Time analysis of structural concrete elements using the equivalent displacement method

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1 Available online at Materials and Strutures 38 (July 5) Time analysis of strutural onrete elements using the uivalent displaement method G. Ranzi 1 and M.A. Bradford (1) Department of Civil Engineering, The University of Sydney, NSW6, Australia () Shool of Civil and Environmental Engineering, UNSW, NSW5, Australia Reeived: 3 May 4; aepted: 4 August 4 ABSTRACT This paper presents a modelling tehnique referred to as the uivalent displaement method (EDM) whih desribes the behaviour in time of strutural onrete elements, suh as reinfored onrete beams and omposite beams with full shear interation, aounting for time effets, suh as reep, of the onrete omponent. The time-dependent behaviour of the onrete is modelled using the algebrai representations, suh as the age-adjusted effetive modulus method (AEMM), while the steel joist and reinforement are assumed to behave in a linear-elasti fashion. The main advantage of the EDM method is that it ruires only one analysis to obtain the deformation state of the strutural system at one step in time based on the AEMM method, instead of the two ruired by available modelling tehniques (i.e. one performed at time t and one performed one step in time at time t). The EDM method is then applied to the analysis of strutural onrete elements using the stiffness analysis and the results obtained based on this modelling tehnique are validated against other modelling methods. The advantages of using the EDM method in design appliations are also illustrated RILEM. All rights reserved. RÉSUMÉ Cet artile présente une méthode appelée méthode de déplaement équivalent (EDM), qui dérit le omportement temporel des éléments de béton, par exemple les poutres en béton armé et les poutres mixtes à interation omplète, tenant ompte des effets temporaux, tel que le fluage du béton. Le omportement temporel du béton est modelé par représentation algébrique, tel que la méthode âge ajusté du module effetif (AEMM), tandis que la poutre d aier et les renforts sont supposés linéaire et élastique. L avantage majeur de la méthode EDM est qu elle n exige qu une seule analyse pour obtenir l état de déformation du système à un point temporel (méthode EDM), au lieu des deux points exigés par les méthodes ourantes, un en temps t et l autre à t. La méthode EDM est ensuite appliquée à l analyse des éléments struturels en béton, utilisant l analyse de raideur, et les résultats obtenus par ette méthode sont vérifiés par d autres méthodes. Les avantages de l utilisation de la méthode EDM dans les appliations du dessein sont aussi illustrés. 1. INTRODUCTION This paper is onerned with the time analysis of strutural members that ontain onrete and whih deform due to reep and shrinage. The behaviour of reinfored onrete beams and omposite beams with full shear interation has been investigated extensively over the last entury; both short- and long-term effets have been onsidered by many researhers, and so it is beyond the sope of this paper to provide an exhaustive review of the literature in this area. Useful referenes that report researh into time effets in onrete strutures an be found in [1-4]. This paper presents a modelling tehnique, referred to as the uivalent displaement method (EDM), whih handles the time analyses of reinfored onrete beams and of omposite beams with full shear interation based on the well-nown age-adjusted effetive modulus method (AEMM) [5], whih lends itself to an algebrai desription. The main advantage of the EDM method is that only one analysis is ruired to obtain the deformation state at time t based on the AEMM method, while usually two analyses are ruired by other modelling tehniques that are desribed in the literature, viz. one instantaneous analysis at time t, where t is the time of first loading, and one analysis arried out at one step in time (i.e. at time t). The EDM modelling tehnique is also apable of handling algebrai representations of the onrete behaviour other than the AEMM. The EDM method originates from the time analysis of the ross-setion as desribed by Ghali and Favre [] and by Gilbert [4]. In this paper, the proedure of the rosssetional time analysis is outlined briefly to better illustrate the priniples of the EDM method. The use of the EDM formulation oupled with the stiffness method of strutural RILEM. All rights reserved. doi:1.1617/1417

