Concept explainers
For the beam and loading shown, (a) express the magnitude and location of the maximum deflection in terms of w0, L, E, and I. (b) Calculate the value of the maximum deflection, assuming that beam AB is a W18 × 50 rolled shape and that w0= 4.5 kips/ft, L = 18 ft, and E = 29 ×106 psi.
Fig. P9.11
(a)
The magnitude and location of the maximum deflection in terms of
Answer to Problem 11P
The location of the maximum deflection
The magnitude and location of the maximum deflection in terms of
Explanation of Solution
Given that:
The length (L) of the beam is
The load
The young’s modulus E is
Calculation:
Sketch the free body diagram of beam as shown in Figure 1.
Find the reactions of the beam.
Take the moment at B.
Find the reaction at B.
Take the section 1-1 at x distance from A as shown in Figure 2.
Consider a section
Sketch the section x-x as shown in Figure 3.
Calculate the intensity of loading w at the section x using similar triangle method as shown below:
Find the shear force using the expression as follows:
Find the shear force using integration:
Find the moment using the relation as follows:
Apply the boundary conditions:
When
When
Substitute 0 for
Write the moment Equation:
Substitute
Integrate the Equation (2).
Integrate the Equation (3).
Apply the boundary condition in
At
Find the
Substitute 0 for x and 0 for
Apply the boundary condition in
At
Find the
Substitute 0 for x and 0 for
Substitute
Differentiate with respect to x in Equation (5).
To find the location of maximum deflection:
Consider the function
Differentiate with respect to z in Equation (7).
Find the value z using Newton-Raphson method as follows:
Show the calculated values of
0.22 | -0.01583 | 0.050908 | 0.53100 |
0.24 | -0.01479 | 0.053504 | 0.51639 |
0.26 | -0.01369 | 0.055796 | 0.50544 |
0.28 | -0.01256 | 0.057792 | 0.49730 |
0.3 | -0.01138 | 0.0595 | 0.49134 |
0.32 | -0.01018 | 0.060928 | 0.48708 |
0.34 | -0.00895 | 0.062084 | 0.48415 |
0.36 | -0.0077 | 0.062976 | 0.48224 |
0.38 | -0.00643 | 0.063612 | 0.48111 |
0.4 | -0.00516 | 0.064 | 0.48056 |
0.42 | -0.00387 | 0.064148 | 0.48039 |
0.44 | -0.00259 | 0.064064 | 0.48045 |
0.46 | -0.00131 | 0.063756 | 0.48059 |
0.48 | -4.2E-05 | 0.063232 | 0.4807 |
0.5000 | 0.0012 | 0.0625 | 0.4806 |
0.52 | 0.002456 | 0.061568 | 0.48010 |
Refer to table: 1.
The value of
Find the value of
Substitute
Therefore, he magnitude of the maximum deflection in terms of
Therefore, the location of maximum deflection is
(b)
The value of maximum deflection.
Answer to Problem 11P
The value of maximum deflection is
Explanation of Solution
Calculation:
Convert
The rolled shape section
The value of
Find the maximum deflection using the relation:
Substitute
Thus, the value of maximum deflection is
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Mechanics of Materials, CE3110
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