Nonlinear finite elements/Homework 7/Solutions
Problem 1: Index Notation
editSolutions
editPart 1
edit- Determine whether the following expressions are valid in index notation. If valid, identify the free indices and the dummy indices.
1)
Invalid. The free indices are on the LHS and on the RHS.
2)
Valid. Both and are free indices.
3)
Valid. The free index is and the dummy index is .
4)
Invalid. The free index is on the LHS and on the RHS.
5)
Valid. The dummy index is . So the sum is a scalar which is equal to .
6)
Invalid. There is one free index on the LHS and no free index on the RHS.
Part 2
editShow the following:
1)
2)
The LHS is
If , then we have
We get the same result if . The only nonzero LHS and RHS occur when and .
- Case 1: , , , .
- Case 2: , , , or
, , , .
- Case 3: , , , .
- Case 4: , , , .
- Case 5: , , , or
, , , .
- Case 6: , , , .
- Case 7: , , , .
- Case 8: , , , or
, , , .
- Case 9: , , , .
Hence the relation is satisfied for all cases.
3)
4)
For ,
For ,
For ,
Hence shown.
Part 3
editThe elasticity tensor is given by
where are Lame constants, is the second order identity tensor, and is the fourth-order symmetric identity tensor. The two identity tensors are defined as
The stress-strain relation is
Show that the stress-strain relation can be written in index notation as
Write the stress-strain relations in expanded form.
Now, in dyadic notation
Therefore
Similarly,
The stress-strain law becomes
Expanding the left hand side, we get
Therefore,
Problem 2: Rotating Vectors and Tensors
editLet ( ) be an orthonormal basis. Let be a second order tensor and be a vector with components
Solution
editPart 1
editWrite out and in matrix notation.
Part 2
editFind the components of the vector in the basis ( ).
The components of are
Therefore
Part 3
editFind the components of the vector in the basis ( ).
The cross product is given by
Therefore,
Part 4
editFind the components of the tensor in the orthonormal basis.
The tensor product is given by
Hence, in matrix notation
Part 5
editRotate the basis clockwise by 30 degrees around the direction. Find the components of , , , and in the rotated basis.
The vector transformation rule is
where are the direction cosines.
In this case, the direction cosines are
Therefore,
Similarly,
Also,
From the handout from Slaughter's book, the tensor transformation rule is
where are the direction cosines.
In matrix form,
Therefore the components of in the rotated basis are give by
Problem 3: More Beams
editPart A
editConsider a beam of length = 100 in., cross-section 1 in. 1 in., and subjected to a uniformly distributed transverse load lbf/in. Model one half of the beam using symmetry considerations.
Part 1
editHinged-Hinged Beam
The boundary conditions are
Compute a plot similar to that shown in Figure 4.3.4 for this case using Beam188 elements. What do you observe?
The result is shown in Table 1.
Table 1. Deflections of a hinged-hinged beam | |
---|---|
Load | at |
1 | -0.520746 |
2 | -1.04086 |
3 | -1.55922 |
4 | -2.07510 |
5 | -2.58768 |
6 | -3.09622 |
7 | -3.60000 |
8 | -4.09835 |
9 | -4.59065 |
10 | -5.07636 |
Part 2
editClamped-Clamped Beam
The boundary conditions are
Compute a plot for this case using Beam188 elements. Comment on your plot.
The result is shown in Table 2.
Table 2. Deflections of a clamped-clamped beam | |
---|---|
Load | at |
1 | -0.103456 |
2 | -0.202476 |
3 | -0.294220 |
4 | -0.377753 |
5 | -0.453387 |
6 | -0.521968 |
7 | -0.584455 |
8 | -0.644185 |
9 | -0.696440 |
10 | -0.745243 |
Listed below is the ANSYS input code for Problem 3A.1 and 3A.2.
/prep7
b = 1
h = 1
et,1,188
sectype,1,beam,rect
secdata,b,h
MP,EX,1,30e6
MP,PRXY,1,0.3
K,1,0,0,0
K,2,50,0,0
k,3,0,50,0
L,1,2,50
latt,1,,1,,3,3,1
LMESH,ALL
!change this section to d,1,all,0 for Problem 3A.2
d,1,uy,0
d,1,uz,0
d,2,rotz,0
nsel,all
sfbeam,all,,pres,10
fini
/solu
nlgeom,on
autots,on
nsubst,10,100,10
outres,all,all
solve
finish
Part B
editPart 1
editSimulate the unrolling of a cantilever beam from Section 4.1.1 of Ibrahimbegovic (1995) and compare your results with the results shown in the paper.
The result is shown in Table 3.
Table 3. Cantilever free-end displacement components | |||
---|---|---|---|
Load | Rotation | ||
-0.040666 | 6.3205 | -3.1287 | |
9.9578 | 0.12729 | -6.2577 |
The deformation plots are shown in Figure 4 and 5.
The ANSYS input code for this problem is listed below.
/prep7
et,1,beam188
sectype,1,beam,rect
secdata,1,1
mp,ex,1,1200
mp,prxy,1,0
l = 10
pi = 4*atan(1)
r = L/2/pi
K,1,0,-r
K,2,-r,0
K,3,0,r
K,4,r,0
K,5,0,-r
k,6,0,0,10
k,7,0,0,0
larc,1,2,7,r,5
larc,2,3,7,r,5
larc,3,4,7,r,5
larc,4,5,7,r,5
latt,1,,1,,6,6,1
lmesh,all
dk,5,all,0
fk,1,mz,-20*pi
/solu
nlgeom,on
cnvtol,f,5,0.001
outres,all,all
arclen,on
nsubst,100
solve
fini
Part 2
editSimulate the clamped-hinged deep circular arch from Example 7.3 of Simo and Vu Quoc (1986) and compare you results with the results shown in the paper.
