36 MR CLERK MAXWELL ON 



so that if/, g, h in the diagram of stress correspond to F, G, H in the original 

 figure, we have 



f = 2-p 2q cos2q<f> g = -p?cosq<p h = - pi sin q<f> . (28). 



Case of a Uniform Horizontal Beam. 



As an example of the application of the condition that the stresses must be 

 such as are consistent with an initial condition of no strain, let us take the case 

 of a uniform rectangular beam of indefinite length placed horizontally with a 

 load = h per unit of length placed on its upper surface, the weight of the beam 

 being k per unit of length. Let us suppose the beam to be supported by vertical 

 forces and couples in a vertical plane applied at the ends; but let us consider 

 only the middle portion of the beam, where the conditions applicable to the ends 

 have no sensible effect. Let the horizontal distance x be reckoned from the 

 vertical plane where there is no shearing force, and let the planes where there 

 is no moment of bending be at distances ± a from the origin. Let y be 

 reckoned from the lower edge of the beam, and let b be the depth of the beam. 



Then, if IT - — j—r is the shearing stress, the total vertical shearing force 



through a vertical section at distance x is 



and this must be equal and opposite to the lveight of the beam and load from 

 to x, which is evidently (// + k)x. 



Hence 



dF 



— =-(h + k)xcj>(jj) where 0(6) - </>(0) = 1 . . (29). 



From this we find the vertical stress 



Q = 5? + J* = -(* + *)*<*) + 5 If. 



The vertical stress is therefore a function of y only. It must vanish at the 

 lower side of the beam, where y — 0, and it must be — h on the upper side of 

 the beam, where y = b. The shearing stress U must vanish at both sides of the 

 beam, or <f>'(y) = 0, when y = 0, and when y = b. 



The simplest form of <p (y) which will satisfy these conditions is 



<Ky) = p (3% 2 - 2^) . 



Hence we find the following expression for the function of stress by integrating 

 (29) with respect to x, 



F = ^ ^ ~ ^ VW ~ 2 ^ + Y • • • • ( 3 °)> 



