Liu, X., Zhang, L. "Structural Theory." Bridge Engineering Handbook. Ed. Wai-Fah Chen and Lian Duan Boca Raton: CRC Press, 2000

Transcription

1 Liu, X., Zhang, L. "Structural Thory." Bridg Enginring Handbook. Ed. Wai-Fah Chn and Lian Duan Boca Raton: CRC Prss, 2000

2 7 Structural Thory Xila Liu Tsinghua Univrsity, China Liming Zhang Tsinghua Univrsity, China 7.1 Introduction Basic Equations: Equilibrium, Compatibility, and Co nstitutiv Law Thr Lvls: Continuous Mchanics, Finit-Elmnt Mthod, Bam Column Thory Thortical Structural Mchanics, Computational Structural Mchanics, and Qualitativ Structural Mchanics Matrix Analysis of Structurs: Forc Mthod and Displacmnt Mthod 7.2 Equilibrium Equations Equilibrium Equation and Virtual Work Equation Equilibrium Equation for Elmnts Coordinat Transformation Equilibrium Equation for Structurs Influnc Lins and Surfacs 7.3 Compatibility Equations Larg Dformation and Larg Strain Compatibility Equation for Elmnts Compatibility Equation for Structurs Contragrdint Law 7.4 Constitutiv Equations Elasticity and Plasticity Linar Elastic and Nonlinar Elastic Bhavior Gomtric Nonlinarity 7.5 Displacmnt Mthod Stiffnss Matrix for Elmnts Stiffnss Matrix for Structurs Matrix Invrsion Spcial Considration 7.6 Substructuring and Symmtry Considration 7.1 Introduction In this chaptr, gnral forms of thr sts of quations rquird in solving a solid mchanics problm and thir xtnsions into structural thory ar prsntd. In particular, a mor gnrally usd mthod, displacmnt mthod, is xprssd in dtail Basic Equations: Equilibrium, Compatibility, and Constitutiv Law In gnral, solving a solid mchanics problm must satisfy quations of quilibrium (static or dynamic), conditions of compatibility btwn strains and displacmnts, and strss strain rlations or matrial constitutiv law (s Figur 7.1). Th initial and boundary conditions on forcs and displacmnts ar naturally includd. From considration of quilibrium quations, on can rlat th strsss insid a body to xtrnal xcitations, including body and surfac forcs. Thr ar thr quations of quilibrium rlating th

3 FIGURE 7.1 Rlations of variabls in solving a solid mchanics problm. σ ij six componnts of strss tnsor for an infinitsimal matrial lmnt which will b shown latr in Sction In th cas of dynamics, th quilibrium quations ar rplacd by quations of motion, which contain scond-ordr drivativs of displacmnt with rspct to tim. In th sam way, taking into account gomtric conditions, on can rlat strains insid a body to its displacmnts, by six quations of kinmatics xprssing th six componnts of strain ( ε ij ) in trms of th thr componnts of displacmnt ( u i ). Ths ar known as th strain displacmnt rlations (s Sction 7.3.1). Both th quations of quilibrium and kinmatics ar valid rgardlss of th spcific matrial of which th body is mad. Th influnc of th matrial is xprssd by constitutiv laws in six quations. In th simplst cas, not considring th ffcts of tmpratur, tim, loading rats, and loading paths, ths can b dscribd by rlations btwn strss and strain only. Six strss componnts, six strain componnts, and thr displacmnt componnts ar connctd by thr quilibrium quations, six kinmatics quations, and six constitutiv quations. Th 15 unknown quantitis can b dtrmind from th systm of 15 quations. It should b pointd out that th principl of suprposition is valid only whn small dformations and lastic matrials ar assumd Thr Lvls: Continuous Mchanics, Finit Elmnt Mthod, Bam Column Thory In solving a solid mchanics problm, th most dirct mthod solvs th thr sts of quations dscribd in th prvious sction. Gnrally, thr ar thr ways to stablish th basic unknowns, namly, th displacmnt componnts, th strss componnts, or a combination of both. Th corrsponding procdurs ar calld th displacmnt mthod, th strss mthod, or th mixd mthod, rspctivly. But ths dirct mthods ar only practicabl in som simpl circumstancs, such as thos dtaild in lastic thory of solid mchanics. Many complx problms cannot b asily solvd with convntional procdurs. Complxitis aris du to factors such as irrgular gomtry, nonhomognitis, nonlinarity, and arbitrary loading conditions. An altrnativ now availabl is basd on a concpt of discrtization. Th finitlmnt mthod (FEM) divids a body into many small bodis calld finit lmnts. Formulations by th FEM on th laws and principls govrning th bhavior of th body usually rsult in a st of simultanous quations that can b solvd by dirct or itrativ procdurs. And loading ffcts such as dformations and strsss can b valuatd within crtain accuracy. Up to now, FEM has bn th most widly usd structural analysis mthod. In daling with a continuous bam, th siz of th thr sts of quations is gratly rducd by assuming charactristics of bam mmbrs such as plan sctions rmain plan. For framd structurs

