Large Strain presented the theory for general motion of a material point. Large-rotation theory follows a similar development, except that the logarithmic strain measure (Equation 3–6) is replaced by the Biot, or small (engineering) strain measure:

(3–36)

where:

A corotational (or convected coordinate) approach is used in solving large-rotation/small-strain problems (Rankin and Brogan([67])). "Corotational" may be thought of as "rotated with". The nonlinearities are contained in the strain-displacement relationship which for this algorithm takes on the special form:

(3–37)

where:

The convected element coordinate frame differs from the original element coordinate frame by the amount of rigid body rotation. Hence [T_n] is computed by separating the rigid body rotation from the total deformation {u_n} using the polar decomposition theorem, Equation 3–5. From Equation 3–37, the element tangent stiffness matrix has the form:

(3–38)

and the element restoring force is:

(3–39)

where the elastic strain is computed from:

(3–40)

is the element deformation which causes straining as described in a subsequent subsection.

The large-rotation process can be summarized as a three step process for each element:

The updated transformation matrix [T_n] relates the current element coordinate system to the global Cartesian coordinate system as shown in Figure 3.3: Element Transformation Definitions.

Figure 3.3: Element Transformation Definitions

[T_n] can be computed directly or the rotation of the element coordinate system [R_n] can be computed and related to [T_n] by

(3–41)

where [T_v] is the original transformation matrix. The determination of [T_n] is unique to the type of element involved, whether it is a solid element, shell element, beam element, or spar element.

The displacement field can be decomposed into a rigid body translation, a rigid body rotation, and a component which causes strains:

(3–42)

where:

{u^d} contains both translational as well as rotational degrees of freedom.

The translational component of the deformational displacement can be extracted from the displacement field by

(3–43)

where:

= translational component of the deformational displacement

[Rn] = current element rotation matrix

{xv} = original element coordinates in the global coordinate system

{u} = element displacement vector in global coordinates

{u^d} is in the global coordinate system.

For elements with rotational degrees of freedom, the rotational components of the deformational displacement must be computed. The rotational components are extracted by essentially "subtracting" the nodal rotations {u} from the element rotation given by {u^r}. In terms of the pseudovectors this operation is performed as follows for each node:

Because of the definition of the pseudovector, the deformational rotations extracted in step 3 are limited to less than 30°, since 2sin(θ /2) no longer approximates θ itself above 30°. This limitation only applies to the rotational distortion (that is, bending) within a single element.

Figure 3.4: Definition of Deformational Rotations

Once the transformation [T] and deformational displacements {u^d} are determined, the element matrices Equation 3–38 and restoring force Equation 3–39 can be determined. The solution of the system of equations yields a displacement increment {Δu}. The nodal rotations at the element level are updated with the rotational components of {Δu}. The global rotations (in the output and on the results file) are not updated with the pseudovector approach, but are simply added to the previous rotation in {u_n-1}.

The large-rotation computations in those elements which support it are activated by the large-deformation key (NLGEOM,ON). Stress stiffening is also be included and contributes to the tangent stiffness matrix (which may be required for structures weak in bending resistance).

Stresses (output as S) are engineering stresses in the rotated element coordinate system (the element coordinate system follows the material as it rotates). Strains (output as EPEL, EPPL, etc.) are engineering strains, also in the rotated element coordinate system. This applies to element types that do not have large-strain capability. For element types that have large-strain capability, see Large Strain.

It has been found in many situations that the use of consistent tangent stiffness in a nonlinear analysis can speed up the rate of convergence greatly. It normally results in a quadratic rate of convergence. A consistent tangent stiffness matrix is derived from the discretized finite element equilibrium equations without the introduction of various approximations. The terminology of finite rotation in the context of geometrical nonlinearity implies that rotations can be arbitrarily large and can be updated accurately. A consistent tangent stiffness accounting for finite rotations derived by Nour-Omid and Rankin([177]) for beam/shell elements is used. The technology of consistent tangent matrix and finite rotation makes the buckling and postbuckling analysis a relatively easy task. The theory of finite rotation representation and update has been described in Large Rotation using a pseudovector representation. The following will outline the derivations of a consistent tangent stiffness matrix used for the corotational approach.

The nonlinear static finite element equations solved can be characterized by at the element level by:

(3–45)

where:

N = number of total elements

= element internal force vector in the element coordinate system, generally see Equation 3–46

[Tn]T = transform matrix transferring the local internal force vector into the global coordinate system

= applied load vector at the element level in the global coordinate system

(3–46)

Hereafter, we shall focus on the derivation of the consistent tangent matrix at the element level without introducing an approximation. The consistent tangent matrix is obtained by differentiating Equation 3–45 with respect to displacement variables {u_e}:

(3–47)

It can be seen that Part I is the main tangent matrix Equation 3–38 and Part II is the stress stiffening matrix (Equation 3–34, Equation 3–61 or Equation 3–64). Part III is another part of the stress stiffening matrix (see Nour-Omid and Rankin([177])) traditionally neglected in the past. However, many numerical experiments have shown that Part III of is essential to the faster rate of convergence. In some cases, Part III of is unsymmetric; when this occurs, a procedure of symmetrizing is invoked.

As Part III of the consistent tangent matrix utilizes the internal force vector to form the matrix, it is required that the internal vector not be so large as to dominate the main tangent matrix (Part I). This can normally be guaranteed if the realistic material and geometry are used, that is, the element is not used as a rigid link and the actual thicknesses are input.

It is also noted that the consistent tangent matrix Equation 3–47 is very suitable for use with the arc-length solution method.