§11.11 Strain Energy 267 B(+8P) Pa A Unloaded beam position Beam loaded with Pa,etc Beam loaded with P,Pa T8. 8 Pe,etc.plus extra load 8P Fig.11.9.Any beam or structure subjected to a system of applied concentrated loads P Pg PC...PN,ctc. If one of the loads,P,is now increased by an amount &P,the changes in deflections will be δa,6bandδc,etc.,as shown in Fig.11,9. Load at A Load at B 86 8a Pa P+8R Extra work o -(8*82)8加 Extension Extension 00+8a b+8b Fig.11.10.Load-extension curves for positions A and B. Extra work done at A (see Fig.11.10) =(PA+δP)δa Extra work done at B,C,etc.(see Fig.11.10) =Paδb,Pcδc,etc. Increase in strain energy total extra work done U=PA6a+PAa+P8ob+Pcoc+... and neglecting the product of small quantities δU=P4δa+Ps6b+Pcδc+... (11.15) But if the loads P+P,Ps,Pc,etc.,were applied gradually from zero the total strain energy would be U+U=∑×load×extension U+δU=支(P4+δP)(a+δa)+Pab+δb)+Pc(c+δc)+.. =Pua+P4δa+δPua+δP4δa+Pab+Paδb+Pcc+Pcc+· Neglecting the square of small quantities (Pa)and subtracting eqn.(11.14), 6U=δP4a+2P48a+Psδb+是Pcc+. or 2δU=δPA4a+Paδa+Pgδb+Pcc+..$11.11 Strain Energy 267 *- Unlooded beam position Beam loaded with > . -__-- PA, PB,pc, e'c 1- -Beam loaded with P,,P,, Pc , etc plus extra load 8% Fig. 11.9. Any beam or structure subjected to a system of applied concentrated loads PA, P,, P, . . . P,, etc. If one of the loads, PA, is now increased by an amount SPAthe changes in deflections will be Sa, Sb and Sc, etc., as shown in Fig. 11.9. Load at A Load at B a o+80 b b+8b Fig. 11.10. Load-extension curves for positions A and E. Extra work done at A (see Fig. 11.10) = (PA+fdPA)da Extra work done at B, C, etc. (see Fig. 11.10) = PBSb, Pc6c, etc. Increase in strain energy = total extra work done .. 6u = PA6a+36PA6a+P,6b+Pc6C+ . . . and neglecting the product of small quantities 6U=PA&l+P,db+Pc6C+ . . (1 1.15) But if the loads PA+ 6PA, PB, Pc, etc., were applied gradually from zero the total strain U + SU = 14 x load x extension energy would be u+6u =3(PA+6PA)(a+ba)+4P,(b+6b)+3Pc(C+6C)+ . . . = +PA a ++PA 6a ++ 6P, a ++SPA ha +:p, b ++P,6b +4Pcc +iP,6C + . . . Neglecting the square of small quantities (f6PAGa) and subtracting eqn. (1 1.14), 6U=+6PAa+3PA6a+3P,6b+4Pc6C+ . . . or 26u = 6PAa+PA6a+Pg6b+PCbC+ . .