4 or structurs constructd using bam columns, structural mchanics givs thm a mor pithy and practical analysis Thortical Structural Mchanics, Computational Structural Mchanics, and Qualitativ Structural Mchanics Structural mchanics dals with a systm of mmbrs connctd by joints which may b pinnd or rigid. Classical mthods of structural analysis ar basd on principls such as th principl of virtual displacmnt, th minimization of total potntial nrgy, th minimization of total complmntary nrgy, which rsult in th thr sts of govrning quations. Unfortunatly, convntional mthods ar gnrally intndd for hand calculations and dvloprs of th FEM took grat pains to minimiz th amount of calculations rquird, vn at th xpns of making th mthods somwhat unsystmatic. This mad th convntional mthods unattractiv for translation to computr cods. Th digital computr calld for a mor systmatic mthod of structural analysis, lading to computational structural mchanics. By taking grat car to formulat th tools of matrix notation in a mathmatically consistnt fashion, th analyst achivd a systmatic approach convnint for automatic computation: matrix analysis of structurs. On of th hallmarks of structural matrix analysis is its systmatic natur, which rndrs digital computrs vn mor important in structural nginring. Of cours, th analyst must maintain a critical, vn skptical, attitud toward computr rsults. In any vnt, computr rsults must satisfy our intuition of what is rasonabl. This qualitativ judgmnt rquirs that th analyst possss a full undrstanding of structural bhavior, both that bing modld by th program and that which can b xpctd in th actual structurs. Enginrs should dcid what approximations ar rasonabl for th particular structur and vrify that ths approximations ar indd valid, and know how to dsign th structur so that its bhavior is in rasonabl agrmnt with th modl adoptd to analyz it. This is th main task of a structural analyst Matrix Analysis of Structurs: Forc Mthod and Displacmnt Mthod Matrix analysis of structurs was dvlopd in th arly 1950s. Although it was initially usd on fuslag analysis, this mthod was provd to b prtinnt to any complx structur. If intrnal forcs ar slctd as basic unknowns, th analysis mthod is rfrrd to as forc mthod; in a similar way, th displacmnt mthod rfrs to th cas whr displacmnts ar slctd as primary unknowns. Both mthods involv obtaining th joint quilibrium quations in trms of th basic intrnal forcs or joint displacmnts as primary unknowns and solving th rsulting st of quations for ths unknowns. Having don this, on can obtain intrnal forcs by backsubstitution, sinc vn in th cas of th displacmnt mthod th joint displacmnts dtrmin th basic displacmnts of ach mmbr, which ar dirctly rlatd to intrnal forcs and strsss in th mmbr. A major fatur vidnt in structural matrix analysis is an mphasis on a systmatic approach to th statmnt of th problm. This systmatic charactristic togthr with matrix notation maks it spcially convnint for computr coding. In fact, th displacmnt mthod, whos basic unknowns ar uniquly dfind, is gnrally mor convnint than th forc mthod. Most gnralpurpos structural analysis programs ar displacmnt basd. But thr ar still cass whr it may b mor dsirabl to us th forc mthod. 7.2 Equilibrium Equations Equilibrium Equation and Virtual Work Equation For any volum V of a matrial body having A as surfac ara, as shown in Figur 7.2, it has th following conditions of quilibrium:

5 FIGURE 7.2 Drivation of quations of quilibrium. At surfac points T =σ n i ji j (7.1a) At intrnal points σ ji, j + F i =0 σ ji = σ ij (7.1b) (7.1c) whr n i rprsnts th componnts of unit normal vctor n of th surfac; T i is th strss vctor at th point associatd with n; σ ji, j rprsnts th first drivativ of σ ij with rspct to x j ; and F i T i is th body forc intnsity. Any st of strsss σ ij, body forcs F i, and xtrnal surfac forcs that satisfis Eqs. (7.1a-c) is a statically admissibl st. Equations (7.1b and c) may b writtn in (x,y,z) notation as σ τ x xy τxz Fx = 0 x y z τyx σy τyz Fy = 0 x y z (7.1d) and τ τ zx zy σz Fz = 0 x y z τ xy = τ, yx tc. (7.1) whr σ x, σ y, and σ z ar th normal strss in (x,y,z) dirction rspctivly; τ xy, τ yx, and so on, ar th corrsponding shar strsss in (x,y,z) notation; and F x, F y, and F z ar th body forcs in (x,y,z) dirction, rspctivly. Th principl of virtual work has provd a vry powrful tchniqu of solving problms and providing proofs for gnral thorms in solid mchanics. Th quation of virtual work uss two

6 FIGURE 7.3 Two indpndnt sts in th quation of virtual work. indpndnt sts of quilibrium and compatibl (s Figur 7.3, whr and rprsnt displacmnt and strss boundary, rspctivly), as follows: compatibl st A u A T i i i i A V V * * * Tu da + Fu dv = σ ε dv ij ij (7.2) quilibrium st or δw xt = δw int (7.3) which stats that th xtrnal virtual work ( δw xt ) quals th intrnal virtual work ( δw int ). Hr th intgration is ovr th whol ara A, or volum V, of th body. Th strss fild σ ij, body forcs F i, and xtrnal surfac forcs T i ar a statically admissibl st that satisfis * * Eqs. (7.1a c). Similarly, th strain fild ε ij and th displacmnt u i ar a compatibl kinmatics st that satisfis displacmnt boundary conditions and Eq. (7.16) (s Sction 7.3.1). This mans th principl of virtual work applis only to small strain or small dformation. Th important point to kp in mind is that, nithr th admissibl quilibrium st σ ij, F i, and * * T i (Figur 7.3a) nor th compatibl st ε ij and u i (Figur 7.3b) nd b th actual stat, nor nd th quilibrium and compatibl sts b rlatd to ach othr in any way. In th othr words, ths two sts ar compltly indpndnt of ach othr Equilibrium Equation for Elmnts For an infinitsimal matrial lmnt, quilibrium quations hav bn summarizd in Sction 7.2.1, which will transfr into spcific xprssions in diffrnt mthods. As in ordinary FEM or th displacmnt mthod, it will rsult in th following lmnt quilibrium quations: { F} = [ k] {} d (7.4)

7 FIGURE 7.4 Plan truss mmbr nd forcs and displacmnts. (Sourc: Myrs, V.J., Matrix Analysis of Structurs, Nw York: Harpr & Row, With prmission.) { F} {} d whr and ar th lmnt nodal forc vctor and displacmnt vctor, rspctivly, whil [ k] is lmnt stiffnss matrix; th ovrbar hr mans in local coordinat systm. In th forc mthod of structural analysis, which also adopts th ida of discrtization, it is provd possibl to idntify a basic st of indpndnt forcs associatd with ach mmbr, in that not only ar ths forcs indpndnt of on anothr, but also all othr forcs in that mmbr ar dirctly dpndnt on this st. Thus, this st of forcs constituts th minimum st that is capabl of compltly dfining th strssd stat of th mmbr. Th rlationship btwn basic and local forcs may b obtaind by nforcing ovrall quilibrium on on mmbr, which givs { F} = [ L]{ P} (7.5) [ ] { P} whr L = th lmnt forc transformation matrix and = th lmnt primary forcs vctor. It is important to mphasiz that th physical basis of Eq. (7.5) is mmbr ovrall quilibrium. Tak a convntional plan truss mmbr for xmplification (s Figur 7.4), on has EA / l 0 EA / l {} k = EA / l 0 EA / l (7.6)

8 FIGURE 7.5 Coordinat transformation. and { F} = r1 r2 r3 r4 {}= d d1 d2 d3 d4 [ L] = { P} = { P} { } { } { } T T T (7.7) whr EA/l = axial stiffnss of th truss mmbr and P = axial forc of th truss mmbr Coordinat Transformation Th valus of th componnts of vctor V, dsignatd by v 1, v 2, and v 3 or simply v i, ar associatd with th chosn st coordinat axs. Oftn it is ncssary to rorint th rfrnc axs and valuat nw valus for th componnts of V in th nw coordinat systm. Assuming that V has componnts v i and v i in two sts of right-handd Cartsian coordinat systms x i (old) and x (nw) having v v i th sam origin (s Figur 7.5), and i, i ar th unit vctors of x i and x i, rspctivly. Thn v = l v (7.8) i ij j v v whr l, that is, th cosins of th angls btwn and axs for i and ji = j i = cos( x j, xi) x i x j j ranging from 1 to 3; and [ α]= ( l ij ) 3 is calld coordinat transformation matrix from th old 3 systm to th nw systm. It should b notd that th lmnts of l ij or matrix [ α] ar not symmtrical, l. For ij lji xampl, l 12 is th cosin of angl from x 1 to x 2 and l 21 is that from x 2 to x 1 (s Figur 7.5). Th angl is assumd to b masurd from th primd systm to th unprimd systm. For a plan truss mmbr (s Figur 7.4), th transformation matrix from local coordinat systm to global coordinat systm may b xprssd as cosα sinα sinα cosα [ α]= cosα sinα sinα cosα (7.9)

