Large Displacement Solutions

File: PVE-4048, Last Updated: March 2010, By: LB

This solar reflector uses a vacuum to pull the front and back surfaces together to focus the reflective surface. The deflected surface shape can be calculated using FEA, but the correct shape can only be computed with large deflection theory.

Stretched membrane heliostat
A stretched membrane heliostat

Surface model stretched membrane heliostat
A surface model of a stretched membrane heliostat reflector (not the same reflector as in the photo)

For this sample, a 0.064" thick 16ft diameter stainless steel reflector is focused with a 0.1 psi vacuum. This reflector is studied first with linear theory:

0.1 psi vacuum applied to heliostat
A 0.1 psi vacuum is applied to create the focus by stretching the membrane

Linear theory results for heliostat
Initial linear theory results - the displacement is wrong (3235 inches!) - the two surfaces are shown passing through each other

What went wrong? The linear theory assumes that the stiffness of the reflector does not change as its shape changes. As a result the only stress computed is a flat panel bending stress. In reality, the application of the vacuum changes the shape from flat to spherical. After a very small deflection, the membrane stress in the deflected spherical shape is much higher than any bending stress.

Mesh for linear theory
Linear theory - no membrane stresses are reported for the mirror.

FEA Analysis for bending stresses
Linear theory - huge bending stresses are reported.

SolidWorks Simulation suggests using large displacement theory to solve the problem:

Solidworks Linear Static message
Large displacement theory is suggested for this study

From the SolidWorks Simulation help files:

The linear theory assumes small displacements... This approach may lead to inaccurate results or convergence difficulties in cases where these assumptions are not valid... The large displacement solution is needed when the acquired deformation alters the stiffness (ability of the structure to resist loads) significantly... The large displacement solution assumes that the stiffness changes during loading so it applies the load in steps and updates the stiffness for each solution step.

This perfectly describes this reflector. The application of a very small vacuum changes the shape from a flat plate to a curved shape. The correct analysis is membrane not bending.

SolidWorks Simulation applies the pressure in steps. The stiffness of the membrane is recalculated after each step. The large displacement solution takes a lot longer to run.

Solidworks Displacement at 3x
Large displacement theory deflection magnified 3x

Large displacement membrane stress plot
Large displacement membrane stress plot

Membrane stresses - the stresses are approximately those of a sphere (where the stress would be uniform across the whole surface).

Minimal stress results for large displacement theory
Minimal stress is shown in the large displacement theory bending stress results. Bending stresses are almost zero except at the fixed edges.


Graph of Deflection vs Location

A plot of the actual deflection vs the deflection for a true sphere shows that the shape is not truly spherical, which matches the membrane stress plot which shows a non uniform stress distribution. The linear theory plot is different in shape and magnitude.

The SolidWorks Simulation help file has useful information on using large displacement solutions.