Users:General FEM Analysis/Analyses Reference/Buckling

General Description

This analysis performs a linear estimation of the buckling load factor. To this purpose the system stiffness K is considered to consist of two parts, the elastic stiffness Kel, which is well known from the field of linear static analysis, and the geometric stiffness part Kgeo which is associated to the current state of stress. The complete stiffness can be determined as K = Kel + Kgeo.

The analysis refers to the load factor γ defined by the user in the input deck. Starting from this load factor a linear dependency of the geometric stiffness with respect to the load augmentation factor λ is assumed, and so the critical load augmentation factor where buckling occurs can be estimated by demanding singularity of the complete stiffness (linearized w.r.t. λ):

As the linear dependency of Kgeo w.r.t. λ is a simplifying assumption, the estimated total load carrying factor γ·λ is the more exact the closer λ is to 1.

static nonlinear analysis

Structure of Equation System

The linear static analysis solves the problem K*U=F, with the linear stiffness matrix K, the unknown displacement fields U and the right hand side vectors F. This system of equations is solved for the unknown displacements by a linear solver. In general, linear solvers may be separated in iterative and direct solvers. Direct solvers transform the stiffness matrix K into an upper diagonal matrix. The displacement field follow by a back substitution with the respective right hand side vector. These back substitution requires only small numerical effort which makes direct solvers very attractive if a couple of displacement fields have to be computed.

Input Parameters

Parameter Description

Example of a Complete Input Block

PC-ANALYSIS 1: STA_GEO_LIN
SOLVER = PC-SOLVER 5
OUTPUT = PC-OUT 1
COMPCASE = LD-COM 1,2,3
DOMAIN = EL-DOMAIN 1

A Full Example

The following example describes a simple cantilever problem discretized by SHELL8 elements. The respective input file can be found here:

http://www.develop.caratplusplus.de/upload_from_server/benchmark_examples/elements/shell8_quad_lin_canti_I/shell8_canti_2x20_elem_load.dat

The problem computes three computation cases (dead load, snow load and pressure load). The boundary conditions are visualized by the figure below.

Simple cantilever problem using SHELL8 elements

The basic goal of each linear static analysis is the computation of the displacement field. For load case 1 this result is depicted in the figure below, whereas the deformation in z-direction is additionally visualized by the color plot.

Displacement of cantilever problem

It can be seen that the support region does not show any deformation whereas the tip region deforms by a value of 8.76.

Often the stress distribution is visualized by color plots. Shell structures require the specification of the layer on which the stresses are computed. The picture below shows the first principle stress on the top of the cantilever.

First principle stress of cantilever problem (top layer)

It can be seen that the stresses show maximum values at the support region of the cantilever. At the tip they are nearly zero. In contrast to analytical results the stresses are not exactly zero in numerical models. The reason is the stress computation at the Gauss points. These points are situated inside the elements and not exactly at the tip of the cantilever.