Failure Prediction in a Layered Shell

In the tree, select .

In the tree, select .

In the tree, select .

In the tree, select .

Load the text file containing the material properties and material strengths.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Set up three rotated coordinate systems.

Browse to the model’s Application Libraries folder and double-click the file failure_prediction_in_a_layered_shell_materialproperties.txt.

Definitions

Rotated System 2 (sys2)

In the toolbar, click

and choose .

In the window for , locate the section.

Find the subsection. In the α text field, type pi/2.

Rotated System 3 (sys3)

Right-click and choose .

In the window for , locate the section.

Find the subsection. In the α text field, type -pi/4.

Rotated System 4 (sys4)

Right-click and choose .

In the window for , locate the section.

Find the subsection. In the α text field, type pi/4.

Geometry 1

Work Plane 1 (wp1)

In the toolbar, click

Work Plane 1 (wp1)>Plane Geometry

In the window, click .

Work Plane 1 (wp1)>Square 1 (sq1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 1e-2.

button in the toolbar.

Materials

Material 1 (mat1)

In the window, under right-click and choose .

Ply 1

Activate option from button.

button in the toolbar.

In the dialog box, in the tree, select the check box for the node .

The layered shell is modeled using four separate shell interfaces located on the same boundary (mesh surface), sharing the degrees of freedom. The stacking of the shells is done using a option. With this option the constraints and loads are transferred to the actual midplane of the shells without modeling it.

As the same degrees of freedom are to be shared by all shell interfaces, set the displacement field to and the displacement of the shell normals to for Shell 2, Shell 3, and Shell 4.

Set the discretization for the displacement field to in order to resemble the benchmark example.

The results given in the benchmark example are at the midplane of each shell layer. Set the to zero for all shells.

In the window for , type Ply 1 in the text field.

In the text field, type shell1.

Click to expand the section. In the z text field, type 0.

Click to expand the section. From the list, choose .

In the window, under click .

In the window for , locate the section.

In the d text field, type th.

From the list, choose .

In the zoffset text field, type 1.5*th.

Choose the orthotropic solid model for the linear elastic material and assign as .

In the window, click .

In the window for , locate the section.

From the list, choose .

Shell Local System 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Ply 2

In the window, under click .

In the window for , type Ply 2 in the text field.

Locate the section. From the list, choose .

Locate the section. In the z text field, type 0.

Click to expand the section. In the text field, type u.

In the text field, type ar.

In the window, under click .

In the window for , locate the section.

In the d text field, type th.

From the list, choose .

In the zoffset text field, type 0.5*th.

Choose the orthotropic solid model for the linear elastic material and assign as .

In the window, click .

In the window for , locate the section.

From the list, choose .

Shell Local System 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Ply 3

In the window, under click .

In the window for , type Ply 3 in the text field.

Locate the section. From the list, choose .

Locate the section. In the z text field, type 0.

Locate the section. In the text field, type u.

In the text field, type ar.

In the window, under click .

In the window for , locate the section.

In the d text field, type th.

From the list, choose .

In the zoffset text field, type -0.5*th.

Choose the orthotropic solid model for the linear elastic material and assign as .

In the window, click .

In the window for , locate the section.

From the list, choose .

Shell Local System 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Ply 4

In the window, under click .

In the window for , type Ply 4 in the text field.

Locate the section. From the list, choose .

Locate the section. In the z text field, type 0.

Locate the section. In the text field, type u.

In the text field, type ar.

In the window, under click .

In the window for , locate the section.

In the d text field, type th.

From the list, choose .

In the zoffset text field, type -1.5*th.

Select the material properties for the orthotropic material from Table 1.

In the window, click .

In the window for , locate the section.

From the list, choose .

Materials

Material 1 (mat1)

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	{Evector1, Evector2, Evector3}	{E1, E2, E3}	Pa	Orthotropic
Poisson’s ratio	{nuvector1, nuvector2, nuvector3}	{nu12, nu23, nu13}	1	Orthotropic
Shear modulus	{Gvector1, Gvector2, Gvector3}	{G, G, G}	N/m²	Orthotropic
Density	rho	7800	kg/m³	Basic

Ply 1 (shell1)

Safety: Tsai-Wu Orthotropic Criterion

In the window, under click .

In the toolbar, click

and choose .

In the window for , type Safety: Tsai-Wu Orthotropic Criterion in the text field.

Locate the section. From the list, choose .

Safety 2, 3, 4, 5, 6, 7

Create six similar nodes by duplicating the node, and replace the failure criterion as given in the table below:


Name	Failure Criterion
Safety 2	Tsai-Hill
Safety 3	Hoffman
Safety 4	Modified Tsai-Hill
Safety 5	Azzi-Tsai-Hill
Safety 6	Norris
Safety 7	Tsai-Wu anisotropic

Select all nodes under , and right click to . Then, go to under , , and and right click to .

