COMSOL 6.4 - Bending of a Simply Supported Composite Laminate

Bending of a Simply Supported Composite Laminate

A composite material is a heterogeneous material formed of two or more constituents integrated together to achieve enhanced structural performance. Owing to the improved strength and reduced weight compared to conventional materials, the range of applications of composite materials spans diverse fields. This necessitates a thorough understanding of the behavior of these materials under various loading conditions.

This verification example analyses a simply supported composite plate subjected to a transverse sinusoidal distributed load. The plate is made of three layers with a cross-ply configuration (0/90/0). The analysis is performed using both the layerwise and the Equivalent Single Layer (ESL) theories, and the results are compared with the 3D elasticity solutions (Ref. 1). The estimates of the in-plane normal stress, the transverse normal stress, and the transverse shear stresses in each layer show an exact match with the reference solution.


	Read more about the Composite Materials Module in COMSOL blog, Introduction to the Composite Materials Module.

Model Definition

The model geometry consists of a square composite plate of 200 mm side length. All the edges of the plate are simply supported. In addition to constraining the out-of-plane translation, such support conditions also constrain the in-plane translation in the edge direction.

For the bending analysis, a sinusoidally varying distributed load is applied on the top surface of the plate in the transverse direction. Because of symmetry in geometry and loading, only one quarter of the plate is modeled, with the symmetry boundary conditions at two edges as shown in Figure 1.

Figure 1: Model geometry showing one quarter of the laminated composite plate.

lamina Material Properties

The homogenized orthotropic material properties (Young’s modulus, shear modulus, and Poisson’s ratio) of the lamina are given in Table 1.

Table 1: Material properties of a lamina.
Material property	Value
{E1,E2,E3}	{25e6,1e6,1e6}(psi)
{G12,G23,G13}	{0.5e6,0.2e6,0.5e6}(psi)
{υ12,υ23,υ13}	{0.25,0.25,0.25}

Thickness and Stacking Sequence

The composite plate considered in this example is rather thick, having a thickness to side ratio of 1/4. Thus the total thickness of the laminate is 50 mm.

The laminate consists of three layers in a cross-ply layup. The stacking sequence of the laminate is shown in Figure 2, and the corresponding fiber orientations are given in Table 2. The fiber orientations are presented with respect to the first axis of the laminate coordinate system as shown in Figure 3.

Figure 2: Stacking sequence of laminated composite plate showing fiber orientation in each layer from bottom to top.

Table 2: Fiber orientation in layered material
Layer number	Fiber orientation in layered material (°)
1	0
2	90
3	0

Finite element mesh

The structure (quarter plate) is discretized using a mapped mesh as shown in Figure 4. The distribution of elements along the edges is taken as 2-by-2 (Ref. 1). Additionally, two mesh elements are used in through thickness direction for each material layer of the plate as given in the reference.

Quadratic serendipity shape functions are utilized in the in-plane direction, while cubic serendipity shape functions are employed in the out-of-plane direction within layerwise theory to accurately capture shear stress profiles. In the Equivalent Single Layer (ESL) theory, quadratic Lagrange shape functions are used.

Figure 3: The laminate coordinate system used in the model.

Figure 4: Finite element mesh with 4 elements.

Nondimensional Stress Expressions

The following nondimensional stress expressions are defined in order to compare to the reference solution:

where h is the laminate thickness, a is the side length, and q0 is the transverse load amplitude. The three point locations — (A, A), (A, B), and (B, A) — on the composite plate are shown below:

Figure 5: Points on the laminated composite plate where different stress values are compared.

Results and Discussion

Figure 6 and Figure 7 show the stress in the fiber direction in one quarter of the composite laminate computed using the layerwise theory and the ESL theory, respectively. The distribution of stress in the fiber direction in the full composite laminate is shown in Figure 8.

Figure 6: The stress in the fiber direction computed using the layerwise theory.

