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

Ω

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 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:(1)

A = \sum A_{i} = \sum (d y_{i} d z_{i})

(2)

I_{Z} = \sum A_{i} (y_{i}^{2} + \frac{1}{12} d y_{i}^{2})

(3)

I_{Y} = \sum A_{i} (z_{i}^{2} + \frac{1}{12} d z_{i}^{2})

Minimum Time Step

The minimum time step for a beam element is determined using the following relation:(4)

Δ t = \frac{a L}{c}

Where,

$c$: Speed of sound, $\sqrt{E / ρ}$
$a = \sqrt{1 + 2 d^{2}} - \sqrt{2} d$
$d = \max (d_{m}, d_{f})$

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: (5)

{\dot{ε}}_{x x} = {\dot{ε}}_{x x}^{m} - y_{i} {\dot{X}}_{z} + z_{i} {\dot{X}}_{y}

(6)

{\dot{ε}}_{y y} = {\dot{ε}}_{z z} = 0

(7)

{\dot{ε}}_{x y} = {\dot{ε}}_{x y}^{m} + z_{i} {\dot{X}}_{x}

(8)

{\dot{ε}}_{x z} = {\dot{ε}}_{x z}^{m} - y_{i} {\dot{X}}_{x}

Where,

$\{{\dot{ε}}_{x x}^{m} {\dot{ε}}_{x y}^{m} {\dot{ε}}_{x z}^{m}\}$: Constant strain rate at local axis x
$\{{\dot{X}}_{x} {\dot{X}}_{y} {\dot{X}}_{z}\}$: Curvature rate

Using material constituent relation for the beam, the stress components

\{σ_{x x} σ_{x y} σ_{x z}\}

are obtained. The generalized stress (force, moment) can be easily computed by:(9)

N_{x} = \iint_{A} σ_{x x} d A

(10)

N_{y} = \iint_{A} σ_{x y} d A

(11)

N_{z} = \iint_{A} σ_{x z} d A

(12)

M_{x} = \iint_{A} (y σ_{x z} - z σ_{x y}) d A

(13)

M_{y} = \iint_{A} z σ_{x x} d A

(14)

M_{z} = \iint_{A} - y σ_{x x} d A