A thin steel beam AB used in conjunction with an electromagnet in a high-energy physics experiment is securely bolted to rigid supports (see figure), A magnetic field produced by coils C results in a force acting on the beam. The force is trapezoidally distributed with maximum intensity q0= 18 kN/m. The length of the beam between supports is L = 200 mm, and the dimension c of the trapezoidal load is 50 mm. The beam has a rectangular cross section with width b = 60 and height h = 20 mm.
Determine the maximum bending stress
The maximum bending stress and the maximum deflection.
Answer to Problem 10.4.40P
The maximum bending stress is
The maximum deflection in the beam is
Explanation of Solution
Given information:
Width of the rectangular cross-section is
The below figure shows the schematic diagram of the beam with parameters.
Figure-(1)
Write the expression for the equilibrium in vertical direction.
Substitute
Here, the reaction force at point A is
There is symmetry in the beam therefore the reaction forces at A and B will be same.
There is symmetry in the beam therefore the moment at A will be equal to moment at B .
Write the expression for the relation between the reaction forces at A and B .
Write the expression for the relation between the moment about A and B .
Here, the moment about A is
The below figure shows the deflection in the beam.
Figure-(2)
Write the expression for the compatibility.
Here, the rotation about point A is
The below figure shows the deflection slope.
Figure-(3)
Write the expression for the slope from figure-(3).
Here, the load is
Write the expression for the deflection for figure-(3).
Write the expression for load.
Write the expression for load from
Write the expression for load from
Write the expression for rotation about A .
Write the expression for the deflection.
The below figure shows the moments at the ends of the beam.
Figure-(4)
Write the expression for the compatibility.
Write the expression for slope for figure-(4).
Write the expression for the deflection for figure-(4).
Write the expression for maximum deflection.
Substitute
Write the expression for the moment about C .
Write the expression for the maximum moment.
Write the expression for moment of inertia.
Here, moment of inertia is
Write the expression for the section modulus.
Write the expression for the normal stress.
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Conclusion:
The maximum bending stress is
The maximum deflection in the beam is
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Chapter 10 Solutions
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