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Time dependent creep and shrinkage analysis of composite beams curved in plan R Emre Erkmen a Mark A Bradfordb aLecturer School of Civil and Environmental Engineering University of Technology Sydney Broadway NSW 2007 Australia bAustralian Laureate Fellow and Scientia Professor Centre for Infrastructure Engineering and Safety Faculty of Engineering The University of New South Wales UNSW Sydney NSW 2052 Australia a r t i c l ei n f o Article history Received 26 September 2009 Accepted 9 August 2010 Keywords Curved composite members Creep Shrinkage Viscoelasticity Wiechert model Partial interaction a b s t r a c t This paper develops a numerical formulation for the time dependent creep and shrinkage analysis of steel concrete composite beams that are curved in plan under conditions of service load The creep behaviour of the concrete is considered by using the viscoelastic Wiechert model in which the aging effect of the concrete is taken into account The curved composite beam model that is developed also accounts for the partial shear interaction at the deck girder interface in the tangential or longitudinal direction as well as in the radial or horizontal direction due to the fl exibility of the shear connectors Models based on the developed formulation are validated by comparisons with sophisticated and com putationally intensive ABAQUS shell element models and with available results reported in the literature The effects of initial curvature and partial interaction on the time dependent behaviour of curved com posite beams are also illustrated in the examples 2010 Elsevier Ltd All rights reserved 1 Introduction Composite steel and concrete beams which are curved in plan are used widely in many engineering applications especially in highway bridge construction Accurate predictions of the compos ite beam defl ections are of paramount importance to the engineer because the serviceability limit state usually reduces to that of pre venting excessive defl ections on which effects of creep and shrink age of the concrete have marked infl uences 1 Creep behaviour can be characterised by the fact that the rate at which the inelastic strains develop depends not only on the current state of the stress and strain but also on the full history of their development Be cause of this the stress strain relations are usually expressed in the familiar convolution integral form 2 Numerical integration of the convolution integral has been used for the time dependent creep and shrinkage analysis of composite beams by several researchers including Bradford and Gilbert 1 Tarantino and Dezi 3 and Ranzi and Bradford 4 The drawback of this approach however is that the whole strain history of the preceding times has to be stored and this creates computational diffi culties Under serviceability conditions the time dependent creep effects of the concrete can be considered by using linear viscoelastic constitutive material models 5 Incremental numerical integration strategies based on viscoelastic constitutive material models that require only the information at the previous confi guration were initially proposed by Herrmann and Peterson 6 and Taylor et al 7 This approach was later generalised to account for the aging effects of the concrete by Bazant and Wu 8 This type of incremental ap proach was adopted for the time dependent creep and shrinkage analysis of steel concrete composite beams by Jurkiewiez et al 9 10 Comparisons between alternative procedures have also been presented in 9 The behaviour of curved composite beams is fundamentally dif ferent from that of straight composite beams because of the pres ence of primary torsion effects and its coupling with bending and the research in this area has been comparatively recent Giussani and Mola 11 developed an analytical formulation for the ser vice stage analysis of elastic composite beams curved in plan by assuming full interaction between the steel girder and the concrete deck However it is well known that the behaviour of composite beams is infl uenced by the fl exibility of the shear connection Erkmen and Bradford 12 reported a formulation that incorporates partial interaction in the tangential or longitudinal and radial or horizontal directions as well as the effects of torsion in the anal ysis of composite curved beams The objective of the present paper therefore is to extend the composite curved beam formulation of Erkmen and Bradford 12 to include the important time depen dent creep and shrinkage response An incremental recurrence type formulation that does not require the storage of the whole strain history of the preceding times is used by adopting the viscoelastic Wiechert model in which the spring and dashpot coeffi cients are selected by best fi tting of the model to the age dependent relaxation curves The relaxation curves for a very stiff material such as concrete are usually diffi cult to obtain from direct experiments and so a numerical interconversion technique is 0045 7949 see front matter 2010 Elsevier Ltd All rights reserved doi 10 1016 pstruc 2010 08 004 Corresponding author Tel 61 2 9385 5026 fax 61 2 9385 9747 E mail address emre erkmen unsw edu au R E Erkmen Computers and Structures 89 2011 67 77 Contents lists available at ScienceDirect Computers and Structures journal homepage adopted herein to obtain the discrete relaxation points from the age dependent creep function The proposed numerical formula tion that is developed is validated by comparing the results with sophisticated and computationally intensive ABAQUS shell ele ment models and with available experimental results reported in the literature The proposed formulation is shown to provide a very effi cient technique for the time dependent analysis of curved com posite beams Representative examples are considered to illustrate the effects of initial curvature and partial interaction on the time dependent behaviour of composite beams curved in plan 2 Kinematic relations 2 1 Basic assumptions Fig 1 shows a composite beam curved in plan for which the following assumptions are made The steel girder is a doubly symmetric I beam that is curved in plan whose response is elastic The deck has a rectangular cross section and has the same ini tial curvature as the girder in the undeformed confi guration Both cross sections remain rigid throughout the deformation i e no cross sectional distortion occurs There is no uplifting between the girder and the deck The radius of curvature is constant along the beam The shear connection between the girder and the deck is fl exible in both the tangential and radial directions Rotations and defl ections are small According to the assumption that plane sections remain normal to the deformed axis shear strains on the cross sections are induced by uniform torsion only 2 2 Deformations and strains A fi xed spatial oxys coordinate system is used to describe the geometry of the undeformed curved beam as shown in Fig 1 The s axis is oriented along the axial direction of the curved composite beam while the axes ox and oy are in the plane of the cross section The material o1x1y1s1coordinate system alters with the deforma tion of the structure The total deformations are considered to re sult from i a rigid translation due to the displacements u s v s and w s of the centroid along the tangential direction of the axes ox oy and os respectively ii rotation of the cross section through an angle s about the axis os iii superimposed warping displacement of the whole cross section due to non uniform tor sion and iv displacement functionsXx s Xs s andXj s due to slip action in the radial and tangential directions in the horizon tal plane and warping of the cross section respectively The slip displacement functionsXx s Xs s andXj s are considered inde pendent of the displacements of the axes 12 13 and are assumed positive for the girder and negative for the deck based on which the total slip displacements usp xi yi s in the x radial and wsp xi yi s in the s tangential directions at a point P xi yi s at the interface of the girder and the deck can be written as 12 usp xi yi s 2Xx s 1 and wsp xi yi s 2Xs s 2x xi yi Xj s 2 wherex x y is the normalised section warping displacement func tion 14 The normal strains and the shear strains can be obtained in terms of the displacement components as 12 e x y s w0 X0s Xxj x u00 xX0sj yv00 yj x 00 v00j X0j 3 c r s 2ra 0 v0j Xj 4 inwhich 0 d ds C cos S sin u0 u0 wj w0 w0 uj jis the initial curvature of the curved beam s centroidal locus about the y axis and r is the perpendicular distance from the mid surface of a plate segment to the point on the cross section with coordinates x y as shown in Fig 1 The plate segment refers to the web or the fl ange of the girder or to the concrete deck In Eq 4 acan be obtained from the Saint Venant solution of the uniform torsion problem 15 For thin rectangular plate segments acan be equated to unity 3 Variational formulation of the equilibrium equations The equilibrium equations can be obtained by means of the principle of virtual work which can be stated as dP Z L Z As deT srsdAds Z L Z Ac deT crcdAds Z L Z b ddT spqshdxds X duT QQ Z L duT qqds 0 5 in which the fi rst two integrals are the internal virtual work due to the deformations of the steel girder and the concrete deck where As and Acare the cross sectional areas of the steel and concrete com ponents respectively L is the span of the beam andes rs ecand rcare the vectors of strain and stress components for the steel gir der and concrete deck respectively The third integral in Eq 5 is the internal virtual work due to the slip at the interface between the girder and the deck in which dspand qshare the vectors of the relative slip and shear stresses between the deck and the girder respectively The interface shear forces applied by the shear connec tors are assumed to be continuously distributed and b is the width of the effective interface surface The last two terms in Eq 5 are the virtual work done by the external concentrated forces Q on the associated conjugate displacements duQand the virtual work done by distributed member forces q on the associated conjugate displacements duq x y s P x y s ii P u x1 y1 o v w s1 wsp usp r o1 1 Fig 1 Coordinate system and displacements 68R E Erkmen M A Bradford Computers and Structures 89 2011 67 77 3 1 Variations of strains From Eqs 3 and 4 the fi rst variation of the normal and shear strains for the both steel girder and concrete deck can be written as dek SBkdh 6 in which index k can be changed to s for the steel girder and c for the concrete deck The matrices S and Bkin Eq 6 are wherejs 0 5jand as 0 5 for the steel girder andjc 0 5jand ac 0 5 for the concrete deck The deformation vector of the se lected origin in Eq 6 can be written as h huvw XiT 9 in which the vectors u huu0u00u000i v hvv0v00v000i w hww0w00i h 0 00i and X h2Xx2X0 x2Xs2X0s 2Xj2X0ji have been used 3 2 Stresses The steel girder is assumed as being linear elastic under service loads and so the normal and shear stresses for the steel can be ob tained from the strains according to rs rs ss Es0 0Gs es cs 10 where Esand Gsare the Young s modulus and shear modulus for steel respectively The stress response of concrete on the other hand can be written in Stieltjes convolution integral form 18 in terms of the relaxation function R t t0 which represents the stress at time t caused by a unit constant strain acting since time t0as rc rc t sc t Z t t1 R t t0 10 00 5 1 mc dec t0 desh t0 dcc t0 11 in whichrcandscare the time dependent concrete normal and shear stresses respectively eshis the stress independent volumetric shrinkage strain and t1 is the time at fi rst loading It should be noted that in Eq 11 the Poisson s ratio of concretemcis assumed as being time independent 16 3 3 Variations of relative displacements at the interface From Eqs 1 and 2 the variation of the vector of the relative displacements due to the slip in Eq 5 can be written as ddsp hduspdwspiT SXBXdh 12 in which the matrices SXand BXare SX 100 01x 13 BX 00000000000000100000 00000000000000001000 00000000000000000010 2 6 4 3 7 5 14 The shear stresses at the interface quspand qwsp with units of force length2 in the radial and tangential directions respectively can be associated with the slip of a point length at the interface by assuming linear elastic behaviour for the shear connectors and so the vector of interface shear stresses in Eq 5 can be written as qsh qusp qwsp qu0 0qw usp wsp 15 wherequandqware the shear stiffnesses of the shear connectors with units of force length3 which are the shear stresses in the radial and tangential directions for a unit slip respectively 3 4 External loads The external concentrated load vector Q in Eq 5 is Q hQxQyQsMexMey MesiT 16 where Qxand Qyare the concentrated radial and vertical forces in the x and y directions Qsis the external concentrated axial force in the s direction Mexand Meyare the external bending moments about the x and y axes and Mesis the torque Similarly the external distributed load vector acting on the beam is q hqxqyqs mexmeymesiT 17 where qx qy qs mex meyand mesare the counterpart actions to Qx Qy Qs Mex Meyand Meswhich are distributed along the beam The external loads are assumed to act along the beam axis and so the displacements uQand uqat the points at which loads Q and q act can be written in terms of the displacements of the origin 4 Incremental step by step solution 4 1 Viscoelastic Wiechert model for concrete creep Computation of the stresses from Eq 11 by using numerical step by step integration requires the retaining of the complete past history of strain To overcome the problem of excessive com puter storage the viscoelastic Wiechert model Fig 2 a which is composed of parallel Maxwell units Fig 2 b is adopted herein The relaxation function can be conveniently approximated in terms of the age dependent modulus of the springs Em t0 and vis cosities of the dashpotsgm t0 in each Maxwell unit m as 8 S 1xyx0 00002ra 7 Bk j0000000010000 jk00ak00 00 1000000 j0000000 jk00 000000 10000j00000000 000000j000000 100000 ak 00000j000000 10000000 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 8 R E Erkmen M A Bradford Computers and Structures 89 2011 67 7769 R t t0 X n m 1 Em t0 exp t t0 Em t0 gm t0 E1 t0 18 where n represents the total number of parallel Maxwell units and E1 t0 is the modulus of the spring that is not coupled to any dash pot The stress strain relations in each Maxwell unit can be written as rm Em t0 rm gm t0 ec esh 19 in which d dt denotes the time derivative andrmis the normal stress at each Maxwell unit The total concrete normal stress is rc X n 1 m 1 rm 20 For the unit with spring modulus E1 t0 the viscosity can be taken as infi nity i e gm t0 1 and thus Eq 19 is also valid when the spring is not coupled to any dashpot Identically the relations be tween shear stress and shear strain can be written as 2 1 m sm Em t0 sm gm t0 cc 21 wheresmis the shear stress at each Maxwell unit The total concrete shear stress is sc X n 1 m 1 sm 22 4 2 Determination of spring moduli For any given number of Maxwell units n the spring moduli can be determined from the method of least squares by minimising the expression 17 U X b j 1 X n m 1 Em t0 e tj t 0 k m E1 t0 R tj t0 2 23 in which km gm t0 Em t0 are the selected retardation times and b 25 sampling points is selected for the analysis herein As sug gested in 8 retardation times are selected as km t1 10m 1 where t1is the age of the concrete at the initiation of loading The minimum ofUin Eq 23 yields to a system n 1 linear algebraic equations from which the spring moduli can be calculated On the other hand the relaxation function can be deduced numerically from a given creep function J t t0 by using the Volterra equation as e t Z t t1 J t t0 dr t0 esh t 24 By assuming that the strain is constant and equal to unity through out the time history the relaxation test can be simulated for a given creep function Thus from Eq 24 the value of the relaxation func tion at the discrete time trunder the loading since age t1can be cal culated as 18 R tr t1 R tr 1 t1 drr 25 where drr 1 desh tr Pr 1 s 1J tr ts 1 2 drs J tr tr 1 2 26 For the fi rst time step the stress increment and the value of the relaxation function are calculated as R t1 t1 dr1 1 J t1 t1 27 In Eq 26 s 1 2 refers to the middle of the time step dts thus the value of the creep function J tr ts 1 2 can be calculated as J tr ts 1 2 J tr ts J tr ts 1 2 28 The procedure in Eqs 25 28 can be repeated for increasing val ues of loading ages i e for t1 t2 t3 t4 From Eq 23 four sets of spring moduli viz Em t1 Em t2 Em t3 and Em t4 corresponding to these loadings are calculated herein based on the selected retarda tion times viz km t1 10m 1 The time dependent spring moduli are then determined based on a cubic polynomial fi t to these four sets of spring moduli as suggested by Bazant and Asghari 17 These convenient computations can be completed prior to the time dependent structural analysis 4 3 Incremental equilibrium equations The fundamental task of numerical viscoelastic analysis at each time increment is to calculate the updated stressesrc t dt and sc t dt from the state variables at the previous time step t By using the solutions of the fi rst order differential equations in Eqs 19 and 21 at discrete times t and t dt assuming that between times t and t dt the time dependent strain increments dec t desh t and dcc t are constant and the spring modulus can be aver aged so that Em 1 2 t Em t dt Em t 2 the stresses in each Maxwell unit at time t dt can be updated as 8 rm t dt rm t e dt km Em 1 2 t 1 e dt km km dt dec t desh t 29 sm t dt sm t e dt km 2 1 mc Em 1 2 t 1 e dt km km dt dcc t 30 By subtracting the virtual work expressions at two neighbouring equilibrium states confi gured at times t and t dt and using Eqs 5 10 12 15 20 22 29 and 30 the incremental equi librium equation can be obtained as Ktdd dFext dF00 dFsh 31 w