To provide: The expression of maximum shearing stress as
The constant k for each orientation.
Answer to Problem 53P
The constant k for orientation (a) is
The constant k for orientation (b) is
Explanation of Solution
Given information:
The beam is a hollow square of side a and thickness t.
The beam is subjected to a vertical shear of V.
Calculation:
Orientation (a)
Sketch the cross section of an extruded beam as shown in Figure 1.
Refer to Figure 1.
The height of the section above neutral axis is
Area of the cross member is
Calculate the area of the member A as shown below.
Calculate the moment of inertia of the beam I as shown below.
Substitute
Calculate the first moment of area as shown below.
Here, A is the area of the section and
Calculate the first moment of area along the neutral axis as shown below.
Substitute
Calculate the shear stress
Here V is the vertical shear.
Substitute
Substitute
Hence, the expression for maximum shearing stress is
Calculate the constant as shown below.
Therefore, the constant k is
(b)
Sketch the cross section of an extruded beam as shown in Figure 2.
Refer to Figure 2.
The height of the section above neutral axis is
Area of the cross member is
Calculate the area of the member A as shown below.
Calculate the moment of inertia I of the beam as shown below.
Substitute
Calculate the first moment of area as shown below.
Here, A is the area of the section and
Calculate the first moment of area along the neutral axis as shown below.
Substitute
Calculate the shear stress as shown below.
Here V is the vertical shear.
Substitute
Substitute
Hence, the expression for maximum shearing stress is
Calculate the constant as shown below.
Therefore, the constant k is
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Chapter 6 Solutions
Mechanics of Materials, 7th Edition
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