problems in which separate bodies or structures may come in contact with each others. Several methods have been developed to handle such problems. One of the techniques that is extensively used to solve contact problems is the penalty method. In this method, large numerical values are introduced into the stiffness matrix of the system to simulate the rigidity between two nodes such that the two have approximately, since the constraint exactly is not satisfied, same displacements. The penalty method seems attractive since it preserves the size and the bandwidth of the stiffness matrix. However, a major difficulty, typically associated with this approach, arises in the selection of the proper penalty values. Very large penalty values cause numerical difficulties, while small penalty values produce inaccuracy. A compromise between the numerical performance and the accuracy of the results is often made. Some researchers try to tackle this difficulty by implementing algorithms that adaptively update the penalty values based on the stiffness changes of the structure throughout the incremental solution.

Another method, commonly used by researchers in contact problems, incorporates the Lagrange multiplier method. Inherently to this approach is the introduction of new variables (Lagrange multipliers) to the problem which in turn increase the size and bandwidth of the matrices involved in the analysis. In addition, special care should be devoted to avoid zero pivots in solving the equations of the system.


Chapter 4 Gap/Contact Problems

4-2

A third approach, used in COSMOSM NSTAR Module, uses a hybrid technique to solve contact problems. This technique does not require assigning penalty values and keeps the matrices size and bandwidth unchanged.

Hybrid Technique for Gap/Contact Problems: General Description

A brief description on the hybrid technique used in solving nonlinear problems involving contact is presented. For the simplicity of the presentation, the contact will be assumed frictionless, however, the technique is general and is applicable to contact problems with friction.

Hybrid Techniques

Matrix methods of structural analysis can be categorized as:

The Displacement Method

In this method, the matrix equation to be solved can be expressed as:

[K] {U} = {R} (4-1)

where

[K] = the stiffness matrix of the structure

{U} = the vector of nodal displacements

{R} = the vector of nodal forces

The unknown quantities in this matrix equation are the nodal displacements while the prescribed quantities are the nodal forces.

The Force Method

In this method, the matrix equation to be solved can be expressed as:

[F] {R} = {U} (4-2)



where

[F] = the flexibility matrix of the structure

{R} = the vector of nodal forces

{U} = the vector of nodal displacements

The unknown quantities in this matrix equation are the nodal forces while the prescribed quantities are the nodal displacements.

The Hybrid Method

In this method, the displacement and the force methods are combined to solve the matrix equation. The displacement method is used where external forces are prescribed while the force method is utilized where the displacements are prescribed.

In general purpose finite element programs, a displacement-based method is used. However, in dealing with nonlinearities, such as contact, a hybrid method can be efficient.

Gap Definition

A gap is defined by two nodes, for example, i and j. The direction of the gap is defined as the line connecting node i to node j. The gap distance is defined as the maximum allowable relative displacement between the two nodes along the gap direction (see Figure 4-1). An open gap has no effect on the response of the structure while a closed gap, if rigid, limits the relative displacements of its two nodes along the gap direction not to exceed the gap distance.

Figure 4-1. Gap Direction

i

1

3

2

g

j

ng

g

g j

j

ji

i

i

n

n

3

3

2

2

1

1

Gap Direction

Gap Direction



4-4

In order to analyze gaps, the force method can be used to calculate the forces at the gap locations. Thus, each gap is replaced by two forces, equal in magnitude but opposite in direction, which are applied to the two nodes connected by a gap.

Rewriting equation 4-2 for the gaps:

[Fg] {Rg} = {Xg} (4-3)

where

{Rg} = Vectors of gap forces

{Xg} = Vectors of relative gap displacements

In order to define [Fg], a unit force is applied in the gap direction and the relative displacements induced in all gaps are determined. This process is repeated for all other gaps in order to obtain [Fg].

Now, consider a configuration where the effect of the gaps is neglected. The

following inequity implies that the ith gap is closed:

Uig = Ui

2 - Ui1 > gi (closed gap) (4-4)

where

gi = Gap distance

Ui1 = Displacement induced by the external force vector {R}

Defining,

xgi = – (Ui

g - gi) (4-5)

By solving equation 4-4, the gap force vector, {Rg}, is obtained. Applying these

forces to the structure, the relative gap displacement of the ith gap will equal xig.

Since, the external force vector {R} will produce {Ug} displacement vector, and the gap forces {Rg} will produce - ({Ug} - gi) then, ({R} + {Rg}) will produce {g}.

Therefore, the displaced shape of the structure will resume a position where the relative displacement for each closed gap remains equal to its prescribed allowable distance.



It should be noted that this method of solution uses no approximation and requires no iterations. However, iterations are used to determine which gaps are closed at a particular time, hence, forming equation 4-3 for those gaps only.

Contact Definition

A contact problem can be considered as a general case of a gap problem for the following:

1. The direction of the normal gap force is not fixed.

2. The point of contact also may change, for example, if the gap is originally between nodes i and j, the structure may displace such that point i comes in contact with another point (see Figure 4-2).

Figure 4-2. Contact Problem

Due to these factors, unlike simple gaps, the convergence and the accuracy of the contact problem will depend on the incremental solution where the forces are applied gradually to enable a node to move slowly on the surface.

In order to consider the contact between two bodies, one body is arbitrarily declared “contactor” (source of contact), while the other is designated as the “target”. The region of contact between the two bodies is governed by the overall problem geometry, applied loads, material properties, and other relevant conditions.

In COSMOSM, the contact problem is defined in accordance with the following procedure:

1. The region of contact of the “contactor” body is established by a series of nodal points to which one-node gap elements should be assigned.

i

j

i'

k

Original Location

Location at Contact

Contacting Surfaces



4-6

2. The region of contact on the “target” body is defined by a series of contact lines (in 2D problems) or surfaces (in 3D problems).

3. The extent of contact between the two bodies is limited to areas defined by the one-node gap elements. With the small-displacement restriction removed, each gap can come in contact with any of the surface segments in that same group.

4. Each surface (line) of the “target” body is assigned a positive and a negative side based on its node connectivity as shown in Figure 4-3. The negative side is where the gap elements are forbidden to enter.

5. Surfaces defined in one group must form one continuous overall surface.

GAP Elements

Two-Node Gap Element (Node-to-Node Gap)

Two-node gap elements are used in 2D and 3D contact problems where bodies are coming in contact with each other due to the application of external forces. The main assumption for the this type of element is that the direction of the normal contact force(s) and the contact points are known in advance and remain unchanged

throughout the analysis. The two-node gap elements are placed between two nodes of the contacting bodies (one node on each body) such that the direction of the gap element, represented by the straight line joining the initial locations of its two nodes (before deformation) (see Figure 4-4a), coincides with the normal contact force (which is normal to the tangent line/plane at the point of contact of the two bodies). Depending on the type of contact problems, the gap element can be specified to be

Figure 4-3. Contact Definition

Contact Elements

Contactor Body

Contact Surface

Target Body

Contact Sub-Surface+



either a compressive gap (to limit the relative contraction between two nodes) or a tensile gap (to limit the relative expansion between two nodes) (see Figure 4-4b). Friction can be considered (only for compressive gaps) for both static and dynamic analyses. The friction force associated with a gap element will lie in the tangent plane.

Figure 4-4a. Two-Node Gap Element Definition

Figure 4-4b. Two-Node Gap Elements

For more information about the element definition, commands, and examples refer to the following:

COSMOSM User Guide Manual (for Element Definition)

COSMOSM Command Reference Manual (for Commands)

Problems NS13-NS18 in this Manual (for Verification Problems)

u 2

After Deformation

1

2

1'

2'

relv

FnFn

Fs

Fs

u1

n

BeforeDeformation

1

2

Tension Gap

1

2

gdist gdist

Compression Gap

n

d ≥ gdist d ≥ gdist

d = Relative position change along direction 12u u n•-( )n =

n



4-8

One-Node Gap Elements

The one-node gap elements are used to establish the motion of a certain node on a “contactor” entity (line or surface represented by those nodes) with respect to a “target” entity (line or surface defined by a number of sub-surfaces).

The main advantage of one-node gap elements over the two-node gap elements are:

1. The user does not need to know the exact location of the point of contact a priori. The program internally will determine that location and apply the contact forces accordingly.

2. The direction of the contact forces is determined by the program based on the deformed shape of the entities in contact.

3. The nodal points on the contacting entities do not need to match each other.

4. “Shrink fit” problems, where a portion of the model is forced to assume a new position, can be handled through this type of gap elements.

The element group definition command for a one-node gap element is the same as a regular two-node gap element with the exception of the changes in options 4 and 5. Option 4 is used to define the dimensionality of the contact problem (2D or 3D) while option 5 is used to define the number of nodes used to construct the “target” entity (2 or 3 nodes for line and 4 or 9 nodes for surfaces) (see Figure 4-5). The “target” contact entity (line or surface), defined as an assembly of sub-entities (sub-lines or sub-surface) through the use of the NL_GS (Analysis > NONLINEAR > CONTACT > Contact Surface) command, is associated with active gap element group and should follow the gap element definition commands. If the problems under consideration has more than one set of contact bodies, then a separate gap element group, “contactor” and “target” must be defined for each potential contact bodies.



Figure 4-5. One-Node Contact Element

Example

Consider a contact “target” surface (Q) which is formed by faces of solid or shell elements (or simply defined by a set of fixed nodes in space) to belong to the “target” body. Also consider nodes i, j, and k (belonging to the “contactor” body) to represent the area that may come in contact with surface (Q) (see Figure 4-6).

Target Sub-line

x

y

Target Node

Contact ElementsContactor

+

3 5

6r

2D Contact

1 2 4

7

"Contactor" Area of Contact

"Target" Area of Contact

Contact Element

Contact Surface

Target Node

Target Sub-surface

Target Surface

3

2 1

4

+

9

56

78

z

r

s

3D Contact

x

y

Figure 4-6. Contact Definition

ContactElements

Q

TargetSurface

321 4

9

5

67 8

k

j

14131211

10

15

i



4-10

The command sequence to specify this case is based on the following points:

1. Define a GAP element group for node-to-surface contact (using the proper option). For this case, each sub-surface is defined by 4 nodes. Activate this element group and the corresponding real constant set.

2. Define 3 gap elements at each of the three nodes i, j, and k.

3. Define 8 sub-surfaces (for example sub-surface 1 is defined by nodes 1, 6, 7, and 2) to form the “target” surface (Q).

The input commands related to this portion of contact definition are shown below:

Geo Panel: Geo Panel: Propsets > Element Group (EGROUP)

EGROUP,3,GAP,1,1,,2,4,,,

Geo Panel: Propsets > Real Constant (RCONST)

RCONST,3,3,,2,1,0.3

Geo Panel: Control > ACTIVATE > Set Entity (ACTSET)

ACTSET,EG,3ACTSET,RC,3

Geo Panel: Meshing > ELEMENTS > Define Element (EL)

EL,101,PT,0,1,iEL,102,PT,0,1,jEL,103,PT,0,1,k

Geo Panel: Analysis > NONLINEAR > CONTACT > Contact Surface (NL_GS)

NL_GS,1,1,6,7,2NL_GS,2,2,7,8,3NL_GS,3,3,8,9,4NL_GS,4,4,9,10,5NL_GS,5,6,11,12,7NL_GS,6,7,12,13,8NL_GS,7,8,13,14,9NL_GS,8,9,14,15,10

In the above commands, the EGROUP (Propsets > Element Group) command specifies gap elements for node-to-surface contact. The “target” sub-surfaces are made of 4-node areas which can displace in space. The RCONST (Propsets > Real

Constant) command specifies a coefficient of friction of 0.3 between the contacting surfaces (represented by gap elements on the “contactor” and the “target” surface Q). The EL (Meshing > ELEMENTS > Define Element) command defines the gap



elements and the NL_GS (Analysis > NONLINEAR > CONTACT > Contact Surface) command specifies the associated contact sub-surfaces that form the “target” surface.

In defining “contactors” and “targets” the following should be observed:

1. Quadratic Sub-surfaces

An interior node of a defined contact surface must be surrounded by four contact sub-surfaces. Figure 4-7 shows examples of valid and invalid definition of contact sub-surfaces.

2. Triangular sub-surfaces

The only restriction is that all sub-surfaces in one group should be triangular defined by same number of nodes.

3. Contact is assumed to be rigid, therefore, the only real constant needed is the coefficient of friction.

(Gap stiffness is also considered).

4. Each sub-surface (sub-line) used to define an overall “target” surface (line) must be defined such that the normal to the sub-surface (sub-line) points towards the positive side of the overall surface (line).

An easier and more efficient way to model (input) a contact element group is described in the following section.

For more information about the element definition, commands, and examples refer to the following:

COSMOSM User Guide Manual (for Element Definition)

COSMOSM Command Reference Manual (for Commands)

Problems NS36-NS40, and NS-99 in this Manual (for Verification Problems)

Valid

Invalid

Invalid

Figure 4-7. Sub-Surfaces Formations



4-12

Automatic Generation of Gap Elements

To facilitate the definition of the entities involved in contact problems, the NL_GSAUTO (Analysis > NONLINEAR > CONTACT > Contact Surface by Geometry) can be used to automatically generate the one-node gap elements on the “contactor” entities and line(s)/surface(s) on the “target” entities. All the entities (contactor or target) must be meshed before issuing this command. The one-node gap elements are created at each node on all contactor entities and the gap line(s)/surface(s) are generated at each edge/face on the target entities. For more details on the NL_GSAUTO (Analysis > NONLINEAR > CONTACT > Contact Surface by

Geometry) command, refer to the COSMOSM Command Reference Manual.

Example

Consider the contact problems between the two objects depicted in Figure 4-8 where the curves, the elements, and the nodes are shown. It has to be noted that the entities involved in the contact problem must be meshed before defining the gap elements.

Figure 4-8a



Figure 4-8b

Figure 4-8c

The contact sequence to specify this case is based on the following:

1. There are two contactors (sources of contact), curves 1 and 2, and two targets curves 14 and 11.

2. Curve 1 and Curve 2 can come in contact with curve 11 and 14.

3. Define a GAP element group (group 2), node-to-line contact, to be associated with the contact between curves 1 and 2 (sources) and 11 and 14 (targets).



4-14

4. Contact nodes are to be created on the contactors.

5. Contact sub-lines are to be created on contact targets.

6. Each sub-line is defined by three nodes.

The input commands related to this portion of contact definition are shown below:


EGROUP,2,GAP,1,0,0,1,3,0,0,

Geo Panel: Control > ACTIVATE > Set Entity (ACTSET)

ACTSET,EG,2

Geo Panel: Analysis > NONLINEAR > CONTACT > Contact Surface by Geometry (NL_GSAUTO)

NL_GSAUTO,0,1,0,14,14,1,1,NL_GSAUTO,0,2,0,11,11,1,1,

Geo Panel: Meshing > ELEMENTS > Merge Element (NMERGE)

EMERGE;

Figure 4-9 shows the elements after contact definition.

Figure 4-9

1

2

3

45 6

7

8

9

10

11 12 13 14 1516 17 18 19 20

21 22 2324

25

26

27

28

29

30

3132

33

34

35

36

37

3839

40 41



Contact/Gaps Enhancement

Triangular Sub-Surfaces for Target Surface

The 3-dimensional contact analysis is modified to allow triangular sub-surfaces to define the target surface. Each triangular sub-surfaces is defined by 3 or 6 nodes and results from a mesh that utilizes one of the following element types:

Tetra4, tetra10, Shell3/3T/3L, and

Solid 8- to 20-node with collapsed nodes

✍ In the case of the solid with collapsed nodes, each sub-surface is defined by 4 or 8 nodes (one node is repeated once or twice)

✍ There is no restriction to the shape or relative positioning of these sub-surfaces, except that the sub-surfaces in one gap group, must all be triangles with the same number of nodes.

Automatic Soft Springs for Contact Source or Target

Six additional constants are added to the gap real constant set (r8 to r13). Each constant represents a stiffness to be used to stabilize a structural part in one of the three global directions (3 for source and 3 for target). The specified stiffness is equally divided among the nodes on the source or the target. This capability eliminates the need for soft springs and provides stability for the structural parts that are unstable if contact was to be ignored.

✍ Make sure to use a different real constant set for each gap group, so that stiffening is not extended to undesired areas.

A New Solution Strategy for Initial Interference

A new feature allows that a gap group be excluded from analysis during certain phases of solution. Whenever this option is turned on for a gap element group, that group will not participate in the solution. However, if the flag is turned back off prior to a restart, same group can come back to life and participate in the analysis.

This feature is particularly useful for problems where considerable initial interference as well as geometric and/or material nonlinearities exist.



4-16

As an example, consider the case of two thick cylinders with an initial interference that can cause some plastic deformation in the cylinders. Different means can be utilized to fit one cylinder inside the other. The Outer cylinder may be heated, the inner cylinder may be cooled, or pressure may be applied to one or both cylinders. It must also be noted that the amount of plastic deformation is greatly dependent on the procedure that is used. To solve this problem, the two cylinders are modeled by their unstressed geometry and contact is defined along the interference area. In The first phase of solution, the contact group need to be ignored (op7=1), the loading or heating is prescribed such that the interference between the cylinders is eliminated. Next, using the restart option, include contact (op7=0), and allow the forces/temperatures to be removed gradually.

✍ This option can be specified for each gap group, independently. [option 7 in the EGROUP command]

✍ This option is not effective for the Node-to-node gaps.

✍ Each gap group can be killed or brought back to life, independent of other gap groups.

Troubleshooting for Gap/Contact Problems

Following is a list of commonly encountered errors during the execution of nonlinear gap/contact problems.

1. During the first step, the program stops with the error message: “Stop, the diagonal term in equation ...., node ...., direction .. is ....." (a zero or negative).

This error usually indicates that the whole model, or portion of it, is externally statically unstable due to improper constraints. It should always be remembered that the gap elements do not alter the stiffness. If a portion of the model is supported only by gap elements, then that portion can be stabilized through the use of soft trusses (see verification problems NS17 and NS18).

2. The program runs successfully, but the postprocessing module shows that the one node gap elements go beyond the contact surface which should have stopped them.

Make sure that you are looking at the structure's iso-scale displaced plot (both the dimensions of the structure and the deformations have the same scale in the plot). The default setting of the postprocessing programs show the deformed



shape with an exaggerated deformation scale. Therefore, try to use a scale factor of 1.0 in plotting the deformation (see the DEFPLOT (Results > PLOT > Deformed Shape) command in COSMOSM Command Reference Manual).

If the gap elements still exceed the contact surface after issuing the above commands, then the conclusion is that the gap elements are not properly closed. The following two possibilities then should be considered:

a. The orientation of the gap surfaces might be wrong. The gap elements are allowed to remain on the positive side of the surface.

b. The original displacement is too large that the displaced location of the gap elements cannot be compared correctly with respect to the contact surfaces. This case usually occurs when one of the bodies is an unconstrained structure supported only by soft trusses. To investigate this possibility, the analyst should calculate the displacements of the soft trusses under the loads. If the resultant displacements are excessive and the gap elements are pushed far beyond the contact surface, it is likely that the gap iterations also will not converge. This situation can be overcome by decreasing the applied load step through modifying the “time” curve, or by using stiffer trusses to support the unattached portion of the model.

3. The program stops with the error message: “Stop, wrong definition for the contact surfaces.”

Check the target surface connectivity and direction. Make sure that each target contact surface is represented by continuous sub-surfaces.

4. In nonlinear dynamic problems, the program converges but the structure behaves erratically after the gaps close.

This condition often occurs due to the assumed perfect rigidity of the closed two-node gaps. To avoid that situation, some flexibility for the contact can be introduced through the third real constant of the two-node gap elements.

5. The program completes one or more steps with some gap elements closed, but finally stops with one of the following error messages:.

a. “Stop, the diagonal term in equation..., node ..., direction ... is zero or negative."

b. “Stop, convergence not achieved for gap elements.”

c. “*** ERROR: Convergence is not achieved in 200 iterations” or “Convergence is not achieved in 100 contact iterations.”



4-18

These errors basically imply difficulties in problem convergence due to:

– System stiffness has deteriorated and become singular or close to singular due to other nonlinearities (geometric or material).

– The load increment is too large.

In either case, reducing the load increment is most likely to solve the problem. However, in case I, if the stiffness has extremely deteriorated, a solution continuation may not be possible.

✍ If friction forces are present, the analysis is nonconservative (dependent on the load application sequence). Therefore, the loads must be applied in increments which resemble the actual load history.

✍ Contact problems which involve large-deflection analysis are likely to require mesh refinement in the regions where contact is expected.

References

1. Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice Hall, 1982.

2. Belytschko, T., and Hughes, T., (eds.) Computational Methods for Transient Analysis, North-Holland, Amsterdam, 1983.

3. Cook, D. R., Malkus, D. S., and Plesha, M. E., “Concepts and Applications of Finite Element Analysis,” Third edition, Wiley, 1989. Kardestuncer, H., “Finite Element Handbook,” McGraw-Hill, 1987.

4. Kulak, R. F., “Adaptive Contact Elements for Three-Dimensional Explicit Transient Analysis,” Comp. Meth. Appl. Mech. Eng., 72, pp. 125-151, 1989.

5. Mazurkiewicz, M., and Ostachowicz, W., “Theory of Finite Element Method for Elastic Contact Problems of Solid Bodies,” Comput. Struct., Vol. 17, pp. 51-59, 1983.

6. Parisch, H., “A Consistent Tangent Stiffness Matrix for Three-Dimensional Non-linear Contact Analysis,” Int. J. Num. Meth. Eng.. Vol. 28, pp. 1803-1812, 1989.



5 Numerical Procedures

Static Analysis

There are different numerical procedures that can be incorporated in the solution of nonlinear problems using the finite element method. A successful procedure must include the following:

• A control technique capable of controlling the progress of the computations along the equilibrium path(s) of the system.

• An iterative method to solve a set of simultaneous nonlinear equations governing the equilibrium state along the path(s).

• Termination schemes to end the solution process.

Additional schemes such as line search, acceleration, and/or preconditioning may be augmented to enhance the solution procedure.

Incremental Control Techniques

Different control techniques have been devised to perform nonlinear analysis. These techniques can be classified as:


Chapter 5 Numerical Procedures

5-2

Force Control

In this strategy, the loads applied to the system are used as the prescribed variables. Each state (point) on the equilibrium path is determined by the intersection of a surface (F = constant) with the path to determine the deformation parameters (Figure 5-1a).

In adapting this technique for finite element analysis, the loads (base motions, prescribed displacements, thermal, gravity, ...) are incrementally applied according to their associated “time” curves.

Displacement Control

In this technique, a point on the equilibrium path is determined by the intersection of a surface defined by a constant deformation parameter (U = constant) with the solution curve (Figure 5-1b).

To incorporate this technique in finite element analysis, the pattern of the applied loads is proportionally incremented (using a single load multiplier) to achieve equilibrium under the control of a specified degree of freedom. The controlled DOF is incremented through the use of a “time” curve.

Arc-Length Control

In this strategy, a special parameter is prescribed by means of a constraint (auxiliary) equation which is added to the set of equations governing the equilibrium of the system. In the geometric sense, the control parameter can be viewed as an “arc length” of the equilibrium path (Figure 5-1c).

To use this technique in finite element analysis, the pattern of the applied loads is proportionally incremented (using a single load multiplier) to achieve equilibrium

Figure 5-1a. Force Control

f *

Fk

uk

Figure 5-1b. Displacement Control

Fk

uk

u*



under the control of a specified length (arc-length) of the equilibrium path. The arc-length will be automatically calculated by the program. No “time” curve is required.

Both Force control and Displacement control will breakdown in the neighborhood of turning points (known as snap-through for force control and snap-back for displacement control) as in Figure 5-2. These difficulties usually are encountered in buckling analysis of frames, rings, and shells. Arc-Length control will successfully overcome these difficulties.

Figure 5-2. Failures of Control Techniques

Figure 5-1c. Arc Length Control

Fk

uk

S*

F

uu

b. Failure of Displacement Control

F

u

f

a. Failure of Force Control

Load

Displacement

c. Failure of Force and Displacement Controls

b, c, e - limit points under force controlf, g - limit points under displacement control

h

g

f

e

d

c

b

'snap-through' under force

control

'snap-back' under

displacement control

a



5-4

Thermal Loading for Displacement/Arc Length Controls

Here, the input temperatures are used as the loading pattern. The output load factor defines the temperature factor for the state of deformation. The input temperature pattern is assumed to be relative to reference:

{T(u)} = LF(u) *{To} + Tref

Where

{T(u)} - Temperature vector associated with displacement {u}

{To} - vector of nodal Temperature pattern

LF(u) - Load Factor obtained for displacement {u}

Tref - Reference Temperature

Iterative Solution Methods

In nonlinear static analysis, the basic set of equations to be solved at any “time” step, t+∆t, is:

t + ∆t{R} - t + ∆t{F} = 0 (5-1)

where

t + ∆t{R} = Vector of externally applied nodal loads

t + ∆t{F} = Vector of internally generated nodal forces

Since the internal nodal forces t+∆t{F} depend on nodal displacements at time t+∆t, t+∆t{U}, an iterative method must be used.

The following equations represent the basic outline of an iterative scheme to solve the equilibrium equations at a certain time step, t+∆t,

(5-2)



(5-3)

(5-4)

(5-5)

where

t+∆t{R} = Vector of externally applied nodal loads

t+∆t{F}(i-1) = Vector of internally generated nodal forces at iteration (i)

{∆R}(i-1) = The out-of-balance load vector at iteration (i)

{∆U}(i) = Vector of incremental nodal displacements at iteration (i)

t + ∆t{U}(i) = Vector of total displacements at iteration (i)

t + ∆t[K] (i) = The Jacobian (tangent stiffness) matrix at iteration (i)

There exists different schemes to perform the above iteration. In the following, a brief description of three methods of the Newton type will be furnished.

Newton-Raphson (NR) Scheme

In this scheme, the tangential stiffness matrix is formed and decomposed at each iteration within a particular step (Figure 5-3a). The NR method has a high convergence rate and its rate of convergence is quadratic. However, since the tangential stiffness is formed and decomposed at each iteration, which can be prohibitively expensive for large systems, it may be advantageous to use another iterative method.



5-6

Figure 5-3a. Newton-Raphson Iterative Method with Force Control, 1D

Modified Newton-Raphson (MNR) Scheme

In this scheme, the tangential stiffness matrix is formed and decomposed at the beginning of each step (or the user-specified reformation interval) and used throughout the iterations (Figure 5-3b).

Load

Displacement

Ut t + ∆t (1)U

(3)(1)∆U

t + ∆t (1)F t + ∆t (2)Ft + Dt

F

t + ∆t (2)Kt + ∆t (1)K

t + ∆t (0)K = K

∆U(2)

∆U

t + ∆tUt + ∆t (3)U

t + ∆t (2)U

Ft

t + ∆tR

1

1

1F(U)

Rt



Figure 5-3b. Modified Newton-Raphson iterative Method with Force Control, 1D

Quasi-Newton (QN) Schemes

Unlike the NR and MNR iterative schemes, the QN family of schemes employs a lower-rank matrix to update the stiffness matrix (or its inverse) to provide secant approximation from iteration (i-1) to iteration (i) (Figure 5-3c). Defining a displacement increment as:

(5-6)

and an increment in the out-of-balance loads;

(5-7)

the updated iterative matrix should satisfy the QN equation:

(5-8)

Load

Displacement

Ut t + ∆t (1)U

(1)∆U

t + ∆tF

tK = K

(3)∆U(2)∆U

t + ∆tUt + ∆t (3)U

t + ∆t (2)U

Ft

t + ∆tR

1

F(U)

Rt

(2)F(1)F t + ∆tt + ∆t



5-8

Figure 5-3c. Quasi-Newton Iterative Method with Force Control, 1D

The BFGS (Broyden-Fletcher-Goldfarb-Shanno) update formula are widely used with the QN algorithm.

A version of the above-mentioned iterative scheme is implemented in COSMOSM.

Line Search Scheme

The performance of an iterative method is very much dependent on the choice of the “optimal” step length (ß) in the direction of increment (search direction). An estimate of such a step can be calculated for BFGS iterative technique by requiring that the projection of the residual load vector in that direction to vanish, i.e.;

(5-9)

and the solution is then expressed as:

(5-10)

Load

DisplacementUt

(2)(1)∆U

t + ∆t F(1)K

(3)∆U∆U

t + ∆tU

Ft

t + ∆tR

1

1

F(U)

(2)K

(1)δ

(2)δ

(3)δ

(1)γ

(2)γ

(3)γ

Rt

(3)Ut + ∆t(2)U

t + ∆t(1)Ut + ∆t



The line search operation is expensive because one search process may involve a large number of recalculating the residual vector. Instead of using the strict criterion of vanishing the residual, many researchers suggested that the following criterion will generally be adequate,

(5-11)

A value of 0.50 is recommended for the tolerance (TOL).

The line search option is supported in conjunction with the BFGS iteration method where its effectiveness is recognized.

Termination Schemes

For any incremental procedure, based on iterative methods, to be effective, practical termination schemes should be provided. At the end of each iteration, a check should be made to test if the iteration converged within realistic tolerances or it is diverging. Very loose tolerance will initiate inaccurate results, while very strict one can needlessly make the computational cost high. On the other side, bad divergence check can end the iterative process when the solution is not diverging or allow the process to continue for searching unrealizable solution.

A number of procedures have been introduced as convergence criteria for terminating an iterative process. In the following, three convergence criteria will be discussed.

Displacement Convergence

This criterion is based on the displacement increments during iterations. It is given by:

(5-12)

where |{α}| denotes the Euclidean norm of {α}, and εd is the displacement tolerance.



5-10

Force Convergence

This criterion is based on the out-of-balance (residual) loads during iterations. It requires that the norm of the residual load vector to be within a tolerance (εf) of the applied load increment, i.e.,

(5-13)

Energy Tolerance

In this criterion, the increment in the internal energy during each iteration, which is the work done by the residual forces through the incremental displacements, is compared with the initial energy increment. Convergence is assumed to reach when the following is satisfied:

(5-

14)

where εe is the energy tolerance.

In addition, a number of schemes have been described as divergence criteria. One of them is based on the divergence of the residual loads. Another is based on the divergence of the incremental energy.

Procedure Activation

The solution techniques implemented in COSMOSM nonlinear module can be accessed through NL_CONTROL (Analysis > NONLINEAR > Solution Control) and A_NONLINEAR (Analysis > NONLINEAR > NonL Analysis Options) commands. For specific input explanation, the user is referred to COSMOSM Command Reference.



Dynamic Analysis

The skeleton of the procedures used for nonlinear dynamic analysis follow the same method used for nonlinear static analysis (Control + Iteration + Termination).

The discretized equilibrium equations of the dynamic system can be written in the form:

(5-15)

where

[M] = Mass matrix of the system

[C] = Damping matrix of the system

t+∆t[K](i) = Stiffness matrix of the system

t+∆t{R} = Vector of externally applied nodal loads

t+∆t{F}(i-1) = Vector of internally generated nodal forces at iteration (i)

t+∆t{∆U}(i)= Vector of incremental nodal displacements at iteration (i).

= Vector of total displacements at iteration (i)

= Vector of total velocities at iteration (i)

= Vector of total accelerations at iteration (i)

Using implicit time integration schemes such as Newmark-Beta or Wilson-Theta methods, and employing a Newton's iterative method, the above equations can be cast in the form:

(5-16)



5-12

where

= The effective load vector

= The effective stiffness matrix

and a0, a1, a2, a3, a4, and a5 are constants of the implicit integration schemes.

Since Displacement and Arc-length controls are adapted for proportional loadings, which is not the case in dynamic problems where loading time histories are prescribed, then, only Force control can be incorporated for dynamic analysis.

Also, all iterative solution strategies discussed in static analysis can also be incorporated for dynamic analysis. However, only MNR and NR methods are available in this version for dynamic analysis. Since, the inertia of a system tends to smoothen its dynamic response more than its static response, convergence is generally expected to be easier than static analysis. No line search is implemented for dynamic analysis.

NL_CONTROL (Analysis > NONLINEAR > Solution Control), NL_INTGR (Analysis > NONLINEAR > Integration Options) and A_NONLINEAR (Analysis > NONLINEAR > NonL Analysis Options) commands are used to activate the required procedure. For specific input explanation and default values, the user is referred to COSMOSM Command Reference.



Rayleigh Damping Effects

To include the effects of damping forces in dynamic analysis, the proportional Rayleigh damping matrix is introduced. The damping matrix of the system [C] is assumed to take the form:

where

[K0] = Initial stiffness matrix of the system

[M] = Mass matrix of the system

α = Rayleigh damping coefficient associated with the stiffness matrix

β = Rayleigh damping coefficient associated with the mass matrix

The NL_RDAMP (Analysis > NONLINEAR > Damping Coefficient) command is used to activate the effect of damping.

Concentrated Dampers

The effects of concentrated dampers is also included in the dynamic analysis. The PD_CDAMP command can be used to define damping between two nodes on the structure.

Base Motion Effects

The dynamic effects of base motion accelerations in global X, Y, and Z directions (may resemble the effects of seismic motions on a structural system) can be utilized in the analysis.

The NL_BASE (Analysis > NONLINEAR > Base Motion Parameter) command is used to activate this procedure.



5-14

Inclusion of Dead Loads in Dynamic Analysis

In many cases, it is desired to start the dynamic analysis of structural systems from the deformed configurations obtained under the effect of dead loads. Such an analysis may be performed as follows:

1. Suppose that the pseudo time to be used for static analysis is Ts. The true dynamic time at a time t > Ts is equivalent to (t-Ts).

2. Define dead loads by associating them with time curves that start from zero (at time 0), reach their final values at the pseudo time Ts, and remain unchanged afterwards throughout the period of the dynamic solution.

3. Define dynamic loads by associating them with time curves that keep a value of zero throughout the static solution pseudo time, and vary as desired during the dynamic solution true time past Ts.

4. Perform a static analysis for the time duration (0 to Ts).

5. Activate the restart flag and run a dynamic analysis from Ts to Td (where Td-Ts represents the actual true time for dynamic analysis).

Note that if Rayleigh damping is used, then the damping matrix calculations will be based on the stiffness at the start of the dynamic run.

Figure 5-4. Time Curves for Loads

tTs Td

Multiplier

0

a) Time Curves for Dead Loads

tTs Td

StaticAnalysis

DynamicAnalysis

StaticAnalysis

DynamicAnalysis

Multiplier

0

b) Time Curves for Dynamic Loads

Dynamic Load



Adaptive Automatic Stepping Technique

In COSMOSM the user has the choice to solve nonlinear problems by directly specifying the load and/or displacement increment to be followed or by letting the program select its own incremental procedure based on user specified parameters.

The adaptive automatic stepping algorithm (Analysis > NONLINEAR > AutoStep Options) provides the following:

Step Size Optimization

The algorithm automatically adjusts the incremental step so that smaller steps are enforced in the region of the most severe nonlinearity while larger steps are allowed when the response tends to be linear. To avoid excessive cut-backs in cases of limit loads or buckling, a minimum step DTMIN is defined. In addition, to prevent convergence on a higher equilibrium path during solution process, especially when path-dependent materials or loads are present, a maximum step DTMAX is used. The definition of DTMIN and DTMAX is optional (COSMOSM will provide default values if not specified) but it is recommended for complex problems.

Safe-guard Against Equilibrium Iteration Failures

The scheme senses the rate of convergence and provides adaptive step adjustment to avoid the termination of the solution process due to:

• Exceeding the number of permissible equilibrium iterations because of lack of convergence.

• Divergence of the incremental residual load and/or energy.

• Gap/contact iterations nonconvergence.

• The presence of a negative term on the diagonal of the stiffness matrix during iterations (due to a large load increment) under force control.

• The presence of a negative term on the diagonal of the stiffness matrix at the beginning of a new step due to local singularity under force control.



5-16

Safe-guard Against Converging to Incorrect Solutions

• Produced by large incremental rotations:

An incremental equivalent rotation at each node is computed from:

The criterion for resetting the time increment is:

• Produced by large incremental creep strain:

For Creep analysis (Element Group Op. 7 = 1) and Viscoelastic material model (Element Group Op. 5 = 8), the effective creep strain at each integration point is computed:

The criterion for resetting time increment is:

Note that if CETOL is not input by the A_NONLINEAR (Analysis > NONLINEAR > NonL Analysis Options) command, the default value is 0.01.

• Produced by large incremental plastic strain:

For Elastoplastic material models (Element Group Op. 5 = 1, 2, 5, 11) an incre-mental effective plastic strain at each integration point is computed:

The criterion for time increment is:



• Produced by unacceptable element status change:

When the Wrinkling Membrane model is used (Element Group Op. 5 = 10), the element status is determined by the state of strain, namely, Taut, Wrinkled, and Slack (details in Chapter 3: Material Models and Constitutive Relations). The program only accepts the following element status changes:

• Encountering a state of equilibrium on a higher path that produces a negative determinant for the deformation tensor.

• Produced by large incremental logarithmic strain in large strain plasticity model. The criterion for time increment is:

< 40% where is the incremental effective logarithmic strain.

The NL_AUTO (Analysis > NONLINEAR > AutoStep Options) command is used to activate this procedure.

J-Integral Evaluation for Nonlinear Fracture Mechanics NLFM

Evaluation of J-integral (the two-dimensional formulation) as a viable fracture criteria for linear elastic and elastic-plastic deformations is implemented in the NSTAR module. The two-dimensional J-integral formulation has been modified to include axisymmetric behavior as well as thermal gradients.

