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【正文】 B carries two loads , a concentrated force P and a couple Mo, acting as shown in (a). Find the shear force and bending moment in the beam at cross sections located as follows: (a) a small distance to the left of the middle of the beam and (b) a small distance to the right of the middle of the beam .Solution The first step in the analysis of this beam is to find the reactions RA and RB. Taking moments about ends A and B gives two equations of equilibrium, from which we find RA=3P/4 – Mo/L RB=P/4+mo/LNext, the beam is cut at a cross section just to the left of the middle, and a freebody diagram is drawn of either half of the beam. In this example we choose the lefthand half of the bean, and the corresponding diagram is shown in (b). The force p and the reaction RA appear in this diagram, as also do the unknown shear force V and bending moment M, both of which are shown in their positive directions. The couple Mo does not appear in the figure because the beam is cut to the left of the point where Mo is applied. A summation of forces in the vertical direction gives V=R – P= P/4M0/LWhich shown that the shear force is negative。 hence, it acts in the opposite direction to that assumed in (b). Taking moments about an axis through the cross section where the beam is cut [(b)] gives M = RAL/2PL/4=PL/8Mo/2Depending upon the relative magnitudes of the terms in this equation, we see that the bending moment M may be either positive or negative . To obtain the stress resultants at a cross section just to the right of the middle, we cut the beam at that section and again draw an appropriate freebody diagram [(c)]. The only difference between this diagram and the former one is that the couple Mo now acts on the part of the beam to the left of the cut section. Again summing force in the vertical direction, and also taking moments about an axis through the cut section , we obtain V= P/4 Mo/L M=PL/8+Mo/2We see from these results that the shear force does not change when the section is shifted from left to right of the couple Mo, but the bending moment increases algebraically by an amount equal to Mo . ( Selected from: Stephen and James M. Gere,Mechanics of materials, Van Nostrand reinhold Company Ltd.,1978.)
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