A computer-aided optimization method of bending beams

Transcription

1 WSES Internatonal Conferene on ENGINEERING MECHNICS, STRUCTURES, ENGINEERING GEOLOGY (EMESEG '8), Heraklon, Crete Island, Greee, July 22-24, 28 omputer-aded optmzaton method of bendng beams CRMEN E. SINGER-ORCI Insttute of Sold Mehans - Romanan ademy 5, Constantn Mlle St., uharest, 4 ROMNI bstrat: Ths paper presents a new general omputer-aded way for optmzaton of bendng beams, based on a omputer-aded method [] of obtanng the nfluene oeffents for any statally determned or undetermned straght beam of a onstant ross seton. The extenson to a non-onstant ross seton s easy to obtan. The model s a p -lumped beam under all the ombnatons of loadng and boundary ondtons, unrelated to how bg s p. The flexblty of ths mathematal model synergstally ompleted by the Mathemata software symbol alulus apabltes, allows us to determne the values of the desgn parameters that optmze the dynam behavor, aordng to predefned rtera. Key-Words: bendng, lumped beam, nfluene oeffents, omputer-aded optmzaton, Mathemata. Introduton To study the bendng behavor of a lumped beam, the nfluene oeffents are to be known [2]. So, for a real mehansm wth p branhes, wth the man amshaft modeled as a ( p + ) -lumped beam under ertan loadng and boundary ondtons, a ( p + ) square nfluene oeffents matrx s to be known. It s well known that the spealty lterature [3] offers omputng methods for only a very small number, p, of onentrated masses and not for any type of boundary ondtons. Thus, fndng a way to ompute nfluene oeffents matrx for a lumped beam wth any fnte number, p, of onentrated masses and n any boundary ondtons, appears to be very hallengng. In ths paper, our response to ths hallenge s presented. We hose to do ths by usng a omputer-aded (C) method startng from the ntal parameters method [, 3, and 4]. 2 Flexblty nfluene oeffents In the lterature, the onept of nfluene oeffents denotes both the stffness nfluene oeffents and the flexblty nfluene oeffents, whh are ntmately related - they desrbe the manner n whh the mehanal system deforms under the fores. We deal only wth the flexblty nfluene oeffents, whh wll be named, the nfluene oeffents [6, 8]. To defne the nfluene oeffents, let us onsder a smple dsrete system, wth no dampng, onsstng of p masses m oupyng the poston x, =, p and beng n equlbrum. Fores F at upon eah mass m (ths an be assumed wthout loosng generalty) so that the masses undergo dsplaements z. Thus, the flexblty nfluene oeffent e s the dsplaement of the pont x due to a unt fore F = appled at x. Note that the flexblty nfluene oeffents e have the approprate unts orrespondng to the type of loadng: [LF] for torson, and [LF - ] for fores. For a lnear system, usng the prnple of superposton, the flexblty nfluene oeffents e allow to obtan the dsplaement at the pont x due to all the fores F ( =, p) atng on the system, as: z = ef,, =, p. () In () and everywhere else n text, the repeated ndex stands for summaton. 3 endng beam equatons, soluton In the usual notatons from the bendng beams theory: T = T( x) : the shear fore; M = M( x) : the bendng moment; z = z( x) : the elast defleton; ISN: ISSN

