. Proceedings of the annual convention . i (^)] m[,.»{^)] rl» (m +1) I [-(^)T ^^ = ^* L [iT^bn] Ml [x Area under -^ Diogram in Length bLIx MaA=—^ L xdx [..(-^)] = 2(m + l) (2o)» Centroid of Areo under .511 Oiogrom in Length bL :Ix Ma x^dx L r. / bL- X xT0 [+n^( bL ^J ^b- ^ _ ,, 2(m + l) r. „ f„ * n x m liLzlil (■^„\*- bL — Logg(m + l)+m — (oai See Mondbooli of Frome Cons fonts, p. 30, Published by Portlond Cement Associotion,Chicogo, Illinois, 1947, for prior publication. FIGURE A3 Equivalent Joi nt Method of Rigid Frame Analysis 41 FORMULAS FOR p,q,u,v. CUBIC VARIATION OF I.

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. Proceedings of the annual convention . i (^)] m[,.»{^)] rl» (m +1) I [-(^)T ^^ = ^* L [iT^bn] Ml [x Area under -^ Diogram in Length bLIx MaA=—^ L xdx [..(-^)] = 2(m + l) (2o)» Centroid of Areo under .511 Oiogrom in Length bL :Ix Ma x^dx L r. / bL- X xT0 [+n^( bL ^J ^b- ^ _ ,, 2(m + l) r. „ f„ * n x m liLzlil (■^„\*- bL — Logg(m + l)+m — (oai See Mondbooli of Frome Cons fonts, p. 30, Published by Portlond Cement Associotion,Chicogo, Illinois, 1947, for prior publication. FIGURE A3 Equivalent Joi nt Method of Rigid Frame Analysis 41 FORMULAS FOR p,q,u,v. CUBIC VARIATION OF I. A2=Areo=MBL^-^. M K. r(m+2)-b(m+l)l ^ MI Area under -=— diagram In length bL. J-x A,= Mb L ,bL (L-x)dx r(nH-2)-b(m + i)1 MI Centrold of area under -=— diogram In length bL Mb L pbLx(L-x)dx Xh = J(m^2)-b(m-H)-1 ■ L 2(m+l)2 J (m-<-2)-b(m + l)2{m+l)^ FIGURE A4 42 Iron and Steel Structures FORMULAS FOR p, q,u,v CUBIC VARIATION OF I X , i Xo bL Aa. ^^^bM^, ^b A A,i N^ ^>^ uL Centroids—^ L Xb A,+ XA2= L(l-u)AX|5 A|+ X Ae u= I- LA -^ m(l-b^)+l^ 2(m-»-l) M<

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