Flexible Angle Beam

Angle beam with elastic properties for deformation

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

Description

The Flexible Angle Beam block models a slender beam with an L-shaped cross-section, also known as an L-beam. The L-beam consists of one horizontal component and one vertical component, which are called beam legs. The L-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 L-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 an L-beam and its cross-section. The reference frame of the beam is located at the midpoint of the intersection line of the mid-planes of the two legs.

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

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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

`End-to-End Width` — Distance between ends of horizontal leg
1 `m` (default) | positive scalar

Distance between the ends of the horizontal leg.

Note

The End-to-End Width must be larger than the Vertical Leg Thickness.

`End-to-End Height` — Distance between ends of vertical leg
1 `m` (default) | positive scalar

Distance between the ends of the vertical leg.

Note

The End-to-End Height must be larger than the Horizontal Leg Thickness.

`Horizontal Leg Thickness` — Distance between faces of horizontal leg
0.1 `m` (default) | positive scalar

Distance between the two faces of the horizontal leg.

`Vertical Leg Thickness` — Distance between faces of vertical leg
0.1 `m` (default) | positive scalar

Distance between the two faces of the vertical leg.

`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: 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.

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]:
$[C] = α [M] + β [K]$ ,
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. A color picker provides an alternative interactive means of specifying a color.