Concept explainers
Use the method of superposition to solve the following problems and assume that the flexural rigidity El of each beam is constant.
9.90 Before the load P was applied, a gap, δ0 = 0.5 mm, existed between the cantilever beam AC and the support at B. Knowing that E = 200 GPa, determine the magnitude of P for which the deflection at C is 1 mm.
Fig. P9.90
Find the magnitude of load P for the given condition using superposition method.
Answer to Problem 90P
The magnitude of load P in the beam is
Explanation of Solution
Given information:
The gap at the point B is
The modulus of elasticity of the material is
The size of the square cross section is
The deflection at point C is
Calculation:
Find the moment of inertia of the square cross section (I) using the relation.
Here, the size of the square cross section is a.
Substitute 60 mm for a.
Consider the portion AB of the beam.
The load P at point C will be converted into a load and moment at point B.
Show the free-body diagram of the superimposed beam AB as in Figure 1.
Loading I:
The downward load P is acting at point B of the beam.
Refer to Case 1 in Appendix D “Beam Deflections and Slopes” in the textbook.
Write the deflection equation for concentrated load acting in a cantilever beam as follows;
Find the deflection at point B due to load P at point B as follows;
Loading II:
The upward reaction
Refer to Case 1 in Appendix D “Beam Deflections and Slopes” in the textbook.
Write the deflection equation for concentrated load acting in a cantilever beam as follows;
Find the deflection at point B due to reaction at point B as follows;
Loading III:
The clockwise moment is acting at point B of the beam.
Refer to Case 3 in Appendix D “Beam Deflections and Slopes” in the textbook.
Write the deflection equation for moment in a cantilever beam as follows;
Find the deflection at point B due to the moment at point B as follows;
Find the resultant deflection at point B as follows.
Substitute
Substitute 0.5 mm for
Consider the beam ABC.
Show the free-body diagram of the superimposed beam ABC as in Figure 2.
Loading IV:
The downward load P is acting at point C of the beam.
Refer to Case 1 in Appendix D “Beam Deflections and Slopes” in the textbook.
Write the deflection equation for concentrated load acting in a cantilever beam as follows;
Find the deflection at point C due to load P at point C as follows;
Loading V:
The upward reaction
Refer to Case 1 in Appendix D “Beam Deflections and Slopes” in the textbook.
Write the slope and deflection equation for concentrated load acting in a cantilever beam as follows;
Find the deflection at point B due to reaction at point B as follows;
Find the slope at point B due to reaction at point B as follows;
The portion BC remains straight.
Find the deflection at point C due to reaction at point B as follows;
Substitute
Find the resultant deflection at point C as follows.
Substitute
Substitute 1 mm for
Solve the Equation (1) and (2).
Therefore, the magnitude of load P in the beam is
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