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Economic Sections
From the flexure formula , it can be seen that the bending stress at the neutral axis,
where , is zero and increases linearly outwards. This means that for a rectangular or circular
section a large portion of the cross section near the middle section is understressed.
For steel beams or composite beams, instead of adopting the rectangular shape, the area may be
arranged so as to give more area on the outer fiber and maintaining the same overall depth, and
saving a lot of weight.
When using a wide flange or I-beam section for long beams, the compression flanges tend to buckle
horizontally sidewise. This buckling is a column effect, which may be prevented by providing lateral
support such as a floor system so that the full allowable stresses may be used, otherwise the stress
should be reduced. The reduction of stresses for these beams will be discussed in steel design.
In selecting a structural section to be used as a beam, the resisting moment must be equal or
greater than the applied bending moment. Note: .
The equation above indicates that the required section modulus of the beam must be equal or
greater than the ratio of bending moment to the maximum allowable stress.
A check that includes the weight of the selected beam is necessary to complete the calculation. Inchecking, the beams resisting moment must be equal or greater than the sum of the live-load
moment caused by the applied loads and the dead-load moment caused by dead weight of the
beam.
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Dividing both sides of the above equation by , we obtain the checking equation
Problem529
General instruction
Assume that the beam in the problem is properly braced against lateral deflection. Be sure to include
the weight of the beam itself.
Problem 529
A 10-m beam simply supported at the ends carries a uniformly distributed load of 16 kN/m over its
entire length. What is the lightest W shape beam that will not exceed a flexural stress of 120 MPa?
What is the actual maximum stress in the beam selected?
Solution 529
Starting at the bottom of Appendix B, Table B-2 Properties ofWide-Flange Sections (W Shapes): SI Units, of text book, the
following are the first to exceed the S above:
Designation Section Modulus
W250 × 149 1,840 × 103 mm3
W310 × 118 1,750 × 103 mm3
W360 Ã — 101 1,690 × 103 mm3
W410 × 100 1,920 × 103 mm3
W460 × 89 1,770 × 103 mm3
W530 × 85 1,810 × 103 mm3
W610 × 82 1,870 × 103 mm3
W690 × 125 3,500 × 103 mm3
Use the lightest section W610 × 82 answer
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Checking:
The resisting S of W610 × 82 is 1,870 × 103 mm3, the S due to
live-load and dead-load is only 1,750.46 × 103 mm3,
therefore, the chosen section is sufficient to resist the combined dead-load and live-load.
Actual bending moment due to dead and live loads:
Actual stress:
answer
Problem 530
Repeat Prob. 529 if the distributed load is 12 kN/m and the length of the beam is 8 m.
Solution 530
From Appendix B, Table B-2 Properties of Wide-Flange
Sections (W Shapes): SI Units, of text book:
Designation Section Modulus
W200 × 86 853 × 103 mm3
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W250 × 67 806 × 103 mm3
W310 Ã — 60 849 × 103 mm3
W360 × 57 897 × 103 mm3
W410 × 54 924 × 103 mm3
W460 × 52 943 × 103 mm3
Use the lightest section W460 × 60 answer
Checking:
(ok! )
Actual bending moment:
Actual stress:
answer
Problem531
General instruction
Assume that the beam in the problem is properly braced against lateral deflection. Be sure to include
the weight of the beam itself.
Problem 531
A 15-ft beam simply supported at the ends carries a concentrated load of 9000 lb at midspan. Select
the lightest S section that can be employed using an allowable stress of 18 ksi. What is the actual
maximum stress in the beam selected?
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Solution 531
From Appendix B, Table B-8 Properties of I-Beam Sections
(S Shapes): US Customary Units, of text book: Use S10 ×
25.4 with S = 24.7 in3 answer
Checking:
(ok! )
Actual bending moment:
Actual stress:
answer
PROBLEM 532
General instruction
Assume that the beam in the problem is properly braced against lateral deflection. Be sure to include
the weight of the beam itself.
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Problem 532
A beam simply supported at the ends of a 25-ft span carries a uniformly distributed load of 1000 lb/ft
over its entire length. Select the lightest S section that can be used if the allowable stress is 20 ksi.
What is the actual maximum stress in the beam selected?
Solution 532
From Appendix B, Table B-8 Properties of I-Beam Sections (S
Shapes): US Customary Units, of text book: Use S15 ×
42.9 with S = 59.6 in3 answer
Checking:
(ok! )
Actual bending moment:
Actual stress:
answer
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Problem 533
A beam simply supported on a 36-ft span carries a uniformly distributed load of 2000 lb/ft over the
middle 18 ft. Using an allowable stress of 20 ksi, determine the lightest suitable W shape beam.
What is the actual maximum stress in the selected beam?
Solution 533
From Appendix B, Table B-7 Properties of Wide-Flange
Sections (W Shapes): US Customary Units, of text book:
Designation Section Modulus
W12 × 120 163 in3
W14 × 99 157 in3
W16 × 89 155 in3
W18 × 76 146 in3
W21 × 73 151 in3
W24 × 68 154 in3
Use W24 × 68 with S = 154 in3 answer
Checking:
(ok! )
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Actual bending moment:
Actual stress:
answer
Problem 534
Repeat Prob. 533 if the uniformly distributed load is changed to 5000 lb/ft.
Solution 534
From Appendix B, Table B-7 Properties of Wide-Flange
Sections (W Shapes): US Customary Units, of text book:
Designation Section Modulus
W12 × 279 393 in3
W14 × 233 375 in3
W24 × 146 371 in3
W27 × 146 411 in3
W30 × 132 380 in3
W33 × 130 406 in3
Use W33 × 130 with S = 406 in3 answer
Checking:
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(ok! )
Actual bending moment:
Actual stress:
answer
Problem 535
A simply supported beam 24 ft long carries a uniformly distributed load of 2000 lb/ft over its entire
length and a concentrated load of 12 kips at 8 ft from left end. If the allowable stress is 18 ksi, select
the lightest suitable W shape. What is the actual maximum stress in the selected beam?
Solution 535
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From Appendix B, Table B-7 Properties of Wide-Flange Sections (W Shapes): US Customary Units,of text book:
Designation Section Modulus
W12 × 96 131 in3
W14 × 90 143 in3
W16 × 77 134 in3
W18 × 76 146 in3
W21 × 68 140 in3
W24 × 62 131 in3
Try W24 × 62 with S = 131 in3
Checking:
At critical section:
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(not ok! )
Try W21 × 68 with S = 140 in3
Checking:
At critical section:
(ok! )
Use W21 × 68 answer
Actual bending moment:
Actual stress:
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answer
Problem 536
A simply supported beam 10 m long carries a uniformly distributed load of 20 kN/m over its entirelength and a concentrated load of 40 kN at midspan. If the allowable stress is 120 MPa, determine
the lightest W shape beam that can be used.
Solution 536
From Appendix B, Table B-2 Properties of Wide-Flange
Sections (W Shapes): SI Units, of text book:
Designation Section Modulus
W310 × 202 3 050 × 103 mm3
W360 × 179 3 120 × 103 mm3
W460 × 144 3 080 × 103 mm3
W530 × 138 3 140 × 103 mm3
W610 × 125 3 220 × 103 mm3 W690 × 125 3 500 × 103 mm3
W610 × 125 has a theoretical mass of 125.1 kg/m while W690 × 125 has a theoretical mass of 125.6
kg/m. Thus, use W610 × 125 with S = 3 220 × 103 mm3. answer
Checking: