Mechanics of international work (or work of deformation) in elastic bodies and systems in equilibrium

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 Excerpt: ...purpose dM x in enabling us to get-jp-, we might have put P equal to zero in the expression for M above, before substituting in the integration, and it is also interesting to note that the value,dM.,..,, of-Tp is the same as the moment at any section due to a load of unity applied vertically at the point B (thus leading to "Fraenkel's Formula.". Of course, also, by putting W= zero in eq. (82) we may obtain the deflection due to P alone and thus check the result already obtained in eq. (78). 41. Example III. Deflection. Beam with Overhang. A continuous prismatic beam (homogeneous and originally straight) rests on two unyielding slip3clrts at the same level as shown in Fig/18., TheJ verticil' force1 P is applied at D, midway between 0 and B. while Jhi extremity of the overhanging portion carries &,vertical, load Q. The length BC equals 0B, =2a. Required the vertical deflection of the point D, neglecting the work of shear (that due to thrust being zero, as before). By eq. (25) this deflection where U See Mr. C. W. Hudson's paper on "Deflection of Beams of Variable Moment of Inertia," in Vol. LI (Dec, 1903) of the Transac. Am. Soc. Civ. Engineers, p. 1. rc M2dx is the total internal work= J gj. This summation will be separated into the three integrals (reckoning x for each segment in the way shown in Fig. 18) for 0D, DB, and BC, respectively, Fig. 18. In this case we note that from ordinary statics the reaction P at 0 is V=---Q, and obtain the following expressions for M and its various derivatives in the three segments, viz., _., Tr IP n. dM x For 0D, M = Vx=-QJ x; and p--g; for points on DB, in which, wnen values are substituted from above relations for M, etc., there results In this equation, if we now make Q equal to zero, we have th...