136 



DEFLECTION OF A BEAM DUE TO SHEAR. Art. 87. 



The deflection due to bending may be calculated from the 



value of i/ m a 

 20000X144 3 



in Art. 82, when L = 

 =0.094 inches. 



ft. = 



3X30000000X7061.7 



la calculating 1, no allowance is made for rivet holes; this 

 will give a result nearer the truth than for a section through 

 rivet holes. 



The deflection due to shear is about 19% of that due to 

 bending; this is, however, in the nature of an estimate, on ac- 

 count of the assumption upon which it is based. 



In beams having rectangular and circular cross sections, 



^ w - -- 



the deflection due to shear is less than in beams having I-shaped 

 cross sections! This deflection is calculated in a different man- 

 ner, because -the shearing stresses are not uniformly distributed 

 over the cross section (75). In the above case of uniform distri- 

 bution of the shearing stresses vertically, the deflection in a 



length dx is dy= ^rdx, but in general it is dy = k-^- dx, in 



Jll% -&B 



which k is a factor depending upon the form of cross section, 

 and is greater than unity, that is, the deflection is^reatejLjthan 

 upon the assumption of uniform distribution of stress over the 

 entire area of the cross section. For rectangular cross sections 

 k = 1.2, and for circular cross sections k = 1.185 1 ; for / beams- 

 assuming the stress distributed over the entire cross section in 

 accordance with equation (22) k about 2.4 for a 3 inch / 

 beam and about 2.0 for a 20 inch / beam. 



For a timber beam of width b and depth 7i, having a span of 

 L ft. and a load P at the middle, 



In the above E B has been put equal to 

 equation shows the following relations : 



L * r\ r . O . Q 



E (78). The last 



3 



100 



12 

 100 



75 

 100 



4.7 . 



100 ' 



16 



1.17 

 100 



Bach's Elaatieitaet und Festigkeit Cliap. IX, 52. 



