HOMOGENEOUS BEAMS 175 



plaster to crack and fall into the room below. The allow- 

 able deflection of a plastered ceiling is usually placed at 

 T&y of the span, or & in. for each foot of span. Where 

 stiffness is lacking in the rafters of a roof, they will be liable 

 to sag, thereby causing unsightly hollows in the surface, in 

 which moisture and snow may lodge. 



The amount of deflection that exists in beams loaded and 

 supported in different ways may be calculated by the for- 

 mulas given in the accompanying table. In using these for- 

 mulas, all the loads should be expressed in pounds and the 

 lengths in inches. The modulus of elasticity is denoted 

 by E, and the moment of inertia of the section by 7. 



EXAMPLE. A 10-in. 35-lb. steel I beam supported at the 

 ends must sustain a uniformly distributed load of 10,000 Ib. 

 The span of the beam is 20 ft., and its moment of inertia is 

 146.4. There is to be a plastered ceiling on its under side, 

 the allowable deflection of which is 3^ in. for each foot of 

 span. Will the deflection of the beam be excessive? 



SOLUTION. The formula of the deflection of a beam of this 



5 WP 



character, from the table, is . The modulus of elas- 

 384 El 



ticity of structural steel is 29,000,000. Substituting the 



values of the example in the formula, the deflection equals 



5 X 10,000 X240 3 



384X29,000,OOOX146.4 -' 



Since the allowable deflection is yfa of the span, the total 

 allowable deflection is 7^X240 = $ in. This is greater than 

 the calculated deflection, and the beam therefore satisfies 

 the required conditions. 



The values for .V and N' t the coefficients of deflection for 

 uniform and center loads, respectively, given in the tables 

 containing the properties of sections of I beams, channels, 



and Z bars, were obtained from the formulas N = -- 



Wl 3 



and N' = -- , in which W equals 1,000 Ib.; /, 12 in.; E, 

 48 El 



29,000,000; and 7, the moment of inertia about the axis 1-1. 

 Therefore, these coefficients represent the deflection, in 



