Theory for Section Stiffness
The Section Stiffness node in the Beam interface allows modeling of beams with non-homogeneous cross sections by, instead of geometrical and material properties, directly entering values for the stiffness matrix S.
For a 3D Euler-Bernoulli beam, the relationship between the section forces and the deformation of the beam is given by
(8-1)
including initial normal force Ni; moments Mixl, Miyl, and Mizl; normal strain εi; and curvatures θsixl, θsiyl, and θsizl. Here, S is a 4-by-4 symmetric matrix. For a homogeneous prismatic beam with isotropic material properties, it is defined as
For a 2D Euler-Bernoulli beam, out-of-plane moments are zero, and Equation 8-1 thus simplifies to
with S now being a 2-by-2 symmetric matrix.
For a 3D Timoshenko beam, the relationship between the section forces and the deformation of the beam is given by
(8-2)
with S being a 6-by-6 symmetric matrix. The equation now also includes the initial shear forces Tiyl and Tizl as well as the initial shear strains γiyl and γizl.
For a 2D Timoshenko beam, out-of-plane forces and moments are zero, and Equation 8-2 thus simplifies to
with S now being a 3-by-3 symmetric matrix.
Using the section forces, the virtual work for the beam becomes
where only the relevant terms are kept, depending on the beam formulation and the space dimension.