Degree of Freedom Numbering

Let us understand this through an example. Consider the plane frame shown in the figure below, lying in the x-y plane. For a plane frame, all joints are considered to be rigid joints, that is, when members bend after loads are applied, the joint rotations are such that the angles between the different members meeting at a joint are the same after applying loads as they were before applying loads. In such a frame, a typical joint may experience linear displacements along x- and y-axes and rotations about the z-axis (ux, uy and rz). That is, by definition, a plane frame lying in the x-y plane before applying loads, must remain in the x-y plane after applying loads. If it experiences displacements which make it move out of the x-y plane, it would be classified as a space frame and not a plane frame. For example, even when all members of the plane frame lie in the x-y plane, applying load along the z-axis makes it a space frame. While the displacements ux, uy and rz are experienced at a typical joint, joints which are supports restrain one or more of these degrees of freedom. Thus the degree of kinematic indeterminacy is three times the number of nodes in the plane frame, less the number of restrained displacement components.

In the Direct Stiffness Method (DSM), every degree of freedom that is free to displace is assigned a number and the total number of displacements that are to be determined is called the degree of kinematic indeterminacy.