   # The weight of a car, moving at a constant speed on a beam bridge, is modeled as a single concentrated load, as shown in Fig. P3.14. Determine the expressions for the vertical reactions at the supports in terms of the position of the car as measured by the distance x , and plot the graphs showing the variations of these reactions as functions of x. FIG.P3.14

#### Solutions

Chapter
Section
Chapter 3, Problem 14P
Textbook Problem
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## The weight of a car, moving at a constant speed on a beam bridge, is modeled as a single concentrated load, as shown in Fig. P3.14. Determine the expressions for the vertical reactions at the supports in terms of the position of the car as measured by the distance x, and plot the graphs showing the variations of these reactions as functions of x. FIG.P3.14

To determine

Find the expression for vertical reaction at the supports in terms of distance x.

Sketch the graph for reactions as a function of x.

### Explanation of Solution

Given information:

The structure is given in the Figure.

Apply the sign conventions for calculating reaction forces and moments using the three equations of equilibrium as shown below.

• For summation of forces along x-direction is equal to zero (Fx=0), consider the forces acting towards right side as positive (+) and the forces acting towards left side as negative ().
• For summation of forces along y-direction is equal to zero (Fy=0), consider the upward force as positive (+) and the downward force as negative ().
• For summation of moment about a point is equal to zero (Matapoint=0), consider the clockwise moment as negative and the counter clockwise moment as positive.

Calculation:

Let the roller at B exerts the vertical reaction By.

Let Ax and Ay be the horizontal and vertical reactions at the hinged support A.

Sketch the free body diagram of the beam as shown in Figure 1.

Use equilibrium equations:

Summation of moments about B is equal to 0.

MB=0Ay(8)+20(13x)=08Ay=260+20xAy=26020x8

Ay=655x2kN

Therefore, the vertical reaction at A is Ay=655x2kN_

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