Concept explainers
Forces are applied at points A and B of the solid cast-iron bracket shown. Knowing that the bracket has a diameter of 0.8 in., determine the principal stresses and the maximum shearing stress at (a) point H, (b) point K.
Fig. P8.72
(a)
The principal stresses and the maximum shearing stress at point H.
Answer to Problem 72RP
The maximum principal stress at point H is
The minimum principal stress at H is
The shear stress at point H is
Explanation of Solution
Given information:
The diameter (d) of the bracket is
Calculation:
Sketch the free body diagram of solid cast iron as shown in Figure 1.
Refer to Figure 1.
Find the value of P at the section containing point H and K.
Find the shear force about y axis as follows:
Find the shear force about x axis as follows:
Find the moment about x axis as follows:
Find the moment about y axis as follows:
Find the moment about z axis as follows:
Find the value of radius (c) using the relation:
Here, d is the diameter of bracket.
Substitute
Find the area (A) of the circular section using the equation:
Here, c is the half of the diameter.
Substitute
Find the moment of inertia (I) of section using the relation:
Substitute
Find the moment of inertia (J) of section using the relation:
Substitute
Find the value of Q for semicircle using the relation:
Substitute
Determine the normal stress at point H using the relation:
Here, P is the centric force, A is the area of circular cross section, I is the moment of inertia, M is the moment, and c is the centroid distance.
Substitute
Determine the shear stress at point H using the relation:
Here, T is the Torque and J is the polar moment of inertia.
Substitute
Sketch the stresses at point H as shown in Figure 2.
Find the average
Here, normal stress at point H.
Substitute
Find the R using the relation:
Here, shear stress at point H.
Substitute
Determine the maximum principal stress
Substitute
Thus, the maximum principal stress at point H is
Determine the minimum principal stress
Substitute
Thus, the minimum principal stress at H is
Determine the maximum shear stress at point H using the relation:
Here,
Substitute
Thus, the shear stress at point H is
(b)
The principal stresses and the maximum shearing stress at point K.
Answer to Problem 72RP
The maximum principal stress at point K is
The minimum principal stress at K is
The shear stress at point K is
Explanation of Solution
Calculation:
Determine the normal stress at point K using the relation:
Substitute
Determine the shear stress at point K using the relation:
Substitute
Sketch the stresses at point K as shown in Figure 3.
Find the average
Here, normal stress at point H.
Substitute
Find the R using the relation:
Here, shear stress at point H.
Substitute
Determine the maximum principal stress
Substitute
Thus, the maximum principal stress at point K is
Determine the minimum principal stress
Substitute
Thus, the minimum principal stress at K is
Determine the maximum shear stress at point K using the relation:
Here,
Substitute
Thus, the shear stress at point K is
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Chapter 8 Solutions
Mechanics of Materials, 7th Edition
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