CREEP AND SHRINKAGE EFFECT ON THE DYNAMIC ANALYSIS OF REINFORCED CONCRETE SLAB-AND-BEAM STRUCTURES

Transcription

1 ECCM 99 Euroean Conference on Comutational Mechanics August 31 Setember 3 München, Germany CREEP AND SHRINKAGE EFFECT ON THE DYNAMIC ANALYSIS OF REINFORCED CONCRETE SLABANDBEAM STRUCTURES Evangelos J. Saountzakis, John T. Katsikadelis Deartment of Civil Engineering National Technical University of Athens, Greece cvsaoun@central.ntua.gr jkats@central.ntua.gr Key words: Bending, ribbed, late, cree, shrinkage, dynamic Abstract. In this aer a solution to the dynamic roblem of reinforced concrete slabandbeam structures including cree and shrinkage effect is resented. The adoted model takes into account the resulting inlane forces and deformations of the late as well as the axial forces and deformations of the beam, due to combined resonse of the system. The analysis consists in isolating the beams from the late by sections arallel to the lower outer surface of the late. The influence of cree and shrinkage effect relative with the time of the casting and the time of the loading of the late and the beams is taken into account. A lumed mass matrix is constructed from the tributary mass areas to the nodal mass oints and a stiffness matrix is comuted using the solution of the corresonding static roblem. Both free and forced vibrations are considered and numerical examles with great ractical interest are resented. The discreancy of the eigenvalues obtained using the resented analysis, which aroximates better the actual resonse of the latebeams system and the corresonding ones using the finite element method and ignoring the inlane forces and deformations necessitates the need of the adoted model.

2 Evangelos J. Saountzakis, John T. Katsikadelis 1 Introduction The interest in reinforced concrete slabandbeam structures has been widesread in recent years due to the economic and structural advantages of such systems. Stiffened reinforced concrete late structures are efficient, economical and functional, while construction using recast beams is the quickest, most familiar and economical method for long riveror valley bridges or for long san slabs. The extensive use of the aforementioned late structures necessitates a rigorous static and dynamic analysis. Although there is an extensive literature on static analysis of these systems [11], to the authors knowledge a rather limited amount of technical literature is available on the dynamic analysis of stiffened late systems limited to evaluating bounds of eigenfrequencies using the Rayleigh Ritz rocedure [11,1] or emloying the finite element method [13]. In all revious work the adoted model for the dynamic analysis of the late beams system consists in isolating the beams from the late and neglecting the shear forces at the interfaces. This assumtion results in considerable discreancies from the actual resonse of the stiffened late. Moreover it does not allow the establishment of these forces, which are necessary for the design of comosite or refabricated structures. In this aer the dynamic analysis of reinforced concrete slabandbeam structures is resented. The adoted structural model is that emloyed by Saountzakis and Katsikadelis [1], which contrary to the models used reviously takes into account the resulting inlane forces and deformations of the late as well as the axial forces and deformations of the beam, due to combined resonse of the system. Using this model, the study of the dynamic behaviour of a stiffened late subjected to a lateral load and to the effects of cree and shrinkage, either for simultaneous or for searate casting of the late and the beams is attemted. The resented method emloys the static solution to establish a flexibility matrix with resect to a set of nodal mass oints in the interior of the late. The mass matrix is constructed by luming the mass areas contributing to the nodal mass oints of the discretized late [14, 15]. The static solution is obtained using the Analog Equation Method (A.E.M.) [16]. The analysis consists in isolating the beams from the late by sections arallel to the lower outer surface of the late. The forces at the interface, which roduce lateral deflection and inlane deformation to the late and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. Both free and forced transverse vibrations are considered and numerical examles with great ractical interest are resented. The discreancy of the eigenfrequencies obtained using the resented analysis, which aroximates better the actual resonse of the latebeam system, and the corresonding ones which ignore the inlane forces and deformations, justify the analysis of the roosed model. Finally, the influence of the time interval between the casting of the late and the beams to the dynamic behaviour of the stiffened late is examined. Statement of the roblem

