Dynamic Relaxation Method with Critical Damping for Nonlinear Analysis of Reinforced Concrete Elements * Anna Szcześniak 1) and Adam Stolarski 2)

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1 Dyamic Relaxatio Method with Critical Dampig for Noliear Aalysis of Reiforced Cocrete Elemets * Aa Szcześiak 1) ad Adam Stolarski ) 1) ) Faculty of Civil Egieerig ad Geodesy Military Uiversity of Techology Warsaw Polad 1) aa.szczesiak@wat.edu.pl ABSTRACT The method for aalysis of the oliear behavior of reiforced cocrete elemets subjected to short-term static loads called as the Dyamic Relaxatio Method is preseted i the paper. I this method the pseudo-dyamic problem ca be reduced to aalysis of a static problem by itroductio the critical dampig ad cosiderig the process as a aperiodic motio that approaches the equilibrium state uder the actig load. For the applicatio of complete form of the method cosideratios i the rage of ielastic properties for the structural materials (Szcześiak 016) as well as the modellig of deformatio processes of reiforced cocrete bar elemets are carried out. To icrease the effectiveess of the method i the post-critical rage of aalysis the path trackig for multiple variables method of solutio is itroduced to the solutio of the system of dyamic oliear equilibrium equatios. After icludig the arc-legth parameter to the solutio the global softeig of the structural elemet i post-critical rage which leads to realizatio of failure mechaism ca be aalyzed. The results of umerical tests are carried out for reiforced cocrete beam ad eccetrically loaded colum. Comparative aalysis of the obtaied results with experimetal (Foley 1998 Lloyd 1996) umerical ad aalytical results preseted i the literature demostrated that the proposed computatioal method is very effective. 1. INTRODUCTION The mai purpose of this work was to develop a computatioal model of reiforced cocrete bar elemets which ca be used to study the behavior of a elemet goig from purely elastic to elastic plastic ad the to stress softeig ad failure of the cross-sectio. To obtai a solutio the modelig rage of the ielastic material properties the deformatio processes of the bar structural elemets ad the umerical solutios must be cosidered. The properties of the structural material were modeled by 1) Adjuct ) Professor

2 assumig a reduced plae compressive/tesile stress state with shear. A elastic plastic material model with hardeig for the reiforcig steel was used. The elastic plastic material model was developed by cosiderig the stress softeig ad degradatio of the deformatio modulus for the cocrete. Aalysis of the structural elemet icludes a descriptio of the behavior of a eccetrically compressed reiforced cocrete elemet modeled as a bar system uder static loadig. The equatios for moderately large displacemets of the bar iclude the ifluece of the logitudial deformatios chages to the rotatio agle of the cross-sectio (curvature) ad isochoric strais. These was used as the basis of theoretical model of the behavior of the structural elemet. A load capacity aalysis techique for a structural system was developed usig the fiite differece method ad the dyamic relaxatio method (DRM) was applied to solve the oliear equilibrium equatios. I this method the problem ca be reduced to aalysis of a pseudo-dyamic process by cosiderig critical dampig as a aperiodic motio that approaches the equilibrium state uder the actig load. The solutio to this problem ca be obtaied usig a iterative process with pseudo-time steps. The DRM is used to aalyze highly oliear problems (Rodriquez et al. 011 Hicz 009). The characteristics of this method were preseted by Uderwood (1983). Rezaiee-Pajad ad Alamatia (010) proposed further modificatios to improve the covergece ad accuracy of the solutios. The arc-legth parameter o the equilibrium path was itroduced to the proposed umerical procedure to icrease of the efficiecy of the method. This is referred to as the arc-legth method ad it simultaeously determies the displacemets ad load parameter by tracig the equilibrium path that cotais the local sigular poits. Riks itroduced the precursor of this method i Riks (197). Subsequet developmets ad iterpretatios of the arc-legth method were preseted by Ramm (1981) Crisfield (1981 ad 1996) Schweizerhof ad Wriggers (1986) amog others. A combiatio of the dyamic relaxatio ad arc-legth methods was preseted by Pasqualio (004). This was the applied to practical aalyses of structural members by Remesh ad Krishamoorthy (1993) Pasqualio ad Estefe (001). I this paper a geeral computatioal algorithm that is based o developed effort aalysis methods for reiforced cocrete elemets was proposed. This techique was implemeted i Fortra to calculate strais stresses iteral forces ad displacemets. To verify the accuracy of the proposed method umerical studies of bar structural elemets were carried out. Obtaied umerical results were compared with the results of experimetal aalytical ad theoretical works from the existig literature.. MODELING OF STRUCTURAL MATERIALS The elastic plastic model with regard to material hardeig was applied for the reiforcig steel. The reduced plae stress state (compressio/tesio with shear) because of the ature of the reiforcig sleder bars i the reiforced cocrete elemet was cosidered.

