ABSTRACT
Rigid jointed frames are often used in buildings. They resist the combined effects of horizontal and vertical loads. They derive their strength from the moment interactions between the beams and the columns at the rigid joints. As a result, the elements are subjected not only to bending but also to axial force. Such elements are referred to as beam-column elements. Their nodal displacements include both translations and rotation (u, v,θ), as shown in Figure 4.1. In total, there are six degrees of freedom
{de} = {u1, v1,θ1, u2, v2,θ2}T (4.1)
corresponding to six nodal loads
{Fe} = {Fx1,Fy1,M1,Fx2,Fy2,M2}T (4.2)
If we assume that the deformations are infinitesimally small, and the material is linear elastic, then the axial displacements of the beam-column element do not interact with the bending deformations. Consequently, the principle of superposition applies, and the displacements, forces, and stiffness matrix of the beam-column element can be obtained by simply adding the respective matrices of a truss element, Equation (2.10), and that of a beam element, Equation (3.30)
[Ke] =
⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
AE/L 0 0 −AE/L 0 0 0 12EI/L3 6EI/L2 0 −12EI/L3 6EI/L2 0 6EI/L2 4EI/L 0 −6EI/L2 2EI/L
−AE/L 0 0 AE/L 0 0 0 −12EI/L3 −6EI/L2 0 12EI/L3 −6EI/L2 0 6EI/L2 2EI/L 0 −6EI/L2 4EI/L
⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(4.3)
Sometimes a designer may specify an internal hinge in a frame, which results in a zero value for the bending moment. To account for the presence of a hinge, the stiffness matrix can be obtained by superimposing the respective matrices of a truss element, Equation (2.10), and that of a beam element with a hinge at its right end, Equation (3.52), or a hinge at its left end, Equation (3.53).