3 ECCM 99, München, Germany Consider a thin reinforced concrete late having constant thickness h, occuying the domain Ω of the x, y lane and stiffened by a set of arallel reinforced concrete beams. The late J may have J holes and is suorted on its boundary Γ = j= Γj, which may be iecewise smooth (Fig.1), while the beams may have oint suorts. For the sake of convenience the x axis is taken arallel to the beams. Let the time of the casting of the beams t bc be the beginning of the time considered t, t bl be the time at initial loading of the beams, t c be the time of the casting of the late and t l be the time at which the late is initially subjected to the lateral load g = g( x, t ), x :{ x, y }, t. corner s t f n beam k L k d k q r = q G j t HOLE beam L d P r = qp? y corner beam 1 G L 1 d 1 corner k (O) o x Figure 1: Two dimensional region O occuied by the late. The solution of the roblem at hand is aroached by isolating the beams from the late by sections in the lower outer surface of the late and taking into account the tractions at the fictitious interfaces (Fig.). These tractions result in the loading of the beam as well as the additional loading of the late. Their distribution is unknown and can be established by imosing dislacement continuity conditions at the interfaces and using the rocedure develoed in this investigation. The integration of the tractions along the width of the beam result in line forces er unit length which are denoted by q x, q y and q z. Taking into account that the torsional stiffness of the 3

4 Evangelos J. Saountzakis, John T. Katsikadelis beam is small, the traction comonent q y, in the direction normal to the beam axis, may be ignored. However, in a more refined model the influence of this comonent may also be considered. The other two comonents q x and q z roduce the following loadings along the trace of each beam. g O q x q z h q x q z h b (a) (b) Figure: Thin elastic late stiffened by beams (a) and isolation of the beams from the late (b). a. In the late (i) A lateral line load q z at the interface. (ii) A lateral line load M / x due to the eccentricity of the comonent q x middle surface of the late. M = q h / is the bending moment. x (iii) An inlane line body force q x at the middle surface of the late. from the b. In each beam (i) A transverse load q z. (ii) A transverse load M b / x due to the eccentricity of q x beam cross section. (iii) An inlane axial force q x. from the neutral axis of the The structural models of the late and the beams are shown in Fig.3. On the base of the above considerations the resonse of the late and of the beams may be described by the following initial boundary value roblems. a. For the late. The late undergoes transverse deflection and inlane deformation. Thus, for the transverse deflection we have 4

5 ECCM 99, München, Germany M q x q z late model q z beam model q x Mb Figure 3: Structural model of the late and the beams. D 4 w ρ w&& N x w x N xy w x y N y w y = g K k= 1 q ( k ) z ( k ) M δ ( y yk ) x in O (1) α w α V = α (a) 1 n 3 β w n 1 n 3 on G β M = β (b) ( w x, ) = ε( x ) & ( x, ) = ε( x ) w (3a,b) where w = w ( x, t ) is the transverse deflection of the late; D( t ) = E ( t ) h / 1( 1 v ) is 3 its time deendent flexural rigidity with E being the elastic modulus and ν the Poisson ratio; N = N ( x, t ), N = N ( x, t ), N = N ( x, t ) are the membrane forces er unit x x y y xy length of the late cross section at time t; ρ = ρh is the surface mass density of the late with? being the volume mass density; δ( y y k ) is the Dirac s delta function in the y direction; ε( x ), ε ( x ) are the initial deflection and the initial velocity of the oints of the middle surface of the late; M n and V n are the bending moment normal to the boundary and the effective reaction along it, resectively, and they are given as xy 5