2 WSES Internatonal Conferene on ENGINEERING MECHNICS, STRUCTURES, ENGINEERING GEOLOGY (EMESEG '8), Heraklon, Crete Island, Greee, July 22-24, 28 θ=θ ( x) = dz : dx the slope (of the elast defleton). For a straght beam of onstant ross seton, the followng equatons between defleton, shear fore, and bendng moment hold: 4 d z f =, 4 dx (2) 3 d z = T, 3 dx (3) dm T dx =, (4) where E s Young's module, I s the moment nerta of the onstant ross seton, and f s an externally appled load. Eqs. (2) wth the approprate boundary ondtons allows us to obtan the defleton z = z( x) for any gven external loadng. The boundary ondtons are to be wrtten for eah range: between any par of external onentrated fores/moments and for eah portons of the beam on whh the dstrbuted fores are appled. The relatonshps (2) and (3) serve as ontnuty ondtons. Lookng for the homogenous soluton of (2) of the form 3 2 x x zx ( ) = + + Cx+ D, 3! 2! the slope, the bendng moment and the shear fore are: 2 x θ ( x) = + x+ C, 2! M ( x) = ( x+ ), T( x) =. Denotng by ndex the values at the pont = = ( ), M = M ( ), T T( ) x z z the ntegraton onstants beome: = T, =, = M, C =θ, D= z, and the homogenous soluton of (2) s: 3 2 x x zx ( ) = T M x z 3! 2! +θ +. (5) Sne the relatonshp (5) onnets the urrent values of the defletons zxto ( ) the loadng parameters at the orgn ( x = ), the method s known as the ntal parameters method. Fg. Typal external loadng Now the partular solutons, typal to eah type of external loadng, are to be added to the homogenous soluton (5). Let us onsder the beam of length l ( x l ) wth typal external loadng, as shown n Fg.. If the onentrated fore P s atng n the seton x =, the approprate partular soluton s: ( ) z x =, f x<, ( x ) 3 z ( x) = P, f x. 3! If the moment Me s atng n the seton x the approprate partular soluton s: ( ) z x =, f x< d, ( x d) 2 z ( x) = Me, f d x. 2! (6) = d, (7) The moments were onsdered to be postve n a lokwse sense and the external fores n the desendent sense of the vertal axs. If the dstrbuted fore p s atng on the porton x ( gh, ) of the beam seton, the approprate partular soluton s:, f x< g 4 ( x g) z ( x) = p, f g x h (8) 4! 4 4 ( x g) ( x h) p, f h< x 4! 4! If the dstrbuted fore wth lnear varaton, ISN: ISSN

3 WSES Internatonal Conferene on ENGINEERING MECHNICS, STRUCTURES, ENGINEERING GEOLOGY (EMESEG '8), Heraklon, Crete Island, Greee, July 22-24, 28 p( x a)( x b) s atng on the porton x ( ab, ) of the beam seton, the approprate partular soluton s: ( ) z x =, f x< a, ( x a) 5 z ( x) = p, f a x b, 5! z ( x) p = b a ( ) ( x a) ( x b) ( x b) (9) 5 5 4, f b< x. 5! 5! 4! Let us onsder that the beam s loaded by: (L) p dstrbuted fores atng n the seton x ( g, h) of the beam and the dstrbuted fore wth lnear varaton, p ( x a )( x b) s atng on the porton x ( a, b), ( =, n ) (L2) P onentrated fores atng n =, n ); ( 2 x = (L3) Me k external moments atng n the seton x = d of the beam ( k =, n k 3 ). Usng the superposton prnple, the general soluton obtaned ombnng the homogenous soluton and the partular solutons orrespondng to onentrated fore, moment and dstrbuted fore, respetvely, s 3 2 x x zx ( ) = T M +θ x+ z 3! 2! + < > < > n 4 4 p( x a x b ) 4! = n2 3 P < x > + 3! = n3 2 + Mk < x dk >, 2! where k = ( ), ( ), ( ) ( ) z = z x θ =θ x M = M x, T = T x, are the ntal parameters, and () n ( ), f, n x α x α < x α> =, f x <α. () The relatonshp () determnes the elast defleton, z = z( x) for a straght beam of onstant ross seton under the external loadng (L), (L2) and (L3) wth respet to the ntal parameters z, θ, M, T. The relatonshp () serves us to determne the flexblty nfluene oeffents usng the defnton: e s the dsplaement of the pont x due to a unt fore F =, appled at x. For that we developed a MTHEMTIC soft to obtan the matrx of the flexblty nfluene oeffents for a lumped beam wth any fnte number, p, of onentrated masses and n any boundary ondtons. 4 C way to aqure the flexblty nfluene oeffents It s mportant to note that the formula () s: () not dependent on the type of external loadng and number and / or knd of boundary ondtons or ntermedate supports and () does not requre, as a separate step, the stat determnaton of the reatons at the supports. So, t s applable wthout any restrtons to the ase of a statally undetermned beam. Our routne an be onentrated n the followng mportant proedural steps that are to be followed: (S) nalyss of the loadng ondtons, n order to establsh the boundary ondtons at the ends of the beam, as well as at the ntermedate supports; (S2) Removal of the ntermedate supports and ther replaement by the ntermedate reatons; the ntermedate reatons wll be regarded as a part of the loadng; (S3) Computaton of the terms appearng neessary n the ondtons establshed for step (S) and wrtng the ondtons. Thus, we obtan a set of algebra equatons n whh the unknowns are the values of the ntal parameters, z, θ, M, T, and the ntermedate reatons; (S4) Obtanng the ntermedate reatons n terms of ntal parameters by solvng the equatons establshed n step (S3); (S5) Determnaton of the defleton, the shear fore, and the bendng moment n terms of values ISN: ISSN

4 WSES Internatonal Conferene on ENGINEERING MECHNICS, STRUCTURES, ENGINEERING GEOLOGY (EMESEG '8), Heraklon, Crete Island, Greee, July 22-24, 28 of the ntal parameters- z, θ, M, T. Thus, the method allows both statally determned and undetermned beams to be treated n the same way. fter analyss, we onluded that all the ombnatons of boundary ondtons an be grouped n followng fve dfferent ases (Type=,5) desrbed n Table. Type Table. The boundary ondtons. The boundary ondtons at ( x =) ( x = l ) ω =, θ = M, T ω =, θ = M, T 2 ω =, θ = M, T 3 ω =, θ M =, T 4 ω =, θ M =, T 5 ω, θ M =, T = ω, θ M =, T = ω =, θ M =, T ω =, θ = M, T ω =, θ M =, T ω, θ M =, T = ω, θ M =, T = a e Fg.2 oundary ondtons. Fg Thus, the method allows both statally determned and undetermned beams to be treated n the same way. Modfyng the flexblty nfluene oeffents for a statally determned or undetermned beam modeled as a p-lumped beam (unrelated of how bg p s) under any ombnatons of loadng and boundary ondtons, we an verfy the stat or/and dynam performane rtera. b d f 2a 2b 2 2d 2e 2f Sne the dsplaement due to all the fores an be alulated from (), ether the shear fore and bendng moment an be alulated from (2)- (3) (after approprate fttng), or the shear fore and bendng moment dagrams an be drawn. Ths way, all the stat performane rtera an be verfed. Wang 5 C quas-optmzaton Our C quas-optmzaton method s an example of usng the C apabltes to obtan a large quantty of data to reeve new qualtatve nformaton. The word "optmzaton" s not used n ts strt mathematal meanng; for nformaton about optmzaton n the lassal meanng see [5], for example. In our vew, the desgn optmzaton an be made omparng and hoosng the best ft from a large enough set of alulated data. If the quantty of data and omparsons s large enough, we an get the ombnaton that suts our needs. The easy way to obtan the nformaton allows us to repeat the alulaton for a large number of varants of the model (the varants dffer by desgn or value of one parameter at least). For eah run, the defned measures are to be alulated and stored for further omparson. y measures we mean the output data that have been hosen to be stored. If the results are numeral solutons, a large storage memory s needed. In ths ase the statstal measures have to be used. Synergstally ombnng the nreased apablty of omputers to obtan, to save and to ompare a large number of dfferent ases wth the Mathemata software symbol alulus apabltes, besde a statstal representaton of the results, we developed a tool for desgn optmzaton: The State Matrx Strategy, a quasoptmzaton tool [9]. The state matrx s a m N N matrx, where: N p s the number of the rtal nodal ponts and N m s the number of the hosen measures. pont n the N m N p spae s named behavor pont. fter eah run a behavor pont s aheved. The behavor map s a set of behavor ponts obtaned when all the desgn parameters over ts utlzaton domans, that s the doman n whh the parameter may vary wthout the state matrx exeedng gven admssble lmts. p ISN: ISSN

