Analysis of Axisymmetric Structures

Application to Circular Reservoirs

Cristiano Yudok Chung Rodrigues

Extended Abstract

October 2009

1

Analysis of Axisymmetric Structures: Application to Circular Reservoirs

Cristiano Yudok Chung Rodrigues

Abstract

The present paper addresses the linear analysis of axially symmetric structures of thin shell subjected to axisymmetric loads. The basic axisymmetric structural element analyzed is the conical shell, and its governing equations were derived and identified. Two methodologies were developed based on the displacement method: one uses analytical solutions and the other the finite element method. The first methodology allows obtaining exact solutions (for the adopted structural theory) only for some cases (e.g., circular and cylindrical shells). On the other hand, the second methodology involves approximations but it is of general application to conical shells. The two formulations are implemented in a computer program with graphical interface allowing the presentation and output of graphical and numerical data. The computer program also considers spherical shells using a “polygonal” approximation by conical shell elements. The capabilities of the program were demonstrated and validated with some representative examples.

Keywords: axisymmetric structures, axisymmetric loading, thin shells, analytical solutions, finite

element method, computer program.

1 Introduction

The present paper addresses the linear analysis of axially symmetric structures subjected to axisymmetric loads. The conical thin shell is adopted as the basic general element of single curvature.

The formulation of two different methods designed to solve axisymmetric structures will be obtained, both based on the displacement method: one using analytical solutions and the other using the finite element method.

A graphical interface computer program will be designed using the obtained methodologies. The goal of the program is to have a numerical and graphical output of the solutions provided by the methodologies with the possibility of several visualization options.

Some examples will be conducted in order to demonstrate the capabilities of the program, to perform an error analysis and validate the solutions obtained.

This paper defines the plane of axisymmetric simplification (PAS) as the plane that contains the axis and the cross-section of revolution that generates the axisymmetric structure.

2 Conical shell theory

One studies a conical shell with a general α angle relative to the horizontal. For α = 0° the element behaves as a circular shell (plate and membrane behavior). For α = 90° the element behaves like a cylindrical shell. If α is different from the angles referred, the element behaves as a conical shell.

Two coordinate systems are adopted: a global system and a local system that depends on the element. A conical shell element can be defined by two nodes: the starting (1) and the ending (2). Figure 2.1 presents the axes corresponding to the two systems of coordinates and the positive stress resultants:

Figure 2.1 – Conical shell element in the PAS

2

2.1 Kinematics

In respect to axisymmetric shells, Kirchhoff’s theory is adopted, which is valid for thin shell elements. This theory considers that the plane cross-sections remain plane, disregarding the deformation by shear stress resultant. In these conditions the rotation of an element can be obtained by the slope of the deflection ( ):

. The displacement field of a conical

shell, in a local coordinate system, is displayed in Figure 2.2:

Figure 2.2 - Displacement field in local coordinate

system

The strain equations obtained for the conical shell are [3]:

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

2.2 Equilibrium equations

The equilibrium equations can be obtained, by a physical reasoning, using an infinitesimal element of a conical shell with all the internal forces and loads considered:

(2.7)

(2.8)

(2.9)

2.3 Constitutive equations

Assuming that the thin conical shell element has a (plane stress) linear-elastic behavior, we can establish the following relationships between stress resultants and strain measures:

(2.10)

(2.11)

(2.12)

(2.13)

3 Analytical solutions

Analytical solutions will be obtained for some types of element.

3.1 Ring

A ring element can be fully represented by one node in the PAS and its analytical solution uses only one node. For this reason the ring element is defined in the global system of coordinates, and has only three degrees of freedom associated to the node. The stiffness matrix will have the corresponding size (see Annex A.1).

3.2 Circular annular shell

The following image represents a general element of a circular annular shell:

Figure 3.1 - General element of circular annular shell

The process to obtain of the stiffness matrix for the element of a circular annular shell can be divided in two topics: plate behavior [4] and membrane behavior [6].

