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More on Nonlinear Analysis
Linear or Non Linear ?

Users often ask the question: "Is this problem Linear or should I consider the use of the Geometric Nonlinear Solver ?". To answer this question we need to go back and look at how both the linear and geometric nonlinear solvers work.

Linear Solver - The linear solver assembles a set of equations [k]{d}={P} and solves these to get the displacements {d}. The stresses and strains etc are then calculated based on these displacements. These equations and hence the solution are based on the original undeformed geometry of the structure (as modelled).

The fact that the linear solution is based on the initial geometry will mean that in some cases the distribution of loads within the structure is wrong as the examples to follow will illustrate.

Nonlinear Solver - The geometric nonlinear solver works in a completely different manner to the linear solver. The load is usually applied gradually in a number of increments. For each load increment the solver iterates, updating the stiffness matrix to reflect the current deformed geometry and resolving, until the external forces are in equilibrium with the internal stresses in the deformed structure.

The important implication of this is that the stress state of the structure depends on the deformed shape. The internal load distribution is a function of deformation and is not based on the initial geometry.

There are two classic problems that can be used to illustrate the difference between the two solvers.

Cantilever Beam - If the deflection of a cantilever beam is small, as it is in most engineering structures, then engineer's theory of bending applies and the solution is linear. If on the other hand the deflections are large we get the situation shown below.

Nonlinear Bending Behaviour

The moment in the beam from a linear static analysis is PL. It is evident from the figure that as the beam deflects the point of force application moves toward the wall and thus the moment is reduced to PL1 (the result in a nonlinear analysis). In addition to this a component of the applied load now acts along the beam and thus the beam will have an axial force. The axial force in the linear static analysis is zero.

Thin Plate or Membrane with Normal Pressure - Another common example of nonlinear behaviour is a thin flat plate or membrane loaded with a normal pressure. When structures of this type are loaded the membrane deflects and the load is carried almost entirely by membrane (tension) action. Typically the bending stiffness of thin plates or membranes is very low or zero and thus there is little bending resistance to the loading. The load state in a loaded membrane is as follows:

Membrane Structure

Structures of this type are normally modelled as flat plate elements in the initial unloaded plane. If we run the linear static solver the analysis will be based on this initial undeformed (flat geometry). Membrane loads cannot develop and the pressure will be reacted by bending in the plate elements only. Since the plate elements have little bending stiffness the deflections will be large and unrealistic.

This problem should be run using the geometric nonlinear solver. This solver continually updates the stiffness matrix to reflect the current deformed geometry of the membrane. As the structure deflects the membrane loads will develop along with the deflections.

So the answer to the question of linear vs nonlinear is that if the geometry of the structure changes appreciably as the structure is loaded and/or the load redistributes itself in the structure as the structure deflects (say from bending to membrane) then you should consider a nonlinear analysis. What this means in practical terms is that if the deflections in a plate structure are equal to or exceed the thickness or depth of the plate or beam then a nonlinear analysis should be considered. Obviously there is no definitive rule that defines linear and nonlinear behaviour. The degree of nonlinearity increases with the magnitude of the applied load and hence the deflection.

There is one other nonlinear effect that we should mention in passing. This is the effect of large deflections on the direction of load application. Consider two cases:

1. If a structure is loaded by a gravity force then this force will always act in the same direction irrespective of the deflection of the structure.
2. When a membrane is loaded by pressure, the pressure load always acts normal to the local surface of the membrane.

In finite element analysis loads whose orientation changes as the structure deflects are often called follower or nonconservative forces. In STRAUS the plate pressures, edge stresses and beam local UDLs follow the deflection of the structure.

For more information please contact us by e-mail: hsh@iperv.it