You can also compute the through-thickness variation of different stress components using the layerwise and ESL theories. Both theories can be used to study thin to moderately thick laminates. The fact that the layerwise formulation has degrees of freedom available at each layer makes it preferable for the analysis of thick laminates. As the thickness to side ratio for the current problem is 1/4, there is good agreement between the layerwise theory and the stress values computed from 3D elastic solutions (Ref. 1) as shown in Figure 9, Figure 10, Figure 11, and Figure 12.

The transverse shear stresses, in Figure 11 and Figure 12, depict the interlaminar stress values at the junction/interface of the two respective layers. It can be seen that the in-plane normal stress, shown in Figure 9, is discontinuous across the layers and having different stress values in two layers at the junction/interface whereas the transverse shear stress is continuous across the layers. This unique value of transverse shear stress at the junction/interface is known as interlaminar stress, which is one of the deciding factors for interfacial failures like delamination.

Figure 7: The stress in the fiber direction computed using the ESL theory.

Figure 8: The stress in the fiber direction in the full laminate computed using the layerwise theory.

Figure 9: Through-thickness variation of normalized in-plane normal stress (SXX).

Figure 10: Through-thickness variation of normalized transverse normal stress (SZZ).

Figure 11: Through-thickness variation of normalized transverse shear stress (SYZ).

Figure 12: Through-thickness variation of normalized transverse shear stress (SXZ).

Notes About the COMSOL Implementation

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Modeling a composite laminate as a layered shell requires a surface geometry, in general referred to as a base surface, and a node which adds an extra dimension (1D) to the base surface geometry in the surface normal direction. You can use the functionality to model several layers stacked on top of each other having different thicknesses, material properties, and fiber orientations. You can optionally specify the interface materials between the layers, and control the number of through-thickness mesh elements for each layer.

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The third direction for the selected coordinate system in the , , or represents the normal direction of the or physics. This is also the direction in which the layer stacking is interpreted from bottom to top, and therefore, it is crucial to know it during modeling. There are two ways to achieve this:

Using physics symbols: Go to the physics settings, find the section, and select the checkbox. Then go to the material feature, for instance, , to see the normal direction represented by green arrows in the geometry.

Using result templates: When a solution dataset is available, use the result template to plot the normal direction.

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From a constitutive model point of view, you can either use the Layerwise (LW) theory based interface, or the Equivalent Single Layer (ESL) theory based node in the interface.

References

1. J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, Second Edition, CRC Press, 2004.

Composite_Materials_Module/Verification_Examples/simply_supported_composite_laminate

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

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

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

Click

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

Geometry 1

Work Plane 1 (wp1)

In the toolbar, click

In the window for , 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 a/2.

Locate the section. In the text field, type a/2.

In the text field, type a/2.

In the window, right-click and choose .

Definitions

Boundary System 1 (sys1)

In the window, under > click .

In the window for , locate the section.

Find the subsection. From the list, choose .

Global Definitions

Material 1 (mat1)

In the window, under right-click and choose .

Layered Material: [0/90/0]

Right-click and choose .

In the window for , type Layered Material: [0/90/0] in the text field.

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


Layer	Material	Rotation	Value	Thickness	Mesh elements
Layer 1	Material 1 (mat1)	0.0	0 rad	thl	2

Click two times.

In the table, enter the following settings:


Layer	Material	Rotation	Value	Thickness	Mesh elements
Layer 2	Material 1 (mat1)	90	1.5708 rad	thl	2
Layer 3	Material 1 (mat1)	0	0 rad	thl	2

Click to expand the section. In the text field, type 0.6.

Locate the section. Click in the upper-right corner of the section.

Materials

Layered Material Link 1 (llmat1)

In the window, under right-click and choose > .

Global Definitions

Material 1 (mat1)

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}	{G12,G23,G13}	N/m²	Orthotropic
Density	rho	1	kg/m³	Basic

In order to have more accuracy in the through-thickness direction, use element defining quadratic variation of displacement field on the base surface and cubic variation along the through-thickness direction.

Layered Shell (lshell)

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Prescribed Displacement 1

In the toolbar, click