9 whr α is th inclind angl of th truss mmbr which is assumd to b masurd from th global to th local coordinat systm Equilibrium Equation for Structurs For discrtizd structur, th quilibrium of th whol structur is ssntially th quilibrium of ach joint. Aftr assmblag, For ordinary FEM or displacmnt mthod For forc mthod (7.10) (7.11) whr F = nodal loading vctor; = total stiffnss matrix; = nodal displacmnt vctor; [ A] = total forcs transformation matrix; { P} = total primary intrnal forcs vctor. It should b notd that th coordinat transformation for ach lmnt from local coordinats to th global coordinat systm must b don bfor assmbly. In th forc mthod, Eq. (7.11) will b adoptd to solv for intrnal forcs of a statically dtrminat structur. Th numbr of basic unknown forcs is qual to th numbr of quilibrium quations availabl to solv for thm and th quations ar linarly indpndnt. For statically unstabl structurs, analysis must considr thir dynamic bhavior. Whn th numbr of basic unknown forcs xcds th numbr of quilibrium quations, th structur is said to b statically indtrminat. In this cas, som of th basic unknown forcs ar not rquird to maintain structural quilibrium. Ths ar xtra or rdundant forcs. To obtain a solution for th full st of basic unknown forcs, it is ncssary to augmnt th st of indpndnt quilibrium quations with lastic bhavior of th structur, namly, th forc displacmnt rlations of th structur. Having solvd for th full st of basic forcs, w can dtrmin th displacmnts by backsubstitution Influnc Lins and Surfacs { F}= [ K]{ D} { F}= [ A]{ P} { } [ K] { D} In th dsign and analysis of bridg structurs, it is ncssary to study th ffcts intrigud by loads placd in various positions. This can b don convnintly by mans of diagrams showing th ffct of moving a unit load across th structurs. Such diagrams ar commonly calld influnc lins (for framd structurs) or influnc surfacs (for plats). Obsrv that whras a momnt or shar diagram shows th variation in momnt or shar along th structur du to som particular position of load, an influnc lin or surfac for momnt or shar shows th variation of momnt or shar at a particular sction du to a unit load placd anywhr along th structur. Exact influnc lins for statically dtrminat structurs can b obtaind analytically by statics alon. From Eq. (7.11), th total primary intrnal forcs vctor P can b xprssd as { } 1 { P}= [ A] { F} (7.12) by which givn a unit load at on nod, th xcitd intrnal forcs of all mmbrs will b obtaind, and thus Eq. (7.12) givs th analytical xprssion of influnc lins of all mmbr intrnal forcs for discrtizd structurs subjctd to moving nodal loads. For statically indtrminat structurs, influnc valus can b dtrmind dirctly from a considration of th gomtry of th dflctd load lin rsulting from imposing a unit dformation corrsponding to th function undr study, basd on th principl of virtual work. This may bttr b dmonstratd by a two-span continuous bam shown in Figur 7.6, whr th influnc lin of intrnal bnding momnt at sction B is rquird. M B

10 FIGURE 7.6 Influnc lin of a two-span continuous bam. FIGURE 7.7 Dformation of a lin lmnt for Lagrangian and Elurian variabls. Cutting sction B to xpos M B and giv it a unit rlativ rotation δ=1 (s Figur 7.6) and mploying th principl of virtual work givs Thrfor, M δ = B P v ( x ) M = B v ( x ) (7.13) (7.14) which mans th influnc valu of M B quals to th dflction vx ( ) of th bam subjctd to a unit rotation at joint B (rprsntd by dashd lin in Figur 7.6b). Solving for vx ( ) can b carrid out asily rfrring to matrial mchanics. 7.3 Compatibility Equations Larg Dformation and Larg Strain Strain analysis is concrnd with th study of dformation of a continuous body which is unrlatd to proprtis of th body matrial. In gnral, thr ar two mthods of dscribing th dformation of a continuous body, Lagrangian and Eulrian. Th Lagrangian mthod mploys th coordinats of ach particl in th initial position as th indpndnt variabls. Th Eulrian mthod dfins th indpndnt variabls as th coordinats of ach matrial particl at th tim of intrst. Lt th coordinats of matrial particl P in a body in th initial position b dnotd by x i ( x 1, x 2, x 3 ) rfrrd to th fixd axs x i, as shown in Figur 7.7. And th coordinats of th particl aftr dformation ar dnotd by ξ i ( ξ 1, ξ 2, ξ 3 ) with rspct to axs x i. As for th indpndnt variabls, Lagrangian formulation uss th coordinats ( x i ) whil Eulrian formulation mploys th coordinats ( ). From motion analysis of lin lmnt PQ (s Figur 7.7), on has ξ i

