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CEEN 111 – Matrix Structural Analysis

Direct Stiffness Method:

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Direct Stiffness Method
A. Introduction The preceding section of the lecture notes provided an introduction to the basic concepts of stiffness analysis and the element stiffness equations. Now that we know how to derive element stiffness equations and we understand the principle of virtual work basis for the stiffnessmethod; we can begin to examine how element stiffness equations can be mechanically assembled to form stiffness equations for analytical models of entire structural systems. This mechanical

assembly of the stiffness equations is suitable for implementation on the computer. We will have to understand, however, that we have only dealt with these stiffness equations in the_______________________________________ and therefore, our models will be limited somewhat (only a temporary dilemma and the coordinate transformation chapter in the lecture notes will remove this limitation). The basic premise of the direct stiffness method is that we can develop element stiffness equations (i.e. load vectors, displacement vectors, and stiffness matrices) and assemble them together in a very systematicmanner to create models of structural systems. In this section of the lecture notes, we will take a quick examination of what we expect the structure stiffness (equilibrium) equations to look like, and then move onto applying the stiffness method using a systematic programmable algorithm.

Lecture Notes

C.M. Foley – Marquette University

CEEN 111 – Matrix Structural Analysis

DirectStiffness Method:

2

B. Features of the Structure Stiffness Equations There are several very important features to the stiffness equations that we should point out at this juncture. First of all, element and structure stiffness matrices are almost always symmetric matrices (in our course, they will always be symmetric). The symmetry property comes from reciprocity of linear elastic systems. This isan important concept, but we won’t dwell on it here in this course. The symmetry allows portions of the stiffness matrices to be developed and subsequently stored rather than all terms. This saves a little computational headache. In general, the structure stiffness matrix is rather weakly populated. That is to say, it has a lot of zeros in it. If the nodes are numbered with subsequent “nice”connection of elements, the stiffness matrix will likely be banded as shown below.

Some programs will actually renumber the nodes to minimize the bandwidth, nb. In general, the difference between node numbers on any given element is a measure of the bandwidth. When really large systems are considered, only storing the upper band of the stiffness equations can save storage requirements and improvealgorithm execution speed.

Lecture Notes

C.M. Foley – Marquette University

CEEN 111 – Matrix Structural Analysis

Direct Stiffness Method:

3

C. Direct Stiffness Method - Algorithm Our examination of the principle of virtual work basis for the stiffness method demonstrated that the method is theoretically sound (assuming our model is adequate), but terribly inefficient inapplication. It would be really nice if we had an algorithm for implementation of the method. The stiffness method can be greatly streamlined and as a result, the process can be programmed. In this section, we will consider an algorithm for implementation of the stiffness method that we will call the direct stiffness method. The algorithm can be outlined as follows: 1. Number the nodes in the analyticalmodel. Use care to number the nodes in a “smart” manner to lessen the workload. 2. Assign an index to the global direction displacements at each node. 3. Set up the stiffness matrix for each element in the structure. Index the rows and columns of the element stiffness matrix using the indices established in step 2. 4. Assemble the global stiffness matrix for the structure using the element stiffness...