Finite Element Analysis of Plate and Shell Structure

The plate and shell structure is a structure in which the dimension in the thickness direction is much smaller than the size in the length and width directions. Among them, the straight surface is called a plate, and the curved surface is called the shell. Because the shell considers the surface curvature, it is theoretically much more complicated than the plate. At the same time, the plate as a special case of the shell. In the practical analysis, the plate can be replaced by the shell element.

1.Shell Structures

The shell structure is widely used in automobile, ships, and aerospace fields because of its excellent lightweight and easy-manufacturing properties. The bodies of the car, aircraft, submarine, and pressure vessels belong to shell structure as shown in the figures below.

2. Shell Theory
The shell theory is based on the elasticity mechanics and some engineering assumptions (such as Kirchhoff hypothesis, Kirchhoff-Love assumption, etc.), and studies the stress distribution, deformation, and stability of shell structures under external forces. The shell theory is a more complex theory in engineering mechanics.

3. Finite Element Analysis
For the plate and shell structures, WELSIM offers efficient solutions to evaluate the characteristics quickly. Today we take a simple case to understand the supported features provided by WELSIM.

3.1 CAD Model Creation and Import
In WELSIM, a simple plate geometry model can be created. The graphical interface is as shown in the figure:

You can also import a STEP-format CAD file that contains complex surfaces. Importing a complex surface model is shown below.

For the imported models, the system needs to know the type of the structure. We will change the Structure Type from Solid to Shell in the Properties window.

As the structure type is set to Shell, there will be new properties of thickness and integration points, and the user can define the thickness of the shell.

3.2 Mesh Generation
The current shell solver supports TRI3 element, so we mesh the domain using linear triangle element. Simply set the parameters, and you’ll quickly generate the mesh. There are 263 nodes and 437 triangular elements created in the mesh:

3.3 Static Analysis
For the static analysis of the shell, the boundary and body conditions currently supported by WELSIM are fixed support, fixed rotation, displacement, force, pressure, body force, acceleration, standard Earth gravity, and rotational velocities. Among them, the fixed rotation is specifically for the shell structure.

Here we apply fixed support and rotation to one edge.

Next, we impose a pressure boundary condition with the magnitude of 1e3 on the shell surface.

The results of the shell analysis supports displacement, rotation, stress, strain, reaction forces, and reaction moment. Among them, the rotation and reaction moment are specifically for the shell structure.

The result of total displacement. The maximum magnitude is 2.997e-5.

The result of the total rotation. The maximum magnitude is 1.797e-4 rad.

The von-Mises stress result shows here as well. The maximum stress is 3.861e6. It can be seen that the stress concentration happens around the location of the curved areas.