The influence of theories on the quality of computer models

Size: px

Start display at page:

Download "The influence of theories on the quality of computer models"

Gervais Parrish
5 years ago
Views:

1 icccbe Nottinghm University Press Proceeings of the Interntionl Conference on Computing in Civil n Builing Engineering W Tizni (Eitor) The influence of theories on the qulity of computer moels Peter Jn Phl Technische Universität Berlin, Germny bstrct The qulity of computer moels epens on the qulity of the theories on which their bstrct moels re bse. Three theories re investigte for the nonliner eformtion n stbility of br structures. The im is to evelop stbility theory tht is s generl s the liner theory for isplcements n forces. The conventionl first orer theory is not suitble. The secon orer theory for Euler column stbility is extene into generl theory for plne frmes. Its rnge of pplicbility is investigte with thir orer nlysis of -br truss. It is shown tht secon orer theory oes not preict snpthrough of the -br truss, n tht consierble numericl problems cn be encountere if thir orer theory is pplie t bifurction points. Keywors: stbility, moel, singulrity, bifurction, relibility Introuction The beneficil use of computers epens on the mpping of the worl to computer moels. In orer to crete computer moel, the relevnt prt of the worl is bstrcte by mentl process. The bstrction is bse on humn unerstning of the worl n les to n bstrct moel of the consiere prt of the worl. The epth n correctness of humn unerstning etermine the qulity of the bstrct moel, which is then implemente s computer moel on specific computer pltform. The ese n efficiency with which the computer moel cn be hnle epen on the qulity of the pltform n of the implementtion. The qulity of computer moels in civil engineering is influence by the theories on which the corresponing bstrct moels re bse. It is well known tht ifferent theories cn be formulte for the sme phenomenon. These theories iffer in the ssumptions on which they re bse n, s consequence, in the extent to which they preict events in the worl completely n correctly. The economy, urbility n sfety of structures therefore epen on the use of suitble theories. The influence of theories on the qulity of computer moels of br structures is trete in this pper. The liner eformtion nlysis of such structures is well unerstoo n supporte by welth of commercil softwre. The sme cnnot be si for the stbility nlysis of br structures. This becomes pprent in the builing coes. While liner nlysis is ccepte s generl concept, lrge number of empiricl rules re given for the stbility of ifferent types of structurl elements n systems. These rules re reily pplie to iniviul members, but not lwys to structures s whole. The evlution of structurl stbility epens strongly on the experience n the personl jugment of the project engineer. Experience shows tht this cn le to errors n serious ccients.

2 In the pst, it prove to be impossible to formulte generl theory for the stbility of br structures tht cn be pplie without computer implementtion. This is ue to the lrge number of rithmetic opertions require for the numericl solution of the complex governing equtions. The question is whether suitble theory for the stbility of br structures cn be formulte n implemente on computers toy. This theory shoul hve generlity similr to tht of the conventionl liner theory for the computtion of isplcements n forces. The investigtion in this pper is limite to structures with liner elstic mteril behviour. Hypotheses for br behviour Theories for br structures re specil cses of the theory of elsticity (Glishnikov et l., 9). Bse on the specific geometry, supports n los tht by efinition re permitte for br structures, hypotheses re postulte for the behviour of these structures n for the simplifictions in its mthemticl formultion. The hypothesis specifying the configurtion for which the equilibrium of the structure is formulte istinguishes stbility nlysis cpble of preicting buckling from eformtion nlysis which cn only preict isplcements n forces. Br structures re in equilibrium in the instnt configurtion. The formultion of the governing equtions for first orer eformtion theory is bse on the ssumption tht the ifference between the reference n instnt configurtions of the structure oes not ffect the equilibrium equtions significntly. The equtions re formulte for the reference configurtion. The resulting stiffness mtrix is constnt n cnnot become singulr. The first orer theory therefore cnnot preict instbility. Conventionl stbility nlysis is bse on ssumptions tht were first introuce in the Euler theory for columns. The equilibrium equtions re formulte for the instnt configurtion, but the reltionship between strins n isplcements is liner. The resulting stiffness mtrix is function of the isplcements n becomes singulr for specific vlues of these isplcements. If isplcement increments beyon the singulr configurtion le to reuction of the equilibrium lo, the structure is clle unstble. Theories bse on this hypothesis cn preict instbility n re clle secon orer theories. generl secon orer theory for plne frmes is erive below. The reltionship between strins n stresses in the theory of elsticity contins nonliner proucts of isplcement erivtives tht re neglecte in secon orer theory. The significnce of these terms increses s the mgnitue of the isplcements n rottions increses reltive to the imensions of the structure. Neglect of the nonliner terms cn le to serious errors in the preiction of instbility, prticulrly in the cse of trusses. It is shown below tht snp-through instbility of regulr -br truss is not preicte by secon orer theory. theory tht ccounts for the nonliner terms in the strin-isplcement reltions n formultes equilibrium for the instnt configurtion is clle thir orer theory. Snp-through of shllow trusses is preicte correctly by thir orer theory. thir orer theory (Roik et l., 97; Crisfiel, 99; Belytschko et l., ; Wriggers, ) is generlly not equivlent to the theory of elsticity. This is ue to ssumptions tht re me in thir orer theories. It is, for exmple, ssume tht the shpe of the section of br oes not chnge uner lo. While this ssumption is pproprite for hot-rolle sections, it is not lwys suitble for thinwlle sections tht re wele of thin steel sheets. It cn become necessry to nlyse structures such s hollow brige sections with other types of structurl elements. They cn, for exmple, be trete s fole plte structures. This topic is not covere in the pper.

