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

## Problem 2: Numerical simulation[edit | edit source]

** Given:**

Consider the cantilever beam shown in Figure~22.

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[edit | edit source]

#### Part 1[edit | edit source]

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

- Plot the axial stress of the centroidal line as a function of distance from the fixed end.
- Plot the axial displacement of the centroidal as a function of distance from the fixed end.
- 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: . The built-in end conditions are . Plots are illustrated in Figure 23, 24, and 25.

#### Part 2[edit | edit source]

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

- Plot the axial stress of the centroidal line as a function of distance from the fixed end.
- Plot the axial displacement of the centroidal as a function of distance from the fixed end.
- 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: . The built-in end conditions are . The effect of large deformations is included in the solution process. Plots are illustrated in Figure 23, 24, and 26.

#### Part 3[edit | edit source]

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

- Plot the axial stress of the centroidal line as a function of distance from the fixed end.
- Plot the axial displacement of the centroidal as a function of distance from the fixed end.
- 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: . The built-in end conditions are . Plots are illustrated in Figure 23, 24, and 25.

#### Part 4[edit | edit source]

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

- Plot the axial stress of the centroidal line as a function of distance from the fixed end.
- Plot the axial displacement of the centroidal 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: . The built-in end conditions are . The effect of large deformations is included in the solution process. Plots are illustrated in Figure 23, 24, and 26.

#### Part 5[edit | edit source]

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 in local coordinate for use as the axial stresses.

Note: stresses given by ANSYS are in global cartesian coordinate. Hence, 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 (m) | max (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