Concept explainers
6.56 and 6.57 A composite beam is made by attaching the timber and steel portions shown with bolts of 12-mm diameter spaced longitudinally every 200 mm. The modulus of elasticity is 10 GPa for the wood and 200 GPa for the steel. For a vertical shear of 4 kN, determine (a) the average shearing stress in the bolts, (b) the shearing stress at the center of the cross section. (Hint: Use the method indicated in Prob. 6.55.)
Fig. p6.56
(a)
The average shearing stress in the bolts.
Answer to Problem 56P
The average shearing stress in the bolts is
Explanation of Solution
Given information:
The diameter of the bolts is
The longitudinal spacing is
The beam is subjected to a vertical shear of
The modulus of elasticity for wood
The modulus of elasticity for steel
Calculation:
Consider the steel is to be the reference material. So modular ratio of steel is
Calculate the modular ratio of timber wood
Here,
Substitute
Total depth of the section d is as follows:
Calculate the moment of inertia for the symmetric section I as shown below.
Here, b is the width of the section and d is the depth of the section.
For steel:
For wood:
Calculate the moment of inertia for the transformed section as shown below.
Substitute 1 for
Calculate the first moment of area as shown below.
For wooden section:
Calculate the first moment of area for the transformed section Q as shown below.
Substitute
Calculate the horizontal shear per unit length q as shown below.
Here V is the vertical shear.
Substitute
Calculate the force acting on the bolt
Here, s is the longitudinal spacing.
Substitute
Calculate the area of bolt
Here,
Substitute
The bolt is subjected to double shear.
Calculate the shearing stress of the bolt
Substitute
Therefore, the average shearing stress in the bolts is
(b)
The shearing stress at the center of the cross section.
Answer to Problem 56P
The shearing stress at the center of the cross section is
Explanation of Solution
Given information:
The diameter of the bolts is
The longitudinal spacing is
The beam is subjected to a vertical shear of
The modulus of elasticity for wood
The modulus of elasticity for steel
Calculation:
Refer to part (a).
Moment of inertia for the transformed section
Calculate the first moment of area as shown below.
For the two steel plates:
Calculate the first moment of area along the neutral axis for the transformed section as shown below.
Substitute
Calculate the horizontal shear per unit length as shown below.
Substitute
Calculate the shearing stress as shown below.
Substitute
Therefore, the shearing stress at the center of the cross section is
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Chapter 6 Solutions
Mechanics of Materials, 7th Edition
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