2 61 G. Ranzi, M.A. Bradford / Materials and Strutures 38 (5) analysis is then outlined, and this modelling proedure is validated against another modelling tehnique, whih is based on the diret stiffness method desribed by Ranzi [6]. Finally, the use of the EDM method for design purposes and its advantages are illustrated. The EDM method has been derived to aount for reep effets, as it is able to aount for the stress history of the onrete omponent. Considerations for shrinage effets, assuming the onrete material to be modelled by means of the algebrai representations, do not ruire the use of the EDM method as the deformation state at time t is independent of the stress history of the onrete (i.e. the stress state at time t ) and it an be easily determined in one analysis by modelling the shrinage effet as a deformation in the onrete while adopting the Young s modulus of the onrete appropriate to the algebrai representation onsidered; for example, using the AEMM method the relevant Young s modulus of the onrete would be taen as the age-adjusted effetive modulus E e. In the modelling onsidered in this paper, it is assumed that plane setions remain plane. For generality, the model is derived about an arbitrary referene axis loated at a distane y below the top fibre of the ross-setion from whih the ross-setional properties of the beam are defined. Also, it is assumed that the onrete behaviour, modelled by means of the AEMM method, is idential in both ompression and tension, as reommended by Gilbert [4] and Bažant and Oh [7] for stress levels in ompression less than about one half of the ompressive strength of the onrete, and for tensile stresses less than about one half of the tensile strength of the onrete; the results obtained using the proposed approah are assumed to be aeptable from a quantitative and qualitative viewpoint when the alulated stresses remain in this stress range.. CROSS-SECTIONAL TIME ANALYSIS The desription of the time analysis arried out on the ross-setion is briefly outlined in this setion, where it is subdivided in the following three stages: instantaneous analysis of the ross-setion; inremental analysis of the ross-setion; post-proessing of the time analysis of the ross-setion. Fig. 1 - Cross-setion and strain diagrams at times t and t. As the EDM method deals only with reep effets, the time analysis of the ross-setion desribed in the following only aounts for these time effets. For generality, a omposite ross-setion (with full shear interation) is onsidered in the following, whih is omprised of a reinfored onrete slab and a steel joist as shown in Fig. 1. The ross-setion is assumed to be loaded by a moment M and an axial fore N at time t and at time t. Referene should be made to [, 4] for a more detailed desription of the time analysis of the ross-setion. The sign onvention adopted in this paper assumes tensile (ompressive) stresses, strains and fores to be positive (negative). For larity, subsripts and have been used throughout this paper to distinguish between ations and ross-setional properties alulated at time t and at time t respetively..1 Instantaneous analysis of the ross-setion The instantaneous analysis of the ross-setion determines the strain diagram at time t as shown in Fig. 1(b), where the generi instantaneous strain at time t, referred to as ε, is defined in terms of the strain in the top fibre of the rosssetion ε and of the urvature ρ alulated at time t as ( y ) ε ( y y) ε = ρ (1) where y is the distane of the referene axis from the top fibre of the ross-setion and y is the vertial oordinate at the ross-setion measured from this referene axis. The strain in the top fibre of the ross-setion ε and the urvature ρ an be determined at time t based on the applied loading M and N as [, 4, 6] AE BE ρ = M N () BE y AE IE ybe ε = M N (3) where AẼ, BẼ, and IẼ are material and ross-setional properties alulated at time t and whih are defined in the appendix.. Inremental analysis of the ross-setion The inremental analysis of the ross-setion determines the hanges in strain whih ours with time due to the time-dependent behaviour of the onrete based on a relaxation proedure. These hanges in strain are expressed in terms of the hange in the strain in the top fibre of the ross-setion and in the urvature, referred to as ε and ρ respetively, whih may be onsidered to be artifiially prevented in the relaxation proedure by the restraining ations M and N ; these an be determined as: AE BE ρ = M N (4) BE y AE ε = M N (5) AE IE BE AE IE BE IE y BE

3 G. Ranzi, M.A. Bradford / Materials and Strutures 38 (5) where AẼ, BẼ, and IẼ are defined in the appendix and represent the material and ross-setional properties alulated at time t, and where ( I ) yb Eeφ M = B E φ (6) e ε ρ ( B y A ) E φ e N = A E φ (7) ε ρ e in whih A, B and I represent the ross-setional properties of the onrete omponent of the ross-setion, E e is the ageadjusted effetive modulus and φ is the reep oeffiient defined as the ratio between the reep strain at time t and the initial strain at time t..3 Post-proessing of the time analysis of the ross-setion The time analysis of the ross-setion is terminated one the instantaneous and the inremental analyses previously desribed are ompleted. At this point it is possible to determine the strain diagram and to alulate the stresses at the ross-setion at time t. The strain diagram at time t is defined from the strain in the top fibre of the ross-setion ε and the urvature ρ, where ε and ρ are shown in Fig. 1(), as ε = ε ε (8) ρ = ρ (9) ρ The inremental analysis enables the alulation of the hange in stresses in the ross-setion due to time effets. For the onrete omponent, whose behaviour is modelled based on the AEMM method, the following hanges in stress our: σ = φ ε y y ρ (1) [ ( ) ] relax E e [ ε ( y y) ρ ] σ restore = Ee (11) while for the elasti materials onsidered (i.e. the steel joist and the reinforement) the hanges in stress determined from the inremental analysis beome σ = E [ ε ( y y) ρ ] (1) α α where E α is the Young s modulus of the material of domain α, where α = s for the steel joist and α = r for the steel reinforement. The total stresses at time t an then be alulated as σ = σ σ σ (13) σ relax α α α restore = σ σ (14) where σ and σ are the stresses in the onrete omponent at times t and t, while σ α and σ α are the stresses at times t and t for the elasti material of domain α (where α = s,r). 3. EQUIVALENT DISPLACEMENT METHOD (EDM) The EDM method was originally developed to obtain the final strain diagram (at time t) by means of one analysis only, therefore avoiding the instantaneous and the inremental analyses of the ross-setion desribed in the previous setion. This is ahieved by introduing a fititious Young s modulus for the onrete E (referred to as the uivalent Young s modulus of the onrete) and a fititious load multiplier λ. The strain in the top fibre of the ross-setion ε and the urvature ρ at time t an then be alulated following Equations () and (3) and using E and λ as: AE BE = λm λ N (15) ρ BE y AE IE ybe = λm λn (16) ε where AẼ, BẼ, and IẼ are the material and rosssetional properties alulated using E as the Young s modulus of the onrete and are defined in the appendix. Using this method, the applied loads (i.e. M and N) are multiplied by a fator ual to λ. Nevertheless, this proedure leads to a orret desription of the deformation state (i.e. strains and displaements) at time t. The expressions for the uivalent Young s modulus of the onrete E and the uivalent load multiplier λ have been derived uating the expressions for the strain in the top fibre of the ross-setion and the urvature alulated at time t based on Equations (15) and (16) to those alulated using Equations (8) and (9) derived for the time analysis of the ross-setion leading to a system of two uations defined in Equations (17) with E and λ as unnowns. These two uations are BE y AE IE y BE ε ε = λm λn AEIE BE AEIE BE AE BE (17) ρ ρ = λm λn and solving Equations (17) simultaneously for E and λ yields ( β β β )( φ χφ ) 1 3 E = E 1 (18) β4 β φ λ = 1 β where 1 1 ( χ) 4 (19) β 1 = A EIE B E (a) β = AE IE BE (b) n n 3 = A EIEn AEnIE n β BE B E () ( 1 φ χφ ) β( 1 χφ χ φ ) ( φ χφ ) β4 = β1 β (d) 1 χφ 3 n

