Concept explainers
a)
Show that the maximum compressive stresses are in the ratio 4:5:7:9.
a)
Explanation of Solution
Given information:
The load act on the point of the bars is P.
Calculation:
At the point A:
Show the cross-sectional diagram of the square bar as in Figure 1.
Here,
Refer to Figure 1.
The maximum compressive stress of the square bar
Here, e is the eccentricity of the load and
The cross-sectional area of the square bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum compressive stress of the square bar
Substitute
Show the cross-sectional diagram of the circular bar as in Figure 2.
Here,
Refer to Figure 2.
The maximum compressive stress of the circular bar
The cross-sectional area of the circular bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum compressive stress of the circular bar
Substitute
Show the cross-sectional diagram of the diamond shape bar as in Figure 3.
Here,
Refer to Figure 3.
The maximum compressive stress of the diamond shape bar
The cross-sectional area of the diamond shape bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum compressive stress of the diamond shape bar
Substitute
Show the cross-sectional diagram of the triangular bar as in Figure 4.
Here,
Refer to Figure 4.
The maximum compressive stress of the triangular bar
The cross-sectional area of the triangular bar
The distance between the centroid from extreme fibre
The eccentricity of the load (e) is
The moment of inertia
Calculate the maximum compressive stress of the triangular bar
Substitute
Calculate the maximum compressive stresses are in the ratio:
Substitute
The four bars shown have the same cross-sectional area.
Hence the maximum compressive stresses are in the ratio 4:5:7:9 is proved.
b)
Show that the maximum tensile stresses are in the ratio 2:3:5:3.
b)
Explanation of Solution
Given information:
The load act on the point of the bars is P.
Calculation:
At the point B:
Refer to Figure 1.
The maximum tensile stress of the square bar
Here, the e is the eccentricity of the load and
The cross-sectional area of the square bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum tensile stress of the square bar
Substitute
Refer to Figure 2.
The maximum tensile stress of the circular bar
The cross-sectional area of the circular bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum tensile stress of the circular bar
Substitute
Refer to Figure 3.
The maximum tensile stress of the diamond shape bar
The cross-sectional area of the diamond shape bar
The eccentricity of the load (e) is
The distance between the centroid from extreme fibre
The moment of inertia
Calculate the maximum tensile stress of the diamond shape bar
Substitute
Refer to Figure 4.
The maximum tensile stress of the triangular bar
The cross-sectional area of the triangular bar
The distance between the centroid from extreme fibre
The eccentricity of the load (e) is
The moment of inertia
Calculate the maximum tensile stress of the triangular bar
Substitute
Calculate the maximum tensile stresses are in the ratio:
Substitute
The four bars shown have the same cross-sectional area.
Hence the maximum tensile stresses are in the ratio 2:3:5:3 is proved.
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Chapter 4 Solutions
Mechanics of Materials, 7th Edition
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