3 (1) where Es 1 E s 3 E s is the deformatio modulus of steel ad s1 1 s E s s (1 ) is the shear modulus s is the scalar multiplier that defies the state of stress accordig to the associated plastic flow rule for the Huber-Mises-Hecky yield criterio is the step umber of the istataeous strai-stress state i subsequet time istat t t 1 t ad t is the time step icremet. A elastic plastic material model for cocrete that cosiders material softeig ad the degradatio of the deformatio modulus was developed. A reduced plae stress state for the compressio/tesio rage with shear ( ) was assumed. The cocrete model describes the icremetal equatios for stresses while cosiderig the limitatios that are a result of the yield coditio. That is 11 dla ft 11 fc Ec11 11 ft dla 11 ft f c dla 11 fc 1 dla fs c fs dla 1 fs () 1 where is the istataeous step of the stress-strai state ad are the kow strais ad strai icremets f c0 f t0 ad f s0 are the iitial compressive tesile ad shear stregths E c0 is the iitial modulus of E deformatio c c (1 c0) is the shear deformatio modulus c 0 is the iitial fc0 f Poisso's ratio c0 ad t0 t0 are the elastic limit strais i compressio Ec0 Ec0 ad tesio fc ad uc are the strai limits for the perfectly plastic flow ad the material softeig rage i compressio ad ut is the strai limit for the material softeig rage i tesio. The yield criterio used i cocrete model is cosistet with experimetal results for the reduced plae stress state. This is also cofirmed by comparig proposed model with that described by Stolarski (1991) which was calibrated with experimetal results.

4 3. FUNDAMENTAL EQUATIONS 3.1 Equatios of motio The equatios of dyamic motio for a reiforced cocrete bar elemet characterized by the uit mass ( ) the uit mass of iertia ( j ) ad the uit dampig coefficiets for the displacemets ( c ) ad rotatios ( c ) were derived. Problem of the dampig coefficiet estimatio was detailed described i the paper of Szcześiak & Stolarski (015). The differetial equilibrium equatios were defied i the global Cartesia coordiate system ({ x ( u) z( w)} ) as show i Fig. 4. They have the form f Ncos Qsi pxsu cu 0 s s Nsi Qcos pzsw cw 0 s s M Q j cf 0 s (3) where N is the iteral logitudial force Q is the trasversal force M is the bedig momet p p } are the exteral loads { u w j } are the iertial { x y logitudial trasverse ad rotatioal forces ds is the legth of the deformed elemet ad is the slope agle. 3. Equatios of iteral equilibrium i the cross-sectio The bar s cross-sectio was discretized to produce a computatioal model. The cross-sectio of the cocrete was divided ito layers that were h - thick ad withi this the areas of the two steel layers were defied A s1 ad A s. The strai states i the differet cross-sectioal layers for a give time step t t 1 t are defied as 11 r e zr zr zk zs1 zs k 1... K 1 1 r (4) If the logitudial strai of the cetral axis ( e ) the chage i the average rotatio agle of the cross-sectio ( ) ad the average agle of the o-dilatatioal strai ( ) were kow the after usig the material models the logitudial force ( N )

5 bedig momet ( M ) ad trasverse force ( Q equilibrium equatios of the cross-sectio. That is (5) where ) ca be determied from the A c k is the cross-sectio area of the cocrete layer ad s1 A is the crosssectio area of the tesile/compressive reiforcig steel. K N 11 k Ac k 11 s 1 As1 11 s As k 1 K M k Ac k zk s As zs s As zs k K Q 1 1 k Ac k 1 s 1 As 1 s As k 1 4. SOLUTION OF THE SYSTEM OF EQUILIBRIUM EQUATIONS 4.1 Numerical solutio of equatios of motio The system of equatios i (5) was solved usig a umerical method ad a discretizatio with respect to time. For this purpose the direct differetial method with respect to time was applied. T I this method acceleratio q ( u i w i i1) ad displacemet velocity T q ( u i w i i1) were approximated i the system of equatios i (3) at time istats t1 t t t t ad t1 t t usig 1 q q q q ad q t (6) Usig these approximatios the system of equatios i (3) becomes (7) where q w T t G( q ) q qg q P 0 ui i i1 is a vector of searched displacemet icremets aipx ( si ) q P aipz ( si ) is a vector of compoets of the a fi[ Mi k( x ) ( )] b1 i xi M b1 i k x bi i xi bi 1 if f 0 displacemet icremets for the total load k ( f ) is a selectio operator 0 if f 0

