# Bridges: Human links and innovations

## The Basic Mathematics of Bridges

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## Activities

2. Students will focus on the construction of a cable-stayed bridge with the objectives of getting practice in measuring angles and linear measurements, and applying the principles of similar triangles.

- a. Compression
- b. Tension
- c. Torsion

3. Students will use a model of the beam bridge to discover the effect of the load on a beam at varying distances.

4. Students could choose to investigate the strength of a model cable (a thin thread) to determine the structure of a suspension bridge of a required load capacity based on the results of their findings.

### Activity 1

Aim: To demonstrate Compression, Tension and TorsionMaterial: Piece of foam material 3" x 3" x 10"

Procedure to demonstrate compression:

1. A weight or load is applied to the piece of foam and students are asked to describe the change in the form of the material.

2. A small portion of about a half of an inch is removed from the center on the top and bottom surface, no deeper than about one inch. Each end of the strip of foam is supported by a thick textbook. The books are placed about five inches apart. A slight pressure is applied to the top middle portion of the foam. Students should observe:

a) The change in the surface of the material in particular in the area of the slit.

What do you notice about the slit on top?

What does the force of compression do to a material?

What kind of force does your body exert on the floor?

Procedure to demonstrate Tension

a) Observe the change at the underside of the foam while a force is applied to the top.

What do you notice about the slit at the bottom while the force was applied?

Try to gently break a thin strip of pine wood, about a foot in length.

Where does the splintering start?

b) Pin two three inch pieces of thin strips of wood at one end forming an inverted V. Place it on the table in an inverted position on the desk. Put a rubber band around the opened end (slightly taut) then press down on the point (vertex) of the V.

What change or changes do you observe?

Procedure to demonstrate Torsion

Material: Strip of rubber or sponge about a foot in length

Hold one end of the strip with the right hand and the other end in the left hand. Then twist the strip in a ringing motion in opposite directions.

What effect does this turning force have on the strip?

### Activity 2

Aim: To show the effect of a moving load on a bridgeMaterial: Two spring-balances, a meter rule

Procedure: Place the balances on a firm table. Place one end of the meter rule on one of the balances and the other end on the other balance. Get a small object that has a weight of about fifty grams starting at the end the rule.

The two balances will act as the abutments of a typical beam bridge while the weight will behave like a moving vehicular load over the bridge. The meter rule represents the beams of the bridge. To begin: places the weight at the one end of the rule, then observe and record the separate weights measured on each balance. Move the load in increments of ten centimeters and record the weights.

Observation:

Inference: If the bridge were to collapse, at what point along the beam is it most likely to occur? Why?

- a) The initial weights of the beam on balance one and on balance two
- b) The changes in the weighs as the moving load changes position

### Activity 3

Aim: To apply the mathematics of proportionStudents will construct a beam bridge proportional to a reasonable dimension of a real beam bridge. Students could visit a local bridge and acquire the basic measurements of the width and length of the span. They can choose a their own materials to construct the bridge. Each project should have a drawn plan that matches the proposed scales. Prior to this exercise, students should be exposed to the principles of finding equivalent proportions. This bridge building exercise should serve as practice in practical applications of the mathematics of proportionality.

### Activity 4

Aim: To construct a cable-stayed suspension bridgeProcedure: Students will identify a particular cable-stayed bridge and ascertain the dimension of the structure. They should draw a scaled rendition of the structure that they will use to construct their model. This activity should provide useful exercises in drawing and identifying the properties of similar right triangles and finding equivalent proportions.

The tower of the bridge forms the vertical side of the right triangle. The design could have five attached cables on each side of the tower. The distance between the points of attachment of preceding cables on the tower should be equal. Likewise, the points of attachment of the cables on the beam of the span should be equidistant.

Students should be able to calculate the length of the remaining cables after the first cable has been installed by applying the proportionality concept. For the more advanced students, the Pythagorean theorem could be utilized.

### Activity 5

Aim: To evaluate the economic effect of a particular bridge.Procedure: Students should select a particular bridge and observe for about an hour the number of vehicles that use the bridge traveling in both directions. Later they should then assume that the bridge is destroyed. They should now show the economic impact on one of the bridge user applying reasonable estimates. For example, find an alternate route to get to his/her original destination. Calculate the additional distance and the increase in the time required to travel, and the cost of the additional volume of gasoline.

Inference: In what ways will this affect the rest of the other drivers who will now share road space with the additional drivers?

### Activity 6

Aim: To measure the arch of a bridge in degreesProcedure: Students should obtain a picture of an arch bridge. They should trace the arch of the bridge on sheet of paper. They should use the arch, which is now considered an arc of a circle, to complete the missing portion of the circle. With the knowledge that a circle has 360°, they should measure the length of the arc that forms the arch of the bridge, and compare it with the length of the total circumference of the circle. They can then write a ratio of the length of the arch of the bridge to the circumference of the circle. This ratio should be considered equivalent to the ratio of the degrees in the arch of the bridge which is unknown but could be called "d ", to the total number of degrees in the full circle, 360°. The two ratios are used as equivalent fractions to calculate the degrees "d " of the arch of the bridge.