Meshed Beam Cross-Sections

Defining the Origin of the Cross-Section

By default, the origin of the cross-section is the origin of the coordinate system used to define the mesh. You can override this default and input the coordinates of the origin directly or specify that the origin coincides with the shear center or centroid of the cross-section. A nondefault origin is particularly useful when the beam node to be used in the actual analysis does not coincide with the origin of the two-dimensional coordinate system.

Defining the Tangent Direction of the Beam Element Centerline for 3-DOF Warping Elements

The beam centerline is the curve that connects the corner nodes of the beam elements to which you assign the properties that you compute with the beam section property generation procedure. By default, Abaqus assumes that the beam centerline tangent direction is parallel to the cross-section normal direction. For 3-DOF warping elements, you can override this default and specify the direction cosines of the beam centerline tangent. Thereby, if the beam centerline tangent is inclined with respect to the cross-section normal direction, you can take into account the effect of this inclination for the beam element cross-section properties.

Defining the Curvatures and the Torsion of the Beam Element Centerline for 3-DOF Warping Elements

By default, Abaqus assumes that the beam centerline is straight and that beam element cross-sections have the same orientation in space. If this is not the case, 3-DOF warping elements can take into account the influence of the curvatures and torsion of the beam centerline on the stiffness properties of the associated beam element cross-sections. For example, you can model a rotor blade that is prebent and pretwisted.

The beam centerline is the curve that connects the beam element nodes. When you consider the projection of the beam centerline onto the plane that is normal to the cross-section 1-direction (or 2-direction), the associated first (or second) precurvature value is the reciprocal radius of the osculating circles. For example, Figure 2 shows the osculating circle of the centerline projection onto the plane that is normal to the 1-direction; the associated first precurvature is $1/R$ .

Precurvature.

The pretorsion is defined as the change in rotation of the beam elements’ cross-section orientations about the beam centerline tangent with respect to the rotation-minimizing frame along the centerline. For example, in Figure 3 the centerline is straight and the rotation-minimizing frames in the cross-section planes are all parallel; the associated pretorsion for the depicted cross-section orientations is $\gamma /\mathrm{\Delta }$ .

Pretorsion.

Requesting Output at Particular Integration Points for 1-DOF Warping Elements

Output to the output database can be recovered during the actual analysis at particular integration points on the cross-section. Requesting output at a large number of cross-sectional points may degrade performance.

Contents of the Output Database

By default, the output database contains field output with warping displacements for one frame for 1-DOF warping elements or for six frames for 3-DOF warping elements. For 1-DOF warping elements, the field output also contains section point strain and stress recovery data for all element integration points. For 3-DOF warping elements, the strain and stress recovery data is contained as strain and stress element output for the six frames. The data in the first, second, and third frames represent the cross-section warping fields for unit beam cross-section forces due to extension, shearing in the 1-axes, and shearing in the 2-axis of the cross-section, respectively. The data in the fourth, fifth, and sixth frames represent the cross-section warping fields for unit beam cross-section torques due to twisting, bending about the 1-axis, and bending about the 2-axis of the cross-section, respectively.

Contents of the jobname.bsp Text File

After the analysis to generate the cross-sectional properties completes, the jobname.bsp text file contains the following lines of data when you use 1-DOF warping elements:

SECTION STIFFNESS
$\left(EA\right)$, ${\left(EI\right)}_{11}$, ${\left(EI\right)}_{12}$, ${\left(EI\right)}_{22}$, $\left(GJ\right)$
SECTION INERTIA 
$\left(\rho A\right)$, ${\left(\rho I\right)}_{11}$, ${\left(\rho I\right)}_{12}$, ${\left(\rho I\right)}_{22}$, $x_{1cm}$, $x_{2cm}$ 
TRANSVERSE SHEAR STIFFNESS 
${\left(GA\right)}_{11}$, ${\left(GA\right)}_{22}$, ${\left(GA\right)}_{12}$ 
CENTROID 
$x_{1c}$, $x_{2c}$ 
SHEAR CENTER 
$x_{1s}$,$x_{2s}$ 
DAMPING, ALPHA=$\left(C_{\alpha }\right)$, BETA=$\left(C_{\beta }\right)$, COMPOSITE=$\left(C_c\right)$ 

The first two lines of data in the jobname.bsp text file correspond to the section property data for an arbitrarily shaped solid general beam cross-section meshed with warping elements (see Defining Linear Section Behavior for Meshed Cross-Sections).

