Text-book of mechanics . ^i « Fig. 17Exercise 33. Write equations for and andconstruct the and diagrams for the loadings shownin Figs. 18 and 19. STRESSES IN BEAMS 31 Exercise 34. In the uniformly loaded beam in Fig. 19find the points at which the is zero. W Q3 M, Iii Fig . 18 TT w w 1 Qx h* h I k 1 Mx / \ : SB ^Xj/ \/^ Fig. 19 Relation between Shearing Force and Bending Mo-ment.— The above discussions and diagrams of , show that whenever an abrupt change in the 32 MECHANICS OF MATERIALS loading occurs a discontinuity in the equations of , res
Text-book of mechanics . ^i « Fig. 17Exercise 33. Write equations for and andconstruct the and diagrams for the loadings shownin Figs. 18 and 19. STRESSES IN BEAMS 31 Exercise 34. In the uniformly loaded beam in Fig. 19find the points at which the is zero. W Q3 M, Iii Fig . 18 TT w w 1 Qx h* h I k 1 Mx / \ : SB ^Xj/ \/^ Fig. 19 Relation between Shearing Force and Bending Mo-ment.— The above discussions and diagrams of , show that whenever an abrupt change in the 32 MECHANICS OF MATERIALS loading occurs a discontinuity in the equations of , results. In any interval between such discontinuities a definiterelation exists between the and the To establish this relation, we may consider the equi-librium of a portion of any beam between two sectionsdx apart and x units from the origin, Fig. 20. , jQx+dQas. From the relations shearing force = resistance to shear,and bending moment = resisting moment, it is evident that we may use Qx and Mx to indicatethe total effect of the internal stresses at the section xunits from the origin, and similarly Mx + dMx andQx + dQx represent the action of the internal stresses atthe section x + dx units from the origin as shown inFig. 20. The load on the element is wdx, where w is the rateof loading, and the line of action of the resultant load, STRESSES IN BEAMS 33 wdx, is located by the distance ndx, where n is somefraction which depends upon the law of change in the element is in equilibrium under the action of theforces shown, we may equate the sum of the momentsof these forces about 0 to zero. Thus, Qxdx + Mx —(Mx + dMx) — wdx {ndx) = o, and as the differentialsof higher order must be omitted we have Qxdx — dMx = o, whence Qx = —-*? • dx Thus, in any interval between concentrated loads(or reactions) or between any such loads and any
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