Aircraft Structures. CHAPER 10. Energy methods. Active Aeroelasticity and Rotorcraft Lab. Prof. SangJoon Shin

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1 Arcraft Structures CHAPER. Energy methods Prof. SangJoon Shn Actve Aeroelastcty and Rotorcraft Lab.

2 - vrtual work prncples ) PVW : entrely equvalent to the equlbrum eqns. However, does not provde any nformaton about the other sets of eqns. { Stran-dsplacement relatonshp Consttutve laws ) PCVW : entrely equvalent to the stran-dsplacement relatonshps sets of eqns. { Equlbrum eqns Consttutve laws Type of forces - In vrtual work prncples, varous categores of forces are clearly defned and used. Internal, external forces Reacton forces : can be elmnated from the formulaton snce the work they perform vanshes when usng knematcally admssble vrtual dsplacements But, when arbtrary vrtual dsplacements are used, the vrtual work does not vansh Become an ntegral part of the formulaton

3 Conservatve forces - The work they perform always vanshes for a closed path dsplacement - Total mechancal energy of the system s preserved - If the externally appled forces are conservatve, they can be derved from a potental further smplfy the calculaton of VW - If the stran energy of an elastc component exsts, the correspondng elastc forces can be derved from ths stran energy further smplfy the calculaton of VW combnaton of PVW Stran energy Potental of external forces Prncple of mnmum total potental energy PVW s always vald PMTPE s lmted to systems nvolvng conservatve forces 3

4 . Conservatve forces r F : poston vector of a partcle : force actng the partcle, depends only upon the poston of the partcle, F F( r) - Fg.. two arbtrary paths ACB, ADB 4

5 . Conservatve forces Defnton F - s conservatve f the work t performs along any path jonng the same ntal and fnal ponts s dentcal W F dr F dr ACB ADB (.) - Work done along path ADB = (-). that along BDA - Work over the closed path ACBDA = W F dr F dr anypath C (.) Potental of a conservatve force - Stoke s theorem W F dr r FdA A r C A : area enclosed by curve C : outward normal to area A (Fg..) F (.3) ( : arbtrary scalar functon) 5

6 . Conservatve forces - Soluton of eqn. F F potental justfed later F x x x - Work done by a conservatve force 3 3 (.4) (.5) r W F dr dr r r r r r ( dx dx dx ) d ( r ) ( r ) x x x r r 3 3 depends only on the poston of ntal/fnal ponts can be evaluated as the dfference between the values of the potental functon W ( r) ( r ) (.6) 6

7 . Conservatve forces Examples of conservatve forces ) Gravty force mgr3 mgx3 F / x mg g x 33 3 x W 3b 3b F dr dx x x x3a x3a ( ) ( ) g 3 3a 3b x3 ) Restorng force of an elastc sprng ku A( u) ku stran energy A FS ku u ub ub A W FS du du Au ( a) Au ( b) u a ua u restorng force Potental elastc force 7

8 . Conservatve forces.. Potental for nternal and external forces - In PVW, a dstncton s made between { Internal forces Externally appled loads - In elastc systems, nternal forces { Stresses actng n a body Elastc forces n structural components Potental of nternal forces = stran energy, deformaton energy, nternal energy WI A (.7) A - Potental of external forces - Total potental energy Φ WE (.8) A (.9) 8

9 . Conservatve forces - Total work done by both nternal and external forces W W W A (.) I E for conservatve systems, the work done by the nternal and external forces = negatve change n total potental energy - Addng an arbtrary constant to the potental fn. wll not alter the work done 9

10 . Conservatve forces.. Calculaton of the potental fns - Potental of nternal forces stran energy, A( ) A A( ) It s convenent to select, undeformed or unstraned state W A[ A( ) A( )] A( ) I A( ) WI - It s cumbersome to compute the work done wthn a sold as the negatve product of the nternal stress component actng through strans or deformatons (.) alternatve approach Eq. (9.9), W I W A( ) WE E (.) f the nternal forces n a sold are conservatve, the work done by the externally appled forces = stran energy stored n a body

11 . Conservatve forces - assumpton the forces are appled slowly, n a quas-steady manner assocated knetc energy s neglgble potental of the externally appled loads, Φ negatve of the work done by the external forces actng through the dsplacements. N P forces, P, const. magntude, lne of acton fxed n space dead loads - Non-conservatve forces ) Aerodynamc force E j j Lft AOA, non-conservatve, cannot be derved from potental N P N O W Pd Q (.3) ) Follower force Const. magntude, but the orentaton of ther lne of acton changes wth the rotaton of structures Ex) thrust of a rocket jet engne