4 61 G. Ranzi, M.A. Bradford / Materials and Strutures 38 (5) and E is the Young s modulus of the onrete at time t, χ is the aging oeffiient, and AẼ n, BẼ n, and IẼ n are defined in the appendix and represent the material and ross-setional properties of the omposite ross-setion exluding the onrete omponent. Equations (18) and (19) highlight how the expressions for E and λ are not dependent on the applied load N and M, and depend only on the geometri and material properties of the ross-setion. The time analysis of the ross-setion an then be arried out by means of the uivalent displaement method (EDM) using Equations (15) and (16) to obtain the orret strain diagram at time t one E and λ have been determined from Equations (18) and (19). The alulation of the stresses at time t is arried out in the usual manner based on the onstitutive laws for the materials onsidered. The stresses in the elasti materials of the rosssetion, suh as the steel joist and the reinforing bars, are alulated based on the elasti onstitutive law σ α (y) = E α ε (y), where σ α and E α are the stress at time t and the Young s modulus of the material of domain α respetively, and ε (y) is the generi strain defined as ε ( y) = ε ( y y) ρ (1) The stresses in the time-dependent materials of the rosssetion, suh as the onrete omponents, are alulated based on the rheologial onstitutive law adopted for the onrete to aount for time effets; onsidering the AEMM method, these an be written as [ ε ( ) ] y y ρ ϕσ σ = E ε ϕσ = E () e in whih ( χ 1) φ ϕ = 1 χφ ; e E E e = (3a,b) 1 χφ where σ and σ are the stresses alulated for the onrete at time t and t based on the onrete strains ε and ε determined at time t and t respetively; in partiular, σ is ual to E ε. ombination with reep effets to yield meaningful results from a quantitative and qualitative viewpoint; nevertheless, ignoring these latter nonlinearities the results might still be meaningful from a qualitative viewpoint, for example in omparing the effets of different ross-setional properties. 4.1 Stiffness analysis by means of the EDM method The EDM method an be applied to the time analysis of strutural systems using the stiffness method. For this purpose, the 6dof stiffness element ommonly defined in strutural analysis textboos, suh as in [8], is utilised, whose degrees of freedoms are the axial and the vertial displaements and the rotations at the element ends as illustrated in Fig., while the stiffness oeffiients are alulated by substituting the Young s modulus of the onrete E with the uivalent Young s modulus E and multiplying the applied loads by λ. Under these onditions, the modelling is appliable for strutural systems with uniform and idential ross-setions. The stiffness analysis based on the EDM method is then able to yield the orret strain and displaement distributions. For the ase of a struture formed by elements of different ross-setions (but still uniform along the length of eah stiffness element), the following modelling tehnique needs to be adopted, as otherwise it would be impossible to apply different load multipliers (whih are defined as a funtion of the geometri and material properties of a partiular rosssetion) to the applied loads depending upon the ross-setion of the element onsidered. For larity, it is assumed that the strutural system onsidered is formed by l = 1,,n el elements, and that it inludes i = 1,,n ross-setions; obviously n n el as usually more than one element is formed by the same rosssetion, and in the extreme ase n = n el all elements have different ross-setions. Also, eah ross-setion is assumed to be formed by j = 1,,m i materials (where i defines the rosssetion onsidered). In this paper it has been assumed for generality that a typial ross-setion is formed by three materials (i.e. the onrete of the slab, and the steel of the joist and reinforement respetively), and therefore m i = APPLICATIONS The EDM method, whih was originally derived for rosssetional analyses, is applied here to the analysis of generi strutural systems using the stiffness method. The results are validated against those obtained based on the time analysis arried out by means of the diret stiffness method, whih is desribed by Ranzi [6]. Furthermore, the EDM method lends itself very well to design appliations. Closed form expressions defining deformation states for an instantaneous analysis an be easily modified to aount for time effets; this beomes partiularly advantageous from a design viewpoint for expressions desribing defletions. It is worth noting that in hogging moment regions it is liely that other onrete nonlinearities would need to be aounted for, suh as raing and tension-stiffening, espeially when shrinage effets are onsidered in Fig. - Nodal displaements and ations of the 6dof stiffness element.

