Chapter 1
A Primer on Matrix Structural Analysis

After reading this chapter, you will be able to:

• Analyze structural members using force and displacement methods
• Derive the stiffness matrix for a 1D bar element
• Assemble elements into a system of equilibrium equations for a structure
• Partition a set of equilibrium equations and solve for unknown displacements and reaction forces
• Write a computer program to analyze a system of 1D bar elements

Matrix structural analysis is a computational technique for analyzing structural systems. As such, the technique requires the system of equilibrium equations to be assembled and solved in an automated fashion to keep the user's input to a minimum. The use of matrices leads to simple algebraic calculations that require little computational resources, even for large problems. Moreover, the models that are used in matrix structural analysis are relatively easy to construct in commercial software. Nevertheless, as with all engineering software, it takes a trained user to ensure that the results obtained from an analysis are logical and, presumably, correct. As a result, the method requires a deep level of understanding of structural mechanics, which is why it is treated as an advanced method of analysis in most engineering programs.

In this chapter, we will look at the theory behind matrix structural analysis, which is rooted in a method known as the displacement method or the direct stiffness method. We will use a statically indeterminate beam to illustrate some of the fundamental differences between the displacement method and the more intuitive force method. We will then use a simple example of an elastic, 1D bar under uniaxial load to come up with a simple process that will allow us to use matrix methods to analyze any type of structural system. From there, we will generalize, giving us a procedure that is directly applicable to other elements, including beams and frames.

1.1 The Force Method

Students who have taken a course on structural analysis have undoubtedly encountered a method for analyzing statically indeterminate structures called the force method or flexibility method. In the force method, a selection of structural forces is specified as redundant because the forces are not needed for stability. Removal of redundant forces results in a structure that is both stable and statically determinant. Using superposition, each redundant force is then introduced to the system and conditions of compatibility are written. The process yields a system of simultaneous equations that may be used to determine the unknown redundant forces.

To illustrate the force method, consider the fixed-fixed beam shown in Figure 1.1, which carries a point load . Using the force method we must first determine the degree of static indeterminacy. In this case, because we have three equilibrium equations (i.e. summing forces in the -direction is zero, summing forces in the -direction is zero, and summing the moments about a point is zero) and six unknown reaction forces, we would say that the beam is statically indeterminate to the third degree (i.e. 6 - 3 = 3). This means that we need three equations of compatibility in addition to our three equilibrium equations to solve for the reaction forces , , , , , and .

Figure 1.1 Statically indeterminate beam with point load at mid-span.

If we choose the reaction forces , , and at the right end of the beam as our redundant forces, the structure, once the redundant forces are removed, is simply a cantilever beam, as shown in Figure 1.2. When we take away the support at the right, the end of the beam displaces (Figure 1.2a). The reaction forces , , and must then be applied to close the fictitious gap in the system, as shown in Figure 1.2b. Using superposition, we can write the compatibility equations, which basically state that the sum of the displacements in , , and directions at the right end of the structure shown in Figure 1.2 must be zero due to the fact that there is a fixed support in our original structure. In other words,

(1.1)

where the components of deflection are shown in Figure 1.2.

Figure 1.2 Beam with (a) the redundant forces removed and (b) the redundant forces applied.

Because the structures in Figure 1.2 are statically determinate, we can analyze the twelve components of deflection that appear in Eq. (1.1). Keep in mind that the externally applied force is known, and the redundant forces , , and are the unknown reaction forces. We can quantitatively evaluate , , and (Figure 1.2a), whereas the deflections in the structure with the redundant forces applied (Figure 1.2b) are in terms of the unknown reaction forces. In particular, from principles of mechanics, we know that , where is the length, is the cross-sectional area, and is the modulus of elasticity. Since this problem is linear elastic and deformations are small, we can calculate the deformations produced by unit forces and then scale the deformations by the unknown reaction forces. We define the flexibility coefficient as the deformation in the beam due to a unit force. For a unit force applied at Redundant 1, this gives

(1.2)

Then, . Note that we give the flexibility coefficient in Eq. (1.2) the subscript (1,1) because it is the deformation in the first direction produced by the redundant force in the first direction . Note that and are zero because the axial force does not produce transverse deformation or rotation, assuming displacements are small.

We use the same procedure to determine the remaining flexibility coefficients. Because bending does not produce axial deformation when deformations are small, we find that flexibility coefficients and are zero. The remaining flexibility coefficients are nonzero and are obtained using methods for analyzing deflections in beams (e.g. the method of virtual work). These flexibility coefficients are left for the reader to derive. Following the derivation, we find

(1.3)

If we express the remaining deflections in terms of flexibility coefficients and their respective unknown forces, and we move the deformations due to load to the right-hand side, then Eq. (1.1) becomes

(1.4)

In matrix form,

(1.5)

If we denote the matrix of flexibility coefficients as , the unknown force vector as , and the known displacement vector as , then the solution to the problem can be obtained by inverting the flexibility matrix, i.e. . Using our flexibility coefficients and solving for the deflections , , and gives

(1.6)

Solving Eq. (1.6) gives

(1.7)

The minus sign on means that the moment acts in the opposite direction from that assumed in Figure 1.2.

Using our equations of equilibrium, we can then determine the reactions , , and . In particular, summing forces in the -direction gives , summing forces in the -direction gives , and summing moments gives .

While the force method is widely used in introductory-level structural mechanics, it cannot be automated in a computer code because governing equations are not unique, i.e. they depend on the forces chosen as redundant. Moreover, the deflections in the compatibility equations are not easily calculated, leading to a process that cannot be scaled to large problems. In addition, once the compatibility equations are solved and the redundant forces are known, it is not trivial to go back to the original structure and analyze deformations and forces at other locations in the structure. In this problem, we may want, for example, to know the deflection at mid-span in the beam. In this case, we would need to invoke a method for evaluating beam deflections, such as the method of virtual work, which requires additional work. For these reasons, the force method is generally not used to analyze large structural systems.

1.2 The Displacement Method

While the force and displacement methods are complementary, the displacement method, unlike the force method, is ideal for computational analysis because it addresses the limitations discussed in the previous section. In the displacement method, we discretize the structural system into elements and nodes. A degree of freedom (DOF) is defined as an unknown displacement or rotation at a node. The degree of kinematic indeterminacy is defined as the total number of unconstrained degrees of freedom in the structure. For the kinematic degrees of freedom, a system of equations is written based on nodal equilibrium. The primary unknown quantities are the displacements, whereas the forces (i.e. the externally applied forces, like in our example) are known. Rather than determine flexibility coefficients, we derive stiffness coefficients, where the stiffness coefficient is the force in direction associated with a unit displacement imposed in direction .

Table 1.1 shows a summary of the force and displacement methods, from which an interesting parallel between the two methods can be seen.

Table 1.1 A comparison between the force and displacement methods.

Method Primary Unknowns Equations Used for the Solution Coefficients of the Unknowns Solution to the System of Equations Force Forces...