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Understanding Geometric Nonlinearities

Impact of Large Deflection on Plate Deformations under Pressure Loading | CAE Associates | FEA Consulting
February 26, 2016 By: Peter Barrett

When performing structural finite element analysis (FEA), there are three types of nonlinearities that the analyst needs to consider including in their simulation. Material nonlinearities are needed to predict plastic strains in metallic parts, cracking or crushing of concrete, or extreme deformation of plastic or rubber materials.  Contact nonlinearities are required to predict change in status and/or sliding friction between assembly parts. The third option, geometric nonlinearity, involves a few different concepts and it is not always easy to identify when it is required.

To better understand all geometric nonlinearities, the following definitions are provided:

Large Deflection: This is probably the most common term used to define geometric nonlinearities. Turning on this option forces the FEA to perform an iterative solution, updating the stiffness matrix based on the incremental nodal displacements at each equilibrium iteration. The deformation may be composed of either rigid body translations and rotations, or significant strains, or a combination of both. Large deflection effects should be included whenever the final geometry or position of the component is significantly different from the initial values. For example, without large deflection active, the displacements of a thin flat plate subjected to external pressure loading will be over-predicted since the in-plane membrane stiffening is not captured by a small deflection analysis (see Figure 1 above).  For these types of problems, if the transverse displacement is on the order of the plate thickness, large deflection effects should be activated. You can never get the wrong answer by turning on the option to include large deflections (NLGEOM is the ANSYS command) and usually it results in a minimal increase in computational effort when material and/or frictional contact nonlinearities exist in your model.

Large Rotations: Large rotations do not have to include large strains, since they can be limited to rigid body rotations. For components undergoing finite rotations, large deflection effects must be included in order to correctly calculate and combine the rigid and flexible deformations and accurately predict the updated geometric stiffness impact on the solution.  As a general rule of thumb, if the angular rotation is greater than 10 degrees, large deformation effects should be included. 

Figure 2: Large Rotation vs. Large Strain

Figure 2 (image on the left) illustrates a single element undergoing large rotation without strain, where large deflection would need to be activated in order to properly capture the response. Pressure loading with large deflections active will follow the displaced geometry as the structure deforms (See Figure 3).

Figure 3: Pressure Loading will follow the Structure with Large Deflections Active

Large Strain:  Large Strain implies the change in shape on the element level, such that individual elements are stretched, squeezed, or sheared in such a way that the final element shape is significantly different from the initial shape. Large Strain is also often referred to as finite strain in structural mechanics text books.  From a practical perspective, large strain effects almost always require the use of a nonlinear material representation which can accurately model the material behavior in the finite strain regime.  For example, when analyzing squeezing and stretching of rubberlike materials, hyperelastic material models (Ogden, Yeoh, Mooney-Rivlin, etc.) are typically used in conjunction with large deformation effects to accurately describe the behavior. Be sure to activate the large deformation option when you are analyzing large strain phenomena, as many finite element codes do not do this automatically. Failure to do so will produce incorrect results. Figure 2 (right image) illustrates a single element undergoing finite strain.

Stress Stiffening: While technically not a geometric nonlinearity, the effects of stress stiffening can be significant, and many times it is lumped together with geometric nonlinearities for convenience. Generally speaking, stress stiffening is the increase (or decrease) in transverse stiffness when a long, thin structure is loaded in tension (or compression) along an axial direction, producing a membrane stress. A thin sheet metal part, a spinning blade or a violin string are some examples.  Even when strains and rigid-body motions are small, significant stiffening (or softening) of the structure can still occur due to membrane stresses, which may play an important role in the accuracy of the analysis.  Stress stiffening is typically important for structures that are small in at least one dimension and/or subject to significant axial or membrane stress.  For these types of problems, if deformations and rotations are small, nonlinear geometric effects can be ignored, and stress stiffening is all that is required to get a correct solution.  However, from a practical perspective, since large deformation effects also include the effects of pre-stress, most people just turn large deformation on and don’t consider stress stiffening independently.

In some large displacement problems, the decision to also include stress-stiffening with large deflection active becomes a convergence tool. In many cases a large deflection solution with stress-stiffening turned on converges faster than when it is not activated. The resulting answers are identical, but turning on pre-stress helps the solution converge better. This works both ways, as I have had a few times where deactivating stress stiffening helped get a nonlinear dynamic buckling beam model to converge more effectively.  

In addition to affecting the transverse response in static analyses, pre-stress effects can change the structure’s natural frequencies and are required in order to perform eigenvalue buckling analyses. Activating the stress stiffening calculation during a static analysis computes the stress induced stiffness matrices that are required input to subsequent eigenvalue buckling simulations or pre-stressed modal analyses.  

Figure 4: Solution Error Caused by Neglecting Large Deflection

So when performing stress analysis, don’t forget to activate the appropriate nonlinear geometric effects when needed, or you may be producing errors you are not aware of.  Figure 4 illustrates a frictional contact and plasticity analysis of a snap fit device where excluding large deflections creates a quite different response. Without close examination or some other check, this error would go unnoticed. Would you be able to tell, just by inspection, that the answer on the right is incorrect?

Can you give me an example of where including geometric nonlinearities had a significant impact on your results?