# Large Displacement Solutions

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.

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:

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.

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

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.

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

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.