Beam Elements for Stress Analysis

1. Introduction

The beam elements are designed for static and dynamic, linear and nonlinear analysis. For nonlinear analysis, finite strains are supported.

Beam elements may contain an initial strain, stress, or force and eccentricities. Special means for computing the cross-section constants are available.

[Note] Note

Beam section-specific data are described with the property general-purpose property description. All other beam data are described within in the elements block (see below).

2. Element Types

The following element types are available:

Table 15. Reissner-Simo Beam Elements for Stress Analysis

Element type Remarks
B2.S.RS A two-node linear prismatic Timoshenko beam element with selective under-integration, according to the Reissner/Simo formulation. The shape functions are compatible with the R2.S rod/cable element and with the element edges of linear shell and continuum elements. This element may not be very accurate for pure bending if only a single element is being used to discretize a thin beam.
B3.S.RS A three-node quadratic prismatic Timoshenko beam element with selective under-integration, according to the Reissner/Simo formulation. The shape functions are compatible with the R3.S rod/cable element and with the element edges of quadratic shell and continuum elements.


Table 16. Cubic Beam Elements for Stress Analysis

Element type Remarks
B2.S.E2 A two-node cubic Timoshenko beam element with 2*6 element-internal degrees-of-freedom. The shape functions are not compatible with those of shell or solid elements, except at the beam ends. A straight beam is assumed for the initial configuration. Full integration is conducted with 5-point Gauss-Legendre-Lobatto quadrature. Thus, stresses are calculated also at the beam ends. This element is accurate for pure bending. The behaviour of a cubic Euler-Bernoulli beam element can be achieved by setting the shear_correction_factors in the beam property to a high value.
B4.S A four-node cubic Timoshenko beam element. The shape functions are compatible with the R4.S rod/cable element and with the element edges of cubic shell and continuum elements. Full integration is conducted with 5-point Gauss-Legendre-Lobatto quadrature. Thus, stresses are calculated also at the beam ends. This element is accurate for pure-bending if it is not curved in the initial configuration and if the nodes are equidistant. The behaviour of a cubic Euler-Bernoulli beam element can be achieved by setting the shear_correction_factors in the beam property to a high value.


3. Required element attributes

beam_orientation x y z

Defines the beam orientation vector in the beam-local x-y plane. The same definition will be used for all elements defined hereafter, until a new beam_orientation option is encountered or until the eltype command is specified.

beam_orientation refnode node-id

Defines the beam reference node node-id (external node identifier). For the definition, see generic beam elements. The reference node defines a vector in the beam local x-y plane. The same definition will be used for all elements defined hereafter, until a new beam_orientation option is encountered or until the eltype command is specified.

pid id

Specifies the beam property identifier id pointing to the beam property table. The same definition will be used for all elements defined hereafter, until a new pid option is encountered or until a new element type is specified.

4. Optional element attributes

beam_offsets x1 y1 z1 x2 y2 z2 ...

Defines the beam eccentricities (or offsets) for all nodes defining the beam, for subsequently defined beam elements (for the definition, see generic beam elements). x, y, and z are defined with respect to the branch reference frame. The same definition will be used for all elements defined hereafter, until a new beam_offsets option is encountered or until the eltype command is specified.

beam_dof_release1 dofs | clear

Release part of the element's connections to the degrees-of-freedom at the first node. The degrees-of-freedom to be released are specified w.r.t. the beam-local coordinate system. dofs is a concatenated string containing the degrees of freedom (1, 2, 3, 4, 5, or 6) to be released. clear removes all releases (i.e., nothing is released). The same definition will be used for all elements defined hereafter, until a new beam_dof_release1 option is encountered or until the eltype command is specified.

Example: beam_dof_release1 456 releases all rotational degrees of freedom.

beam_dof_release2 dofs | clear

Release part of the element's connections to the degrees-of-freedom at the second node. The degrees-of-freedom to be released are specified w.r.t. the beam-local coordinate system. dofs is a concatenated string containing the degrees of freedom (1, 2, 3, 4, 5, or 6) to be released. clear removes all releases (i.e., nothing is released). The same definition will be used for all elements defined hereafter, until a new beam_dof_release2 option is encountered or until the eltype command is specified.

Example: beam_dof_release2 4 releases the rotational degree-of-freedom of the beam end node around the beam-local x-direction (the beam axis).

initial_strain_xx e | initial_stress_xx s | initial_force_x f

initial_strain_xx defines an initial strain e for subsequently defined beam elements. The initial strain is assumed constant along the element x-axis. The same definition will be used for all elements defined hereafter, until a new initial_strain_xx option is encountered or until the eltype command is specified.

initial_stress_xx defines an initial stress s for subsequently defined beam elements. The initial stress is assumed constant along the element x-axis. The same definition will be used for all elements defined hereafter, until a new initial_stress_xx option is encountered or until the eltype command is specified.

initial_force_x defines an initial force f acting at the centroid for subsequently defined beam elements. The initial force is assumed constant along the element x-axis. The same definition will be used for all elements defined hereafter, until a new initial_force_x option is encountered or until the eltype command is specified.

group eid

Defines the element group number gid (a non-negative integer number). The default group number is 0. The same definition will be used for all elements defined hereafter, until a new group option is encountered or until the eltype command is specified.

non_structural_mass v

Defines the non-structural mass per unit length. Default is 0. The same definition will be used for all elements defined hereafter, until a new non_structural_mass option is encountered or until the eltype command is specified.

5. Stresses and Strains

Forces, moments, stresses, and strains are stored on database are all expressed in the element-local coordinate system. For linear analysis, the strains and stresses stored on the database are the "engineering" stresses and strains. For nonlinear analysis, the strain is the Green-Lagrange strain, while the stress is the Cauchy stress.

Stresses and strains are evaluated at the Gauss integration points and may be extrapolated to the nodes defining the beam.