Product: ABAQUS/Standard
The hybrid beam elements in ABAQUS/Standard are designed to handle very slender situations, where the axial stiffness of the beam is very large compared to the bending stiffness; and so a mixed method, where axial force is treated as an independent unknown, is required. For the shear beams mixed elements are provided where the transverse shear forces are also treated as independent unknowns. This section discusses the basis of these mixed methods.
The internal virtual work of the beam can be written
, and write 
is a Lagrange multiplier introduced to impose the constraint 
A linear combination of these expressions is 
(where L is the element length), then the beam is flexible axially and the mixed formulation is unnecessary. Otherwise, we assume that an inverse of the first equation above defines 
from 
:
and the second multiplied by 
, the Newton contribution of the element becomes
is 
is taken as an independent value at each integration point in the element. We choose as 
, where 
is a small value. With this choice, by ensuring that the variables are eliminated after the displacement variables of each element, the Gaussian elimination scheme has no difficulty with solving the equations.In the mixed elements that allow transverse shear (B21H, B22H, B31H, B32H), the transverse shear constraints are imposed by treating the shear forces as independent variables, using the following formulation. The internal virtual work associated with transverse shear is
and 
are shear forces on the section, and 
and 
are variations of transverse shear strain. The virtual work can also be written by introducing independent shear force variables 
and 
, as 
are Lagrange multipliers. As in the axial case, we take a linear combination of these two forms, 
will be defined later. This gives 
ABAQUS treats transverse shear elastically, so 
, where 
is constant. Then the Newton contribution is
We now define 
and choose 
, where 
is a small value compared to , to give
