ANALYSIS OF STRESS IN BEAMS 57 



section is moved from this position parallel to itself a distance dx, 

 say to the position EFGH in the figure, the intensity of the stress at 

 P is increased by the amount 



- dx I I- 



The difference between the normal stresses acting on these two 

 adjacent cross sections tends to shove the point P in a direction 

 parallel to the axis of the beam, and this tendency is resisted by 

 a shearing stress of intensity q at P, also parallel to the axis of 



the beam. Therefore, since the resultant normal stress on the area 



* 



/2 



BCEF is I dp-dF, and the resultant shearing stress on the area 



AB CD is qbdx, h 



/t 

 dp dF = qbdx. 



Substituting the value of dp from equation (27), 



whence 



(28) q = 



Formula (28) applies to any cross section bounded by parallel sides. 



In Article 23 it Was proved that whenever a shearing stress acts 

 along any plane in an elastic solid, there is always another shearing 

 stress of equal intensity acting at the same point in a plane at right 

 angles to the first. Consequently, formula (28) also gives the intensity 

 of the stress at any point P in a direction perpendicular to the neutral 

 axis of the section. 



For a rectangular cross section 



and hence 



