Finite Element Modelling of truss/cable structures

Transcription

1 Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures

2 1 Fnte Element Analyss of prestressed structures o ensure stablty of a truss/cable structure prestresses have to be appled n the cables to prevent cable slackng after loadng. Desgnng the structure wth the fnte element method means that these prestresses have to be taken nto account. In the followng, the weghted resdual formulaton of a truss element s elaborated to ndcate the locaton of the prescrbed prestress. Incorporatng the prestress essentally mples that the problem s nonlnear and that a Newton-Raphson teratve procedure s used to determne the nodal dsplacements and afterwards the truss stresses. f e 2 qs) f e 1 s l e s l e Fg. 1.1 : Inhomogeneous truss element n undeformed and deformed state 1.1 Equlbrum equaton and weghted resdual formulaton he axal equlbrum of a truss n three-dmensonal space s descrbed by a dfferental equaton, whch results from consderng force equlbrum of an nfntesmal part of the truss. he summaton of the varaton of the axal cross-sectonal force σa n and the volume force q must be zero n each pont of the truss. he spatal drecton of the truss axs s specfed by the unt vector n, the axal stress s σ and the cross-sectonal area s A. he local coordnate along the truss axs s s. he axal force per unt of length at pont s s q. Because the dfferental equaton s not a good startng pont for FE analyss, an equvalent formulaton s used. he summaton of forces whch has to be zero s multpled by a weghtng functon ws) and ths product s ntegrated over the truss length. When the value of ths ntegral s zero for each weghtng functon, the equlbrum equaton s automatcally satsfed n each pont of the truss axs. hs approach, so the converson of a dfferental equaton to an ntegral equaton, s called the weghted resdual formulaton. he frst term n wegthed resdual ntegral s ntegrated by parts to get rd of the dfferentaton of the stress σ. hs results n the so-called weak form of the weghted resdual

3 1 ntegral, whch comprses the nternal force term f w, u), wth the axal stress σ, and the external force term f e w), wth the volume load q and the end forces N. he stress-stran relaton,.e. the materal law, wll be used and thus the nternal load term s only a functon of the dsplacement u of the cross-sectons of the truss. dσa n) ds sl s sl s + q s { dσa n) w ds } + q ds ws) ds σa n)ds N 2 N 1 + f w, u) f e w) ws) sl s w q ds ws) 1.2 Iteratve soluton procedure he nternal load term s n essence a functon of the cross-sectonal dsplacement u. Due to possble large deformatons, non-lnear materal behavor and prestresses, ths relaton s non-lnear. o solve the dfferental c.q. ntegral) equaton, an teratve procedure must be used, n ths case the Newton-Raphson algortm, whch s llustrated n the fgure below for a smplfed one-dmensonal case. he fnal soluton u f where f u) equals f e s reached n a number of teraton steps. Wth a known approxmate value u for u f the nternal force f can be calculated and the tangent stffness K df du can be determned. An teratve dsplacement δu s solved from K δu f e f where f e f r s the resdual. A new approxmate value for the fnal soluton s u. When, after some teraton steps, the resdual s small enough accordng to a so-called convergence crteron, a good approxmaton for u f s reached. f u) K f e r f δu u u u f u Fg. 1.2 : Newton-Raphson teraton procedure

4 2 We have an approxmate value for all unknown varables, denoted as ) : σ, A and n. her values n the equlbrum state can than be wrtten as the sum of the approxmaton and a devaton δ ), whch can than be substtuted n the equaton. When t s assumed that the devaton s small, lnearzaton can be done, resultng n an equaton, whch s lnear n the devatons δ ). As can be seen below, we do not nclude the devaton of the cross-secton, whch s consdered to be almost zero. Because the current state s unknown, the ntegral s transformed to the ntal state, wth coordnate s and element length. he teratve change of the stress and the axal drecton can both be expressed n the teratve cross-sectonal dsplacement δ u. Here, m s a unt vector perpendcular to n and λ s the stretch rato. δσ dσ dλ δσ A n + n n dδ u) ; δ n ) dσ dλ A n dδ u) + f e w) σ A δ n f e w) [ m m 1 ] dδ u) λ σ A n m σ A n σa 1 ) m dδ u) λ Vectors are now wrtten n components wth respect to an orthonormal vector bass { e 1, e 2, e 3 }. Column notaton s used to wrte w w 1 e 1 + w 2 e 2 + w 3 e 3 [ ] e 1 w 1 w 2 w 3 e 2 w ẽ e 3 and furthermore we have ẽ ẽ I where I s the 3 3 unty matrx. he weghted resdual ntegral s now : dσ dw ñ dλ A f e w ) ) n dδũ) + dw σa) ñ dw m σa 1 λ) m dδũ) In each element a local coordnate ξ s ntroduced, whch has a value between -1 and 1. Integraton and dfferentaton s then done n terms of ths coordnate.