The J-integral provides estimations for the stress intensity factors in a nonlinear environment. For an elastic fracture assessment, the stress intensity factors can be determined from the J-integral parameters:

where:

KI, KII = Stress intensity factor for modes I and II

JI, JII = J-integral values for modes I and II

J = Total J-integral value = JI + JII



5-18

= plane stress

= plane strain and axisymmetric

I, II = First and second (opening and shearing) crack modes

E, υ = Modulus of Elasticity and Poisson's ratio

The J-integral parameters are path-independent. They can be obtained from any arbitrary closed path which starts from one crack surface, travels around the crack tip and ends on the other crack surface.

Since this path can be taken well away from the crack tip singularity, it requires significantly less mesh refinement than other fracture assessment techniques.

Definition

Referring to Figure 5-5. The two-dimensional J-integral is defined by:

S = an arbitrary path (J path) surrounding the crack tip

= Outward normal to curve S

X = Crack axis

Y = Normal to crack axis

w = Strain energy density =

εij = Infinitesimal strain tensor

σij = Cauchy stress tensor

= The traction vector defined as: Ti = σij nj

= Displacement vector



Modification for Temperature

A = The area enclosed within Contour S

β = Coefficient of thermal expansion

δ = Cronecker delta

T = Temperature

Figure 5-5. Coordinates and Curves for Definition of the J-Integral

Axisymmetric Formulation

CrackSurface

CrackAxis

x

X

y

Y

n

J Path (Counter Clockwise) (Curve S)

dA

x, y = Global Cartesian coordinates

n

A

_n

= Areas enclosed within the contour s

= Outward normal to curve s

= Crack coordinatesX, Y



5-20

Rc = The crack tip radius

r = Radius

A = Area enclosed within the Contour S

The Requirements in Selection of the Path

1. The J-integral path should not pass through elements at the crack tip.

2. In case of elastic-plastic analysis, the J-integral path can pass through plastically deformed regions.

3. More than one path can be selected. Different paths around the same crack tip should render similar results.

Requirements for JI and JII Evaluation

In addition to the combined mode parameter J, the J-integral values for modes I, and II (crack opening, and shearing), can be evaluated. For this case:

1. The entire area around the crack tip must be included in modeling.

2. The J-path must be symmetric with respect to the crack axis. This infers that the finite element mesh inside the J-integral path (around the crack tip) must also be symmetric with respect to the crack axis.

3. In the case of elastic-plastic analysis, the J-integral path can only pass through elastically deformed regions.

4. Axisymmetric analysis and/or temperature gradients are not available.

Symmetric Modeling

If due to symmetry only half of a crack is modeled, then the total J-integral equals to JI, and JII = 0. Notice that the J value which is output in this case, is twice the value which is obtained from the path.



Specifications

Presently, the J-integral can be evaluated for two dimensional homogeneous structures (with cracks) which are modeled using PLANE2D (plane strain, plane stress, or axisymmetric) elements. Effects of thermal loading on crack parameters is also included.

Although, the presented J-integral formulation is valid for deformation theory of plasticity, the flow theory of plasticity (available in NSTAR) can be employed so long as the loading is proportional and there is no unloading. Note that the likelihood of violating these conditions is greater in presence of thermal gradients than for purely mechanical loading.

J-Integral Path Definition

The J-integral path is defined by a number of segments each representing a side of a finite element. The J-integral is evaluated by taking the path through the Gauss points which are closest to the selected element side, within the element.

Each J-integral segment is defined by two nodes and a global element number. J-segments must be defined in the proper (Counter Clockwise) order.

Figure 5-6 Part of a Finite Element Mesh Including a Crack Notch

17 16 15 14 13

Start of the J Path

2i

1

20

19

End of the J Path

Crack Axis

X

CrackTip

J Segments

3 4 5 6 7

8

9

10

11

1218

mkj



5-22

In the model shown in Figure 5-6, 20 segments must be input to define the selected J-path. Segment number 3, for example, is defined by nodes j and k and element m. (Segment 2 is defined by nodes i and j and element m). [Command J_INTDEF (Analysis > NONLINEAR > J INTEGRAL > Define Path)]

In case of an axisymmetric analysis and/or thermal loading, elements inside the J-integral path must also be input. No order is required for input of these elements. [Command J_INTELEM (Analysis > NONLINEAR > J INTEGRAL > Define Element)]

Special care must be given to avoid merging of the nodes along the two crack free surfaces.

Frequencies and Mode Shapes in a Nonlinear Environment

Frequency analysis of structures which include nonlinear effects (Geometric and/or material) is implemented in the NSTAR module.

The natural frequencies and mode shapes for a nonlinear structure are determined by performing a frequency analysis using the nonlinear stiffness matrix which is obtained from a nonlinear step-by-step solution.

Moreover, by performing several frequency analyses at different stages of a nonlinear solution, the variations of the structure's natural frequencies with respect to the level of loading (or time in the case of viscoelastic, or creep behavior) can be detected.

The solution procedure is as follows:

1. The NSTAR module determines and stores the current nonlinear stiffness and the mass matrices on files (the current stiffness matrix means the stiffness calculated at the last solution step).

2. The DSTAR module can then perform a frequency analysis of the structure using the latest nonlinear stiffness matrix calculated by NSTAR.



3. The nonlinear solution can then be continued by using the RESTART option, followed by another frequency analysis, followed by another nonlinear restart, and so on.

✍ The preparation of files for a frequency analysis must be requested prior to running NSTAR. [Command A_NONLINEAR (Analysis > NONLINEAR > NonL Analysis Options)].

✍ A frequency analysis can be performed, at any time, after a nonlinear run is successfully completed. [Command R_FREQUENCY (Analysis > FREQBUCK > Run Frequency)]

✍ All the information pertaining to the frequency as well as the nonlinear analysis must be input prior to the first nonlinear run (start from time zero).

✍ Prior to running for frequencies, it is necessary to request the frequency analysis to be based on the nonlinear results. [Command A_FREQUENCY (Analysis > FREQBUCK > Frequency Options)].

Buckling Analysis in a Nonlinear Environment

Buckling analysis of structures with inclusion of the nonlinear effects such as material nonlinearities, and gaps is implemented in the NSTAR and DSTAR modules. The buckling load parameter and mode shape is determined by performing a buckling analysis using the nonlinear stiffness matrix which is obtained from a nonlinear step-by-step solution.

Buckling analyses at different stages of a nonlinear solution, yield different buckling loads. As long as the structure is in the pre-buckling state, the buckling load parameter (which is obtained from a buckling analysis) must be greater than one. As the structure approaches a buckling state (during the nonlinear analysis), the buckling load parameter approaches one. In the post-buckling state, the buckling load parameter is usually less than one.

It must be noted, however, that at any stage of a nonlinear step-by-step solution, when the structure stiffness matrix becomes singular, the buckling analysis will predict a buckling state, i.e., the buckling load parameter equals one.



5-24

✍ The preparation of files for a buckling analysis must be requested prior to running NSTAR [Command A_NONLIN].

✍ A buckling analysis can be performed, at any time after a nonlinear run (if the nonlinear run is not successfully completed, the results of the last successful step will be used). [Command R_BUCK].

✍ All the information pertaining to the buckling as well as nonlinear analysis must be input prior to the first nonlinear run (start from time zero).

✍ Prior to the first buckling run, it is necessary to request the buckling analysis to be based on the nonlinear results. [Command A_BUCK].

Release of Global Prescribed Displacements

Prescribed displacements are the effect of unknown external forces that can be determined from the reaction forces; Each applied force is equal in magnitude but opposite in direction to the reaction force that is obtained for the degree of freedom with a prescribed displacement.

Considering a node for which displacement is prescribed, it is sometimes desired to release that node once a certain level of displacement/loading is reached. In addition, for certain problems, it may be desired to keep the pre-release force acting on the released node.

In another case, it may be desired to prescribe displacement(s) only after certain level of loading or displacement is reached. For example, consider a structure that is heated first. Next, certain nodes on it's boundary are secured at their heated position. And finally, the structure is cooled and it may be subjected to other loading conditions.

✍ The release periods are specified as those areas on the time curve (associated with a prescribed displacement) where the curve value is greater than 1.E8, as well as, the in-between areas, i.e., areas where one curve value is greater and one smaller than 1.E8.

✍ To request to keep the pre-release forces on released degrees of freedom, a flag (RFKEEP) is added to the A_NONLINEAR command.



✍ The release of a prescribed displacement may be sudden (dynamic) or slow (static).

In the dynamic case, the solution can be obtained in the following steps:

a. First the static response due to prescribed displacement(s) is found.

b. Next, a release must be specified for the time duration of the dynamic analysis.

c. The analysis type needs to be changed to dynamic.

d. Restart of the solution, yields the proper dynamic response.

In any case, displacements and forces may be prescribed for the same d.o.f. (forces become effective whenever the node is released.

Defining Temperatures Versus Time Relative to a Reference Temperature

If nodal temperatures are associated with a time curve that has a value of zero at time zero, the curve is assumed to prescribe relative temperatures (relative to reference) instead of prescribng Cotal temperatures:

Trel( n,t) = Curve_value(t) *( Tn - Tref ) Where: Trel( n,t) - Relative temperature at node n and time t

Tn - Input temperature at node n

Tref - Reference temperature specified by the TREF command

This assumption is made to provide ease of use for certain cases where otherwise each node requires a separate curve to specify its total temperature.



5-26

Modified Central Difference Technique for Dynamic Time Integration

A “Modified” Central Difference (explicit) technique is implemented into NSTAR for the integration of response (in time) in a dynamic analysis. As an explicit technique, Central Difference can be used to investigate the dynamic response of structures that are subjected to shock loading or (high impact) collisions. Here, the term “modified” has been used to represent the changes that are made to the Standard Central Difference to make it more practical and suitable for use. Since this method is only “Conditionally” Stable, it usually requires very small time increments which yield a great number of solution steps. Furthermore, it is not easy to guess a small-enough time increment for a general dynamic problem.

To avoid these difficulties and to speed up the solution, the following modifications are made.

1. Sub-Steps within Each Solution Step:

In order to reduce the impracticality and the length of time that results from too many solution steps, a special arrangement has been made:

As usual, End_Time and Time_Inc, specified in the TIMES command, are used to define:

NSTEP = number of steps for output & graphs

= End_Time / Time_Inc.

The dynamic analysis, however, is performed such that:

DSTEP = number of steps for dynamic integration

= NSTEP * isub

isub = number of sub-steps to be specified [Default=100]

This means that isub number of (sub-) steps are performed within each solution step. As a result, the time increment that is used for dynamic integration, will be different from Time_Inc:

dt = time increment for dynamic integration

= Time_Inc / isub

As an example, consider a certain model for which a dynamic step-by-step solution is to be performed. Moreover, lets assume that for this analysis to be accurate, at least a million solution steps are required. Here, by setting isub to be 1000, the number of external solution steps can be reduced to 1000.



2. Evaluation of Critical Time Increment:

Since Central difference is only conditionally stable, i.e., time increment should be smaller than a critical value, it is helpful to obtain this critical time value:

dt(critical) = Tn/π

where Tn is the system's smallest period.

To evaluate dt(critical), an iterative technique (Forward Iteration) is implemented which evaluates the highest frequency of the finite element system. This calculation is done once at the start of the solution (using the linear stiffness matrix) and the critical time increment is printed in the output file.

3. Mass and Damping:

In order to use central difference in its most efficient way, i.e., requiring no matrix decompositions, mass and damping matrices need to be diagonal. Thus, a Lumped mass matrix is used with a modified Rayleigh damping of the form:

[C] = β[M]

4. Checks for Convergence of Nonlinear Solution:

While the above technique of dividing total dynamic steps into steps and sub-steps is 100% accurate for a linear structure (nstep=1 and isub=1000, or nstep=100 and isub=10, or nstep=1000 and isub=1 are all equivalent), the same may not be true for nonlinear problems where material and/or geometric nonlinearities are to be considered. This is due to the fact that the structure stiffness is assumed to remain constant during the sub-step calculations. In other words, the stiffness is updated at intervals separated by isub number of (sub-) steps. To avoid divergence due to changes of stiffness during the sub-step phases of solution, another check is added to ensure that the out of balance forces remain small. Using norms of vectors:

| |{F´t}| - |{Ft}| | < toln * |{Ft}|

where toln is a preset tolerance and:

{Ft} = obtained internal force vector at time t

{F´t} = expected internal force vector at time t, assuming [Kt] is constant

5. Auto-Stepping for Nonlinear dynamic problems:

Auto-Stepping with considerations for the dynamic accuracy is available with the central difference technique. The Minimum/Maximum time increments are internally adjusted, based on the time increment that is specified by the TIMES command:



5-28

Maximum Time Inc. = specified Time_Inc.

6. Gaps and Contact:

The central difference method is extended to include Gaps and Contact algorithm with or without Friction. For this case, however, a stiffness should be assigned to the gap elements [default=1.5E7]. (Spring-Damper is excluded.)

7. Other capabilities:

Other capabilities include prescribed global displacements as dynamic excitations, local boundary conditions, and reaction force calculation. (Non-zero local prescribed displacements, and constraint equations with non-zero right hand sides are not available.)

8. Inclusion of dead loads in the dynamic analysis:

Similar to other dynamic methods, it is possible to perform a dynamic analysis, using central difference, following a static analysis. This feature maybe more useful here than it is for the unconditionally stable techniques; in this case, static modes can not be estimated by modes with zero masses or modes with too small periods. (See the section on guidelines.)

Advantages

The two-phase solution (steps and sub-steps), in absence of matrix decompositions, results in a solution procedure which is faster than implicit dynamic integration techniques such as Newmark or Wilson. The solution speed can help solve problems that need too many solution steps such as shock or high impact.

Disadvantages

The technique, being conditionally stable, imposes a limit on the size of the time increment. Since this limit depends on the smallest period of the finite element assemblage:

1. All degrees of freedom require to have non-zero masses; a zero mass means a zero period which means a zero value for critical time increment.

2. Systems having either very small masses or very large stiffnesses at some degrees of freedom relative to others, may not be appropriate to be analyzed by central difference.



✍ Specify some mass for all degrees of freedom; otherwise the program will assign an approximate value (1% of the minimum mass used for any degree of freedom) to all degrees of freedom with no mass. (If the reaction force flag is on, and you get a warning message about degrees of freedom with zero masses, you can ignore it)

✍ In order to exclude the static modes from dynamic analysis, never use small masses or large stiffness. Instead, use one of the two following procedures:

a. First perform a static analysis raising the static forces in a very small time duration (1.e-7 seconds), then keeping the static forces constant, restart to perform a dynamic solution.

b. To minimize the dynamic effects due to static loads, the time curve, associated with static loads, can be defined such that, loads are raised from zero to their proper value in a very short time (smaller than dynamic time inc.), and are kept constant thereafter.

✍ When nonlinearities (material and/or geometric) exist, Auto-Stepping can be used with central difference.

✍ When there is possibility of contact/impact, it must be noted that gaps or contact elements act as stiff springs which can result in a reduction of the critical time increment; This effect is not included in the evaluation of the critical time increment and must be considered by the user in selecting a proper time inc.

For best results, first perform a trial run to find the approximate time of the impact. Next, you can divide solution time into two or more portions (using restart), for which different time increments (and/or different number of sub-steps) are used (smaller time incs. and/or larger isubs may be required during the impact phase). Make sure for every duration you re-issue the NL_AUTO command and change the maximum time increment to the time increment you are requesting in the TIMES command.

✍ Friction is based on the generalized model (friction may be sliding or non-sliding), regardless of the type that is specified in the input.

✍ Note that the variation of loads during the sub-step phase is assumed to be linear, i.e., linear interpolation is used between the times of two consecutive steps.


5-30

✍ It must also be noted that, for this technique, the accelerations and velocities are always one step behind the displacements (and stresses). As a result, there is an error in display of time in the xy-plots of accelerations or velocities (time coordinates should be reduced by a time increment).

✍ If the solution requires many reductions of the time increment due to:

“Out of Balance Loads Diverging”

solution time as well as the number of steps may be reduced by selecting a smaller time increment.

✍ If the selected time increment is larger than critical, or nonlinearities exist in the model (impact in particular), to assure convergence, it is helpful to compare results between two runs for which two different time increments are used.

Combination of Force Control and Displacement/Arc-Length Control Methods

Definitions:

Parametric Loads are loads that represent a loading pattern. They are not defined by time curves and their actual value at a state of deformation, is defined by the load factor.

Non-Parametric Loads are loads are loads with known magnitudes defined by multipliers and time curves.

Assumption:(for displacement or arc-length analysis)

All loads (concentrated forces, pressures, temperatures, centrifugal, gravity) that are associated with time curve number 1 are considered to be parametric loads. Loads that are associated with a time curve >1 are considered as non-parametric (defined by the time curve).

✍ For displacement control, time curve 1 describes the displacement of the controlled degree of freedom. For the arc length, time curve 1 is ignored.



A combined analysis of parametric and non-parametric loads can be performed in two ways:

• Using displacement control, and including both types of loads in the analysis, simultaneously.

• Starting with force control to find deformation under non-parametric loads (curve 1 should have zero values during this phase), then restart with either displacement or arc-length control and obtain the load factor and response due to parametric loads while other loads are kept constant.

Artificial Bulk Viscosity

Artificial bulk viscosity can be used to provide response continuity for shock waves. The bulk viscosity damps out the response slightly, replacing the shock discontinuities with rapidly varying but continuous response.

The use of the bulk viscosity will not affect regions of the model away from the shock front, while maintaining the jump conditions across the shock transition.

This technique introduces an additional viscous term, q, to relax the element internal pressure. Two variations are implemented:

1. Quadratic Form, used for strong shocks

q = ρ l {C0 l (δεκκ/δt)2 − C1 C2 δεκκ/δt } if δεκκ /δt < 0 q = 0 if δεκκ /δt >= 0

2. Linear Form, used for weak shocks [Strain rate << Wave speed]

q = ρ l {− C1 C2 δεκκ/δt }

Where:

εκκ = Trace of the strain tensor = ε11 + ε22 + ε33

δ /δt = Variation with respect to time

ρ = Density

l = Equivalent Length = (Volume)(1/3) for 3D

= (Area)(1/2) for 2D

C0 = Dimensionless Constant [default = 1.5]

C1 = Dimensionless Constant [default=0.06]



5-32

C2 = Material Wave Propagation Speed

[default = {(Κ+ 4G/3)/ρ }(1/2) , Κ = Bulk Modulus, G = Shear Modulus]

The above formulation can be activated, using Option 7 (Option 7 =2 or 3) in the EGROUP command, for the SOLID, TETRA4, TETRA10, PLANE2D, & TRIANG elements. Parameters C0, C1, & C2 can be specified using command:

MPROP, ,CREEPC,…

References

1. Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice Hall, 1982.

2. Bathe, K J., and A. Cimento, “Some Practical Procedures for the Solution of Nonlinear Finite Elements Equations,” Comput. Meth. Appl. Mech. Eng., 22:59-85, 1980.

3. Grisfield, M. A., “A Fast Incremental/Iterative Solution Procedure That Handles 'Snap-through', “Comput. Struct., 13:55-62, 1980.

4. Grisfield, M. A., Finite Elements and Solution Procedures for Structural Analysis, Vol. I: Linear Analysis, Pineridge Press Limited, U.K., 1986.

5. Geradin, M., S. Idelsohn, and M. Hogge, “Computational Strategies for the Solution of Large Nonlinear Problems via Quasi-Newton Methods,” Comput. Struct., 13:73-81, 1981.

6. Geradin, M., M. Hogge, and S. Idelsohn, in T. Belytschko and T. Hughes (eds.), “Implicit Finite Element Methods,” Computational Methods for Transient Analysis, North-Holland, Amsterdam, 1983,chap. 4, pp. 417-471.

7. Mathies, H., and G. Strang, “The Solution of Nonlinear Finite Element Equations,” Int. J. Numer. Meth. Eng., 14:1613-1626, 1979.

8. Riks, E., “An Incremental Approach to the Solution of Snapping and Buckling Problems,” Int. J. Numer. Meth. Eng., 15:529-551, 1979.

9. Tsai, C. T., and A. N. Palazotto, “Nonlinear and Multiple Snapping Responses of Cylindrical Panels Comparing Displacement Control and Riks Method,” Comput. Struct., 41:605-610, 1991.



10. Zienkiewicz, O. C., “Incremental Displacement in Nonlinear Analysis,” Int. J. Numer. Meth. Eng., 3:587-588, 1971.

11. Zienkiewicz, O. C., and Taylor, R. L., The Finite Element Method, Vol. II, Fourth edition, McGraw-Hill, 1991.



5-34

6 Element Library

Introduction

In this chapter, general information regarding the different element types, their geometric dimensions and dimensional behavior are presented. The different material models available for use with the different element groups are also furnished. Tables for real constants for NSTAR elements are also provided.


Chapter 6 Element Library

6-2

Table 6-1. Element Library: General Information

COSMOSM Element

Name Description

Number of DOF

/Node

ElementDimensional

Behavior

TRUSS2D Plane Truss 21D/ 2D (XY-Plane)

TRUSS3D Space Truss 3 1D/ 2D/ 3D

BEAM2D Plane Beam 31D/ 2D (XY-Plane)

BEAM3D Space Beam 6 1D/ 2D/ 3D

IMPIPE Immersed Pipe 6 2D/ 3D

SPRING Axial and/or Torsional Spring 3 to 6 1D/ 2D/ 3D

PLANE2D4 to 8-node (Plane Stress, Strain, Axisymmetric)

2 2D (XY-Plane)

TRIANG3 to 6-node (Plane Stress, Strain, Axisymmetric)

2 2D (XY-Plane)

SHELL3 3-node Triangular Thin Shell 6 2D/ 3D

SHELL4 4-node Quadrilateral Thin Shell 6 2D/ 3D

SHELL3T 3-node Triangular Thick Shell 6 2D/ 3D

SHELL6 6-node Triangular Thin Shell 6 2D/ 3D

SHELL6T 6-node Triangular Thick Shell 6 2D/ 3D

SHELL4T 4-node Quadrilateral Thick Shell 6 2D/ 3D

SHELL3L 3-node Triangular Composite Shell 6 2D/ 3D

SH3LL4L 4-node Quadrilateral Composite Shell 6 2D/ 3D

SOLID 8 to 20-node Continuum Brick 3 3D

TETRA4 4-node Continuum Tetrahedron 3 3D

TETRA10 10-node Continuum Tetrahedron 3 3D

GAP Gap/Contact with Friction 1/2/3* 1D/ 2D/ 3D

MASS Concentrated Mass 6 1D/ 2D/ 3D

BUOY Immersed Spherical Mass 6 1D/ 2D/ 3D

GENSTIF General Stiffness 6 3D

• According to the contact nodes



Table 6-2a. Element Library: Linear Material Model Information

COSMOSM Element Name

Elastic Viscoelastic IsotropicIsotropic Orthotropic Composite

TRUSS2D • •

TRUSS3D • •

BEAM2D • •

BEAM3D • •

IMPIPE •

SPRING •

PLANE2D • • •

TRIANG • • •

SHELL3 •

SHELL4 •

SHELL3T • •

SHELL6 •

SHELL6T • •

SHELL4T • •

SHELL3L • • •

SHELL4L • • •

SOLID • • •

TETRA4 • • •

TETRA10 • • •

• Available



6-4

Table 6-2b. Element Library: Nonlinear Material Model Information

COSMOSM Element Name

ElasticCurved

Description

Elastic-Plastic Hyperelastic

CreepClassical

CreepExp

Mooney-Rivlin

Ogden Blatz-

Ko

TRUSS2D • • •

TRUSS3D • • •

BEAM2D •

BEAM3D •

IMPIPE

SPRING •

PLANE2D • • • • • •

TRIANG • • • • • •

SHELL3

SHELL4

SHELL3T • • •

SHELL6

SHELL6T • • •

SHELL4T • • •

SHELL3L

SH3LL4L

SOLID • • • • • •

TETRA4 • • • • • •

TETRA10 • • • • • •

• Available



Table 6-2c. Element Library: Nonlinear Material Model Information

COSMOSMElement

Name

Plastic Bilinear + Curve

Description

PlasticElastic-

Perfectly Plastic

Drucker-Prager

PlasticCon-crete

FabricMem-brane

Failure of Com-posites

User-Defined von

MisesIso

vonMises

Kin

TRUSS2D • • • •

TRUSS3D • • • •

BEAM2D • •

BEAM3D • •

IMPIPE

SPRING

PLANE2D • • • • • * •

TRIANG • • • • • * •

SHELL3

SHELL4

SHELL3T • • • +

SHELL6



SHELL3L •

SH3LL4L •

SOLID • • • • •

TETRA4 • • • • •

TETRA10 • • • • •

• Available* Plane Stress+ Membrane



6-6

Table 6-3. Element Library: Real Constants

COSMOSMElement

Name

RC Constants

No. RC Description

TRUSS2D 4

RC1RC2RC3RC4

Cross sectional areaCross sectional perimeter (for thermal analysis only)Initial axial force (with large displacement options)Initial axial strain (with large displacement options)

TRUSS3D 4

RC1RC2RC3RC4

Cross sectional areaCross sectional perimeter (for thermal analysis only)Initial axial force (with large displacement options)Initial axial strain (with large displacement options)

BEAM2D 8

RC1RC2RC3RC4RC5RC6RC7RC8

Cross-sectional areaMoment of inertiaDepth (diameter for circular cross section)End release code at node 1End release code at node 2Shear factor in the element y-axisTemperature difference in the element y-axisPerimeter (for thermal analysis only)

BEAM3D14 to

27

RC1RC2……

Cross sectional areaMoment of inertia… … …Temperature difference in the element y-axisPerimeter (for thermal analysis only)… … …(Refer to the elements chapter in COSMOSM User Guide for more information)



Table 6-3. Element Library: Real Constants (Continued)

COSMOSMElement

Name

RC Constants

No. RC Description

IMPIPE 17

RC1RC2RC3RC4RC5RC6RC7RC8RC9RC10RC11RC12—RC13RC14RC15RC16RC17

Outside diameter of the pipePipe thicknessFlexibility factorEnd-release codeInternal pressureInternal fluid densityGlobal Z-coordinate of the pipe internal fluid's free surfaceMass of internal fluid/hardwareExternal insulation densityThickness of external insulationCoefficient of buoyant forceCoefficient of axial strain correction due to external hydrostatic and hydrodynamic pressuresCoefficient of added massCoefficient of fluid inertia forceCoefficient of normal dragCoefficient of tangential dragPrestrain

SPRING 2RC1RC2

Axial stiffness (for linear material only)Rotational stiffness (for linear material only)

PLANE2D 2RC1RC2

Thickness (for plane stress option only)Material angle (for linear orthotropic material only)

TRIANG 2RC1RC2

Thickness (for plane stress option only)Material angle (for linear orthotropic material only)

SHELL3/4/6SHELL3T/4T/6T

6

RC1RC2RC3RC4RC5RC6

ThicknessTemperature gradientUnused constant for this elementUnused constant for this elementNormal prestress value for element x and y directionsNormal prestrain value for element x and y directions



6-8

Table 6-3. Element Library: Real Constants (Concluded)

COSMOSMElement

Name

RC Constants

No. RC Description

SHELL3L/4L2+3NL

RC1RCxx

The number of real constants to be entered depends on the number of layers (NL). Refer to the elements chapter in COSMOSM user guide for more information.

TETRA4R 9 RC1-9X,Y,Z coordinates of three points (for orthotropic materials only)

SOLID 9 RC1-9X,Y,Z coordinates of three points (for orthotropic materials only)

GAP(Node-to-Node)

7

RC1RC2RC3RC4RC5

RC6RC7RC8RC9RC10RC11RC12RC13

Allowable relative displacement between two nodes Coefficient of frictionSpring stiffnessPreload in the gapMaximum allowable distance beyond which gap responds as perfectly rigidDamper constantDamper constantSource Stiffness in X-directionSource Stiffness in Y-directionSource Stiffness in Z-directionTarget Stiffness in X-directionTarget Stiffness in Y-directionTarget Stiffness in Z-direction

MASS 7

RC1RC2RC3RC4RC5RC6RC7

Mass in the global Cartesian X-directionMass in the global Cartesian Y-directionMass in the global Cartesian Z-directionRotary inertia about the global Cartesian X-directionRotary inertia about the global Cartesian Y-directionRotary inertia about the global Cartesian Z-directionThermal capacity (in units of heat energy)

BUOY 11

RC1RC2RC3RC4RC5RC6RC7RC8RC9RC10RC11

Mass in the global Cartesian X-directionMass in the global Cartesian Y-directionMass in the global Cartesian Z-directionRotary inertia about the global Cartesian X-directionRotary inertia about the global Cartesian Y-directionRotary inertia about the global Cartesian Z-directionOutside diameter of the buoyCoefficient of buoyant forceCoefficient of added massCoefficient of fluid inertia forceCoefficient of drag


7 Commands and Examples

Command Summary

Table 7-1. Frequently Used Commands for Nonlinear Analysis

Command/Menu Path

A_NONLINEAR Analysis > NONLINEAR > NonL Analysis Options

ACTSET Control > ACTIVATE > Set Entity

CONTACT > Analysis > NONLINEAR >

CURDEF LoadsBC > FUNCTION CURVE > Time/Temp Curve

INITIAL LoadsBC > LOAD OPTIONS > Initial Cond

J INTEGRAL Analysis > NONLINEAR >

NL_AUTO Analysis > NONLINEAR > AutoStep Options

NL_BASE Analysis > NONLINEAR > Base Motion Parameter

NL_CONTROL Analysis > NONLINEAR > Solution Control

NL_INTGR Analysis > NONLINEAR > Integration Options

NL_NRESP Analysis > NONLINEAR > Response Options

NL_PLOT Analysis > NONLINEAR > Plot Options

NL_PRINT Analysis > NONLINEAR > Print Options

NL_RDAMP Analysis > NONLINEAR > Damping Coefficient

MPC LoadsBC > FUNCTION CURVE > Material Curve

MPCTYP LoadsBC > FUNCTION CURVE > Material Curve Type

R_NONLINEAR Analysis > NONLINEAR > Run NonL Analysis

RESTART Analysis >

TIMES LoadsBC > LOAD OPTIONS > Time Parameter

EKILL and ELIVE Analysis > NONLINEAR >ELEMENT_BIRTH/DEATH


Chapter 7 Commands and Examples

7-2

The above represent some of the most frequently used commands needed to perform nonlinear analysis using the NSTAR module. Information regarding analysis type, direct time integration, initial conditions, damping, time and temperature curves associated with different loading conditions and material property sets, and numerical solution procedures are provided using these commands. In addition, both line and surface contact problems may be considered for analysis with the nonlinear module. Parameters required to define material models are discussed in Chapter 3. Command descriptions are presented in COSMOSM Command Reference Manual.

NSTAR can handle geometric, material, and contact nonlinearities. The temperature dependency of material properties can also be handled.

As a guide to the users, a brief outline describing the application of specific commands required to set up different categories of nonlinear problems is presented.

Elastoplastic Analysis

In the following, only the commands essential for this nonlinear analysis case are listed.

Command (Path) Intended Use

EGROUP(Propsets > Element Group) Option 5 of this command

specifies the use of von Mises elastoplastic model with an isotropic or a kinematic hardening rule for this element group.

MPROP(Propsets > Material Property) Using this command the

required material properties (EX, ETAN, NUXY, and SIGYLD) for defining the bilinear elastoplastic stress-strain curve are input.



MPCTYP and MPC (LoadsBC>FUNCTION CURVE>Material Curve Type, Material Curve) These commands are used for

curve description of elastoplastic stress-strain curve.

TIMES (LoadsBC > LOAD OPTIONS > Time Parameter) The command is used to define

the starting time, the ending time and the time step increment for all nonlinear analysis cases. This command is not needed if the Arc-length Control technique is used.

ACTSET, TC, Curve_Label (Control > ACTIVATE > Set Entity) Commands to activate time

curves (previously defined by the CURDEF command) and define the accompanying forces and/or pressure and/or prescribed displacements. Note that it is a good practice to deactivate the curves immediately after their use to avoid errors or confusion of any kind. This command is not needed if the Arc-length Control technique is used.

FND, FPT, FCR, FSF, FCT, FRG (LoadsBC > STRUCTURAL > FORCE >) See ACTSET, TC,

Curve_Label above.PEL, PCR, PSF, PRG(LoadsBC > STRUCTURAL > PRESSURE >) See ACTSET, TC,

Curve_Label above.DND, DPT, DCR, DSF, DCT, DRG (LoadsBC>STRUCTURAL>DISPLACEMENT>) See ACTSET, TC,

Curve_Label above.ACTSET, TC, 0 (Control > ACTIVATE > Set Entity) See ACTSET, TC,

Curve_Label above.



7-4

A_NONLINEAR***(Analysis > NONLINEAR > NonL Analysis Options) This command specifies the

nonlinear option (S = STATIC, or D = DYNAMIC) and some other relevant options and parameters.

NL_CONTROL***(Analysis > NONLINEAR > Solution Control) This command specifies the

numerical procedure to be used in nonlinear analysis. It defines the Control technique, the iterative method, and their associated input.

NL_AUTO***(Analysis > NONLINEAR > AutoStep Options) This command is used to

activate the adaptive automatic stepping option in nonlinear structural analysis. Delimiters for the step size can be specified.

NL_NRESP(Analysis > NONLINEAR > Response Options) This command is used to select

nodes for which the displacement response is to be saved for X-Y plot purposes.

ACTSET, TC, Curve_Label NL_PLOT(Analysis > NONLINEAR > Plot Options) This command is used to define

sets of steps for which the deformations and stresses are to be plotted.

ACTSET, TC, Curve_Label NL_PRINT(Analysis > NONLINEAR > Print Options) This command controls the

output quantities to be written in the output file.

ACTSET, TC, Curve_Label R_NONLINEAR(Analysis > NONLINEAR > Run NonL Analysis) This command runs NSTAR

which is the nonlinear module of COSMOSM.

As it is evident, the commands required to completely describe the finite element model and other properties should be defined before issuing the solution command [R_NONLINEAR (Analysis > NONLINEAR > Run NonL Analysis)].



** Load curves essential for the incremental solution of nonlinear static problems are represented by (pseudo) time curves. These curves are defined using the CURDEF (LoadsBC > FUNCTION CURVE > Time/Temp Curve) command.

*** If the default options (and/or parameters) are satisfactory, this com-mand may be omitted.

Geometrically Nonlinear Analysis

The special commands required to set up the geometrically nonlinear analysis available in NSTAR are as follows:


EGROUP(Propsets > Element Group) Option 6 of this command sets

the flag on for geometrically nonlinear analysis formulation to be associated with this element group.

TIMES(LoadsBC > LOAD OPTIONS > Time Parameter) See the Elastoplastic Analysis

section.ACTSET, TC, Curve_Label (Control > ACTIVATE > Set Entity) See the Elastoplastic Analysis

section.FND, FPT, FCR, FSF, FCT, FRG (LoadsBC > STRUCTURAL > FORCE >) See the Elastoplastic Analysis

section.PEL, PCR, PSF, PRG (LoadsBC > STRUCTURAL > PRESSURE >) See the Elastoplastic Analysis

section.DND, DPT, DCR, DSF, DCT, DRG (LoadsBC > STRUCTURAL > DISPLACEMENT >) See the Elastoplastic Analysis

section.ACTSET, TC, 0 (Control > ACTIVATE > Set Entity) See the Elastoplastic Analysis

section.



7-6

A_NONLINEAR(Analysis > NONLINEAR > NonL Analysis Options) See the Elastoplastic Analysis

section.NL_CONTROL(Analysis > NONLINEAR > Solution Control) See the Elastoplastic Analysis

section.NL_AUTO(Analysis > NONLINEAR > AutoStep Options) See the Elastoplastic Analysis

section.R_NONLINEAR(Analysis > NONLINEAR > Run NonL Analysis) See the Elastoplastic Analysis

section.

Again, it is assumed that the full model has been properly constructed with the help of GEOSTAR commands.

Elastoplastic Large Displacement Analysis


EGROUP(Propsets > Element Group) Options 5 and 6 are both turned

on to specify the elastoplastic model and the geometric nonlinear option.

MPROP(Propsets > Material Property) See the Elastoplastic Analysis

section.MPCTYP and MPC(LoadsBC>FUNCTION CURVE > Material Curve Type, Material Curve) See the Elastoplastic Analysis

section.TIMES(LoadsBC > LOAD OPTIONS > Time Parameter) See the Elastoplastic Analysis


section.FND, FPT, FCR, FSF, FCT, FRG(LoadsBC > STRUCTURAL > FORCE >) See the Elastoplastic Analysis

section.



PEL, PCR, PSF, PRG(LoadsBC > STRUCTURAL > PRESSURE >) See the Elastoplastic Analysis

section.DND, DPT, DCR, DSF, DCT, DRG (LoadsBC>STRUCTURAL>DISPLACEMENT> See the Elastoplastic Analysis

section.ACTSET, TC, 0 (Control > ACTIVATE > Set Entity) See the Elastoplastic Analysis

section.A_NONLINEAR(Analysis>NONLINEAR>NonL Analysis Options) See the Elastoplastic Analysis




section.

Nonlinear Dynamic Analysis

The nonlinear dynamic analysis based on direct time integration methods requires the following special commands:


EGROUP(Propsets > Element Group) Options 5 and 6 both control the

material models and the geometric nonlinear analysis flags.

MPROP(Propsets > Material Property) See the Elastoplastic Analysis

section.MPCTYP and MPC(LoadsBC>FUNCTION CURVE>Material Curve Type, Material Curve) See the Elastoplastic Analysis

section.



