General Flexible Beam

Slender extrusion with elastic properties for deformation

Libraries:
Simscape / Multibody / Body Elements / Flexible Bodies / Beams

Description

The General Flexible Beam block models a slender beam of constant, general cross-section that 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 flexible beam is an extrusion of its cross-section. The beam cross-section, defined in the xy-plane, is extruded along the z-axis. You can use this block to create flexible beams with simply or multiply connected cross-sections. For example, you can create the beam shown in the figure by entering these values for the Cross-section in the block's dialog box: {[-0.25,-0.50;0.25,-0.50;0.25,0.50;-0.25,0.50],[-0.15,-0.40;0.15,-0.40;0.15,-0.05;-0.15,-0.05],[-0.15,0.05;0.15,0.05;0.15,0.40;-0.15,0.40]}.

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

${\bar{S}}^{b} = [\begin{matrix} S^{b} & 0 & 0 & 0 \\ 0 & H_{x}^{b} & - H_{x y}^{b} & 0 \\ 0 & - H_{x y}^{b} & H_{y}^{b} & 0 \\ 0 & 0 & 0 & H_{z}^{b} \end{matrix}]$

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.
$H_{x y}^{b}$ 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

${\bar{M}}^{c} = [\begin{matrix} m^{c} & 0 & 0 & 0 & 0 & 0 \\ 0 & m^{c} & 0 & 0 & 0 & 0 \\ 0 & 0 & m^{c} & 0 & 0 & 0 \\ 0 & 0 & 0 & i_{x}^{c} & - i_{x y}^{c} & 0 \\ 0 & 0 & 0 & - i_{x y}^{c} & i_{y}^{c} & 0 \\ 0 & 0 & 0 & 0 & 0 & i_{z}^{c} \end{matrix}]$

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.
$i_{x y}^{c}$ 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

$[\begin{matrix} {\bar{F}}_{z} \\ {\bar{M}}_{x} \\ {\bar{M}}_{y} \\ {\bar{M}}_{z} \end{matrix}] = {\bar{S}}^{b} [\begin{matrix} {\bar{ε}}_{z} \\ {\bar{κ}}_{x} \\ {\bar{κ}}_{y} \\ {\bar{κ}}_{z} \end{matrix}]$

where:

${\bar{F}}_{z}$ is the axial force along the beam.
${\bar{M}}_{x}$ is the bending moment about the x-axis.
${\bar{M}}_{y}$ is the bending moment about the y-axis.
${\bar{M}}_{z}$ is the torsional moment about the z-axis.
${\bar{ε}}_{z}$ is the axial strain along the beam.
${\bar{κ}}_{x}$ is the bending curvature about the x-axis.
${\bar{κ}}_{y}$ is the bending curvature about the y-axis.
${\bar{κ}}_{z}$ is the torsional twist about the z-axis.

Ports

Frame

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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

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Geometry

Cross-section — Cross-section coordinates specified on the `XY` plane
[0.5 0.5; -0.5 0.5; -0.5 -0.5; 0.5 -0.5] `m` (default) | N-by-2 matrix | M-by-1 or 1-by-M cell array of N-by-2 matrices

Coordinates used to specify the boundaries of a beam cross-section. Specify the beam cross-section using one of the following methods:

Use an N-by-2 matrix of xy coordinates to specify a simply connected section. Each row gives the [x,y] coordinates of a point on the cross-section boundary. The points connect in the order given to form a closed polyline. To ensure that the polyline is closed, a line segment is always inserted between the last and first points.
Use an M-by-1 or 1-by-M cell array of N-by-2 matrices of xy coordinates to specify a multiply connected section. The first entry in the cell represents the outer boundary and subsequent entries specify the hole boundaries.

Note

To properly define the cross-section of beams, any two boundaries should not intersect, overlap, or touch.

Additionally, each individual boundary should have:

No repeated vertices.
No self-intersections.
At least three non-collinear points.

Length — Extrusion length of beam
10 `m` (default) | positive scalar

The beam's length. 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

Type — Method to use to specify stiffness and inertia properties
`Calculate from Geometry` (default) | `Custom`

Method to use to specify the stiffness and inertia properties, specified as Calculate from Geometry or Custom.

When you set the parameter to Calculate from Geometry, the block calculates the stiffness and inertia properties based on the specified density, Young’s modulus, and Poisson’s ratio or shear modulus. Set the parameter to Custom to manually specify the stiffness and inertia properties.

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.

Dependencies

To enable this parameter, under the Stiffness and Inertia section, set Type parameter to Calculate from Geometry.

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.

Dependencies

To enable this parameter, under the Stiffness and Inertia section, set Type parameter to Calculate from Geometry.

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.

Dependencies

To enable this parameter, under the Stiffness and Inertia section, set Type parameter to Calculate from Geometry.

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.

Dependencies

To enable this parameter, under the Stiffness and Inertia section, set Type parameter to Calculate from Geometry and set the Specify parameter to Young's Modulus and Poisson's Ratio.

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.

Dependencies

To enable this parameter, under the Stiffness and Inertia section, set Type parameter to Calculate from Geometry and set the Specify parameter to Young's and Shear Modulus.

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: EA
Flexural Rigidity: [EI_x, EI_y]
Cross Flexural Rigidity: EI_xy
Torsional Rigidity: GJ

The mass sectional properties are computed as follows:

Mass per Unit Length: ρA
Mass Moment of Inertia Density: [ρI_x, ρI_y]
Mass Product of Inertia Density: ρI_xy
Polar Mass Moment of Inertia Density: ρI_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:
$[I_{x}, I_{y}] = [\int_{A} {(y - y_{c})}^{2} d A, \int_{A} {(x - x_{c})}^{2} d A]$ ,
I_xy — Centroidal product moment of area:
$I_{x y} = \int_{A} (x - x_{c}) (y - y_{c}) d A$ ,
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.

Geometric Centroid — Centroid of beam cross section
[0 0] `m` (default) | 2-by-1 array

Centroid of the beam cross section, specified as a 2-by-1 array. The array specifies the x and y coordinates of the centroid with respect to the beam reference frame.