12 . Prncple of mnmum total potental energy - System represented by N generalzed coord. - If the system s conservatve, stran energy potental of the externally appled loads q { q, q,..., q } T N A Aq ( ) ( q) Infntesmal ncrement - VW done by the nternal forces external forces N A A A A da dq dq... dqn dq q q q q N N d dq dq... dqn dq q q q q W W I E N A( q) ( q) N A WI A( q) dq q N WE ( q) dq q (.4) (.5)

13 . Prncple of mnmum total potental energy - Comparng Eq.(9.4) and (.5) Q I A q, Q E q (.6) Q I Q E - PVW :, by ntroducng Eq.(.6) A ( A ) q q q q (.7) W where, s total potental. Prncpal 4 : a system s n statc equlbrum f the sum of the VW done by the nternal and external forces vanshes for all arbtrary vrtual dsplacements, W (.8) 3 N q, Eq.(.7) q q (.9)

14 . Prncple of mnmum total potental energy Prncple 8 : A conservatve system s n equlbrum f vrtual changes n the total PE vansh for all vrtual dsplacements. Prncple of statonary TPE - Knematcally admssble vrtual dsplacements are used reacton forces are elmnated from the formulaton. Arbtrary vrtual dsplacements are used reacton forces must be treated as externally appled loads. - Graphcal llustraton of Prncple 8 (Fg..3) TPE s statonary at ponts A, B and C. 4 Fg..3 Total potental energy.

15 . Prncple of mnmum total potental energy - Increments n TPE by Taylor seres n the neghborhood of statc equlbrum 3 > for all TPE s mnmum at equlbrum stable (A) TPE cannot ncrease wthout an external source of E = neutrally stable (B) N N N j q j qqj d dq dqdq dq N N d dqdq j q qj < unstable (C) released PE s converted to KE, leadng to spontaneous moton of the system Prncple 9 : A conservatve system s n a stable state of equlbrum f the TPE s a mn. w.r.t. changes n the generalzed coord. 5

16 . Prncple of mnmum total potental energy.. Non-conservatve external forces - If the externally appled loads are not conservatve nc W W W AW L E E Prncple : A system s n equlbrum f vrtual changes n the stran energy equal the VW done by the externally appled loads for all arbtrary vrtual dsplacements. - If externally appled forces are a mxture of conservatve non-conservatve forces W W W c nc E E E ( A ) W nc E VW done by the non-conservatve forces 6

17 .3 Stran energy n sprngs - Stran energy functon of deformaton of the structure deformaton feld functon of sprng A AE ( ) rectlnear sprng torsonal rotatonal sprng dplcement feld generalzed coord..3. Rectlnear sprngs - prmary lumped propertes stffness constant unstretched length : u F F F( ), uu F( ) F( u u ) - force appled to the sprng :, force n the sprng : consttutve behavor : : extenson FS 7

18 .3 Stran energy n sprngs Lnearly elastc sprng - Relatonshp between an appled load and the resultng extenson s ( F k) lnear sprng s lnear k : stffness constant, unt : force/length, N/m - Stran energy n the sprng A W u Fdu u k du k d k F E u u (.) : postve-defnte fn. of,.e. A> for any (+) or (-) vanshes only when - nternal force n the sprng F S A u k (-) : force n the sprng opposes the externally appled force. - consttutve law : straght lne n the force vs. extenson plot (Fg..5) stran energy (A) : shaded area under the curve 8

19 .3 Stran energy n sprngs Fg..5. Consttute law a lnearly elastc sprng Fg..6. Consttute law a nonlnearly elastc sprng - Complmentary stran energy (A ), stress energy : shaded area to the left of the straght lne, force energy F F F F F ( ) (.) A u u df df df F k k F A F k A k A A F, A A F (.3) 9

20 .3 Stran energy n sprngs Nonlnearly elastc sprng - metals(alumnum, copper) slght amount of nonlnearly elastc behavor pror to yeld pont elastomers qute obvous nonlnearly elastc behavor - analytcal models, the smplest form F u : ref. force, : ref. dsplacement F F tanh u (.4) -Fg.(.6) alumnum, no sharp transton from lnear to nonlnear behavor F F sech sech u u u F :, u : stffness of the sprng at zero elongaton