6 Evangelos J. Saountzakis, John T. Katsikadelis M n w w = D( v ) (4) n t w Vn = D[ w v n ( 1 ) ] (5) s n t Finally, a i, β i ( i = 1,, 3 ) are functions secified on the boundary Γ. The boundary conditions (a,b) are the most general linear boundary conditions for the late roblem including also the elastic suort. It is aarent that all tyes of the conventional boundary conditions (clamed, simly suorted, free or guided edge) can be derived form these equations by secifying aroriately the functions a i and β i (e.g. for a clamed edge it is a 1 = β 1 = 1, a = a = β = β = ). 3 3 If the stresses are ket within the limits corresonding to the normal service conditions, assuming linear relationshi between cree and the stress causing the cree and denoting by t = t l t c, Trost s theory [17] gives the tangent modulus of elasticity as E E l ( t ) = (6) 1 χϕ ( t, t ) where E l is the tangent modulus of elasticity of the late at time t ; χ is an ageing coefficient deending on strain develoment with time; ϕ( t, t ) is the cree coefficient related to the elastic deformation at t days which is defined as [18] ϕ( t, t ) = φ β( f ) β( t ) β ( t t ) (7) RH cm c where ϕ RH, β( f cm ) and β( t ) are factors deending on the relative humidity, the concrete strength and the concrete age at loading, resectively, which are defined as ( ) 3 ( ) ϕ RH = 1 1 RH / 1 /. 1 h (8a) ( ) β f cm = / f (8b) cm ( ) = 1 ( 1. /. t ) β t where RH is the relative humidity of the ambient environment in %; h = A / u is the notional size of the late in mm; A is the area of the late cross section; u is the late erimeter in contact with the atmoshere; f cm is the mean comressive strength of concrete in (8c) 6

7 ECCM 99, München, Germany N/mm at the age of 8 days. Moreover, β c ( t t ) in eqn.(7) is a coefficient for the develoment of cree with time, which is estimated from β [ ] ( t t ) = ( t t ) / ( β t t ) c H where β H is a coefficient deending on the relative humidity RH, given as 18 β H = 15. [ 1 (. 1RH) ] h 5 15 (1) Since linear late bending theory is considered, the comonents of the membrane forces N x, N y, N xy do not deend on the deflection w. They are given as. 3 (9) N C u x = x ν v y (11a) N y u = C ν x v y (11b) N xy 1 u = C ν y v x (11c) where C( t ) = E ( t ) / ( 1 ) ν ; u = u ( x, t ) and v = v ( x, t ) are the dislacement comonents of the middle surface of the late arised from both the line body force q x and the temerature distribution T( x, t ) due to the late shrinkage. The latter are established by solving indeendently the lane stress roblem, which is described by the following quasistatic (inlane inertia forces are ignored) boundary value roblem (Navier s equations of equilibrium) 1 u 1 v v v u x x 1 1 v v 1 ( ) vy 1 G q y y v T x δ k α 1 v x u y x v 1 v T α y 1 v y = = in Ω (1a) (1b) γ 1un γ N n = γ 3 (13a) on Γ δ u δ N = δ (13b) 1 t t 3 7

8 Evangelos J. Saountzakis, John T. Katsikadelis in which G ( t ) = E ( t ) / ( 1 ν ) is the shear modulus of the late; α is the linear coefficient of thermal exansion; N n, N t and u n, u t are the boundary membrane forces and dislacements in the normal and tangential directions to the boundary, resectively; γ i, δ i ( i = 1,, 3 ) are functions secified on Γ. Assuming that cree and shrinkage are indeendent, the temerature distribution T( x, t ) is given as [18] where ε s ( t t c ) is the shrinkage strain calculated from T( x, t ) = εs( t t c ) / α (14) ε ( t t ) = ε ( f ) β β ( t t ) (15) s c s cm RH s c where ε s ( f cm ), β RH are factors deending on the concrete strength and the relative humidity, resectively, which are defined as β RH ( 1 ( RH / 1) ) = 3. 5( 1 ( RH / 1) ) ε [ ] ( f ) = β ( f ) s cm sc cm (16a),for 4% RH 99% (stored in air),for RH 99% (immersed in water) (16b) where β sc is a coefficient deending on tye of cement. Moreover, β s ( t t c ) in eqn.(15) is a coefficient for the develoment of shrinkage with time, which is estimated from [ ] ( t t ) = ( t t ) / (. 35h t t ) β s c c c b. For each beam. Each beam undergoes transverse deflection and axial deformation. Thus, for the transverse deflection we have. 5 (17) E b I b d 4 wb 4 dx wb Mb ρbw&& b Nb = q z in Lk k K x x, = 1,,..., (18) a1wb av = a3 (19a) wb b1 b M = b3 at the beam ends x =, l (19b) x w b ( x, ) = e( x ) w& b ( x, ) = e( x ) (a,b) 8

9 ECCM 99, München, Germany where wb = wb ( x, t ) is the transverse deflection of the beam; I b is its moment of inertia; Nb = Nb( x, t ) is the axial force at the neutral axis; V, M are the reaction and the bending moment at the beam ends, resectively; ρ b is the surface mass density of the beam; a i, b i ( i = 1,, 3 ) are coefficients secified at the boundary of the beam; e( x ), e( x ) are the initial deflection and the initial velocity of the oints of the neutral axis of the beam. It is worthnoting that the initial deflection and velocity of the oints of the middle surface of the late ε( x ), ε ( x ) must comly with the initial deflection and velocity of the oints of the neutral axis of the beam e( x ), e( x ) since the late and the beams are assumed to be firmly bonded together. Finally, Eb = Eb( t) is the time deendent tangent modulus of elasticity of the beam given as [17] E b Ebl ( t ) = (1) 1 χϕ ( t, t ) where E bl is the tangent modulus of elasticity of the beam at time t bl ; χ is an ageing coefficient deending on strain develoment with time; tb = tbl tbc; ϕ( t, t b ) is the cree coefficient related to the elastic deformation at t b days, which is defined as [18] b ϕ( t, t ) = φ β( f ) β( t ) β ( t t ) () b RH cm b cb b where ϕ RH, β( f cm ), β( t b ) and β cb ( t t b ) are cree coefficients for the beam similar to those for the late. It is aarent that all tyes of the conventional boundary conditions (clamed, simly suorted, free or guided edge) can be derived from eqns (19a,b) by secifying aroriately the coefficients a i, b i (e.g. for a simly suorted end it is a1 = b = 1, a = a3 = b1 = b3 = ). Since linear beam bending theory is considered, the axial force of the beam does not deend on the deflection w b. The axial deformation of the beam arised from both the inlane axial force q x and the temerature distribution Tb ( x, t ) due to the beam shrinkage is described by solving indeendently the following quasistatic (axial inertia forces are neglected) boundary value roblem i.e. ub Tb Eb Ab qx α x = x in L k, k = 1,,..., K (3) c u c N = c at the beam ends x =, l (4) 1 b 3 where N is the axial reaction at the beam ends. Similarly to the late, the temerature distribution T b for the beam is given as 9

10 Evangelos J. Saountzakis, John T. Katsikadelis Tb = εsb( t tbc ) / α (5) where ε ( t t ) = ε ( f ) β β ( t t ) (6) sb bc sb cm RH sb bc whereε s ( f cm ), β RH, β sb ( t t bc ) are shrinkage coefficients for the beam similar to those for the late. Eqns. (1), (1a), (1b), (18), (3) constitute a set of five couled artial differential equations including seven unknowns, namely w, u, v, wb, ub, qx, qz. Two additional equations are required, which result from the continuity conditions of the dislacements in the direction of the z and x axes at the interfaces between the late and the beams. These conditions can be exressed as u w = w (7) h w hb = ub x It must be noted that the couling of eqns (1) and (1a,b), as well as of eqns (18) and (3) is nonlinear due to the terms including the unknown membrane and axial forces, resectively. b w x b (8) 3 Solution Procedure The solution of the dynamic roblem requires the integration of the set of eqns. (1), (1a,b), (18) and (3) subjected to the rescribed boundary and initial conditions. An analytic solution of this roblem is out of question. Therefore, the recourse to a numerical solution is inevitable. The method resented by Katsikadelis and Kandilas [14] is emloyed in this investigation. According to this method the domain O occuied by the late is discretized by establishing a system of M nodal oints on it, corresonding to M mass cells, to which masses are assigned according to the lumed mass assumtion. Care is taken so that nodal oints are laced on the traces of the beams (Fig. 4). Subsequently, the stiffness matrix as well as the load vector with resect to the nodal oints are established. This rocedure leads to the tyical equation of motion for the stiffened late [ m]{ w& } [ k]{ w} = { g} (9) where { w} is a vector including the deflections at the M domain nodal oints, [ m ] is the mass matrix, [ ] g is the nodal load vector. k is the stiffness matrix and { } 1