11 For Lagrangian formulation, th Lagrangian strain tnsor is 1 2 ( ) ε ij = u i, j + u j, i + u r, i u r, j (7.15) whr u and all quantitis ar xprssd in trms of ( ). ij, = ui xj x i For Eulrian formulation, th Eulrian strain tnsor is 1 Eij = ui j+ uj i+ ur iur j 2 ( ) / / / / (7.16) whr u and all quantitis ar dscribd in trms of ( ). i/ j= ui ξj ξ i If th displacmnt drivativs u ij, and u i/ j ar not so small that thir nonlinar trms cannot b nglctd, it is calld larg dformation, and th solving of u i will b rathr difficult sinc th nonlinar trms appar in th govrning quations. If both th displacmnts and thir drivativs ar small, it is immatrial whthr th drivativs in Eqs. (7.15) and (7.16) ar calculatd using th ( x i ) or th ( ξ i ) variabls. In this cas both Lagrangian and Eulrian dscriptions yild th sam strain displacmnt rlationship: 1 2 ( ) ε ij = E ij = u i, j + u j, i (7.17) which mans small dformation, th most common in structural nginring. For givn displacmnts ( u i ) in strain analysis, th strain componnts ( ε ij ) can b dtrmind from Eq. (7.17). For prscribd strain componnts ( ε ij ), som rstrictions must b imposd on it in ordr to hav singl-valud continuous displacmnt functions u i, sinc thr ar six quations for thr unknown functions. Such rstrictions ar calld compatibility conditions, which for a simply connctd rgion may b writtn as ε + ε ε ε =0 ij, kl kl, ij ik, jl ji, ik (7.18a) or, xpanding ths xprssions in th (x, y, z) notations, it givs ε ε ε x y xy + = y x x y 2 ε 2 2 y ε ε z yz + = z y y z εz εx εzx + = x z z x 2 εyz ε ε zx xy εx + + = x x y z y z (7.18b) 2 ε ε zx xy εyz εy + + = y y z x z x 2 εxy εyz ε zx εz + + = z z x y x y

12 Any st of strains ε ij and displacmnts u i, that satisfis Eqs. (7.17) and (7.18a) or (7.18b), as wll as displacmnt boundary conditions, is a kinmatics admissibl st, or a compatibl st Compatibility Equation for Elmnts For ordinary FEM, compatibility rquirmnts ar slf-satisfid in th formulating procdur. As for quilibrium quations, a basic st of indpndnt displacmnts can b idntifid for ach mmbr, and th kinmatics rlationships btwn mmbr basic displacmnts and mmbr nd displacmnts of on mmbr can b givn as follows: { } = [ L] {} d T { } [ L] d (7.19) whr is lmnt primary displacmnt vctor, and hav bn shown in Sction For plan truss mmbr, { } = { }, whr is th rlativ displacmnt of th mmbr (s Figur 7.5). It should also b notd that th physical basis of Eq. (7.19) is th ovrall compatibility of th lmnt Compatibility Equation for Structurs For th whol structur, on has th following quation aftr assmbly procss: {} T { }= [ A] { D} (7.20) { } { D} whr = total primary displacmnt vctor; = total nodal displacmnt vctor; and [ A] T = th transposition of [ A] dscribd in Sction A statically dtrminat structur is kinmatically dtrminat. Givn a st of basic mmbr displacmnts, thr ar a sufficint numbr of compatibility rlationships availabl to allow th structur nodal displacmnts to b dtrmind. In addition to thir application to sttlmnt and fabrication rror loading, thrmal loads can also b considrd for statically dtrminat structurs. Extrnal forcs on a structur caus mmbr distortions and, hnc, nodal displacmnts, but bfor such problms can b solvd, th rlationships btwn mmbr forcs and mmbr distortions must b dvlopd. Ths will b shown in Sction Contragrdint Law During th dvlopmnt of th quilibrium and compatibility rlationships, it has bn noticd that various corrsponding forc and displacmnt transformations ar th transposition of ach othr, as shown not only in Eqs. (7.5) and (7.19) of lmnt quilibrium and compatibility rlations, but also in Eqs. (7.11) and (7.20) of global quilibrium and compatibility rlations, although ach pair of ths transformations was obtaind indpndntly of th othr in th dvlopmnt. Ths spcial sts of rlations ar trmd th contragrdint law which was stablishd on th basis of virtual work concpts. Thrfor, aftr a particular forc transformation matrix is obtaind, th corrsponding displacmnt transformation matrix would b immdiatly apparnt, and it rmains valid to th contrary. 7.4 Constitutiv Equations Elasticity and Plasticity A matrial body will produc dformation whn subjctd to xtrnal xcitations. If upon th rlas of applid actions th body rcovrs its original shap and siz, it is calld an lastic matrial, or

13 FIGURE 7.8 Sktchs of bhavior of lastic and plastic matrials. on can say th matrial has th charactristic of lasticity. Othrwis, it is a plastic matrial or a matrial with plasticity. For an lastic body, th currnt stat of strss dpnds only on th currnt stat of dformation; that is, th constitutiv quations for lastic matrial ar givn by σ = F ( ε ) ij ij kl (7.21) F ij whr is calld th lastic rspons function. Thus, th lastic matrial bhavior dscribd by Eq. (7.21) is rvrsibl and path indpndnt (s Figur 7.8a), in which cas th matrial is usually trmd Cauchy lastic matrial. Rvrsibility and path indpndnc ar not xhibitd by plastic matrials (s Figur 7.8b). In gnral, a plastic matrial dos not rturn to its original shap; rsidual dformation and strsss rmain insid th body vn whn all xtrnal tractions ar rmovd. As a rsult, it is ncssary for plasticity to xtnd th lastic strss strain rlations into th plastic rang whr prmannt plastic stain is possibl. It maks th solution of a solid mchanics problm mor complicatd Linar Elastic and Nonlinar Elastic Bhavior Just as th trm linar implis, linar lasticity mans th lastic rspons function of Eq. (7.21) is a linar function, whos most gnral form for a Cauchy lastic matrial is givn by F ij σ = B + C ε ij ij ijkl kl (7.22) B ij whr = componnts of initial strss tnsor corrsponding to th initial strain-fr stat (i.., ε ij = 0), and C ijkl = tnsor of matrial lastic constants. If it is assumd that B ij = 0, Eq. (7.22) will b rducd to σ = C ε ij ijkl kl (7.23) which is oftn rfrrd to as th gnralizd Hook s law. For an isotropic linar lastic matrial, th lastic constants in Eq. (7.23) must b th sam for all dirctions and thus C ijkl must b an isotropic fourth-ordr tnsor, which mans that thr ar only two indpndnt matrial constants. In this cas, Eq. (7.23) will rduc to σij = λεkkδ ij +2µεij (7.24) whr λ and µ ar th two matrial constants, usually calld Lam s constants; = Kronckr dlta and ε kk = th summation of th diagonal trms of ε ij according to th summation convntion, which mans that, whnvr a subscript occurs twic in th sam trm, it is undrstood that th subscript is to b summd from 1 to 3. δ ij

14 If th lastic rspons function F ij in Eq. (7.21) is not linar, it is calld nonlinar lastic, and th matrial xhibits nonlinar mchanical bhavior vn whn sustaining small dformation. That is, th matrial lastic constants do not rmain constant any mor, whras th dformation can still b rvrsd compltly Gomtric Nonlinarity Basd on th sourcs from which it ariss, nonlinarity can b catgorizd into matrial nonlinarity (including nonlinar lasticity and plasticity) and gomtric nonlinarity. Whn th nonlinar trms in th strain displacmnt rlations cannot b nglctd (s Sction 7.3.1) or th dflctions ar larg nough to caus significant changs in th structural gomtry, it is trmd gomtric nonlinarity. It is also calld larg dformation, and th principl of suprposition drivd from small dformations is no longr valid. It should b notd that for accumulatd larg displacmnts with small dformations, it could b linarizd by a stp-by-stp procdur. According to th diffrnt choic of rfrnc fram, thr ar two typs of Lagrangian formulation: th total Lagrangian formulation, which taks th original unstraind configuration as th rfrnc fram, and th updatd Lagrangian formulation basd on th latst-obtaind configuration, which ar usually carrid out stp by stp. Whatvr formulation on chooss, a gomtric stiffnss matrix or initial strss matrix will b introducd into th quations of quilibrium to tak account of th ffcts of th initial strsss on th stiffnss of th structur. Ths dpnd on th magnitud or conditions of loading and dformations, and thus caus th gomtric nonlinarity. In bam column thory, this is wll known as th scond-ordr or th P ffct. For dtaild discussions, s Chaptr Displacmnt Mthod Stiffnss matrix for lmnts In displacmnt mthod, displacmnt componnts ar takn as primary unknowns. From Eqs. (7.5) and (7.19) th quilibrium and compatibility rquirmnts on lmnts hav bn acquird. For a statically dtrminat structur, no subsidiary conditions ar ndd to obtain intrnal forcs undr nodal loading or th displacd position of th structur givn th basic distortion such as support sttlmnt or fabrication rrors. For a statically indtrminat structur, howvr, supplmntary conditions, namly, th constitutiv law of matrials constructing th structur, should b incorporatd for th solution of intrnal forcs as wll as nodal displacmnts. From structural mchanics, th basic stiffnss rlationships for a mmbr btwn basic intrnal forcs and basic mmbr nd displacmnts can b xprssd as { P} = [ k] { } (7.25) [ ] [ k] EA l whr is th lmnt basic stiffnss matrix, which can b trmd for a convntional plan truss mmbr (s Figur 7.4). Substitution of Eqs. (7.19) and (7.25) into Eq. (7.5) yilds { F} = [ L][ k] [ L] {} d T = [ k] {} d (7.26) whr T [ k] = [ L][ k] [ L] (7.27)

