Force: Beam

Description

The stiffness properties for the beam are derived using Timoshenko beam theory. The beam axis is assumed to be along the x-axis of the J Reference_Marker. The x-axis of the J Reference_Marker is also defined to be the neutral axis of the undeformed beam. The beam is assumed to undergo small rotational deflections; large rotations are not supported.

Format

<Force_Beam
     id                  = "integer"   
   [ label               = "string" ]     
     i_marker_id         = "integer"     
     j_marker_id         = "integer"     
     length              = "real"     
     E                   = "real"     
     G                   = "real"     
     area                = "real"     
     ixx                 = "real"     
     iyy                 = "real"     
     izz                 = "real"     
     ASY                 = "real"     
     ASZ                 = "real"     
     cratio              = "real"     
     preload_x           = "real"     
     preload_y           = "real"     
     preload_z           = "real"     
     preload_tx          = "real"     
     preload_ty          = "real"     
     preload_tz          = "real"
</Force_Beam>

Attributes

id

Element identification number (integer>0). This number is unique among Force_Beam elements and uniquely identifies the element.

label

A string describing the Force_Beam element.

i_marker_id

Specifies the Reference_Marker at which the force is applied. This is designated as the point of application of the force.

j_marker_id

Specifies the Reference_Marker at which the reaction force and moment is applied. This is designated as the point of reaction of the force. The x-axis of j_marker_id defines the neutral axis of the beam. The y- and z-axes should be oriented along the principal axes of the cross section (area products of inertial are zero). See Comment 2.

length

Specifies the free length of the beam. This is the distance from the origin of j_marker_id to the origin of i_marker_id. The corresponding vector must lie along the x-axis of j_marker_id. See Comment 2 below.

E

Specifies the Young's modulus of the beam material. The beam is assumed to be homogeneous in its material properties. E > 0.

G

Specifies the modulus of elasticity of the beam. This is related to the Young's modulus and POISSON's ratio by the formula:

G = E/2(1+v), where v is POISSON's ratio. G > 0.

area

Specifies the area of the cross section that is orthogonal to the neutral axis of the beam. This is assumed to be constant along the length of the beam. area > 0

ixx

Specifies the torsional stiffness shape factor for the cross section.

For circular sections ixx is equal to the polar moment of inertia . For non-circular sections the torsional stiffness constant is not equal to the polar moment of inertia. It's usually much smaller because of warping effects associated with torsion. See Comment 11 for more information. ixx > 0.

iyy

Defines the second moment of inertia of the beam cross sectional area about an axis on the cross section that is parallel to the y-axis of j_marker_id. iyy > 0.

izz

Defines the second moment of inertia of the beam cross sectional area about an axis on the cross section that is parallel to the z-axis of j_marker_id. izz > 0

ASY

Specifies the shear area ratio in the y direction for Timoshenko beams. This quantity accounts for shear deflection in the Y direction. This is defined as:

(1)

Q_y is the first moment of the cross-sectional area to be sheared by a force in the y direction. l_z is the cross section dimension in the z direction. I_yy is the area moment of inertia about the beam y-axis. To neglect shear deformation in the y-direction, set ASY=0. See Comment 14 for more information. ASY ≥ 0

ASZ

Specifies the shear area ratio in the z direction for Timoshenko beams. This quantity accounts for shear deflection in the Z direction. This is defined as:

(2)

Q_z is the first moment of cross-sectional area to be sheared by a force in the z direction. l_y is the cross section dimension in the y direction. I_zz is the area moment of inertia about the beam z-axis.

To neglect shear deformation in the z-direction, set ASZ=0. See Comment 14 for more information. ASZ > 0.

cratio

Defines the damping ratio for the beam. The beam damping matrix is calculated by multiplying the beam stiffness matrix with the cratio. In other words:

[C] = cratio * [K]

A value of 0.01 (or 1%) is typically used for cratio.

cratio≥ 0.

preload_x, preload_y, preload_z

Define a force in the X, Y, or Z direction(s) that is applied as a preload to the beam. The preload force that is specified will be applied to the beam element in addition to any forces computed along the X, Y, or Z directions during the simulation.

You may specify preload in one, two, or all three directions.

The default for preload in any direction is zero; in other words, the beam is not preloaded by default.

preload_tx, preload_ty, preload_tz

Define a moment about the X, Y, or Z axes that is applied as a preload to the beam. The preload moment that is specified will be applied to the beam element in addition to any moments computed about the X, Y, or Z directions.

You may specify preload about one, two, or all three directions.

The default for preload about any direction is zero; in other words, the beam is not preloaded by default.

Example

The Force_Beam definition for these specifications:

<Force_Beam
     id                  = "7"
     i_marker_id         = "37"
     j_marker_id         = "47"
     length              = "57.55"
     E                   = "200000."
     G                   = "76923.08."
     area                = "314.1593"
     ixx                 = "15707.96"
     iyy                 = "7853.982"
     izz                 = "7853.982"
     ASY                 = "1.2"
     ASZ                 = "1.2"
     cratio              = "0.001">
</Force_Beam>

Comments

1. The Timoshenko beam includes the effect of the shear stresses on transverse deformation; the Euler-Bernoulli beam does not. A constant shear over the beam height is assumed.

   Figure 1 shows the definitions associated with a Force_Beam element.

2. The x-axis of the J_Reference_Marker defines the neutral axis of the beam. The y- and z-axes of the J_Reference_Marker define the principal axes of the cross section. In other words, those directions along which the products of inertia I_yz = 0.
3. At zero deformation, the I_Reference_Marker is offset along the x-axis of J_Reference_Marker by a distance equal to the free length of the beam. It has the same orientation as J_Reference_Marker.

   The following forces act on the beam.

4. Axial forces S1 and S7.
   • Shear forces S2, S3, S8 and S9.
   • Bending moments S5, S6, S11 and S12.
   • Twisting moments S4 and S10.

   
   
   Figure 1. The definition of Force_Beam

5. The three angular deflections, AX(I,J), AY(I,J) and AZ(I,J) must remain small at all times. The angles AX(), AY() and AZ() lose physical significance otherwise, and the beam theory is no longer valid. Small means < 10 degrees.
6. The stiffness matrix is calculated from the inputs according to the following formulae:
   (3)
   
   

   (r_z/l) and (r_y/l) are measures of the slenderness of the beam. For long and slender beams, this ratio is quite small, thus and can be taken as zero. This is accomplished by setting A_sy and A_sz to zero.

   The stiffness matrix (see Przemieniecki, J.S., "Theory of Matrix Structural Analysis", Dover Publications, Inc., ISBN 0-486-64948-2, pages 70-82) can be computed from the physical properties as:

   (4)

   $$K=\left[\begin{array}{cccccc}\frac{EA}{l}& 0& 0& 0& 0& 0\\ 0& \frac{12E{I}_{ZZ}}{l^3\left(1+{\Phi }_Y\right)}& 0& 0& 0& \frac{-6E{I}_{ZZ}}{l^2\left(1+{\Phi }_Y\right)}\\ 0& 0& \frac{12E{I}_{YY}}{l^3\left(1+{\Phi }_Z\right)}& 0& \frac{6E{I}_{YY}}{l^2\left(1+{\Phi }_Z\right)}& 0\\ 0& 0& 0& \frac{G{I}_{XX}}{l}& 0& \\ 0& 0& \frac{6E{I}_{YY}}{l^2\left(1+{\Phi }_Z\right)}& 0& \frac{\left(4+{\Phi }_Z\right)E{I}_{YY}}{l\left(1+{\Phi }_Z\right)}& 0\\ 0& \frac{-6E{I}_{ZZ}}{l^2\left(1+{\Phi }_Y\right)}& 0& 0& 0& \frac{\left(4+{\Phi }_Y\right)E{I}_{ZZ}}{l\left(1+{\Phi }_Y\right)}\end{array}\right]$$
7. The forces acting at I and J are equal and opposite. Since there is a separation between J and I and the force does not act along the separation vector, the torque acting on body I is not the same as the torque acting on body J.
8. The sign convention for the forces and torques is as follows:
   • A positive force tends to repel the I and J Reference_Markers. A negative force tends to attract the I and J Reference_Markers.
   • A positive torque tends to rotate the I Reference_Marker in a counterclockwise direction, relative to the J Reference_Marker. Thus, a positive value of TX tends to increase the value of AX, TY of AY and TZ of AZ.
9. Force_Beam is a linear element. If you wish to define a nonlinear force relationship, then use either the Force_Field or the Force_Vector_TwoBody modeling element.
10. Force_Beam does not model warping, non-homogeneous material, curved beams or beams with non-uniform cross sections.
11. Non-uniform cross sections can be modeled by a group of beams, each having different cross sectional properties.
12. Figure 2 shows an exploded view of a Force_Beam acting between two bodies. The exploded view is used to explain the implementation details of a Force_Beam.
    • Equations 1-4 show how the deformations are calculated.
    • Equations 5-6 defines the force implementation.
    • The last four equations, 7-10, show the forces and torques generated by the Force_Beam element are applied to the two bodies.

    
    
    Figure 2. Implementation of a Force_Beam

13. When the cross section of the beam is not circular, torsional loading on the beam causes the beam to twist. The twisting causes the end sections to warp and radial lines do not necessarily remain straight. The cross section resisting torsion is thus reduced. The torsional shape factor has been calculated for commonly used cross sections.

    For more information, refer to "Formulas for Stress and Strain", Raymond J. Roark and Warren C. Young, McGraw-Hill Book Company. ISBN 0-07-053031-9; Table 20, pages 290-296.

14. For beams with a small span/depth ratio, in other words, with (r_z > 0.3L) or (r_y > 0.3L) shear stresses can cause significant deflection and may in fact dominate the bending effects. The two factors ASY and ASZ in the Force_Beam statement account for shear deflection. The table below shows typical values for ASY and ASZ for cast iron.

    Cross Section

    ASY or ASZ

    Circular

    1.2

    Rectangular

    1.111 (10/9)

    Thin walled, hollow circular section

    2.0

    To obtain ASY and ASZ for other shapes or materials, refer to "Formulas for Stress and Strain", Raymond J. Roark and Warren C. Young, McGraw-Hill Book Company. ISBN 0-07-053031-9; Table 15, pages 200-201.