5 G. Ranzi, M.A. Bradford / Materials and Strutures 38 (5) Nevertheless, a larger number of elasti materials an be easily handled by the EDM method, as this would simply affet the alulations of AẼ n, BẼ n, and IẼ n that are defined in the appendix. For eah element l, the Young s moduli of its rosssetion are divided by λ.i (whih is ual to λ alulated using Equation (19) based on the ross-setional properties of the element onsidered) as E ji. EDM E ji = (4) λ. i where E ij is the Young s modulus of the j-th material at the i-th ross-setion (as the l-th element onsidered is formed by the i- th ross-setion) and E ij.edm is the Young s modulus of the j-th material at the i-th ross-setion to be adopted in the EDM method. Obviously, elements formed by the same rosssetion would have idential values of E ij.edm. The strutural analysis based on the EDM method is then arried out applying the atual loads (without any modifiations) to the struture. In this proedure, the uivalent Young s modulus E.i alulated from Equation (18) for the onrete omponent of the i-th ross-setion also needs be modified, similarly to the other stiffness moduli, aording to Equation (4). The results obtained from the stiffness analysis based on the EDM method have been validated against those alulated using the diret stiffness method in the ases of a simply supported beam, of a propped antilever and of ontinuous beams. The numerial examples arried out below are based on a omposite ross-setion whose properties are speified in Table 1, where also the material properties of the steel joist, of the reinforement and of the onrete omponent are defined, while the applied loads onsist of a sustained uniformly distributed load of 5N/m (whih represents its self-weight). The time-dependent properties of the onrete (to aount for reep) have been alulated in aordane with [9] based on the data outlined in Table Simply supported beam A simply supported beam 1m long has been modelled by means of the diret stiffness approah and of the uivalent displaement method using one element. Fig. 3 outlines the strain in the top fibre of the ross-setion, the urvature and the defletion alulated along the beam at time t by means of the diret stiffness method and of the uivalent displaement method, whih are shown to math exatly Propped antilever beam A propped antilever beam was modelled by means of the diret stiffness approah and of the uivalent displaement method using one element. The beam is 1m long. The strain in the top fibre of the ross-setion, the urvature and the defletion along the beam alulated at time t by means of the diret stiffness method and of the uivalent displaement method are plotted in Fig. 4, where these are again shown to be in perfet agreement Continuous beam A ontinuous four-span omposite beam was modelled by Table 1 - Composite ross-setional and material properties Steel joist Setion 1WB455 Flange 5mm x 4mm Web 11mm x 16mm E s 1 MPa Reinforement A r 85 mm Loation at mid-height of slab E r 1 MPa Conrete omponent b 35 mm d 3 mm f 3 MPa E MPa at 8 days MPa at 1 days s.5 t 8 days t f 1 days RH 7% h 31.3 φ with t = 8days and t f = 1 days χ with t = 8days and t f = 1 days NOTE: material properties of the onrete have been alulated in aordane with [9] strain.e E-5 -.E-5-3.E-5-4.E-5 (a) strain in the top fibre of the ross-setion urvature (m -1 ) 8.E-5 6.E-5 4.E-5.E-5.E (b) urvature v (m) E -.E-4-4.E-4-6.E-4-8.E-4 () defletion EDM n.(6) Fig. 3 - Simply supported beam subjet to a uniformly distributed load at time t = 1 days ( = Diret stiffness method EDM = Equivalent displaement method). means of the diret stiffness approah and of the uivalent displaement method using one element for eah span. Eah span is 1m long. The left support is fixed and the others are

6 614 G. Ranzi, M.A. Bradford / Materials and Strutures 38 (5) strain urvature (m -1 ) 4.E-5.E-5.E E-5-4.E-5 (a) strain in the top fibre of the ross-setion 6.E-5 4.E-5.E-5.E -.E-5-4.E-5-6.E-5-8.E-5 (b) urvature v (m).e -.E-4-4.E-4-6.E-4-8.E-4 () defletion EDM - n. (3) strain urvature (m -1 ) v (m) 5.E-5 4.E-5 3.E-5.E-5 1.E-5.E -1.E-5 -.E-5 (a) strain in the top fibre of the ross-setion 4.E-5.E-5.E -.E E-5-6.E-5-8.E-5 (b) urvature 5.E-5.E -5.E E-4 -.E-4 -.E-4-3.E-4-3.E-4 () defletion Fig. 4 - Propped antilever beam subjet to a uniformly distributed load at time t = 1 days ( = Diret stiffness method EDM = Equivalent displaement method). roller supports. The variation along the beam of the strain in the top fibre of the ross-setion, the urvature and the defletion alulated at time t by means of the diret stiffness method and of the uivalent displaement method are plotted in Fig. 5. To outline the aduay of the uivalent displaement method in modelling strutural systems with elements of different ross-setions, the ontinuous beam analysed in Fig. 5 was modified. The two left spans of the ontinuous beam were still modelled with the ross-setion speified in Table 1, while the third and fourth spans (i.e. the two spans on the right hand-side of the beam) were modelled with the ross-setion defined in Table 1 halving the depth of the web from 11mm to 56mm; for this ase, the variation along the beam of the strain in the top fibre of the rosssetion, the urvature and the defletion alulated at time t by means of the diret stiffness method and of the uivalent displaement method are plotted in Fig. 6. Figs. 5 and 6 highlight how the results obtained by means of the EDM method and the diret stiffness method are in perfet agreement. 4. Design appliations for the EDM method The formulation of the EDM method is very onvenient for design appliations. In fat, it is possible to extend the appliability in the time domain of expressions desribing displaements and/or strains derived in losed form for Fig. 5 - Continuous beam subjet to a uniformly distributed load at time t = 1 days ( = Diret stiffness method EDM = Equivalent displaement method). instantaneous analyses. This proedure is illustrated in the following for the expressions for the defletion along a simply supported beam and a propped antilever respetively, subjeted to a uniformly distributed load, while modelling the onrete behaviour by means of the AEMM method Simply supported beam The ommon expression used to determine the defletion along a simply supported beam subjeted to a uniformly distributed load at time t is defined as: [1] 4IEˆ 3 3 ( L Lz z ) wz v = (5) where v is the defletion along the beam length at time t, w is the uniformly distributed load, L is the length of the beam analysed, and IÊ is defined in the appendix. The appliability of this expression for the defletion in the time domain is arried out simply by multiplying Equation (5) by λ and substituting the elasti modulus of the onrete E with E in the alulation of the rosssetional properties as v 3 3 ( L Lz z ) wz = λ (6) 4IEˆ where v is the defletion at mid-span at time t, and IÊ is defined in the appendix.

7 G. Ranzi, M.A. Bradford / Materials and Strutures 38 (5) Fig. 3() highlights how the defletions alulated using Equation (6) for a simply supported beam 1m long subjeted to a uniformly distributed load of 5N/m (whih represents its self-weight) oinide with those obtained using the diret stiffness method and the stiffness analysis based on the EDM method. The ross-setional and material properties were those defined previously in Table 1. A similar proedure an be applied to the expression desribing the maximum defletion at mid-span; in this instane, the defletion at time t defined as 4 5 wl v = (7) 384 IEˆ an be modified to aount for reep effets as v strain urvature (m -1 ) v (m) 8.E-5 6.E-5 4.E-5.E-5.E -.E-5-4.E-5 (a) strain in the top fibre of the ross-setion.e-4 1.E-4 5.E (b) urvature () defletion 4 5 wl = λ (8) 384 IEˆ.E -5.E E-4 -.E-4 -.E-4.E-4.E -.E E-4-6.E-4-8.E-4-1.E-3 Fig. 6 - Continuous beam (with varying ross-setion) subjet to a uniformly distributed load at time t = 1 days ( = Diret stiffness method EDM = Equivalent displaement method). 4.. Propped antilever The defletion along a propped antilever subjeted to a uniformly distributed load an be determined at time t as: [1] ( 3L 5Lz z ) wz v = (9) 48IEˆ Similarly to the ase of the simply supported beam, the appliability of Equation (9) in the time domain is arried out as follows: v ( 3L 5Lz z ) wz = λ (3) 48IEˆ The defletions alulated using Equation (3) for a propped antilever 1m long subjeted to a uniformly distributed load of 5N/m (whih represents its selfweight), whose ross-setional properties are speified in Table 1, are shown in Fig. 4() to be idential to those alulated at time t based on the diret stiffness approah and on the stiffness analysis oupled with the EDM method. 5. CONCLUSIONS This paper has proposed a unique modelling tehnique, referred to as the uivalent displaement method (EDM), for the time analysis of reinfored onrete beams and omposite beams with full shear interation whih is able to aount for the reep effets of the onrete slab. The steel joist and the reinforement of the onrete slab have been assumed to behave in a linear-elasti fashion, while the time-dependent behaviour of the onrete has been modelled based on the ageadjusted effetive modulus method (AEMM). The onrete behaviour is assumed to be idential in both ompression and tension, as reommended by Gilbert [4] and Bažant and Oh [7] for stress levels in ompression and tension less than about one half of the ompressive and tensile onrete strength respetively; for higher stresses, whih are more liely to our when shrinage effets are onsidered in ombination with reep effets, other material nonlinearities suh as raing and tension-stiffening should be aounted for. The main advantage of this method is that the deformation state (i.e. strains and displaements) at time t an be obtained by means of one analysis only, instead of the two analyses (i.e. one at time t and one at time t) usually ruired by the AEMM method. This method originates from the time analysis of the ross-setion and it has been applied in the analysis of strutural systems using the stiffness method. The EDM method has then been oupled with a onventional strutural stiffness analysis in order to be able to predit the deformation state of a strutural system. The results obtained from the stiffness analysis based on the EDM method have been validated against those alulated using the diret stiffness method presented by Ranzi [6], and these have been shown to math perfetly; this has been arried out for a simply supported beam, a propped antilever and two ontinuous beams. Finally, the onveniene of the EDM method for design purposes has been demonstrated. In this appliation, it has been shown that the appliability of the expressions desribing the deformation state derived in losed form for instantaneous analyses an be easily extended in the time domain to aount for reep effets.

