Users:General FEM Analysis/Analyses Reference/Static Linear

Revision as of 06:47, 3 September 2010

General Description

The linear static analysis is applied to analyze displacements, strains or stresses of structures. This analysis is valid for linear static mechanical problems without any nonlinearity (geometrically nonlinear, contact, material nonlinear, ...). Static nonlinear problems are can be solved by the static nonlinear analysis whereas transient problems are solved by dynamic analysis methods.

The static linear analysis formulates structural equilibrium on the initial configuration without consideration of deformations. Hence, it is only valid for problems showing small displacements.

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 in the SVN repository under

carat20/examples/benchmark_examples/elements/shell8_quad_lin_canti_I/shell8_canti_2x20_elem_load.dat

http://www.develop.caratplusplus.de/upload_from_server/benchmark_examples/

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.