21 .3 Stran energy n sprngs - Stran energy complementary stran energy - n contrast to the lnearly elastc sprng,, however, - elastc force n the sprng - Fg.(.7), A Fdu Fu tanh dufu ln(cosh ) F F arctanh( ) ( arctanh ln ) A df F u F df u F F F F A A A A F A F Fu ln(cosh ) Ftanh u u (.5) upper stran energy or potental mddle force-extenson relatonshp : softenng sprng, decreasng stffness at hgher extensons Fg..7. Nonlnear sprng wth the consttutve law gven by eq.(.4). Top fgure : stran energy; mddle fgure : force; bottom fgure: stffness. Sold lne: nonlnear sprng; dashed lne: lnear sprng.

22 .3 Stran energy n sprngs.3. Torsonal sprngs - Angular moton,, under the acton of an externally appled torque, M (Fg..9) - lnearly elastc torsonal sprng : - : unt k Nm/ rad, N m/ deg M k Fg..9. Tosonal sprng subjected to a moment, M..3. Bars - stran energy e : bar elongaton EA A ke e L (.9)

23 .4 Stran energy n beams.4. Beam under axal loads - Beam subjected only to axal loads (Fg. 5.6) - nfntesmal slce, left force dsplacement - nfntesmal slce, rght force dsplacement - Left force, axal force N, dsplacement from to, work : Nu -, (-) due to that dsplacement and force are counted postve n opposte drectons -rght force, work : - total work : - external work : u u du dx dx u du Nu dx dx du N dx N dx dx dwe Ndx Sdx (.33) 3

24 .4 Stran energy n beams : stran energy densty functon potental of the axal force, a S N a( ) S Internal force n the beam (.34) - total stran energy by the axal force dstrbuton - n terms of the axal force A( ) a( ) dx S dx L L (.35) L N A( ) dx A ( N ) S a( N ) N S " total stress E " " complementary E " (.36) : stran energy densty functon complementary stran energy densty 4

25 .4 Stran energy n beams.4. Beam under transverse loads 5 - Beams subjected only to transverse loads (Fg. 5.4) - left force rotaton : - rght force rotaton : - work by bendng moment at left force : (-) due to that rotaton and moment are counted postve n opposte drectons - work by bendng moment at rght force : - total work : - external work : du dx du dx d u dx M 3 M 3 dx du M 3 dx M 3 3dx dx M M 3 du dx du d u 3 dx dx dx sectonal curvature c dwe M 33dx H333dx Fg Beam subjected to transverse loads. (.37)

26 .4 Stran energy n beams a( ) c potental of the bendng moment : E - Total stran by the bendng moment dstrbuton or or H : stran energy densty fn (.38) a( 3) c M3 H333 A( ) a( ) dx H dx L L c L c du 33 dx Au ( ( x)) H dx M AM ( ) dx A( M) L 3 3 c 3 H 33 M a( M3) : H 3 c 33 stress E densty fn 3 Internal moment n the beam (.39) (.4) (.4) 6

27 .4 Stran energy n beams.4.3 Beam under torsonal loads - crcular cylndrcal beam subjected to torson - rotaton of the left force : - rotaton of the left force : M d dx dx - work by the torque at the left force : M (-) due to that rotaton and torque are counted postve n opposte drectons - work by the torque at the rght force : - total work : - external work : d M M dx dx d M dx M dx dx sectonal twst rate dwe M dx H dx (.4) 7

28 .4 Stran energy n beams potental of the torque: a( ) H a( ) M : stran energy densty fn H (.43) (.44) - Total stran energy by the torque dstrbuton or or A( ) a( ) dx H dx M AM ( ) dx A( M) L L L H M a( M) : H stress E densty fn total complementary stran E stored (.39) (.4) (.4) 8

29 .4 Stran energy n beams.4.4 Relatonshp wth VW - nternal VW by a bendng moment Eq.(9.69) However, n Sec..4, stran energy stored n beam s - nternal VW : bendng moment s assumed to reman constant whle undergong a curvature E M : dw M dx, dwe M dx I 3 I 3 3 dw dw M dx factor dfference 3 3 dw M dx M d dx M dx E

30 .4 Stran energy n beams - Stran energy stored n beam : bendng moment s assumed grow n proporton to the curvature 3 3 dw M dx k d dx k dx - Same reasonng for torson Internal, external VW : Stran energy : E - When computng VW and CVW : vrtual dsplacements do not affect the forces or stresses n the system Stran energy stored n the structure : nternal forces and moments ncrease n proporton to the deformaton dwe dwi M dx dwe H dx factor dfference M dx 3 3 3

31 .5 Stran energy n solds.5. 3-D sold - Sec , work done by the constant, external stress - Then, f the stresses ncrease n proporton to the deformatons - Hook s law, C W W E E T V T V dv dv Eq.(9.76) E C ( )( ) (.46) (.4) 3

32 .5 Stran energy n solds W E a( ) E [( )( 3) ( 3 3) V ( )( ) ( 3 3 )] dv a ( ) dv A ( ) V : stran E densty fn for a 3-D sold -more compact form I, I E a( ) [( ) I ( ) I] ( )( ) : frst nvarants of the stran tensor, Eqs.(.86) - Hook s law s a lnear relatonshp (.48) T a( ) C (.49) a( ) a( ) - complementary stran E densty a( ) [ 3 ( 3 3) E ( )( )] 3 3 (.5) 3

33 .5 Stran energy n solds S (.) S E ( ) ( ) ( ) T a( ) S (.) (.5) 33

34 .5 Stran energy n solds.5. 3-D beams - Eq.(9.78) : nternal W done by const. stress results n 3-D beams - W done by the same stress resultants when they ncrease n proporton to the deformaton L WE ( N M M ) dx Hook s law, sectonal consttutve laws, Eq.(6.) A S H H H dx - complementary stran E usng the complance form, Eq.(6.3) assumng that the orgn must be located at the secton s centrod (.53) L c c c c ( ) (.54) L N H H H A M M M M dx S H H H c c c where, H H H H c c c

35 .6 Applcatons to trusses and beams.6. Applcaton to trusses 3-bar, hyperstatc truss (Fg..6) - bar length : L L L 3, L cos L e u cos u sn, e u, - bar elongatons : Eq.(9.7), e u cos u A ke k EA L 3 sn - bar stran E :, Eq.(.9), (bar stffness) EAcos EA EAcos A e e e L L L EA [( u cos u sn ) cos u L u u 3 ( cos sn ) cos ] EA u cos sn cos u L 3 Fg..6. Smple 3-bar truss 35

36 .6 Applcatons to trusses and beams - potental of externally appled load, total potental A APu P Pu - D.O.F. s, PMTPE, Eq.(.7) EA 3 u cos P u L EA u L ( sn cos ) u - Matrx form two lnear eqn.s for the generalzed coord. 3 z cos L sn cos u EA PL u, u 3 EAcos u P 36

37 .6 Applcatons to Truss and Beams.6.3 Applcatons to beams - Potental of the externally appled loads p ( x ) beam under a dstrbuted transverse load,, Fg. 5.4 L p ( x ) u ( x ) dx - Total Potental E of the beam...from Eq.(.9) (.58) L L c du 33 d x A H dx p u dx u ( ( x )). now, a functon of another functon functonal Eq.(.4) Beam problems are nfnte dmensonal or contnuous problems snce determnaton of the transverse dsplacement feld, u ( x ) planar truss w/ N unknowns, fnte dmensonal, dscrete 37

38 .6 Applcatons to Truss and Beams Mnmzaton of the TPE of fnte dmenson standard calculus functonal calculus of varatons Reducton of nfnte # of D.O.F fnte #..by choosng specfc u ( x ) functons for Chap. 3-D beam under complex loadng condton dstrbuted loads,, P P P 3 concentrated loads,, p ( x ) p( x) p3( x) dstrbuted moment,, Q Q Q 3 concentrated moment,, L q ( x ) q( x) q3( x) L p u dx Pu L q dx Q L 3 3 L L du du pudx Pu L q dx Q L dx dx L L du du pudx Pu L q dx Q L dx dx (.59) 38

39 .6 Applcatons to Truss and Beams du Euler-Bernoull assumpton, 3 Q3 3 L dx du Q L 3, Q L dx 3 Q du dx du dx 3 L 39

40 .8 Prncple of mnmum complementary energy Sec Prncple of Vrtual Work Prncple of Mnmum Total Potental Energy two assumptons ---- nternal forces are conservatve stran Energy external forces are also conservatve potental of the externally appled loads Fgure.7 -- consttutve relaton shp nd assumpton not shown stran energy 4

41 .8 Prncple of mnmum complementary energy Prncple of mnmum complementary energy Prncple of complementary vrtual work two assumptons ---- complementary stran energy functon prescrbed dsplacements can be derved from a potental Sec The potental of the prescrbed dsplacements - 3- bar truss, prescrbed dsplacement at B drvng force D, unknown quantty 4 Fg..8 Three-bar truss wth prescrbed dsplacement - Prncple of complementary vrtual work, Eq.(9.57) W ' E D now t s assumed that the prescrbed dsplacement can be derved from a potental, '( D) D ' potental of the prescrbed dsplacement or dslocaton potental ' ' WE D D'( D) D (.) (.)

42 .8 Prncple of mnmum complementary energy.8. Consttutve laws for elastc materals A ke stran energy for a bar, bar forces EA k L Ae () F e ke A' F k complementary stran energy, A A' elongaton A() e ke lnearly elastc materal,,, e k : complance AF ( ) F F k A'( F) F k 4

43 .8 Prncple of mnmum complementary energy - elastc, but not lnear Eq.(.3) ' A() e A( F) ef dfferentate, Regroupng A A' de df Fde edf e F A A' F dee df e F - bracketed terms must vansh F A() e A'( F) e e, F (.3).. Some consttutve laws {n stffness} form {n complance} 43 - exstence of the {stran energy functon} assumpton of consttutve law {complementary counterpart}

44 .8 Prncple of mnmum complementary energy.8.3 Prncple of mnmum complementary energy Prncple of Complementary Vrtual Work. W W W ' ' ' E I 3-bar truss, Fg.8 W e F e F e F ' I A A B B C C - Assumng elastc materal, exstence of complementary stran energy functon Eq. (.3b) ' ' ' ' AA( FA) AB( FB) AC( FC) I A B C FA FB FC W F F F A A A A ' ' ' ' A B B 44 A A A A total complementary stran energy ' ' ' ' A B C

45 .8 Prncple of mnmum complementary energy - Prescrbed dsplacement at B..can be derved from a potental WE' ' ( D) - Prncple of Complementary Vrtual Work W W W A ( A ) ' ' ' ' ' ' ' E I ' - total complementary energy, - Statement ' ' ' A (.4) ' (.5) Prncple (Prncple of statonary complementary energy) A conservatve system undergoes compatble deformatons f and only f the total complementary energy vanshes for all statcally admssble vrtual forces - Statonary = mnmum value for stable equlbrum Prncple of mnmum complementary energy 45

46 .8 Prncple of mnmum complementary energy Prncple (Prncple of Mnmum complementary energy) A conservatve system undergoes compatble deformatons f and only f the total complementary energy s a mnmum wth respect to arbtrary changes n statcally admssble forces. 46

47 .8 Prncple of mnmum complementary energy Example.8 Three-bar truss wth prescrbed dsplacement only relevant equlbrum eqn: at jont O complementary stran energy, frst n terms of F A, F B, and F C three bar forces are expressed n terms of one, say F C Potental of the prescrbed dsplacement Total complementary potental E F F, F cos F F cos P A C A B C F A FB FC A kacos kb kc cos F cos F C C FC kfc A kacos kb kc cos kkk A B C cos D, DF, F F cos, F cos B B C C kfc A FC cos, kkk cos A B C 47

48 .8 Prncple of mnmum complementary energy PMCE kf C cos, F kkk cos C A B C Ths yelds F A, F B, and F C kk cos A C ka kc FA FC kb, FB D kb k k dsplacement at O: extenson of the bar B u k k e k B A C B 48

49 .8 Prncple of mnmum complementary energy.8.4 The prncple of least work total complementary energy=system s complementary energy + potental of the prescrbed dsplacement f prescrbed dsplacement =, total complementary energy = complementary stran energy Prncple of least work Prncple 3 (Prncple of least work) In the absence of prescrbed dsplacement, a conservatve system undergoes compatble dsplacements f and only f the complementary stran energy s mn wth respect to arbtrary changes n statcally admssble forces. Prncple 4 (Prncple of least work) In the absence of prescrbed dsplacement, a lnearly elastc system undergoes compatble deformatons f and only f the stran energy s a mnmum wth respect to arbtrary changes n statcally admssble forces. 49

50 .8 Prncple of mnmum complementary energy Example.9. Three-bar truss wth tp load only relevant equlbrum eqn: at jont O stran energy, frst n terms of F A, F B, and F C three bar forces are expressed n terms of one, say F C Prncple of least work, prncple 4 F F, F cos F F cos P A C A B C F A FB FC A kacos kb kc cos P F cos F C C FC A kacos kb kc cos A F P F cos cos C C FC FC kacos kb kc cos can be solved for the bar force,, and the equlbrum eqn then yeld the other bar forces FA F kk cos F k k, P P k P k C A C B A C 5 PMCE: derve the same condton n a more abstract but also systematc manner