11 ECCM 99, München, Germany The mass matrix [ m ] is diagonal. Its elements m i are readily comuted by luming the mass area contributing to the i nodal oint. The elements g i of the load vector { g } are comuted from the mean value of g( x ) on each element. The stiffness matrix [ k ] with resect to the M domain nodal oints is obtained by inverting the flexibility matrix [ f ]. The latter is established by solving the static roblem by working as follows. The tyical element f ij is comuted as the static deflection at oint i due to a unit load at oint j. It is aarent that M(M1) static solutions are required since the flexibility matrix is symmetric. The static roblem results from the same equations i.e. eqns. (1), (1a,b), (18), (3) under the rescribed boundary conditions by neglecting the inertia forces. The solution is achieved using the Analog Equation Method (AEM) as this is resented in detail by Saountzakis and Katsikadelis [16]. nodal oints mass cells Figure 4: Discretization of the late. (a) Forced vibrations For forced vibrations it is [ k] = [ f ] integration scheme. The initial conditions in this case are { w( )} { }, (b) Free vibrations For free vibrations, it is { g } = { }. By setting 1 and eqn. (9) can be solved using any time ste = ε {&( )} { } w = ε (3) 11

12 Evangelos J. Saountzakis, John T. Katsikadelis { w( t )} = { } W e i ω t (31) we obtain the following tyical eigenvalue roblem 1 [ f ][ m] [ I] { W} = ω i (3) from which the M eigenfrequencies and mode shaes can be established. 4 Numerical examles On the basis of the analytical and numerical rocedures resented in the revious sections, a comuter rogram has been written and reresentative examles have been studied to demonstrate the efficiency and the range of alications of the develoed model. In all the examles treated, the numerical results have been obtained using 54 constant boundary elements and 16 constant domain rectangular cells. The following data have been used for the numerical results: Concrete C5/3, f cm =5N/mm, RH=4%, t =t b =8days, E l =E bl =E c8 =3.55kN/mm, h=.m, v=.154, β sc =5 (normal or raid hardening cement), a=1 5 o C. Examle 1 (a) Free vibrations The free vibrations of a rectangular late with side lengths a = 18. m and b = 9. m, stiffened by a beam of rectangular cross section 1. m h b laced along its xaxis of symmetry (Fig. 5) have been studied. The edges of the late along the xaxis are free, while the other two are simly suorted according to the transverse boundary conditions. Moreover, u = u at x=, a while N = N at y=±b / according to the inlane boundary x y = x xy = conditions. In this examle both the late and the beam are simultaneously casted (t bc =t c =). In Table 1 the first ten comuted eigenfrequencies along with the corresonding mode shaes and their contours of the late beam system with h b =.6m for various instants are resented. Moreover, in Table the fundamental eigenfrequency for various values of the beam height h b and for various instants are resented as comared with those obtained from a FEM solution [13], which cannot include the inlane forces and deformations, using 16 square elements. From both tables it is worthnoting that the obtained eigenfrequencies are decreased with time due to the redominant action of cree comared to shrinkage. As exected, the influence of cree action is more significant in the early age of concrete. Moreover, the discreancy of the results obtained from the FEM solution ignoring the inlane forces and deformations is remarkable. 1

13 ECCM 99, München, Germany y Free SS b a Free SS x Figure 5: Plan of the stiffened late. (b) Forced vibrations The forced vibrations of the examined stiffened late have been studied when subjected to a suddenly alied uniformly distributed load q of infinite duration. In Table 3 the maximum values, w max, of the deflection at the center and at the middle of the free side of the late for various instants t are shown together with the corresonding static ones, w st, for the calculation of the dynamic magnification factor D = max R( t ), R( t ) = w( t ) / w st. Examle The casting of the beam of the stiffened late of Examle 1 receded the casting of the late at a time interval of T=t c t bc days. In Table 4 the time develoment of the fundamental eigenfrequency of the stiffened late for the height of the stiffening beam h b =.6m and for different values of the time interval T is resented. From the obtained results it is easily concluded that the reinforced concrete stiffened late is not influenced from the time interval between the casting of the late and the beams. 5 Conclusions The dynamic analysis of reinforced concrete slabandbeam structures including cree and shrinkage effect has been studied. A realistic model has been adoted, which contrary to other aroaches, takes into account the inlane forces and deformations of the late as well as the axial forces and deformations of the beams. The main conclusions that can be drawn from this investigation are a. The roosed model ermits the study of the dynamic behaviour of a stiffened late including the oosed effects of cree and shrinkage either for simultaneous or for searate casting of the late and the beams. b. The evaluated eigenfrequencies are decreased with time due to the redominant action of cree comared to shrinkage. c. The influence of cree action is more significant in the early age of concrete. d. The evaluated eigenfrequencies of the late beams system are found to exhibit considerable discreancy from those of other models, which neglect inlane and axial forces and deformations. 13

