Concept explainers
The beam AB supports two concentrated loads P and Q. The normal stress due to bending on the bottom edge of the beam is +55 MPa at D and +37.5 MPa at F. (a) Draw the shear and bending-moment diagrams for the beam. (b) Determine the maximum normal stress due to bending that occurs in the beam.
Fig. P5.62
(a)
Draw the shear and bending-moment diagrams for the beam.
Explanation of Solution
Given information:
The normal stress due to bending at the point D is
The normal stress due to bending at the point F is
Determine the section modulus (S) of the rectangular beam section using the equation.
Here, the width of the beam is b and the depth of the beam is h.
Substitute 24 mm for b and 60 mm for h.
Determine the bending moment at point D
Here, the normal stress at point D is
Substitute 55 MPa for
Determine the bending moment at point F
Here, the normal stress at point F is
Substitute 37.5 MPa for
Show the free-body diagram of the region FB as in Figure 1.
Determine the vertical reaction at point B by taking moment about point F.
Show the free body diagram of the region DEFB as in Figure 2.
Determine the magnitude of the load Q by taking moment about the point D.
Show the free body diagram of the entire beam as in Figure 3.
Determine the magnitude of the load P by taking moment about the point A.
Determine the vertical reaction at point A by resolving the vertical component of forces.
Shear force:
Show the calculation of shear force as follows;
Show the calculated shear force values as in Table 1.
Location (x) m | Shear force (V) N |
A | 3600 |
C (Left) | 3600 |
C (Right) | 360 |
E (Left) | 360 |
E (Right) | –1800 |
B | –1800 |
Plot the shear force diagram as in Figure 4.
Bending moment:
Show the calculation of the bending moment as follows;
Show the calculated bending moment values as in Table 2.
Location (x) m | Bending moment (M) N-m |
A | 0 |
C | 720 |
E | 900 |
B | 0 |
Plot the bending moment diagram as in Figure 5.
Refer to Figure 5;
The maximum absolute bending moment is
(b)
The maximum normal stress due to bending.
Answer to Problem 62P
The maximum normal stress due to bending is
Explanation of Solution
Given information:
Determine the section modulus (S) of the rectangular beam section using the equation.
Here, the width of the beam is b and the depth of the beam is h.
Substitute 24 mm for b and 60 mm for h.
The maximum absolute bending moment is
Determine the maximum normal stress
Substitute
Therefore, the maximum normal stress due to bending is
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Chapter 5 Solutions
Mechanics of Materials, 7th Edition
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