# Prestressed Linear Analysis

The response of a structure is affected by its initial state and this is in turn affected by the various preloading/prestressing applied to the structure, prior to the analysis of interest. Examples of prestressed linear analysis include analysis of rotorcraft blades under centrifugal preloading, pillar-like structures under compressive preloading, preload arising from the pretensioning of bolts on a structure, etc. OptiStruct can be used to take into account such preloading or prestressing effects.

• Linear static

Linear statics, Normal modes, Complex eigenvalue, direct frequency response, modal frequency response, direct transient response, and modal transient response analyses. Preloading for a Component Mode Synthesis (CMSMETH) subcase is also supported.

Prestressing Effect and Prestressed Stiffness Matrix

A prestressed stiffness matrix $\overline{K}$, instead of the original stiffness matrix $K$ of the unloaded structure, is used in the prestressed linear analysis to account for the prestressing effect.

When the prestressing subcase is a linear static one, the prestressing is captured or defined by a geometric stiffness matrix, ${K}_{\sigma }$ which is based on the stresses of the preloading static subcase. And this geometric stiffness matrix is augmented with the original stiffness matrix $K$ to form the prestressed stiffness matrix $\overline{K}$.(1) $\overline{K}=K+{K}_{\sigma }$
Depending on preloading conditions, the resulting effect could be a weakened or stiffened structure.
When the prestressing loadcase is a nonlinear quasi-static subcase, the prestressed stiffness matrix $\overline{K}$ does not only include the geometric stiffness matrix, ${K}_{\sigma }$ also accounts for the changes of $K$ due,
• Converged contact status
• Instantaneous elastic property
• Spin softening effect from centrifugal load
• Load stiffness effect from pressure and other follower forces

carried over from the prestressing loadcase to the prestressed loadcase. If the prestressing loadcase is an LGDISP quasi-static subcase, $\overline{K}$ is calculated based on the deformed configuration; otherwise, it is calculated based on the initial configuration.

The stiffness calculations in linear analysis preloaded with small displacement NLSTAT analysis can be controlled via PARAM, KSMNL4PL.

## Prestressed Linear Analysis Types

Below are the different prestressed linear analysis types.

### Static Analysis

Prestressed static analysis is governed by the following equation, where, $f$ is the load vector and $u$ is the displacement.(2) $\overline{K}u=f$

While linear static subcases can have prestressing, nonlinear static subcases under prestressing are not supported.

### Normal Modes Analysis

Prestressed eigenvalue analysis is governed by the following equation, where, $M$ is the mass matrix, $A$ is the eigenvector and $\lambda$ are the eigenvalues.(3) $\left(\overline{K}-\lambda M\right)A=0$

Prestressed eigenvalue analysis is currently supported by AMSES, AMLS and the Lanczos Method. However, if the specified preload is greater than the first critical buckling load, an appropriate error will be reported for AMSES/AMLS runs.

### Complex Eigenvalue Analysis

Prestressed Complex Eigenvalue Analysis is governed by the following equation, where, $M$ is the mass matrix, $A$ is the eigenvector, $C$ is the viscous damping matrix, ${C}_{GE}$ is the element structural damping matrix, ${\alpha }_{f}$ is the coefficient of the extra stiffness matrix, $\omega$ is the loading frequency, $g$ is the global structural damping coefficient, and ${K}_{f}$ is the extra stiffness matrix by direct matrix input.(4) $\left[{\omega }^{2}M+\omega C+\left(\overline{K}\left(1+ig\right)+i\sum {C}_{GE}+{\alpha }_{f}{K}_{f}\right)\right]A=0$

In addition to Prestressed Complex Eigenvalue Analysis implementation, Brake Squeal Analysis can be performed via the STATSUB(BRAKE) command, wherein the contribution of friction to the stiffness matrix is automatically included. For further information, refer to Brake Squeal Analysis in the User Guide.

### Direct Frequency Response Analysis

Prestressed direct FRF analysis is governed by the following equation, where, $M$ is the mass matrix, $u$ is the complex displacement vector, ${C}_{GE}$ is the material damping matrix, $C$ is the viscous damping matrix that includes the Area Matrix for fluid-structure coupling, $f$ is the loading vector, and $g$ is the structural damping.(5) $\left[\overline{K}+ig\overline{K}+i{C}_{GE}+i\omega C-{\omega }^{2}M\right]u=f$

### Modal Frequency Response Analysis

Prestressed modal FRF analysis is governed by the following equation, where, $M$ is the mass matrix, $u$ is the complex displacement vector and $d$ is its corresponding modal value, ${C}_{GE}$ is the material damping matrix, $C$ is the viscous damping matrix that includes the Area Matrix for fluid-structure coupling, $A$ is the set eigenvectors which includes normal modes and residual vectors, $f$ is the loading vector, $\omega$ is the loading frequency, and $g$ is the structural damping coefficient.(6) $\begin{array}{l}\left[{A}^{T}\overline{K}A+ig{A}^{T}\overline{K}A+i{A}^{T}{C}_{GE}A+i\omega {A}^{T}CA-{\omega }^{2}{A}^{T}MA\right]d={A}^{T}f\\ u=Ad\end{array}$

### Direct Transient Response Analysis

Prestressed direct transient analysis is governed by the following equation, where, $M$ is the mass matrix, $u$ is the displacement vector, $C$ is the viscous damping matrix - which also includes structural damping in an approximate way, and $f\left(t\right)$ is the transient load.(7) $M\stackrel{¨}{u}+C\stackrel{˙}{u}+\left(\overline{K}\right)u=f\left(t\right)$

### Modal Transient Response Analysis

Prestressed modal transient analysis is governed by the following equation, where, $M$ is the mass matrix, $u$ is the displacement vector; $d$ is its corresponding modal value, $C$ is the viscous damping matrix - which also includes structural damping in an approximate way, $A$ is the set eigenvectors which includes normal modes and residual vectors, and $f\left(t\right)$ is the transient load.(8) $\begin{array}{l}{A}^{T}MA\stackrel{¨}{u}+{A}^{T}CA\stackrel{˙}{u}+{A}^{T}\overline{K}Au={A}^{T}f\left(t\right)\\ u=Ad\end{array}$

### Component Mode Synthesis (CMSMETH) Subcase

Prestressed Component Mode Synthesis (CMS) uses ($\overline{K}$) as the stiffness matrix, with other aspects such as the mass and damping matrices remaining unchanged in order to calculate the static and normal modes for both flexible body generation (as an input to multibody dynamics) and direct matrix input generation (for external superelements). The primary effect of a preloaded/prestressed CMS analysis will be a frequency shift in the results.

## Results

All results that are supported for regular structural linear analyses are also available in the corresponding prestressed linear analyses.

It is important to note that, while the prestressed linear analysis includes the effects of preloading as a weakening or a stiffening of the structure, the results from the prestressed analysis do not include the preloading results. For example, the displacements from prestressed linear static analysis do not include the preloading displacements. In order to get the overall deflection/stresses of the structure, the displacements/stresses from the prestressed linear analyses have to be carefully superposed with the preloading displacements/stresses while post-processing. Particularly, while post-processing complex results from prestressed direct FRF, the correct approach would be to first obtain the complex results for a certain phase and then superpose the appropriate preloading result. Any other superposing approach would lead to incorrect results.