112 



SHEARING STRESSES IN SOLID BEAMS. Art. 75. 



This shows that the unit shearing stress at the neutral axis 

 of a rectangular cross section is 50% greater than the mean 

 value. This corresponds with what was found in the numerical 

 example above. 



When v = y<>h, s s as was found above from general con- 

 siderations. 



For a circular cross section equation (22) becomes, for 

 points on the circumference, 



8 



r 



77 / 2 

 j r s 



2r cos 

 sin 0' 



' r sin <' r cos <j>' d <' = 



r 



I si 

 J 0' 



sin <' cos 2 <f>' d 



cos 2 0' / 



I = \-jrr* (68) and for stress at the neutral axis, 

 constant over the full width, so that 



= 0, and s 8 is 



T90 



s =^2- / sin <' 

 JO 



This shows that the unit shearing stress at the neutral axis of 

 a circular cross section is 33 1-3% greater than the mean value. 

 Equation (22) is not applicable to an I section on account 

 of the sudden change in width at the junction of the flange and 

 web. For such a section the usual assumption of uniform distri- 

 bution of the shearing stresses, over the section of the Web only, 

 is on the side of safety. These stresses must be zero at the upper 

 and lower edges of each flange, but since the flanges take some 

 stress, the effect is to approximate to a uniform distribution of 

 stress vertically Thus the actual maximum unit stress will be 

 about the same as the unit stress on the assumption. 



According to the above investigation, the 

 shearing stress is the greatest at the neutral axis 

 and becomes zero at the upper and lower edges. 

 The law of variation between these extremes de- 

 pends upon the cross section. For a rectangular 

 cross section, the strrsscs vary as the ordinates to 



3 <y 

 a parabola whose middle ordinate is ^-j. This is 



Fig. 85. is nitrated in Fig. So. 



76. Resultant of Bending and Shearing Stresses. Principal 

 Stresses in Beams. The bending and shearing stresses in beams 

 and girders have so far been considered separately, and it has 