3 Secon orer stbility theory of plne frmes Figure shows the isplcements n the stress resultnts cting on the instnt configurtion of n element of br. The equilibrium of the element is formulte for its instnt configurtion, but the components of the forces cting on the sections re referre to the reference coorinte system. Ĉ ˆD Ĉ ˆD q y n x Figure. Reference n instnt configurtions of br element. The equilibrium equtions re: F T M y + nx = + qy = F + T = The xil strin ε n the bening strin ε b re referre to the instnt coorinte system n re liner functions of the isplcement erivtives: u v ε = ε b = y The expressions for norml force N, bening moment M n sher force Q referre to the instnt coorinte system re tken from first orer theory: u v M v N = σ = E M = y b EI Q EI σ = = = The rottions re ssume to be smll so tht force F in figure cn be pproximte by force N. The stress resultnts re substitute into the equilibrium equtions: 4 u u v E + nx = EI F q 4 y = If it is ssume tht xil forcef is compressive n known (s for Euler columns) the following isplcement function stisfies the homogeneous governing eqution: εx εx P v = csin + ccos + c + c4 with ε : = n P = F > EI The free prmeters c i re chosen so tht the bounry conitions t the en noes x = n x = re stisfie. This metho is use to etermine the bening stiffness coefficients of br of length. The xil stiffness is tken from first orer theory. For smll vlues of the chrcteristic number ε, numericl problems re voie by replcing the exct stiffness mtrix K with the pproximte mtrix K ˆ : h h h h h h h h h h h h h h4 h h5 h h ˆ 4 h h 5 K = K = h h h h h h h h h h h h h h h h h h h h

4 E EI h = = ( cos ε) εsinε h = ε EI ε EI ε 6EI h = sin h ( cos ) h ε = ε = ε 6 EI ε EI ε 4EI h 4 = (sin cos ) h 5 ( sin ) h4 ε ε ε = ε ε = ε EI h 5 = + ε 6 Consier plne frme subjecte to lo pttern q with lo fctor λ leing to lo λq. Let the lo be pplie in steps. In ech step, set the xil forces in the brs equl to the force t the en of the previous step. The stiffness of the brs is compute s shown bove if the xil force is compressive, otherwise the first orer stiffness is use. The system equtions re ssemble n solve, n new vlues for the xil forces re compute. The solution is iterte with the new xil forces until these no longer chnge significntly. Then the next lo step of the eformtion nlysis is compute. s the lo fctor is increse, one or more of the coefficients of the igonl mtrix D in T ecomposition K = LDL of the system stiffness mtrix with left tringulr mtrix L cn become negtive. This implies tht the stiffness mtrix hs psse singulr point in the lst lo step. The lo fctor t the singulr point cn be etermine, for exmple by intervl bisection. n exct solution of the governing equtions of secon orer theory for the stbility of pinne portl frmes is known. The metho propose in this pper les to result tht is within. percent of this solution. The presente metho of stbility nlysis offers the significnt vntge tht it cn be pplie to structures of rbitrry complexity. The metho is reily extene to spce frmes. It therefore hs generlity tht is comprble to liner eformtion nlysis. The metho cn be pplie without ny necessity to istinguish between instbility with or without sie-swy. 4 Secon n thir orer stbility nlyses of -br-truss In orer to illustrte the eficiencies of secon orer theory, the regulr -br truss in figure is nlyse by both secon orer n thir orer theory. x u P C u C^ P u L L L L B B h h Figure. Displcement of the pex of -br truss. Let the isplcement of the pex in the xil irection of brs C n BC be v n v.the xil strins ε, ε B n br forces N,NB for secon orer theory re liner functions of the isplcements: E v = (u+ hu ) ε = (u + hu ) N = (u + hu ) L L L E v = ( u + h u ) ε = ( u + h u ) N = ( u + h u ) B B B L L L x

