# Beam Elements (TYPE3)

Radioss uses a shear beam theory or Timoshenko formulation for its beam elements.

- No cross-section deformation in its plane.
- No cross-section warping out of its plane.

With these assumptions, transverse shear is taken into account.

This formulation can degenerate into a standard Euler-Bernoulli formulation (the cross section remains normal to the beam axis). This choice is under user control.

## Local Coordinate System

The properties describing a beam element are all defined in a local coordinate system.

This coordinate system can be seen in Figure 1. Nodes 1 and 2 of the element are used to define the local X axis, with the origin at node 1. The local Y axis is defined using node 3, which lies in the local XY plane, along with nodes 1 and 2. The Z axis is determined from the vector cross product of the positive X and Y axes.

The local Y direction is first defined at time $$t=0$$ and its position is corrected at each cycle, taking into account the mean rotation of the X axis. The Z axis is always orthogonal to the X and Y axes.

## Beam Element Geometry

- $$A$$
- Cross section area
- ${I}_{x}$
- Area moment of inertia of cross section about local x axis
- ${I}_{y}$
- Area moment of inertia of cross section about local y axis
- ${I}_{z}$
- Area moment of inertia of cross section about local z axis

## Minimum Time Step

Where,

$c$ is the speed of sound: $\sqrt{E/\rho}$,

$a=\frac{1}{2}\mathrm{min}\left(\sqrt{\mathrm{min}\left(4,1+\frac{b}{12}\right)}\cdot {F}_{1},\sqrt{\frac{b}{3}}\cdot {F}_{2}\right)$,

${F}_{1}=\sqrt{1+2{d}^{2}}-\sqrt{2}d$

${F}_{2}=\mathrm{min}\left({F}_{1},\sqrt{1+2{d}_{s}{}^{2}}-\sqrt{2}{d}_{s}\right)$

$b=\frac{A{L}^{2}}{\mathrm{max}\left({I}_{y},{I}_{z}\right)}$

$d=\mathrm{max}({d}_{m},{d}_{f})$

${d}_{s}=d\cdot \mathrm{max}\left(1,\sqrt{\frac{12}{b}}\cdot \sqrt{1+\frac{12E}{\frac{5}{6}Gb}(1-{I}_{shear})}\right)$

## Beam Element Behavior

- Membrane or axial deformation
- Torsion
- Bending about the z axis
- Bending about the y axis

### Membrane Behavior

- $E$
- Elastic modulus
- $l$
- Beam element length
- ${\upsilon}_{x}$
- Nodal velocity in x direction

### Torsion

- $G$
- Modulus of rigidity
- ${\dot{\theta}}_{x}$
- Angular rotation rate

### Bending About Z-axis

Where,

${\varphi}_{y}=\frac{144\left(1+v\right){I}_{z}}{5A{l}^{2}}$,

$\upsilon $ is the Poisson's ratio.

The factor $${\varphi}_{y}$$ takes into account transverse shear.

### Bending About Y-axis

Where, ${\Phi}_{z}=\frac{144\left(1+\nu \right){I}_{y}}{5A{l}^{2}}$.

Like bending about the Z axis, the factor ${\Phi}_{z}$ introduces transverse shear.

## Material Properties

- Elastic
- Elasto-plastic

### Elastic Behavior

The elastic beam is defined using material LAW1 which is a simple linear material law.

The cross-section of a beam is defined by its area $$A$$ and three area moments of inertia ${I}_{x}$, ${I}_{y}$ and ${I}_{z}$.

### Elasto-plastic Behavior

A global plasticity model is used.

However, this model also gives good results for the circular or ellipsoidal cross-section. For tubular or H cross-sections, plasticity will be approximated.