7-8


section.

ACTSET, TC, Curve_Label (Control > ACTIVATE > Set Entity)

FND, FPT, FCR, FSF, FCT, FRG (LoadsBC > STRUCTURAL > FORCE >)

PEL, PCR, PSF, PRG (LoadsBC > STRUCTURAL > PRESSURE >)

ACTSET, TC, 0 (Control > ACTIVATE > Set Entity) See the Elastoplastic Analysis

section.A_NONLINEAR(Analysis>NONLINEAR>NonL Analysis Options) The analysis option should be

switched to “D” for dynamic.NL_CONTROL(Analysis > NONLINEAR > Solution Control) The Force control option with

MNR or NR methods must be used. No line search is performed for dynamic analysis.

NL_RDAMP(Analysis > NONLINEAR > Damping Coefficient) This command is used to

incorporate Rayleigh proportional damping in the dynamic analysis.

NL_BASE(Analysis>NONLINEAR>Base Motion Parameter) This command is used to

incorporate the effects of base motion accelerations in the dynamic analysis.

NL_INTGR(Analysis > NONLINEAR > Integration Options) This command can be used to

choose the direct implicit time integration schemes. The user can select either Newmark-Beta or Wilson-Theta methods. If this command is not issued, the Newmark-Beta method with defaults values is incorporated in the analysis.



NL_AUTO(Analysis > NONLINEAR > AutoStep Options) See the Elastoplastic Analysis

section.NL_NRESP(Analysis > NONLINEAR > Response Options) This command is used to select

nodes for which the displacement, velocity, and acceleration responses are to be saved for X-Y-plotting purposes.

NL_PLOT(Analysis > NONLINEAR > Plot Options) See the Elastoplastic Analysis

section.NL_PRINT(Analysis > NONLINEAR > Print Options) See the Elastoplastic Analysis


section.

Linear Dynamic Analysis (Time-History)

The nonlinear module NSTAR may also be used for the solution of linear dynamic problems using time integration methods. Special commands required to set up this case are listed below.



section.ACTSET, TC, Curve_Label(Control > ACTIVATE > Set Entity) See the Elastoplastic Analysis


section.PEL, PCR, PSF, PRG (LoadsBC > STRUCTURAL > PRESSURE >) See the Elastoplastic Analysis

section.



7-10

ACTSET, TC, 0 (Control > ACTIVATE > Set Entity) See the Elastoplastic Analysis

section.A_NONLINEAR(Analysis>NONLINEAR>NonL Analysis Options) See the Nonlinear Dynamic

Analysis section.NL_CONTROL(Analysis > NONLINEAR > Solution Control) See the Nonlinear Dynamic

Analysis section.NL_RDAMP(Analysis > NONLINEAR > Damping Coefficient) See the Nonlinear Dynamic

Analysis section.NL_BASE(Analysis>NONLINEAR>Base Motion Parameter) See the Nonlinear Dynamic

Analysis section.NL_INTGR(Analysis > NONLINEAR > Integration Options) See the Nonlinear Dynamic

Analysis section.NL_AUTO(Analysis > NONLINEAR > AutoStep Options) See the Elastoplastic Analysis


section.

Analysis Including Temperature Loading



section.ACTSET, TC, Curve_Label (Control > ACTIVATE > Set Entity) These commands are used to

activate time curves and define the accompanying loading temperature. Again, it is a good practice to deactivate the curves immediately after their use to avoid errors or confusion of any kind.



NTND, NTPT, NTCR, NTSF, NTCT, NTRG (LoadsBC > THERMAL > TEMPERATURE >) See ACTSET, TC,

Curve_Label above.ACTSET, TC, 0 (Control > ACTIVATE > Set Entity) See ACTSET, TC,

Curve_Label above.A_NONLINEAR(Analysis>NONLINEAR>NonL Analysis Options) In this command, the special

loading flag must include the character “T''.

NL_CONTROL(Analysis > NONLINEAR > Solution Control) See the Elastoplastic Analysis



section.

Structural Analysis with Temperature-Dependent Material Properties

The relevant commands needed to set up this analysis for problems with temperature-dependent material properties are:


EGROUP(Propsets > Element Group) Options 5 and 6 both control the

material models and the geometric nonlinear analysis flags (refer to COSMOSM Command Reference Manual for a list of different material models with temperature-dependent parameters).

ACTSET, TP, Curve_Label (Control > ACTIVATE > Set Entity) These commands are used to

activate temperature curves and define the associated temperature-dependent-



7-12

material properties. Again, it is a good practice to deactivate the curves immediately after their use to avoid errors or confusion of any kind.

MPROP(Propsets > Material Property) See ACTSET, TP,

Curve_Label above.EX, SIGYLD, ETAN,....., etc.

ACTSET, TP, 0 (Control > ACTIVATE > Set Entity) See ACTSET, TP,

Curve_Label above.TIMES(LoadsBC > LOAD OPTIONS > Time Parameter) See the Elastoplastic Analysis

section.ACTSET, TC, Curve_Label(Control > ACTIVATE > Set Entity) See the Analysis Including

Temperature Loading section.NTND, NTPT, NTCR, NTSF, NTCT, NTRG (LoadsBC > THERMAL > TEMPERATURE >) See the Analysis Including

Temperature Loading section.ACTSET, TC, 0(Control > ACTIVATE > Set Entity) See the Analysis Including

Temperature Loading section.A_NONLINEAR(Analysis > NONLINEAR > NonL Analysis Options) See the Analysis Including

Temperature Loading section.NL_CONTROL(Analysis > NONLINEAR > Solution Control) See the Elastoplastic Analysis



section.



Elastic Creep Analysis

The special commands essential for setting up the input for this type of analysis are given below:


EGROUP(Propsets > Element Group) Option 7 of this command sets

the flag for creep analysis with this element group.

MPROP(Propsets > Material Property) Use CREEPC or CREEPX

according to the creep law used in the analysis to define creep constants.




section.PEL, PCR, PSF, PRG(LoadsBC > STRUCTURAL > PRESSURE >) See the Elastoplastic Analysis

section.DND, DPT, DCR, DSF, DCT, DRG(LoadsBC > STRUCTURAL > DISPLACEMENT >) See the Elastoplastic Analysis

section.ACTSET, TC, 0(Control > ACTIVATE > Set Entity) See the Elastoplastic Analysis

section.A_NONLINEAR Command (Analysis>NONLINEAR>NonL Analysis Options) See the Elastoplastic Analysis



section.



7-14

R_NONLINEAR(Analysis > NONLINEAR > Run NonL Analysis) See the Elastoplastic Analysis

section.

Static Analysis Using Displacement Control Technique



section.

Define a Pattern of Loads (no association with time curve).ACTSET, TC, Curve_Label = 1(Control > ACTIVATE > Set Entity) This curve will be associated

with the controlled degree of freedom used in the solution process using the Displacement Control technique.

FND, FPT, FCR, FSF, FCT, FRG(LoadsBC > STRUCTURAL > FORCE >) —

PEL, PCR, PSF, PRG (LoadsBC > STRUCTURAL > PRESSURE >) —

A_NONLINEAR(Analysis>NONLINEAR>NonL Analysis Options) See the Elastoplastic Analysis

section.NL_CONTROL(Analysis > NONLINEAR > Solution Control) The Displacement Control

option with MNR or NR iterative methods is selected. Also, the controlled degree of freedom is specified.


section.NL_NRESP(Analysis > NONLINEAR > Response Options) This command is used to select

nodes for which the displacement response is to be saved for XY-plotting



purposes. X-Y-plots of the load factor multiplier (LFACT), on the Y-axis, versus the nodal displacement components (UX, UY, UZ, ...), on the X-axis, of the selected nodes can be provided.




section.

Static Analysis Using Arc-Length Control Technique

Define a Pattern of Loads (no association with time curve).


FND, FPT, FCR, FSF, FCT, FRG(LoadsBC > STRUCTURAL > FORCE >) Forces and pressures defined:

ACTSET,TC,1 will be considered. (No time curves need to be defined).

PEL, PCR, PSF, PRG (LoadsBC > STRUCTURAL > PRESSURE >)

A_NONLINEAR(Analysis > NONLINEAR > NonL Analysis Options) See the Elastoplastic Analysis

section.NL_CONTROL(Analysis > NONLINEAR > Solution Control) The Arc-Length Control option

with MNR or NR iterative methods is selected. Also, the parameters required for this control are input.



7-16


section.NL_NRESP(Analysis > NONLINEAR > Response Options) See the Static Analysis Using

Displacement Control Technique section.




section.

Examples

The following are examples of nonlinear analyses.

Elastoplastic Nonlinear Analysis Example

An example of elastoplastic, nonlinear, static analysis is described. This includes the descriptions of the steps required to set up and solve the problem, in detail from GEOSTAR.

Statement of the Problem

A cantilever metal sheet is subjected to a uniform pressure along the edge as shown in Figure 7-1. Investigate the elastoplastic response of this metal sheet using 4-node, 2D plane stress elements. The material of the sheet is assumed to obey the von Mises yield criterion.



Figure 7-1. Problem Sketch

GivenE = EX = 21,000 N/mm2

L = 30 mm

σy = 10 N/mm2

ν = 0.3

ET = 5,000 N/mm2

t = 1 mm

GEOSTAR Input

1. Define the element group. For this example, the 2D plane stress element is selected.


Element group > 1Element category > Area

Element type (for area) > PLANE2D

9

L

E

E

T

σ

ε

σy

Stress - Strain Curve

10

13

1 2 8

t

p

Time Curve

L

4 5 6

7 8 9

1 2 31 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

Problem Sketch



7-18

OP1:S/F flag > Solid

OP2:Integr Type > QM6

OP3:Type > Plane Stress

OP4:Stress direction > Global Cartesian

OP5:Mat > von Mises (isotropic)

OP6:Disp. > Small

OP7:Material creep > No

OP8:Strain plasticity > Small

2. The plasticity model used in the analysis is based on von Mises yield criterion with bilinear isotropic hardening rule. Define the bilinear stress-strain curve (Figure 7-1) by EX, ETAN and SIGYLD options in command MPROP (Propsets > Material Property). EX defines Young's Modulus, ETAN defines the Tangent Modulus (ET) and SIGYLD defines the Yield Stress σy.

Geo Panel: Propsets > Material Property (MPROP)Material property set > 1Material property name > EX

Property value > 21000

Material property name > ETAN


Material property name > SIGYLD

Property value > 10

The default value of NUXY is 0.3. Therefore, it is not necessary to specify Poisson's ratio unless other than the default value is required.

3. Define the thickness of the plane stress element.


Associated element group > 1Real constant set > 1

Start location of the real constants > 1No. of real constants to be entered > 2

RC1: Thickness > 1.0

RC2: Material angle (beta) > 0.0

4. Define the geometry of the model. Change the view to X-Y using the Viewing (Binocular) icon.

Geo Panel: Geometry > SURFACES > Draw w/ 4 Coord (SF4CORD)



Surface > 1Keypoint 1 XYZ-coordinate value > 0,0,0

Keypoint 2 XYZ-coordinate value > 30,0,0



5. Use the Auto Scale icon to readjust the model scale. Define the elements and nodes through mesh generation.

Geo Panel: Geo Panel: Meshing > PARAMETRIC MESH > Surfaces (M_SF)

Beginning surface > 1Ending surface > 1Increment > 1Number of nodes per element > 4

Number of elements on first curve > 3

Number of elements on second curve > 3

Spacing ratio for first curve > 1.0

Spacing ratio for second curve > 1.0

6. Define displacement constraints using the DCR (LoadsBC > STRUCTURAL > DISPLACEMENT > Define Curves) command.

Geo Panel: LoadsBC > STRUCTURAL > DISPLACEMENT > Define by Curves (DCR)

Beginning curve > 2Displacement label > AL: All 6 DOF

Accept defaults ...

7. Define the starting time, final time and time increment using the TIMES (LoadsBC > LOAD OPTIONS > Time Parameter) command.

Geo Panel: LoadsBC > LOAD OPTIONS > Time Parameter (TIMES)

Starting time > 0.0

Final time > 8Time increment > 1

8. Now define the load versus time curve using the CURDEF (LoadsBC > FUNCTION CURVE > Time/Temp Curve) command.

Geo Panel: LoadsBC > FUNCTION CURVE > Time/Temp Curve (CURDEF)



7-20

Curve type > Time

Curve number > 1

Start point > 1Time value for point 1 > 0.0

Function value for point 1 > 0Time value for point 2 > 1.0

Function value for point 2 > 9Time value for point 3 > 2.0

Function value for point 3 > 10

Time value for point 4 > 8.0


Time value for point 5 > 9.0


Time value for point 6 >

9. A number of time-load curves can be defined. However, the curve associated with the applied load must be activated prior to the definition of the load. Since you just defined time curve 1, it is currently active.

10. Define pressure loading using the PCR (LoadsBC > STRUCTURAL > PRESSURE > Define Curves) command.

Geo Panel: LoadsBC > STRUCTURAL > PRESSURE > Define Curves (PCR)

Beginning curve > 1Pressure magnitude > -1Ending curve > 1Increment > 1Pressure at the end of direction 1 > -1Pressure direction > Normal direction

11. By default, COSMOSM will write displacement output for all the nodes defined in the problem. But here, we are only interested in the displacements of the free edge of the cantilever plate. Therefore, define the group of nodes on the edge to be considered for displacement output. Remember that this step only affects the output file (.OUT) and not the files used for postprocessing.

Geo Panel: Analysis > OUTPUT OPTIONS > Set Nodal Range (PRINT_NDSET)

Number of groups > 1

Beginning node of group 1 > 1Ending node of group 1 > 4



12. Define the number of nodes for which the response curve is required to be generated. Use the NL_NRESP (Analysis > NONLINEAR > Response Options) command for this purpose. This command specifies a time-history plot file to be created for the defined set of nodes.

Geo Panel: Analysis > NONLINEAR > Response Options (NL_NRESP)

Starting location [1] > 1

Node 1 > 1Node 2 > 2Node 3 > 3Node 4 > 4

13. Next, define the time step number for which the displacements and stresses are to be saved for later graphic processing. Here, results for time step number 1 through 8 are desired.

Geo Panel: Analysis > NONLINEAR > Plot Options (NL_PLOT)

Set 1 beginning step > 1Set 1 ending step > 8Set 1 step increment > 1

14. Finally, having completed the description of the model and the specification of the desired postprocessing data, the nonlinear analysis will be performed.

Geo Panel: Analysis > NONLINEAR > Run NonL Analysis (R_NONLINEAR)

15. This command runs the NSTAR module and gives displacement as well as stress output. You can examine the output file using the EDIT (File > Edit...) command or your favorite text editor.

Postprocessing

16. Plot the deformed shape at time step 8.

Geo Panel: Results > PLOT > Deformed Shape (DEFPLOT)

Step number > 8Beginning element > 1Ending element > 9Increment > 1Deformation scale flag >Scale factor > 97.258057



7-22

17. Use the sliding scale button to set the scale to 0.5. Animate the deflected shape for time steps 1 through 8.

Geo Panel: Results > PLOT > Animate (ANIMATE)

Beginning step number > 1Ending step number > 8Step increment > 1Animation type > Two Way

Delay number > 0Scale factor > 97.258057

Number of frames > 9Save and play as AVI > No

AVI file name > test.aviNumber of iterations > 1

18. Plot von Mises stresses at time step 8:

Geo Panel: Results > PLOT > Stress (STRPLOT)

Time step number > 8Component > VON

Layer number > 1Coordinate system > 0Stress flag > Nodal stress

Face of element > Top face

Select contour option ...

Plot type > Color filled contour

Accept defaults ...

19. Plot the variation of von Mises stresses along a line segment in the sheet.

Geo Panel: Results > PLOT > Path Graph (LSECPLOT)

Node > 2Node > 14

Node > 14

The resulting plot is shown in Figure 7-2. The TRANSLATE command (or the Translate icon) is used to adjust the relative position of the plotted figures.

20. Animate the von Mises stresses for steps 1 through 8.


Beginning step number > 1Ending step number > 8



Step increment > 1Animation type > Two way Animation

Delay number > Scale factor > 97.1118

21. Generate a time history plot:

Geo Panel: Display > XY PLOTS > Activate Post-Proc (ACTXYPLOT)

Graph Number >1X_variable > Time

Y_variable > UY

Node number > 2Graph color > 12

Graph line style > Solid

Graph symbol style > 0Graph id > 2N

Geo Panel: Display > XY PLOTS > Plot Curves (XYPLOT)

Plot graph > Yes

The resulting plot is shown in Figure 7-3.

Figure 7-2. The von Mises Stress Plot at Time Step 8



7-24

Figure 7-3. Vertical Displacement Versus Time at Node 2

Large Displacement Nonlinear Analysis Example

An example of Large Displacement Static analysis is described in this section. This includes the description of the steps required to set up and solve the problem, in detail. Here, the problem of a cantilever beam under uniformly distributed load is considered.

Statement of the Problem

Find the static response of a cantilever beam under uniform loading that causes large displacements. Use 2D plane stress elements. The cantilever is shown in Figure 7-4.



Figure 7-4. A Cantilever Beam Under Uniform Loading

GivenL = 10 in

b = 1 in

ν = 0.2

h = 1 in

E = 12,000 psi

p = 10 lb/in

GEOSTAR Input

1. Define the element group. For this example, the PLANE2D with plane stress option is selected.


Element group > 1Element category > Area

Element type (for area) > PLANE2D

OP1:S/F flag > Solid

L b

h

p/2

4 5321

Problem Sketch

1

12

18

2

19

11

17

28

2 D Model

p

t

Time Curve

5

100



7-26

OP2:Integr type > QM6

OP3:Type > Plane Stress

OP4:Stress direction > Global Cartesian

OP5:Mat > Linear Elastic

OP6:Disp. formulation > Updated Lagrangian

OP7:Material creep > No

OP8:Strain plasticity > Small

2. Define material properties (EX, NUXY).

Geo Panel: Propsets > Material Property (MPROP)

Material property set > 1Material property name > EX


Material property name > NUXY

Property value > 0.20

3. Define the thickness of the plane stress elements.


Associated element group > 1Real constant set > 1

Start location of the real constants > 1No. of real constants to be entered > 2

RC1: Thickness > 1.0

RC2: Material angle (beta) > 0.0

4. Define the geometry of the model. Change the view to X-Y using the Viewing icon.

Geo Panel: Geometry > SURFACES > Draw w/ 4 Coord (SF4CORD)

Surface > 1Keypoint 1 XYZ-coordinate value > 0,0,0,

Keypoint 2 XYZ-coordinate value > 10,0,0,



5. Define the elements and nodes through mesh generation.

Geo Panel: Meshing > PARAMETRIC MESH > Surfaces (M_SF)

Beginning surface > 1Ending surface > 1



Increment > 1Number of nodes per element > 8

Number of elements on first curve > 5

Number of elements on second curve > 1

Spacing ratio for first curve > 1.0

Spacing ratio for second curve > 1.0

6. Define displacement constraints on the boundary using the DCR (LoadsBC > STRUCTURAL > DISPLACEMENT > Define Curves) command.

Geo Panel: LoadsBC > STRUCTURAL > DISPLACEMENT > Define Curves (DCR)

Beginning curve > 3Displacement label > AL: All 6 DOF

Accept defaults ...

7. Define the starting time, final time and time increment for the solution using the TIMES (LoadsBC > LOAD OPTIONS > Time Parameter) command.

Geo Panel: LoadsBC > LOAD OPTIONS > Time Parameter (TIMES)

Starting time > 0.0

Final time > 100

Time increment > 1

8. Define the load versus time curve using the CURDEF (LoadsBC > FUNCTION CURVE > Time/Temp Curve) command.

Geo Panel: LoadsBC > FUNCTION CURVE > Time/Temp Curve (CURDEF)

Curve type > Time

Curve number > 1

Start point > 1

Time value for point 1 > 0Function value for point 1 > 0Time value for point 2 > 100

Function value for point 2 > 5Time value for point 3 > <CR>



7-28

9. A number of time-load curves can be defined. However, the curve associated with an applied load or pressure must be activated prior to the definition of that load. Note that the last defined time curve is currently active.

10. Define pressure loading using the PCR (LoadsBC > STRUCTURAL > PRESSURE > Define Curves) command.

Geo Panel: LoadsBC > STRUCTURAL > PRESSURE > Define by Curves (PCR)

Beginning curve > 1Pressure magnitude > -1.0

Accept defaults ...

Geo Panel: LoadsBC > STRUCTURAL > PRESSURE > Define by Curves (PCR)

Beginning curve > 2Pressure magnitude > 1.0

Accept defaults ...

11. By default, COSMOSM will write displacement output for all the nodes defined in the problem. But here, we are only interested in the displacements, of the free edge of the cantilever beam. Therefore, define the group of nodes on the edge to be considered for displacement output. Remember that this step only affects the output file (.OUT) and not the files used for postprocessing.

Geo Panel: Analysis > OUTPUT OPTIONS > Set Nodal Range (PRINT_NDSET)

Number of groups > 1

Beginning node of group 1 > 17

Ending node of group 1 > 17

12. Define the number of nodes for which the response curve is required to be generated. Use the NL_NRESP (Analysis > NONLINEAR > Response Options) command for this purpose. This command specifies a time-history plot file to be created for the defined set of nodes.

Geo Panel: Analysis > NONLINEAR > Response Options (NL_NRESP)

Starting location > 1

Node 1 > 17



13. Define the time step number for which displacements and stresses are to be saved for later graphic examination. Here, results for time step number 10 through 100 with increment 10 are desired.

Geo Panel: Analysis > NONLINEAR > Plot Options (NL_PLOT)

Set 1 beginning step > 10

Set 1 ending step > 100

Set 1 step Increment > 10

14. Define the type of analysis by the A_NONLINEAR (Analysis > NONLINEAR > NonL Analysis Options) command. Theoretically, after each time step increment, the structural system should be in the equilibrium state. So, generally it is advisable to keep the equilibrium option on. When the time step size is very small and the loading is smooth, this equilibrium option can be turned off. For this example, the number of time steps between equilibrium is set to 100 to check equilibrium only at time step 100.

Geo Panel: Analysis > NONLINEAR > NonL Analysis Options (A_NONLINEAR)

Analysis option > SSteps between reforming stiffness > 1Steps between eqlbm. iterations > 100

Max equilibrium iterations > 20

Accept defaults ...

15. Finally, having completed the description of the model and the specification of postprocessing data, the nonlinear analysis will be performed.

Geo Panel: Analysis > NONLINEAR > Run NonL Analysis (R_NONLINEAR)

16. This command runs the NSTAR module and gives displacement as well as stress output. You can examine the output file using the EDIT (File > Edit...) command or your favorite text editor.

Postprocessing

1. Plot the deformed shape at time step 100.

Geo Panel: Results > PLOT > Deformed Shape (DEFPLOT)

Accept defaults ...

Geo Panel: LoadsBC > STRUCTURAL > DISPLACEMENT > Plot (DPLOT)



7-30

Accept defaults ...


Figure 7-5. Deformation at Time Step 100

2. Generate a displacement contour plot. Scale the resulting plot by a factor of 0.7 using the sliding button.

Geo Panel: Results > PLOT > Displacement (ACTDIS, DISPLOT)

Time step number > 100

Component > URES


Plot Type > Color filled contour

Accept defaults ...


Accept defaults ...

Geo Panel: LoadsBC > STRUCTURAL > PRESSURE > Plot (DPLOT)

Accept defaults ...




Figure 7-6

3. Animate the resultant displacements for steps 10 through 100 with increment 10:


Beginning step number > 10

Ending step number > 100

Step increment > 10

Animation type > Two Way

Delay number > 0Scale factor > 0.13701

Save and play as AVI > No

AVI file name > test.aviNumber of iterations > 1

4. Plot von Mises Stresses at time step 100:

Geo Panel: Results > PLOT > Stress (STRPLOT)

Time step number > 100

Component > VON

Accept defaults ...


Plot type > Color filled contour

Beginning element > 1Ending element > 5Increment > 1Shape of model > Deformed shape

Scale factor > Default



7-32



Figure 7-7

5. To generate a time history curve:

Geo Panel: Display > XY PLOTS > Activate Post-Proc (ACTXYPOST)

Graph number > 1X_variable > Time

Y_variable > UY

Node number > 17

Graph color > 12

Graph line Style > Solid

Graph symbol > 0

Graph id > 17N

Geo Panel: Display > XY PLOTS > Plot Curves (XYPLOT)

Plot graph > Yes




Figure 7-8Time history.


7-34

8 Verification Problems

Introduction

In the following, a comprehensive set of verification problems are provided to illustrate the various features of the nonlinear analysis module (NSTAR). The problems are carefully selected to cover a wide range of applications in different fields of nonlinear analyses.

The input files for the verification problems are available in the “...\Vprobs\ Nonlinear” folder. Where “...” refers to the COSMOSM installation folder. For example the input file for problem NS1 is available in the file “...\Vprobs\Nonlinear\NS1.GEO” and the input file for problem ND1 is “...\Vprobs\Nonlinear\ND1.GEO.” NS in the problem name refers to Nonlinear Static, and ND refers to Nonlinear Dynamic.


8-2

Nonlinear Static Analysis



NS1: Elastoplastic Compression of a Composite Pipe Assembly

TYPE:

Static, Plastic Analysis, Truss Elements.

REFERENCE:

Crandall, Dahl and Lardner, “An Introduction to the Mechanics of Solids,” Second Edition, McGraw-Hill, 1972, pp. 277-280

PROBLEM:

Two coaxial pipes, the inner one of 1020 CR steel and cross-sectional area As, and the outer one of 2024-T4 aluminum alloy and of area Aa, are compressed between heavy, flat end plates, as shown below. Determine the load-deflection curve of the assembly as it is compressed into the plastic region by an axial force P.

Figure NS1-1

Y

Z

X

1 2

2

1

Finite Element Model

σ

σys

σ ya

E

E

s

a

ETs

ETa

ε

P

t

Stress-Strain Curve

P

L

Problem Sketch

1299860

103


Chapter 8 Verification Problems

8-4

GIVEN:

As = 7 in2

Aa = 12 in2

L = 10 in

σys = 86,000 psi

σya = 55,000 psi

ν = 0.3

Es = 26.875 x 106 psi

Ea = 11 x 106 psi

ETS = 41,322 psi

ETa = 52,632 psi

COMPARISON OF RESULTS:

Figure NS1-2



NS2: Nonlinear Analysis of a Cable Assembly

TYPE:

Static, Plastic Analysis, Truss Elements.

REFERENCE:

Crandall, Dahl and Lardner, “An Introduction to the Mechanics of Solids,” Second Edition, McGraw-Hill, 1972 page 356.

PROBLEM:

A chain hoist is attached to the ceiling through three tie rods as shown in the sketch. The tie rods are made of cold-rolled steel with yield strength σy, and each has an area A.1. What is the load-deflection relation when the deflections are elastic in all three

rods?2. What is the value of the load at which the three rods become plastic?

Figure NS2-1

Strain, ε

Stress, σ

σy

E t

E

θ θ1 3

2

X

P

Y

4

1 2 3

Problem Sketch and Finite Element Model

Stress-Strain Curve

L



8-6

Ultimate load (lb).

Figure NS2-2

GIVEN: COMPARISON OF RESULTS:

A = 1 in2

L = 100 in

θ = 30°σy = 30,000 psi

E = 30 x 106 psi

Et = 0.606 x 106 psi

P (lb)Displacement at Node 4 (inch)

Theory COSMOSM

8000 0.0225991 0.011599

16000 0.0231981 0.023198

24000 0.0347972 0.034797

32000 0.0463962 0.046396

40000 0.0579953 0.057995

48000 0.0695943 0.069594

56000 0.0811934 0.081193

64000 0.0927924 0.092792

Theory COSMOSM

Pu (lb) 81961.52 82100.0



NS3: Static Collapse of a Truss Structure

TYPE:

Static, Plastic Analysis, Truss Elements, Force control, BFGS iterations, Line search.

REFERENCE:

Bathe, Klaus-Jurgen, Ozdemir, H., and Wilson, E., “Static and Dynamic Geometric and Material Nonlinear Analysis,” Report No. UCSESM 74-4, University of California, Berkeley, 1974, pp. 111-112.

Figure NS3-1

Strain, ε

Stress, σ

σy

E t

E

Stress-Strain Curve

4

23

1

2

1 3

36 in

P

4

48 in

Problem Sketch

P

t

Load_Time Curve

24

24



8-8

GIVEN:

A = 1 in2

E = 3x104 ksi

σy = 30 ksi

Et = 300 ksi


Elastic-plastic displacement of the truss at node 4 is plotted.

Figure NS3-2



NS4: Truss with Temperature Dependent Material Properties

TYPE:

Nonlinear static thermal analysis using truss elements.

PROBLEM:

A one dimensional structure consists of truss elements with various material properties. Determine deflections and Thermal stresses in the structure, due to a uniform rise of temperature to:a. 100° F

b. 200° F

Figure NS 4-1

GIVEN:

A = 1 in2

α = 0.00001 1/°F

E = 30,000 ksi

L = 4 in

Problem Sketch

L

1 2 3 4 5X

Y


30000.

E(ksi)

30000.

20000.

150 200Temperature

E = 30000. ksi

Element 1

Elements 2 & 4

E(ksi)

27500.

125 200Temperature

Element 3

Elements Elastic Modulus - Temperature Curves

E(ksi)

Temperature8



8-10

ANALYTICAL SOLUTION:

1. T = 100° FAt this temperature all the elements have the same modulus of elasticity, therefore

u(i) = 0

stress = E α T = 30 ksi

2. T = 200 ° F

For this case, first we assume that all nodes are constrained, and later release these nodes. If:

e = Thermal strain

u(i) = Deflection at node i (i = 1,5)

S(j) = Stress at element j (j = 1,4)

Assuming that all nodes are fixed: e = αT = 0.00001*200 = 0.002 in/in

S(1) = E e = 30000* 0.002 = 60 ksi

S(2) = E e = 20000* 0.002 = 40 ksi

S(3) = E e = 27500* 0.002 = 55 ksi

S(4) = E e = 20000* 0.002 = 40 ksi

Releasing the nodes:S(1) = 60-30000 u(2) / 10

S(2) = 40-20000 [u(3) - u(2)] / 10

S(3) = 55-27500 [u(4) - u(3)] / 10

S(4) = 40+20000 u(4) / 10

However, equilibrium is only satisfied if the stresses in all elements are equal, thus yielding a solution to the above equations.

u(2) = 0.0042857 in

u(3) = 0.0007143 in

u(4) = 0.0035714 in

stress = 47.14286 ksi


Same results are obtained using COSMOSM nonlinear module, NSTAR.



NS5: Elastic-Plastic Static Analysis of a Metal Sheet

TYPE:

Nonlinear Static Analysis, Plasticity, 2D Isoparametric Plane Stress Elements (4 node).

PROBLEM:

Investigate elastoplastic response of a metal sheet using nine 4-node 2D plane stress elements. Material of the sheet is assumed to obey the von Mises yield criterion.

Figure NS5-1

2 8

Load - Time Curve

10

9

13

L

E

E

T

σ

ε

σ y


Y

L

4 5 6

7 8 9

1 2 31 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16


X

1

t

p



8-12

GIVEN:

E = 21,000 N/mm2

ET = 5,000 N/mm2

ν = 0.3

σy = 10 N/mm2

L = 30 mm

Thickness = 1 mm


Figure NS5-2



NS6: ElastoPlastic Analysis of a Thick Walled Tube

TYPE:

Nonlinear Static Analysis, Plasticity, 2D Axisymmetric Elements (8 node).

REFERENCE:

Hodge, P. G., and White, G. H., “A Quantitative Comparison of Flow and Deformation Theories of Plasticity,” J. Appl. Mech., Vol. 17, pp. 180-184, 1950.

PROBLEM:

Investigate elastoplastic response of a thick-walled cylinder using sixteen 4-node 2D axisymmetric elements. The material is assumed to obey the von Mises yield criterion.

Figure NS6-1

x


Problem Sketch

z

y

H1 2 3

R

1 rad

x

4

3 523

22

211 4

1

2

R1

R2

pP

R2

E

σ

σy


TE

ε

11.5

t

p

7.5

13

1 9 20

Load - Time Curve

12

12



8-14

GIVEN:

G = 0.333333E5 psi

ν = 0.3

σy = 17.32 psi

R2 = 2 in

H = 1 in

E = 86666 psi

ET = 0 psi

R1 = 1 in


Figure NS6-2



NS7: Large Deflection Analysis of an Infinitely Long Plate

TYPE:

Nonlinear Static Analysis, Large Displacement, 2D Plane Strain (8 node) elements.

REFERENCE:

Timoshenko, S., Woinowsky-Kreiger, S., “Theory of Plates and Shells,” pp.422-423.

PROBLEM:

Study the large displacement response of an infinitely long rectangular plate using 2D plane strain elements.1

MODELING HINTS:

Due to symmetry, only a half of the plate is modeled.

Figure NS7-1

1

q

h

q

t

Load_Time Curve

2 3 4 5 6 7 8 9 10

5

4

53

52

51


y

X

CL

Problem Sketch

2b

1.0

20

CL

CL



8-16

GIVEN:

E = 10.92 E6 N/mm2

ν = 0.3

h = 1 mm

b = 100 mm

q = 1 N/mm2

COMPARISON O F RESULTS:

Figure NS7-2

Theory COSMOSM

Maximum Bending Stress (Node 53) 12730 12778

Midspan Deflection(Node 1) 1.526 1.5255



NS8: Static Large Displacement Analysis of a Cantilever Beam

TYPE:

Nonlinear Analysis - Large Displacement.NS8A) PLANE2D elements

NS8B) BEAM2D elements

REFERENCE:

Holden, J. T., “On the Finite Deflections of Thin Beams,” Int. J. Solid Structure, Vol. 8, pp. 1051-1055, 1972.

PROBLEM:

Investigate large displacements of a cantilever beam using five 8-node 2D plane stress elements.

Figure NS8-1

4 5321

1

2

3

4

5

26

27

28

Plane2D (8 node) Finite Element Model

1 2 3 4 5 6

1 2 3 4 5

Beam2DFinite Element Model

Lb

p/2

Problem Sketch

h

p/2

t

Load - Time Curve

p

10

100



8-18

GIVEN:

L = 10 in

E = 12,000 psi

ν = 0.2

p = 10 lb/in

h = 1 in

b = 1 in

COMPARISON OF RESULTS

Figure NS8-2

Element Type Node No. Maximum Deflection

PLANE2D 27 7.17

BEAM2D 6 7.05



NS9: Static Large Displacement Analysis of a Spherical Shell

TYPE:

Nonlinear Static Analysis, Large Displacement, 2D Isoparametric Axisymmetric (8 node) Elements.

REFERENCE:

Stricklin, J. A., “Geometrically Nonlinear Static and Dynamic Analysis of Shells of Revolution,” High Speed Computing of Elastic Structures, Proceedings of the Symposium of IUTAM, University of Liege, pp. 383-411, August 1970.

PROBLEM:

Investigate large displacements of a spherical shell using ten 8-node 2D axisymmetric elements.

Figure NS9-1

4

5

1

2

34

56

78910

53

52

51


p/2π

12

3

θ

h

R

P

wo

H

p

t

Load - Time CurveProblem Sketch

100

200



8-20

GIVEN:

E = 10 x 106 lb/in2

ν = 0.3

h = 0.01675 in

R = 4.76 in

P = 100 lb

H = 0.0859 in


Figure NS9-2



NS10: Large Displacement Analysis of a Fixed Beam with Concentrated Load

TYPE:

Nonlinear Static Analysis, Large Displacement.NS10A) 2D Plane Stress (8-node) Elements

NS10B) 2D Beam Elements.

REFERENCE:

Noor, A. K. and Peters, J. M., “Reduced Basis Technique for Nonlinear Analysis of Structures,” AIAA J., Vol 18, No. 4, Apr. 1980, Article No. 79-0747R, pp. 455-462.

PROBLEM:

Investigate large displacements of a fixed-fixed beam subject to a concentrated load at mid-span using 8-node 2D plane and BEAM2D elements.

MODELING HINTS:

Due to symmetry, only a half of the beam is modeled.

Figure NS10-1

5

p

t

Load - Time Curve

Lb

h

p

Problem Sketch

4321

1

2

3

4

21

22

23

Plane2D Finite Element Model

p/2

L/2

X

Y Y

1 2 3 4 5

1 2 3 4

Beam2D Finite Element Model

p/2

L/2

X

3113.6

100

NS10A NS10B



8-22

GIVEN:

E = 20.684 x 1010 N/m2

b = 0.0254 m

h = 3.175 x 10-3 m

L = 0.508 m

Iz = 0.6774E-10 in4


Figure NS10-2



NS11: Simply Supported Rectangular Plate, Large Deflection Analysis

TYPE:

Nonlinear Analysis, SHELL3 Elements.

REFERENCE:

Levy, S., “Bending of Rectangular Plates with Large Deflections,” Technical Note, National Advisory Committee for Aeronautics. No. 846, 1942.

PROBLEM:

Calculate the large deflection response of a simply supported square plate subjected to uniformly distributed pressure.

MODELING HINTS:

Due to symmetry, only a quarter of the plate is modeled.

Figure NS11-1


X

16

y

z

1 4

13

a

b

h

Problem Sketch

K = qa /Eh44

500

150

30

4 12 22

Time Curve

t



8-24

GIVEN:

E = 1.0 x 107 psi

ν = 0.3162

h = 0.12 in

a = b = 24 in

q = Uniform applied pressure per unit area = 0.00625 kpsi

k = Load factor

BOUNDARY CONDITIONS:

All edges are simply supported.


Figure NS11-2



NS12: Large Deflection Analysis of a Cantilever

TYPE:

Nonlinear Static Analysis, Large Displacement.NS12A) SHELL3 Elements

NS12B) 2D Beam Elements

REFERENCE:

Ramm, E., “A Plate/Shell Element for Large Deflections and Rotations,” in Formulations and Computational Algorithm in Finite Element Analysis. [M.I.T. Press, 1977].

PROBLEM:

A cantilever beam subjected to a concentrated end moment.

Figure NS12-1

b

h

Problem Sketch

L

u

v

M

1

Shell Model

2 3 4 5

Beam Model

M/2

M/2

1 2 3 4 5 6

7 8 9 10 11 12

Z Y

X

6

M

X

M

t

Time Curve80

6.28518



8-26

GIVEN:

L = 100 in

I = 0.01042 in4

A = 0.5 in2

ν = 0

h = 0.5 in

M = Concentrated End Moment = 2 in-lb

E = 12,000 psi


Figure NS12-2



NS13: Analysis of Simply Supported Beam Using Gap Elements

TYPE:

Static, Nonlinear, Gap-friction elements.

PROBLEM:

The beam is modeled using BEAM2D elements. Material of the beam is assumed to be elastic, and large deformations are not considered. Two gap elements are used to model the supports. Two soft truss elements are used to avoid stiffness singularities.

Figure NS13-1

GIVEN:

h = 10 in

b = 1.2 in

Iz = 100 in4

E = 30 x 106 lb/in2

P = 1000 lbs


The result is in good agreement with the solution obtained using the theory of structures. (At time step 1) Theory COSMOSM

Max. Displacement (in) 0.00694 0.006944

Reactions (lb) 500 500

Lb

h

p

Problem Sketch


p

t

Time Curve

p

1

5

4635

4

3

2

1 2

Y

X



8-28

NS14: Analysis of a Propped Cantilever Beam Using Gap Elements

TYPE:

Static, Nonlinear, Gap-friction elements.

PROBLEM:

The beam is modeled using BEAM2D elements. Material of the beam is assumed to be elastic, and large deflections are not considered. A gap element is used to model the simply supported end.

Figure NS14-1

GIVEN:

L = 100 in

h = 10 in

Iz = 100 in4

E = 30 x 104 lb/in

b = 1.2 in

P = 1,000 lbs


The result is in good agreement with the solution obtained using the theory of structures. (At time step 1) Theory COSMOSM

Max. Displacement (in) 0.003040 0.003038

Reactions (lb) 312.5 312.5

Lb

h

p

Problem Sketch


p

t

Time Curve

p

1

3

221 3

4

Y

X

Gap Element



NS15: Nonlinear Analysis of a Cantiliver Beam with Gaps Under Multiple Loading Conditions

TYPE:

Nonlinear Static Analysis, Beam and Gap-friction elements.

PROBLEM:

The beam is modeled using BEAM2D elements. Five gap elements with zero gap distances are used. Ten different load cases were selected, and the analysis was performed in 10 solution steps. The material of the beam was assumed to be elastic; deformations were assumed to be small.

Figure NS15-1

L

b

h

p

Problem Sketch

Fa

Fb

Fc F

dFe

L1 2L

2 L2

L2

FInite Element Model

16

pFa F

bFc F

dFe

15141312

Y

X11



8-30

GIVEN:

Ebeam = 30 x 106 psi

b = 1.2 in

IZ = 100 in4

h = 10 in

L2 = 50 in

L1 = 100 in

Figure NS15-2


The state of gaps at any time agrees with the beam deformed shape at that time. The results can be compared with the solution obtained from linear static analysis, where the gaps are removed but the nodes which are connected to closed gaps are fixed.

Time Curves

4 6 8

-1000

Fe

t

-2000

15 6 8

-1000

Fd

t3

t5 7 8

-1000

Fb

10

9

100

p

t11

F

104 7

-1000

a

t

Fc

t4 6 9

-1000

-2000



OBTAINED RESULTS:

Applied Forces

Time Step No.

Forces in Gap

No. 1 No. 2 No. 3 No. 4 No. 5

Fa = - 1000 1 312.50

Fa = - 1000Fe = - 1000

2 390.38 498.02 482.14

Fa = - 1000Fe = - 2000

3 353.17 1075.4 928.57

Fd = - 1000 4 564.45 453.12

Fc = - 1000 5 586.54 442.31

Fb = - 1000 6 631.25 425.00

Fd = - 1000Fe = - 1000

7 351.81 1338.7 318.55

Fa = - 1000Fc = - 2000

8 361.84 1197.4 842.11

Fa = - 1000Fb = - 1000

91206.3 275.00

p = - 10010 2050.0 6100. 1900.0



8-32

NS16: Cantilever Beam on Elastic Foundation

TYPE:

Nonlinear Static Analysis, Beam and Gap-friction elements.

PROBLEM:

The geometry of the beam and loading conditions are similar to problem NS15. Thirty gap elements are used to model the interface between the beam and the ground. To account for the ground elasticity, one end of each gap is connected to a truss with equivalent stiffness. The analysis was performed for ten steps.

Figure NS16-1

b

h

pFa

Fb

Fc F

dFe

Problem Sketch

50 70 50 50 50

150 100 50

(All dimensions are in inches)

BeamCross

Section

61

pFa

Fb

Fc F

dFe

1 2 30 31

9162

32


Y



GIVEN:

Ebeam = 30 x 106 psi

b = 1.2 in

h = 10 in

ktruss = 30 x 105 lb/in

Iz = 100 in4

Figure NS16-2


Again, the results may be verified if the gaps are removed and truss elements which connect to closed gaps were directly attached to the beam. The deflection of the beam at any step confirms the state of the gaps.

Load - Time Curves

4 6 8

-1000

Fe

t

-2000

15 6 8

-1000

Fd

t3

5 7 8

-1000

Fb

t10

9

100

p

t11

4 6 9

-1000

Fc

-2000

4 7 10

-1000

Fa

tt



8-34

OBTAINED RESULTS:

AppliedForces

TimeStep Forces in Gap Elements

Fa = - 1000 1 2)20.31 3)103.97 4)256.07 5)368.06 6)248.73

7)43.96

Fa = - 1000 2 2)20.34 3)103.04 4)253.43 5)363.72 6)246.17

Fe = - 1000 7)53.97

25)57.377 26)257.93 27)381.11 28)257.88 29)45.84

Fa = - 1000 3 2)20.34 3)103.85 4)255.50 5)366.79 6)247.16

Fe = - 2000 7)44.31

25)123.92 26)514.84 27)759.15 28)513.76 29)91.37

Fd = - 1000 4 20)65.66 21)257.02 22)378.34 23)256.08 24)45.57

Fc = - 1000 5 15)71.25 16)256.41 17)376.50 18)254.86 19)45.38

Fb = - 1000 6 10)81.11 11)255.88 12)373.62 13)252.88 14)45.0

Fd = - 1000 7 20)62.93 21)238.71 22)347.15 23)237.66 24)115.48

Fe = - 1000 25)116.94 26)242.32 27)354.30 28)242.14 29)44.81

Fa = - 1000 8 2)20.39 3)103.30 4)253.42 5)362.56 6)242.65

Fc = - 2000 7)47.88

15)152.33 16)511.01 17)749.14 18)507.36 19)90.51

Fa = - 1000 9 2)19.87 3)100.18 4)245.31 5)350.23 6)237.94

Fb = - 1000 7)84.09 9)1.236 10)89.21 11)249.6 12)366.8

13)249.47 14)45.29

p = - 100 10 14)115.77 15)496.20 16)826.44 17)991.33 18)1035.2

19)1031.2 20)1025.2 21)1032.4 22)1040.3 23)1004.3

24)846.45 25)504.95 26)60.48



NS17: Simply Supported Beam Subjected to Pressure from a Rigid Parabolic Shaped Piston

TYPE:

Static, Nonlinear, Gap-friction elements, BEAM2D and PLANE2D elements.

MODELING:

The shape of the piston was simulated through gap distances. In order to avoid singularities in the structure stiffness, two soft truss elements were used to hold the piston. The problem was analyzed in one hundred steps, gradually increasing the pressure load.

Figure NS17-1

120 in 120 in60 in

p y = (x/100)3

h

Problem Sketch110

101

10

1t

p

Time Curve

b

p

24 25

23

22

91

2 3 4 5 6 7 8

g1 g7

10

11


Y

X

KK



8-36

GIVEN:

Gap Distancesg1 = g7 = 0.027 in

g2 = g6 = 0.008 in

g3 = g5 = 0.001 in

g4 = 0 in

Iz = 100 in4

h = 10 in

b = 1.2 in

E = 30 x 106 psi

Episton = 30 x 108 psi


The forces of gaps at any time were in good agreement with the total force applied to the pis-ton at that time. The deformed shape of the beam at any solution step confirmed with the forces and location of closed gaps at that step.

OBTAINED RESULTS:

Applied Forces

Time Step

Closed Gaps Pressure

Forces COSMOSM (lb) Total Force

(Theory) (lb)Gap Forces Total

p < 13 4 4 p = 13 777.1 777.1 780

13 < p < 14 5 3,4,5 p = 14 423.5,206.7,206.9

836.9 840

14 < p < 33 24 3,5 p = 33 986.3,986.3

1972.0 1980

33 < p < 36 27 2,3,4,5 p = 36 13.50,13.5 2153.1 2160

36 < p < 58 27 2,6 p = 58 1062.6,1062.6

3467.0 3480

58 < p < 63 54 1,2,6,7 p = 63 215.4,215.4,1668.,1668

3766.8 3780

63 < p 100 15,21 p = 109 3258.8,3258.8

6517.6 6540



NS18: Static Analysis of Sheets Connected by Friction Elements

TYPE:

Static, Nonlinear, Gap elements with friction.

MODELING:

The sheets are modeled using PLANE2D plane stress elements. In order to avoid any stiffness singularities soft truss elements are used to hold the upper sheet. In addition, truss elements were also used in the gap locations; these additional trusses were found to be essential for a large deflection analysis, where the stiffness had to be reformed at every step. The normal force was kept constant, while the horizontal force was increased gradually, in one hundred steps.

Figure NS18-1

Fy

xF

26 27 28 29 30

21 22 23 24 25

16 17 18 19 20

11 12 13 14 15

6 7 8 9 10

1 2 3 4 5

33 34Fy

Fx

31


X

2 in

2 in

4 in

Problem Sketch

500

1000

Fy

Fx

Time Curves

T

T



8-38

GIVEN:

E = 30 x 106 psi

ν = 0.3

Thickness = 1 in

Friction Coefficient = 0.5


In both cases (using small or large deformation theory), the structure starts slipping as soon as the horizontal force exceeds the normal force times coefficient of friction (step = 101, Fx > 500.), i.e., the horizontal deflection becomes large. For this case, the force distributions among the gap elements are the same whether small or large deformation theory is used.

OBTAINED RESULTS:

1. Small Displacement AnalysisGap forces (in pounds) at step 101:

2. Large Displacement Analysis

Gap forces (in pounds) at step 101:

Gap No. Gap Forces Friction Force

1 31.08 -15.5

2 160.28 -80.14

3 264.0 -132.0

4 361.9 -180.9

5 182.8 -91.4

Total 1000.96 -499.94

Gap No. Gap Forces Friction Force

1 31.08 -15.5

2 160.28 -80.14

3 264.0 -132.0

4 361.9 -180.9

5 182.8 -91.4

Total 1000.96 -499.94



NS19: Large Displacement Analysis of a S.S. Circular Plate

TYPE:

Nonlinear Static Analysis, Large Displacement, 2D Isoparametric (8 node) Axisymmetric Elements.

REFERENCE:

Timoshenko, S., Woinowsky-Kreiger, S., “Theory of Plates and Shells,” p. 411.

PROBLEM:

Investigate the large displacement behavior of a simply supported circular plate with radically movable edges using 2D axisymmetric elements.

Figure NS19-1

1 2 3 4 5 6 7 8 9 10

1

2

35

4

53

52

51


q

h

2a

Problem Sketch

Time Curve

q

t

a

CL

X

1.0

0.5

0.250.1

70605040



8-40

GIVEN:

a = 100 cm

h = 1 cm

E = 2 x 108 N/cm2

ν = 0.25

q = 400 N/cm2


Figure NS19-2

Theory COSMOSM

Maximum Deflection (cm) 7.88 7.91

Principal Stress(at node 53) (N/cm2) -1.1646 x 106 -1.177 x 106



NS20: Large Deflection Analysis of a Fixed-Fixed Shallow Arch

TYPE:

Nonlinear Static Analysis, Large Displacement, 2D Beam elements.

REFERENCE:

Bathe, K. J., and Bolourchi, S., “Large Displacement Analysis of 3D Beam Structures,” Int. J. for Numerical Methods in Engineering, Vol. 14., 1979, page 977.

PROBLEM:

Find the static response of a spherical arch under apex load due to large displacement effect using 2D Beam elements.

Figure NS20-1

3541

β

L

P

H

2

10

13

P/2

W

Problem Sketch

1112

h

b

Beam Cross-Section

p

t

Time Curve

34

34




8-42

GIVEN:

E = 1.0 x 107 psi

ν = 0.2

h = 0.1875 in

L = 34 in

A = 0.1875 in2

Iz = 0.00055 in4

R = 133.114 in

β = 7.3397°


Figure NS20-2

COSMOS/MBATHE & BOLOURCHI

TIME STEP

w



NS21: Elastoplastic Small Displacement Analysis of a Cantilever Beam with Tip Moment

TYPE:

Nonlinear Static Analysis, Plasticity, BEAM2/3D Elements, and beam-section-definition.

REFERENCE:

Timoshenko, S. P, and Gere, James M., “Mechanics of Materials,” pp. 289-316.

PROBLEM:

Investigate the elastoplastic response of a cantilever beam subjected to an end moment. Both elastic-perfectly plastic and elastic-linear strain hardening models are studied.

Figure NS21-1

LB

H

Problem Sketch

M

E

T

σ

ε

σy

Stress-Strain Curve

E

case-2

case-1ρH/2

σy

σ y

M


1 2 3 4 5

1 2 3 4 5

6 7 8 9 10

6 7 8 9 10

11

M

t

Time Curve

15,000

30



8-44

GIVEN:

The problem sketch is shown in Figure NS21-1. Ten (10) BEAM2/3D elements are used in the analysis.E = 30E6 psi

ν = 0

σy = 5,000 psi

L = 90 in

H = 3 in

B = 1 in

Case–1:ET = 0

Case–2:ET = 3E6 psi


Analytical solutions in the elastoplastic range:

Case1:

Case 2:

where:δ = Displacement

K = Curvature

Ky = Curvature at the yield point

Fig NS21-2 and NS21-3 show load-deflection curves of the beam with respect to the stress-strain curve-1 and -2. Analytical solutions are also included.



Figure NS21-2

Figure NS21-3

L0AD

LB*IN

LIMIT LOAD = 11250 LB*IN

DISPLACEMENT (INCH)

1 ANALYTICAL SOLUTION2 COSMOS/M

0.9 10.8

12

1 ANALYTICAL SOLUTION2 COSMOS/M

L0AD

LB*I

N

DISPLACEMENT (INCH)



8-46

NS22: Clamped Square Plate with Pressure Loading

TYPE:

Nonlinear Static Analysis, Large Displacement, 3D Isoparametric (20 node) Solid Elements.

REFERENCE:

Timoshenko, S., Woinowsky-Kreiger, S., “Theory of Plates and Shells,” pp. 422-423

PROBLEM:

Carry out large displacement analysis of a clamped square plate using 3D (20 node) solid elements.

MODELING HINT:

Only a quarter of the plate is modeled due to symmetry.

Figure NS22-1

516

109

15

4

11

1314

12

3

67

812

h

2a 2a


Y

X

q

q

t

Time Curve

20,000

20



GIVEN:

a = 1 m

h = 0.01 m

q = 2 x 104 N/m2

E = 10.92 x 1010 N/m2

ν = 0.3


Figure NS22-2

Theory COSMOSM

Central Deflection (m) 0.01594 0.015938



8-48

NS23: Large Displacement Static Analysis of a Clamped Sandwich Plate

TYPE:

Nonlinear Static Analysis, Large Displacement.NS23A) 4-node Composite Shell Elements.

NS23B) 3D Isoparametric (20 Node) Solid Elements

REFERENCE:

Schmit, L. A., Monforton, G. R., “Finite Deflection Discrete Element Analysis of Sandwich Plates and Cylindrical Shells with Laminated Faces,” AIAA Journal, Vol. 8, No. 89., pp. 1454-1461.

PROBLEM:

Perform Large Displacement analysis of a clamped sandwich plate subject to uniform pressure using 3D (20 node) solid elements and 4-node composite shell elements.

MODELING HINTS:

Only an eighth of the plate is modeled due to symmetry for 20-node composite shell elements and one quarter of the plate is modeled for 4-node composite shell elements. The properties of the core material are adjusted to match the linear solution given in the reference.

Figure NS23-1

2a 2a

Problem Sketch and Finite Element Model (20-node solid)

1

3 3

4

5

7

6

8

t f

tf

t c

q

t

Time Curve

14

64.615



)

Properties of the Core:

Exc = Eyc = 1 E-12 psi; Ezc = 34,500 psi

νxy = νxzc = νyzc = 0

Gxy = 0 psi; Gxzc = Gyzc = 50,000 psi

t = 1 in

Figure NS23-2

GIVEN:

a = 25 in

q = 64.615385 psi

Properties of the Face Sheets:

Ef = 10.5 x 106 psi

νf = 0.3

tf = 0.015 in

Gxzf = Gyzf = 0.1 psi


(At time step 14

Schmit and Monforton

COSMOSM

SHELL4T SOLID3D

Central Deflection

(in)2.48 2.6804 2.5285



8-50

NS24: Plate Subjected to TriangularTemperature Loading

TYPE:

Linear Static thermal analysis using PLANE2D 8-node planes tress elements.

REFERENCE:

Heldenfels, R. R., and Roberts, W. M., “Experimental and Theoretical Determination of Thermal Stresses in a Flat Plate,” Technical Note 2769, NACA, Washington, Aug. 1952.

PROBLEM:

A flat plate is subjected to a triangular thermal loading along two parallel edges. Determine the thermal stresses in the plate.

MODELING HINT:

Due to symmetry, a quarter of the plate is modeled.

GIVEN:

a = 18 in

b = 12 in

T = 150°E = 10.4E6 psi

ν = 0

α = 12.7E-6/° F


The stresses along the vertical middle axis are obtained and compared to those from the reference in the table below. On the whole good agreement is observed, although as seen, some discrepancy exists near the top edge, where it appears that a finer mesh is required. The two values of the stresses come from joining elements. When the two values differ by a large amount convergence has not been achieved.

Figure NS24-1

b b

T1

T1

a

a

Problem Sketch



Figure NS24-2

Node Ref COSMOSM

σx σy σxy σx σy σxy

1 9403 - 6.6 0.0 10858 - 3.95 - 264.6

21 3687 - 768 0.0 2604.5 - 1954 +18.6

3051.3 - 2003 - 104.4

41 573 - 2352 0.0 121.3 - 3966 - 16.3

241.2 - 3951 - 29.3

61 - 952 - 4084 0.0 - 1124 - 5487 - 10.6

- 1078 - 5479 - 19.7

81 - 1573 - 5631 0.0 - 1604 - 6680 - 7.2

- 1580 - 6676 - 10.1

101 - 1727 - 6855 0.0 - 1716 - 7570 - 5.

- 1701 - 7568 - 6.9

121 - 1688 - 7718 0.0 - 1668 - 8186 - 2.9

- 1660 - 8185 - 4.2

141 - 1611 - 8115 0.0 - 1594 - 8547 - 1.2

- 1590 - 8546 - 2.3

161 - 1577 - 8392 0.0 - 1560 - 8667 0.05



8-52

NS25: Analysis of a Hollow Thick-Walled Cylinder Subjected to Temperature and Pressure Loading

TYPE:

Static analysis, 2D axisymmetric element.

REFERENCE:

Timoshenko, S. P. and Goodier, “Theory of Elasticity,” McGraw-Hill Book Co., New York, l961.

Figure NS25-1


(At node No. 8)


The hollow cylinder in plane strain is subjected to two independent loading conditions.1. An Internal pressure 2. A steady state axisymmetric temperature distribution

given by the equation:

T(r) = Ta {[ln (b/r)] [ln (b/a)]}Where Ta is the temperature of the inner surface.

GIVEN:

E = 30 x 106 psi

a = 1 in

b = 2.0 in

ν = 0.3

α = 1.0 x10-6/degree

Pa = 100 psi

Ta = 100°

Stresses TheoryCOSMOSM

4-Node Element (Problem S19) 8-Node Element

Radial (psi) - 398.34 - 396.78 - 398.45

Tangenient (psi) - 592.47 - 597.00 - 596.03

21 31.0

x

y

2.0

T(r)

Ta

Pa

4 5 6 7

Problem Sketch Finite Element Model

35 51

17




1 2 3 4 5

6 8 9 107

Y

CL

1 2 3 4

Figure NS26-1

NS26: Thermal Stress Analysis of a Plate,Temperature Dependent Material Properties

TYPE:

Nonlinear static thermal analysis using:NS26A) PLANE2D 4-node plane stress elements

NS26B) Solid 8-node elements

PROBLEM:

A flat plate consists of different material properties through it’s length. Determine deflections and thermal stresses in the plate due to uniform changes of temperature equal to 100° F and 200 °F.

GIVEN:

t = 0.1 in

L = 4 in

ν = 0

α = 1 x 10-5/° F

E = 30E3 ksi


Since the effect of Poisson’s ratio is neglected, problem reduces to one dimension and the solution can be obtained as follows:Max. Stresses in x-direction (ksi): T = 100° F T = 200° F

Theory - 30 - 48

COSMOSM PLANE2D (Node 5) - 30 - 48

COSMOSM SOLID - 30 - 48

Problem Sketch

L

X

Elements 1 & 2

E(ksi)

30000.

20000.

150 200Temperature

Elements 3 & 4

Material Properties

E(ksi)

30000.

Temperature

8



8-54

1000.

P (psi)

0.1

0.2

0.4

0.6

0.8

1.0

-0.2

-0.4

-0.6

1.1

4x 10ε c

Figure NS27-1

NS27: Uniaxial Creep Strain in a Bar (Cyclic Loading)

TYPE:

Nonlinear static analysis, creep, using:NS27A) Truss elements NS27B) PLANE2D plane stress elements NS27C) Solid elements

REFERENCE:

Harry Kraus, “Creep Analysis,” Wiley-Interscience, New York, (1980)

PROBLEM:

A bar is subjected to a cyclic loading as shown in Figure NS27-1. Determine the creep strain in the bar as a function of time.

MODELING HINT:

The creep material properties are also given in Figure NS27-1. A time increment of 0.1 hr. is used for the analysis. The same results were obtained for all models (using different element types); see Figure.

Theoretical Loading

Applied Loading

-1000.

2.9

900.

3.9 4.8

time (hr)

Negative Creep Origin

Positive Creep Origin

Theoretical

COSMOSM

2.1

2.9 3.7 4.83.9time (hr)

ε = 6.4x10 σ t in/inc 4.4 0.75-18

Problem Sketch

P



NS28: Creep Analysis of a Cylinder Subject to Cyclic Internal Pressure

TYPE:

Nonlinear static analysis, creep, PLANE2D axisymmetric elements.

REFERENCE:

Mark D. Snyder, Klaus-Jurgen Bathe, “Formulation and Numerical Solution of Thermo-Elastic-plastic and Creep Problems,” report 8248-23 (1977)

PROBLEM:

A thick walled cylinder is subjected to a cyclic loading as shown in figure. Determine the variation of effective (von Mises) stress in the inner and outer boundaries of the cylinder as a function of time.

MODELING HINTS:

This problem is modeled using three 8-node two-dimensional axisymmetric elements. Forty-eight solution steps with a time increment of 0.1 hour is used. The finite element model plus the selected properties are shown in Figure NS28-1. A graphic representation of the results is given.

Figure NS28-1



8-56

GIVEN:

E = 20E + 6 psi

ν = 0.3

εc = 6.4E-18 σ4.4 t in/in

∆t = 0.1 hr


Figure NS28-2

p (psi)

365

0

-365

2.2 2.6

time (Hr)

Load - Time Curve

200

400

600

800

1000

1200

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 5.2 5.64.8

Y = 0.166 in

Y = 0.244 in

time (Hr)

Effective Stress (psi)

Reference

COSMOS/M*



NS29: Creep Analysis of a Cantilever Beam

TYPE:

Nonlinear static analysis, Creep, PLANE2D plane stress elements.

REFERENCE:

Mark D. Synder, Klaus-Jurgenm Bathe, “Formulation and Numerical Solution of Thermo-Elastic-Plastic and Creep problems,” Report 8448-3, (1977)

PROBLEM:

A cantilever beam is subjected to a constant end moment as shown in figure NS29-1. Determine the variation of the bending stress along the depth of the beam after a steady state is reached.

MODELING HINTS:

Due to symmetry, the upper half of the beam is modeled using two rows of 4 8-node plane stress elements. The nodes which are located on the neutral axis of the beam are assumed to have no elongation; 100 steps of solution with a time increment of 2 hrs. are used. The finite element model plus the selected proportions are shown in Figure NS29-1. A graphical representation of the results is given in Figure NS29-2.

GIVEN:

E = 30E6 psi

ν = 0.3

εc = 6.4E-18 σ3.15 t in/in

∆t = 2 hr

L = 40 in

b = 0.3 in

h = 4 in


Bending stress distribution in the y-direction (ksi)



8-58

Figure NS29-1

Figure NS29-2

L

h

b

M

Y

X

P = 2250 lb

Problem Sketch


Min-lb

2.0time (hr)

6000

Load - Time Curve

54321

37

35

33

h/2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

2.

1. Theoretical

ADINA

COSMOSM

σ (ksi)xx

Y (in) (Distance from N. Axis)

Steady State (step 100)

Linear Elastic(step 1)



NS30: Beam Supported by Nonlinear Springs (Flexible Gaps)

TYPE:

Nonlinear static analysis using beam and gap elements.

PROBLEM:

A beam is supported by two nonlinear springs at its ends. The springs have different properties in tension and compression, as shown below. Determine the deflection of the springs due to a normal force acting on the center of the beam.

MODELING HINTS:

The nonlinear spring are modeled using 4 gaps (2 tensile and 2 compressive).

Figure NS30-1

U

Deflection Versus Force for Nonlinear Spring

f (lbs)

0.5

- 500

-4.

250

F (lbs)

1000

-10000

10. 20.

Applied Force Versus Time

t

1 2 3

63

4

5

5

X

F


1 2

4

Problem Sketch



8-60

GAP PROPERTIES:

Based on the properties of the springs, the gaps properties are defined:ki = 1,000 lbs/in (i = 1,4)

fi = 500 lbs (i = 1,2)

fi = 250 lbs (i = 3,4)

gi = 0 in (i = 1,2)

gi = -1 E-8 in (i = 3,4)


Since no moment is taken by the two ends of the beam, the beam is simply supported. This means that, regardless of the beam properties, the spring force is always half of the applied force. Thus the following represents the deflection versus spring force.

Figure NS30-2

Same results are obtained by COSMOSM except that the negative spring forces are taken by gaps No. 1 and 2, while positive forces are shared by gaps No. 3 and 4.

-1000.

-2000.

-3000.

-4000.

-5000.

-4.0

1000.

Us

Fs

0.5

Spring Force - Deflection



NS31: Crushing of a Pipe between Two Anvils

TYPE:

Nonlinear static analysis, plasticity, large displacements, gaps, 2D plane strain 8-node elements.

NS31) Force control without equilibrium iterations

NS31A) Displacement control with Newton-Raphson iterations, adaptive automatic stepping

NS31B) Large strain plasticity theory, force control with Newton-Raphson iterations, adaptive auto-stepping.

REFERENCE:

ABAQUS, Example problems manual, page 4.2.8.1, Rev. 4.5.

PROBLEM:

A long straight pipe is crushed between two flat, frictionless anvils by gradually pushing down on the anvil. Determine the variation of the relative anvil displacement with respect to the total force/length applied to the anvil (Shear force on the pipe section).

Figure NS31-1

GIVEN:

Yield stress = 35,000 psi

Thickness = t = 0.349 in

Length = L = 1 in

20

40

60

.002 .004 .006 .008

Stressksi

Strain

Stress - Strain CurveProblem Sketch

t

L



8-62

Figure

Poisson ratio = 0.3

Ext. diameter = D = 4.5 in

Modulus of elasticity = 30E6 psi

MODELING HINTS:

NS31 and NS31A

Due to symmetry, only a quarter of the pipe is modeled. Two rows of eight PLANE2D plane strain 8-node elements are used to model the pipe. These elements are allowed to undergo large deflection (total Lagrangian) and plasticity (von Mises). Another group of eight elements of the same type are used to model the anvil. These elements, however, are defined to be elastic and close to rigid. Gap elements are used to model contact between the anvil and the pipe section. Two soft truss elements are added to hold the anvil to prevent singularity of the stiffness matrix.

NS31B

The large strain plasticity analysis is performed by using the von Mises plasticity model and the updated Lagrangian formulation. The anvil is modeled by a very stiff BEAM element which is used as a contact line as well. The displacement is prescribed onto the BEAM element along the vertical direction until the designed value. No soft spring is required. The equivalent force is obtained by coupling the second node of the BEAM element to the first and the reaction force along the vertical direction is recorded.

Figure NS31-3

DEF STEP: 50 = 1

PRESCRIBEDISPLACEMENT

UY

CONTACTLINE

ORIGINAL AND FINAL DEFORMED SHAPES (NS31B)

NS31-2



LOADING:

NS31

Half of the total force/length on the section, divided by the width of the mesh of the anvil (equal to the radius of the pipe), produces a state of constant pressure (uniform displacement) on the anvil. Thus: total force/length = 2. * radius * pressure.Small time increments and therefore a large number of solution steps (792 steps) were used. Equilibrium iterations were not performed.

NS31A

The adaptive automatic stepping option is utilized along with displacement control.

NS31B

The adaptive automatic stepping option is utilized along with force control.


Figure NS31-4



8-64

Figure NS31-5

Figure NS31-6

FORCE/UNIT

LENGTH

(LB/IN)

RELATIVE ANVIL DISPLACEMENT (IN)FORCE-DISPLACEMENT RESPONSE OF THE ANVIL (NS31B)



NS32: Uniaxial Elastoplastic STRAIN in a Bar (Cyclic Loading)

TYPE:

Nonlinear Static Analysis, plasticity based on the kinematic hardening rule for cyclic loading conditions:

NS32A) TRUSS elements

NS32B) PLANE2D plane stress elements

NS32C) SOLID elements

NS32D) BEAM elements

NS32E) SHELL elements

NS32F) Combined Kinematic & Isotropic Hardening (RK=0.5)

REFERENCE:

Owen, Dr. J., Hinton, E., “Finite Elements in Plasticity: Theory and Practice,” Pineridge Press Limited, Swansea, U. K., (1980).

PROBLEM:

A bar is subjected to a cyclic loading condition. Investigate the effects of plasticity and stress reversals on the uniaxial stress-strain curve, based on kinematic hardening rule; compare with the isotropic hardening rule. Compare also with the case of combined kinematic & isotropic hardening.

Figure NS32-1

p

Problem Sketch

1816

40

62

p (kips)

1.52.2

2.199

2.22.199

2.2 2.199

Cyclic Loading

StepNo.

2 60

38



8-66

GIVEN:

Modulus of elasticity = 30E6 psi

Tangential modulus = 30E5 psi

Yield stress = 1,599 psi

NOTES:

Equilibrium iterations are performed every other step, so that at the beginning of each load reversal equilibrium iterations are suppressed. If we run the same problem using the isotropic hardening approach, the same results are obtained as long as the loading is not reversed. After a stress reversal occurs, the strain-stress relations will remain linear elastic, unless the magnitude of the load at its peak is increased. For this case, the load curve was selected (as shown in Figure NS32-2) such that the magnitude of maximum and minimum strains are equal.For the case of combined kinematic & isotropic hardening, the load curve of Figure NS32-1 is used. Here only half of the hardening (RK=0.5) is used towards expansion of the yield radius (the other half is used to displace the center of the yield surface based on the kinematic hardening assumption). Thus at the end of loading:

Radius of the yield surface=1.5e3+(2.2e3-1.5e3)/2=1.85e3 psi

Start of yield in the 1st reversal= 2.2e3-2*Radius=-1.5e3 psi

And at the end of one cycle of loading, reversing, and unloading:Radius of the yield surface= 1.85e3+(2.2e3-1.5e3)/2=2.2e3 psi

Start of yield for the 2nd cycle= -2.2e3+2*Radius=2.2e3 psi

Figure NS32-2

18

p (kips)

1.52.2

2.199

3.283.279

3.28

32step no.

44

4.28



RESULTS:

Figure NS32-3



8-68


NS33: Elastoplastic Analysis of a Thick Walled Tube (Cyclic Loading)

TYPE:

Nonlinear static analysis, plasticity, PLANE2D axisymmetric 8-node elements.

REFERENCE:

Mark D. Snyder, Klaus-Jurgen Bathe, “Formulation and Numerical Solution of Thermo-Elastic-Plastic and Creep Problems,” Report 8244-3, (l977).

PROBLEM:

A thick-walled cylinder is subjected to cyclic internal pressure. Plane strain conditions are assumed in the direction of the axis of cylinder. Investigate the radial deflection of the outer boundary of the cylinder; compare with isotropic hardening rule.

Figure NS33-1

1 rad

py


x

323

211

1

2

3

4

ProblemSketch

pP

step no.

p (ksi)

1525

24.9

25.24.9

25. 24.9

20

221 8 3432

10



8-70

GIVEN:

E = 30E6 psi

ν = 0.3

ET = 30E5 psi

σy = 30E3 psi

Thickness = 1 in

Int. Diameter = 2 in


Figure NS33-2



Figure NS33-3



8-72

NS34: 3D Extension/Compression Tests on Mooney-Rivlin Model

TYPE:

Nonlinear static analysis, Mooney-Rivlin hyperelastic material model, large deflections using total Lagrangian formulation, displacement control, Newton-Raphson iterations.

NS34) One 3D SOLID element

NS34A) One 4-node PLANE2D element

REFERENCE:

Frank J. Marx, “Hyperelastic Elements (STIF84, STIF86),” ANSYS Revision 4.3 Tutorial, 1987.

PROBLEM:

A 3D sheet of material (2x2x1 inch) is subjected to biaxial equal loadings in the X and Y directions. Investigate the behavior of Mooney-Rivlin hyperelastic material for various ratios of A/B.

MODELING HINTS:

Due to symmetry, a quarter of the sheet is modeled using one solid 8-node element or one 4-node PLANE2D element. Loading is applied along the two sides such that the sheet will expand or compress equally in the X and Y directions.Test cases are performed for different values of Mooney-Rivlin constants (B varies from 0.2A to 2A).



Figure NS34-1. (Solid Element)

Figure NS34-2

U

U

y

x

Finite Element



8-74

NS35: Inflation of a Simply Supported Circular Plate, Mooney-Rivlin Material

TYPE:

Nonlinear static analysis, hyperelastic material model, large deflections using total Lagrangian formulation, Newton-Raphson iterations.

NS35) PLANE2D axisymmetric 8-node elements, Mooney-Rivlin model

NS35A) PLANE2D axisymmetric 8-node elements, Mooney-Rivlin model with adaptive automatic stepping

NS35B) PLANE2D axisymmetric 8-node elements, Ogden model

NS35C) PLANE2D axisymmetric 8-node elements, Mooney-Rivlin model with u/p formulation, constraint equations, and auto-stepping.

NS35D TRIANG axisymmetric elements, Mooney-Rivlin model with u/p formulation

NS35E TETRA4 axisymmetric elements, Ogden model with u/p formulation

REFERENCE:

Frank J. Marx, “Hyperelastic Elements (STIF84, STIF86),” ANSYS Revision, 4.3 Tutorial, 1987.

PROBLEM:

A simply supported circular flat plate is subjected to an external pressure varying from 0. to 50 psi. The plate is made of an isotropic incompressible material of the Mooney type.

MODELING HINTS:

NS35 through NS35B

One row of ten PLANE2D axisymmetric 8-node elements, in the radial direction, is used to model the plate. A Poisson’s ratio of 0.49 is defined to approximate the incompressibility of the material. Since the displacements increase rapidly at low pressures (0-8 psi), the problem requires a slow initial loading. A very large load increment leads to negative diagonal terms or divergence.



Unloading will mirror the loading perfectly. It can be achieved by reversing the sequence of load application. A rapid load decrease yields the same result and may lead to termination due to negative diagonal terms or lack of convergence.

NS35C

Two rows of 20 PLANE2D axisymmetric 4-node elements in the radial direction are used to model the disk as shown in Figure NS35-4. The u/p formulation is used for the current case with a Poisson’s ratio equal to 0.4999, which corresponds to the ratio of bulk modulus to shear modulus (K/G) equal to 5000. From the result, it is seen that the error of the volume ratio (V/V0) is kept within 1%. In order to satisfy the hinged boundary condition along the edge, two linear constraint equations are prescribed:

UX(61) + UX(63)= 0

UY(61) + UY(63)= 0

Figure NS35-1

NS35D

One row of ten TRIANG axisymmetric 6-node elements, in the radial direction, is used to model the plate. The Poisson’s ratio of the material is 0.4999.

1

2

3 5

4

Finite Element Mesh (NS35 thru NS35B)

centerof plate

8

96

53

52

51x

CL

D

Y

Section I-I

Problem Sketch

I

I



8-76

NS35E

TETRA4 solid elements are used to model this problem with a Poisson’s ratio of 0.4995. The material model is Ogden hyperelastic model. Axisymmetry boundary conditions are applied to the model.

GIVEN:

Mooney-Rivlin constants:

A = 80

B = 20

Poisson’s Ratio in x-y:

ν = 0.49 (NS35 through NS35B)

= 0.4999 (NS35C)

D = 15 in

Ogden constants:

α1 = 2, µ1 = 2A = 160

α2 = -2, µ2 = -2B = -40

ν = 0.4975


Finite Element Mesh (NS35D)



Figure NS35-2



8-78

Figure NS35-3

Figure NS35-4

DEFORM ED SHAPE PLOT AT PRESSURE = 36 PSI

DEF STEP: 16 = 1

PRESSURE

HINGED BOUNDARY

CONDITIONS

ORIGINAL AND FINAL DEFORMED SHAPES (NS35C)



Figure NS35-5

Figure NS35-6



8-80

NS36: Initial Interference between Two Thick Hollow Cylinders

TYPE:

Nonlinear static analysis using PLANE2D axisymmetric and contact (node to line gap) elements.

REFERENCE:

Timoshenko, S. P., and Goodier, J. N., “Theory of Elasticity,” McGraw-Hill, New York (1970).

PROBLEM:

A cylinder is first compressed and then placed inside another cylinder. Evaluate the deformed shape and the forces at the interface.

MODELING HINTS:

The two cylinders are modeled separately, each based on its unstressed geometry, using PLANE2D axisymmetric elements. The interface is modeled with contact (node to line) elements; nodes are selected on the outer boundary of the inner cylinder, while a line is used to define the inner surface of the outer cylinder. A one step solution is performed; no external forces are defined.

Figure NS36-1



Figure NS36-2

NOTES:

In order to obtain accurate results when a contact problem exists, the contacting nodes must be located inside the contacting surface. If a node moves to a position where a normal to the surface does not exist, the program assumes the point is not in contact with the surface (unless a step-by-step solution is performed in which deformations are induced gradually). Therefore, in this problem we define a height for the outer cylinder which is slightly more than the height of the inner cylinder.

Figure NS36-3

GIVEN:a1 = 20 inb1 = 22 in h1 = 1 in a2 = 21.25 in b2 = 24.25 in h2 = 1.01 inModulus of elasticity = 30E6 psiPoisson’s ratio = 0.3

COMPARISON OF RESULTS:Theory COSMOSM Difference

Contact Loca-tion (radius) 21.570915 in 21.57546 in 0.02%

Pressure at the Interface 57.231 ksi 57.756 ksi 0.92%



8-82

NS37: Contact Between Two Solid Cubes

TYPE:

Nonlinear static analysis using 8-node solid and contact (node to surface gap) elements.

NS37) 8-node solid elementsNS37A) TETRA10 elements, automatic soft springs

PROBLEM:

A cube is pushed against a second larger cube which stands against a rigid wall. Verify that the two cubes will displace together by using contact elements.

MODELING HINTS:

NS37:

Each cube is modeled by one solid 8-node element. The interface is modeled by defining nodes on the smaller cube and one 4-noded surface on the larger cube. Soft truss elements are used to avoid stiffness singularities.

NS37A:

Each cube is modeled using TETRA10 elements. The interface in modeled by gap elements on the smaller cube and 6-noded sub-surfaces on the large cube. To stabilize the smaller cube in the global x-direction, a soft stiffness of 100 lb/in is defined for the contact source in that direction (gap real constant).

GIVEN:

E = 30E6 psiυ = 0p = 4E6 psi


U = u1 + u2

Ui = P li / Ai Ei

U1 = (4E6 x 1) / (1 x 30E6)

U2 = (4E6 x 1.4) / [(1.4 x 1.4) x 30E6]

U = 0.22857 in

Figure NS37-1




NS37:

Deflection at the free end:

Figure NS37-2

NS37A

Here since each cube is defined with several elements, the contacting surfaces undergo translation and bending. As a result, the displacements obtained are about 8% higher than those predicted by the 8-node solid model.

δ, inch

Theory 0.22857

COSMOSM 0.22857

Figure NS37-3



8-84

NS38: Contact Between Two Parallel S.S. Circular Plates

TYPE:

Nonlinear static analysis using 8-node PLANE2D axisymmetric and contact (node to line gap) elements.

NS38A) Without friction

NS38B) With friction

PROBLEM:

Two parallel simply supported circular plates stand 0.5 inches apart. Pressure loading is applied to the upper plate. Verify that the plates come in contact and therefore calculate the displacement at the center of the lower plate. Compare with regular gap elements.

MODELING HINTS:

The two plates are modeled separately, using PLANE2D axisymmetric elements. The interface is modeled with contact (node to line gap) elements. A one step solution is performed for each case.

Figure NS38-1

101103

31

h

Problem Sketch

R

t

CL

CL

h

P


P

X



GIVEN:

R = 10 in

t = 0.1 in

h = 0.5 in

Pressure = 6 psi

Poisson’s ratio = 0.3

Coefficient of friction = 0.7

Modulus of elasticity = 10.5E + 6 psi


Note that for this problem the results obtained from the two methods (contact and regular gaps) are almost identical.

Figure NS38-2

Vertical Disp. (inch)(center of the lower plate)

No Friction With Friction

Contact (Node to line) 1.7347 1.7268

Regular Gaps 1.7377 1.7308



8-86

NS39: Contact Between Two Parallel Simply Supported Circular Plates

TYPE:

Nonlinear static analysis using contact (node to surface gap) elements and:NS39A) 4-node shell elements

NS39B) 20-node solid elements

NS39C) 3-node thick shell elements, Lower plate = target

NS39D) 3-node thick shell elements, Upper plate = target

PROBLEM:

Same as problem NS38, except here shell or solid elements are used to model a quarter of the plates. As a result, contact is modeled using node to surface gaps. The goal is to compare the results with those obtained in problem NS38.

Figure NS39-1

MODELING HINTS:

For NS39A and NS39B:A quarter of each plate is modeled using:a) 80, 4-node shell elements b) 50, 20-node solid elementsA hole with a small radius (0.05 inches) is assumed at the center of each plate so that triangular elements are not needed in modeling of plates.



Contact between the two plates is modeled using 3 different groups of gaps:1. Nodes nearest to the center where only vertical motion is allowed are connected

by regular (node to node) gaps.2. Along the two axis of symmetry (θ = 0. and θ = 90.) where motion is to remain

in planes XZ and YZ, two groups of node to line gaps are used.

3. The last group of gaps (node to surface) considers contact between the inside nodes of the upper plate and 4-noded (8-noded when 20-node solid elements are used) surface segments on the lower plate.

For NS39C and NS39D:

Contact between the two plates is modeled using a node-to-surface gap element group. In both cases, target is defined by triangular 3-noded sub-surfaces.

GIVEN:R = 10 in

t = 0.1 in

h = 0.5 in

Coeff. of friction = 0.7

Pressure = 6 psi

Modulus of elasticity = 10.5E+6 psi



*The results obtained from PLANE2D axisymmetric and solid 20-node elements are almost identical.

**Shear deformation is neglected based on the Linear shell theory. Therefore, when shell elements are used in modeling of plates, friction is not considered.

Vertical Disp. (inch)(center of the lower plate)

No Friction With Friction

Axisymmetric PLANE2D(2D Analysis) 1.7347 1.7268

20-node Solid Element(3D Analysis) (at node No. 1001) 1.7340 1.7262*

4-node Shell Element(3D Analysis) (at node No. 101) 1.7260 — **

3-node Shell ElementsCased C and D) 1.733 —



8-88

Figure NS39-2

Figure NS39-3

Figure NS39-4



Figure NS39-6

Figure NS39-5

Figure NS39-7



8-90

NS40: Bending and Inflation of a Simply Supported Circular Plate Split

into Halves Through Its Thickness

TYPE:

Nonlinear static analysis using 8-node PLANE2D and contact (node to line gap) elements; Mooney-Rivlin hyperelastic material model, large deflections using total Lagrangian formulation, Newton-Raphson iterations.

REFERENCE:

Frank J. Marx, “Hyperelastic Elements (STIF84, STIF86),” ANSYS Revision, 4.3 Tutorial, 1987.

PROBLEM:

A simply supported circular plate is split into halves through its thickness. The lower half is then subjected to an external pressure varying from 0 to 50.0 psi. The plate is made of an isotropic incompressible material of the Mooney type.The behavior of this plate as a whole was studied in problem NS35. When the plate is sliced, the two slices bend together through contact. Therefore, this problem is useful in testing the accuracy of the contact elements when large rotations exist.

MODELING HINTS:

The two slices of the plate are modeled separately, using PLANE2D axisymmetric elements. The interface is modeled using contact (node to line gap) elements. Pressure is increased gradually, similar to problem NS35.

Figure NS40-1



Figure NS40-2

GIVEN:

Mooney-Rivlin Constants:

A = 80

B = 20

ν = 0.49


While in low stresses, the split plate is weaker, in higher stresses when the two slices undergo large deflections as well as large rotations, the response of the sliced plate is expected to be close to that of the uncut plate.By studying the results obtained for this problem the following can be observed:1. The contact line gradually deforms from flat to nearly a quarter of a circle.

Thus, the contact force (normal to the line) near the support undergoes about 90 degrees change in direction; regular gaps can not be used for this problem.

2. At very low pressures (0., to 0.1 psi) there is up to 32% increase in response due to the slicing of the plate.

3. At higher pressures (.1, to 28. psi) the response of the sliced plate is higher but remains within 2% of that of the uncut plate.

4. When pressure exceeds 28 psi, the sliced plate starts to weaken again. It begins to deform at a faster rate in comparison to the uncut plate, and therefore, the two responses start to deviate.

Figure NS40-3



8-92

NS41: Nonlinear Elasticity of a Cantilever Beam

TYPE:

Nonlinear static analysis, nonlinear elasticity.NS41A) PLANE2D elements

NS41B) SOLID elements

NS41C) SHELL4T elements

NS41D) BEAM elements

NS41E) TETRA10 elements

REFERENCE:

The theoretical solution is carried out based on thin beam theory.

PROBLEM:

Determine the deflection of a cantilever beam under an end moment as shown in Figure NS41-1. The material is nonlinear elastic as shown in Figure NS41-2.

Figure NS41-1

t

45

43

41

54321 PLANE2D

(NS41A)

X

Y

1

2

3 87

26

25

SHELL4T(NS41C)

X

Y

5

6

4

1 2 3 4 5

1 2 3 4

BEAM(NS41D)

Finite Element Models

7000.

2000.

1.0 1.001 2.0

M (lb.- in.)

Moment - Time Curve

SOLID (NS41B)

Y

X

Z

11

6

5 85

81

86

90

TETRA10(NS41E_

Y

X

Z

236

228

M/2

L

M/2h

Problem Sketch

b



GIVEN:

L = l00 in

h = 2 in

b = 1 in

ν = 0


1. Linear elasticity (M = 2000 lb-in)

2. Nonlinear elasticity (M = 7000 lb-in)

Deflection atFree End (inch)

Rotation atFree End

Theory - 5.0 0.1

COSMOSM

PLANE2D - 5.0001 0.1000

SOLID - 4.9991 0.1000

SHELL4T - 5.0000 0.1000

BEAM - 5.0001 0.1000

TETRA10 - 4.9998 0.1000

Deflection atFree End (inch)

Rotation atFree End

Theory - 7.5 0.15

COSMOSM

PLANE2D - 7.5279 0.1506

SOLID - 7.5275 0.1506

SHELL4T - 7.4791 0.1496

BEAM - 7.4340 0.1487

TETRA10 - 7.5156 0.1504

Figure NS41-2

60.003E6

3E3

0.001 2.001

σ psi

ε(Strain)

(Stress)

Strain_Stress Curve



8-94

NS42: Elastoplastic Analysis of a Simply Supported Plate, Elastic-Perfect Plastic Case

TYPE:

Nonlinear static analysis, SHELL4T elements.

PROBLEM:

Determine the response of a simply supported plate under uniform pressure. The pressure load (p) is varied well into the plastic range.

Figure NS42-1

GIVEN:

a = b = l0 in

h = 0.l in (thickness of plate)

E = l0E6 psi

ν = 0.3

ET = 0.0

σY = 36,000 psi

2. 7. 10.t

4.05

5.8055.4

ρ, psi

Load - Time Curve

Hinge

2a

X

Y

Hinge73 81

9


12b




The variation of pressure (p) versus central deflection is shown in the next figure. The result from COSMOS7 is also enclosed.

Figure NS42-2



8-96

NS43: Large Displacement Response of a Cylindrical Shell Under a Conc. Load

TYPE:


REFERENCE:

Horrigmoe, G., “Finite Element Instability Analysis of Free-Form Shells,” Report No. 77-2, the Norwegian Institute of Technology, The University of Trondhem, Norway (l977).

PROBLEM:

Determine the response of a shallow cylindrical shell under a concentrated load at the center of the shell. The curved edges are free and the straight edges are hinged and immovable.

Figure NS43-1

Free Edge

θ

Y

Z

21

25

5

X

θ

_P2b

h

Problem S ketch

_P , (KN)

19 58t

2.1282.2156

Load - Time Curve

Hinge



GIVEN:

R = 2,540 mm

b = 254 mm

h = l2.7 mm

θ = 0.l rad

E = 3l02.75 N/mm2

ν = 0.3


The force (p) - central deflection curve is shown in the next figure and compared with the result from reference.

Figure NS43-2

Reference

COSMOS/M

CENTRAL DEFLECTION (mm)

FORCE

(*E3), N



8-98

NS44: Buckling and Post Buckling of a Simply Supported Plate

TYPE:


REFERENCE:

Timoshenko and Woinosky-Krieger, “Theory of Plates and Shells,” McGraw-Hill Book Co., 2nd Ed., pp. 389.

PROBLEM:

Find the buckling load and post buckling behavior of a simply supported isotropic plate subjected to inplane uniform pressure p applied at x = -a and x = a.

Figure NS44-1

Z

Y

X5

h

1

2a

25

2a

21

Fe


SimplySupported

SimplySupported

P

P



Figure NS44-2

GIVEN:

a = 20 in

h = l in

E = 3E4 psi

ν = 0.3

NOTE:

The post buckling behavior is obtained by applying a transverse force (Fe) at the center of the plate at the first stage. The inplane pressure (p) is then applied and the transverse load is reduced at this stage. The buckling load is obtained by decreasing the inplane pressure to zero at the final stage.

P lb./in.

Time

67.8

75.

5 15 30 60 65 75

0.5

1.0

5 20 60 75

Fe, lb.

25 30Time



8-100


The inplane pressure - central deflection curve is shown in the next figure. The theoretical buckling load is 67.78 lb/in.

Figure NS44-3

75

70

65

60

INPLANE

PRESSURE

(LB/IN)

BUCKLING LOAD = 67.78 LB/IN

CENTRAL DEFLECTION, INCH

0.0 0.40.1 0.50.2 0.60.3 0.7 0.8



NS45: Large Displacement Response of a Cylindrical Shell Under a Uniform Load

TYPE:

Nonlinear static analysis, SHELL4 elements.

REFERENCE:

Sabir, A. B., and Lock, A. C., “The Application of Finite Elements to the Large Deflection Geometrically Nonlinear Behavior of Cylindrical Shells,” Variational Methods in Engineering, South Hampton University Press (l973).

PROBLEM:

Determine the response of a cylindrical shell under normal uniform pressure (p). All edges are clamped and immovable.

Figure NS45-1

Clamped

θ

Z

739

X

θ

2L

1

Clamped

81

P


Yt

2.25

3.0

6 18 21

Pressure - Time Curve

1.5

(10 ), N/mm2-3P

R

h



8-102

GIVEN:

R = 2,540 mm

L = 254 mm

h = 3.l75 mm

θ = 0.l rad

E = 3l02.75 N/mm2

ν = 0.3


The pressure (p) - central deflection curve is shown in the next figure and compared with the result from reference.

Figure NS45-2

Reference

COSMOS/M

CENTRAL DEFLECTION (mm)

2 Nmm

FORCE

(*E3),



NS51: Thermo-Plasticity in a Thick-Walled Cylinder

TYPE:

Nonlinear static analysis, thermo-plasticity, temperature-dependent yield stress, PLANE2D axisymmetric 8-node elements

REFERENCE:

Mark D. Snyder, Klaus-Jurgen Bathe, “Formulation and Numerical Solution of Thermo-Elastic-Plastic and Creep Problems,” Report 8244-3, (1977).

PROBLEM:

A thick-walled cylinder is subjected to varying internal pressure and temperature, as shown in the figure. Plane strain conditions are assumed in the direction of the axis of the cylinder. Determine:1. Response at the outer surface of the cylinder2. Residual stress distributions through the wall thickness when pressure is

reduced to zero (step = 10).

Figure NS51-1



8-104

Figure NS51-3

RESULTS:

The response and stress plots are identical to those obtained by ADINA program.

Figure NS51-2



Figure NS51-4

Figure NS51-5



8-106

Figure NS51-6



NS52: Thermo-Plasticity in a Bar

TYPE:

Nonlinear static analysis, thermo-plasticity, temperature-dependent material properties.

NS52) Truss elements

NS52A) PLANE2D plane stress elements

NS52B) Solid elements

REFERENCE:

Mark D. Snyder, Klaus-Jurgen Bathe, “Formulation and Numerical Solution of Thermo-Elastic-Plastic and Creep Problems,” Report 8244-3, (1977).

PROBLEM:

A bar is subjected to varying pressure and temperature, as shown in the figure. Determine strain in the bar.

Figure NS52-1



8-108

Figure NS52-2

RESULTS:

Similar results are obtained for different element types. Since there is no load reversal, using kinematic hardening law for plasticity yields identical results.



Figure NS52-3

Figure NS52-4



8-110

NS61: Bearing Capacity for a Strip Footing

TYPE:

Nonlinear static analysis, Drucker-Prager Elastoplastic material model, small deflection, Newton-Raphson.

NS61A) PLANE2D (plane strain)


REFERENCE:

Joseph E. Bowels, “Foundation Analysis and Design,” McGraw-Hill Book Co., 2nd. Ed (1977), pp. 113-124.

PROBLEM:

Find bearing capacity for a strip footing with width B sitting at the ground surface subjected to a uniform pressure P and soil parameters shown below.

Figure NS61-1

FINITE ELEMENT MESH

LINE OF SYMMETRY

B/2p



GIVEN:

B = 120 in

Soil parameters: ρ = 1.7961E - 4 lb - sec2/in4

φ = 27.27°C = 1 psi

E = 3E4 psi

υ = 0.3

NOTE:

Due to symmetry, half of the soil is modeled. Because the bearing capacity of foundations depends on the self-weight of the soil, an acceleration of gravity in the y-direction (-386.4 in/sec2) is applied to simulate this effect. When the applied pressure approaches the limit load (bearing capacity of foundations), the soil “bulging” takes place adjacent to the footing. This phenomenon induces difficulties in achieving the limit load. Reducing the load increment by taking a couple of RESTART procedures and using a regular (not modified) Newton Raphson iterations will help solve these problems.


The deformed shape plots for PLANE2D (plane strain) and SOLID models are shown in Figures NS61-2 and NS61-3.

Ultimate Bearing Capacity qu (psi)

Terzaghi (1943) 90.0

Hansen (1970) 93.0

COSMOSM (plane strain) 88.5

COSMOSM (SOLID) 95.0



8-112

Figure NS61-2

Figure NS61-3



NS62: Bearing Capacity for Round Footing

TYPE:

Nonlinear static analysis, Drucker-Prager Elastoplastic material model, small deflection, Newton-Raphson iterations, PLANE2D (Axisymmetric) elements.

REFERENCE:

Joseph E. Bowles, “Foundation Analysis and Design,” McGraw - Hill Book Co., 2nd.Ed (1977), pp. 113-124.

PROBLEM:

Find bearing capacity for a round footing with radius R sitting at the ground surface subjected to a uniform pressure p and soil parameters shown below.

GIVEN:

R = 6 in

Soil parameters:

ρ = 1.7961E-4 lb • sec2/in4

φ = 27.27 °c = 1 psi

E = 3E4 psi

υ = 0.3

NOTES:

The finite element mesh is as same as that shown in Figure NS61-1 except the replacing of B/2 by R. All the precautions mentioned in Problem NS61 are also needed in this problem.



8-114


The deformed shape plot is shown in Figure NS62-1.

Figure NS62-1

Ultimate BearingCapacity qu (psi)

Terzaghi (1943) 75.0

Hansen (1970) 63.0

COSMOSM 68.5

= 0.8053 yU

DEFORMED SHAPE PLOT (p = 68 PSI)

PLANE2D (AXISYMMETRIC)

DEF STEP: 19 = 19



NS63: Bending and Inflection of a Simply Supported Plate

TYPE:

Nonlinear static analysis, Mooney-Rivlin hyperelastic material model, large deflection, deformation-dependent pressure, local boundary conditions, and auto-stepping.

NS63A) SHELL4T with regular shell analysis

NS63B) SHELL4T with membrane element analysis

REFERENCE:

T. J. R. Hughes and E. Carnoy, “Nonlinear Finite Element Shell Formulation Accounting for Large Membrane Strains,” Nonlinear Finite Element Analysis of Plates and Shells. AMD Vol. 48, pp. 193-208, ASME, New York.

PROBLEM:

A simply supported circular plate is subjected to a uniform pressure. The plate is made of an isotropic incompressible material of the Mooney-Rivlin type (see Problem NS35). Both regular and membrane shells are tested and results are compared.

GIVEN:

r = 7.5 in Mooney-Rivlin constants:

h = 0.5 in A= 80 psi

B= 20 psi

NOTES:

The circular plate is modeled by SHELL4T elements (shown in Figure NS63-1). Due to axisymmetry, only a 5-degree wedge of the plate is modeled and local boundary conditions along the line A-C are applied to remain the axisymmetry, i.e.,

Uθ = 0

θ r = 0

For the membrane element analysis, a tiny prestress (0.01 psi) is adapted to prevent singularity. An auto-stepping algorithm is used to control the load increment such that the convergence and accuracy of the solution are insured especially for the membrane element analysis.



8-116


The plots of normal deflection at center versus pressure are shown in Figure NS63-2. The result of membrane element analysis is almost identical to the result of regular shell because the membrane strain becomes dominant in this type of rubber analysis.

Figure NS63-1

Figure NS63-2

Raduis

Plane View

Finite Element Mesh (10 SHELL3/4T)

SimplySupported

Y

AZ

C

B

NormalPressure

5 Degrees

X

PRESSURE (PSI)

C

E

N

T

R

A

L

D

E

F

L

E

C

T

I

O

N

(IN)

HUGHES: SHELL

COSM OS/M (REGULAR M EM BRANE)

∆

∆∆

∆

∆

∆

∆

∆

∆

∆∆



NS64: Failure and Strain Response of a [0,90,90,90,90,0] Graphite-Epoxy Laminate

TYPE:

Nonlinear static analysis, Tsai-Wu failure criterion, small deflection, Newton-Raphson iterations, SHELL4L elements.

REFERENCE:

Stephen W., Tsai and H. Thomas Hahn, “Introduction to Composite Materials,” Technomic Publishing Co., Inc. (1980), pp. 308.

PROBLEM:

Find the FPF (First Ply Failure) and UF (Ultimate Failure) stresses for a 6-layer T300/5208 cross-ply laminate under the uniaxial tensile load. The geometry of the laminate is shown in Figure NS64-1.

Figure NS64-1. Geometry of a 6-Layer T300/5208 Cross-Ply Laminate

GIVEN:

a = b = 10 in

6h = 1 in

Nx > 0 (tensile loading)

E1 = 26.27E6 psi

x

y

bN x

a

N x



8-118

E2 = 1.49E6 psi

ν12 = 0.28

G12 = G13 = G23 = 1.04E6 psi

F1T = 2.17E5 psi

F1C = 2.17E5 psi

F2T = 5.81E5 psi

F2C = 3.57E4 psi

F12 = 9.87E3 psi

where F1T and F1C are tensile and compressive strengths in the 1st material direction; F2T and F2C are tensile and compressive strengths in the 2nd material direction; F12 is the shear strength in the material 1st-2nd plane.

NOTES:

Due to symmetry, a quarter of the laminate is modeled.


Reference:

R = 38.03E3 psi for 90° ply

R = 69.09E3 psi for 0° ply

where R = failure stress

COSMOSM

FPF stress = 40E3 psi on 90° ply; along direction 2UF stress = 75E3 psi

DISCUSSION:

In the COSMOSM failure analysis, a two-stage failure algorithm is used, i.e., the failure occurs in the transverse direction first and then the fiber direction. In the current example, the stress can still be carried by the 90° ply (2,3,4 and 5) in the fiber direction after FPF. The laminate is then degraded to the point where the secondary failure occurs on the 0° ply in the transverse direction. Finally UF is reached and the largest load the laminate can carry is determined.



NS65: Failure and Strain Response of a Symmetric Angle-Ply Laminate of Graphite-Epoxy

TYPE:

Nonlinear static analysis, Tsai-Wu failure criterion, small deflection, Newton-Raphson iterations, thermal analysis, SHELL4L elements.

REFERENCE:

Stephen W., Tsai and H. Thomas Hahn, “Introduction to Composite Materials,” Technomic Publishing Co., Inc. (1980), pp. 362-363.

PROBLEM:

Find the FPF (First Ply Failure) and UF (Ultimate Failure) stresses for a symmetric AS/3051 angle-ply laminate under a uniaxial tensile load.

GIVEN:

where F1T and F1C are tensile and compressive strengths in the 1st material direction; F2T and F2C are tensile and compressive strengths in the 2nd material direction; F12 is the shear strength in the material 1st-2nd plane.

NOTES:

Due to symmetry, a quarter of the laminate is modeled.

a = b = 10 m G12 = G13 = G23 = 7.1 Gpa4h = 1 m α1 = 0.18E-6 / °KNx > 0 (tensile loading) α2 = 22.5E-6 /°K

< 0 (compressive loading) F1T = 1447 MpaE1 = 138 Gpa F1C = 1447 MpaE2 = 8.96 Gpa F2T = 51.7 Mpaν12 = 0.3 F2C = 206 Mpa T-T0 = 1000° K F12 = 93 Mpa



8-120


The comparison is also shown in Figure NS65-1.

Figure NS65-1. Comparison of Results of a Symmetric AS/3051 Angle-Ply Laminate

DISCUSSION:

The behavior of angle-ply laminates is different from that of cross-ply laminates (see NS64). No progressive failure occurs in angle-ply laminates under the uniaxial loading. The FPF stress is therefore equal to the UF stress.

Nx/h at UF (Mpa) Tensile/Compressive

Theta (degrees) Reference (from Test) COSMOSM

0.015.030.045.090.0

1520.0/-1521.0879.0/-648.0

482.0193.0

54.0/-214.0

1482.0/-1482.01128.5/-666.0

465.5/-308200.0/-230.052.0/-210.0

1500

1000

-1500

-1000

500

-500

0

Reference (test)

Reference (computation)

COSMOS/M

15 9075604530

a

b 1N 1N 1

+φ

-φ

+φ-



NS66: Static Equilibrium Position and Relaxation of a Cable-Buoy System

TYPE:

Nonlinear hydrostatic and hydrodynamic analysis, large deflection, auto-stepping algorithm, IMPIPE, BUOY, and GAP elements.

REFERENCE:

John W. Leonard, “Tension Structures - Behavior and Analysis,” McGraw-Hill Book Company, pp. 197-201.

PROBLEM:

NS66A) Find the static equilibrium position of a cable-buoy system (see Figure NS66-1). The system is completely immersed and subjected to the gravity and buoyant forces.

NS66B) The cable-buoy system was then released. Find the dynamic response of the system.

Figure NS66-1. Geometry of a Cable-Buoy System

SEA BED

B

A

X

SWL

BUOY

H

X

Z

CABLE



8-122

GIVEN:

Cable:Total length of riser = 71.56 in

A = 51.1 in

B = 42.1 in

EA = 4.8 lb

Mass = 11.443E-3 lb/ft

Outside diameter = 0.163 in

Coefficient of normal drag = 1.2

Coefficient of tangential drag = 0.02

Coefficient of added mass and inertia = 1

Buoy:Mass = 0.025 lb

Outside diameter = 2 in

Coefficient of drag = 0.5

Coefficient of added mass and inertia = 0.5

Water depth = 200 in (assumed)

Water density = 0.9366E-4 lb • sec.2/in4

RESULTS:

1. The static equilibrium position of the system from the current analysis is shown in Figure NS66-2 and compared with the reference one.

2. For the dynamic analysis, Figures NS66-3 and NS66-4 illustrate the compari-sons of the horizontal and vertical position histories of the buoy; Figure NS66-5 shows total velocity histories of the buoy, and Figure NS66-6 shows the tension histories of the cable.



NOTES:

1. Problem NS66A is a static problem, so all parameters regarding hydrodynamic analysis are assigned as zero.

2. To obtain the static equilibrium position of the immersed structure, it is sug-gested the loading start from a small value with an auto-stepping algorithm and a regular Newton-Raphson method is used.

3. To simulate the sea bed contact for the cable-buoy system, several gap elements are arranged near the sea bed. The sea bed contact is detected by those ‘closed’ gaps.

4. Problem NS66B starts from an initial configuration which is the static equilib-rium position of the structure, and during the analysis acceleration of gravity is unchanged.

5. Tension in Figure NS66-6 is the force in the element x-direction (‘Positive’ sign means tension) excluding the hydrostatic pressure.

Figure NS66-2. Static Equilibrium Position of the Cable-Buoy System

X

COSMOS

Z

REFERENCE

GAP



8-124

Figure NS66-3. Comparison of the Horizontal Position Histories of the Buoy

Figure NS66-4. Comparison of the Vertical Position Histories of the Buoy

REFERENCECOSMOS/M V

ERTICAL

POSITION

(IN)

TIME SEC

REFERENCECOSMOS/M

VERTICAL

POSITION

(IN)

TIME SEC



Figure NS66-5. Comparison of Total Velocity Histories of the Buoy

Figure NS66-6. Comparison of the Tension Histories of the Cable

REFERENCECOSMOS/M

VERTICAL

POSITION

(IN/SEC)

TIME SEC



8-126

NS67: Static Equilibrium Position and Dynamic Analysis of a Steep S

Model Riser with Fixed Top End

TYPE:

Nonlinear hydrostatic and hydrodynamic analysis, large deflection, IMPIPE and BUOY elements.

REFERENCE:

Rumbod Ghadimi, “A Simple and Efficient Algorithm for the Static and Dynamic Analysis of Flexible Marine Riser,” Computer & Structures, Vol.29, No.4, pp. 541-555, 1988.

PROBLEM:

NS67A) Find the static equilibrium position of a steep S model riser. The riser is partially immersed and subjected to the gravity and buoyant forces as shown in Figure NS67-1.

NS67B) Find the dynamic response of a steep S model riser in (NS67A). The riser is subjected to the hydrodynamic force due to the wave.

Figure NS67-1. Static Equilibrium Position of the Steep S Model Riser

SEA BED

SWL

BUOY B H

MODELRISER

A

Z

X Y

C



GIVEN:

Riser:Total length of riser = 3.97 m

A = 1.48 m

B = 2.58 m

C = 0.79 m, EA = 3.012E3 N

EI = 9.88E-5 N • m2

Mass = 0.0836 Kg/m

Outside diameter = 0.006 m



Buoy:Mass = 0.1 Kg

Outside diameter = 0.1 m

Coefficient of drag = 1.5


Water depth = 2.13 m

Water density = 1000. Kg/m3

Wave height = 0.255 m

Wave period = 1.69 sec

RESULTS:

1. The static equilibrium position of the riser is shown in Figure NS67-1. 2. For the dynamic analysis, the node near SWL (ND = 6) is selected:

The horizontal and vertical displacement histories are shown in Figures NS67-2 and NS67-3. The results from the reference are listed below for comparison:

The period of dynamic response 1.69 sec

Amplitude of horizontal displacement 100.0 mm

Amplitude of vertical displacement 20.0 mm



8-128

NOTES:

1. The wave length in (NS66B) is not input but computed in the program automat-ically.

2. Please refer to NS66 for more notes.

Figure NS67-2. Horizontal Displacement History of Node-6

Figure NS67-3. Vertical Displacement History of Node-6

VERTICAL

DISPLACEMENT

M

TIME

TIME INCREMENT = 0.005 SEC

HORIZONTAL

DI

SPL

M

TIME

TIME INCREMENT = 0.005 SEC



NS68: Steady-State Analysis of a Cable Towing a Submerged Body

TYPE:

Nonlinear hydrostatic analysis, large deflection, IMPIPE elements.

REFERENCE:

Anthony R. Rizzo, “FE Analysis Simulates Undersea Structures,” Mechanical Engineering, pp. 51-54, August 1991.

PROBLEM:

Determine the peak tension in a towed cable and the depth of the towed body as functions of the ship’s speed (see Figure NS68-1).

Figure NS68-1

GIVEN:

Cable:Total length of cable = 1000 ft

E = 144E7 psf

ν = 0.35

Specific gravity = 1.5

Outside diameter = 1 in


Coefficient of tangential drag = 0.02

SHIP SPEEDCABLE

TOWEDBODY

SWL



8-130

Towed body:Weight in water = 200 lb

Characteristic area = 1 ft2

Coefficient of drag = 1

Water depth = 2000 ft

Water density = 1.9908 lb sec2/ft4

Ship speed = 0.5, 1, 1.5, and 2 ft/sec

MODELING HINTS:

1. The towed body is represented as two forces acting on the cable end: a vertical force representing the weight of the towed body in water and a horizontal force representing the drag force on the towed body (see Figure NS68-2).

2. The ship motion is simulated with a uniform current past the cable (see Figure NS68-2).

3. Consider the current velocity as a function of time, such that a static analysis is performed instead of a dynamic analysis.

4. In order to use cable option (Op. No. 1 = 1) in IMPIPE elements, a small pre-strain (Real constant-17) is applied to eliminate the difficulty at the beginning of the analysis.

Figure NS68-2

DRAG FORCE

SWL

WEIGHTIN WATER

CURRENTVELOCITY

Z

X



RESULTS:

Figure NS68-3 shows the cable configuration corresponding to different current velocities. Figures NS68-4 and NS68-5 show the depth of the towed body and the peak tension of the cable as a function of the tow velocity. Results from the current analysis and the reference one are very coincident.

Figure NS68-3

Figure NS68-4

SWL

ORIGINAL CONFIGURATION

VELOCITY = 0.5 FT/SEC

1.0

1.5

2.0

SHIP SPEED FT/SEC

DEPTH

FT



8-132

Figure NS68-5

SHIP SPEED FT/SEC

PEAK

CABLE TENSION

( LB)



NS69: 3D Large Deflection Analysis of a 45° Circular Bend

TYPE:

Nonlinear static analysis, large deflection, 3D BEAM elements.

REFERENCE:

Klaus-Jürgen Bathe and Saïd Bolourchi, “Large Displacement Analysis of Three-Dimensional Beam Structures,” IJNME, Vol. 14, pp. 961-986, 1979.

PROBLEM:

Find the large displacement response of a cantilever 45-degree bend subjected to a concentrated end load.

GIVEN:

The bend has an average radius R, cross-section area H*B and lies on the X-Y plane as shown in Figure NS69-1. The concentrated tip load P is applied into the z-direction.R = 100 in

H = 1 in

B = 1 in

E = 1E7 psi

ν = 0

Figure NS69-1

8

2

5

B

Beam Cross-section

R=100"

8 Equal Straight BEAM Elements

1

9

76

43

1"

1"

z

A

z

y

x

FixedEnd



8-134

RESULTS:

Figure NS69-2 shows the deformed configurations of the bend at various load levels. The tip coordinates predicted by COSMOSM and ADINA are listed in Table NS69-1 for comparison.

Table NS69-1. Tip Coordinates of a 45-Degree Circular Bend.

Figure NS69-2

P (lb) ADINA (X, Y, Z) (in) COSMOSM (X, Y, Z) (in)

300 (22.5, 59.2, 39.5) (21.9, 59.0, 40.2)

600 (15.9, 47.2, 53.4) (15.2, 47.4, 53.5)

ORIGINAL CONFIGURATION

Y

P = 600 LB

P =300 LB

A

A

Z

FINAL CONFIGURATION

X

A

P = 0 lb



NS70: Static Analysis of an Elastic Dome

TYPE:

Nonlinear static analysis, large deflection, 3D BEAM elements.

REFERENCE:

B. A. Izzuddin, A. S. Elnashai and P. J. Dowling, “Large Displacement Nonlinear Dynamic Analysis of Space Frames,” Eurodyn 90, Bochum, W. Germany, June 1990.

PROBLEM:

Find the static response of an elastic dome subjected to a concentrated load at the crown point.

GIVEN:

The top and side views of an elastic dome are shown in Figure NS70-1 and Figure NS70-2. Six 3D BEAM elements per member are used to model the dome with the beam cross-section area as H*B for all the members where H = 1.22 m and B = 0.76 m.E = 20690 MN/m2

ν = 0.3

Figure NS70-1

12.57M

10.385M

21.115

6.283M

Y

XZ



8-136

Figure NS70-2

NOTES:

Only one quarter of the dome is modeled due to symmetry. Appropriate boundary conditions and cross-section modulus are used along the lines of symmetry. The displacement control algorithm is used in the iteration procedure to avoid convergence problems.

RESULTS:

Figure NS70-3 illustrates load-displacement curves of the elastic dome where ref-1, -2 and -3 represent the following references:

ref-1: Izzuddin, et. al. (1990),

ref-2: Kondoh, et. al. (1986),

ref-3: Shi and Atluri (1988).

“Load Factor” (L.F.) represents the applied load (P) divided by the critical load (Pcr) where Pcr = 123.8 MN.

4.55m

1.55m

P

12.18m

24.38m

Z

XY



Figure NS70-3

REF-1 AND -2REF-3COSMOS/M

123

DISPLACEMENT (M)

LOAD

FACTOR



8-138

NS71: Large Deflection Analysis of a Cantilever Beam with Different Beam Cross-Sections

TYPE:

Nonlinear static analysis, large deflection, BEAM2/3D elements, and beam-section-definition.

REFERENCE:

Ramm, E., “A Plate/Shell Element for Large Deflections and Rotations, in Formulations and Computational Algorithm in Finite Element Analysis,” M.I.T. Press, 1977.

PROBLEM:

A cantilever beam with different cross-sections subjected to an end moment.

GIVEN:

The problem sketch is shown in Figure NS71-1. Five (5) BEAM2/3D elements are used in the analysis.

E = 12,000 psi

ν = 0

L = 100 in

A and I varied

Beam-Section-Type:1 (Rectangular), (NS71A)

2 Circular), (NS71B)

3 (Pipe), (NS71C)

4 (Box), (NS71D)

5 (I-Section), (NS71E)

6 (Trapezoidal), (NS71F)

0 (User-defined) (NS71G)



Figure NS71-1

RESULTS:

The normalized analytical solution is:

The numerical results by COSMOSM are close to the above solution.

Problem S ketch

L

M

Beam Model

Y

1 2 3 4 5 6 Y

M

XX

Y

H

Y

B

z

H = 1.0 in.B = 1.0 in.

zR

R = 0.5642 in.

Y

B

HTH

TB

Y

z

4

H = 2.0 in.B = 2.0 in.

TB = 0.134 in.TH = 0.134 in.

H

Y

B

TH

TBz

5

H = 5.0 in.B = 2.25 in.

TB = 0.1 in.TH = 0.1 in.

1 2

Beam S ection Types

T

Y

z

7

123

4

5

6

78 9 10

11

12

13

14

15

1617

R

R = 4.0 in.T = 0.2 in.

(17 pts. used)Y

z

B1

B2

H

Z r

H = 1.3333 in.

Y r

6

B1 = 1.0 in.B2 = 0.5 in

DT

Y

z

D = 1.1416 in.T = 0.4842 in.

3



8-140

NS72: Uniformly Loaded Elastoplastic Plate

TYPE:

Nonlinear static analysis, Plasticity, Kinematic Hardening, and SHELL4T element.

REFERENCE:

Foster Wheeler Corporation, “Intermediate Heat Exchanger for Fast Flux Test Facility: Evaluation of the Inelastic Computer Program,” prepared for Westinghouse ARD, Livingston, N. J., 1972.

PROBLEM:

Verify the accuracy of the result of an elastoplastic flat plate in the case of uniformly non-proportional stressing. Kinematic hardening is used.

GIVEN:

The geometry and material model is shown in Figure NS72-1 where 2*2 SHELL4T elements are used. Two edges are simply supported and all rotations are constrained to be zero.E = 30E6 psi

ν = 0.3

σy = 30, 000 psi

ET = 1.5E6 psi

L = 1 in

H = 0.001 in



Figure NS72-1

Loading History:

1. The plate is loaded into the plastic range in uniaxial tension in the x-

direction, unloaded slightly, and reloaded.2. Biaxial loading then follows, with σx and σy prescribed, as shown in Figure

NS72-2, so that the effective stress remains constant as 40,000 psi.

Figure NS72-2

Strain,

Geometry Model (2x2 SHELL4T)

L

L

H = THICKNESS

ε x

E

E t

1

Stress, σ

σ y

ε

1

Stress-Strain Curve

σx



8-142

Loading History:

RESULTS:

The σx versus ε x plot is shown in Figure NS72-3. The agreement with analytical solution is very close.

Figure NS72-3

Loading Step σx psi

1 30,000

2 32,500

3 35,000

4 37,500

5 40,000

6 40,500

7 40,000

7 through 84 (σx2 + σy

2 – σx σy)1/2 = 40,000

ANALYTICAL SOLUTIONCOSMOS/M

STRESS

KSI

STRAIN * E-3ε x

σx



NS73: Viscoelastic Rod/Bar Subjected to Constant Axial Load

TYPE:

Nonlinear static analysis, Time domain linear viscoelasticity, auto-stepping.NS73A-B) PLANE2D elements

NS73C) SOLID elements

NS73D) TRUSS elements

NS73E) BEAM elements

NS73F) SHELL4T elements

REFERENCE:

R. M. Christensen, “Theory of Viscoelasticity – an Introduction,” 2nd Ed., 1982, pp. 1-76.

PROBLEM:

A rod/bar is fixed in the axial direction on one end and a constant axial load is suddenly applied to the other end (shown in Figure NS73-1). The rod/bar is made up of a linear viscoelastic material.

GIVEN:

The material model is shown in Figure NS73-2 where the linear viscoelastic material is represented by a combination of linear springs and a dashpot. The extensional relaxation function is

where

Extensional relaxation moduli:

E∞ = 1,000 psi

E1 = 9,000 psi



8-144

Extensional relaxation time:

τ1E = 1 sec

Bulk modulus:

K = 100,000 psi (K0 = K∞)

The time-dependent material behavior inside the code is approximated with a generalized Maxwell model:

Shear relaxation modulai:

G0 = 3K0 E0 / (9K0 – E0) = 3370.7865 psi

G∞ = 3K∞ E∞ / (9K∞ – E∞) = 333.7041 psi

G1 = G0 – G∞ = 3037.0824 psi

Shear relaxation time: τ1G

The input material parameters are:

E0 = E∞ + E1 = 10,000 psi

υ0 = (3K0 – E0) / 6K0 = 0.4833

g1 = Gi / G0 = 0.9010

τ1G = 0.9899 sec

L = 10 in



For a rod:

D = 1 in

For a bar:

B = 3.14159 in

H = 0.25 in

Such that

A = π/4 in2

A Heaviside loading function is shown in Figure NS73-2. To capture the instantaneous material behavior, a tiny time step (0.001 sec) is used in the beginning along with the auto-stepping algorithm during which the rod/bar is relaxed toward its long time behavior. Tolerance for creep strain increment CETOL = 5E-4.

Figure NS73-1

P lane 2

B

L

P (t)

Truss and Be am

P (t)

2 D-P lane S tre ss (2 x2 ) SHELL4 T (2 x2 )

B

H



8-146

Figure NS73-2

RESULTS:

Analytical Solution:

For a prescribed stress problem, the strain is determined by the current value and past history of stress:

where J(t) is termed creep function. Taking the form of the generalized Kelvin model:

where:

Mate rial Mode l

F

E 1

E 8

η1

τE

1=

η1

E 1

= Dashpot

E 1

η1

= Linear SpringLoading Function

P(t)

P t = 100

0.001 60t (se c)



Comparison:

Instantaneous behavior:υ0 = 0.4833

COSMOSM:

υ (t = 0.001 sec) = – ε22 / ε11 = 0.4833

Long term behavior:

υ∞ = 0.4983

COSMOSM:

υ (t = 50 sec) = – ε22 / ε11 = 0.4983

Time histories of the axial strain from COSMOSM along with the analytical solution are plotted in Figure NS73-3 for comparison.

Figure NS73-3

Reference

COSMOS/M

AXIAL

STRAIN

TIME



8-148

NS74: Transient Thermal Loading of a Viscoelastic Slab

TYPE:

Transient thermal analysis, Nonlinear static analysis, Thermal loading, Time domain linear viscoelasticity, Temperature-time shift, PLANE2D elements, auto-stepping.

REFERENCES:

1. Carslaw, H. S., and Yeager, J. C., “Conduction of Heat in Solids,” Clarendon Press, Oxford, 1959.

2. Williams, M. L., Landel, R. F., Ferry, J. D., J. American Chemical Society, V77, pp. 3701, 1955.

PROBLEM:

A viscoelastic slab under plane strain restraint in all directions in its plane is subjected to a temperature loading on its faces (see Figure NS74-1). Investigate the response of the slab corresponding to different time values.

GIVEN:

Model:

The half-thickness (H/2 = 1 in) of the slab is modeled with a row of 8-node PLANE2D (plane strain) elements (see Figure NS74-1) for the heat transfer analysis. The same elements are then used for the nonlinear static analysis under the temperature loading where plane strain condition is imposed in the y-direction and symmetry is imposed about x = 0.

Loading:

The outside face of the slab is prescribed a uniform temperature. The temperature-time curve is a Heaviside function with T0 = 100 °F (Figure NS74-1).



Material:

1. Thermal material properties:Thermal conductivity:

k = 1 Heat/sec • in °F

Specific heat:

c = 1 Heat / (lb • sec2/in) °F

Density:

ρ = 1 lb sec2/in4

2. Elastic material properties:

E = 10,000 psi

ν = 0.4833

α = 1E-5/ °F

3. Viscoelastic material properties:

Relative shear relaxation modulus:

g1 = 0.9

Shear relaxation time:

τ1G = 0.9899 sec

4. Temperature-time shift function:

Williams-Landell-Ferry approximation is applied:

c1 = 4.92

c2 = 215° Fθ0 = 70° F



8-150

Figure NS74-1

ANALYSIS AND RESULTS:

Analysis 1:

The transient heat transfer analysis is performed. To capture the rapid temperature changes at the start of transient, a tiny time step (0.0005 sec) is used in the beginning. The time step is then increased gradually for a time period of 6.0 sec, during which the slab is allowed to reach its thermal equilibrium condition.

Analysis 2:

Reading the additional data from the second input file, the nonlinear static analysis is performed. The temperature loading is prescribed by reading the temperature distribution previously written in “.HTO” file. An auto stepping algorithm is used with tolerance for creep strain increment CETOL = 5E-5. Note: During the auto-stepping, temperatures at any time are obtained by interpolation.

Plain Strain

PROBLEM SKETCH

H

Temperature Loading on Surfaces 5 and 6UY = 0 on Surfaces 7 & 5UZ = 0 on Surfacss 8 & 10

FINITE ELEMENT MODEL: 10+1 PLANE 2D

H/2

X

Y

Z

65

10

9

8

7

Plain Strain

X

y

z

Line of Sym-metry

Temp.Loading

Ts

TIME

TEMPERATURE



Analytical Solution and Comparison:

1. Temperature distribution at t = 1 sec:

2. The stress and strain distributions at various times during the analysis are shown in Figure NS74-2 and Figure NS74-3.

3. Long term behavior:

4. The stress and strain distributions at the end of nonlinear static analysis (t = 6.0 sec) can be compared with those of an elastic slab with material properties to be the long-term viscoelastic material properties.

E = E∞ = 1,000 psi

ν = ν∞ = 0.4983

εxx = (σxx – νσyy – νσzz) / E + α T

εyy = (σyy – νσxx – νσzz) / E + α T

Symmetry:

σyy = σzz

Plane strain:

εyy = εzz = 0

Unrestrained condition in the x-direction:

σxx = 0

so:

x, in Node Analytic Solution°F COSMOSM°F

0.9 51 98.2 98.276

0.7 31 95.0 94.994

0.5 16 92.2 92.203

0.2 5 89.5 89.506

0 4 89.2 88.963

COSMOSMσyy = – E α T / (1 – ν) = 1.9932 psi –1.995 psiεxx = (1 + ν) α T / (1 – ν) = 2.9864E – 3 2.986E3



8-152

Figure NS74-2

Figure NS74-3

yy(psi)

X, INCH

σ

1234

T=0.009 secT=0.1 secT=1.2 secT=6.0 sec

3

1

2

4

xx

X, INCH

3

4

2

1ε

(*E-3)

1234

T=0.009 secT=0.1 secT=1.2 secT=6.0 sec



NS75: Transient Response of a Viscoelastic Cylinder Under Torsional Oscillation

TYPE:

Nonlinear static analysis, Displacement control algorithm, Time domain linear viscoelasticity, BEAM3D elements, auto stepping.

REFERENCES:

Christensen, R. M., “Theory of Viscoelasticity – an Introduction,” 2nd Ed., 1982, pp. 48-52.

PROBLEM:

A viscoelastic right circular cylinder has one end fixed and the other end prescribed an harmonic twisting angle (see Figure NS75-1a and NS75-1b). Find the twisting moment required to produce the given oscillation

Figure NS75-1a

4

3

2

PROBLEM SKETCH FINITE ELEMENT MODEL

y

x

z

R1 Ro

H

e (t)

1

1

45

θ (t)



8-154

Figure NS75-1b

GIVEN:

Four (4) BEAM3D elements are used to model the cylinder (see Figure NS75-1) where H = 1.0 in, r0 = 0.5 in and ri = 0.125 in The prescribed twisting angle is an harmonic function having the form:

Θ (t) = θ sin wt

w/2π = 1/T

where T = 50 sec and θ = 2.075E – 3 rad. The twisting angle-time curve is shown in Figure NS75-1.

Material properties:

E = 10,000 psi

ν = 0.4205

G = 3,520 psi

t

T

TWISTING ANGLE-TIME CURVE

θ



Shear relaxation moduli:

Tolerance for creep strain increment CETOL = 5.E-5

RESULTS:

The torque-twisting angle plot is shown in Figure NS75-2 for a total period of time 100 sec (two cycles). The analytical solution is also enclosed for comparison. It is noted that two or three cycles of twisting are generally required for the transient part of the response to become ignorable in comparison with the steady-state response. It is important that this transient response does not arise from inertia effects at this low frequency, but rather due to the fading memory nature of the material.

Figure NS75-2

i gi τiG, sec

1 0.2832 1.5E–5

2 0.1528 1.5E–4

3 0.1403 1.5E–3

4 0.1114 1.5E–2

5 0.0869 1.5E–1

6 0.0438 1.5

7 0.0338 1.5E1

8 0.0057 1.5E2

1ST CYCLE

2ND CYCLE

ANALYTICALSOLUTION

COSMOS/M

ANGLE OF TWISTING (RAD *E-3)

TORQUE

LB*IN



8-156

NS76: Extension of an Ogden Hyperelastic Bar/ Sheet

TYPE:

Nonlinear static analysis, displacement control algorithm, Ogden hyperelastic material model

NS76A) PLANE2D–plane stress elements


NS76C) PLANE2D–plane strain elements

NS76D) SHELL4T elements

PROBLEM:

NS76A-B, D)An Ogden hyperelastic bar is subjected to an axial load (see Figure NS76-1), find the response of the nominal stress versus principal stretch.

NS76C) An infinitively long sheet of Ogden hyperelastic material is subjected to a uniform extension load (see Figure NS76-1), find the response of the nominal stress versus principal stretch.

Figure NS76-1

L

D

PROBLEM SKETCH 1

L

E-E

PROBLEM SKETCH 2

DISPLACEMENT - TIME CURVE

50

Ux

00 50

E-E



GIVEN:

Problem–1:

L = 0.1 m

B = 0.01 m

H = 0.005 m

Modeled with 10*1 PLANE2D–plane stress, 20*2 SOLID and 1 x 1 SHELL4T elements.Problem–2:

L = 10 cm

B = 0.5 cm

Modeled with 10*1 PLANE2D–plane strain elements.Ogden material model constants:

For PLANE2D–plane strain and SOLID elements, ν = 0.499, for PLANE2D – plane stress and SHELL4T ν is not needed to input (ν = 0.5 is assumed internally).A displacement control algorithm is used along the axial direction by following the time curve as shown in Figure NS76-1 where the maximum displacement is five (5) times the initial length (L).

RESULTS:

Analytical solution:

Problem 1:

2nd p. k. stress in the principal direction:

i αi µi, MPa

1 1.3 0.618

2 5.0 0.001245

3 –2.0 –0.00982



8-158

whereλ1 = L/L0

λ3 = λ1-0.5

L = Final length

L0 = Initial length

Nominal Stress t1p:

t1p = λ1 = F/A0 = p

F = Force

A0 = Initial area

Problem 2:

whereλ1 = L/L0

λ2 = λ1-1

λ3 = 1

Nominal Stress t1p:

t1p = λ1 = F/A0 = p

The nominal stress-principal stretch curves for both examples are shown in Figure NS76-2 and Figure NS76-3. The results by using the Mooney-Rivlin hyperelastic material model are also enclosed for comparison where Mooney-Rivlin material constants: C1 = 0.11026 MPa and C2 = 0.09708 MPa. It is noted that the results by Ogden and Mooney–Rivlin models are close only in a small range of principal stretch.



Figure NS76-2

Figure NS76-3

OGDEN1 ANALYTICAL2 COSMOS/M PLANE STRESS3 COSMOS/M 3D SOLID4 MOONEY-RIVLIN

NOMINAL

STRESS

PRINCIPAL STRETCH

1 2 &3

4

OGDEN1 ANALYTICAL2 COSMOS/M PLANE STRESS3 MOONEY-RIVLIN

NOMINAL

STRESS

PRINCIPAL STRETCH

1 &2

4



8-160

NS77: Snap-Through/Snap-Back of a Thin Hinged Cylindrical Shell Under a Central Point Load

TYPE:

Nonlinear static analysis, SHELL4 elements, Arc-length control, automatic stepping.

NS77A) Ignore the effect of gravity (structure is assumed weightless)

NS77B) Include the effect of gravity

REFERENCE:

Crisfield, M. A., “A Fast Incremental/Iterative Solution Procedure That Handles Snap-Through,” Computers & Structures, Vol. 13, pp. 55-62, 1981.

PROBLEM:

Determine the snap-through/snap-back response of a shallow cylindrical shell under a concentrated load (P) at the center of the shell. The curved edges are free and the straight edges are hinged and immovable.

GIVEN:

R = 2,540 mm

b = 254 mm

θ = 0.10 rad

ν = 0.30

h (thickness) = 6.35 mm

P (reference load) = 10 N

MODELING HINTS:

Due to symmetry, a 4 x 4 mesh is used to model a quarter of the shell.Arc-length Control Information:

Max. load parameter (approx. value) = 100Max. Displacement (approx, value) = 30 Max. number of arc steps = 50Desired average number iterations/step = 5Initial load parameter = 10



Unloading check flag = 0Arc-length step adjustment coefficient = 0.5Automatic Stepping Information:

Min. (arc) step increment = 1.E-8Max. (arc) step increment = 30 for NS77A and 5 for NS77BAdditional Hints for NS77B:

The central point force is associated with time curve 1 and the acceleration of gravity is associated with time curve 2.First, force control is used to obtain displacements under gravity loading. During this phase, curve 2 is raised to 1.0 while curve 1 is kept at zero.Next, control is changed to the Arc-length algorithm with active restarting to find the response under the effect of the central force. (Time curve definitions are ignored during this phase).

RESULTS:

The curve of the load multiplier factor (LFACT) versus central deflection is shown in the next two figures.

Figure NS77A-1

p

2b

R

θ

1



8-162

Figure NS77B-1



NS78: Multiple Snap-through/Snap-back of a Thick Hinged Cylindrical Shell

Under a Central Point Load

TYPE:

Nonlinear static analysis, SHELL4T elements, Arc-length control, automatic stepping.

REFERENCE:

Tsai, C. T., and Palazotto, A. N., “Nonlinear and Multiple Responses of Cylindrical Panels Comparing Displacement Control and Riks Method,” Computers & Structures, Vol. 41, pp. 605-610, 1991.

PROBLEM:

Determine the multiple snap-through/snap-back response of a shallow cylindrical shell under a concentrated load (P) at the center of the shell. The curved edges are free and the straight edges are hinged and immovable.

GIVEN:

R = 2,540 mm

b = 254 mm

θ = 0.20 rad

ν = 0.30

h (thickness) = 12.70 mm

P (reference load) = 10 N

MODELING HINTS:

Due to symmetry, a 4 x 12 mesh is used to model a quarter of the shell.Arc-length Control Information:

Max. load parameter (approx. value) = 1,000Max. Displacement (approx. value) = 150 mm Max. number of arc steps = 140Desired average number iterations/step = 5



8-164

Initial load parameter = 10Unloading check flag = 0Arc-length step adjustment coefficient= 0.50

Automatic Stepping Information:

Min. (arc) step increment = 1 x 10E-8Max. (arc) step increment = 150

RESULTS:

The curve of the load multiplier factor (LFACT) versus central deflection is shown in the next figure.

Figure NS78-1

p

2b

R

θ

1



NS79: Large Displacement Nonlinear Static Analysis of a Cantilever Beam

Subjected to a Prescribed End Rotation

TYPE:

Nonlinear static analysis, large displacement, BEAM2D elements, prescribed displacement associated with time curve, Force control, automatic stepping.

REFERENCE:

Ramm, E., “A Plate/Shell Element for Large Deflection and Rotations,” in Formulations and Computational Algorithms in Finite Element Analysis. [M.I.T. Press, 1977].

PROBLEM:

Determine the deformed shape of the beam.

GIVEN:

L = 100 in

I = 0.01042 in4

A = 0.50 in2

E = 12,000 psi

ν = 0

h = 0.5 in

Prescribed end rotation = 6.28 rad

Automatic Stepping Information:

Min step increment = 1.E-8

Max. step increment = 0.02

RESULTS:

The deflected shape of the beam is shown in Figure NS79-2. Also, the horizontal and vertical displacements at the tip of the beam are shown in Figure NS79-3.



8-166

Figure NS79-1

Figure NS79-2

Lb

h

Problem Sketch

p

t

Time Curve

1 2 3 4 5

1 2 3 4 5


θ

θ6 7 8 9 10

6 7 8 9 10

11

1

1

DEFLECTED SHAPE OF THE BEAM



Figure NS79-3



8-168

NS80: Rubber O-Ring Squeezed Between Two Parallel Steel Plates

TYPE:

Nonlinear static analysis, Contact with friction, Mooney-Rivlin model, Large Deflection formulation, Displacement Control, PLANE2D Axisymmetric elements.

PROBLEM:

A rubber O-ring is squeezed between two steel plates. The plates travel a relative distance of 0.0347" towards each other. Determine stresses in the ring, and total force required:a) Coefficient of friction = 0.01

b) Coefficient of friction = 0.5

MODELING:

Due to symmetry, half of the ring cross section and one of the plates are modeled. Two soft springs are used to hold the plate, to prevent singularity of the stiffness matrix. Uniform pressure with a magnitude of one is applied to the top surface of the plate.While in part (a), the O-ring expands (slides), in part (b), friction is large enough to prevent the ring from sliding. Therefore, in part (b), contact with generalized friction is required. Part (a) can use either general or sliding friction options. However, the sliding option is faster in dealing with sliding problems.

Figure NS80-1

0. 3"

0.3"Top Steel Plate

Bottom Steel Plate . 139"

0. 3475= 1. 123"/



PROPERTIES:

1. O-Ring:C1 = 175 psi

C2 = 10 psi


2. Plates:

Young’s Modulus = 30E6 psi


3. Soft Trusses

Young’s Modulus = 1 psi

Area = 1 in

Length = 0.3 in

RESULTS:

In both cases, 12 solution steps are used to attain the prescribed plates’ relative displacement.

Case (a) Case (b)

Fric. Opt. = 2 Fric. Opt. = 1 Fric. Opt. = 1

Load Factor (per radian) 36.067 36.067 46.124

Load Factor (per radian) LF • D/8 14.90 lb/rad 14.90 lb/rad 19.056 lb/rad

Analysis Time (seconds) 592 sec 671 sec 452 sec



8-170

Figure NS80-2

Figure NS80-3



NS81: Extension of a Blatz-Ko Hyperelastic Bar

TYPE:

Nonlinear static analysis, displacement control algorithm, Blatz-Ko hyperelastic material model

NS81A) PLANE2D plane stress elements


REFERENCE:

J. K. Knowles and Eli Sternberg, “On the Ellipticity of the Equations of Nonlinear Elastostatics for a Special Material,” Journal of Elasticity, Vol. 5, Nos. 3-4, 1975, pp. 341-361.

PROBLEM:

A Blatz-Ko hyperelastic bar is subjected to an axial load (see Figure NS81-1). Determine the variations of nominal stress (force per unstressed area) versus principal stretch.

MODELING HINTS:

A displacement control algorithm is used along the axial direction where the maximum displacement equals the initial length (L).

GIVEN:

L = 0.1 m

B = 0.01 m

H = 0.005 m

E = 2.0732 Mpa

RESULTS:

The analytical solution for 2nd P.K. stress in the principal direction has the following form:

S1 = µ λ1-2 (−λ1

-2 + J)



8-172

where

J = λ1 λ2 λ3

λi = Stretch ratio in the i-th direction

λ1 = L/L0 (final length/initial length)

since

S2 = S3 = 0 → λ2 = λ3 = λ1-1/4

thus

S1 = µ λ1-2 (-λ1

-2 + λ1-1/2)

The nominal stress is:

F/A = S1 λ1

where F and A represent the applied force and the initial area respectively. Figure NS81-3 shows the nominal stress versus principalstretch curve.

Figure NS81-1

PROBLEM SKETCHE-E

L

E-E

P



Figure NS81-2

Figure NS81-3

DISPLACEMENT - TIME CURVE

DISPLACEMENT/L

o

ANALYTICALCOSMOS/M (PLAIN STRESS)

PRINCIPAL STRETCH

NOMINAL

STRESS



8-174

NS82: Extension of a Blatz-Ko Hyperelastic Sheet

TYPE:

Nonlinear static analysis, displacement control algorithm, Blatz-Ko hyperelastic material model, PLANE2D plane strain elements.

REFERENCE:


PROBLEM:

An infinitely long sheet of Blatz-Ko hyperelastic material is subjected to a uniform extension load (see Figure NS82-1). Determine the variation of nominal stress versus principal stretch.

GIVEN:

L = 10 cm

B = 0.5 cm

E = 2.0732 Mpa

MODELING HINTS:

The sheet was modeled with 10*1 PLANE2D-plane strain elements. Displacement is controlled in the direction of the applied force with a maximum displacement equal to the initial length (L).

RESULTS:


S1 = µ λ1-2 (-λ1

-2 + J)

where

J = λ1 λ2 λ3





since

S2 = 0, λ3 = 1 → λ2 = λ1-1/3

thus

S1 = µ λ1-2 (-λ1

-2 + λ1-2/3)

The nominal stress:

F/A = S1 λ1

where F and A represent the applied force and the initial area respectively. Figure NS81-2 shows the nominal stress versus principalstretch curve.

Figure NS82-1

Figure NS82-2

B P

L

PROBMEM STRETCH

NOMINAL

STRESS

ANALYTICALCOSMOS/M (PLAIN STRAIN)

PRINCIPAL SKETCH



8-176

NS83: Biaxial Tension of a Blatz-KoHyperelastic Sheet

TYPE:

Nonlinear static analysis, displacement control algorithm, Blatz-Ko hyperelastic material model, PLANE2D-plane stress elements.

REFERENCE:


PROBLEM:

A square sheet of Blatz-Ko hyperelastic material is subjected to a uniform extension load in both X- and Y- directions (see Figure NS83-1). Determine the variation of nominal stress versus principal stretch.

GIVEN:

L = 10 cm

B = 0.1 cm

E = 2.0732 Mpa

MODELING HINTS:

The sheet was modeled with one PLANE2D plane stress element. Displacement is controlled such that the maximum displacement equals the initial length (L).

RESULTS:


S1 = µ λ1-2 (-λ1

-2 + J)

where

J = λ1 λ2 λ3





since

S2 = 0, λ1 = λ2 → λ3 = λ1-2/3

thus

S1 = µ λ1-2 (-λ1

-2 + λ1-4/3)

The nominal stress:

F/A = S1 λ1

where F and A represent the applied force and the initial area. Figure NS83-2 shows the nominal stress versus principalstretch curve.

Figure NS83-2

Figure NS83-1

PROBMEM STRETCH

L

L

P

P

NOMINAL

STRESS

ANALYTICALCOSMOS/M (PLAIN STRESS)

PRINCIPAL STRETCH



8-178

NS84: Pure Bending of a Stretched Rectangular Membrane

TYPE:

Nonlinear static analysis, wrinkling membrane material model, small deflection, coupling degree of freedom, PLANE2D plane stress and BEAM2D elements.

REFERENCE:

Richard K. Miller and coworkers, “Finite Element Analysis of Partly Wrinkled Membranes,” Computers and Structures, Vol. 20, No. 1-3, pp 631-639, 1985.

PROBLEM:

Consider a rectangular membrane which is uniformly pretensioned with normal stress σo in the y-direction and with axial load P = σo th in the x-direction, as shown in Figure NS84-1. After pretensioning, an in-plane bending moment M is applied along the edges as shown. As M is increased, a band of vertical wrinkles of width b forms along the lower edge of the membrane as the normal strain εxx in this region becomes compressive. Find the nonlinear moment-curvature curve.

GIVEN:

L/2 = 10 in

h = 5 in

t = 0.1 in (thickness)

σo = 1 psi

P = σ0 ht = 5 lb

E = 3.E7 psi

ν = 0.3

MODELLING:

A finite element model of this problem (only half of the membrane) was created by using PLANE2D plane stress elements as shown in Figure NS84-2. The symmetric boundary conditions are applied to the left edge of the model. The right edge is attached to a column of very stiff BEAM2D elements on which the external loads P and M are applied. In order to satisfy the displacement compatibility, some degrees of freedom are coupled along the interface (CPDOF).

Figure NS84-1

TAUT REGIONh

b

oσ

oσ

Y

X

WRINKLED REGIONM

P

M

P



The time-curves of the pretensioning force P and σo and the bending moment M are shown in Figure NS84-3.


If K denotes the overall curvature of the membrane acting as a beam, then the analytical solution of the moment-curvature relation is given by:

where

The relation shows that for excessively large loads with > 1, the entire surface is wrinkled and instability results. the membrane then collapses.The numerical solution from COSMOSM is shown in Figure NS84-4 for comparison. Note that the curvature K =θ/L where θ is the rotation of the stiff beam.

Figure NS84-2

STIFF BEAM

P M

PRETENSION

PRETENSION



8-180

Figure NS84-3

Figure NS84-4

PRETENSION

BENDING MOMENT

TIME

VALUE

ANALYTICALCOSMOS/M

K

M

_

_



NS85: Shape-finding and Loading Analysis of a Suspended Membrane

TYPE:

Nonlinear static analysis, wrinkling membrane material model, large deflection, prescribed non-zero displacement, SHELL4T membrane elements, auto-stepping, geometry updating.

REFERENCE:

Hideki Magara and Kiyoshi Okamura, “A study on Modeling and Structural Behavior of Membrane Structures,” Shells, Membranes and Space Frames, proceedings IASS Symposium, Osaka, 1986, Vol. 2, pp. 161-168.

PROBLEM:

Modeling and mechanical characteristics of a suspended wrinkling membrane in the reference configuration and under the external loads are studied. The surface geometry of the membrane is in equilibrium under the specified boundary condition prestresses. After the equilibrium is reached, the external snow loads are uniformly distributed on the surface. Compare the numerical results to the experimental results.

GIVEN:

Surface geometry:

z = x2/400 + y2/400

span = 160 in

thickness = 0.1 in

E = 2.47E3 psi

ν = 0.39

prestress = 2 psi

snow loads = 520 lb



8-182

MODELLING:

A finite element model of this problem was created by using SHELL4T membrane elements as shown in Figure NS85-1.1. Shape-finding analysis:

The analysis starts with a flat surface geometry. Prescribed displacements are applied to the nodes along boundary-1 to -4 to satisfy the equation shown above. An auto-stepping algorithm is used with a tiny initial time increment to control the ill-conditioned stiffness in the beginning of the analysis. Several runs are performed until the stress distribution is almost uniform, i.e., the discrepancy is within one percent tolerance of 2.0 psi. It is noted that each run starts with a surface geometry obtained from the previous run (A_NONLINEAR) without the use of the RESTART option.

2. Loading analysis:

Adopting the surface geometry from the shape-finding analysis, a pressure loading is then applied to it as specified by the associated time curve. The surface geometry is no longer updated at the end of analysis.


1. Shape-finding analysis:As illustrated in Figure NS85-2, the curves representing the bracing of the model show good agreement between analytical, experimental, and numerical results. Figure NS85-3 shows the three dimensional view of the final shape of the membrane.

2. Loading analysis:

The load-displacement curve in the saddle point and the load-tension curves in the bracing and hanging are shown in Figures NS85-4 and NS85-5 respectively. It is noted that in the load-tension curves, an increase in loading results in an increase in tension in hanging but a greater decrease in the bracing direction due to a partially wrinkled region along the bracing. This phenomena is found in both experimental and numerical results.



Figure NS85-1

Figure NS85-2

BOUNDARY - 4HANGING

MEMBRANE

BOUNDARY - 1 BOUNDARY - 2

BOUNDARY - 3

BRACING

Z X

Y

Z

COORDINATE

(inch)

X COORDINATE (inch)

ANALYTICALEXPERIMENTCOSMOS/M

ALONG Y = 0



8-184

Figure NS85-3

Figure NS85-4

Main DEF Step:13 = 1

DEFORMED SHAPE WITH PRESTRESS = 2.0 PSI

Y

X

DI

SPLACEMENT

LOAD ( PSI )

(inch)

EXPERIMENTCOSMOS/M

IN SADDLEPOINT



Figure NS85-5

TENSION

( PSI )

LOAD ( PSI )

EXPERIMENTCOSMOS/M

Y

X

BRACING

HANGING

HANGING

BRACING



8-186

NS86: Nonlinear Elasticity of a Bar

TYPE:

Nonlinear static analysis, nonlinear elastic material model (unsymmetric behavior in tension and compression), small deflection, displacement control algorithm.

NS86A) TRUSS3D

NS86B) BEAM3D

NS86C) SPRING (axial)

NS86D) PLANE2D (plane stress)

NS86E) SOLID

NS86F) SHELL4T

PROBLEM:

A nonlinear elastic bar is subjected to a uniform pressure loading in the x-direction as shown in Figure NS86-1. The stress-strain curve is given in Figure NS86-2. Note that the material shows stronger resistance in compression than in tension. the displacement in the x-direction is controlled such that the path of loading follows the curve as shown in Figure NS86-3. Verify the corresponding load factor for each element type listed above.

GIVEN:

L = 10 in

B = 1 in

t = 1 in (thickness)

ν = 0.3 (required for 2- and 3-dimensional elements)


Figure NS86-4 shows the load-displacement curve for PLANE2D plane stress elements. Other types of elements show similar results. Remember that in the 2- and 3-dimensional elements, a ratio R is used to calculate the elastic modulus by interpolation. R is defined as:



In the current case, R = 1 for the time in the range of (0,2) and R = -1 for the time in the range of (2,4).

Figure NS86-1

Figure NS86-2

PROBMEM SKETCH AND FINITE ELEMENT MODEL (PLANE2D)

L

PRESSUREB

STRAIN

STRESS COMPRESSION

TENSION

(psi)



8-188

Figure NS86-3

Figure NS86-4

DISPLACEMENT

(IN)

COMPRESSION

TENSION

TIME

LFACT

UX (INCH)



NS87: Rubber Cylinder Pressed Between Two Plates

TYPE:

Nonlinear static analysis, Mooney-Rivlin hyperelastic material model, large strain and large deflection, prescribed displacements, coupling degrees of freedom, reaction force calculation, auto-stepping, frictionless contact, PLANE2D plane strain, displacement-pressure (u/p) elements.

PROBLEM:

A plane strain rubber cylinder is pressed between two frictionless plates, see Figure 87-1. Determine the force-deflection curve for the cylinder and the distribution of the von Mises stresses when the applied displacement equals one-half of the initial diameter of the cylinder.

REFERENCE:

T. Sussman and K. J. Bathe, “A Finite Element Formulation for Nonlinear Incompressible Elastic and Inelastic Analysis,” Computers & Structures, Vol. 26, No. 1/2, pp. 357-409, 1987.

GIVEN:

Diameter of cylinder D = 0.4 m

Mooney-Rivlin constants A = 0.293 MPa

Mooney-Rivlin constants B = 0.177 Mpa

Poisson’s ratio ν = 0.4999 (K/G = 5000)

Prescribed displacement ∆ = 0.2 m

MODELLING:

A finite element mesh of 8-noded PLANE2D displacement-pressure elements are used with 64 elements (Figure 87-2). Due to symmetry, only one quarter of the cylinder is meshed. Along the outer surface of the cylinder, the gap elements are generated in conjunction with a rigid frictionless target surface at the bottom. The load is applied by prescribing the displacements at the top of the mesh. A displacement of ∆ = 0.2 m is applied.



8-190


The resulting force-deflection curve and von Mises stress band plot are shown in Figure 87-3 and 87-4 respectively. The solutions by ADINA are enclosed for comparison. They are in close agreement.

Figure NS87-1

Figure NS87-2

Plain S train Rubber Cylinder

∆

D = 0.4M

FRICTIONLESSCONTACT

A = 0.293 MPAB = 0.177 MPA

COMPRESSION

PRESCRIBED

DISPLACEMENT ∆ / 2

TARGET SURFACE



Figure NS87-3

Figure NS87-4

ADINACOSMOS/M

∆ (M)

FORCE

Y

(IN)



8-192

NS88: Torsion of a Rubber Cylinder

TYPE:

Nonlinear static analysis, Mooney-Rivlin hyperelastic material model, large strain and large deflection, prescribed rotation, coupling degrees of freedom in local coordinate system, reaction force calculation, auto-stepping, SOLID, displacement-pressure (u/p) elements.