15 is calld th lmnt stiffnss matrix, th sam as in Eq. (7.4). It should b kpt in mind that th lmnt stiffnss matrix [ k ] is symmtric and singular, sinc givn th mmbr nd forcs, mmbr nd displacmnts cannot b dtrmind uniquly bcaus th mmbr may undrgo rigid body movmnt Stiffnss Matrix for Structurs Our final aim is to obtain quations that dfin approximatly th bhavior of th whol body or structur. Onc th lmnt stiffnss rlations of Eq. (7.26) is stablishd for a gnric lmnt, th global quations can b constructd by an assmbling procss basd on th law of compatibility and quilibrium, which ar gnrally xprssd in matrix notation as { F}= [ K]{ D} (7.28) [ ] whr K is th stiffnss matrix for th whol structur. It should b notd that th basic ida of assmbly involvs a minimization of total potntial nrgy, and th assmbld stiffnss matrix [ K] is symmtric and bandd or sparsly populatd. Eq. (7.28) tlls us th capabilitis of a structur to withstand applid loading rathr than th tru bhavior of th structur if boundary conditions ar not introducd. In othr words, without boundary conditions, thr can b an infinit numbr of possibl solutions sinc stiffnss matrix [ K] is singular; that is, its dtrminant vanishs. Hnc, Eqs. (7.28) should b modifid to rflct boundary conditions and th final modifid quations ar xprssd by insrting ovrbars as { F}= [ K]{ D} (7.29) Matrix Invrsion It has bn shown that sts of simultanous algbraic quations ar gnratd in th application of both th displacmnt mthod and th forc mthod in structural analysis, which ar usually linar. Th cofficints of th quations ar constant and do not dpnd on th magnitud or conditions of loading and dformations, sinc linar Hook s law is gnrally assumd valid and small strains and dformations ar usd in th formulation. Solving Eq. (7.29) is, namly, to invrt th modifid stiffnss matrix [ K]. This rquirs trmndous computational fforts for larg-scal problms. Th quations can b solvd by using dirct, itrativ, or othr mthods. Two stps of limination and backsubstitution ar involvd in th dirct procdurs, among which ar Gaussian limination and a numbr of its modifications. Ths ar som of th most widly usd sts of dirct mthods bcaus of thir bttr accuracy and small numbr of arithmtic oprations Spcial Considration In practic, a varity of spcial circumstancs, ranging from loading to intrnal mmbr conditions and supporting conditions, should b givn du considration in structural analysis. Initially strains, which ar not dirctly associatd with strsss, rsult from two causs, thrmal loading or fabrication rror. If th mmbr with initial strains is unconstraind, thr will b a st of initial mmbr nd displacmnts associatd with ths initial strains, but nvrthlss no initial mmbr nd forcs. For a mmbr constraind to act as part of a structur, th gnral mmbr forc displacmnt rlationships will b modifid as follows: ( ) { } = [ ] {} { 0} F k d d (7.30a)