5 WSES Internatonal Conferene on ENGINEERING MECHNICS, STRUCTURES, ENGINEERING GEOLOGY (EMESEG '8), Heraklon, Crete Island, Greee, July 22-24, 28 So, the nfluene of any parameter on the behavor of the studed model an be observed and analyzed. Generally, we an onsder: desgn hanges and/or number of onstrutve parameters; hanges n the mass dstrbuton, the dampng level, the mean frequeny; supplementary desgn ondtons; hanges n the dynam model; any partular parameter relevant for the studed ase. The strategy for the omputer-aded quasoptmzaton of a bendng beam follows the followng steps:. defne the problem geometr and mehan desrpton of the bendng beam,. dentfy all the parameters that an hange ther value; determne ther utlzaton domans; hoose the approprate desgn parameters set,. defne the obetve - a set of stat and/or dynam rtera that has to be fulflled, v. defne the rtal ponts approprate for the problem and for the obetve; the results/measures obtaned n these ponts have to be stored, v. obtan the state matrx that arres the nformaton about eah aeptable varant; v. get (automatally) the optmal ombnaton of values of the desgn parameters for whh the obetve s reahed. We developed a MTHEMTIC soft for ths quas-optmzaton for a lumped beam wth any fnte number, p, of onentrated masses and n any boundary ondtons that allow us to hoose the optmal ombnaton of values of the desgn parameters for whh the obetve s reahed. In appendx t s gven as example a beam wth a fxed end and three smple supports under sx onentrated fores. Its 6x6 matrx nfluene oeffents were omputed. The postons of these onentrated fores are determned n order to avod resonane aordng to egenvalues rtera. for a statally determned or undetermned beam modeled as p-lumped beam (unrelated of how bg p s) under any the ombnatons of loadng and boundary ondtons, we an verfy the stat or/and dynam performane rtera. In onluson, the bendng beams under any of the ombnatons of loadng and boundary ondtons an be automatally alulated after the C determnaton of the flexblty nfluene oeffents. The optmzaton proposed here s performed usng a new desgn optmzaton method (The State Matrx Strategy, a quas-optmzaton tool), that allows to hoose the optmal ombnaton of values of the desgn parameters for the obetve - a set of rtera that have to be fulflled. Referenes: [] Esnger ora, C. E., general omputeraded method of obtanng the nfluene oeffents, Pro. of the Romanan ademy, Seres, Vol. 8, No., 28. [2] Case, J., Chlver, L., Strength of Materals and Strutures, Elsever, 999. [3] Soare, M. V., a, C., Ille, V., Strength of Materals and Elastty Theory, Edtura Tehna, 983. [4] Esnger ora, C. E., Computer-ded Elasto- Dynam nalyss of ranhed Systems Contanng Mehansms wth Nonlnear Knemats, en- Guron Unversty of the Negev, Israel, 996. [5] Popesu, H., Chrou, V., Optmum desgn of strutures (n Romanan), Edtura ademe, uharest 98. [6] Waldhaw,.C., Mehanal Vbratons wth pplatons, John Wlley and Sons, 984. [7] Tmoshenko S., Strength of Materals, Kreger Publshng Company, 976. [8] Yao Wen-Juan, Ye Zh-mng, nalytal soluton for bendng beam subet to lateral fore wth dfferent modulus, J. ppl. Maths. and Mehs, Vol. 25, No., 24, pp.7-7. [9] C.E. Esnger ora,.sandler, Computer- ded nalyss and Desgn of ranhed Mehansms, Pro. of the EUROSIM'95 Smulaton Congress - Venna, pp , Elsever, Conlusons Handlng the flexblty nfluene oeffents. ISN: ISSN

6 WSES Internatonal Conferene on ENGINEERING MECHNICS, STRUCTURES, ENGINEERING GEOLOGY (EMESEG '8), Heraklon, Crete Island, Greee, July 22-24, 28 PPENDIX-Study ase We present some results for the beam shown n the followng fgure The length of the beam s m. The dstanes from of the smple supports (, 2, and 3) are: {2, 5, 8}[m]. The beam s loaded by the fores: {25,, 2, 3, 2, 2} [N]. The omputed matrx of the nfluene oeffents for ths beam s: ased on ths matrx of the nfluene oeffents we reeved from the program: - the ara of the bendng moment dagram: 7475 and - the frequenes vetor :{.66, 3.95, 9., 22.7, 5.6, 86.53} for the followng postons of the loadng: {x, x 2, x 3, x 4, x 5, x 6 }={.5,.5, 3.5, 6.5, 8.5, 9.5}. ISN: ISSN