# Finite element calculations

Basics

FEM-Design can perform the following calculations:

- linear static analysis for all structure types,
- static analysis according to second order theory for spatial structures, global stability analysis-buckling shapes and critical loads for spatial structures,
- dynamic analysis-vibration shapes and eigen frequencies for all structure types,
- seismic calculation-response spectra method for 3D models,
- non-linear static analysis-supports resisting only compression,
- cracking analysis-tracking of the cracking process.

# Static analysis

The linear static analysis is the solution of the equation:

K u = Q

linear, inhomogenous equation system with constant coefficients, which is derived from the displacement method, where:

- K is the the coefficient matrix of the system, the so called stiffness matrix,
- Q is the matrix of the load vectors, derived from the loads of every load cases,
- u is the matrix of the displacement of nodes.

FEM-Design contains two equationsystem-solvers. One of them is the so called frontal type, the other is the SKYLINE type solver. Both methods are optimized for the available memory, and they contain very efficient node numbering opti- mization for minimizing of on one hand the front width, on the other hand the band width.

The results of the linear static analysis are always the node displacements, reaction forces and internal forces or stresses of elastic elements.

Note, that an average value is taken for the internal forces of elements in nodes of region or plate where more elements of the same kind are connected, in order to avoid discontinuities that does not exist in reality but generated by the finite element method during the calculation.

# 2nd order analysis

Calculation of structures based on the linear theory mean that the equilibrium conditions are determined according to the shape of the structure before loading. In case of larger deformations the results would be more accurate if the change of structure geometry was taken into consideration.

In case of flexible, elastic structures the approximate solution for this problem is the second order theory which gives satisfying accurate results for practice. In this theory the deformations during the loading are only taken into consideration in the relationship of membrane forces and bending moments. For example, at a straight bar the normal force influences the bending moments because of the deflections perpendicular to the bar, and it modifies of course the deflections. Consequently, the stiffness matrix of the system is a linear function of the normal internal forces (in case of plane plate, membrane forces):

[K + KG (N)] u = Q

where:

- K is the original (linear) stiffness matrix,
- u is the matrix of the node displacement,
- Q is the matrix calculated from the loads,
- KG is the geometrical stiffness matrix. N in the argument means the distribution of the normal (or membrane) forces of the structure.

Since the stiffness is a function of the normal force distribution, the calculation has to be performed in two steps. First, the normal forces of the elements have to be calculated by using the K matrix. In the second step KG can be determined according to the previously calculated N, then the modified displacements, internal forces and stresses can be calculated by the [K + KG] matrix.

It is possible, that the N normal force distribution calculated from the loads happens to result in a singular [K + KG] modified coefficient matrix, which means that the equation system can not be solved. This phenomenon occurs if the load is larger then the critical load of the system which makes it lose stability.

Stability analysis

At description of second order theory it was pointed out that the resultant stiffness of the system depends on the normal force distribution. In case of linear elastic structures the geometrical stiffness matrix is a linear function of normal forces and consequently of loads:

KG (λN) = λ KG (N)

The structure loses its loadbearing capability if the normal forces decrease the stiffness to zero, i.e. the resultant stiffness matrix becomes singular:

det [K + λ KG (N)] = 0

It is an eigenvalue calculation problem, and the smallest λ eigenvalue is the critical load parameter.

The calculation has to be performed in two steps. First, the normal forces of the elements has to be calculated by using the K matrix. In the second step KG and the λ parameter can be determined. The critical load is the product of the load and the λ parameter. The above mentioned eigenvalue problem is solved by the so called Lanczos method in FEM-Design. The results of the calculations are as many buckling shapes as the user required and the matching λ critical load parameters.

# Linear dynamic

If the loads acting on a structure vary quickly, the node displacements of the structure also vary as a function of time. In this case the outer loads - according to the d’Alambert theorem - should be extended by the distributed inertial forces which are proportional to the acceleration of the points of the structures. This results the following basic equation, if the dumping of the structure is ignored:

K u = Q - M u''

where:

- M is the diagonal mass matrix of the structure
- u'' is the matrix of the node acceleration (second derivative of the node displacements and rotations).

If the structure is unloaded, i.e. the free oscillation is analysed, all points of a structure with statically determined supports move periodically, according to the following equation:

u = A sin (ωt)

If Q = 0, it results in the following eigenvalue problem:

[K - ω2 M] A = 0

where:

- ω is the eigen angular frequency and
- A is the matching vibration shapes, or amplitude distribution.

The eigenvalue problem is solved by the so called Lanczos method in the 3D and by the subspace iteration method in the 2D modules of FEM-Design.