Concept explainers
Two steel bars (Es = 200 GPa and αs = 11.7 × 10-6/°C) are used to reinforce a brass bar (Eb =105 GPa, αb = 20.9 × 10-6/°C) that is subjected to a load P = 25 kN. When the steel bars were fabricated; the distance between the centers of the holes that were to fit on the pins was made 0.5 mm smaller than the 2 m needed. The steel bars were then placed in an oven to increase their length so that they would just fit on the pins. Following fabrication, the temperature in the steel bars dropped back to room temperature. Determine (a) the increase in temperature that was required to fit the steel bars on the pins, (b) the stress in the brass bar after the load is applied to it.
Fig. P2.55
(a)
The required temperature increment to fit the steel bars on the pins.
Answer to Problem 55P
The required temperature increment to fit the steel bars on the pins is
Explanation of Solution
Given information:
The magnitude of load
The coefficient
The coefficient
The young’s modulus
The young’s modulus
Determine the temperature change require to expand steel bar by this amount.
Here,
Substitute
Thus, the required temperature increment to fit the steel bars on the pins is
(b)
The normal stress in the brass bar after load applied.
Answer to Problem 55P
The normal stress due to steel is
The normal stress due to brass is
Explanation of Solution
Calculation:
After the assembly of the bars, a compressive force is developed in the brass bars and a tensile force is developed in the steel bars.
Determine the area
Here, b is the width of the steel bar , d is the depth of the steel bar, and t is the thickness of the bar.
Substitute
Determine the deformation of steel
Here, F is the compressive force, L is the length of the bar, and
Substitute
Determine the area
Here, b is the width of the brass bar, d is the depth of the brass bar.
Substitute
Determine the deformation of brass
Here,
Substitute
The deformation of bar and steel is equal to the initial amount of misfit.
Here,
Substitute
Find the stresses due to fabrication for steel using the relation:
Substitute
Find the stresses due to fabrication for brass using the relation:
Substitute
The stresses should be added the stress due to the
Consider
Find the force developed in steel bar
Substitute
Find the force developed in brass bar
Substitute
Find the force P as follows:
Here,
Substitute
Substitute
Substitute
Find the normal stress
Substitute
Find the normal stress
Substitute
The stress due to fabrication added.
Find the total stress using the relation:
Here,
Substitute
Thus, the normal stress due to steel is
Find the total stress using the relation:
Substitute
Thus, the normal stress due to brass is
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Chapter 2 Solutions
MECHANICS OF MATERIAL (LL)
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