# 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.

The /PROP/TYPE18 formulation also assumes that the internal virtual work rate is associated with the axial, torsional and shear strains. The other assumptions are:
• 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.

Deformations are computed with respect to the local coordinate system displaced and rotated to take into account rigid body motion. Translational velocities $V$ and angular velocities $\Omega$ with respect to the global reference frame are expressed in the local frame.

## Beam Element Geometry

The five pre-defined standard sections can be used with input Isect:
= 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:

$A=\sum {A}_{i}=\sum \left(d{y}_{i}\text{\hspace{0.17em}}d{z}_{i}\right)$
${I}_{Z}=\sum {A}_{i}\left({y}_{i}{}^{2}+\frac{1}{12}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d{y}_{i}{}^{2}\right)$
${I}_{Y}=\sum {A}_{i}\left({z}_{i}{}^{2}+\frac{1}{12}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d{z}_{i}{}^{2}\right)$

## Minimum Time Step

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

$\text{Δ}t=\frac{aL}{c}$

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

## 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:

${\stackrel{˙}{\epsilon }}_{xx}={\stackrel{˙}{\epsilon }}_{xx}^{m}-{y}_{i}{\stackrel{˙}{X}}_{z}+{z}_{i}{\stackrel{˙}{X}}_{y}$
${\stackrel{˙}{\epsilon }}_{yy}={\stackrel{˙}{\epsilon }}_{zz}=0$
${\stackrel{˙}{\epsilon }}_{xy}={\stackrel{˙}{\epsilon }}_{xy}^{m}+{z}_{i}{\stackrel{˙}{X}}_{x}$
${\stackrel{˙}{\epsilon }}_{xz}={\stackrel{˙}{\epsilon }}_{xz}^{m}-{y}_{i}{\stackrel{˙}{X}}_{x}$

Where,
$\left\{{\stackrel{˙}{\epsilon }}_{xx}^{m}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˙}{\epsilon }}_{xy}^{m}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˙}{\epsilon }}_{xz}^{m}\right\}$
Constant strain rate at local axis x
$\left\{{\stackrel{˙}{X}}_{x}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˙}{X}}_{y}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˙}{X}}_{z}\right\}$
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:

${N}_{x}=\iint {}_{A}\text{\hspace{0.17em}}{\sigma }_{xx}\text{\hspace{0.17em}}dA$
${N}_{y}=\iint {}_{A}\text{\hspace{0.17em}}{\sigma }_{xy}\text{\hspace{0.17em}}dA$
${N}_{z}=\iint {}_{A}\text{\hspace{0.17em}}{\sigma }_{xz}\text{\hspace{0.17em}}dA$
${M}_{x}=\iint {}_{A}\left(y{\sigma }_{xz}-z{\sigma }_{xy}\right)dA$
${M}_{y}=\iint {}_{A}\text{\hspace{0.17em}}z{\sigma }_{xx}\text{\hspace{0.17em}}dA$
${M}_{z}=\iint {}_{A}\text{\hspace{0.17em}}-y{\sigma }_{xx}\text{\hspace{0.17em}}dA$