6 i is the ode umber ad i 1 i 1 i 0 i 1 are the segmet umbers of the iteral spatial divisio 1 bi ui ai[ Ni1 cosi1 Ni0 cosi0 Qi1 sii1 Qi 0 sii0] 1 1 qg bi wi ai[ Ni1 sii1 Ni0 sii0 Qi1 cosi1 Qi 0 cosi0] q P is a vector b fii1 a fi( Mi Mi Qi1 si1) 1 of compoets of the displacemet icremets for the load parameter from the m previous time step ad b i i mi Cit t j ai b i fi mi Cit ji C fit ad t a fi Ci - the dampig factor for displacemets i mai ode C fi - the ji C fit dampig factor for the rotatios of the segmet divisio. The iitial coditios for time step t 0 0 have the form q 1 0 (8) The iterative procedure termiates whe the solutios of displacemets i subsequet time istats have coverged i.e. q q (9) The time step t is determied so that the umerical itegratio is stable ad has the form t rmi{ t( l) t( b) t( f) } (10) cs where t( l) si mi is the critical time step for the logitudial elastic Ec0 s max Es vibratio problem bedig problem s t 1 i b si ( ) ( ) mi Bcs is the critical time step for the elastic t cs ( f ) s i mi is the critical time step for the elastic isochoric c si1 si0 wave propagatio si r is a safety factor for the time step ( si ) cs Acs is the uit mass of the elemet cs is the desity of the reiforced cocrete Bcs Ec0Jcs is the bedig rigidity of the reiforced cocrete cross-sectio i the elastic rage is the momet of iertia of the ucracked (elastic) Jcs reiforced cocrete cross-sectio ad s max is the largest total reiforcemet ratio for the etire reiforced cocrete elemet.

7 (11) where 4. Solutio to the costraits equatio The costrait equatio has the form (Wriggers 008) T m m f( q ) q q l 0 l is the icremet (parameter) of the arc legth o the solutio path q u w T is a vector of the searched displacemets m m is the icremet of the curret load parameter ad qm q qm is the icremet of the curret vector of ukow displacemets q i relatio to the last coverget values of the load parameter m ad displacemet vector assumed accuracy i the previous load step m. q m m obtaied with the The load parameter was determied by directly solvig the costraits i T Equatio (11). The sig of the differetiator g( q) l q m qm was checked ad the solutios were determied accordig to T l q m qm 1 m if g ( q) 0 ad T 1 m m qm l q if g ( q) 0 The the solutio that is the smallest "distace" away from the solutio obtaied i load step m 1 was selected. That is the smallest agle (or the largest cosie of the agle) betwee the solutios i steps m ad m 1 qm1qm1 qmqm1 mi( 1 ) miarccos arccos l l (1) Aalyzig the coditios of the solutio to the costraits i Equatio (11) it follows that this criterio satisfies (13) T T m sig( l q m q m ) abs( l q m qm) 4.3 Coceptual algorithm for solvig the exteded system of equatios The costraits i Equatio (11) together with the system of equatios of motio i (7) form a exteded system of equatios. This system for ay time istat t t ca be iteratively solved. This exteded system of equatios ca be used to determie the searched displacemet vector ( q ) ad the load parameter ( ) for the oliear equilibrium path cotaiig the local limit poits. This exteded system was solved usig a two-stage procedure. First i the predictio stage the load parameter (which depeds o the displacemet values from the previous iterative step) was determied usig the costraits i (11). Secod i the correctio stage the ew displacemet

8 values i the ext iterative step were determied usig the system of motio equatios i (7). Fig. 1 shows the iterative scheme for solvig the exteded system of equatios. Fig. 1 Iterative scheme for solvig the system of exteded equatios (14) The solutio after iteratios is 1 q q 1 Δq The iteratios are termiated after reachig a predefied accuracy ( G ) that is G (q ) G (15) O the basis of carried out umerical experimets it was determied that the 3 6 accuracy should satisfy G This method allows us to aalysis of the static behavior of a physically ad geometrically oliear structural elemet which was described by system of dyamic equilibrium equatios after applyig the critical dampig. Figures ad 3 schematically show load ad displacemet i pseudo time fuctio for the dyamic relaxatio method before ad after applyig critical dampig. I the elastic rage usig the critical dampig allows determiig the displacemet values that correspod to the equilibrium state of the udamped vibratios. Whereas i ielastic rage the displacemets determied after itroducig critical dampig are usually lower tha displacemet values correspodig to the equilibrium state of the udamped vibratio.