5 The equilibrium equtions for the instnt configurtion re: + u + u h + u h + u N + NB = P N + NB = P L L L L Normlise vribles re efine s follows: ui P i L isplcement : wi = lo : pi = shpe fctor: m = h E h h The governing equtions of.orer theory re obtine by substitution: m w + w w = p (+ w )w = p The secon orer equtions hve two explicit solutions: p w = w = ( ± + p ) (m + w ) The isplcement of the pex of the truss becomes infinite for m + w =. The expression for w with the plus sign les to the first singulr point. The verticl isplcement n the lo t the singulr point re: wc = m p c = ( + ( m ) ) The strins in the brs for thir orer theory re erive with the full expressions of the nonliner theory of elsticity. The length of the eforme br is enote by L: ε= v, + v, + v, = (L L ) L h h ε = w(m + w) + w( + w ) ε b = w( m+ w) + w( + w ) L L These strins re substitute into the equilibrium equtions for the instnt configurtion to obtin the governing equtions of the -br truss for thir orer theory: m w + ww + w (w + w ) = p w (+ w ) + (+ w )(w + w ) = p The thir orer governing equtions contin ll terms of the secon orer governing equtions. In ition, they contin terms with fctor (w + w ) ue to the nonliner terms in the strinisplcement reltions. The thir orer equtions hve two solutions for the specil cse p =. The first solution is inepenent of the shpe fctor m: w = w (+ w )(+ w ) = p The secon solution yiels rel roots for m.77: p p w = m w = m m The first solution les to singulr point where shllow -br trusses snp through: wc = w c =.465 pc = pc =.849 The secon solution les to singulr point where steep trusses bifurcte: wc = wc = + m pc = pc = m m The snp-through point tht is preicte by thir orer theory is not preicte by secon orer theory. The bifurction point is preicte by both theories. The vlues of the criticl lo re in goo greement for smll vlues of shpe fctor m (steep trusses). The vlues for m =. re pc =.98 for secon orer n p c =.9799 for thir orer theory. The exmple emonstrtes tht the secon orer theory use to nlyse the stbility of frmes in engineering prctice cn le to unrelible computer moels for some types of structures such s the very simple -br truss. It shoul be replce by thir orer theory.

5 Numericl solutions t bifurction points The thir orer governing equtions for -br truss

if lo p n isplcement w re chosen s inepenent vribles cubic eqution is obtine for w : w +

The smllest bsolute vlue of lo p is of interest for buckling.

6 5 Numericl solutions t bifurction points The thir orer governing equtions for -br truss cnnot be solve explicitly for the isplcements w,w s functions of the los p,p.if lo p n isplcement w re chosen s inepenent vribles cubic eqution is obtine for w : w + ((+ w ) + m ) w p = p = ( + w )(w + (+ w ) ) Figure shows isolines of w n p for equilibrium configurtions of the truss. The cubic eqution hs one rel root outsie the brrier n three rel roots insie. The smllest bsolute vlue of lo p is of interest for buckling. n enlrgement of the tip of the brrier is shown on the right hn sie of figure. Only t the very tip of the brrier re the vlues of w insie n outsie the brrier equl. Figure. Bifurction of -br truss with shpe fctor m =.5.

7 n exct lo pth strting t the reference configurtion (p,w ) = (,) must pss exctly through the tip in orer to rech the interior of the brrier. Relible numericl methos for entry into the brrier re not known. The exmple shows tht even thir orer theory oes not gurntee totlly relible computer moels. 6 Conclusions The investigtion shows tht the secon orer theory of structurl stbility cn be generlise s bsis for computer moels. The rnge of ppliction of secon orer theory is, however, limite. If phenomen such s snp-through points re to be etecte, thir orer theory is require. This theory is reily formulte, but ifficulties cn rise in the numericl solution of the equtions t bifurction points. Further reserch on this topic is require. References GLISHNIKOV, V., DUNISKI, P., PHL, P.J., 9. Geometriclly Nonliner nlysis of Plne Trusses n Frmes. Stellenbosch: SUN MeDI. ROIK, K., CRL, J., LINDNER, J, 97. Biegetorsionsplobleme gerer ünnwniger Stäbe. Berlin: Wilhelm Ernst & Sohn. CRISFIELD, M.., 99. Non-liner Finite Element nlysis of Solis n Structures. Chichester: John Wiley & Sons. BELYTSCHKO, T., LIU, W.K., Morn, B.,. Nonliner Finite Elements for Continu n Structures. Chichester: John Wiley n Sons. WRIGGERS, P.,. Nichtlinere Finite-Element-Methoen. Berlin: Springer-Verlg.

Sturm-Liouville Theory

LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory