A fast incremental-iterative procedure for ultimate strength analysis of composite cross-sections of arbitrary shape

Transcription

1 Computational ethods and Eperimental easurements XV 363 fast inremental-iterative proedure for ultimate strength analsis of omposite ross-setions of arbitrar shape C. G. Chiorean Tehnial Universit of Cluj-apoa Fault of Civil Engineering Romania bstrat new omputer method for bi-aial ultimate strength analsis of omposite steel-onrete ross-setions of arbitrar shape subjeted to aial fore and biaial bending moments is developed. n inremental-iterative proedure based on ar-length approah is proposed in order to determine in a unitar formulation both interation diagrams and moment apait ontours overoming the diffiulties and inauraies of the previousl published methods. This proedure adopts a tangent stiffness strateg for the solution of the non-linear equilibrium equations thus resulting in a high rate and unonditionall onvergene. n objet oriented omputer program to obtain the ultimate strength of omposite ross-setions under ombined biaial bending and aial load was developed. Eamples run and omparisons made have proved the effetiveness and time saving of the proposed method of analsis. Kewords: omposite ross-setions ultimate strength ar-length method bi-aial bending. 1 Introdution In reent ears some methods have been presented for the ultimate strength analsis of various onrete and omposite steel-onrete setions suh as retangular L and T -shape polgonal and irular under biaial moments and aial loads [1 5]. mong several eisting tehniques two are the most ommon; the first onsists of a diret generation of points of the failure surfae b varing the position and inlination of the neutral ais and imposing a strain distribution orresponding to a failure ondition. This tehnique generates the failure surfae ISS X (on-line) WIT Transations on odelling and Simulation Vol WIT Press doi:1.2495/ce11321

2 364 Computational ethods and Eperimental easurements XV through 3D urves (Fig. 1) that are diffiult to plot having limited appliabilit in the pratial appliations. The seond approah is based upon the solution of the non-linear equilibrium equations aording to the lassial ewton s sheme to obtain either the strain equilibrium plane or the loation and inlination of the neutral ais. In general these methods generates plane interation urves and gives fast solutions but are sensitive to the origin of the loading aes and some problems in onvergene ma arise partiularl when the initial or starting values of variables are not seleted properl and under large aial fores [2]. These methods an be further distinguished b how the plot the interation urves. These urves ma be omputed indiretl b epliit omputation of the moment-urvature response and the failure of the ross-setion orresponds to the top of the moment-urvature diagram when an of the materials reahes its predefined maimum allowable strain [3 5]. This method an be used under an loading mode but is rather time-onsuming and the basi equations of equilibrium are not alwas satisfied [3]. On the other hand to determine diretl the interation diagrams or load ontours the eat loation of the neutral ais ma be determined b solving three oupled nonlinear sstem of equations b an iterative approah [1 2 4]. Failure surfae Triplets of stress resultant along a straight line Interation diagram for =onstant oment-apait ontours (=onstant) Interation diagram for =onstant (3D urves) Figure 1: Failure surfae and interation diagrams. These algorithms are not straightforward to implement starting or initial values are not alwas simpl to hoose and problems of onvergene ma arise espeiall when strain softening of the onrete in ompression is taken into aount and the ma beome unstable near the state of the pure ompression. There eist three different methods to generate plane interation urves for rosssetions under biaial bending: (1) interation urves for a given bending moment ratio [1] (2) load ontours for a given aial load [2] and (3) generate triplets of stress-resultants on the failure surfae b etending an arbitrar oriented straight line [4]. The plane interation urves generated b these methods are depited in Fig. 1. The main objetive of the present paper is to present a new formulation b whih the biaial interation diagrams and moment apait ontours of an arbitrar omposite ross-setion an be determined. Essentiall the inelasti WIT Transations on odelling and Simulation Vol WIT Press ISS X (on-line)

3 Computational ethods and Eperimental easurements XV 365 ross-setion analsis method emploed herein uses the aura of the fibber element analsis through the use of path integral approah for numerial integration of the ross-setional nonlinear harateristis and addresses its effiien and modelling shortomings both to failure surfae generation proedure overoming the diffiulties and inauraies of the previousl proposed methods and to postproessing proedure of the aial fore and bending moments obtained at a ross-setion level in order to hek diretl that the fulfil the ultimate limit state ondition. 2 athematial formulation 2.1 ssumptions and problem definition Consider the arbitrar ross-setion shape subjeted to the ation of the eternal bending moments about eah global aes and aial fore as shown in Figure 2. It is assumed that plane setion remains plane after deformation. This implies perfet bonding between the steel and onrete omponents of a omposite ross setion. Thus resultant strain distribution orresponding to the urvatures about global aes ={ } and the aial ompressive strain an be epressed at a generi point in a linear form as: (1) eutral ais Compresion side Tension side C ( ) Interior boundar Eterior boundar u 2 2 tan Figure 2: odel of arbitrar omposite ross-setion. The onstitutive relations for onrete under ompression are represented b a ombination of a seond-degree parabola and a straight line as depited in Fig. 3(a). The parameter represents the degree of onfinement in the onrete and allows for the modelling of reep and onfinement in the onrete b simpl varing the rushing strain and respetivel. The tensile strength of onrete is negleted. multi-linear elasto-plasti stress-strain relationship both in tension and in ompression is assumed for the strutural steel and the steel WIT Transations on odelling and Simulation Vol WIT Press ISS X (on-line)

4 reinforing bars (Fig. 3(b)). t ultimate strength apait the equilibrium is satisfied when the eternal fores are equal to the internal ones and in the most ompressed or tensioned point the ultimate strain is attained. These onditions an be represented mathematiall in terms of the following nonlinear sstem of equations as: (a) f''(1-) f'' u Stress f Strain 2 2 '' 2 f f ' ' 1 u f f Figure 3: Stress-strain relationships: (a) for onrete in ompression; (b) for steel. ; ; u d d d (2) and in whih represent the unknown. In the Eqs. (2) the first three relations represent the basi equations of equilibrium for the aial load and the biaial bending moments respetivel given in terms of the stress resultants. The last equation represents the ultimate strength apait ondition; that is in the most ompressed or most tensioned point the ultimate strain is attained and in whih and represent the oordinates of the point in whih this ondition is imposed. The oordinates of the onstrained point an be alwas determined for eah inlination of the neutral ais defined b the parameters and and u represents the ultimate strain either in most ompressed onrete point or in most tensioned reinforement steel fibre. Under the above assumptions the problem of the ultimate strength analsis of omposite ross-setions an be formulated as: With strain distribution orresponding to a failure ondition find the ultimate resistanes so as to fulfil the basi equations of equilibrium and one of the following linear onstraints: ) ( ; ) ( L L b L L a (3) where represents the given aial fore and bending moments respetivel. (b) 366 Computational ethods and Eperimental easurements XV ISS X (on-line) WIT Transations on odelling and Simulation Vol WIT Press

5 Computational ethods and Eperimental easurements XV 367 (a) (b) ( ) ( ) ( ) ( ) ( ) Figure 4: General solution proedures. (a) Interation diagrams for given bending moments; (b) oment-apait ontours for given aial fore and bending moment. The general solution proedure is organized around solving the nonlinear sstem (2) for one of two linear onstraints defined b the Eqs. (3ab) Corresponding to eah linear onstraints we an define a point on the failure surfae as: (I) when the onstraints (a) are injeted in the nonlinear sstem (2) a point on the failure surfae is defined omputing the aial resistane assoiated to a failure riterion and for a fied value of bending moments ( ); (II) when onstraints (b) are used the point is assoiated to a fied aial load () and a given bending moment about ais. ll these situations are graphiall illustrated in Figure ethod of solution n inremental-iterative proedure based on ar-length onstraint equation is proposed in order to determine the biaial strength of an arbitrar omposite steel-onrete ross setion aordingl to the alread desribed situations. The failure diagrams orresponds either to maimum strains attained at the outer ompressed point of the onrete setion (i.e. u equal to the ompressive strain at failure) or to maimum strains attained in most tensioned reinforement steel fibre (i.e. u equal to the tensile steel strain at failure). Consider an irregular omposite setion as shown in Figure 2. The global -aes of the ross setion ould have their origin either in elasti or plasti entroid of the ross-setion. For eah inlination of the neutral ais defined b the parameters and the farthest point on the ompression side (or the most tensioned steel bar position) is determined (i.e. the point with o-ordinates ). We assume that at this point the failure ondition is met and onsequentl the aial ompressive strain ε an be epressed as: u (4) WIT Transations on odelling and Simulation Vol WIT Press ISS X (on-line)

6 368 Computational ethods and Eperimental easurements XV Thus resultant strain distribution orresponding to the urvatures and an be epressed in linear form as: u (5) In this wa substituting the strain distribution given b the Eq. (5) in the basi equations of equilibrium the unknown ε together with the failure onstraint equation an be eliminated from the nonlinear sstem (2). Thus the basi equations of equilibrium together with the linear onstraints Eqs. (3(a)) (3(b)) forms a determined nonlinear sstem of equations (i.e. 5 equations and 5 unknowns): d d d L L 1 2 and the solutions an be obtained iterativel following an approah outlined in the net setions Interation diagrams for given bending moments In this ase introduing the onstraints (3a) in the sstem (6) the problem of the ultimate strength analsis of ross-setion an be epressed mathematiall as: d ; d ; d (7) in whih aial load and urvatures and represents the unknowns and represents the load parameter defining the intensit of the bending moments. If we regard the urvatures as independent variables in aial fore equation the urvatures and the load amplifier fator are given b solving the following nonlinear sstem of equations: d (8) d This an be rewritten in terms of non-linear sstem of equations in the following general form: int et F Φ f f (9) To traverse a solution path a proper parametrization is needed. ommon setting of a ontinuation proess is to augment the equilibrium equations (9) with a onstraint [6]. In this ase the urvature-moment onstraint an be defined b equation g in the following form: int et f f H Φ (1) g Φ (6) WIT Transations on odelling and Simulation Vol WIT Press ISS X (on-line)

7 Computational ethods and Eperimental easurements XV 369 In this proedure ommonl alled ar-length method these equations are solved in a series of steps or inrements usuall starting from the unloaded state ( ) and the solution to (1) is referred to as equilibrium path. Instead of solving Eqs. (1) diretl an indiret solution sheme for the onstraint equation ma be introdued. ording to the indiret ar-length tehnique [6] the iterative hanges of urvature vetor for the new unknown load level is written as: k 1 k 1 1 et T F KT f Φ K F ΦT (11) where F represents the out-of-balane fore vetor (Eq. 9) and K T represents the tangent stiffness matri of the ross-setion: int int F K T int int (12) Φ in whih the partial derivatives are with respet to the strains and stresses evaluated at urrent iteration k. ssuming the strain distribution given b the Eq.(5) the oeffiients of the stiffness matri an be smboliall evaluated as: k k k k int int int int d d ET d d ET d d ET d d d d d ET d where the oeffiients k ij are epressed in terms of the tangent modulus of elastiit E t. Thus the inremental urvatures for the net iteration an be written as: Φk 1 Φk Φ (14) This proedure is iterated until onvergene upon a suitable norm is attained. ssuming that a point ( ) of the equilibrium path has been reahed the net point ( + + ) of the equilibrium path is then omputed updating the loading fator and urvatures as: Φ k 1 (13) (15) Φ Φk 1 In this wa with urvatures and loading fator known the aial fore resistane is omputed based on the resultant strain distribution orresponding to the urvatures and through Equation (7) and the ultimate bending moments and are obtained saling the referene eternal moments and through urrent loading fator given b Equation (15). Graphial WIT Transations on odelling and Simulation Vol WIT Press ISS X (on-line)

