# Nonlinear finite elements/Homework 5/Solutions/Problem 2

## Problem 2: Numerical simulation

Given:

Consider the cantilever beam shown in Figure~22.

 Figure 22. Cantilever beam with point load at the free end.

The beam is made of 6061-T6 aluminum alloy. The Young's modulus of the beam is 69 MPa, the Poisson's ratio is 0.33, the tensile yield strength is 275 MPa.

The length of the beam is 5 m, the width and the thickness are both 10 cm. A point load of 10 kN is applied at the free end as shown in the figure.

### Solution

#### Part 1

Perform a linear three-dimensional finite element analysis of the beam using 20 elements along the length and 3 elements across the thickness. Then,

1. Plot the axial stress of the centroidal line as a function of distance from the fixed end.
2. Plot the axial displacement of the centroidal as a function of distance from the fixed end.
3. Plot the transverse displacement of the centroidal line as a function of distance from the fixed end.

A SOLID45 (3-D Structural) element is used for the model of this problem. This element has the following degree of freedoms: ${\displaystyle u_{x},u_{y},u_{z}}$ . The built-in end conditions are ${\displaystyle u_{x}=u_{y}=u_{z}=0}$ . Plots are illustrated in Figure 23, 24, and 25.

 Figure 23. Axial displacements as a function of distance from fixed end. Figure 24. Transverse displacements as a function of distance from fixed end.

#### Part 2

Perform a nonlinear three-dimensional finite element analysis of the beam using 20 elements along the length and 3 elements across the thickness. Then,

1. Plot the axial stress of the centroidal line as a function of distance from the fixed end.
2. Plot the axial displacement of the centroidal as a function of distance from the fixed end.
3. Plot the transverse displacement of the centroidal line as a function of distance from the fixed end.

A SOLID45 (3-D Structural) element is used for the model of this problem. This element has the following degree of freedoms: ${\displaystyle u_{x},u_{y},u_{z}}$ . The built-in end conditions are ${\displaystyle u_{x}=u_{y}=u_{z}=0}$ . The effect of large deformations is included in the solution process. Plots are illustrated in Figure 23, 24, and 26.

 Figure 25. Maximum stresses as a function of distance from fixed end: Linear. Figure 26. Maximum stresses as a function of distance from fixed end: Nonlinear.

#### Part 3

Perform a linear finite element analysis of the beam using 4 beam elements. Then,

1. Plot the axial stress of the centroidal line as a function of distance from the fixed end.
2. Plot the axial displacement of the centroidal as a function of distance from the fixed end.
3. Plot the transverse displacement of the centroidal line as a function of distance from the fixed end.

A BEAM3 (2-D Beam) element is used for the model of this problem. This element has the following degree of freedoms: ${\displaystyle u_{x},u_{y},\theta _{z}}$ . The built-in end conditions are ${\displaystyle u_{x}=u_{y}=\theta _{z}=0}$ . Plots are illustrated in Figure 23, 24, and 25.

#### Part 4

Perform a nonlinear finite element analysis of the beam using 4 beam elements. Then,

1. Plot the axial stress of the centroidal line as a function of distance from the fixed end.
2. Plot the axial displacement of the centroidal as a function of distance from the fixed end.
3. Plot the transverse displacement of the centroidal line as a function of distance from the fixed end.

A BEAM3 (2-D Beam) element is used for the model of this problem. This element has the following degree of freedoms: ${\displaystyle u_{x},u_{y},\theta _{z}}$ . The built-in end conditions are ${\displaystyle u_{x}=u_{y}=\theta _{z}=0}$ . The effect of large deformations is included in the solution process. Plots are illustrated in Figure 23, 24, and 26.

#### Part 5

Comment on your results.

The axial and transverse displacement are not different by much between each analysis. Figure 23 shows that axial displacement is non zero in the nonlinear case while it is zero in the linear case.

The axial stress for each analysis yields very small values in comparison to maximum stress (Figure 25 and 26). In theory, the axial stress should be zero for each case. However, due to errors such as round off errors or the interpolations between elements, ANSYS computes non zero value for this stress. The stresses for SOLID45 are illustrated as Von Mises stress since ANSYS does not provide ${\displaystyle {\hat {\sigma }}_{x}}$  in local coordinate for use as the axial stresses.

Note: stresses given by ANSYS are in global cartesian coordinate. Hence, ${\displaystyle \sigma _{x}}$  does not represent axial stress.

There are significant difference between the maximum stresses obtained using BEAM3 and SOLID45 near the built-in end. SOLID45 model has constraints in all degree of freedoms at the fixed end. BEAM3 allows the fixed section to distort. Its results rely on the fact that the cantilever beam is comparatively long at considerable distance from the terminals. As one can see from Figure 25 and 26 that further from the built-in end, the stresses value from BEAM3 becomes closer to the value obtained using SOLID45 element.

A comparison of the displaceemnts is shown in the table below.

Element type max ${\displaystyle u_{x}}$  (m) max ${\displaystyle u_{y}}$  (m)
Linear SOLID45 2.03E-07 -0.71455
Nonlinear SOLID45 -5.98E-02 -0.70209
Linear BEAM3 0.0 -0.72464
Nonlinear BEAM3 -6.01E-02 -0.71023

The ANSYS input file listing for the linear and nonlinear 3-D finite element analysis is given below.

/prep7 yield = 275e6 l = 5 b = 10/100 h = 10/100 P = 10e3 et,1,45 mp,ex,1,69e9 MP,PRXY,1,0.33 K,1,0,0 K,2,l,0 K,3,l,h K,4,0,h L,1,2,20 L,2,3,4 L,3,4,20 L,4,1,4 A,1,2,3,4 esize,,4 vext,1,,,,,b VMESH,ALL NSEL,S,LOC,X,0,0 D,ALL,ALL,0 NSEL,S,LOC,X,l,l NSEL,R,LOC,Y,h,h F,ALL,FY,-P/5 NSEL,ALL FINISH 

Use the following commands according to the analysis type.

 Linear Nonlinear /SOLU /SOLU solve ANTYPE,0 fini NLGEOM,1 NSUBST,100,1000,10 /STATUS,SOLU SOLVE fini

The postprocessing code is shown below

/post1 lpath,350,281 pdef,ux,u,x pdef,uy,u,y pdef,seqv,s,eqv /output,beam45_c,txt prpath,ux,uy,seqv /output,, lpath,287,275 pdef,ux,u,x pdef,uy,u,y pdef,seqv,s,eqv /output,beam45_t,txt prpath,ux,uy,seqv /output,, fini 

The input listing for the linear and nonlinear finite element analysis of the beam problem is given below.

/prep7 yield = 275e6 l = 5 b = 10/100 h = 10/100 P = 10e3 et,1,3 keyopt,1,6,1 r,1,b*h,1/12*b*h**3,h mp,ex,1,69e9 MP,PRXY,1,0.33 n,1,0,0 n,2,l/4,0 n,3,2*l/4,0 n,4,3*l/4,0 n,5,l,0 e,1,2 e,2,3 e,3,4 e,4,5 D,1,ALL,0 F,5,FY,-P FINISH 

Use the following commands according to the analysis type.

Linear Nonlinear
/SOLU /SOLU
solve ANTYPE,0
fini NLGEOM,1
NSUBST,100,1000,10
/STATUS,SOLU
SOLVE
fini
/post1 etable,sigaxi,ls,1 etable,sigmax,nmisc,1 etable,sigmin,nmisc,2 /output,beam2,txt prnsol,u pretab,sigaxi pretab,sigmax pretab,sigmin /output,, fini