Problem Solving with Linear Systems of Equations Using Linear Algebra Techniques

Objectives:

• (1) Set up a system of equations to solve a problem.
• (2) Recognize unique solutions, no solution and infinite solutions as results.

Introduction

In the mid-1800s, a large number of ironwork bridges were constructed as railways crossed the continents. One of the popular designs for trusses, rigid triangular support structures, was the Warren truss with vertical supports.^[1] The center span of the 1926 Bridge of the Gods (Figure 5.1) over the Columbia River is a nice example.

FIGURE 5.1: Bridge of the Gods Warren Truss with Vertical Supports

Warren trusses supported carrying heavy loads. The civil engineering technique “Method of Joints” is used to analyze the forces acting on the truss. Individual parts of the truss are connected with rivets, rotatable pin joints, or welds that permit forces to be transferred from one member of the truss to another.

Figure 5.2 shows a truss that is fixed at the lower left endpoint pi, and can move horizontally at the lower right endpoint p₄. The truss has pin-joints ^at Pi! P2i Рз, and p₄. A load of 10 kilonewtons (kN) is placed at joint рз the forces on the members of the truss have magnitude /i, /2, /3, /4, and /5 as indicated in the figure. The stationary support member has both a horizontal force F and a vertical force F2 (see Figure 5.3); the horizontally movable support member has only a vertical force F3.^[2]

FIGURE 5.2: A Warren Truss with Vertical Supports

If the truss is in static equilibrium, the forces at each joint must sum to the zero vector. If there were net nonzero forces, the joint would move—the truss would not be in static equilibrium. Therefore, the corresponding components of the vector must also be zero; i.e., the sum of the horizontal components of the forces at each joint must be zero and the sum of the vertical components must be zero.

FIGURE 5.3: Force Vector Components

A system of linear equations will model the forces. We will build this model and solve its linear system later in the chapter.

• [1] 'See Flank Griggs’ “The Warren Truss,” Structure, July, 2015, for a brief history of theWarren truss; available at https://www.structuremag.org/?p=8715.
• [2] Variants of this problem appear in many textbooks; see, e.g., Burden and Faires[BurdenFaires2005], Fox [Fox2011Maple].