5 3 1 ξ 1 2 dξ ; d ) 2 d ) dξ ξ1 dw dσ dξ ñ dλ A 2 ) n dδũ) dξ + dξ ξ1 fe e w ) dw dξ σa) ñ dξ ξ1 dw dξ m σa 1 λ 2 ) m dδũ) dξ dξ Both the wegtng functon and the teratve dsplacement n a cross-secton ξ of the element are nterpolated between ther values n the begn and end pont, the element nodes. hs lnear nterpolaton s done wth nterpolaton functons whch are lnear functons of the local coordnate. hey are stored n a matrx N. Dfferentaton of ths matrx leads to the socalled B-matrx. he nodal values of weghtng functon and teratve dsplacement are stored n colums w e and δũ e respectvely. hey are not a functon of ξ, so can be placed outsde the ntegrals. e ; δũ Nδũ w Nw e ; N ξ) ; N ξ) N 1 N 2 N N 1 N 2 dn 1 ; N 1 N 2 dξ B Substtuton leads to w e ξ1 dσ B ñ dλ A 2 ) n B dξ δũ e + w fe e w e e ) w ξ1 B σa) ñ dξ e ξ1 B σa m 1 λ 2 ) m B dξ δũ e In our applcatons all elements are homogeneous, mplyng that they are also cylndrcal. Integraton over the element can now be carred out analytcally and results n the element stffness matrx K e e and the element nternal forces. [ e dσ w dλ A 1 ) M L + σa 1 ) 1 ] M N δũ e f e λ ew e ) w e [σa) Ṽ ] w e K e δũ e w e e e w e e Connectng all elements n the structure mples that the element equatons have to be combned. hs assemblng s an admnstratve procedure, done accordng to the requrement that

6 4 n a structural node where elements are connected ther nternal forces are added and ther nodal dsplacements are the same. he assembled weghted resdual equaton has columns w, δũ, and wth all nodal values of weghtng functon, teratve dsplacement, external e force and nternal force, respectvely. hs equaton has to be satsfed for all values of w, leadng to the requrement that the teratve dsplacements have to satsfy a set of algebrac equatons. w K δũ w e w w K δũ e r he teratve dsplacements must be solved from the lnear teratve equaton system. However, the tangent stffness matrx s snglar, so ths s not possble. he sngularty comes from the fact that rgd body movement s stll possble, so teratve dsplacements exst wthout a need for external fores. o prevent ths, proper boundary condtons have to be appled, whch prevent the rgd body dsplacements. When ths s done, the teratve dsplacements can be solved and a new approxmate soluton s reached. Convergence has to be checked by evaluatng the new resdual forces. K δũ BC s r } r e δũ [K ] 1 r r < ε r?? u u + δu 1.3 Internal stresses he frst part of the stffness matrx s proportonal to the materal stffness dσ dλ and s thus referred to as the materal stffness matrx. he second part s proportonal to the nternal stress σ as s denoted as the stress stffness matrx. Because m m s a shortcut for I ññ, ths can also be wrtten slghtly dfferent, as s often done n lterature. he nternal force s also a functon of the nternal stress. dσ K e dλ A 1 ) M L l + σa 1 ) 1 M N λ dσ dλ A 1 ) B n ñ B + σa 1 ) B m l m B [ dσ dλ A 1 ) σa 1 ) ] B ñ ñ B + σa 1 ) B B l l e σa) Ṽ σa) B ñ he nternal stress s the addton of the ntal stress σ and the stress resultng from the axal stran σ m. Intally the stress s the prescrbed prestress σ. dσ K e dλ A 1 ) B n ñ B + σ + σ m )A 1 ) B m l m B

7 5 [ dσ dλ A 1 ) σ + σ m )A 1 ) ] B ñ ñ B + σ + σ m )A 1 ) B B l l e σ + σ m )A) Ṽ σ + σ m )A) B ñ ) dσ K e 1 dλ A )B ñ ñ B + 1 σ A [ ) )] dσ 1 1 dλ A σ A B ñ ñ B + B m m B σ A 1 ) B B e σ A Ṽ σ A B ñ he frst part of the total stffness matrx s proportonal to the materal stffness C dσ dλ and the second part s proportonal to the prestress σ as s the case wth the nternal forces. Because C s much larger than σ, the dsplacement due to prestressng wll be proportonal to σ. [KC) + Kσ + σ m )] δũ e σ + σ m )