14 BROOMALL : 



the beam a vertical shear is assumed to act, a stress neces- 

 sary for equilibrium in conjunction with the horizontal 

 stresses. This vertical shear is assumed to produce a uniform 

 shearing unit stress over the cross section of the beam. These 

 horizontal stresses and vertical stresses are all that we need 

 assume as far as the equilibrium of the external forces is con- 

 cerned. This is the Common Theory of Flexure. 



In the development of the theory of stresses probably one 

 of the first steps in advance was the recognition of the exist- 

 ence of horizontal shear. The necessity for its presence is 

 easily seen, for example in the case of a number of boards 

 lying face to face and made use of to act as a beam. Natur- 

 ally such a composite beam has little strength or stiffness until 

 the boards are spiked or bolted together so that they cannot 

 slide upon one another. When this sliding is prevented the 

 structure becomes nearly as strong and stiff as a solid beam 

 of the same size. 



But the existence of horizontal shear, by the principles of 

 statics, necessarily requires the presence at each point in the 

 material of an equal vertical shear of opposite character, in 

 order to prevent the rotation of the infinitesimal elements 

 around their own axes. In the same way the existence of 

 horizontal shear might have been predicted from the known 

 existence of the external applied vertical shear. However, 

 when we calculate the value of the horizontal shear in a 

 beam by the usual method, it is found to be zero at the upper 

 and lower surfaces and a maximum at the neutral axis. This 

 being so, it follows, since the vertical shear at each level must 

 have the same value as the horizontal, that the vertical shear 

 is not uniformly distributed over the section, but is zero at 

 top and bottom and a maximum at the neutral surface. In a 

 rectangular beam the value of the horizontal and vertical 

 shears at the neutral axis is 50 per cent, greater than the 

 value found by assuming the shear distributed equally over 

 the vertical cross section. 



Having now recognized and calculated the shears, the 