If you requested output at particular integration points in the two-dimensional cross-section model generation, the jobname.bsp text file contains the following additional lines:

SECTION POINTS
section point label, 2D element number, integration point number 
E, $G_1$, $G_2$, $\alpha$, $-\left(x_1-x_{1c}\right)$, $\left(x_2-x_{2c}\right)$, $\left(\frac{\partial \mathrm{\Psi }}{\partial x_1}-\left(x_2-x_{2s}\right)\right)$, $(\frac{\partial \mathrm{\Psi }}{\partial x_2}+(x_1-x_{1s})$ 
...

where the set of two data lines is repeated for as many section points as requested.

When you use 3-DOF warping elements, the jobname.bsp text file contains the following lines of data:

SECTION STIFFNESS, FULL COUPLING
${\left(K\right)}_{11}$, ${\left(K\right)}_{22}$, ${\left(K\right)}_{33}$, ${\left(K\right)}_{44}$, ${\left(K\right)}_{55}$, ${\left(K\right)}_{66}$, ${\left(K\right)}_{12}$, ${\left(K\right)}_{13}$
${\left(K\right)}_{14}$, ${\left(K\right)}_{15}$, ${\left(K\right)}_{16}$, ${\left(K\right)}_{23}$, ${\left(K\right)}_{24}$, ${\left(K\right)}_{25}$, ${\left(K\right)}_{26}$, ${\left(K\right)}_{34}$
${\left(K\right)}_{35}$, ${\left(K\right)}_{36}$, ${\left(K\right)}_{45}$, ${\left(K\right)}_{46}$, ${\left(K\right)}_{56}$
SECTION INERTIA 
$\left(\rho A\right)$, ${\left(\rho I\right)}_{11}$, ${\left(\rho I\right)}_{12}$, ${\left(\rho I\right)}_{22}$, $x_{1cm}$, $x_{2cm}$ 

In this case, the cross-section properties might represent a fully coupled, linear elastic behavior according to the 21 values of the symmetric 6 × 6 stiffness matrix. In general, the elastic response to the different beam section strains, transverse shear, axial stretch, bending, and torsion cannot be separated; and no unique centroid and shear center locations can be defined. Depending on the material properties of the two-dimensional cross-section mesh and the options that are defined for the analysis procedure, some coupling terms might vanish, allowing for a transformation into the representation of the cross-section properties that is used with 1-DOF warping elements. Typically, this is the case only if the following conditions are met:

• The beam element centerline has zero curvature and torsion.
• The beam element centerline tangent direction is aligned with the cross-section normal direction.
• All elements have isotropic elasticity or orthotropic elasticity with one of the elasticity tensor principal axes aligned with the cross-section normal direction.

The corresponding properties are then printed into the jobname.dat data file.

The cross-sectional property information written to the jobname.bsp text file is read into the general beam section definition in the subsequent beam analysis in Abaqus/Standard as described below.

Contents of the Flexible Body Interface File for 3-DOF Warping Elements

When you use 3-DOF warping elements, a flexible body interface (.fbi) file is generated that contains

• the cross-section mesh and its mass and stiffness properties,
• recovery information for reconstructing the in-plane and out-of-plane warping displacements, and
• the associated strain and stress based on beam element cross-section forces and torques.

The .fbi file uses a binary format, and you can assign the file to Simpack Nonlinear SIMBEAM element cross-sections. Simpack expects .fbi file data in SI units. By default, Abaqus assumes that the cross-section mesh is defined in SI units. You can override the default and specify the units used for the cross-section definition. The data are then converted automatically to SI units before it is written to the .fbi file.