The inputs are: , , , and . We assume a square cross section. Then
Therefore, .
The deformed shape (unconverged) for a load of 905 is shown in Figure 6.
The load-displacement curve (up to the last converged solution) is shown in Figure 7.
The buckling load is 900.925 compared to 905.28 in Simo and Vu Quoc.
The ANSYS file use for the calculations is shown below.
/prep7
!*
!* Total load
!*
load = 905.0
!*
!* Element type
!*
et,1,beam188
keyopt,1,1,0
keyopt,1,2,0
keyopt,1,3,0
keyopt,1,4,0
keyopt,1,6,0
keyopt,1,7,0
keyopt,1,8,0
keyopt,1,9,0
keyopt,1,10,0
keyopt,1,11,0
keyopt,1,12,0
!*
!* Beam cross-section type
!*
sectype, 1, beam, rect, , 0
secoffset, cent
secdata, 0.34641, 0.34641, 0,0,0,0,0,0,0,0
!*
!* Material properties
!*
mptemp,,,,,,,,
mptemp,1,0
mpdata,ex,1,,8.3e8
mpdata,prxy,1,,0.33
!*
!* Keypoints
!*
k, 1, 0.000, 0.000, 0.000
k, 2, -95.372, -30.071, 0.000
k ,3, 95.372, -30.071, 0.000
k ,4, 0.000, 100.000, 0.000
!*
!* Arcs
!*
larc,3,4,1,100,
larc,4,2,1,100,
!*
!* Element size = 20 elements per arc
!*
lesize,all, , ,20, ,1, , ,1,
!*
!* Mesh the arcs
!*
lmesh, all
!*
!* Plot the nodes
!*
nplot
/pnum,node,1
/number,0
/replot
!*
!* Apply displacement BCs
!*
!* Hinged end
!*
d, 22, ux, 0
d, 22, uy, 0
d, 22, uz, 0
!*
!* Clamped end
!*
d, 1, all, 0
!*
!* Apply load
!*
f, 2, fy, -load
finish
!*
!* Solve
!*
/solu
antype, static
nlgeom, on
!autots, on
!solcontrol, on
!*
!* Load step 1
!*
time, 1.0
! f, 2, fy, -load
nsubst,100,0,0
kbc, 0
neqit, 100
outres, ,1
arclen,on,100.0,0.0
lswrite
solve
finish
!*
!* See solution
!*
/post26
!*
!* Save solution in variables 2 and 3
!*
nsol, 2, 2, u, x ! Save ux at node 2
nsol, 3, 2, u, y ! Save uy at node 2
!*
!* Scale solution
!*
prod, 4, 1, , , Load, , ,load ! Scale time to get load
prod, 5, 2, , , , , ,-1 ! Make disp +ve
prod, 6, 3, , , , , ,-1 ! Make disp +ve
prvar, 4, 5, 6 ! Print load, ux, uy
!*
!* Plot solution
!*
/axlab, x, Deflection
/axlab, y, Load
/grid, 1
xvar, 5
plvar, 4 ! plot ux vs load
/noerase
xvar, 6
plvar, 4 ! plot uy vs load
/erase
Here is another version of solution to this problem.
The ANSYS input code for Problem 3B.2 is listed below.
/prep7
A = 1
I = A/100
E = 1e6/I
nu = 0.3
et,1,beam188 ! Element type - BEAM188
sectype,1,beam,asec
secdata,A,I,,I,,2*I
mp,ex,1,E
mp,prxy,1,nu
pi = 4*atan(1)
phi = 35/2/180*pi
x = 100*cos(phi)
y = 100*sin(phi)
k,1,0,0,0
k,2,x,-y,0
k,3,0,100,0
k,4,-x,-y
k,5,0,0,100
larc,2,3,1,100
larc,3,4,1,100
latt,1,,1,,5,5,1
lesize,all,,,40
lmesh,all
dk,2,all,0
dk,4,ux,0
dk,4,uy,0
dk,4,uz,0
fk,3,fy,-900
/solu
nlgeom,on
nsubst,100,0,0
outres,all,all
arclen,on
solve
finish
Part 3
editSimulate the buckling of a hinged right-angle frame under both fixed and follower loads from Example 7.4 of Simo and Vu Quoc (1986) and compare your results with those shown in the paper.
The force-displacement diagram is shown in Figure 10. The deformation is illustrated in Figure 11 and 12.
The ANSYS input code for Problem 3B.3 is listed below.
/prep7
et,1,188
E = 7.2e6
I = 2
A = 6
sectype,1,beam,asec
secdata,A,I,,I,,2*I
mp,ex,1,7.2e6
mp,prxy,1,0.3
k,1,0,0,0
k,2,0,120,0
k,3,23,120,0
k,4,26,120,0
k,5,120,120,0
k,6,200,0,0
k,7,0,200,0
l,1,2,5
l,2,3,1
l,3,4,2
l,4,5,4
latt,1,,1,,6,6,1
lmesh,1
latt,1,,1,,7,7,1
lmesh,2,4
d,1,ux,0
d,1,uy,0
d,1,uz,0
d,18,ux,0
d,18,uy,0
d,18,uz,0
esel,s,elem,,7,8
!replace the line below with f,15,fy,-40000 for fixed load
sfbeam,all,,pres,40000/2
esel,all
fini
/solu
nlgeom,on
outres,all,all
arclen,on
nsubst,200
solve
fini
Warning: The arc length method no longer converges with Ansys 13. Try using the stabilization option instead of arclen, on:
stabilize, constant, energy, 0.001, anytime, 0