Clamped Square Isotropic Plate with Uniform Pressure Load

This example shows how to calculate the deflection of a structural plate under a pressure loading.

The partial differential equation for a thin isotropic plate with a pressure loading is

$\nabla^{2} (D \nabla^{2} w) = - p,$

where $D$ is the bending stiffness of the plate given by

$D = \frac{E h^{3}}{12 (1 - ν^{2})},$

and $E$ is the modulus of elasticity, $ν$ is Poisson's ratio, $h$ is the plate thickness, $w$ is the transverse deflection of the plate, and $p$ is the pressure load.

The boundary conditions for the clamped boundaries are $w = 0$ and $w^{'} = 0$ , where $w^{'}$ is the derivative of $w$ in a direction normal to the boundary.

Partial Differential Equation Toolbox™ cannot directly solve this fourth-order plate equation. Convert the fourth-order equation to these two second-order partial differential equations, where $v$ is the new dependent variable.

$\nabla^{2} w = v$

$D \nabla^{2} v = - p$

You cannot directly specify boundary conditions for both $w$ and $w^{'}$ in this second-order system. Instead, specify that $w^{'}$ is 0, and define $v^{'}$ so that $w$ also equals 0 on the boundary. To specify these conditions, use stiff "springs" distributed along the boundary. The springs apply a transverse shear force to the plate edge. Define the shear force along the boundary due to these springs as $n \cdot D \nabla v = - k w$ , where $n$ is the normal to the boundary, and $k$ is the stiffness of the springs. This expression is a generalized Neumann boundary condition supported by the toolbox. The value of $k$ must be large enough so that $w$ is approximately 0 at all points on the boundary. It also must be small enough to avoid numerical errors due to an ill-conditioned stiffness matrix.

The toolbox uses the dependent variables $u_{1}$ and $u_{2}$ instead of $w$ and $v$ . Rewrite the two second-order partial differential equations using variables $u_{1}$ and $u_{2}$ :

$- \nabla^{2} u_{1} + u_{2} = 0$

$- D \nabla^{2} u_{2} = p$

Create a PDE model for a system of two equations.

model = createpde(2);

Create a square geometry and include it in the model.

len = 10;
gdm = [3 4 0 len len 0 0 0 len len]';
g = decsg(gdm,'S1',('S1')');
geometryFromEdges(model,g);

Plot the geometry with the edge labels.

figure
pdegplot(model,'EdgeLabels','on')
ylim([-1,11])
axis equal
title 'Geometry With Edge Labels Displayed'

PDE coefficients must be specified using the format required by the toolbox. For details, see

The c coefficient in this example is a tensor, which can be represented as a 2-by-2 matrix of 2-by-2 blocks:

$[\begin{array}{cccc} c (1) & c (2) & \cdot & \cdot \\ \cdot & c (3) & \cdot & \cdot \\ \cdot & \cdot & c (4) & c (5) \\ \cdot & \cdot & \cdot & c (6) \end{array}]$

This matrix is further flattened into a column vector of six elements. The entries in the full 2-by-2 matrix (defining the coefficient a) and the 2-by-1 vector (defining the coefficient f) follow directly from the definition of the two-equation system.

E = 1.0e6; % Modulus of elasticity
nu = 0.3; % Poisson's ratio
thick = 0.1; % Plate thickness
pres = 2; % External pressure

D = E*thick^3/(12*(1 - nu^2));

c = [1 0 1 D 0 D]';
a = [0 0 1 0]';
f = [0 pres]';
specifyCoefficients(model,'m',0,'d',0,'c',c,'a',a,'f',f);

To define boundary conditions, first specify spring stiffness.

k = 1e7;

Define distributed springs on all four edges.

bOuter = applyBoundaryCondition(model,'neumann','Edge',(1:4),...
                                     'g',[0 0],'q',[0 0; k 0]);

Generate a mesh.

generateMesh(model);

Solve the model.

res = solvepde(model);

Access the solution at the nodal locations.

u = res.NodalSolution;

Plot the transverse deflection.

numNodes = size(model.Mesh.Nodes,2);
figure
pdeplot(model,'XYData',u(:,1),'Contour','on')
title 'Transverse Deflection'

Find the transverse deflection at the plate center.

numNodes = size(model.Mesh.Nodes,2);
wMax = min(u(1:numNodes,1))

wMax = -0.2763

Compare the result with the deflection at the plate center computed analytically.

wMax = -.0138*pres*len^4/(E*thick^3)

wMax = -0.2760

Partial Differential Equation Toolbox Documentation

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