## 29 Matrix Methods

In this method, a set of simultaneous equations that describe the load-deformation characteristics of the structure under consideration are formed. These equations are solved using the matrix algebra to obtain the load-deformation characteristics of discrete or finite elements into which the structure has been subdivided. The matrix method is ideally suited for performing structural analysis using a computer. In general, there are two approaches for structural analysis using the matrix analysis. The first is called the flexibility method in which forces are used as independent variables and the second is called the stiffness method, which employs deformations as the independent variables. The two methods are also called the force method and the displacement method, respectively.

### 2.9.1 Flexibility Method

In the flexibility method, the forces and displacements are related to one another by using stiffness influence coefficients. Let us consider, for example, a simple beam in which three concentrated loads W^ W2, and W3 are applied at sections 1, 2, and 3, respectively, as shown in Figure 2.69. The deflection at section 1, A1, can be expressed as

where F11, F12, and F13 are the flexibility coefficients, defined as the deflection at section 1 due to unit loads applied at sections 1, 2, and 3, respectively. Deflections at sections 2 and 3 are similarly given as

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