51 .9 Energy theorems Properly constraned elastc body subjected to varous concentrated loads and couples P,,... N j, dsplacement Q j,,... M, rotaton j Fg..4. Elastc body subjected to varous loads.9. Clapeyron s theorem Eq.(.) ---- stran energy stored n the body=work done by the external forces as they are ncreased quas-statcally from zero to fnal values. N A W Pdu Q d E j j j M - lneary elastc -----appled loads are proportonal to the dsplacements, N M A W P Q E j j j P Q u j j (.7) 5

52 .9 Energy theorems ---- Clapeyron s theorem useful for evaluatng the stran energy as well as computng the deflecton,, at the pont of applcaton of a load, P Eq.(.3) ----dfference by a factor of ½. load P s assumed to reman constant dfference n the Example.3 nature of the appled loadng..9. Castglano s frst theorem Eq.(.) ---- AA P N Dead loads Prncple of mnmum total potental energy statonarty of the total energy, Eq.(.7) N A A P P j j j j j 5 P A Castglano s frst theorem (.8)

53 .9 Energy theorems * All theorems are vald only for elastc structures Clapeyron s theorem Castglano s nd theorem further lmted to lnearly elastc structures.9.3 Crott-Engesser theorem Clapeyron s and Castglano s frst theorems Prncple of mnmum total potental energy Parallel developments based on prncple of mmmum complementary energy A - Eq (.4): ' ' ' 53 N ' P N ' ' ' ' A A P P : drvng forces requred to obtan the prescrbed dsplacements

54 .9 Energy theorems - Statcally admssble stress feld Prncple of mnmum complementary energy ' A A ' ( P ) ' ' N ' A A P P P P P j j j j j ' A : Crott-Engesser theorem (.9) P can be appled to multple appled loads.9.4 Castglano s nd theorem - In the dervaton of the Crott-Engesser theorem, exstence of complementary energy s assumed for elastc materal If lnearly elastc, A ' A 54 A : Castglano s nd theorem (.) P prescrbed deflecton

55 .9 Energy theorems.9.5 Applcatons of energy theorems Castglano s nd theorem. also useful for hyper statc problems - cantlevered beam wth a tp support a prescrbed tp dsplacement, whch s requred to vansh P :drvng force, Reacton force at the support Castglan s nd theorem, A P Compatblty equaton at the tp support Prncple of least work (Prncple 3) Example.4 55

56 .9 Energy theorems.9.6 The dummy load method Is t possble to use Castglano s nd theorem to compute the deflecton at a pont where no load s appled? st step a fcttous or dummy load,, s appled to the structure at the pont where the dsplacement s to be computed. ˆ A = nd step. By castglano s nd theorem last step... = lmˆ = lm A (.) A' f elastc, but nonlnear, must be used nstead of. A 56

57 .9 Energy theorems Example.9 Tp deflecton of a cantlevered beam 4 3 ˆ A pl L = c H = lmˆ p L 4 8H c 33 or, can be obtaned by takng the lmt before carryng out the ntegratons M M M = dx dx L L c c H 33 H 33 (.) 57

58 .9 Energy theorems.9.7 Unt load method revsted Prncple of complementary vrtual work dummy load method Unt load method (?) Castglano s nd theorem Prncple of mnmum complementary energy - dummy load method stran energy n an sostatc beam A L H 3 c 33 dx ( x ) 3 bendng moment dstrbuton generated by the externally appled loads and dummy load 58

59 .9 Energy theorems - Castglano s nd theorem lm M 3 3 A (.3) 3 3 = lm lm L dx c H33 = bendng moment due to externally appled loads only lm Ρ 3 Mˆ 3 = bendng moment due to a unt load only Eq. (.3) unt load method, Eq.(9.83) MM ˆ L 3 3 = dx c (.4) H33 59

60 .9 Energy theorems M 3 s dentcal for unt Dummy load method However, ˆM 3 has a dfference dummy load method..bendng moment actng n the structure subjected to a unt dummy load unt load method. any statcally admssble bendng moment dstrbuton n equlbrum wth unt load not necessarly the actual bendng moment dstrbuton actng n the structure subjected to the unt load more versatle can results n a sgnfcant smplfcaton of the procedure 6

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