3

Adopting Kirchhoff’s theory, in respect to circular plates of constant thickness subjected to uniform transverse load results in the

following expression for the deflection:

(3.1)

The longitudinal displacement for the circular annular plate, considering no load in this

direction, , is:

(3.2)

These solutions lead to the (plate and membrane) stiffness matrices presented in Annex A.2.1 and A.2.2. The matrix for the shell behavior can easily be obtained because the plate and membrane behaviors are independent (see Annex A.2.3). The vector of the fixed-end forces can be obtained using a particular solution, in this case, considering {B} a null vector (see Annex A.2.4). The element of circular shell can be considered a particular case of the element of the circular annular shell in which one of the nodes has the radius coordinate of zero. This requires a limit analysis to avoid division by zero or logarithmic of zero errors.

3.3 Cylindrical shell

The next figure represents a general element of a cylindrical shell:

Figure 3.2 - General element of a cylindrical shell

The process to obtain the stiffness matrix for the element of cylindrical shell can be divided in two steps.

First, considering that there are no restrictions to longitudinal displacements in the cylindrical shell, (hence, ), the general solution for

the bending behavior is [4]:

(3.3)

The parameter represents the inverse of the

characteristic length and the ’s are functions of the dimensionless quantity (see Annex A.3.1). From here it’s possible to obtain the stiffness matrix for the bending behavior of the cylindrical shell (see Annex A.3.2).

Secondly, the implementation of the membrane behavior for the cylindrical shell can be obtained with the superposition of two

systems displayed in Figure 3.3. System , is

a pure membrane solution (only ) which has non null transverse nodal displacements due to Poisson’s effect, which is canceled by

adding the bending solution of system . The

resulting membrane stiffness matrix is in Annex A.3.3.

Finally, one needs to compute the terms of the stiffness matrix coupling transverse and longitudinal degrees of freedom (see Annex A.3.4) and perform a correction to the previously determined bending stiffness matrix to account for null nodal longitudinal displacements (see Annex A.3.5).

The complete elementary stiffness matrix for the cylindrical shell can be found in Annex A.3.6. The vector of the fixed-end forces (obtained using a particular solution) is in Annex A.3.7.

4

Figure 3.3 – Configuration of systems for the implementation of the membrane solution

4 Finite element method for a conical shell element

The next figure represents a general finite element of a conical shell:

Figure 4.1 - General finite element of a conical shell

The functions of approximation of the displacement field are polynomial. The longitudinal displacement uses linear Lagrange functions and the transverse displacement uses Hermit functions to allow continuity of the function and its slope. The displacements field is:

(4.1)

(See Annex A.4.1)

The strains field can be obtained using the following expression:

(4.2)

With:

(4.3)

Matrix A is a differential operator obtained using the strain equations (see Annex A.4.2).

The stresses field can be obtained using the following expression:

(4.4)

Where is the constitutive matrix (see

Annex A.4.3).

The shear stress resultant is obtained by equilibrium (2.9), because Kirchhoff’s theory is used and it disregards the deformation by shear stress resultant.

The stiffness matrix and the equivalent forces vector are given by (one uses Gaussian

quadrature for and analytical integration

for - see Annex A.4.4):

(4.5)

(4.6)

The variable p represents the load applied to the element, which can be uniform in the longitudinal direction and trapezoidal in the transverse direction:

(4.7)

5

5 Global System

Both methodologies, analytical solutions and the finite element method, are defined in a local coordinate system, except for the ring element. Given the possibilities of directions and orientation, it is necessary to transform the system of coordinates to a global system in order to allow interaction of different elements.

The nodal loads can be assigned to the degrees of freedom associated to a node (uniform load distributed on the perimeter, or concentrated if the radius is null).

The elements defined by the two methodologies, analytical solutions and the finite element method, are compatible, because both are based on nodal displacements.

