Concept explainers
For the beam and loading shown, determine (a) the equation of the elastic curve, (b) the deflection at point B, (c) the deflection at point D.
Fig. P9.44
(a)
Find the equation of the elastic curve.
Answer to Problem 44P
The equation of the elastic curve is,
Explanation of Solution
Show the free-body diagram of the beam AD as in Figure 1.
Determine the vertical reaction at point C by taking moment about point A.
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. It is expressed as follows:
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
Boundary condition 2:
At the point A;
Substitute
Boundary condition 3:
At the point C;
Substitute
Substitute
Therefore, the equation of the elastic curve is
(b)
Find the deflection at point B of the beam.
Answer to Problem 44P
The deflection at mid-point B of the beam is
Explanation of Solution
Refer to part (a);
Refer to Equation (4);
At point B;
Substitute
Therefore, the deflection at point B of the beam is
(c)
Find the deflection at point D of the beam.
Answer to Problem 44P
The deflection at mid-point D of the beam is
Explanation of Solution
Refer to part (a); Equation (4);
At point D;
Substitute
Therefore, the deflection at point D of the beam is
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Chapter 9 Solutions
Mechanics of Materials, CE3110
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