The structure is idealized as the composition of a number of finite elements rather than as a continuum of pieces.

The structure’s external forces (or loads) are represented by a column matrix (or vector) R for each node 1,2, … etc..

The structure’s corresponding nodal displacements are represented by a column matrix r for each node 1, 2, … etc.

The structure’s internal end forces are represented by a column matrix Q for each members a, b, … etc.

The structure’s corresponding internal nodal displacements are represented by a column matrix q for members a, b, … etc.

where the superscript a, b, c, … etc. denote the element while the subscript i denotes the near-end node and i and the far-end node of the element.

For each element, the internal nodal displacements caused by external forces (loads) are threefold: (1) rotation at the near-end node i, (2) rotation at the far-end node j, and (3) elongation k (shortening) of the element.

In matrix notation.

in which is defined as the element flexibility matrix.

For a structure consisting of a, b, … etc. elements

In matrix form

where f is a diagonal matrix with element flexibility matrices as diagonal constituents.

Alternatively, for each element a, b, … etc. the internal forces caused by nodal displacements are threefold: (1) moment at the near-end node I, (2) moment at the far-end node j and (3) tension (compression) in the element itself.

.

In matrix notation

in which is defined as the element stiffness matrix.

For a structure consisting of a, b, … etc. elements we have

In matrix form

where k is a diagonal matrix with elemental stiffness matrices as it constituents.

It can be readily shown that he element flexibility matrix is the inverse of the element stiffness matrix.

The principle of virtual work states that for an elastic structure in equilibrium, the external virtual work done in deforming the structure must be equal to the internal virtual work done by the structure in resisting deformation.

For statically determinate structures, the internal forces ( i= 1, 2, … , m) can be expressed in terms of the external nodal forces (loads) ( j= 1, 2, …, n) by using equilibrium conditions alone

In matrix form

where b is the structure force transformation matrix (m, n)

The structure nodal displacements q are related to the member internal forces Q through the flexibility matrix f

Substituting for Q in above eqution

Using the principle of virtual work

Substituting for Q and q in the right hand side of the above equation

The above equation solves for nodal displacements r in terms of nodal external forces R

where F is defined as the total flexibility matrix of structure

For statically indeterminate structures, the external virtual work done in deforming the structure must include the work done by the forces in the redundant elements as well

For equilibrium, the external and internal forces must balance

Taking the transpose of both sides of the above equation

Recall that

Substituting for Q in the right side of the above equation

Substituting these equations into the equation of virtual work

Expanding the right side of the above equation and comparing virtual forces on the left side and right side of the equation, we have

In matrix form

For structures on rigid supports

Therefore

The above equation is the expression for the redundant forces X.

The nodal displacements are given by

or

where is the flexibility matrix of the redundant structure

With the redundant forces X found, the member forces are solved by equilibrium

Substituting for X in the above equation

or

where is the force transformation matrix of the indeterminate structure.

In the analysis of statically indeterminate structures by the Force Method, recall that

This says that the internal forces Q are linearly related to a set of applied external forces R and a set of unknown forces X associated with the redundant members of the structure.

If the redundant elements are removed one by one, the structure remains stable up to the point at which the last remaining redundant element is removed. What remains once the redundant elements are removes is the notion of the primary structure.

Structures in practice are usually highly redundant in their configurations. Redundant elements (bars) add strength and stability to the structure usually at an increment to the overall cost of the structure.

For a structure, the member deformations q should be consistently related to the nodal displacements r such that the elements of the structure fit together at the nodal points

In matrix form

where a is the structure displacement transformation matrix a (m, n).

From the principle of virtual work

Recall and taking the transform of both sides

Substituting for Q and q in the right hand side of the equation of virtual work

Recall and

Substituting for Q and q in the equation of virtual work

where K is defined as the total stiffness matrix of the structure

In the approach known as the Direct Stiffness Method, K is assembled progressively from scratch by substituting and adding the normalized stiffness matrices (i=1,2, …) of each element in the correct matrix locations of the structure total stiffness matrix K.

Solve for displacements r using the external forces (loads) R using

It can be readily shown, that he structure stiffness matrix K is the inverse of the structure stiffness matrix F

In the displacement method, the irrelevancy of redundancy enables the use of the same procedures for analyzing statically-determinate and statically-indeterminate structures. This fact taken together with the Direct Stiffness Method of assembling K gives the Displacement Method some real advantages over the Displacement Method.

Last updated: 02/12/2007