154 



SHEAR AND MOMENT DIAGRAMS. 



Art. 91. 



It should be remembered that the condition of perfect fixity 

 at the ends of beams is seldom found in practice, except in 

 beams continuous over a support as in a double cantilever. 



In certain constructions, the greatest deflection, and not 

 the maximum fiber stress will govern. 



Some cases not shown in the table may be gotten by simply 

 turning the figure upside down, making loads of reactions and 

 reactions of loads. 



The student should, whenever possible, avoid using formu- 

 las in numerical problems; the fundamental principles should 

 be applied. 



91. Shear and Moment Diagrams. The shear and mo- 

 ment diagrams, Figs. 94 to 112, are laid out from horizontal 

 lines, the ordinates on opposite sides of these lines having oppo- 

 site signs. It is usual to consider a shear acting upward on the 

 part to the left of a section as positive, and such shears are laid 

 out above the base line. For the same reason, bending moment? 

 which make a beam concave on its upper surface (as in an ordi- 

 nary beam) are positive, but are laid out below the base line. 



These diagrams are convenient in showing how the shear 

 and moment vary along a beam, and may be constructed by 

 laying out ordinates calculated from the general equations given, 

 or by the graphic methods of chapter V . They are particularly 

 instructive in showing that where the shear passes through zero, 

 the moment is a maximum (or minimum), and that the points 

 of zero moment are points of contra-flexure (89). The area of 

 the shear diagram to the left (or right) of a section, is equal to 

 the ordinate of the moment diagram at the section (74). 



The shear and moment diagrams may usually be drawn 

 after several ordinates have been laid out, because their char- 

 acter is usually apparent upon inspection of the general equa- 

 tions for 8 and M; the moment diagram for uniform load, for 

 example, is always bounded by a parabola. 



92. Shears and Moments in Cantilever Beams. Fig. 94. 

 If the origin is taken at the free end, Px = M x and M max = PL 

 evidently. If the origin is taken at the fixed end M X = P(L #) 

 or M X = M A R l x = PLPx = P(Lx), the sign of M A being 

 evident. This shows that if there is a moment at a support, it 

 must not be neglected, if that point is in that part of the struc- 