Materials

Material 1 (mat1)

Enter the material properties for the Tsai-Wu Anisotropic criterion as shown in Equation 1 and Equation 2.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Tensile strengths	{sigmats1, sigmats2, sigmats3}	{Sigmats1, Sigmats2, Sigmats3}	Pa	Orthotropic strength parameters, Voigt notation
Compressive strengths	{sigmacs1, sigmacs2, sigmacs3}	{Sigmacs1, Sigmacs2, Sigmacs3}	Pa	Orthotropic strength parameters, Voigt notation
Shear strengths	{sigmass1, sigmass2, sigmass3}	{Sigmass23, Sigmass13, Sigmass12}	Pa	Orthotropic strength parameters, Voigt notation
Second rank tensor, Voigt notation	{F_s1, F_s2, F_s3, F_s4, F_s5, F_s6}	{1/Sigmats1-1/Sigmacs1, 1/Sigmats2-1/Sigmacs2, 1/Sigmats3-1/Sigmacs3, 0, 0, 0}	1/Pa	Anisotropic strength parameters, Voigt notation
Fourth rank tensor, Voigt notation	{F_f11, F_f12, F_f22, F_f13, F_f23, F_f33, F_f14, F_f24, F_f34, F_f44, F_f15, F_f25, F_f35, F_f45, F_f55, F_f16, F_f26, F_f36, F_f46, F_f56, F_f66} ; F_fij = F_fji	{1/(Sigmats1Sigmacs1), -0.5sqrt(1/((Sigmats1Sigmacs1)(Sigmats2Sigmacs2))), 1/(Sigmats2Sigmacs2), -0.5sqrt(1/((Sigmats1Sigmacs1)(Sigmats3Sigmacs3))), -0.5sqrt(1/((Sigmats2Sigmacs2)(Sigmats3Sigmacs3))), 1/(Sigmats3*Sigmacs3), 0, 0, 0, 1/Sigmass23^2, 0, 0, 0, 0, 1/Sigmass13^2, 0, 0, 0, 0, 0, 1/Sigmass12^2}	m²·s^4/kg²	Anisotropic strength parameters, Voigt notation
Density	rho	7800	kg/m³	Basic
Young’s modulus	{Evector1, Evector2, Evector3}	{207e9,7.6e9,7.6e9}	Pa	Orthotropic
Poisson’s ratio	{nuvector1, nuvector2, nuvector3}	{0.3,0,0}	1	Orthotropic
Shear modulus	{Gvector1, Gvector2, Gvector3}	{5e9,5e9,5e9}	N/m²	Orthotropic
Loss factor for orthotropic Young’s modulus	{eta_Evector1, eta_Evector2, eta_Evector3}	{0,0,0}	1	Orthotropic
Loss factor for orthotropic shear modulus	{eta_Gvector1, eta_Gvector2, eta_Gvector3}	{0,0,0}	1	Orthotropic

Ply 1 (shell1)

Fixed Constraint 1

In the toolbar, click

and choose .

Select Point 1 only.

Apply a nodal tensile load of 15 N as an edge load. The load is shared by all shell midplanes, hence it is divided by 4 in order to keep a total value of 15 N.

Edge Load 1

In the toolbar, click

and choose .

Select Edge 4 only.

In the window for , locate the section.

From the list, choose .

Specify the Ftot vector as


Ftotal/4	x
0	y
0	z

Now select and nodes under , and right click to . Then go to , , and ; and right click to .

Ply 2 (shell2)

To enforce a fixed x-direction translation on Node 2, apply the displacement u0 in the x direction to Point 2 of shell2, and the displacement -u0 in the x direction to the same point of shell3. Also add a node under shell3 for the additional degree of freedom u0.

In the window, under click .

Prescribed Displacement/Rotation 1

In the toolbar, click

and choose .

Select Point 2 only.

In the window for , locate the section.

Select the check box.

In the u 0 x text field, type u0.

Ply 3 (shell3)

In the window, under click .

In the toolbar, click

and choose .

Prescribed Displacement/Rotation 1

Select Point 2 only.

In the window for , locate the section.

Select the check box.

In the u 0 x text field, type -u0.

button in the toolbar.

In the dialog box, in the tree, select the check box for the node .

In the table, enter the following settings:

Global Equations 1

In the toolbar, click

and choose .

In the window for , locate the section.


