Chapter 9: Linear Perturbation Analysis
In many engineering applications, the linear behavior of a structure based on a prior linear or nonlinear
preloaded status is of interest. The linear perturbation analysis is designed to solve a linear problem
from this preloaded stage. Typically, in the nonlinear analysis, the Newton-Raphson procedure is used
(see Nonlinear Structural Analysis (p. 193)). The tangent matrix from the Newton-Raphson analysis can
be used in the linear perturbation analysis in order to obtain the preloaded solution, since the linear
stiffness matrix without preloading would not give an accurat e solution.
Generally speaking, the linear perturbation analysis can be any analysis type. However, the program
currently supports only linear perturbation static analyses, linear perturbation modal analyses, linear
perturbation buckling analyses, and linear perturbation full harmonic analyses.
Most current-technology elements are supported in a linear perturbation analysis; see Elements Under
Linear Perturbation in the Element Reference.
The following linear perturbation topics are available:
9.1. Understanding Linear Perturbation
9.2. General Procedure for Linear Perturbation Analysis
9.3. Considerations for Load Generation and Controls
9.4. Considerations for Perturbed Stiffness Matrix Generation
9.5. Considerations for Rotating Structures
9.6. Example Inputs for Linear Perturbation Analysis
9.7. Where to Find Other Examples9.1. Understanding Linear Perturbation
The linear perturbation analysis can be viewed as an iteration on top of a base (or prior) linear or non-
linear analysis. During the linear perturbation process, all of the linear or nonlinear effects from the
base analysis are taken into account and are “frozen” so that the perturbation loads can generate
structural results (such as deformation, stresses, and strains) linearly by using the "frozen" solution
matrices and material properties. The linear or nonlinear effects from the base analysis are also carried
over to the stress expansion pass, if applicable. However, for any downstream analysis, such as a linear
dynamic analysis, only linear effects are accounted for.
If the linear or nonlinear effects from the base analysis are not of interest, there is no need to perform
a linear perturbation analysis; a simple one-step linear or nonlinear analysis can serve that purpose.
Two key points are of fundamental importance for carrying out a linear perturbation analysis:
- The total tangent stiffness matrix from the prior solution (the base analysis) must be obtained for
the current linear perturbation analysis. This matrix is regenerated in the first phase of the linear
perturbation procedure. - The total perturbation loads must be established. This load vector is calculated in the second phase
of the linear perturbation procedure.
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