Concept explainers
Two cover plates are welded to the rolled-steel beam as shown. Using E = 200 GPa, determine (a) the slope at end A, (b) the deflection at end A.
Fig. P9.107
(a)
Find the slope
Answer to Problem 107P
The slope
Explanation of Solution
Given information:
The elastic modulus (E) is
The section of the beam is
The dimension of the top plate and bottom plate is
Calculation:
Refer Appendix C, “Properties of Rolled steel shapes”.
The moment of inertia (I) for the given section is
The depth of the section (D) is
The width of the section (b) is
Use moment area method:
Consider from bottom.
Calculate the neutral axis
Substitute
Top plate:
Calculate the area of the top plate
Since the dimension of the top plate is
Calculate the depth of neutral axis (d) using the formula:
Substitute
Calculate the product of
Substitute
Calculate the moment of inertia (I) using the formula:
Here, b is the width the top plate and h is the height of the top plate.
Substitute 200 mm for b and 12 mm for h.
Bottom plate:
Top plate:
Calculate the area of the bottom plate
Since the dimension of the bottom plate is
Calculate the depth of neutral axis (d) using the formula:
Substitute
Calculate the product of
Substitute
Calculate the moment of inertia (I) using the formula:
Here, b is the width the top plate and h is the height of the top plate.
Substitute
Tabulate the calculated values and compute the moment of inertia (I) as in Table 1.
Segments | Area, A | Depth, d (mm) | ||
Top plate | 2400 | 209 | 28,800 | |
Bottom plate | 2400 | 209 | 28,000 | |
Summation |
Take the greater value of moment of inertia from the three segments is
Calculate the moment of inertia (I) using the relation:
Substitute
Show the free body diagram of beam by considering the point load as in Figure 1.
Draw the moment diagram of the above beam as in Figure 2.
Calculate the moment
Calculate the ratio of
Substitute
Calculate the area
Here,
Substitute 0.6 mm for
Calculate the moment
Calculate the ratio of
Substitute
Calculate the area
Here,
Substitute 2.1 m for
Calculate the area
Substitute
Show the free body diagram of beam by considering the uniformly distributed load as in Figure 3.
Draw the moment diagram of the above beam as in Figure 4.
Calculate the moment
Calculate the ratio of
Substitute
Calculate the area
Here,
Substitute
Show the tangent slope and deflection at point A related to reference tangent as in Figure 5.
Since the support C has fixed support, the slope
Calculate the slope at the end A related to the fixed end C
Substitute
Calculate the slope at the point A
Substitute 0 for
Thus, the slope
(b)
Find the deflection
Answer to Problem 107P
The deflection
Explanation of Solution
Given information:
The elastic modulus (E) is
The section of the beam is
The dimension of the top plate and bottom plate is
Calculation:
Calculate the deflection at end A related to the fixed end C
Substitute
Calculate the deflection at the point A
Substitute 0 for
Thus, the deflection
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Chapter 9 Solutions
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