# Flexible Rectangular Beam

Rectangular beam with elastic properties for deformation

**Library:**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.

Flexible beams are assumed to be made of a homogeneous, isotropic, and linearly elastic
material. You can specify the beam's density, Young’s modulus, and Poisson’s ratio or shear
modulus in the **Stiffness and Inertia **section of the block
dialog box. Additionally, 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.

## Ports

### Frame

`A`

— Connection frame

frame

Frame by which to connect the beam in a model. In the undeformed configuration, this frame
is at half the beam length in the -*z* direction relative to the origin of
the local reference frame.

`B`

— Connection frame

frame

Frame by which to connect the beam in a model. In the undeformed configuration, this frame
is at half the beam length in the +*z* direction relative to the origin of
the local reference frame.

## Parameters

### Geometry

`Type`

— Choice of solid or hollow cross-section

`Solid`

(default) | `Hollow`

Choice of solid or hollow cross-section:

Select

`Solid`

to model a beam with a solid cross-section.Select

`Hollow`

to model a beam with a hollow cross-section.

`Width`

— Distance between outer vertical faces of beam

1 `m`

(default) | positive scalar

Distance between the outer vertical faces of the beam.

`Height`

— Distance between outer horizontal faces of beam

1 `m`

(default) | positive scalar

Distance between the outer horizontal faces of the beam.

`Thickness`

— Wall thickness of hollow cross-section

0.1 `m`

(default) | positive scalar

Wall thickness of the hollow cross-section. This parameter specifies the distance between the inner and outer surfaces of the beam.

#### Dependencies

To enable this parameter, set **Type** to
`Hollow`

.

`Length`

— Extrusion length of beam

10 `m`

(default) | positive scalar

Extrusion length of the beam. The beam is modeled by extruding the specified
cross-section along the *z*-axis of the local reference frame. The
extrusion is symmetric about the *xy*-plane, with half of the beam
being extruded in the negative direction of the *z*-axis and half in
the positive direction.

### Stiffness and Inertia

`Density`

— Mass per unit volume of material

2700 `kg/m^3`

(default) | positive scalar

Mass per unit volume of material—assumed here to be distributed uniformly throughout the beam. The default value corresponds to aluminum.

`Specify`

— Elastic properties in terms of which to parameterize the beam

`Young's Modulus and Poisson's Ratio`

(default) | `Young's and Shear Modulus`

Elastic properties in terms of which to parameterize the beam. These properties are commonly available from materials databases.

`Young's Modulus`

— Ratio of axial stress to axial strain

70 `GPa`

(default) | positive scalar

Young's modulus of elasticity of the beam. The greater its value, the stronger the resistance to bending and axial deformation. The default value corresponds to aluminum.

`Poisson's Ratio`

— Ratio of transverse to longitudinal strains

0.33 (default) | scalar in the range [0, 0.5)

Poisson's ratio of the beam. The value specified must be greater than or equal to
`0`

and smaller than `0.5`

. The default value
corresponds to aluminum.

`Shear Modulus`

— Ratio of shear stress to engineering shear strain

26 `GPa`

(default) | positive scalar

Shear modulus (or modulus of rigidity) of the beam. The greater its value, the stronger the resistance to torsional deformation. The default value corresponds to aluminum.

`Derived Values`

— Calculated values of mass and stiffness sectional properties

button

Calculated values of the mass and stiffness sectional properties of the beam. Click
**Update** to calculate and display those values.

The properties given include **Centroid** and **Shear
Center**. The centroid is the point at which an axial force extends (or
contracts) the beam without bending. The shear center is that through which a transverse
force must pass to bend the beam without twisting.

The stiffness sectional properties are computed as follows:

**Axial Rigidity**:*E**A***Flexural Rigidity**: [*E**I*_{x},*E**I*_{y}]**Cross Flexural Rigidity**:*E**I*_{xy}**Torsional Rigidity**:*G**J*

The mass sectional properties are computed as follows:

**Mass per Unit Length**:*ρ**A***Mass Moment of Inertia Density**: [*ρ**I*_{x},*ρ**I*_{y}]**Mass Product of Inertia Density**:*ρ**I*_{xy}**Polar Mass Moment of Inertia Density**:*ρ**I*_{p}

The equation parameters include:

*A*— Cross-sectional area*ρ*— Density*E*— Young's modulus*G*— Shear modulus*J*— Torsional constant (obtained from the solution of Saint-Venant's warping partial differential equation)

The remaining parameters are the relevant moments of area of the beam. These are calculated about the axes of a centroidal frame—one aligned with the local reference frame but located with its origin at the centroid. The moments of area are:

*I*_{x},*I*_{y}— Centroidal second moments of area:$$\left[{I}_{x},{I}_{y}\right]=\left[{\displaystyle \underset{A}{\int}{(y-{y}_{c})}^{2}}dA,{\displaystyle \underset{A}{\int}{(x-{x}_{c})}^{2}}dA\right]$$,

*I*_{xy}— Centroidal product moment of area:$${I}_{xy}={\displaystyle \underset{A}{\int}(x-{x}_{c})}(y-{y}_{c})dA$$,

*I*_{p}— Centroidal polar moment of area:$${I}_{P}={I}_{x}+{I}_{y}$$,

where *x*_{c} and
*y*_{c} are the coordinates of the
centroid.

### Damping

`Type`

— Type of damping method

`Proportional`

(default) | `Uniform Modal`

| `None`

Damping method to apply to the beam:

Select

`None`

to model undamped beams.Select

`Proportional`

to apply the proportional (or Rayleigh) damping method. This method defines the damping matrix [*C*] as a linear combination of the mass matrix [*M*] and stiffness matrix [*K*]:$$[C]=\alpha [M]+\beta [K]$$,

where

*α*and*β*are the scalar coefficients.Select

`Uniform Modal`

to apply the uniform modal damping method. This method applies a single damping ratio to all the vibration modes of the beam. The larger the value, the faster vibrations decay.

`Mass Coefficient`

— Coefficient of mass matrix

0 `1/s`

(default) | nonnegative scalar

Coefficient, *α*, of the mass matrix. This parameter defines damping
proportional to the mass matrix [*M*].

#### Dependencies

To enable this parameter, set **Type** to
`Proportional`

.

`Stiffness Coefficient`

— Coefficient of stiffness matrix

0.001 `s`

(default) | nonnegative scalar

Coefficient, *β*, of the stiffness matrix. This parameter defines
damping proportional to the stiffness matrix [*K*].

#### Dependencies

To enable this parameter, set **Type** to
`Proportional`

.

`Damping Ratio`

— Damping ratio for uniform modal damping method

0.01 (default) | unitless nonnegative scalar

Damping ratio, *ζ*, applied to all beam vibration modes in the
uniform modal damping model. The larger the value, the faster beam vibrations decay.

Use

*ζ*= 0 to model undamped beams.Use

*ζ*< 1 to model underdamped beams.Use

*ζ*= 1 to model critically damped beams.Use

*ζ*> 1 to model overdamped beams.

