MOMENT AND SHEAR IN BEAMS 57 



R 1 ; between the loads P x and P 2 the shear equals R^ P x ; between 

 the loads P 2 and P 3 the shear equals RI P P 2 ; between the loads 

 P 3 and P 4 the shear equals 7^ P t P 2 P 3 ; and between load P 4 

 and the right reaction the shear equals R P 1 P 2 P 3 P 4 = 

 R 2 . At load P 2 the shear changes from positive to negative. Diagram 

 (c) is called a shear diagram. It will be seen that the maximum 

 ordinate in the moment diagram comes at the point of zero shear. 



The bending moment at any point in the beam is equal to the 

 algebraic sum of the shear areas on either side of the point in question. 

 From this we see that the shear areas on each side of P 2 must be equal. 

 This property of the shear diagram depends upon the principle that the 

 bending moment at any point in a simple beam is the definite integral of 

 the shear between either point of support and the point in question. 

 This will be taken up again in the discussion of beams uniformly loaded 

 which will now be considered. 



Moment and Shear in Beams: Uniform Loads. In the beam 

 loaded with a uniform load of w Ibs. per lineal foot shown in Fig. 38, 

 the reaction J? = R 2 = y^ w L. At a distance x from the left support, 

 the bending moment is 



which is the equation of a parabola. 



The parabola may be constructed by means of the. force and equi- 

 librium polygons by assuming that the uniform load is concentrated at 

 points in the beam, as is assumed in a bridge truss, and drawing the 

 force and equilibrium polygons in the usual way, as in Fig. 38. The 

 greater the number of segments into which the uniform load is divided 

 the more nearly will the equilibrium polygon approach the bending 

 moment parabola. 



The parabola may be constructed without drawing the force and 

 equilibrium polygons as follows : Lay off ordinate m n = n p = bend- 

 ing moment at center of beam = ^ w L 2 . Divide a p and b p into the 

 same number of equal parts and number them as shown in (b). Join 

 the points with like numbers by lines, which will be tangents to the 