1

To solve a structure subjected to a load consisting of several axisymmetric shell elements, it is necessary to assembly the global stiffness matrix and the global vector

of forces from the elementary matrices and vectors, leading to:

The system of linear equations is solved using the method of Gauss elimination in order to

obtain the displacement vector .

6 Examples and error analysis

The following examples have the purpose of demonstrating the capabilities of the program, validating the methodologies used and performing an error analysis.

To evaluate the error the following error norm is defined:

1 However, to obtain the stiffness matrix and the

forces vector in the finite element method, integration along the along the variable of revolution, θ, is used and in the analytical solutions it is not. Therefore, to have compatibility between elements of both methodologies, in the analytical solutions it is necessary to multiply the degree of freedom corresponding to the respective terms of the stiffness matrix and the vector of forces by a factor of , were corresponds to the radius of

the node associated to the degree of freedom.

With:

– Analytical solution

– Finite element method solution

In respect to the finite element method, all the examples use 6 points of Gauss in the integration of the stiffness matrix.

The following data was used in all examples:

Poisson coefficient: ν=0,2

Elasticity module: E=30e6 kPa

6.1 Cylindrical shell

The following figure represents the structure and load configuration:

Figure 6.1 - Structural and load configuration of the cylindrical shell example

Figure 6.2 displays the displacements and stress resultants diagram using a number set of elements and a solution of reference. The diagrams and are not represented, because they are very similar to the deflection

and , respectively.

a)

0

1

2

3

4

5

6

7

0 0,0005 0,001

z

w

1

2

5

10

20

50

100

200

500

Sol. Ref.

Number of elements

6

b)

c)

d)

Figure 6.2 – Displacements and stress resultants diagrams of the cylindrical shell example a) Deflection: w; b) Membrane force: Nx; d) Bending moment: Mx; f) Shear force: Vx;

Figure 6.3 - Displacements and stress resultants error of the cylindrical shell example

Observing the graphics of the results and the error analysis chart presented in Figure 6.3, it’s possible to see that the displacements converge faster and that the error decreases at a faster rate, followed by flexural and membrane stress resultants and finally the shear stress resultant. This is explained by the fact that the finite element method is based on the approximation of the displacements field, the flexural and membrane stress resultants are dependent on the derivatives of the displacements, and the shear stress resultant is obtained by equilibrium using the bending moments. Thus, the more indirect process of achieving results will generate larger errors.

Observing the error chart, it’s possible to notice that when more than 200 elements are used, the error is dominated by numerical precision.

6.2 Circular reservoir

The following figure represents the structure and load configuration:

Figure 6.4 - Structural and load configuration of the circular reservoir example

0

1

2

3

4

5

6

7

-100 0 100

z

Nx

0

1

2

3

4

5

6

7

-50050

z

Mx

0

1

2

3

4

5

6

7

-100 -50 0 50

z

Vx

1

2

5

10

20

50

100

200

500

Sol. Ref.

Number of elements

1,0E-05

1,0E-04

1,0E-03

1,0E-02

1,0E-01

1,0E+00

1,0E+01

1,0E+02

1 10 100 1000

Erro

r(%

)

Number of elements

Error Analysis

w Nθ Mx Mθ Vx

7

For this example, the following elements were used: an arc with 100 finite elements, a ring element (1), a circular shell element (2) and a cylindrical shell element (3).

The following graphics represent the displacements and stress resultants diagrams:

a)

b) deformed state

c) Nx

d) Nθ

8

e) Mx

f) Nθ

g) Vx

Figure 6.5 - Displacements and stress resultants diagrams of the circular reservoir example

a) Orientation and configuration of the elements and arc; b) Deformed state of the structure; Membrane stress resultant: c) Nx and d) Nθ; Bending moment: e) Mx and f) Mθ; g) Shear stress resultant: Vx;

The following graphics show details of some stress resultants diagrams at the junction of the arc and the cylindrical shell:

a) Mx

9

b) Vx

Figure 6.6 – Details of the stress resultants diagram at the junction of the arc and the cylindrical shell

a) Bending moment: Mx; b) Shear stress resultant: Vx;

7 Conclusions

The analysis of axisymmetric linear thin shells under axisymmetric loading was addressed. Two different formulations, one based on analytical solutions and the other using finite element method, were developed and incorporated in a computer program.