#### Dependencies

To enable this parameter, set **Type** to ```
Uniform
Modal
```

.

**Data Types: **`double`

### Discretization

`Number of Elements`

— Number of beam finite elements

1 (default) | positive integer

Number of finite elements in the beam model. Increasing the number of elements always improves accuracy of the simulation. But practically, at some point, the increase in accuracy is negligible when there are many elements. Additionally, a higher number of elements increases the computational cost and slows down the speed of the simulation.

### Graphic

`Type`

— Graphic to use in the visualization of the beam

`From Geometry`

(default) | `None`

Choice of graphic used in the visualization of the beam. The graphic is by default the
geometry specified for the beam. Change this parameter to
`None`

to eliminate this beam altogether from the model
visualization.

`Visual Properties`

— Parameterizations for color and opacity

`Simple`

(default) | `Advanced`

Parameterization for specifying visual properties. Select
`Simple`

to specify color and opacity. Select
`Advanced`

to add specular highlights, ambient shadows, and
self-illumination effects.

`Color`

— True color as [R,G,B] vector on 0–1 scale

[0.5 0.5 0.5] (default) | three-element vector with values constrained to 0–1

RGB color vector with red (R), green (G), and blue (B) color amounts specified on a 0–1 scale. You can also specify a color by using the color picker.

#### Dependencies

To enable this parameter, set:

**Type**to`From Geometry`

**Visual Properties**to`Simple`

`Opacity`

— Surface opacity as scalar number on 0–1 scale

1.0 (default) | scalar with value constrained to 0–1

Graphic opacity, specified on a scale of 0–1. An opacity of `0`

corresponds to a completely transparent graphic and an opacity of `1`

to a completely opaque graphic.

#### Dependencies

To enable this parameter, set:

**Type**to`From Geometry`

**Visual Properties**to`Simple`

`Diffuse Color`

— True color as [R,G,B,A] vector on 0–1 scale

[0.5 0.5 0.5] (default) | three- or four-element vector with values constrained to 0–1

True color under direct white light specified as an [R,G,B] or [R,G,B,A] vector on a
0–1 scale. An optional fourth element specifies the color opacity also on a scale of
0–1. Omitting the opacity element is equivalent to specifying a value of
`1`

.

#### Dependencies

To enable this parameter, set :

**Type**to`From Geometry`

.**Visual Properties**to`Advanced`

.

`Specular Color`

— Highlight color as [R,G,B,A] vector on 0–1 scale

[0.5 0.5 0.5 1.0] (default) | three-element or four-element vector with values constrained to 0–1

Color of the specular highlights, specified as an [R,G,B] or [R,G,B,A] vector on a 0–1
scale. The optional fourth element specifies the color opacity. Omitting the opacity
element is equivalent to specifying a value of `1`

.

#### Dependencies

To enable this parameter, set:

**Type**to`From Geometry`

**Visual Properties**to`Advanced`

`Ambient Color`

— Shadow color as [R,G,B,A] vector on 0–1 scale

[0.15 0.15 0.15 1.0] (default) | three-element or four-element vector with values constrained to 0–1

Color of shadow areas in diffuse ambient light, specified as an [R,G,B] or [R,G,B,A]
vector on a 0–1 scale. The optional fourth element specifies the color opacity. Omitting
the opacity element is equivalent to specifying a value of `1`

.

#### Dependencies

To enable this parameter, set:

**Type**to`From Geometry`

**Visual Properties**to`Advanced`

`Emissive Color`

— Self-illumination color as [R,G,B,A] vector on 0–1 scale

[0.0 0.0 0.0 1.0] (default) | three- or four-element vector with values constrained to 0–1

Surface color due to self illumination, specified as an [R,G,B] or [R,G,B,A] vector on
a 0–1 scale. The optional fourth element specifies the color opacity. Omitting the
opacity element is equivalent to specifying a value of `1`

.

#### Dependencies

To enable this parameter, set:

**Type**to`From Geometry`

.**Visual Properties**to`Advanced`

.

`Shininess`

— Highlight sharpness as scalar number on 0–128 scale

75 (default) | scalar with value constrained to 0–128

Sharpness of the specular light reflections, specified as a scalar number on a 0–128 scale. Increase the shininess value for smaller but sharper highlights. Decrease the value for larger but smoother highlights.

#### Dependencies

To enable this parameter, set:

**Type**to`From Geometry`

**Visual Properties**to`Advanced`

### Frames

`Show Port A`

— Show port A for connection to other blocks

on (default) | off

Select to expose the **A** port.

`Show Port B`

— Show port B for connection to other blocks

on (default) | off

Select to expose the **B** port.

`New Frame`

— Create custom frame for connection to other blocks

button

Click the Create button to open a pane for creating a new body-attached frame. In this pane, you can specify the name, origin, and orientation for the frame.

To name the custom frame, click the text field of the

**Frame Name**parameter. The name identifies the corresponding port on the beam block and in the tree view pane of the Mechanics Explorer.To select the

**Frame Origin**for the custom frame, use one of the following methods:**At Reference Frame Origin**: Make the new frame origin coincident with the origin of the reference frame of the undeformed beam.**Based on Geometric Feature**: Make the new frame origin coincident with the center of the selected undeformed geometry feature. Valid features include surfaces, lines, and points. Select a feature from the visualization pane, then click**Use Selected Feature**to confirm the location of the origin. The name of the origin location appears in the field below this option.

To define the orientation of the custom frame, under the

**Frame Axes**section, select the**Primary Axis**and**Secondary Axis**of the custom frame and then specify their directions.Use the following methods to select a vector for specifying the directions of the primary and secondary axes. The primary axis is parallel to the selected vector and constrains the remaining two axes to its normal plane. The secondary axis is parallel to the projection of the selected vector onto the normal plane.

**Along Reference Frame Axis**: Selects an axis of the reference frame of the undeformed beam.**Based on Geometric Feature**: Selects the vector associated with the chosen geometry feature of the undeformed beam. Valid features include surfaces and lines. The corresponding vector is indicated by a white arrow in the visualization pane. You can select a feature from the visualization pane and then click**Use Selected Feature**to confirm the selection. The name of the selected feature appears in the field below this option.

`FrameN`

— Edit or delete existing custom frame

frame name

Frames that you have created. `N`

is a unique identifying
number for each custom frame.

Click the text field to edit the name of an existing custom frame.

Click the Edit button to edit other aspects of the custom frame, such as origin and axes.

Click the Delete button to delete the custom frame.

#### Dependencies

To enable this parameter, create a frame by clicking **New
Frame**.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using Simulink® Coder™.

## Version History

**Introduced in R2020a**

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