8 616 G. Ranzi, M.A. Bradford / Materials and Strutures 38 (5) LIST OF SYMBOLS A, A r, A s = area of the onrete omponent, of the reinforement and of the steel joist respetively B, B r, B s = first moment of area of the onrete omponent, of the reinforement and of the steel joist respetively b = width of the slab d = depth of the slab E, E = elasti moduli for the onrete alulated at time t and at time t respetively E e, E = age-adjusted effetive modulus and uivalent Young s modulus for the onrete E r, E s = elasti moduli of the reinforement and of the steel joist respetively f = ylinder ompressive strength h = notational size of the member (mm) I, I r, I s = seond moment of area of the onrete omponent, of the reinforement and of the steel joist respetively RH = relative humidity of ambient environment (%) s = oeffiient whih depends on the type of ement t = age of the onrete at loading (days) t f = age of the onrete (days) at the time onsidered φ = reep oeffiient defined as the ratio between the reep strain at time t and the initial strain at time t χ = aging oeffiient ACKNOWLEDGEMENTS The authors wish to than Assoiate Professor Peter Ansourian from the University of Sydney for translating the English abstrat in Frenh. REFERENCES [1] CEB (Comité Euro-International du Béton), CEB Design Manual on Strutural Effets of Time-Dependent Behaviour of Conrete, edited by Chiorino, M.A., Napoli, P., Mola, F. and Koprna, M. (Georgi Publishing, Saint-Saphorin, Switzerland, 1984). [] Ghali, A. and Favre, R., Conrete strutures: stresses and deformations, nd ed. (E&FN Spon, London, 1994). [3] Gilbert, R.I., Time-dependent analysis of omposite steelonrete setions, Journal of Strutural Engineering, ASCE 115(11) (1989) [4] Gilbert, R.I., Time effets in onrete strutures, Elservier Siene Publishers, Amsterdam, The Netherlands, [5] Bažant, Z.P., Predition of onrete reep effets using ageadjusted effetive modulus method, ACI Journal 69(4) (197) [6] Ranzi, G., Partial interation analysis of omposite beams using the diret stiffness method, PhD Thesis, The University of New South Wales, Sydney, Australia, 3. [7] Bažant, Z.P. and Oh, B.H., Deformation of progressively raing reinfored onrete beams, ACI Journal 81(3) (1984) [8] Weaver, W. and Gere, J.M., Matrix analysis of framed strutures, 3 rd edition (Chapman & Hall, 199). [9] CEB-FIB, Model Code 199: Design Code (Thomas Telford, London, UK, 1993). [1] Gere, J.M., Mehanis of materials, 5 th ed. (Broos/Cole, 1) 5. APPENDIX AẼ = A E AẼ n AẼ = A E AẼ n AẼ = A E AẼ n AẼ n = A s E s A r E r BẼ = B E BẼ n (properties alulated about the arbitrary referene axis) BẼ = B E BẼ n (properties alulated about the arbitrary referene axis) BẼ = B E BẼ n (properties alulated about the arbitrary referene axis) BẼ n = B s E s B r E r (properties alulated about the arbitrary referene axis) IẼ = I E IẼ n (properties alulated about the arbitrary referene axis) IÊ = I E IẼ n (properties alulated about the entroid) IÊ = (AẼ IẼ - BẼ )/AẼ IẼ = I E IẼ n (properties alulated about the arbitrary referene axis) IÊ = I E IẼ n (properties alulated about the entroid) IÊ = (AẼ IẼ - BẼ )/AẼ IẼ = I E IẼ n (properties alulated about the arbitrary referene axis) IẼ n = I s E s I r E r (properties alulated about the arbitrary referene axis)