It’s important to understand the concept of axisymmetric condition, because it allows adopting certain simplifications. All problems can be represented in the plane of axisymmetric simplification (PAS), which simplifies the problem, making it similar to solving a frames problem in a two-dimensional plane.

The conical shell is the general element of single curvature when it comes to axisymmetric structures. Defining the angle of the conical shell to the horizontal as a variable it’s possible to obtain some particular cases, such as: cylindrical shell and circular shell including the annular type. In the PAS the representation of a conical shell element is of a line segment define by two nodes, this aspect is very important because any kind of geometry can be polygonized constituting an approximation of the structure, which gets better as more elements are used. In this line of thought, it was implemented in the program the use of spherical shells that are obtained by approximation of conical shells.

It’s possible to observe, in the results of the cylindrical shell example, that the solutions of the finite element method converge to the results that use analytical solutions, verifying that two different types of methodologies can achieve the same goal.

Observing the graphics of the results, and ignoring the cases of significant discontinuity of stress resultants between elements, in most cases, using five finite elements is enough to obtain a satisfactory convergence of the results. However, to obtain a good approximation of the displacements and force fields it’s advisable to choose around 20 finite elements.

Both methodologies were found to be compatible and, furthermore, they complement each other enhancing the functionality of the program. The analytical solutions have some advantages, as they require just one element to achieve exact results, making it very efficient. However, the finite element method also has some advantages, because it allows more complex structural configurations and more general kinds of applied loading.

8 References:

[1] Advanced Finite Element Methods (ASEN 6367), Aerospace Engineering Sciences, University of Colorado at Boulder, Spring 2009

[2] Azevedo, A. F. M., "Método dos Elementos Finitos", Faculdade de Engenharia da Universidade do Porto - Portugal, Abril de 2003

[3] Flügge, W., "Stresses in shells", Berlin : Springer, 1973

[4] Ghali, A., “Circular storage tanks and silos”, Wiley & Sons, 1979

[5] Kelkar, Vasant S.; Sewell, Robert T., "Fundamentals of the analysis and design of shell structures", Englewood Cliffs: Prentice-Hall, 1987

[6] Martins, J. A. C., “Teoria elástica linear de cascas finas”, Abril 1989

[7] Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T., “Numerical Recipes in FORTRAN: The Art of Scientific Computing”, Cambridge University Press, second edition, 1992, Cambridge, England.

[8] Ross, C. T. F., "Finite element programs for axisymmetric problems in engineering", Chichester : Ellis Horwood, 1984

[9] Timoshenko, S.; Woinowsky-Krieger, S., “Theory of plates and shells”, Auckland : McGraw-Hill, 1981

10

A Annex

A.1 Ring element

A.2 Circular annular plate

A.2.1 Matrix of the plate behavior

With:

A.2.2 Matrix of the membrane behavior

A.2.3 Elementary stiffness matrix

A.2.4 Vector of fixed-end forces

With:

11

A.3 Cylindrical Shell

A.3.1 Parameters of and ’s

A.3.2 Matrix of the plate behavior

A.3.3 Matrix of the membrane behavior

A.3.4 Matrix of the plate-membrane behavior

A.3.5 Matrix of the plate behavior corrected

A.3.6 Elementary stiffness matrix

A.3.7 Vector of fixed-end forces

12

A.4 Finite element method

A.4.1 Displacement field

A.4.2 Differential operator

A.4.3 Constitutive matrix

A.4.4 Vector of equivalent forces