# Beam Elements (/PROP/BEAM, /PROP/INT_BEAM)

The two beam elements available in Radioss are used on one-dimensional structures and frames. It carries axial loads, shear forces, bending and torsion moments (contrary to the truss that supports only axial loads).

## Classical Beam (/PROP/BEAM)

The default formulation is based on the Timoshenko formulation; therefore, transverse shear strain is taken into account. This formulation can degenerate into the standard Euler-Bernoulli formulation, where transverse shear energy is neglected.

In Radioss, the beam geometry is defined by its cross-section area and by its three cross-section area moments of inertia. The area moments of inertia around local Y-axis and Z-axis are for bending and they can be calculated using:

The area moment of inertia regarding the local X-axis is for torsion. It can simply be obtained by the summation of ${I}_{y}$ and ${I}_{z}$ . The torsion model is only valid for full cross-section where the warping is neglected.

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

- $c$
- Speed of sound
- $a=\frac{1}{2}\mathrm{min}\left(\sqrt{\mathrm{min}\left(4,1+\frac{b}{12}\right)}\cdot {F}_{1},\sqrt{\frac{b}{3}}\cdot {F}_{2}\right)$
- ${F}_{1}=\sqrt{1+2{d}^{2}}-\sqrt{2}d$
- ${F}_{2}=\mathrm{min}\left({F}_{1},\sqrt{1+2{d}_{s}{}^{2}}-\sqrt{2}{d}_{s}\right)$
- $b=\frac{A{L}^{2}}{\mathrm{max}\left({I}_{y},{I}_{z}\right)}$
- $d=\mathrm{max}({d}_{m},{d}_{f})$
- ${d}_{s}=d\cdot \mathrm{max}\left(1,\sqrt{\frac{12}{b}}\cdot \sqrt{1+\frac{12E}{\frac{5}{6}Gb}(1-{I}_{shear})}\right)$

- $0.01{A}^{2}<{I}_{y}<100{A}^{2}$
- $0.01{A}^{2}<{I}_{z}<100{A}^{2}$
- $0.1\left({I}_{y}+{I}_{z}\right)<{I}_{x}<10\left({I}_{y}+{I}_{z}\right)$

- $12{I}_{y}{I}_{z}={A}^{4}$
- ${I}_{x}={I}_{y}+{I}_{z}$

This model also provides good results for circular or ellipsoidal cross-section. For thin-walled cross-sections, the global plasticity model may provide incorrect results. It is not recommended to use a single beam element per line of frame structure. The mass is lumped onto the nodes; therefore, to get a correct mass distribution, a fine mesh is required. This is especially true when dynamic effects are important.

Moreover, in Radioss beam element, the moment does not vary along the beam length. The moment is supposed constant and is evaluated at the beam center, as is the stress.

`X`rotation. For instance, a beam with one node completely blocked, if an axial rotational velocity

`V`is imposed to the other node, then the beam will rotate at a speed of

`V`, but the local system will rotate at a speed of V/2. This may lead to a bad interpretation of results, especially the shear forces and bending moments.

## New Beam (/PROP/INT_BEAM)

The cross-section of the element is defined using up to 100 integration points (Figure 2). The element properties of the cross-section, that is area moments of inertia and area, are computed by Radioss as: