Beams AB, BC, and CD have the cross section shown and are pin-connected at B and C. Knowing that the allowable normal stress is +110 MPa in tension and –150 MPa in compression, determine (a) the largest permissible value of w if beam BC is not to be overstressed, (b) the corresponding maximum distance a for which the cantilever beams AB and CD are not overstressed.
Fig. P5.89
(a)
The largest permissible value of w for the condition that the beam BC is not overstressed.
Answer to Problem 89P
The largest permissible value of w is
Explanation of Solution
Given information:
The allowable normal stress of the material in tension is
The allowable normal stress of the material in compression is
Calculation:
Show the free-body diagram of the section BC as in Figure 1.
Determine the vertical reaction at point C by taking moment about point B.
Determine the vertical reaction at point B by resolving the vertical component of forces.
Show the free-body diagram of the section AB as in Figure 2.
Determine the vertical reaction at point A by resolving the vertical component of forces.
Determine the moment at point A by taking moment about the point A.
Show the free-body diagram of the section CD as in Figure 3.
Determine the vertical reaction at point D by resolving the vertical component of forces.
Determine the moment at point D by taking moment about the point D.
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) kN |
A | |
B | 3.6w |
C | –3.6w |
D |
Plot the shear force diagram as in Figure 4.
Location of the maximum bending moment:
The maximum bending moment occurs where the shear force changes sign.
Refer to Figure 4;
Use the similar triangle concept.
The maximum bending moment occurs at a distance of
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) kN-m |
A | |
B | 0 |
Max BM | 6.48w |
C | 0 |
D |
Plot the bending moment diagram as in Figure 5.
Show the free-body diagram of the T-section as in Figure 6.
Determine the centroid in y-axis
Here, the area of the section 1 is
Refer to Figure 4;
Substitute
Determine the moment of inertia (I) using the equation.
Here, the depth of the section 1 is
Substitute 12.5 mm for
Refer to Figure 4;
Tension at Points B and D:
Refer to Figure 5;
Determine the moment at points B and D using the relation.
Substitute 110 MPa for
Compression at Points B and C:
Refer to Figure 5;
Determine the moment at points B and D using the relation.
Substitute –150 MPa for
Tension at maximum bending moment:
Refer to Figure 5;
Determine the maximum moment using the relation.
Substitute 110 MPa for
Compression at maximum bending moment:
Refer to Figure 5;
Determine the maximum moment using the relation.
Substitute –150 MPa for
Refer to the calculated distribution loads; the smallest value controls the design.
Refer to Figure 5;
Equate the maximum bending moment calculated and the maximum bending moment in the tension side.
Therefore, the largest permissible value of w for the condition that the beam BC is not overstressed is
(b)
The maximum distance a for the condition that the beams AB and CD are not overstressed.
Answer to Problem 89P
The maximum distance a for the condition that the beams AB and CD are not overstressed is
Explanation of Solution
Refer to Part (a), Figure 4;
The maximum bending moment in the beams AB and CD occurs at the ends A and D.
The calculated maximum bending moment at the points A and D is as follows:
The maximum allowable compression moment at the points A and D is as follows:
Equate the values;
Refer to the answer of the part (a);
Substitute
Therefore, the maximum distance a for the condition that the beams AB and CD are not overstressed is
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Chapter 5 Solutions
Mechanics of Materials, 7th Edition
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