16 FIGURE 7.9 Plan truss with skwd support. or { F} = [ k] {}+ d { RF 0 } (7.30b) whr { } { RF 0} = [ k] { d0} (7.31) ar fixd-nd forcs, and d a vctor of initial mmbr nd displacmnts for th mmbr. 0 It is intrsting to not that a support sttlmnt may b rgardd as an initial strain. Morovr, initial strains including thrmal loading and fabrication rrors, as wll as support sttlmnts, can all b tratd as xtrnal xcitations. Hnc, th corrsponding fixd-nd forcs as wll as th quivalnt nodal loading can b obtaind which maks th convntional procdur dscribd prviously still practicabl. For a skwd support which provids a constraint to th structur in a nonglobal dirction, th ffct can b givn du considration by adapting a skwd global coordinat (s Figur 7.9) by introducing a skwd coordinat at th skwd support. This can prhaps b bttr dmonstratd by considring a spcific xampl of a plan truss shown in Figur 7.9. For mmbrs jointd at a skwd support, th coordinat transformation matrix will taks th form of cosαi sinα sinα α [ α]= i cos i i cosαj sinαj sinαj cosαj (7.32)

17 FIGURE 7.10 coordinat. Plan truss mmbr coordinat transformation. (a) Normal global coordinat; (b) skwd global whr α i and α j ar inclind angls of truss mmbr in skwd global coordinat (s Figur 7.10), say, for mmbr 2 in Figur 7.9, α i = 0 and α = j θ. For othr spcial mmbrs such as inxtnsional or variabl cross sction ons, it may b ncssary or convnint to mploy spcial mmbr forc displacmnt rlations in structural analysis. Although th dvlopmnt and programming of a stiffnss mthod gnral nough to tak into account all ths spcial considrations is formidabl, mor important prhaps is that th application of th mthod rmains littl changd. For mor dtails, radrs ar rfrrd to Rfrnc. [5]. 7.6 Substructuring and Symmtry Considration For highly complx or larg-scal structurs, on is rquird to solv a vry larg st of simultanous quations, which ar somtims rstrictd by th computation rsourcs availabl. In that cas, spcial data-handling schms lik static condnsation ar ndd to rduc th numbr of unknowns by appropriatly numbring nodal displacmnt componnts and disposition of lmnt forc displacmnt rlations. Static condnsation is usful in dynamic analysis of framd structurs sinc th rotatory momnt of inrtia is usually nglctd. Anothr schm physically partitions th structur into a collction of smallr structurs calld substructurs, which can b procssd by paralll computrs. In static analysis, th first stp of substructuring is to introduc imaginary fixd innr boundaris, and thn rlas all innr boundaris simultanously, which givs ris to a subsqunt analysis of ths substructur sris in a smallr scal. It is ssntially th patitioning of Eq. (7.28) as follows. For th rth substructur, on has Cas ( α) : Introducing innr fixd boundaris Kbb Kbi 0 Kib K α ii Di ( r) ( r) ( r) Fb = α F i (7.33) Cas ( β) : Rlasing all innr fixd boundaris K K bb ib K K bi ii ( r) ( r) ( r) Db β Di Fb = β 0 (7.34) whr subscripts b and i dnot innr fixd and fr nods, rspctivly.

18 Combining Eqs. (7.33) and (7.34) givs th forc displacmnt rlations for nlargd lmnts substructurs which may b xprssd as (7.35) which is analogous to Eq. (7.26) and F F r K r K r r. And thrby th convntional procdur is still valid. b b bi ii Fi Similarly, in th cass of structural symmtry of gomtry and matrial, propr considration of loading symmtry and antisymmtry can giv ris to a much smallr st of govrning quations. For mor dtails, plas rfr to th litratur on structural analysis. Rfrncs r r [ Kb] { Db } = { Fb } ( ) ( ) ( r) r { } ( ) = { ( ) } [ ( ) ][ ( ) 1 ] { ( ) } 1. Chn, W.F. and Salb, A.F., Constitutiv Eqs. for Enginring Matrials, Vols. 1 & 2, Elsvir Scinc Ltd., Nw York, Chn, W.F., Plasticity in Rinforcd Concrt, McGraw-Hill, Nw York, Chn, W.F. and T. Atsuta, Thor y of Bam-C olumns, Vol. 1, In-Plan Bhavior and Dsign, McGraw- Hill, Nw York, Dsai, C.S., El mntary Finit El mnt Mthod, Prntic-Hall, Englwood Cliffs, NJ, Myrs, V.J., Matrix Analysis of Structurs, Harpr & Row, Nw York, Michalos, J., Thor y of Structural Analysis and Dsign, Ronald Prss, Nw York, Hjlmstad, K.D., Fundamntals of Structural Mchanics, Prntic-Hall Collg Div., Uppr Saddl Rivr, NJ, Flming, J.F., Analysis of Structural Systms, Prntic-Hall Collg Div., Uppr Saddl Rivr, NJ, Dadppo, D.A. Introduction to Structural Mchanics and Analysis, Prntic-Hall Collg Div., Uppr Saddl Rivr, NJ, 1998.