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Those who are beginning to venture into the demanding world of nonlinear analysis, will find that things are much more complicated than simple linear analysis. The following is a "no maths" explanation of some of the important aspects.
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Convergence
Nonlinear problems are generally solved by an iterative procedure and the level of convergence reached tells us the likely error in the solution. Not all nonlinear problems are guaranteed to converge but often convergence can be helped along by modifying some of the parameters. An example of this is to reduce the amount of initial load into the structure until it has converged, then to progressively increase the load, instead of applying it all at once. Strand7 allows you to set any desired load level for each increment or time step of a nonlinear solution. Alternatively, the automatic load stepping option can be used (in the Solver Defaults) and this instructs Strand7 to reduce the load automatically if convergence becomes difficult to achieve.
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There are a number of ways to check that convergence has been reached. Strand7 uses two convergence criteria:
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Displacements norm: if the structure moves by less than a certain amount
between iterations, then it is assumed that it is not going to move much more and hence
has converged. The tolerance printed by the Strand7 solver is the norm of the incremental displacements in the
current iteration divided by the norm of the total displacements so far. The algorithm
used is called the Updated Lagrangian and is described in many texts on
nonlinear finite element analysis. With this algorithm, the displacements of the current
iteration are added to the total displacements, to obtain the current deformation.
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Residual forces norm: if the out-of-balance forces at the end of the current iteration
are less than a certain amount, then it is assumed that the deformed structure is in
equilibrium. The tolerance printed by the Strand7 solver is the norm of the current out-of-balance
forces divided by the norm of all the externally applied forces.
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A fundamental difference between geometrically linear and geometrically nonlinear analysis is that in linear analysis equilibrium is satisfied on the initial undeformed configuration, whereas in
nonlinear analysis equilibrium must be satified in the deformed configuration. To achieve
final equilibrium in a nonlinear analysis, we solve the problem many times, constantly
adjusting the applied forces based on the current state of equilibrim, and modifying the
geometry based on the current displacements, until we reduce the residual forces to an
acceptable level.
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To illustrate the concept of residual forces, consider the following example. A tip-loaded
cantilever experiences a deflection as shown in the following figure.
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The linear analysis indicates no axial movement in the beam, i.e. no shortening. If we
take the deformed beam as the current configuration and apply the tip load in reverse, a
linear analysis shows that the beam does not return to its original position. A free body
diagram of deformed element 6 of the linear analysis, indicates that the internal loads
are not in equilibrium with the external loads. This is mostly because the length of the
element has apparently increased. At the end of the first iteration, the nonlinear solver sees an
element which has stretched, but no external force to cause the stretching. The residual
loads are the difference between the applied loads and the internal loads. In this case
most of the residual loads arise from Faxial marked below.
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If you actually run this type of analysis in Strand7, you will find that at the end of
the second iteration, the residual loads are quite large. This is due to the apparent axial
strain of the beam after the initial application of the load as shown in the free body
diagram. Because the axial stiffness of a typical beam is much greater than its bending
stiffness, any slight error in the axial strain will generate a large residual load. In
this case, most of the residual load is associated with setting the current length of the
beam back to its correct length. Typical residual force norms reported by Strand7 for
this problem are as shown in the following table.
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If all you are concerned with is the deformed shape, then the residual forces norm can be
relaxed and convergence dictated by the displacement norm only. If however, you are
concerned with the stresses and forces/reactions in the structure, then it is important that
the residual forces norm be enforced to a tight tolerance.
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[Kg]
The Strand7 nonlinear solver panel has a check box for selecting [Kg]. This is only
relevant to nonlinear geometric analysis. [Kg] stands for geometric stiffness matrix. It is sometimes also called the stress stiffness matrix. It is a measure of the change in the lateral stiffness of an element due to membrane forces.
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[Kg] itself does not alter the final equilibrium of the structure. This means that a
converged solution should give identical results irrespective of whether [Kg] is included or
not. It can however aid convergence in problems where there are membrane loads. The
elastica problem shown in the figure below will benefit from the inclusion of [Kg] during the
first couple of increments. After that, the membrane loads are reduced as is the effect of
[Kg].
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For problems where convergence is difficult to achieve, it is sometimes useful to run with and without [Kg] to gauge its effect.
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