FEA mesh convergence and singularity in Connecting Lug

FEA Assignment-1 Q-1: Truss Analysis

1 Introduction A truss structure shown in Figure 1 is analysed analytically and using FEA in Abaqus and results are

compared in the first part of this question. Second part of the question involves comparing the results

of Abaqus/Explicit analysis with Abaqus/Standard analysis. The truss structure has a fixed support at

A and a roller support at E. A point load of 66kN is applied at the tip of the cantilever like structure.

The truss members are made of steel which has the following properties: Youngs Modulus (E)= 200E9

Pa, Poisons Ratio ()= 0.3 and density ()= 7800 kg/m3. The cross-sectional area of each member is

3.125E-3 m2.

Figure 1: 2D Truss Structure

2 Procedure

2.1 Analytical Solution Force in each truss member is found by creating a free-body diagram of the truss structure and doing

joint breaking analysis as shown in Figure 2. All the forces are calculated in terms of point load applied

(P=66kN). Strain energy (U) of the truss is then calculated using Equation-1. Where Fi is force in

member i, Li is the length of member i, and Ai is the cross-sectional area of member i which is same

for all members in this case.

=1

22

Equation 1

The vertical deflection of the point C can be calculated using the principle of Work-Energy which is

essentially given as shown in Equation-2.

=2

Equation 2

Figure 2: Analytical internal force calculations of truss structure.

2.2 Abaqus/Standard

Figure 3: Geometry sketch for truss structure created in Abaqus

SI units are used throughout the model created in Abaqus. A sketch was drawn as shown in Figure 3

to create a two-dimensional planar deformable wire shape part. Material and cross sections are

applied to the part.

The model has 3 degrees of freedom in the horizontal direction (x,U1), vertical (y,U2) and rotational

around Z-axis (UR3). Boundary conditions (BC) are applied in order to replicate the fixed and roller

support at A and E joint respectively. The node A is constrained in U1 and U2 and node E is constrained

in U1 only in the initial step. A concentrated force of 66000 N is applied at node C vertically facing

down (CF2).

The model is seeded with edge seeds for all different edges with their corresponding sizes. As a result

7 elements are created as shown in Figure 4. T2D2 elements are created after selecting Standard

library, linear geometric order and Truss family as element type. The job is created and submitted for

Abaqus to do FE analysis.

Figure 4: Mesh on the truss

2.3 Abaqus/Explicit In order to investigate the dynamic response of the truss structure, Abaqus Explicit model was run by

replacing the static loading step with dynamic loading of same magnitude over 0.01 seconds while

keeping the same BCs. The density has to be applied in the material to run the dynamic model

successfully.

3 Results and Discussion

3.1 Analytical solution and FEA comparison Force in each member calculated analytically and through FE calculations are tabulated in Table 1. The

forces estimated by FEA is are at almost 100% accuracy with the analytical solution.

Table 1: Forces and part of strain energy in each member

2

= 1.23 + 14

The vertical displacement at node C due to the load applied is calculated using Eqation-2 and

found to be 9.35mm. The vertical displacement probed through FEA is also found to be

9.35mm as shown in Figure 5.

Member Fi (P) Fi (N) Li (m)Abs Stress

(Pa)

Force

(Stress*A)

AB 2.4P 1.58E+05 1.80 1.45E+13 5.07E+07 1.58E+05

BC 2.4P 1.58E+05 1.80 1.45E+13 5.07E+07 1.58E+05

CD 2.6P 1.72E+05 1.95 1.84E+13 -5.49E+07 -1.72E+05

BD 0.0P 0.00E+00 0.75 0.00E+00 0.00E+00 0.00E+00

AD 2.6P 1.72E+05 1.95 1.84E+13 5.49E+07 1.72E+05

DE 4.8P 3.17E+05 1.80 5.78E+13 -1.01E+08 -3.17E+05

AE 0.0P 0.00E+00 0.75 0.00E+00 0.00E+00 0.00E+00

Analytical FEA

Figure 5: Deflected structure showing maximum deflection

The results are very close to each other for analytical and FE calculations which was initially expected

since the meshing used in FEA is 1 element per member. The strategy used in the analytical

calculations also considers 1 member as an element. Although the results are satisfying the

expectations from analytical solution, more investigation should be done with finer mesh to check for

any stress concentrations. Also a 3D model of the truss structure might improve the quality of the

analysis.

3.2 Abaqus/Standard and Abaqus/Explicit comparison At 0.01s the maximum deflection at point C is 17.8 mm in negative y direction as shown in Figure 6.

Figure 7 shows the displacement history of point C from 0s to 0.01s.

Figure 6: Contour of vertical displacement of truss structure

Figure 7: Displacement history of point C over time period from 0s to 0.01s

The maximum principle stress in the truss occurs at member ED as predicted by both standard and explicit models. However, the stress value estimeated by dynamic model is more than double than that of predictade by static model.

Figure 8: Maximum stress in the truss structure using Static analysis (Top) and Dynamic analysis (Bottom)

The larger displacement and stress values in dynamic model than in static model could be due to the

fact that the dynamic model creates impulse because of the suddenly applied force. The impulse

creates extra stress in the members on top of already existing static stresses.

Q-2: Mesh convergence and stress singularity in connecting Lug.

1 Introduction

Figure 9: Connecting Lug Problem Description

FE analysis is to be conducted of a connecting lug with dimensions shown in Figure 9 and the mesh

convergence and stress singularity are to be probed in this task. The connecting lug is supported at

the left end with fixed support. 50MPa of pressure is applied at the hole through bolt in the problem.

Material used in the lug is steel with its Youngs Modulus being 200GPa and Poissons ratio of 0.3.

2 Procedure The sketch is drawn as shown in Figure 10 and extruded for 0.04m depth. Steel material is applied

with its elastic properties.

Figure 10: Sketch for the model

Distributed pressure over the bottom half of the hole is used to illustrate the bolt-lug interaction in

order to reduce the complexity. Moreover, uniform pressure is applied neglecting the variaton of

pressure around the hole. 5.0E+07 Pa of pressure is applied in the bottom half of the hole as shown

in Figure 12. Figure 12 also illustrates the boundary condition (ENCASTRE) applied at the left face of

the lug working as a fixed support.

Figure 11: Partitioned Lug

Figure 12: Boundary condition and applied pressure

In order to create better suited mesh for this case, the lug is divided in 10 partitions as shown in Figure

11. These partitions aide in having more elements around the hole where there can be stress

concentrations due to geometrical changes. Five different meshes are created using C3D20R 3D stress

family, Quadratic geometric order and Hex reduced integration element with different global element

sizes. Table 2 and Figure 13 to Figure 17show the meshes used.

Table 2: Different Meshes

Figure 13: Mesh 1 with 216 elements


Mesh

Global Seed

Element Size

Number of

Elements

Mesh 1 0.012 216

Mesh 2 0.008 630

Mesh 3 0.004 6120

Mesh 4 0.003 11934

Mesh 5 0.002 49040


Figure 16: Mesh 4 with 11934 element


3 Results and Discussion

Figure 18: Maximum Principal Stress Contour for Mesh 1


Figure 18 to Figure 22 show the contours of maximum principal stresses on the lug with different

meshes. Location where the maximum stress occurs is same for all meshes however the area over the

stress occur gets smaller as the mesh becomes finer. This suggests that the stress is due to stress

concentrations and finer meshes are better to analyse these concentrations. Finer meshes can capture

the big stresses occurring in very small areas as shown in Figure 22. The values of maximum stresses

are tabulated in Table 3.

Table 3: Max Principal stresses and vertical deflection for different meshes

As shown in Figure 23, the Maximum vertical deflection of the connecting lug converges from Mesh 3

to Mesh 4. Therefore the mesh with 6120 elements would be ideal to use since it gives very accurate

displacement while it only takes 50s to compute it compared to 107s and 915s for Mesh 4 and Mesh

5.

Actually, the change in vertical deflection from Mesh 1 to Mesh 5 is very insignificant and if only

vertical deflection is to be computed then the Mesh 1 would serve well as well taking significantly less

computation time. However if the maximum principal stress in taken into account then Mesh 1 and

Mesh 2 do not give satisfactory results and Mesh-3 is the best option.

Mesh

Number of

Elements

Computation

Time(s)

Maximum

Principal

Stress(Pa)

Maximum

Vertical

Deflection(m)

Mesh 1 216 28 1.98E+08 -1.273E-04

Mesh 2 630 29 2.41E+08 -1.274E-04

Mesh 3 6120 50 3.05E+08 -1.278E-04

Mesh 4 11934 107 3.30E+08 -1.278E-04

Mesh 5 49040 915 3.93E+08 -1.279E-04

Figure 23: Max Vertical Deflection vs Number of Elements convergence graph

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FEA mesh convergence and singularity in Connecting Lug