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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 à— 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 à— 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: 

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(ok! )