# Linear Buckling Analysis

The problem of linear buckling in finite element analysis is solved by first applying a reference level of loading, ${f}_{ref}$ , to the structure.

A linear static analysis is then carried out to obtain stresses which are needed to form the geometric stiffness matrix ${K}_{\sigma}$ , corresponding to ${f}_{ref}$ . The buckling loads are then calculated by solving an eigenvalue problem:

Where, $K$ is the stiffness matrix of the structure and $\lambda $ is the multiplier to the reference load. The solution of the eigenvalue problem generally yields $n$ eigenvalues $\lambda $ , where $n$ is the number of degrees-of-freedom (in practice, only a subset of eigenvalues is usually calculated). The vector $A$ is the eigenvector corresponding to the eigenvalue.

The eigenvalue problem is solved using a matrix method called the Lanczos method. Not all eigenvalues are required. Only a small number of the lowest eigenvalues are normally calculated for buckling analysis.

The lowest eigenvalue ${\lambda}_{\alpha}$ is associated with buckling. The critical or buckling load is:

If
${f}_{ref}$
is applied to a preloaded structure and
`STATSUB`(PRELOAD) of the referential linear
static loadcase points to a nonlinear static analysis, the stiffness matrix
$K$
in the buckling eigenvalue problem is the prestressed
stiffness matrix used in the referential linear static loadcase. Accordingly, the
buckling load
${f}_{\sigma}$
is interpreted as a load on the preloaded structure
instead of the unloaded structure.

## Input

To run a linear buckling analysis, an EIGRL Bulk Data Entry needs to be given because it defines the number of modes to be extracted. The EIGRL card needs to be referenced by a METHOD statement in a SUBCASE in the Subcase Information section. In addition, it is necessary to use a STATSUB card to reference the appropriate referential static loading, ${f}_{ref}$ SUBCASE. STATSUB cannot refer to a subcase that uses Inertia Relief.

Buckling analysis will ignore , CELAS1, CELAS2,
CMASS1, CMASS2, CONM1,
CONM2, CBUSH, CVISC,
CDAMP1, CDAMP2, CGAP,
CGAPG, PLOTEL, CWELD,
CSEAM, CFAST, JOINTG,
MPC, and RBE3 elements. These elements can
be used in buckling analysis, but they do not contribute to the geometric stiffness
matrix,
${K}_{\sigma}$
. By default, the contribution from the rigid
elements to the geometric stiffness matrix is not included. You have to add
PARAM,`KGRGD`,YES to the
Bulk Data Entry section to include the contribution of rigid elements to the
geometric stiffness matrix.

The geometric stiffness matrix ${K}_{\sigma}$ calculation for linear buckling considers the temperature-dependent material updated via TEMP(LOAD) or TEMP(MAT).

In addition, through the EXCLUDE Subcase Information Entry, you may decide to omit the contribution of other elements to the geometric stiffness matrix, effectively allowing you to control which parts of the structure are analyzed for buckling. The excluded properties are only removed from the geometric stiffness matrix, resulting in a buckling analysis with elastic boundary conditions. This means that the excluded properties may still be showing movement in the buckling mode.

Buckling analysis with a referential static loading subcase using inertia relief is not supported by default. In such cases, the stiffness matrix is positive semi-definite and the buckling eigenvalue solution ends in singularity. PARAM,INRELBCK,1 or PARAM,INRELBCK,2 can be used to attempt Buckling Analysis based on inertia relief.

## Linear Buckling and Offset Elements

Some one-dimensional and shell elements can use offset to "shift" the element
stiffness relative to the location determined by element's nodes. For example, shell
elements can be offset from the plane defined by element nodes by means of
`ZOFFS`. In this case all other information, such as material
matrices or fiber locations for the calculation of stresses, are given relative to
the offset reference plane. Similarly, shell results, such as shell element forces,
are output on the offset reference plane.

`ZOFFS`on the element card.

Furthermore, in a fully nonlinear approach, additional instability points may be present on the limit load path.