GENERAL SOLUTION OF STRUCTURAL SYSTEMS
The starting point is always joint equilibrium.
Or,
where A is a load-force transformation matrix and is a function of the geometry of the structure only.
From the element force-deformation equation,
, the joint equilibrium equation can be written as
From the compatibility equation,
joint equilibrium can be written in terms of joint displacements as
(The element deformation-displacement transformation matrix, B, is a function of the geometry of the structure. Of greater significance, however, is the fact that the matrix B is the transpose of the matrix A defined by the joint equilibrium.)
Therefore, the general joint equilibrium can be written as:
The global stiffness matrix K is given by one of the following matrix equations:
It is of interest to note that the equations of equilibrium or the equations of compatibility can be used to calculate the global stiffness matrix K.
The standard approach is to solve Equation 
for the joint displacements and then calculate the member forces from:
It should be noted that within a computer program, the sparse matrices A, B, k and K are never formed because of their large storage requirements. The symmetric global stiffness matrix K is formed and solved in condensed form.
(The C matrix is known as the compliance matrix and can be considered to be the most fundamental definition of the material properties because all terms can be evaluated directly from simple laboratory experiments. Each column of the C matrix represents the strains caused by the application of a unit stress).
Reference
Three-Dimensional Static and Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward L. Wilson
