Geometric Nonlinearity
You can use the beam interface for modeling problems with large displacements and rotations, but small strains. A so-called co-rotational formulation is used. The displacement of each individual beam element is decomposed into a rigid body translation and rotation, and a local response of the rotated element which is linear.
The assumption that the individual element behaves linearly implies that you must use a fine mesh if the curvature of the deformed beam is large. The difference in rotation between the endpoints of the individual element must not be larger than it would be possible to analyze it using linear theory.
Different coordinate systems are needed for describing the beam configurations. The initial configuration of the beam can be described by a triad of orthogonal unit vectors . The first vector is parallel to the beam, and the second and third vectors point in the local y and z directions, respectively. The origin of the local system is taken to be the midpoint of the element. This system translates and rotates with the rigid body motion of the beam, and the new directions of the axes are called .
The rotation of the beam is represented by rotation vectors, θ. The rotation at the midpoint is approximated as the arithmetic mean of the rotations at the nodes,
The rigid rotation is then represented by a rotation matrix Rr, corresponding to this midpoint rotation. It is given by
where is the skew symmetric representation of the midpoint rotation vector.
The axis directions of the co-rotated coordinate system can now be computed as
The position of a point on the rigidly rotated axis of the element can be obtained as
where the local coordinate ξ ranges from 0 to 1, and Xi denotes original node coordinates. xM is the midpoint position, computed as the average of the two nodes,
In addition to the rigid body motion described so far, there are the local deformational displacements with respect to the local rotated beam axes. The deformational displacement can be computed as the difference between the current position and the rigid body position
Here, and in the following, an overline denotes a deformational quantity. The deformational rotation at the nodes is approximated by
These local deformations are interpolated by the same shape functions as described in the previous section:
The total displacement and rotation vectors can be expressed in term of the rigid motion of the local axes, followed by the deformational motion relative to these axes.
A deformational rotation matrix can be defined as
where is the skew-symmetric representation of the deformational rotation vector.
The total rotation vector is computed from a total rotation matrix, R. The total rotation matrix is first composed from the rigid body rotation and the incremental rotation.
The total rotation vector can now be extracted from the total rotation matrix. The magnitude of the rotation vector is first computed as
The full rotation vector is then computed as
To avoid singularity problems when the angle is close to zero, the gamma function is actually used in the expressions, since