# Integrated Beam Elements (TYPE 18)

Beam type /PROP/TYPE18 uses a shear beam theory or Timoshenko formulation like /PROP/TYPE3, but the section inputs (area, inertia) can be default values and can also be discretized by sub-sections; numerical integrations are used to calculate internal forces.

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

Using these assumptions, transverse shear is always considered.

## Local Coordinate System

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

This coordinate system is the same as /PROP/TYPE3 and 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.

In case Node 3 is not defined, Nodes 1 and 2 of the element are used to define the local X axis, with the origin at node 1. Local XY plane is defined using local X axis and Global Z axis (or Global Y axis, if local X is parallel to global Z). The local Y axis is determined from the vector cross product of the positive Z and X axes.

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

## Beam Element Geometry

`I`

_{sect}:

- = 1
- Pre-defined rectangular section
- = 2
- Pre-defined circular section
- = 3
- Pre-defined rectangular section with Gauss-Lobatto quadrature
- = 4
- Pre-defined circular section with Gauss-Lobatto quadrature
- = 5
- Pre-defined circular section

Sub-sections can be used as input as well, in that case, moments of inertia and area, are computed by Radioss as:

## Minimum Time Step

The minimum time step for a beam element is determined using the following relation:

- $c$
- Speed of sound, $\sqrt{E/\rho}$
- $a=\sqrt{1+2{d}^{2}}-\sqrt{2}d$
- $d=\mathrm{max}\text{\hspace{0.17em}}\text{\hspace{0.17em}}({d}_{m},{d}_{f})$

## Beam Element Behavior

The internal force and moment are computed in /PROP/TYPE18 by numerical integration as for a shell element, where each integration point i is computed:

- $\left\{{\dot{\epsilon}}_{xx}^{m}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\dot{\epsilon}}_{xy}^{m}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\dot{\epsilon}}_{xz}^{m}\right\}$
- Constant strain rate at local axis x
- $\left\{{\dot{X}}_{x}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\dot{X}}_{y}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\dot{X}}_{z}\right\}$
- Curvature rate

Using material constituent relation for the beam, the stress components $\left\{{\sigma}_{xx}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma}_{xy}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma}_{xz}\right\}$ are obtained. The generalized stress (force, moment) can be easily computed by: