   # For the beam shown: (a) determine the distance a for which the maximum positive and negative bending moments in the beam are equal; and (b) draw the corresponding shear and bending moment diagrams for the beam.

#### Solutions

Chapter
Section
Chapter 5, Problem 54P
Textbook Problem
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## For the beam shown: (a) determine the distance a for which the maximum positive and negative bending moments in the beam are equal; and (b) draw the corresponding shear and bending moment diagrams for the beam. (a)

To determine

What could be the value of a when the positive and negative bending moments are equal.

### Explanation of Solution

Sign conversion:

Apply the sign convention for calculating the equations of equilibrium as below.

• For the horizontal forces equilibrium condition, take the force acting towards right side as positive (+) and the force acting towards left side as negative ().
• For the vertical forces equilibrium condition, take the upward force as positive (+) and downward force as negative ().
• For moment equilibrium condition, take the clockwise moment as negative and counter clockwise moment as positive.

Apply the following sign convention for calculating the shear and bending moments.

• When the portion of the beam considered is left of the section, then the external force acting to the left are considered as positive.
• When the portion of the beam considered is right of the section, then the external force acting to the right are considered as positive.
• When the portion of the beam considered is left of the section, then the external force acting upward are considered as positive.
• When the portion of the beam considered is right of the section, then the external force acting downward are considered as positive.
• When the portion of the beam considered is left of the section, then the clockwise moments are considered as positive.
• When the portion of the beam considered is right of the section, then the counterclockwise moments are considered as positive.

Calculation:

Show the free-body diagram of the entire beam as in Figure 1.

Consider the section BCD.

Find the vertical reaction at point D by taking moment about the point B.

+MBBCD=060(20a)+Dy(40a)=0Dy=60(20a)(40a) (1)

Consider the entire beam.

Find the moment at point A by taking moment about the point A.

+MA=0MA60(20)+Dy(40)=0MA1200+60(20a)(40a)(40)=0MA=12002400(20a)(40a) (2)

Find the horizontal reaction at point A by resolving the horizontal equilibrium.

+Fx=0Ax=0Ax=0

Find the vertical reaction at point A by resolving the vertical equilibrium.

+Fy=0Ay60+Cy=0Ay60+60(20a)(40a)=0Ay=6060(20a)(40a) (3)

The maximum positive bending moment occurs at the point C

(b)

To determine

Plot the shear force and bending moment diagrams.

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