Concept explainers
Find the magnitude and location of the largest downward deflection of the beam.
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Answer to Problem 61P
The location of the largest downward deflection is
The largest downward deflection of the beam is
Explanation of Solution
Given information:
The modulus of elasticity of the material is
Calculation:
Show the free-body diagram of the beam AD as in Figure 1.
Write the singularity equation for load intensity as follows;
Integrate the equation to find the shear force.
By definition, the change in bending moment with respect to change in distance is shear force.
Integrate the equation to find the bending moment.
Write the second order differential equation as follows;
Here, the moment at the corresponding section is
Substitute
Integrate the equation with respect to x;
Integrate the Equation (2) with respect to x.
Boundary condition 1:
At the point D;
Substitute 16 ft for x and 0 for M in Equation (1).
Boundary condition 2:
At the point A;
Substitute 18.84375 kips for
Boundary condition 3:
At the point D;
Substitute 18.84375 kips for
Refer to Appendix C “The properties of the Rolled-Steel Shapes” in the textbook.
The moment of inertia of the
At point A;
Substitute
At point B;
Substitute
At point C;
Substitute
At point D;
Substitute
The slope changes from negative to positive in the section BC.
The maximum deflection occurs where the slope changes sign. i.e.,
Substitute 0 for
Solve the equation;
Therefore, the location of the largest downward deflection is
At largest deflection point;
Substitute
Therefore, the largest downward deflection of the beam is
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