A wood beam in a historic theater is reinforced with two angle sections at the outside lower corners (see figure). If the allowable stress in the wood is 12 M Pa and that in the steel is 140 M Pa, what is ratio of the maximum permissible moments for the beam before and after reinforcement with the angle sections? See Appendix F Table F-5(b) for angle section properties. Assume that ew= 12 GPa and E3=210 GPa.
The ratio of maximum permissible moment of a beam before and after reinforcement.
Answer to Problem 6.2.16P
The ratio of the maximum permissible moment before and after reinforcement is
Explanation of Solution
Given: .
Moment of Inertia,
Distance from the neutral axis, d=31mm..
Width, b=240mm..
Height, h=480mm..
Calculation: .
The maximum moment if no angle section used is calculated as:.
b- The width of section..
h- Height of section..
The location of the neutral axis from the lower end is stated as:.
Young modulli of steel and wood are
Substitute
The distance of the neural axis from the top..
Substitute h=480mm and
Calculate the moment of Inertia for wood section..
Substitute b=240mm, h=480mm and
The moment of Inertia of the steel section..
Substitute
The maximum moment based on allowable stress in steel..
Substitute
The maximum moment based on the allowable stress on top..
The maximum moment based on the allowable stress in wood on bottom..
Substitute the value
Substitute
The lower magnitude of the moment among the three values for the cross section is to be safe. Hence, the maximum allowable moment is
Therefore the ratio of the maximum permissible moment before and after reinforcement is
Substitute the value
Therefore, the ratio of the maximum permissible moment before and after reinforcement is
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Chapter 6 Solutions
Mechanics of Materials (MindTap Course List)
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