Forces on Bridges with MATLAB

#### Unit Purpose:

By analyzing a bridge in terms of the vector forces acting upon it, the student develops an understanding of the roles that physics and mathematics play in the design of a bridge. Building a mathematical model of a given bridge type provides hands-on experience with rigid structures. The mathematics of solving systems of equations is addressed beginning with a system of 3 variables, and extended with a system with 17 unknowns. Throughout this process, the application of forces via vectors is experienced.

#### Unit Objective:

To develop a mathematical model of the forces on a bridge which is constructed of simple trusses.

#### Unit Overview:

The following concepts will be covered:
• The stability of various geometric shapes
• The mathematical relationship between the members and the joints of a plane truss bridge
• The representation of a vector in terms of its horizontal and vertical components
• The set-up and solution of a system of linear equations
• The use of MATLAB to solve a system of equations

#### Materials:

• tongue depressors
• brass paper fasteners
• drill
• graphing calculator
• MATLAB

#### Mathematical Concepts:

Geometrical shapes, inductive reasoning, sine and cosine functions in a right triangle, vectors (magnitude and direction), systems of equations, matrices (optional)

#### Links to State Mathematics Outcomes:

Students will demonstrate their ability to apply algebraic concepts in the real world in problem solving situations using, for example, algebraic expressions, equations, inequalities, matrices, tables, and graphs. In addition, college-intended students will demonstrate their ability to use matrices to solve linear systems and will demonstrate facility with algebraic transformations.

In addition, students will demonstrate their ability to apply trigonometry to problem solving situations with triangles, and to explore real world phenomena using the sine and cosine functions.

#### Student Outcomes:

The student will be able to:
• compare the rigidity of rectangular and triangular shapes in bridge building;
• use data analysis to determine a linear relationship;
• use vectors and systems of equations to analyze the forces acting on a bridge;
• and evaluate the load-bearing capacities of bridges made from like materials.