How to include in-plane displacements in the PDE solver?

Hi there,

There is an example available for calculating the deflection w of a uniformly loaded, isotropic plate using the PDE solver (not the pdetool):

Clamped, Square Isotropic Plate With a Uniform Pressure Load

The boundary conditions are expressed as:

k = 1e7; % spring stiffness
bOuter = applyBoundaryCondition(pdem,'Edge',(1:4), 'g', [0 0], 'q', [0 0; k 0]);

So I wonder,

1. How can plane stresses sigma_x and sigma_y be calculated?
2. How can in-plane displacements (perpendicular to w) be included as boundary conditions in the formulation of the problem?

Also, the solution to the elliptical PDE:

res = solvepde(pdem);

outputs something like:

res = 

    StationaryResults with properties:

      NodalSolution: [100x2 double]
         XGradients: [100x2 double]
         YGradients: [100x2 double]
         ZGradients: [0x2 double]
               Mesh: [1x1 FEMesh]

I assume that XGradient and YGradient are the dw/dx and dw/dy respectively. My other question is then, how can the second derivative d2w/dx2 and d2w/dy2 can be estimated?

Thanks!

Answer 1

[Vaibhav Awale]

Hi Henry,

I will try to answer the questions that I know.

1) You can refer to this example . PDE app is used in this case to find plane stress and strain. However, you can do it using PDE toolbox command line approach as well.

2) I am not sure what you mean by in plane displacement at boundary.

3) Looking at the way the problem is formulated, res.NodalSolution consists of solution to [u1, u2] where u1 = w u2 = \Delta^2 w = v

and XGradient and YGradient are gradients in x and y direction respectively for u1 and u2.

I hope this answers your queries.

Regards,

Vaibhav

[Henry1993x]

Thanks a lot Vaibhav, your answer helps me a lot. Best!