8 37 Computational ethods and Eperimental easurements XV representation of the present method aording to Equation (1) is depited in Figure 5. It is important to note that the stiffness matri of ross-setion K T given b the Equations (13) ould beome negative definite or singular during the iterative proess therefore the above proedure based on ar-length onstraint equation is essential to overome these diffiulties. Load path (or ) det K T = det KT> det KT< det KT> Equilibrium path (or ) Interation urve det KT= (balane point) det KT< Figure 5: Geometrial representation of the present method oment apait ontour for given aial fore and bending moment In this ase injeting the linear onstraints (3b) in the nonlinear sstem (6) and arranging the sstem aordingl with the deoupled unknowns we obtain: d d d (16) in whih bending moment and urvatures and represents the unknowns. Following a similar approah as presented above the urvatures are obtained solving the first two equations and then with this strain distribution the bending moment resistane about ais is omputed with the last equation of the sstem. 2.3 Evaluation of tangent stiffness and stress resultant Based on Green's theorem the integration of the stress resultant and stiffness oeffiients over the ross-setion will be transformed into line integrals along the perimeter of the ross-setion. For this purpose is neessar to transform the variables first so that the stress field is uniform in a partiular diretion given b the urrent position of the neutral ais [7]. This is ahieved b rotating the referene aes to oriented parallel to and perpendiular to the neutral ais respetivel as shown in Fig. 2. Thus the stress field is uniform in diretion parallel with the neutral ais and strains and the internal fores arried on b the ompressive onrete and strutural steel an be obtained b the following epressions: WIT Transations on odelling and Simulation Vol WIT Press ISS X (on-line)

9 Computational ethods and Eperimental easurements XV 371 int int int dd dd dd sin os dd int os int dd os sin dd sin os int int sin (17) where int int and int are the internal aial fore and bending moments about the and ais respetivel and an be omputed based on the Green s path integral approah. The tangent stiffness matri oeffiients are omputed in the same wa. In order to perform the integral of a determined side of the ontour polgonal or irular of the integration area the interpolator Gauss- Lobatto method is used. Though this rule has lower order of aura than ustomar Gauss-Legendre rule it has integration points at eah ends of interval and hene performs better in deteting ielding. However beause the stress filed is defined b a step funtion and there is no ontinuit in the derivative the polnomial interpolation an produe important integration errors. In this ase an adaptive quadrature strateg an be applied. In this ontet of the adaptivit quadratures the Lobatto integration sheme has another advantage over the Legendre integration sheme observing that the point orresponding to the left end in one interval is the same as the point orresponding to right end in the net. Consequentl the ost of evaluating a Lobatto rule is redued b about one integrand evaluation omparing with Legendre rule. The steel bars are assumed disrete points with area sj o-ordinates sj sj and stress f sj. 3 Computational eample Based on the analsis algorithm just desribed a omputer program SEP has been developed to stud the biaial strength behaviour of arbitrar onrete-steel ross setions. In order to demonstrate the validit aura unonditionall onvergene and time saving of the analti proedure developed here the interation diagrams and moment apait ontours of a retangular rosssetion with asmmetriall plaed strutural steel (Fig. 6(a)) are determined and ompared with the numerial proedure developed in [2]. Charateristi strength for onrete in ompression is f =31.79 pa and the stress-strain urve whih onsists of a paraboli and linear- part was used in the alulation with the rushing strain =.2 and ultimate strain u =.35. The Young modulus for all steel setions was 2GPa while the maimum strain was su =1%. The ield strength of steel reinforing bars is f =42 Pa whereas for the strutural steel the following values has been onsidered f flange =255 Pa f web =239 Pa. In order to demonstrate the unonditionall onvergene of the algorithms developed in the urrent paper the ross-setion has been analsed drawing the interation diagrams and moment apait ontours for aial loads near the aial load apait onsidering both geometri and plasti entroid of the ross-setion. Convergene problems have been eperiened b the Chen et al. [2] in this portion of the moment apait ontour when the geometrial entroid of the ross-setion has been taken as the referene loading aes. WIT Transations on odelling and Simulation Vol WIT Press ISS X (on-line)

10 372 Computational ethods and Eperimental easurements XV Figure 6(b) presents the moment apait ontours obtained b the present algorithm and those obtained b the Chen et al. [2] for aial load =3k onsidering as referene loading aes geometri (GC) and plasti entroid (PC) respetivel. (a) (b) Figure 6: (a) Retangular setion with asmmetriall plaed strutural steel. (b) oment apait ontour with aial load =3k. o onvergene problems have been eperiened b the proposed approah even when the geometri entroid has been hosen as referene aes or the strain-softening of the onrete in ompression has been taken into aount (=.15) and a maimum of just two iterations has been required during the inremental-iterative proess. s it an be seen the results obtained in the urrent paper and those reported in [2] agree losel in both ases. The effet of the strain-softening is not taken into aount in the method proposed in [2]. However it is important to note that although laimed the method proposed b Chen does not generate genuinel plane moment-apait urves. The method proposed in [2] fails in some irumstanes to draw the moment apait ontour under a fied aial load and in order to overome some divergenes aial load value is slightl adjusted. Figure 7(a) shows the omplete interation diagrams under uniaial bending moment about ais with and without the effet of the strain softening of the onrete. s it an be seen near the aial load apait multiple solutions eists in the - spae when the strain softening is modelled. This an be learl observed on the moment-ultimate urvature diagrams (Fig. 7(b)) when multiple snap-through phenomenons our in these situations. When the strain-softening is ignored (=) the moment-ultimate urvature urve is flat in that region (Fig.7.b) that eplains the uniqueness of the solution in the - spae in this ase (Fig.7(a)). Figure 8 shows the omparative interation diagrams for =15 and 3 respetivel ( =tan() ). The bending moments are omputed about aes - whih pass through the geometri entroid of the ross-setion. WIT Transations on odelling and Simulation Vol WIT Press ISS X (on-line)

11 Computational ethods and Eperimental easurements XV 373 Figure 7: (a) Interation diagrams under uniaial bending moment about ais; (b) Bending moment-ultimate urvature variation. Bending moment about Y-ais Bending moment about X-ais Figure 8: Biaial interation diagrams. 4 Conlusions new omputer method based on inremental-iterative ar-length tehnique has been presented for the ultimate strength analsis of omposite steel-onrete ross-setions subjeted to aial fore and biaial bending. Comparing the algorithm presented in the urrent paper with the eisting methods it an be onluded that the proposed approah is general and omplete an determine both interation diagrams and moment apait ontours and of great importane it is fast the diagrams are diretl alulated b solving at a step just two oupled nonlinear equations and assures onvergene for an load ase even near the state of pure ompression or tension and is not sensitive to the initial/starting values how the origin of the referene loading aes is hosen or to the strain softening effet for the onrete in ompression. Furthermore the proposed method as is formulated an be applied to provide diretl the ultimate resistanes of the ross-setion supposing that one or two omponents of the setion fores are known without the need of knowing in advane the whole WIT Transations on odelling and Simulation Vol WIT Press ISS X (on-line)

12 374 Computational ethods and Eperimental easurements XV interation diagram or moment apait ontour. The method has been verified b omparing the predited results with the established results available from the literature. It an be onluded that the proposed numerial method proves to be reliable and aurate for pratial appliations in the design of omposite steelonrete beam-olumns and an be implemented in the advaned analsis tehniques of 3D omposite frame strutures. knowledgement The writer gratefull aknowledges the support from Romanian Researh Foundation (CCSIS- Grant PII-IDEI o. 193/28) for this stud. Referenes [1] Rodrigues J.. & ristizabal-ohoa J.D. Biaial interation diagrams for short RC olumns of an ross setion Journal of Strutural Engineering SCE 125(6): [2] Chen S.F. Teng J.G. Chan S.L. Design of biaiall loaded short omposite olumns of arbitrar setion Journal of Strutural Engineering SCE 127(6) [3] Sfakianakis.F. Biaial bending with aial fore of reinfored omposite and repaired onrete ross setions of arbitrar shape b fibber model and omputer graphis dvanes in Engineering Software [4] Rosati L. armo F. Serpieri R. Enhaned solution strategies for the ultimate strength analsis of omposite steel-onrete setions subjet to aial fore and biaial bending Computer methods in applied mehanis and engineering 197(9-12) [5] Charalampakis.E. Koumousis V.K. Ultimate strength analsis of omposite setions under biaial bending and aial load dvanes in Engineering Strutures 39(11) [6] Crisfield.. on linear finite element analsis of solids and strutures Wile Chihester [7] Chiorean C.G. fast inremental-iterative proedure for inelasti analsis of RC ross-setions of arbitrar shape ta Tehnia apoensis WIT Transations on odelling and Simulation Vol WIT Press ISS X (on-line)