   # Determine the axial forces, shears, and bending moments at points A and B of the structure shown.

#### Solutions

Chapter
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Chapter 5, Problem 10P
Textbook Problem
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## Determine the axial forces, shears, and bending moments at points A and B of the structure shown. To determine

Find the axial force, shear, and bending moment at points A and B of the beam.

### 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 axial forces, 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.

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

+Fx=050+Dx=0Dx=50kN

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

+MD=050sin30°(24)+100cos30°(24)Cycos30°(18)+100cos30°(12)+100cos30°(6)=0600+2078.46115.59Cy+1039.230+519.615=0Cy=271.8kN

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

+Fy=0100+Cy100100+Dy=0100+271.8200+Dy=0Dy=28.2kN

Pass the sections aa and bb at points A and B respectively.

Show the sections aa and bb as in Figure 2.

Consider section bb:

Consider the left side of the section bb for calculation of internal forces.

Show the free-body diagram of the left side of the section bb as in Figure 3.

Find the axial force at point B by resolving the horizontal equilibrium.

+Fx=050cos30°100sin30°+271.8sin30°QB=043.3050+135.9QB=0QB=129.2kN

Find the shear at point B by resolving the vertical equilibrium.

+Fy=050sin30°100cos30°+271

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