Name	f(u,ut,utt,t) (1)	Initial value (u_0) (1)	Initial value (u_t0) (1/s)	Description
u0		0	0

Locate the section. Click

In the dialog box, type displacement in the text field.

In the tree, select .

Use a single quadrilateral element.

Mesh 1

Free Quad 1

In the toolbar, click

and choose .

Select Boundary 1 only.

Distribution 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type 1.

Switch off the generation of default plots, since each Shell interface will generate three plots by default.

Study 1

In the window, click .

In the window for , locate the section.

Clear the check box.

In the toolbar, click

Results

In the window, expand the node.

Cut Point 3D 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.5e-2.

In the text field, type 0.

Failure indices in Ply 1

In the toolbar, click

In the window for , type Failure indices in Ply 1 in the text field.

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
shell1.emm1.sf1.f_im	1	Tsai-Wu orthotropic failure index, middle
shell1.emm1.sf2.f_im	1	Tsai-Hill failure index, middle
shell1.emm1.sf3.f_im	1	Hoffman failure index, middle
shell1.emm1.sf4.f_im	1	Modified Tsai-Hill failure index, middle
shell1.emm1.sf5.f_im	1	Azzi-Tsai-Hill failure index, middle
shell1.emm1.sf6.f_im	1	Norris failure index, middle
shell1.emm1.sf7.f_im	1	Tsai-Wu anisotropic failure index, middle

Failure indices in Ply 1

In the window, expand the node, then click .

In the window for , type Failure indices in Ply 1 in the text field.

Point Evaluation 2, 3, 4

Create three similar nodes by duplicating the node, and replace the word shell1 in the by shell2, shell3, and shell4 for , , and , respectively. Rename point evaluation nodes and tables appropriately.

Safety factors in Ply 1

In the toolbar, click

In the window for , type Safety factors in Ply 1 in the text field.

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
shell1.emm1.sf1.s_fm	1	Tsai-Wu orthotropic safety factor, middle
shell1.emm1.sf2.s_fm	1	Tsai-Hill safety factor, middle
shell1.emm1.sf3.s_fm	1	Hoffman safety factor, middle
shell1.emm1.sf4.s_fm	1	Modified Tsai-Hill safety factor, middle
shell1.emm1.sf5.s_fm	1	Azzi-Tsai-Hill safety factor, middle
shell1.emm1.sf6.s_fm	1	Norris safety factor, middle
shell1.emm1.sf7.s_fm	1	Tsai-Wu anisotropic failure index, middle

Safety factors in Ply 1

In the window, under click .

In the window for , type Safety factors in Ply 1 in the text field.

Point Evaluation 6, 7, 8

Create three similar nodes by duplicating the node and replace the word shell1 in the by shell2, shell3, and shell4 for , , and , respectively. Rename them appropriately.

Stresses in Ply 1

In the toolbar, click

In the window for , type Stresses in Ply 1 in the text field.

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
shell1.Sl11	N/m^2	Second Piola-Kirchhoff stress, local coordinate system, 11 component
shell1.Sl22	N/m^2	Second Piola-Kirchhoff stress, local coordinate system, 22 component
shell1.Sl12	N/m^2	Second Piola-Kirchhoff stress, local coordinate system, 12 component

Point Evaluation 10, 11, 12

Stresses in Ply 1

In the window, under click .

In the window for , type Stresses in Ply 1 in the text field.

Create three similar nodes by duplicating the node, and replace the word shell1 in the by shell2, shell3, and shell4 for , , and , respectively. Rename them appropriately.

To visualize von Mises stress in the layered shell, use four different plots for four shells in the . Modify the Z component in the node for each surface in order to visualize it better.

von Mises Stress in Stack of Shells

In the toolbar, click

In the window for , type von Mises Stress in Stack of Shells in the text field.

Click to expand the section. From the list, choose .

In the text area, type von-Mises Stress (MPa).

Surface 1

Right-click and choose .

In the window for , locate the section.

In the text field, type round(shell1.mises).

From the list, choose .

Deformation 1

Right-click and choose .

In the window for , locate the section.

In the text field, type w+1.5e-3.

Locate the section. Select the check box.

In the associated text field, type 1.

Surface 2, 3, 4

Create three similar nodes by duplicating the node, and replace the word shell1 in the by shell2, shell3, and shell4 for , , and , respectively. Replace the choice of color table in the subsequent nodes, and also replace the Z component field in the corresponding node with the following choices in the table:


Name	Choice of color table	Z component field expression
Surface 2	Cyclic	w+0.5e-3
Surface 3	Disco	w-0.5e-3
Surface 4	Thermal	w-1.5e-3

von Mises Stress in Stack of Shells

In the window, click .

In the window for , locate the section.

From the list, choose .