# FEA General Theory

## General Information

FEA, Finite Element Analysis, means the calculation performed with finite element method (FEM). FEA is a numerical techinque for solving boundary value problems approximately. It is used for example to optimize material usage and examine strength of structures.

## Yield criterion

Structural analysis is based on strength of materials which is examining the mechanical behaviour of solid bodies under loading. Yield criteria are models for determining when material is yielding or breaking.

The most known yield criteria are

- Maximum Principal Stress Criterion - Suitable for brittle materials, e.g. cast iron.
- Maximum Shear Stress Criterion (Tresca) - Suitable for ductile materials, e.g. common structural steels.
- Maximum Distortion Energy Criterion (Von Mises) - Suitable for ductile materials, e.g. common structural steels.

## Interpolation-based finite element method

The basic equations used in finite element analysis are formulated with interpolation. Due to that the results of analysis will be approximate. For example the used geometry and quality of element mesh have effect on the results accuracy. The finite element analysis for three-dimensional solid or shell structures is possible only with interpolation-based finite element method.

The analyzed part is described with element mesh that consists of three-dimensional, planar or one-dimensional elements e.g. tetrahedron, brick, shell or beam elements. These elements are used to represent the real part as accurately as possible but often the element mesh describes the real part only approximately. This causes errors to analysis results. The errors can be minimized by using more dense and homogenous element mesh.

## Linear and non-linear analysis

The nature of Finite Element Analysis can be linear or non-linear. Linear analysis is usually remarkably faster and easier to perform than non-linear. On the other hand, non-linear analysis gives more realistic results in difficult problems than linear.

Linear analysis is suitable for the service limit state (SLS) check for stiffness and displacement response of structure. It can be also used to analyze the general stress state of structure. Linear analysis is **not** suitable for ultimate limit state check (ULS) because the behaviour of structure in the ultimate state is always non-linear in real life.

Next things cause non-linearity to analysis:

- Material: when the yield strength of material is exceeded its stiffness changes. Because of that the relationship between stresses and deformations becomes non-linear.
- Large displacements or strains in geometry.
- Contacts that lead to dependency between boundary conditions and loading.

Linear finite element analysis includes many assumptions. Due to that the analysis results should always be interpreted with care. These assumptions are:

- The displacements and strains of structure is assumed to be small with respect to the size of structure. For plate structures e.g. the deformations smaller than the plate thickness are considered small.
- The rotational displacements are considered small. For example the rotation of 10 degrees causes error of 1% and the rotation of 30 degrees causes error of 10%.
- The material model is linear elastic. Stresses are linearly dependent on deformations.
- The boundary conditions should be independent of loading.

Vertex FEA structural analysis supports linear Finite Element Analysis.