9 Fig. Load ad displacemet vs. pseudo time for the dyamic relaxatio method with critical dampig i elastic rage Fig. 3 Load ad displacemet vs. pseudo time for the dyamic relaxatio method with critical dampig i ielastic rage

10 5. CONCLUSIONS The paper presets a behavior aalysis method for the bedig ad compressio of reiforced cocrete elemets subjected to short-term static loads. The structural system aalysis techique was developed usig assumptios from the fiite differece method. The cetral axis of the structural elemet was discretized ad the equilibrium equatios ad geometrical relatioships i differetial form were writte. A joit-actio rule was determied for the active cocrete ad steel layers of the discretized crosssectioal model. A dyamic process that ca describe the static problem by itroducig critical dampig was aalyzed. Whe this procedure is used to solve a system of oliear equilibrium equatios it is referred to as the DRM. Usig DRM the equatios of motio ca be recursively solved i subsequet pseudo-time istats for each ode of the spatial divisio. There was o eed to solve the system of algebraic equilibrium equatios. The DRM was improved by itroducig a additioal costrait equatio to the system of equilibrium equatios accordig to the assumptios of the arc legth method. This method is called the DRM+AL ad ca be used to solve the oliear equilibrium equatios i the post-critical rage. Whe modelig the behavior of structural materials the reduced plae compressive/tesile states with shear stress was cosidered. Equatios derived from the theory of moderately large displacemets i a bar system form the basis of structural behavior descriptio assumig ifiitesimal deformatios. This assumptio is appropriate whe describig strai ad stress states i the cocrete ad steel layers of commoly used reiforced cocrete elemets. The verificatio of developed method umerical tests o a reiforced cocrete beam ad eccetrically loaded colum were carried out ad idicate to high effectiveess of the method for a post-critical aalysis. REFERENCES Crisfield M.A. (1981) A fast icremetal/iterative solutio procedure that hadles sapthrough Computers ad Structures Crisfield M.A. (1996) No-Liear Fiite Elemet Aalysis of Solids ad Structures. Essetials Wiley. Foley C. M. ad Buckhouse E. R. (1998) Stregtheig Existig Reiforced Cocrete Beams for Flexure Usig Bolted Exteral Structural Steel Chaels Structural Egieerig Report MUST-98-1 Marquette Uiversity Milwaukee Wiscosi. Hicz K. (009) Noliear aalysis of cable et structures suspeded with arches with block ad tackle suspesio system takig ito accout the frictio of the pulleys Iteratioal Joural of Space Structures 4 (3) Lloyd N. A. Raga B. V. (1996) Studies o High-Stregth Cocrete Colums uder Eccetric Compressio ACI Structural Joural Techical Paper 93-S59 November-December

11 Pasqualio I. P. ad Estefe S. F. (001) A oliear aalysis of the buckle propagatio problem i deepwater pipelies Iteratioal Joural of Solids ad Structures Pasqualio I. P. (004) Arc-legth method for explicit dyamic relaxatio XXI ICTAM 15-1 August Warsaw Polad e-book: SM1S Ramm E. (1981) Strategies for tracig the oliear respose ear limit poits. I W. Wuderlich E. Stei ad K. J. Bathe editors Noliear Fiite Elemet Aalysis i Structural Mechaics Spriger Berli Heidelberg New York. Rezaiee-Pajad M. ad Alamatia J. (010) The dyamic relaxatio method usig ew formulatio of fictitious mass ad dampig Structural Egieerig ad Mechaics 34 (1) Riks E. (197) The applicatio of Newtos method to the problem of elastic stability Joural of Applied Mechaics Schweizerhof K. ad Wriggers. P. (1986) Cosistet liearizatio for path fallowig methods i oliear fe-aalysis Computer Methods i Applied Mechaics ad Egieerig Stolarski. A. (1991) Model of dyamic deformatio of cocrete Archives of Civil Egieerig XXXVII (i Polish). Szcześiak A. Stolarski A. (016) A simplified model of cocrete for aalysis of reiforced cocrete elemets Bulleti of MUT Vol. LXV No Szcześiak A. ad Stolarski A. (015) Aalysis of critical dampig i dyamic relaxatio method for reiforced cocrete structural elemets. Mathematical ad Numerical Approaches 13th Iteratioal Coferece Dyamical Systems - Theory ad Applicatios December 7-10 Lodz: Uderwood P. (1983) Dyamic relaxatio i: Computatioal Methods for Trasiet Aalysis 1 i Computatioal Methods i Mechaics edited by: T. Belytschko T. J. R. Hughes Amstedram Wriggers P. (008) Noliear Fiite Elemet Methods Spriger Berli Germay.