# Flexible Rectangular Beam

**Libraries:**

Simscape /
Multibody /
Body Elements /
Flexible Bodies /
Beams

## Description

The Flexible Rectangular Beam block models a slender beam with a rectangular cross-section that can be solid or hollow. The rectangular beam can have small and linear deformations. These deformations include extension, bending, and torsion. The block calculates the beam cross-sectional properties, such as the axial, flexural, and torsional rigidities, based on the geometry and material properties that you specify.

The geometry of the rectangular beam is an extrusion of its cross-section. The beam
cross-section, defined in the *xy*-plane, is extruded along
the *z*-axis. To define the cross-section, you can specify
its dimensions in the **Geometry** section of the block dialog
box. The figure shows a solid beam and a hollow beam. The reference frame is located at the
centroid of the beam.

This block supports two damping methods and a discretization option to increase the accuracy of the modeling. For more information, see Overview of Flexible Beams.

### Stiffness and Inertia Properties

The block provides two ways to specify the stiffness and inertia properties for a beam. To model a beam made of homogeneous, isotropic, and linearly elastic material, in the **Stiffness and Inertia** section, set the **Type** parameter to `Calculate from Geometry`

. Then specify the density, Young’s modulus, and Poisson’s ratio or shear modulus. See the **Derived Values** parameter for more information about the calculated stiffness and inertia properties.

Alternatively, you can manually specify the stiffness and inertia properties, such as flexural
rigidity and mass moment of inertia density, by setting the **Type**
parameter to `Custom`

. Use this option to model a beam that is made
of anisotropic materials. This option decouples the mechanical properties from the beam
cross section so you can specify desired mechanical properties without capturing the details
of the exact cross section, such as fillet, rounds, chamfers, and tapers.

The manually entered stiffness properties must be calculated with respect to the frame located at the bending centroid. The frame must be in the same orientation as the beam reference frame. The stiffness matrix is

$${\overline{S}}^{b}=\left[\begin{array}{cccc}{S}^{b}& 0& 0& 0\\ 0& {H}_{x}^{b}& -{H}_{xy}^{b}& 0\\ 0& -{H}_{xy}^{b}& {H}_{y}^{b}& 0\\ 0& 0& 0& {H}_{z}^{b}\end{array}\right]$$

where:

$${S}^{b}$$ is the axial stiffness along the beam.

$${H}_{x}^{b}$$ is the centroidal bending stiffness about the

*x*-axis.$${H}_{y}^{b}$$ is the centroidal bending stiffness about the

*y*-axis.$${H}_{z}^{b}$$ is the torsional stiffness.

$${H}_{xy}^{b}$$ is the centroidal cross bending stiffness.

The manually entered inertia properties must be calculated with respect to a frame located at the center of mass. The frame must be in the same orientation as the beam reference frame. The mass matrix includes rotatory inertia

$${\overline{M}}^{c}=\left[\begin{array}{cccccc}{m}^{c}& 0& 0& 0& 0& 0\\ 0& {m}^{c}& 0& 0& 0& 0\\ 0& 0& {m}^{c}& 0& 0& 0\\ 0& 0& 0& {i}_{x}^{c}& -{i}_{xy}^{c}& 0\\ 0& 0& 0& -{i}_{xy}^{c}& {i}_{y}^{c}& 0\\ 0& 0& 0& 0& 0& {i}_{z}^{c}\end{array}\right]$$

where:

$${m}^{c}$$ is the mass per unit length.

$${i}_{x}^{c}$$ is the mass moment of inertia density about the

*x*-axis.$${i}_{y}^{c}$$ is the mass moment of inertia density about the

*y*-axis.$${i}_{z}^{c}$$ is the polar mass moment of inertia density.

$${i}_{xy}^{c}$$ is the mass product of inertia density.

The beam block uses the classical beam theory where the relation between sectional strains and stress resultants is

$$\left[\begin{array}{c}{\overline{F}}_{z}\\ {\overline{M}}_{x}\\ {\overline{M}}_{y}\\ {\overline{M}}_{z}\end{array}\right]={\overline{S}}^{b}\left[\begin{array}{c}{\overline{\epsilon}}_{z}\\ {\overline{\kappa}}_{x}\\ {\overline{\kappa}}_{y}\\ {\overline{\kappa}}_{z}\end{array}\right]$$

where:

$${\overline{F}}_{z}$$ is the axial force along the beam.

$${\overline{M}}_{x}$$ is the bending moment about the

*x*-axis.$${\overline{M}}_{y}$$ is the bending moment about the

*y*-axis.$${\overline{M}}_{z}$$ is the torsional moment about the

*z*-axis.$${\overline{\epsilon}}_{z}$$ is the axial strain along the beam.

$${\overline{\kappa}}_{x}$$ is the bending curvature about the

*x*-axis.$${\overline{\kappa}}_{y}$$ is the bending curvature about the

*y*-axis.$${\overline{\kappa}}_{z}$$ is the torsional twist about the

*z*-axis.

## Ports

### Frame

## Parameters

## References

[1] Shabana, Ahmed A. *Dynamics of Multibody Systems*. Fourth edition. New York: Cambridge University Press, 2014.

[2] Agrawal, Om P., and Ahmed A. Shabana. “Dynamic Analysis of Multibody Systems Using Component Modes.” *Computers & Structures* 21, no. 6 (January 1985): 1303–12. https://doi.org/10.1016/0045-7949(85)90184-1.

## Extended Capabilities

## Version History

**Introduced in R2020a**