Nonlinear finite element analyses confront users with many choices. An understanding of the fundamental concepts of
nonlinear finite element analysis is necessary if you do not want to use the finite element program as a black box.
The purpose of this manual is to describe the numerical methods included in Radioss.

The historical shell element in Radioss is a simple bilinear Mindlin plate element coupled with a reduced integration scheme using one integration point.
It is applicable in a reliable manner to both thin and moderately thick shells.

Kinematic constraints are boundary conditions that are placed on nodal velocities. They are mutually exclusive for each degree
of freedom (DOF), and there can only be one constraint per DOF.

The stability of solution concerns the evolution of a process subjected to small perturbations. A process is considered
to be stable if small perturbations of initial data result in small changes in the solution. The theory of stability
can be applied to a variety of computational problems.

A large variety of materials is used in the structural components and must be modeled in stress analysis problems.
For any kind of these materials a range of constitutive laws is available to describe by a mathematical approach the
behavior of the material.

Explicit scheme is generally used for time integration in Radioss, in which velocities and displacements are obtained by direct integration of nodal accelerations.

The performance criterion in the computation was always an essential point in the architectural conception of Radioss. At first, the program has been largely optimized for the vectored super-calculators like CRAY. Then, a first parallel
version SMP made possible the exploration of shared memory on processors.

As for the four node shell element, a simple linear Mindlin Plate element formulation is
used. Likewise, the use of one integration point and rigid body motion given by the time
evolution of the local reference frame is applied. There is no hourglass mode in case of one
integration point.

Local Reference Frame

The local reference frame for the three node shell element is shown in Figure 1.

The vector normal to the plane of the element is defined as:(1)

The method used to calculate the membrane behavior and the membrane strain rates is exactly
the same as that used for four node shell elements (Membrane Behavior).

Bending Behavior

The bending behavior and calculation of the bending strain rates (or curvature rates) is
the exact same method used for four node shell elements (Bending Behavior).

Strain Rate Calculation

The strain rate calculation for the three node shell is the same as the method used for the
four node shell. However, only three nodes are accounted for. This makes the vectors and
matrices smaller. The overall membrane strain rate is calculated by:(15)