Resisting Moment

The moment of resistance of a beam is the sum of the moments about the neutral axis of all the stresses in the fibers composing the section. The safe resisting moment of any beam section is equal to the product of the safe fiber stress and the moment of inertia divided by the distance from the neutral axis to the extreme fibers. If I is the moment of inertia, c the distance in inches from the neutral axis to the extreme fibers, and S the safe fiber stress in pounds per square inch, the resisting moment M1 = I S/c; but since I/c= Q, the section modulus (see page 82), M1 = QS; that is, the safe resisting moment is equal to the safe fiber stress multiplied by the section modulus. To obtain the safe unit fiber stress, the modulus of rupture of the material (see Tables VIII, IX, X) is divided by the required factor of safety.

Example

What is the safe resisting moment of a Northern yellow-pine beam 10 in. wide X 12 in, deep, using a factor of safety of 4?

Solution

In the formula M1 = QS, the section modulus

Q for a rectangular section (by Table XII) is bd2/6, or, for the given section, Q =(10x12x12) / 240. From Table IX, the modulus of rupture for Northern yellow pine is 6,000 lb.; the desired factor of safety being 4, the safe unit fiber stress S is 6,000 / 4 = 1,500 lb. Then QS = 1,500 X 240 = 360,000, the safe resisting moment, or M1, of the beam section in inch-pounds.

A beam will safely support a given load when the safe resisting moment M1, in inch-pounds, is equal to, or greater than, the bending moment Malso in inch-pounds. Economical design requires that the safe resisting moment be bat little, if any, in excess of the bending moment, having regard, also, to the nearest commercial, or stock, sizes.

Deflection

Stillness in beams is as important as strength. Lack of stiffness causes vibrations, springy floors, deflection, or sagging, producing plaster cracks in ceilings. To prevent excessive deflection, shallow beams must be avoided. The deflection of beams carrying plastered ceilings should not exceed 1/360 of the span. Usually this limit is not exceeded when the depth of wooden beams is at least 1/16 the span. In dwellings, the full floor load being seldom realized, and as bridging is used between joists, their depth may, with safety, be 14 in. for a span as great as 22 ft. The depth of rolled-steel beams should not be less than 1/24 the span, and that of plate girders not less than 1/15 If doubt exists as to the stiffness of a beam, its deflection should be calculated by formulas from Table XXVI, and if found excessive, the load should be diminished, or the size of beam increased.

Example

A 6" X10" white-oak beam, uniformly loaded with 5,000 lb., has a span of 16 ft. 8 in. What is the deflection?

Solution

From Table XXVI, the deflection = 5Wl3 /384 EI; from Table IX, E = 1,100,000; I =bd3/12 =(6x1000)/12 = 500.

Hence, deflaction=(5x5,000x8,000,000) / (384x1100000x500)= .94 in