PROBLEM:

A solid rubber cylinder shown in Figure 88-1 is constrained in the axial direction and is twisted by the applied moment. Determine the moment-rotation curve and the axial force-rotation curve for the cylinder.

REFERENCE:

T. Sussman and K. J. Bathe, “A Finite Element Formulation for Nonlinear Incompressible Elastic and Inelastic Analysis,” Computers & Structures, Vol. 26, No. 1/2, pp. 357-409, 1987.

GIVEN:

Radius of cylinder R = 0.05 m

Length of cylinder L = 0.1 m

Mooney-Rivlin constant A = 3.E5 Pa

Mooney-Rivlin constant B = 1.5E5 Pa

Poisson’s ratio ν = 0.4999 (K/G = 5000)

Prescribed rotation θ = 0.1 rad

MODELING:

The mesh layout of 64 collapsed 8-node SOLID elements are shown in Figure 88-2. The displacement-pressure formulation is used. On the far end of the cylinder, all of the nodes are fixed. On the near end, the angular displacement is prescribed at the center node of a beam. Rigid links are then placed between the beam and the nodes on the near end of the cylinder to cause the entire cross-section to rotate by this amount.




The moment-rotation curve and the axial force-rotation curve are shown in Figure 88-3 and Figure 88-4. Analytical solutions are enclosed for comparison.

Figure NS88-1

Figure NS88-2

RL

M

N

N

M

RIGID LINK TO BEAM ELEMENTS

PRESCRIBED ROTATION ATCENTER MODE

FIXED BOUNDARY

8-Node S olid Elements

Z

Y

X



8-194

Figure NS88-3

Figure NS88-4

ANALYTICAL SOLUTIONCOSMOS/M M

0MENT

N-M

ROTATION (RAD) ROTATION (RAD)


AXIAL

FORCE

(N)

ROTATION (RAD)



NS89: Thick-Walled Cylinder Subjected to a Con-stant Radial Displacement Rate at its Inner Wall

TYPE:

Nonlinear static analysis, elastoplastic material model, large strain, large deflection, displacement-dependent-pressure loading, displacement control algorithm, PLANE2D axisymmetric, displacement-pressure (u/p) elements.

PROBLEM:

A long, thick-walled cylinder, as shown in Figure 89-1, is subjected to a constant radial displacement rate at the inner wall of the cylinder. The cylinder is made of an elastoplastic, nearly incompressible material. Compare the numerical solutions against the analytical solutions for the reaction pressure and the normal stress along the radial direction at the point initially half-way through the cylinder wall versus the radius of the inner wall.

REFERENCE:

W. Prager and P. G. Hodge, “Theory of Perfectly Plastic Solids,” John Willy and Sons, New York, 1951.

GIVEN:

Initial inner radius A0= 10 mmInitial yield stress σyo= 50 MPa

Initial outer radius B0= 20 mmTangent modulus ET= 0 (perfect-plasticity)

Elastic modulus E = 25000 MPsInner radius ratioA/A0= 3

Poisson’s ratio ν = 0.499

MODELLING:

The finite element model is built using five 8-noded PLANE2D axisymmetric elements. Plane strain boundary conditions in the cylinder axial direction are applied. The mesh is also shown in Figure 89-1. The displacement-pressure (u/p) formulation is employed because of the nearly incompressible condition for the material. The cylinder is expanded by applying the internal pressure. The cylinder reaches a limit state within the very small strain, after which the pressure decreases rapidly as the cylinder expands. In order to handle such kind of instability, a displacement control algorithm is used. The cylinder is finally expanded to three times its initial radius (inner). Under such large deformation, the displacement-dependent-pressure loading is employed and the finite strain plasticity theory is applied to the element formulation.



8-196


The analytical solution for the normal stress along the radial direction is:

where

As R0 = A0, σxx = -P

where P is the reaction pressure.

Figure 89-2 shows the normalized reaction pressure versus the normalized inner radius. The reaction pressure is the load factor computed from the displacement control algorithm. Figure 89-3 shows the normalized normal stress in the radial direction at the point initially half-way through the cylinder wall (R0/A0 = 1.5) versus the normalized inner radius. Note that the large displacement increment in Figure 89-2 and Figure 89-3. Nevertheless, the numerical solutions are in excellent agreement with the analytical solutions.

Figure NS89-1

PR

ES

SU

RE

Y

ZX

2A

2B

Y

X

P

A

B



Figure NS89-2

Figure NS89-3


A / AO

−σ xx /K


A / AO

xx /K−σ



8-198

NS90: Pressurization of a Sphere with Elastoplastic Material

TYPE:

Nonlinear static analysis, elastoplastic material model, large strain, large deflection, displacement-dependent-pressure loading, arc-length method, local boundary conditions.

NS90A) SHELL4T elements

NS90B) SOLID, U/P elements

NS90C) TETRA4, U/P elements

NS90D) TETRA10, U/P elements

PROBLEM:

The problem discussed in this example involves the pressurization of a sphere with elastoplastic material behavior. The strains are very large so that for the elastoplastic case, rigid plasticity analysis provides an accurate comparative result. Compare the numerical solutions of SHELL4T, SOLID, TETRA4, and TETRA10 elements against the analytical solutions for the reaction pressure versus the mean radius.

GIVEN:

Initial mean radius R0 = 0.1 m

Thickness t = 0.001 m

Young’s modulus E = 2.E11 N/m2


Initial yield stress σyo = 2.5E8 N/m2

Tangent modulus ET = 0 (perfect-plasticity)

MODELING:

The problem geometry is shown in Figure NS90-1. A sector of 3.6 x 1.0° (for SHELL4T) and 1.0 x 1.0 degree (for SOLID, TETRA4, and TETRA10 elements) are used for modeling. The meshes consist of 2 x 2 elements (for SHELL4T) and 2 x 2 x 4 (through the thickness) elements (for SOLID, TETRA4, and TETRA10) as shown in Figure NS90-2A and Figure NS90-2B respectively. Axisymmetric boundary conditions are applied. The sphere is expanded by applying the internal pressure. The sphere reaches a limit state within the very small strain, after which the pressure decreases rapidly as the sphere expands. In order to handle such kind of



instability, an arc-length method is used. The sphere is finally expanded as twice of its initial mean radius. Under such large deformation, the displacement-dependent-pressure loading is employed and the finite strain plasticity theory is applied to the element formulation.


Figure NS90-3 shows the normalized reaction pressure versus the normalized radial displacement. The reaction pressure is the load factor computed from the arc-length method.

Figure NS90-1

Figure NS90-2A

3.6 DEGREE

R

THICKNESS

S HELL4T Elements (2 x 2)

1.0 DEGREE

3.6 DEGREE

1.0 DEGREE

R



8-200

Figure NS90-2B

(b) SOLID Elements (2 x 2 x 4)

(c) TETRA4 Elements (2 X 2 X 4) (d) TETRA10 Elements (2 x 2 x 4)



Figure NS90-3

Exact Rigid-Plasticity

COSMOSM Results



8-202

NS91: Upset Forging of a Cylindrical Billet

TYPE:

Nonlinear static analysis, elastoplastic material model, large strain and large deflection, prescribed displacements, coupling degrees of freedom, reaction force calculation, auto-stepping, sticking friction contact, PLANE2D and TRIANG axisymmetric, displacement-pressure (u/p) elements.

REFERENCE:

G. G. Weber, A. M. Lush, A. Zavaliangos, and L. Anand, “An Objective Time-Integration Procedure for Isotropic Rate-independent and Rate-dependent Elastic-plastic Constitutive Equations,” International Journal of Plasticity, Vol. 6, pp. 701-744, 1990.

PROBLEM:

As a simple metal-forming example, the prototypical problem of isothermal upset forging of a cylindrical billet was solved as shown in Figure 91-1. The dies were modeled as being rigid, with sticking friction acting to prevent sliding between the billet and the die faces when they are in contact. This friction causes the billet to barrel, with the material near the corners folding over to come in contact with the dies. Consequently, this example problem exhibits the realistic features of inhomogeneous deformation, with variable rates of straining at material points and time varying die contact geometry.

GIVEN:

Diameter D = 2 mm

Height H = 3 mm

Young’s modulus E = 25000 Mpa


Initial yield stress σyo = 50 MPa

Tangent modulus ET = 0 (perfect-plasticity)



MODELING:

Figure 91-2 shows the finite element mesh containing 107 4-noded PLANE2D and 119 6-noded TRIANG axisymmetric, displacement-pressure elements. Near the corner where the roll-over is expected to occur, the elements are triangular in shape to accommodate this deformation mode. Symmetry in the problem allowed only a quarter of the billet to be modeled. The die face was modeled as a rigid surface and the external surface of the model was covered with gap elements to model the contact conditions.


Figure 91-2 also shows the deformed finite element mesh superposed on the undeformed mesh after a height reduction of 60%. The billet is seen to have expanded radially by a considerable amount. Five elements have folded over and come in contact with the die. Figure 91-3 shows the history of total die force versus die displacement for the node located at the center of the global coordinate system (node 1 for PLANE2D and node 23 for TRIANG). Note that jumps in die force occur in the calculated result whenever new nodes came in contact with the die.

Figure NS91-1

MIDDLELINE

DIE FACE

AXIS

OUTERSURFACE



8-204

Figure NS91-2

Original and deformed mesh using PLANE2Daxisymmetric elements

Original and deformed mesh using TRIANGaxisymmetric elements



Figure NS91-3

REFERENCE

COSMOSM Results



8-206

NS92: Comparison of Model Prediction with Cyclic Uniaxial Compression Test Data

TYPE:

Non-linear static analysis using TRUSS3D element with concrete material model.

PROBLEM:

To compare the model results in a cyclic uniaxial compression test with some experimental data.

REFERENCES:

Moussa, R. A., and Buyukozturk, O., “A Bounding Surface Model for Concrete,” Nuclear Engineering and Design 121, pp. 113-125, 1990.Soon, K. A., “Behavior of Pressure Confined Concrete in Monotonic and Cyclic Loadings,” Thesis Submitted in Partial Fulfillment for the Degree of Doctor of Philosophy, Department of Civil Eng. M.I.T., Cambridge, MA, June 1987.

MODELING HINTS:

The uniaxial test is modeled using a series of truss elements and applying compression force at one end while constraining the other end in X-direction.

GIVEN:

Ultimate Compression Strength f'c = 1000 N/cm2

Ultimate Strain εu = 0.002

Area of Concrete = 1 cm2

RESULTS:

Figure 92-1 shows the model results (solid line), the results from Reference 1 (circles), and the experimental results (solid squares) from Reference 2.



Figure NS92-1

NO

RM

AL

IZE

D D

ISP

LA

CM

EN

T

NORMALIZED FORCE



8-208

NS93: Reinforced Concrete Truss Analysis

TYPE:

Non-linear static analysis using TRUSS2D element with concrete material model.

PROBLEM:

A two dimensional reinforced concrete truss. The truss is simply supported and vertical concentrated forces are applied at the top cord. The load factor is required such that the maximum vertical displacement is 3 cm.

MODELING HINTS:

The truss members have a typical R.C. cross-section of 0.3 x 0.3 m2 with 4 rebars 2.5 cm diameter each. The truss members are modeled using two TRUSS2D elements superimposed on each other; the first is to simulate the concrete section and the second to simulate the steel rebars. The steel and the concrete are assumed to be fully bonded.

GIVEN:

Ultimate Compression Strength fc' = 2000 T/m2


E (Steel) = 2.0E7 T/m2

Area of Concrete = 0.09 m2

Area of Steel = 4 x 4.9 E-4 m2 = 19.6 E-4 m2

RESULTS:

Figure 93-1 shows the linear and the nonlinear solution of the problem. Figure 93-2 shows the damage factor for each member.



Figure NS93-1

Figure NS93-2

DISPLACEMENT

FORCE



8-210

NS94: Splitting Tension of Concrete Cylinder

TYPE:

Nonlinear static analysis using concrete material model.

PROBLEM:

Simulating the concrete split test on a cylinder.

REFERENCES:

Han, D. J. and Chen, W. F., “Constitutive Modeling in Analysis of Concrete Structures,” Journal of Engineering Mechanics, Vol. 113, No. 4 April 1987.

MODELING HINTS:

Three different models are used in this problem. The three models are generated using different types of elements (plane strain 4-node PLANE2D, 6-node TRIANG, and SOLID element) as shown in Figure 94-1. In all cases the concrete material model is chosen. In order to simulate the concrete split test, the load is applied as a pressure on a width 0.5 inch. Only quarter of the problem is modeled due to the double symmetry of the problem.The results are compared with the finite element analysis using a five parameter model developed by Han and Chen [Reference 1] as shown in Figure 94-2.

GIVEN:

Ultimate Compression Strength fc' = 4.54 ksi


Cylinder Diameter = 6 in

Load is applied on a 0.5 inch width strip



Figure NS94-1

Figure NS94-2

FORCE

12

PLANE 2D & SOLIDTRIANGHAND & CHEN



8-212

NS95: Stress Intensity Factor of Three Point Bend Specimen

TYPE:

J-integral evaluation for a mode I crack, 8-node PLANE2D plane-stress elements, linear elastic material, small deflection.

REFERENCE:

Gross and Strawley, 1972, obtained a solution in the form:

PROBLEM:

A simply supported beam with a through-thickness edge crack at the center is subjected to a point load as shown in the Figure NS95-1. Determine the mode-I stress intensity factor.

MODELING HINTS:

Due to symmetry, one half of the model is used for analysis. Symmetric boundary conditions are enforced along the vertical axis of symmetry. The Finite element mesh is illustrated in Figure NS95-2. Notice that in this case J paths are symmetric (extends from 0 to π) and the crack axis makes a 90° angle with the global x axis.


KI Solution Error

Reference 3.3541 —

STAR Crack Element (800 elements) 3.05867 8.81%

NSTAR J-Integral (200 elements) 3.236 3.5%



Figure NS95-1. Geometry and Properties

Figure NS95-2. Finite Element Mesh and a J-Path

a

B

S S

W = S/2

P/2 P/2

P

Three Point Bend Test for Mode-1 Fracture Toughness

40

80

B = 0.5E = 30E6v = 0.28

Model for Analysis

P/2 = 0.5



8-214

NS96: Slant-Edge-Cracked Plate, Evaluation of Stress Intensity Factors by Using the J-Integral

TYPE:

J-integral evaluation for a combined mode crack, 8-node PLANE2D plane-stain elements, linear elastic material, small deflection.

NS96A) Calculate the total J-integral parameter

NS96B) Calculate the individual J-integral parameters for modes I and II

REFERENCE:

Bowie, O. L., “Solutions of Plane Crack Problems by Mapping Techniques,” in Mechanics of Fracture I, Methods of Analysis and Solutions of Crack Problems (Ed G. C. Sih), pp. 1-55, Noordhoff, Leyden, Netherlands, 1973.

PROBLEM:

A rectangular plate with an inclined edge crack is subjected to uniform uniaxial tensile pressure at the ends. The crack starts from the middle of one side and inclines at an angle towards the opposite side. Evaluate the crack stress intensity factors.

MODELING HINTS:

While for evaluation of the total J-integral parameter, any reasonable mesh is acceptable, to evaluate J values at modes I and II, a symmetric mesh with respect to the crack axis is required. Figures NS96-2A and NS96-2B illustrate the mesh for parts (A) and (B). Special attention was given to avoid merging of nodes along the crack free surfaces, and to pick the proper node defining the start and end of a J path, accordingly.


E' = E/(1 - ν2)

Error Error Error

Reference 1.85 — 0.880 — 2.05 —

J Path Part (a) — — — — 1.991 3%

J Path 1 Part (b) 1.79 3% 0.873 1% 1.99 3%

J Path 2 Part (b) 1.79 3% 0.883 0.3% 1.99 3%



Figure NS96-1. GeomeProperEdge-C

a

φ

w

h

σ

σ

h

GIVEN:

σ = 1 psi

H = 2.5 inch

w = 2.5 inch

a = 1 inch

E = 30 x 106 psi

ν = 0.3

φ = 45°thickness=1 inch

try and ties for Slant-racked Plate Figure NS96-2. Finite Element Mesh and J-Paths

(A) Regular Mesh (B) Symmetric Mesh about Crack Axis



8-216

NS97: Single-Edge-Cracked Plate Subjected to Remote Uniform Tension and a

Thermal Gradient Across the Plate Width

TYPE:

Nonlinear static analysis using 8-node PLANE2D plane stress elements, von Mises elastoplastic model, small deflection, J-integral evaluation.

REFERENCE:

V. Kumar, B. I. Schumacher, M. D. German, “Development of a Procedure for Incorporating Secondary Stresses in the Engineering Approach,” in Advances in Elastic-Plastic Fracture Analysis, EPRI NP-3607, Project 1237-1, Section 7, August 1984.

PROBLEM:

A single-edge cracked plate (SECP) in plane stress (Figure NS97-1) is subjected to a thermal gradient across its width given by:

T (x,t) = T (t) [125 + 400x – 100x2]

The plate is then subjected to a uniformly increasing pressure in the longitudinal direction. Evaluate the J-integral parameter for different stages of solution as the pressure is risen from zero to 15 ksi.

MODELING HINTS:

The uniaxial elastoplastic stress-strain curve (Figure NS97-2), beyond the yield point, is defined by the Ramberg-Osgood stress-strain relationship:

ε/ε0 = σ/σ0 + α (σ/σ0)n

with α = 0.5, and n = 5 (ε0, σ0 are the yield stress and strain). The finite element mesh, and the loading time histories are given in Figure NS97-3. Also, assuming that the thermal gradients, in this case, do not have a considerable effect on the loading proportionality, the von Mises yield criteria (based on the flow theory of plasticity) is employed for the modeling of plasticity.




Figures NS97-4A and NS97-4B demonstrate the variations of J-integral parameter with respect to the applied pressure. The J-integral value at zero pressure is due to the effects of temperature alone. The obtained graphs are in good agreement with those given in the reference.

GIVEN:

b = 2 in

a/b = 0.25

L/b = 4

E = 30 x 103 ksi

Nu = 0.3

Yield Stress = 60 ksi

Tref = 70°F

Thermal Coefficient of Expansion = 7.3 x 10-6 in/in°F

Figure NS97-1.Single-Edge Cracked Plate (SECP) UnderRemote Uniform Tension

a

X

b

c

L

∆ / 2

8∆ / 2

L

σ

8σ

T2

T1

y



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Figure NS97-1.NS97-2.Uniaxial Elastoplastic Stress-Strain Curve(Ramberg-Osgood Model)

Figure NS97-3. Finite Element Mesh and a Few J-Paths

STRESS

(PSI)

STRAIN

T0

1

1.E-5t

σ∞

60

1.E-5t

60

(ksi)

0

Time histories of σσσσσand T for plans stress S ECP

∞0

Finite Element Meshand 4 J Paths



Figure NS97-4. J-integral Versus Applied Pressure (plotted on small scale)

Figure NS97-5.J-Integral Versus Applied Pressure (plotted on large scale)(Plane Stress SECP, Mechanical and Thermal Loading)

PRESSURE (KSI)

J

INTEGRAL

in-Kip/in2

PRESSURE (KSI)

J

INTEGRAL

in-Kip/in2



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NS98: Circumferentially Cracked Cylinder Subjected to a Uniform Axial Tension and a Thermal Gradient in the Radial Direction

TYPE:

Nonlinear static analysis using 8-node PLANE2D Axisymmetric elements, von Mises elastoplastic model, small deflection, J-integral evaluation.

REFERENCE:

V. Kumar, B. I. Schumacher, M. D. German, “Development of a Procedure for Incorporating Secondary Stresses in the Engineering Approach,” in Advances in Elastic-Plastic Fracture Analysis, EPRI NP-3607, Project 1237-1, Section 7, August 1984.

PROBLEM:

A cylinder containing an axisymmetric crack (Figure NS98-1) is first subjected to a thermal gradient in the radial direction:

T (x,t) = T (t) [125 + 100x – 6.25x2]

Where x is the distance from the inner surface. The cylinder is then loaded by a uniformly applied tensile pressure at its ends. Evaluate the J-integral parameter for different stages of solution as the pressure is risen from zero to 27 ksi.

MODELING HINTS:

The elastic-plastic behavior is defined by the Ramberg-Osgood stress-strain law, given in problem NS97. The finite element mesh, and the loading time histories are shown in Figures NS98-2 and NS98-3. Also, assuming that the thermal gradients, in this case, do not have a considerable effect on the loading proportionality, the von Mises yield criteria (based on the flow theory of plasticty) is employed for modeling of plasticity.


Figure NS98-4 demonstrates the variation of J-integral parameter with respect to the applied pressure. The J value at zero pressure is due to the effects of temperature alone. The obtained graph is in good agreement with that given in the reference.



GIVEN:

Ri/b = 10

a/b = 0.25

Ri = 80 inch

L = 120 inch

E = 30 x 103 ksi

Nu = 0.3

Yield Stress = 60 ksi

Tref = 70° F

Thermal Coefficient of Expansion = 7.3 x 10-6 in/in °F

ET = 4.25 x 103 ksi

Figure NS98-2. Time Histories of σÏ and To

Figure NS98-1.Circumferentially Cracked Cylinderin Tension

r

ca

T1

T2 Ri

R0

b

LCσ∞

L

L

σ∞

T0

1

1.E-5t

σ∞

90

1.E-5t

90

(ksi)

00



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Figure NS98-3

Figure NS98-4. J-Integral Versus Applied Pressure

Finite Element Model(not to scale)

A A

A A

J-IN

TEG

RA

L (I

N-K

IP/I

N

)2

PRESSURE (KSI



NS99: Initial Interference Between Two Thick Hollow Cylinders with Elastic-plastic Behavior

TYPE:

Plasticity, large displacement analysis using PLANE2D axisymmetric and contact (node to line gap) elements, Thermal loading.

PROBLEM:

Similar to problem NS36, except that considerable plastic strains are developed during the process of fitting. To fit one cylinder inside the other, the outer cylinder is first heated 100 °F and then cooled 100 °F.

MODELING HINTS:

Here, the analysis is performed in two steps. First the outer cylinder is heated while the gap element group is excluded from analysis. Next, the gap group option is changed to bring the gaps back into consideration, temperatures are gradually reduced back to normal, and the analysis is continued using the restart option.

Figure NS99-1



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GIVEN:

Modulus of Elasticity = 30.E6 psi

Poisson's Ratio = 0.3

Yield Stress = 3.E5 psi

Tangent Modulus = 1.E6 psi

Thermal Coefficient = 0.001 1/°F

Figure NS99-2

Figure NS99-3


* Notice the great reduction in interface pressure due to elastoplastic behavior.

Linear Material Small Displacement

Elastoplastic Material Large Displacement

Contact Location(radius) 21.5709 in. 21.4133 in.

Pressure at the Interface 57.231 ksi 29.486 ksi



NS100: Detection of Buckling Load Based on State of Deformation for a Cylindrical Shell

TYPE:

Large displacement analysis, using SHELL4 elements, Automatic Stepping, and buckling analysis based on the deformed geometry.

PROBLEM:

A shallow cylindrical shell is subjected to pressure loading along the two flat edges. Study the accuracy of the predicted buckling pressure by performing buckling analyses at different levels of deformation.

MODELING HINTS:

Due to symmetry, a 4x4 mesh is used to model a quarter of the shell. Starting from time zero, first a nonlinear analysis is performed to solve for a pressure loading of 0.1 N/mm2, followed by a buckling analysis. Next, the nonlinear analysis is restarted to solve for pressure = 0.2 N/mm2, followed by a second buckling analysis. This procedure is repeated several times, raising the pressure a magnitude 0.1 N/mm2 each time.

GIVEN:

Radius = 2540. mm

Width = 254. mm

Theta = 0.1 rad

Shell Thickness = 6.35 mm

Modulus of Elasticity = 3102.75 N/mm2

Poisson's ratio = 0.3


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Figure NS100-1

Figure NS100-2

CONCLUSIONS:

Figure NS100-3 shows the response at the center of the shell with respect to time (or pressure). It is evident from this curve that there is no perfect buckling behavior for this problem. While the pressure never drops, the graph demonstrates two states of rising and falling of displacement-rates. The predicted buckling pressure at low displacements is close to the point in between these two states (where the displacement path changes). As deformations become larger, the evaluated buckling pressure also becomes larger (A negative buckling pressure, in this case, indicates that the buckling solution is invalid). Considering the fact that the buckling load factor (eigenvalue) at the time of buckling must equal one, we conclude that no



buckling has occurred during this analysis, which is also supported by the results of the nonlinear analysis.

Figure NS100-3

Pressure P(N/mm)

Buckling Parameter

(Eigenvalue)

Buckling Pressure = p x Eigenvalue (N/mm2)

0.1 4.389 0.4389

0.2 2.213 0.4426

0.3 1.517 0.4551

0.4 1.281 0.5124

0.5 1.806 0.903

0.6 3.741 2.2446

0.7 -10.06 ----



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NS101: Plasticity in the State of Pure Shear, Comparison of Tresca

and von Mises Yield Criterion

TYPE:

Plasticity, Tresca yield criterion, cyclic loading conditions, kinematic hardening.NS101A) PLANE2D Plane Stress Elements

NS101B) Solid 8-node Elements

NS101C) Combined Kinematic & Isotropic Hardening (RK=0.5)

PROBLEM

A square plate is subjected to in-plane pressure along two normal edges, while it is fixed along the other two edges. The loading is such that a state of pure shear is created throughout the plate (compression is applied on one edge while tension is applied on the other one).Obtain the response as the maximum shearing stress in the plate is raised to equal the tensile yield stress, reversed to reach the same magnitude in the opposite direction, and reversed back to the original magnitude.Compare the solutions based on Tresca and von Mises criteria.Compare also with the case of combined kinematic and isotropic hardening using Tresca yield criterion.

MODELING HINTS:

Here, the principal directions and the global directions coincide. To study the principal shearing stresses, a local Cartesian coordinate is defined by a 45° rotation of the global coordinates; the stresses are requested to be output in this local system.

Figure TL10-1



Figure NS101-2. Pressure Time History

Given:

Modulus of Elasticity =1.E6 psi

Poisson’s Ratio = 0.3

Yield Stress = 1.E3 psi

Tangent Modulus = 1.E4 psi


Yield Criteria

Tresca von Mises

Pressure at Start of Yielding 0.5 ksi 0.577 ksi

Maximum Displacement in X- or Y-direction 0.7525 in 0.641 in

Maximum Shearing Strain 0.1496 in/in 0.1281 in/in



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Figure NS101-3. Response Based on Tresca Yield Criteri0n

Figure NS101-4. Maximum Shearing Stress in Plate

RESPONSE BASED ON TRESCA YIELD CRITERION

MAXIMUM SHEARING STRESS IN PLATE



Figure NS101-5. Response Based on von Mises Yield Criterion

Figure NS101-6. Response Based on Tresca Yield Criterion ( Combined Kinematic & Isotropic Hardeniing RK=0.5)



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NS102: THERMAL BUCKLING OF A SIMPLY SUP-PORTED PLATE

TYPE:

Nonlinear static analysis, uniform thermal loading, initial imperfection (small transverse point force).

NS102A) Using Displacement Control

NS102B) Using Arc-Length Control

REFERENCE:

Timoshenko and Woinosky-Krieger, “Theory of Plates and Shells,” McGraw-Hill Book Co., 2nd Ed., pp.389.

PROBLEM:

Find the point of buckling and the post buckling behavior of a simply supported plate due to uniform rise of temperature. The plate is restricted from in-plane motion along two parallel supports (Figure NS102).

Figure NS102 Finite Element Model

rollers

uniformtemperaturerise

simplysupported



MODELING HINTS:

A quarter of the plate is modeled using shell elements. A uniform thermal loading of 1.0 oF is applied as the (unit) temperature pattern.A small transverse point force is used as an imperfection to enable a post buckling solution. This force is not associated with time-curve 1, so that it can be given a fixed value, independent of the load (temperature) factor.First, a one-step force control solution is performed to obtain displacements under the transverse force. During this phase, time-curve 1 is kept at zero value, and time-curve 2 (prescribing the transverse force) is raised to 0.01 (and kept constant thereafter).Next, the solution is restarted with the control changed to the Displacement/Arc-length control. In the case of Displacement Control, time-curve 1 is used to define displacement of the controlled degree of freedom (central node, transverse direction). In the case of the Arc-length Control, time curves are not used.

GIVEN:

a = 20. in

h = 1. in

E = 3.E4 psi

Nu = 0.3

Alp = 1.E-4 1/ oF


The behavior of plate under temperature loading is similar to the behavior of plate under in-plane uniform pressure along two parallel supports (without restraining the in-plane motion). The in-plane normal stress at the buckling temperature is equal to the stress under in-plane buckling pressure:

Where Tb,rel is the temperature relative to the temperature at buckling, and Pb is the buckling in-plane pressure.Figures NS102A-1 and NS102B-1 show the response at the center of plate versus the applied temperature, using Displacement and Arc-length controls, respectively. Good agreement with the buckling temperature, based on the reference, is observed

Tb rel,Pb

E Alp×-------------------=



8-234

Figure NS102A-1. Central Deflection Vs. Temperature Using Displacement Control

Figure NS102B-1. Central Deflection Vs. Temperature Using Arc-length Control



NS103: FREE ROTATION OF A TRIANGLAR SEC-TION ABOUT One Tip

TYPE:

Nonlinear static analysis, prescribed displacement in cylindrical coordinate system (Uy=R * Theta), Plane2d Plane-stress elements.

Figure NS103-1. Finite Element Model

PROBLEM:

A triangular section is fixed at one tip, and is rotated 360 degrees about that tip. Find response by prescribing the angle of rotation for one of the free tips. Verify that the structure remains stress-free throughout analysis.

MODELING HINTS:

As the structure undergoes large rotations (regardless of the fact that there are no strains), a large displacement analysis is a must.Here the rotation is defined by prescribing Uy (=R * Theta, Theta in Radians) for a node, in a local cylindrical system that is centered at the fixed tip.



8-236

RESULTS:

Figure NS103-2 shows displacements at different solution steps. Listing or plotting of stresses at any step shows that stresses are negligible (too close to zero).

Figure NS103-2 Displaced Positions Due to Prescribed Rotations

.



NS104: Buckling & Post Buckling of a Simply Sup-ported Orthotropic Plate

TYPE:

Nonlinear static analysis, Orthotropic SHELL4 elements, large displacement.

Figure NS104-1. Finite Element Model

REFERENCE:

Timoshenko and Woinosky-Krieger, “Theory of Plates and Shells,” McGraw Hill Book Company, 2nd Ed.

PROBLEM:

A simply supported plate is subjected to in-plane uniform pressure applied in the X-direction. Investigate the drop in magnitude of the buckling pressure when the material strength is lowered in the y-direction while it is kept constant in the X-direction (Ey < Ex). Compare the results with the isotropic case (Ex = Ey).



8-238

Figure NS104-2-A In-Plane Pressure

Figure NS104-2-B Central Force

MODELING HINTS:

Due to symmetry, a quarter of the plate is modeled using SHELL4 elements. In order to obtain the post buckling behavior, a small transverse force is applied at the center of the plate (initial imperfection).

Given:

a = 20 in

h = 1 in

In-P

lane

Pre

ssur

e (lb

/in)

100

1.0Time(0,0)

Cen

tral

For

ce

1.0

1.0Time(0,0)

0.1



Nu = 0.3

Ex = 3.0E+004 psi

10 < Ey < 3.0E+004


A number of cases are run for different values of Ey. The results are shown in Fig. NS104-3.

Figure NS104-3 Variation of Buckling Pressure with Respect to Orthotropic Intensity

Also for the case when Ey / Ex = 0.5, a graph of the central displacement versus the applied pressure is given by Fig. NS104-4



8-240

Figure NS104-4 Buckling of a S.S.Orthotropic Plate Under In-Plane Pressure(Ey/Ex=0.5)



NS105: Piercing of a Thick-Walled Cylinder with Residual Stresses from Cyclic Internal Pressure

TYPE:

Nonlinear static analysis, Death of elements, Plasticity, Kinematic Hardening, PLANE2D axisymmetric elements.

PROBLEM:

A thick-walled cylinder is first subjected to cyclic internal pressure. At the end of the loading cycle, due to plastic straining of the material, considerable residual stresses are present in the cylinder (same as problem NS33).Next, the cylinder is pierced into two cylinders circumferentially at a radius of 1.35”, Investigate the stress redistribution and drop in the residual stresses after piercing in both cyliders.

Figure NS105-1

MODELING HINTS:

8-noded PLANE2D axisymmetric elements are used to model the cylinder. To model the piercing, one element (element no. 8, r = 1.33” to 1.38”) is removed from the analysis (killed using the EKILL command).

SOLUTION PROCEDURES:

The solution is performed in 2 stages:• During the first phase of the solution, all elements are considered to be alive.

Using the von Mises yield criteria with kinematic hardening, the cylinder is subjected to a complete cycle of loading and unloading.



8-242

• The solution is restarted with no external forces; the variation of results is due only to the killing of element no. 8 prior to the restart of solution.

NOTE:

In order to plot stresses correctly for a solution step in which some elements were killed, you need to create and activate an element selection set that excludes the killed elements before activating a stress plot.

COMPARISON OF THE RESULTS

A comparison of the stresses in the cylinder segments, before and after piercing, shows considerable variation in the magnitude and distribution of the residual stresses.

Figure NS105-2 Residual Stresses in the Cylinder After One Cycle of Loading



Figure NS105-3 Residual Stresses After the piercing of the Cylinder (Element no. 8 is dead)



8-244

NS106: Assembly of Two Cantilever Beams, At-tached by a) Welding b) Cables

TYPE:

Nonlinear static analysis, Birth of Elements, Large Deformation, Contact, PLANE2D elements. The attachment is modeled by:

NS106A- PLANE2D elements (for welding)

NS106B- Nonlinear Spring elements (for cables)

PROBLEM:

Two cantilever beams stand 0.5 inches apart. First, the beams are forced closer near the free ends, such that welding [cables for part (b)] can be applied to attach the beams in that area. Next, the assembly is subjected to a moment at the attached joint. Determine the response of the assembly and maximum stresses in the beams and the weld [or cables]. Contact between the two beams need also be considered in the analysis.

MODELING HINTS:

8-noded PLANE2D elements are used to model the beams. And:NS106A- Two 8-node PLANE2D elements are used to model the welding.

NS106B- Five Nonlinear (tension-only) spring elements are used to model the cables.

Contact is modeled by a node-to-line Gap group. Rigid beam elements are used on the free sections to allow for direct application of moments.

NOTES:

The elements that are used to model the welding (or cables) are treated as non-existent (killed) at the start of the solution.



Figure NS106A-1 Finite Element Model of Two Cantilever Beams Attached by Welding

Figure NS106B-1 Finite Element Model of Two Cantilever Beams Attached by Cables

SOLUTION PROCEDURES:

The solution is performed in 3 stages:

1. At the start of the solution, the attachment elements are not considered in the analysis (EKILL command). vertical displacements are applied to the nodes to bring the free ends together.

2. The imposed displacements are released, while the “killed” elements are brought to life (ELIVE command). A restart allows the assembly to regain equilibrium under no external forces.

0.5”

Cantilevers



8-246

3. The solution continued while the end moments are gradually applied: from zero to the desired magnitude.


A comparison of the two cases shows that while the stresses maybe high in the attachment material, under the same loading conditions, the welded assembly deforms less than the tied-by-cable assembly. The reason is obvious, however, as the latter is not capable of passing any shear stresses.

Figure NS106A-2 Deformation Plots for the Welded Assembly



Figure NS106B-2 Deformation Plots for the Cable Tied Assembly



8-248

NS107: Uniaxial Tests on a Nitinol Cube Specimen

TYPE:

Nonlinear static analysis, Nitinol superelastic material model, Large strain, Large deflection, Force control method with cyclic prescribed displacements, Tetra4 elements.

NS107A) Using a linear flow rule

NS107B) Using an Exponential flow rule

NS107C) Constant Stress Flow

REFERENCE:

Auricchio, F., “A Robust Integration-Algorithm for a Finite-Strain Shape-Memory-Alloy Superelastic Model,” International Journal of Plasticity, vol. 17, pp. 971-990, 2001.

Figure NS107-1

PROBLEM:

Nitinol cube specimens (1x1x1 mm3), with different material properties, are analyzed under uniaxial conditions.A cyclic displacement is prescribed on (and normal to) one the faces, while the boundary conditions are such that a uniaxial state of stress is maintained.

MODELING HINTS:

Fig. NS107-1

Due to symmetry, only a quarter of the cube (1.x.5x.5) is modeled. The finite element mesh is illustrated in Figure NS107-1 and the displacement time history is given in Figure NS107-2. Also, in part (c), in order to obtain unloading at zero stress, the yield stress for unloading is approximated to 1. MPa, since a zero yield stress is unacceptable.



Figure Fig. NS107-2

Given:

Ex = 50,000. MPa

Nuxy = 0.3

Part (a) :

SIGT_S1= 520 MPa

SIGT_F1=600 MPa

SIGT_S2=300 MPa

SIGT_F2=200 MPa

SIGC_S1= 700 MPa

SIGC_F1=800 MPa

SIGC_S2=400 MPa

SIGC_F2=250 MPa

Part (b): Same as Part (a) with the addition of:

BETAT_1= 250 MPa

BETAT_2=20 MPa

BETAC_1=250 MPa



8-250

BETAC_2=20 MPa

Part (c):

SIGT_S1= 500 MPa

SIGT_F1=500 MPa

SIGT_S2=1. MPa

SIGT_F2=1. MPa

SIGC_S1= 700 MPa

SIGC_F1=700 MPa

SIGC_S2=1. MPa

SIGC_F2=1. MPa

RESULTS:

The Stress-displacement graphs are constructed to show that they are in good agreement with the response curves given in reference. Graphs of stress versus time are also presented in Figures NS106-8.





8-252

(B)





8-254




8-256


NS108: Three-Point Bending Test

TYPE:

Nonlinear static analysis, Nitinol superelastic material model, Large strain, Large deflection, Displacement Control method, Tetra10 elements.

NS108A) Ultimate plastic strain =0.092

NS108B) Ultimate plastic Strain= 0.15

NS108C) Ultimate plastic strain =0.092 (Exponential Flow rule)

REFERENCE:

Auricchio, F., Taylor, R.L., and Lubliner, J., “Shape-Memory-Alloys: Macromodeling and Numerical Simulations of the Superelastic Behavior,” Computer Methods in Applied Mechanics and Engineering, vol. 146, pp. 281-312, 1997.

PROBLEM:

A three-point bending test is performed on a Nitinol wire of circular cross section with diameter d=1.49 mm. The wire is 20 mm long and it is simply supported at both ends.

• Obtain the graph of the applied force versus deflection for the mid-span section of the beam.

• Verify that by increasing the ultimate plastic strain for the material a closer match to the experimental results can be obtained.

MODELING HINTS:

• Due to symmetry, only half of the beam with half of the cross section is modeled. The finite element mesh is illustrated in Figure NS108-1.

• The node for displacement control is selected to be same as the node where the force is applied. This node is displaced, in the direction of the force, a maximum of 5.2 mm, and then is brought back to zero.



8-258

a)

Figure Figure NS108-1 – Finite Element Mesh

Given:

Ex = 60,000. MPa

Nuxy = 0.3

Part (a):

SIGT_S1=SIGC_S1= 637 MPa

SIGT_F1=SIGC_F1= 735 MPa


SIGT_F2=SIGC_F2= 245 MPa

Eul = 0.092 mm/mm

Part (b): Same as Part (a), except:

Eul = 0.15 mm/mm

Part (c): SIGT_S1=SIGC_S1= 637 MPa

SIGT_F1=SIGC_F1= 918.5 MPa




SIGT_F1=SIGC_F1= 245 MPa,

BETAT_1=BETA_C1=204 MPa

BETAT_2=BETAC_2= 16.3 MPa

Eul = 0.092 mm/mm

RESULTS:

Figures NS108-2, NS108-3, and NS108-4 demonstrate the load-displacement graphs for parts: (a), (b), and (c). Ns108-2, and NS108-4 show close agreement with results given in reference. Figure NS108-3 shows better agreement with the experimental data.

a)

Figure Figure NS108-2: Part (a)



8-260

a)

Figure Figure NS108-3: Part (b)



a)

Figure Figure NS108-4: Part (c)



8-262

Nonlinear Dynamic Analysis



ND1: Time History of a Cantilever Beam with Tip Mass

TYPE:

Linear Dynamic Analysis, Elastic Material, Beam elements.

REFERENCE:

Biggs, J., “Introduction to Structural Dynamics,” McGraw-Hill, New York, 1964, pp. 43-49.

PROBLEM:

Solve for the displacement time history of the free end of the cantilever shown in the figure below. The driving force is a triangular load pulse, and the displacement is required at time 0.085 sec.

Figure ND1-1

GIVEN:

L = 13 in

b = 1.30 in

h = 0.125 in

E = 29 x 106 psi

m = 0.001 lb sec2/in

L b

h

F

Problem Sketch

1

F

t

Time CurveFinite Element Model

12

td

F1

m

Cross Section



8-264

F1 = 10 lb

td = 0.066 sec


Calculated data for theoretical values:Iz = bh3/12 = 0.000211588 in4

K = 3EI/(L3) = 8.37878 lb/in

ω = (K/m)1/2 = -91.5357 rad/sec

The dynamic load factor (DLF) for t ≥ td is given by:DLF = 2/ω td [2 sin ω (t – td /2) – sin ωt – sin ω (t – td)] = -1.3178853

∆ stat = F1/K = 1.1934905 in

∆ dynamic = ∆ sat x DLF = 1.57288418 in


Displacement at the tip:(Time step No. 170)

Figure ND1-2

δ (in)

Theory 1.57288

COSMOSM 1.57290



Figure ND1-3

Figure ND1-4



8-266

ND2: Transient Response of a Dropped Container Using Gap elements

TYPE:

Nonlinear dynamic analysis, Truss Element, Concentrated Masses, Gap element.

REFERENCE:

Thomson, W. T., “Vibration Theory and Applications,” Prentice Hall, Inc., Englewood Cliffs, N.J., 1965. pp. 110-112.

PROBLEM:

A mass m is packaged in a box (M), and dropped through a height h. Obtain the maximum amplitude attained by the mass m and the time at which it occurs.

ASSUMPTIONS:

To analyze the collision, a tensile gap element is used (it can only resist tension). A stiff truss connects this gap to the ground. This truss is used to account for the elasticity of the ground. A soft truss element is also used to avoid rigid body motion of the masses. The mass of the box is large compared to m. The stiffness of gap (kg) is large compared to k.

Figure ND2-1

dstatic

h

Problem Sketch

3

k

2

1

2

3

Y

1

Kgap, A

Gap Element

X

M

M>>m

Kgap>>k


h

Mass = M

m

k

m



GIVEN:

k = 1973.92 lb/in

g = 386 in/sec2

m = 0.5 lb sec2/in

gap distance = h = 1 in

ASSUME:

kg = 1.5E7 lb/in

M = 50 lb sec2/in

ksoft = 1 lb/in


Max. Displacement at Node 3 Time of Occurrence

Theory 1.5506 in 0.1 sec

COSMOSM 1.5487 in 0.095 sec



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ND3: Time History Analysis of a SDOF System

TYPE:

Truss Element, Concentrated Mass, Gap-friction.

REFERENCE:

“Elements of Vibration Analysis,” L. Meirovitch, McGraw-Hill, p. 21-23.

PROBLEM:

A mass spring system is placed on a surface with friction and is subjected to a step loading. Determine the amplitude decay due to friction and the time at which the system rests.

ASSUMPTIONS:

A compressive gap element with a gap distance of zero and a friction coefficient of 0.01 is used to model the surface. A soft truss element is used along with the gap element to avoid singularity of the structure stiffness.

Figure ND3-1

M

k, L

F(t) µ

Fo

time

Problem Sketch

Force

Load Time Curve

1

y

2

µ , Kgap

3


1

Gap FrictionElement

M F(t) fricF

W

Free Body Diagram

2



GIVEN:

F0 = l N

k = l N/m

M = l kg

W = Mg = 10 N

Coefficient of friction = µ = 0.01

THEORETICAL SOLUTION:

Based on the theory of vibration, an SDOF system subjected to a constant force oscillates with a constant amplitude around the static equilibrium position. The frequency of oscillations is equal to the natural frequency of the system. The effect of friction is to reduce the amplitude of motion by:

Amplitude decay in one cycle = 4 f = 4 Ff /k = 4 x 0.1/1 = 0.4 m

whereFf = µ W = 0.1 N

W = Mg = 10 N (weight)

While the response frequency remains the same.


The magnitude of friction force at any time remains less than or equal to Ff = 0.1 N, and it is opposite in direction to the velocity of mass M. The frequency of response agrees with theory.

Amplitude Decay Theory COSMOSM

First Cycle 0.4 m 0.40111 m

Second Cycle 0.4 m 0.39895 m

Last Half Cycle 0.2 m 0.2004 m

Time to Rest 15.70796 sec 15.7 sec



8-270

Figure N

Figure ND3-2

D3-1



k, L

Prob

Figure ND4-1

ND4: Time History Analysis of a SDOF System with COULOMB Damping

TYPE:

Truss Element, Concentrated Masses, Gap-Friction.

REFERENCE:


PROBLEM:

A mass spring system is placed on a frictionless surface, and is subjected to a step loading. Show that if a second mass is added on top of the first one, with a high coefficient of friction between surfaces of the two masses, the two masses will oscillate together. Also, investigate the amplitude and frequency of response of masses and the magnitude of the friction force.

ASSUMPTIONS:

A compressive gap element with a gap distance of zero and a friction coefficient of 0.01 is used to model the surface between the two masses. A soft. truss element is used along with the gap element to avoid singularity of the structure stiffness.

M F(t)

µ Fo

Time

lem Sketch

m

F(t)

Load_Time Curve M1

y

x

3

Gap-Friction Element


1

m

2



8-272

GIVEN:

k = l. N/m

m = 0.5 kg

M = l. 5 kg

w = mg = 5 N

F = l N

Coefficient of friction= µ = 0.1


The effect of friction in this problem is to increase the total mass of the system which, in turn, increases the natural period. No loss of energy or change in amplitude of oscillations occurs.

ω1 = [k / (M + m)]1/2 = 1 / (2)1/2 = 0.7071 rad/sec

In order for the two masses to move together, if the acceleration of either mass is shown by a (t), then a friction force:

Fs(t) = m.a(t) = 0.5x a(t) < µ.w = 0.5 N

is required to produce this acceleration for mass m.


The results obtained yield the following conclusions:1. The displacement, velocity and acceleration of the two masses are identical at

any time.2. The masses respond harmonically reaching a maximum displacement of two

times the static deflection (F/k=1).

3. The response period, which is equivalent to the natural period of the system for this case, is increased by a factor of

α = [(M + m) / M]1/2 = (2/15)1/2 = 1.547

4. The friction force applied to node 3 (mass m) is equal to 1/2 the acceleration of the system at any time. The friction force applied to node 2 (mass M) has the same magnitude as the one at node 3 in the opposite direction.



ND5: Time History Analysis of a SDOF System

TYPE:

Truss Element, Concentrated Masses, Gap-Friction.ND5) Newmark Method

ND5A Finite Difference Method

REFERENCE:


PROBLEM:

Consider problem ND4, this time assuming that mass M is placed on a surface with friction. Show that if a second mass is added on top of the first one, with a high coefficient of friction between surfaces of the two masses, the two masses will oscillate together. Also, investigate the amplitude and frequency of response of masses and the magnitude of the friction force.

ASSUMPTIONS:

Two gap elements are used to model the interface of the two masses and the lower mass and the ground. Soft truss elements are used along with the gaps to avoid singularity of the structure stiffness.

GIVEN:

F = l N k = l N/m

M = l. 5 kg W = Mg = 15 N

m = 0.5 kg w = mg = 5 N

µ1 = 0.1 µ2 = 0.005

Figure ND5-1



8-274


The effect of friction between the two masses is to increase the total mass of the system which, in turn, increases the natural period. Some energy loss is caused by the Friction between mass M and the ground, which causes and amplitude decay in the harmonic response of the system.

ω1 = [k / (M + m)]1/2 = 1/(2)1/2 = 0.7071 rad/sec

In order for the two masses to move together, if the acceleration of either mass is shown by a (t), then a friction force:

Fs(t) = m.a(t) = 0.5x a(t) < µ w = 0.5 N

is required in gap No.2 to produce this acceleration.

The amplitude decay in one cycle is:

4 f = 4 Ff/k = 4 x 0.1/1.0 = 0.4 m

where:

Ff = µ2 = (W + w) = 0.1 N


The results obtained yield the following conclusions:1. The displacement, velocity and acceleration of the two masses are identical at

any time.2. The masses respond harmonically with an amplitude decay of 0.4 meters at each

cycle.

3. The response period, which is equivalent to the natural period of the system for this case, is increased by a factor of

α = [(M + m) / M)1/2 = (2/15)1/2 = 1.547

4. The friction force applied to mass M from ground, always opposes the velocity at this point.

5. The friction force applied to mass M from mass m is opposite in direction and equal in magnitude to 1/2 the acceleration of the system at any time.

Figure ND5-2



ND6: Two DOF with Friction Dynamic Analysis

TYPE:

Truss Elements, Concentrated masses, Gap-Friction.

PROBLEM:

Investigate the Response of the symmetric system shown below when one of the masses is excited by a step loading.

Figure ND6-1

GIVEN:

k = l N/m

Fo = l N

M = 1 kg

W = 10 N

Coefficient of friction = µ = 0.005

ω1 = (k / M)1/2 = 1.0 rad/sec

ω2 = (3k / M)1/2 = 1.732 rad/sec

M

k

F(t) = F

Problem Sketch

M

k k

o

1 2


M

1

y

2

W

x

5

1

M

WFs F's

2

3

3

4

6

F(t) = Fo

Fs, F's = Friction Forces



8-276

Figure ND6-2

Figure ND6-3



Figure ND6-4



8-278

ND7: Elastic-Plastic Small Displacement Dynamic Analysis

TYPE:

Nonlinear Dynamic Analysis, Plasticity, 2D Isoparametric (8-node) Plane Stress Elements.

ND7) Newmark Method

ND7A) Finite Difference Method

REFERENCE:

Nagarajan, S., Popov, E. P., “Elastic-Plastic Dynamic Analysis of Axisymmetric Solids,” Computers & Structures, Vol. 4, pp. 1117-1134.

PROBLEM:

Evaluate the dynamic response of the simply supported beam shown subject to a uniformly distributed step pressure as depicted in the figure. Use Newmark method to carry out the time integration.

MODELING HINTS:

Only a quarter of the beam is modeled because of the symmetry. Due to the presence of plasticity and heavy loads full integration is used to get reliable results.

SOLUTION PARAMETERS FOR ND7A:

For this case a large time step increment with 500 sub-steps (within each step) is selected.Convergence tolerance is set to 0.05 and automatic stepping is used.



Figure ND7-1

GIVEN:

E = 3 x 107 psi

ν = 0.3

L = 30 in

h = 2 in

b = 1 in

ET = 0

σy = 5 x 104 psi

ρ = 0.733 x 10-3 lb sec2/in4

po = Static Collapse Load = 444.44 lb/in

h/2 4 5321

4

5

31

32

33


p

t

Time Curve

.75 Po

3

1

2

L/2

6

Lb

h

Problem Sketch

0.75 Po psi



8-280


Displacement at the middle of the span (in).

Figure ND7-2

Figure ND7-3



ND8: Large Displacement Dynamic Response of a Cantilever Beam

TYPE:

Nonlinear Dynamic Analysis, Large Displacements.ND8A) 2D Isoparametric (8 node) Plane Stress Elements

ND8B) BEAM Elements

REFERENCE:

Bathe, K. J., Ozdemir, H., Wilson, E. L., “Static and Dynamic Geometric and Material Nonlinear Analysis,” Report No. UC SESM 74-4, Structural Engineering Laboratory, University of California Berkeley, California, February 1974.

PROBLEM:

Determine the dynamic response of the cantilever beam shown subject to the uniformly distributed step pressure given in the figure. Use Newmark method to carry out the time integration.

Figure ND8-1

h

ND8A • PLANE2 D Model

p

t

Time Curve

Finite Element Models

b

4 5321

Problem Sketch

1

2

3

4

5

26

27

28

p/2

p/2

ND8B • Beam Model

L

1 2 3 4 5 6

1 2 3 4 5



8-282

GIVEN:

L = 10 in

h = 1 in

b = 1 in

E = 12,000 psi

ν = 0.2

p = 2.85 lb/in

ρ = 1 x 10-6 lb sec2/in4


Figure ND8-2



ND9: Large Displacement Dynamic Analysis of a Spherical Shell

TYPE:

Nonlinear Dynamic Analysis, Large Displacements, 2D Isoparametric (8 node) Axisymmetric Elements.

REFERENCE:

Bathe, K. J., Ozdemir, H., Wilson, E. L., “Static and Dynamic Geometric and Material Nonlinear Analysis,” Report No. UC SESM 74-4, Structural Engineering Laboratory, University of California Berkeley, California, February 1974.

PROBLEM:

Investigate the dynamic response of the spherical cap shown subject to a step load applied at the apex. Use Newmark method to carry out the time integration.

Figure ND9-1

θ

h

R

P

wo

H

p

t

Time Curve

Problem Sketch

5253

10

9

87

65

4321

3

2

1

51


P/2π



8-284

GIVEN:

R = 4.76 in

H = 0.0859 in

h = 0.01576 in

θ = 10.9°E = 10 x 106 psi

P = 100 lb

ν = 0.3

ρ = 0.245 x 10-3 lb sec2/in4


Figure ND9-2



ND11: Small Displacement Dynamic Analysis of a Simply Supported Plate

TYPE:

Linear Dynamic Analysis, 3D Isoparametric (20 node) Elements.

REFERENCE:

Bathe, K. J., Ozdemir, H., Wilson, E. L., “Static and Dynamic Geometric and Material Non-linear Analysis,” Report No. UC SESM 74-4, Structural Engineering Laboratory, University of California Berkeley, California, February 1974.

PROBLEM:

Determine the dynamic response of a simply supported plate subject to a step load at the center. Use Newmark method to carry out the time integration.

Figure ND11-1

MODELING HINT:

Due to symmetry only one quarter of the plate is modeled.

2b

2a

10

0.006t

P

Time Curve

0

P

t


51



8-286

GIVEN:

a = 20 in

b = 30 in

t = 1 in

E = 3 x 104 psiν = 0.25ρ = 3 x 10-4 lb sec2/in4

P = 1 lb


At the center of plate (node 51):

Figure ND11-2



ND12: SDOF System with Nonlinear Damping Subjected to a Step Loading

TYPE:

Nonlinear dynamic analysis using Spring-damper Elements (gaps).

REFERENCE:

Ray W. Clough, Joseph Penzien, “Dynamic of Structures,” McGraw-Hill, New York (1975).

PROBLEM:

Determine the response for:ND12A) Coulomb damping (p = 1)

ND12B) Damping is proportional to a power of velocity less than 1 (p = 0.5)

ND12C) Damping is proportional to a power of velocity greater than 1(p=1.5)

MODELING HINT:

The spring-damper is modeled using 2 gaps (one tensile and 1 compressive.) A soft truss element is used along with the gap element to avoid singularity of the structure stiffness.



8-288

Figure ND12-1

GIVEN:

m = 1 lbs sec/in2

k = 1 lbs/in

c = 0.1 lbs sec/in


The exact theoretical results are evaluated for case a. This case can also be solved using a modal time-history analysis with a modal damping of 0.05. In all cases, the response frequency remains the same, while some differences are observed in the maximum response values.

Displacement Peaks

Theory) COSMOSM

Case a Case a Case b Case c

First Maximum 1.8547 1.8534 1.8317 1.8692

First Minimum 0.2699 0.2711 0.3205 0.2372

Second Maximum

1.6239 1.6234 1.5435 1.6759

Second Minimum 0.4669 0.4674 0.5779 0.3977

Problem Sketch

1.

F(lbs)

time

Y

4

221

FX


3

Load - Time Curve

1

K

M

c, p 0 0



ND13: Two-Degree of Freedom System with Spring Dampers

TYPE:

Dynamic direct time integration analysis using spring-damper elements (gaps).

REFERENCE:

COSMOSM Advanced Dynamic Module (using Concentrated Dampers).

PROBLEM:

Determine and compare the response peaks for each mass

MODELING HINTS:

The spring-dampers are modeled using 6 gaps (3 tensile and 3 compressive). Soft truss elements are used along with gap elements to avoid singularity of the structure stiffness.

Figure ND13-1

4

2

f(t) = F

X


5

34

Problem Sketch

M

K

F(t)

M

K K

c c c0 00 0

1

1 2 3

876



8-290

GIVEN:

M = 1 lbs sec/in2

K = 1 lbs/in

c = 0.1 lbs sec/in

F = 1 lbs


Same problem is solved using a modal time-history analysis using truss and concentrated damper elements. A comparison of the peaks of response for each mass is given in the following table:

Node 2Modal Time History Direct Integration

Time Displ. Time Displ.

First Maximum 2.7 1.067 2.7 1.067

First Minimum 6.6 0.3014 6.6 0.3012

Second Maximum 9.4 1.017 9.4 1.017

Second Minimum 12.6 0.4233 12.6 0.423

Third Maximum 15.9 0.9005 15.9 0.9007

Third Minimum 18.9 0.4663 18.9 0.4659

Node 3Modal Time History Direct Integration

Time Displ. Time Displ.

First Maximum 3.4 0.8349 3.4 0.8348

First Minimum 6.1 -0.06234 6.1 -0.06278

Second Maximum 9.7 0.6115 9.7 0.6115

Second Minimum 12.7 0.04310 12.7 0.04268

Third Maximum 15.7 0.5578 15.7 0.5580

Third Minimum 19.0 0.1434 19.0 0.1432



ND14: SDOF System with Rayleigh Damping Subjected to Base Excitation

TYPE:

Nonlinear dynamic analysis, TRUSS2D and Mass elements, Elastoplastic Material.

REFERENCE:

Mario, P., “Structural Dynamics,” Third Edition, Van Nostrand, 1991.

PROBLEM:

Determine the response of the mass.

Figure ND14-1

Problem Sketch

M

K u

c 0 0

u. .

g

1 2

R

Disp (in)

- 15 K

1.215

15 K

- 1.215

t (sec)

A (t)

20

- 10

0.45

1.1

1.2 1.4 2.0

X



8-292

GIVEN:

E (truss) = 12.35 kips/in2

A (truss) = 1 in2

L (truss) = 1 in

M = 0.2 kips sec2/in

α = Damping coefficient associated with stiffness matrix = 0.010

β = Damping coefficient associated with mass matrix = 0.755

∆ t = Time increment = 0.10 sec

A(t) = Time curve of the base excitation acceleration

Base acceleration multiplier in X-direction = – 5 (in/sec2)

Figure ND14-2



ND15: Wave Propagation in a Bar

TYPE:

Linear dynamic analysis, Truss Elements, Short Duration rectangular pulse, Finite Difference technique.

REFERENCE:

Ray W. Clough, Joseph Penzien, "Dynamics of Structures," McGraw-Hill, p.364-367.

PROBLEM:

A bar is subjected to a rectangular pulse with a duration of 0.001 seconds. Determine the response of the bar and the maximum base reaction force. Compare results using finite difference and Newmark.

Figure ND15-1

Figure ND15-2

F

Finite Element Mesh



8-294

GIVEN:

L = 100 in

Ex = 12000 psi

Density = 0.001 lb sec/in/in3

CONCLUSIONS:

C = wave velocity = (Ex/Dens)1/2 = 3464.1 in/secT = travel time per unit distance = 1./C = 0.000288675 secThus the time for the wave to return to it's original location: T' = 2 * L * THowever, at this time, the wave hits in the opposite direction. It will take another T' for the wave to reverse itself to it's original direction. Thus, the first period of motion in time is two times T' (which also equals the first natural period of the bar):


Similar results are obtained for response from finite difference and Newmark. However, more steps where required to obtain an accurate base reaction force using the Newmark technique. The following table shows a comparison of the maximum reaction force from several runs.

Dynamic Integration Technique

Number of Solution Steps

Maximum Reaction Force Run-Time

Finite DIfference (ISUB=10) 1293 932.7 lb 170 sec

Newmark 1293 859.0 lb 180 sec





Figure ND15-3

Figure ND15-4

Figure ND15-5



8-296

ND16: Dynamic Response of a Cantilever Beam to Release of a Prescribed Tip Rotation

TYPE:

Nonlinear Dynamic Analysis, release of a prescribed rotation, Plasticity, Large Displacement, BEAM2D Elements, beam-section definition.

ND16A) Linear Elastic Analysis

ND16B) Large Displacement, Elasto-Plastic Analysis

ND16C) The tip rotation is released but the tip moment is kept active

PROBLEM:

A cantilever beam is first subjected to a prescribed tip rotation. The tip is then suddenly released. Investigate the response of the beam due to the initial tip rotation.

MODELING HINTS:

This problem is solved in two steps. First the tip rotation is prescribed by performing a static analysis (In the nonlinear case, the beam undergoes considerable plastic deformation during this phase). Next the rotation is released and analysis is restarted to perform a linear/nonlinear dynamic analysis.

Figure ND16-1



Figure ND16-3

GIVEN:

E = 30E6 psi

n = 0

sy = 5,000 psi

L = 90 in

H = 3 in

B = 1 in

ET = 3E6 psi

Density = 0.001 lb sec/in/in3

CONCLUSIONS:

In case A, the beam vibrates linearly about its undeformed position. In case B, the beam undergoes considerable plastic deformation during the first (static) solution phase. As a result, much of the dynamic response is damped out and the beam oscillates about a deformed position. In case C, since the applied moment is kept acting at the tip, no release takes place and no dynamic response is observed.

Figure ND16-2

Figure ND16-4



8-298

Figure ND16-5

Figure ND16-6

Figure ND16-7



ND17: Long Thick-Walled Cylinder Subjected to a Rectangular Pressure Pulse

TYPE:

Nonlinear Dynamic analysis, Artificial Bulk Viscosity, Nonlinear Elasticity, Finite Difference Technique, PLANE2D axisymmetric elements.

PROBLEM:

A long thick-walled cylinder is subjected to an internal pressure pulse. The material of the cylinder is nonlinear elastic (Fig.ND17-2). Study the effects of artificial bulk viscosity on the solution. Compare with cases of Rayleigh damping and no damping.

Figure ND17-1 The Finite Element Model of the Problem

MODELING HINTS:

4-noded PLANE2D axisymmetric elements are used to model the cylinder. Since the cylinder is considered to be long, all displacements in the y-direction are fixed.

GIVEN:

Ri = 5 in

RO = 200 in

n = 0.35

r = 0.0001 lb.Sec2/in4



8-300

Artificial Bulk Viscosity Constants:

Co = 1.5

C1 = 0.06

C2 = 8012.3 in/Sec

Rayleigh Damping (used for comparison)

a = 0.

β = 1000. 1/Sec

Figure ND17-2 Stress-Strain Property Curve


Figures ND17-3a/b/c show graphs of radial stresses at two different locations:1. Radius =Ri,

2. Radius = (Ri +RO)/2.

Usinga) Artificial bulk viscosity

b) Rayleigh damping

c) No damping

Considering the use of Rayleigh damping, although it effectively damps the oscillations following the peak, it also heavily damps the magnitude of the shock front as the it moves outward in the radial direction. Applying Artificial Bulk Viscosity to the analysis, the response following the peak is effectively damped, while the intensity of the shock front is mostly preserved.



Thus by comparing the three cases, it is easy to conclude that the use of artificial bulk viscosity helps to reduce the stress variations at the shock location, damps the oscillations following the shock, while it also preserves the shock intensity (to a sufficient degree) as the shock travels along the thickness of cylinder.

Figure ND17-3 (a)

Figure ND17-3 (b)



8-302

Figure ND17-3 (c)


Index

Aadaptive step adjustment 5-15arc-length 1-5, 2-3, 5-3arc-length control 3-12, 3-40, 5-2,

5-12, 7-3, 7-15, 8-160, 8-163, 8-198automatic stepping 1-6, 5-15, 7-4,

8-61, 8-153, 8-160, 8-163, 8-165, 8-181, 8-189, 8-192, 8-202, 8-225

automatic stepping algorithm 1-6, 5-15

automatic-adaptive stepping 3-12automatic-stepping 8-121, 8-143,

8-148auto-stepping 8-115axisymmetric 3-2, 3-40, 5-17, 5-20,

8-74

Bbandwidth 4-1base acceleration 8-292base excitation acceleration 8-292base motion 5-13, 7-8beam-section-definition 8-43, 8-

138BFGS 5-8, 8-7Blatz-Ko hyperelastic 8-171, 8-

174, 8-176Blatz-Ko strain energy density

function 3-15bounding surface 3-37, 3-38bulk relaxation 3-68

Ccable-like structures 1-2cable-type behavior 3-6, 3-7Cauchy 3-25Cauchy-Green deformation tensor 3-9

Cauchy-Green strain tensors 3-66classical power law for creep 3-30coefficient of friction 8-168compressible polyurethane foam

type rubbers 3-15compression strength 3-37compressive gaps 4-7concrete 8-206, 8-208, 8-210concrete model 3-37concrete ultimate strength 3-39contact 1-4, 4-1, 4-3, 4-5, 4-6, 4-9,

4-12, 4-13, 8-168contact (node to line gap) 8-80, 8-

84, 8-90, 8-223contact (node to surface gap) 8-

82, 8-86contact surface 4-11, 4-17contactor 4-5, 4-8, 4-9, 4-11, 4-13

control technique 5-1, 7-3controlled degree of freedom 7-14convergence 5-5, 5-15coupling 8-178, 8-189, 8-192, 8-202crack tip 5-18, 5-20creep 1-6, 8-54, 8-55, 8-57creep analysis 1-2, 5-16, 7-13creep constant 3-31, 7-13creep curve 3-30creep laws 3-30, 3-60creep model 3-30, 3-32creep strain 3-64cyclic loading 8-228

Ddamage coefficient 3-37, 3-38damping coefficient 8-292damping matrix 5-13, 5-14deformation tensor 5-17deformation-controlled loading 1-

5deformation-dependent

pressure 8-115deviatoric strain 3-28displacement control 3-40, 5-2, 5-

3, 7-14, 8-61, 8-72, 8-153, 8-156, 8-163, 8-168, 8-171, 8-174, 8-176, 8-186, 8-195

COSMOSM Advanced Modules I-1

Index

displacement increment 5-15displacement vector 4-4displacement-pressure (u/p) 8-189,

8-192, 8-195, 8-202displacement-pressure

formulation 3-19divergence 5-10, 5-15Drucker-Prager 8-110, 8-113Drucker-Prager model 3-20, 3-21dynamic analysis 5-14, 5-26, 7-8

Eeffective strain 3-6elastic creep analysis 3-30elastoplastic 7-2, 7-16, 8-195, 8-198,

8-202, 8-291elastoplastic model 5-16, 7-2element group 7-17, 7-25energy tolerance 5-10equibiaxial test 3-67equilibrium equations 5-4equilibrium iterations 8-61, 8-66experimental data 3-69, 3-70

Ffabric tension structures 3-35failure criterion 3-3failure index 3-3, 3-4fitting problems 1-4flexibility matrix 4-3Flow Rule 3-28flow theory of plasticity 5-21force control 5-3, 5-12, 5-15, 8-7, 8-

61, 8-165force-controlled loading 1-5frequencies 5-22friction 4-2friction force 4-7

Ggap 4-3, 4-5, 4-6, 4-12, 4-16, 8-59, 8-

121, 8-266gap displacements 4-4gap force 4-4gap iterations 4-17, 5-15gap-friction 8-27, 8-28, 8-29, 8-32,

8-35, 8-37, 8-268, 8-271, 8-273, 8-275

gear-tooth contacts 1-4Generalized Maxwell model 3-33geometric nonlinear analysis 7-5,

7-7, 7-11geometry updating 8-181

HHuber-von Mises model 3-17, 3-18hybrid method 4-3hybrid technique 4-2hydrodynamic 8-121, 8-126hydrostatic 8-121, 8-126, 8-129hydrostatic pressure 3-37hyperelastic material 3-8, 8-74

Iincremental procedure 5-15isotropic hardening rule 7-18isotropic material model 3-1iterative method 5-1, 5-4, 5-5, 5-8,

7-4

JJacobian 5-5J-integral 3-32, 5-17, 5-18, 5-20, 5-

21, 8-212, 8-214, 8-216, 8-220J-segments 5-21

Kkinematic hardening 3-19, 8-140, 8-

228kinematic hardening rule 8-65Kirchhoff 3-25

LLagrange multiplier method 4-1Lagrangian 3-25Lagrangian strain tensor 3-65laminated composite material 3-3large deflection 2-3, 8-72, 8-74, 8-

115, 8-121, 8-126, 8-129, 8-133, 8-135, 8-138, 8-168, 8-181, 8-189, 8-192, 8-195, 8-198, 8-202

large displacement 1-2, 2-2, 7-24,

8-15, 8-17, 8-19, 8-21, 8-25, 8-39, 8-41, 8-46, 8-48, 8-61, 8-165, 8-223, 8-225, 8-281, 8-283, 8-296

large strain plasticity 3-66, 8-61line search 5-1, 5-9, 7-8, 8-7load curves 7-5load multiplier 5-2, 7-15local boundary conditions 8-115locking 3-10logarithmic strain 3-28logarithmic strains 3-25

Mmaterial model utility 3-43material models 6-1material nonlinearities 1-2mixed finite element

formulation 3-8MNR 5-7, 5-12, 7-8, 7-14, 7-15Modified Newton-Raphson

(MNR) 3-40Mooney-Rivlin 3-11, 3-14, 3-67, 3-

69, 3-70, 8-168Mooney-Rivlin hyperelastic 8-72,

8-90, 8-115, 8-189, 8-192Mooney-Rivlin strain energy 3-9

NNewmark-Beta 5-11, 7-8Newton's iterative method 5-11Newton-Raphson 3-11, 3-18, 3-21,

3-40, 5-5, 8-61, 8-72, 8-74, 8-90, 8-110, 8-113, 8-117, 8-119

Nitinol 3-25nodal displacements 4-2, 4-3nodal forces 4-3node-to-line contact 4-13node-to-surface contact 4-10nonlinear dynamic analysis 5-11nonlinear elastic 8-92, 8-186nonlinear SPRING 3-7NR 5-7, 5-12, 7-8, 7-14, 7-15numerical procedures 5-1

OOgden 3-67, 3-69, 8-74

I-2 COSMOSM Advanced Modules


Ogden hyperelastic material 8-156Ogden model 3-12, 3-14, 3-74one-node gap 4-8, 4-12orthotropic material model 3-2out-of-balance load 5-5, 5-7, 5-10

Ppenalty approach 3-8penalty finite element

formulation 3-8penalty method 4-1penalty values 4-1plane strain 3-2, 3-67plane stress 3-2, 3-40plastic analysis 8-3, 8-5, 8-7plastic material models 3-65plastic strain 3-64, 5-16plasticity 1-2, 3-18, 3-23, 3-66, 8-11,

8-13, 8-61, 8-65, 8-69, 8-140, 8-223, 8-228, 8-278, 8-296

Poisson's ratio 3-5postbuckling 2-3preconditioning 5-1prescribed displacement 8-165prescribed non-zero

displacement 8-181principal strain 3-64, 3-66principal stretch ratio 3-9, 3-66

QQuasi-Newton (QN) 5-7, 5-8

RRayleigh damping 5-13, 7-8residual load vector 5-8restart flag 5-14Riks method 8-163rubber-like material 3-8, 3-12rubber-like materials 3-12

Ssecant material matrix 3-5secant modulus 3-5shape-finding analysis 3-35shear relaxation 3-68, 3-78shrink fit 4-8

slack 5-17small deflection 8-110, 8-113, 8-

117, 8-119, 8-178, 8-186snap-back 5-3, 8-160, 8-163snap-through 5-3, 8-160, 8-163snap-through buckling 1-2, 1-5softening 1-2spring-damper 8-287, 8-289stiffening 1-2stiffening behavior 1-2stiffness matrix 4-1, 4-2, 5-5, 5-7strain energy 3-8strain hardening 3-38, 3-40strain softening 3-37, 3-40stress intensity 5-17stress-strain curve 3-6, 3-18, 7-2, 7-

18stretch ratio 3-65subroutine UMODEL 3-44

Ttangent modulus 7-18tangential stiffness matrix 5-5, 5-6target 4-5, 4-8, 4-9, 4-10, 4-11, 4-12,

4-13, 4-17taut 5-17temperature gradients 5-20temperature-dependent material properties 7-11, 8-107

temperature-dependent yield stress 8-103

temperature-time shift 8-148tension cracks 3-37termination schemes 5-1, 5-9thermal analysis 8-9, 8-50, 8-53, 8-

119, 8-148thermal gradients 5-17thermal loading 8-148, 8-223thermal strain 3-64thermoplastic analysis 3-18thermo-plasticity 1-2, 8-103, 8-107threaded connections 1-4time curve 1-6, 5-14, 7-3, 7-20, 7-27time-dependent material properties 1-6

total Lagrangian formulation 2-2,

3-11, 3-14, 8-72, 8-74, 8-90total strain 3-6, 3-64Tresca yield criterion 8-228Tresca-Saint Venant yield

criteria 3-21Tsai-Wu failure 3-3, 3-4, 8-117, 8-

119two-node gap 4-6, 4-17types of strain output 3-64

UU/P formulation 3-19UMODEL 3-43, 3-44uniaxial compression test 3-38, 3-

39, 3-40uniaxial creep law 3-31uniaxial tension test 3-38uniaxial test 3-67unsymmetric behavior 8-186updated Lagrangian

formulation 2-2

Vviscoelastic material model 5-16viscoelastic model 3-30, 3-32, 3-77viscoelasticity 3-67, 8-143, 8-148,

8-153volumetric strain 3-6, 3-28von Mises 7-22, 8-216, 8-220von Mises Yield Criterion 8-228von Mises yield criterion 3-19, 3-

22, 7-16, 7-18

WWilson-Theta 5-11, 7-8wrinkling membrane 3-35, 3-36, 5-

17, 8-178, 8-181

XX-Y plot 7-4X-Y-plot 7-9, 7-15

Yyield criterion 3-27yield Stress 7-18

COSMOSM Advanced Modules I-3

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Geometrically Nonlinear Analysis