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The required pair of structural steel channels.

Answer to Problem 57P
The required pair of structural steel channels is
Explanation of Solution
The Figure (1) shows the free body diagram of the beam.
Figure (1)
Here, the reaction force at point
Write the net moment at point O.
Here, the reaction force at point C is
Write the expression for the net force on the beam.
Here, the net reaction force at point O is
Refer to Table 3-1 “Singularity (Macaulay) functions”.
Write the expression for load intensity acting on the beam.
Here, the load intensity on the beam is
Write the expression for the moment.
Here, the net bending moment is
Substitute the value from Equation (III) to Equation (IV).
Write the equation for the deflection across the beam.
Here, the modulus of elasticity is
Substitute the value from Equation (V) to Equation (VI).
Integrate Equation (VII) with respect to
Integrate the Equation (IX) with respect to
Write the expression for the bending stress in the beam.
Here, the bending stress is
Write the expression for deflection at the midspan using similar triangles.
Here, the deflection at the mid-span is
Write the expression for the area moment of inertia.
Here, the moment of inertia of single channel is
Conclusion:
Refer to Table A-5 “Physical Constants of Materials”; obtain the properties of modulus of elasticity for carbon steel as,
Substitute
Substitute
Substitute 0 for
Substitute
Substitute
Substitute
Substitute
The obtain value of second area of moment is
Refer to Table A-7 “Properties of StructuralSteel Channels”. Select two steel channels of
Substitute
Substitute
Substitute
Since the deflection at the midspan and the maximum stress in the beam is lying within the limits, so the selected pair of channels are significant.
Thus, the pair of structural steel channels is
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Chapter 4 Solutions
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
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