14 Evangelos J. Saountzakis, John T. Katsikadelis O m contours modeshaes t=3 t=1 t=3 t= O O O O O O O O O O Table 1: Eigenfrequencies Ω = ω ρ of the late of Examle 1 for various instants t (days). 14

15 ECCM 99, München, Germany Age of Stiffening beam Concrete t 1.x. 1.x.6 1.x. (Days)??? FEM AEM FEM AEM FEM Table : Fundamental eigenfrequency of the late of Examle 1 for various values of the beam height h b and for various instants t. Age of Center Middle of the free side Concrete t (Days) Dynamic w max Static w st D= max R(t) Dynamic w max Static w st D= max R(t) Table 3: Deflections w (m) at the center and at the middle of the free side of the late of Examle 1 for various instants t. Age of Concrete t Time interval T (Days) T=3 T=1 T=3 T= Table 4: Fundamental eigenfrequency of the late of Examle for various instants t and for various values of the time interval T. 15

16 Evangelos J. Saountzakis, John T. Katsikadelis e. The adoted model ermits the evaluation of the inlane shear forces at the interface between the late and the beams, the knowledge of which is very imortant in the design of refabricated late beams structures (estimation of shear connectors). f. Reinforced concrete slabandbeam structures are not influenced from the time interval between the casting of the late and the beams. References [1] A.R. Kukreti and Y. Rajaaksa: Analysis Procedure for Ribbed and Grid Plate Systems used for Bridge Decks, Journal of Structural Engineering, ASCE, 116, No., (199), [] A.R. Kukreti and E. Cheraghi: Analysis Procedure for Stiffened Plate Systems Using an Energy Aroach, Comuters & Structures, 46, No. 4, (1993), [3] Y.K. Cheung: Finite Stri Method in Analysis of Elastic Plates with Two Oosite Sides Simly Suorted Ends, Proceedings of the Institution of Civil Engineers, 4, (1968), 17. [4] I.P. King and O.C. Zienkienwicz: Slab Bridges with Arbitrary Shae and Suort Conditions: A General Method of Analysis Based on Finite Element Method, Proceedings of the Institution of Civil Engineers, 4, (1968), 936. [5] M. Tanaka and A.N. Bercin: A Boundary Element Method Alied to the Elastic Bending Problem of Stiffened Plates, Boundary Element Method XIX, (1997), 31. [6] C. Hu and G.A. Hartley: Elastic Analysis of Thin Plates With Beam Suorts, Engineering Analysis with Boundary Elements, 13, (1994), 938. [7] J.B. de Paiva: Boundary Element Formulation of Building Slabs, Engineering Analysis with Boundary Elements, 17, (1996), [8] S.F. Ng, M.S. Cheung and T. Xu: A Combined Boundary Element and Finite Element Solution of Slab and SlabonGirder Bridges, Comuters & Structures, 37, (199), [9] M.S. Cheung, G. Akhras and W. Li: Combined Boundary Element / Finite Stri Analysis of Bridges, Journal of Structural Engineering, 1, (1994), [1] E.J. Saountzakis and J.T. Katsikadelis: Analysis of Plates Reinforced with Beams, Proc. 5 th National Congress of H.S.T.A.M., Ioannina, 73 August, (1998). [11] D.W. Fox and V.G. Sigillito: Bounds for Frequencies of Rib Reinforced Plates, J. Sound Vib., 69, (198), 497. [1] D.W. Fox and V.G. Sigillito: Bounds for Eigenfrequencies of a Plate with an Elastically Attached Reinforcing Rib, Int. J. Solids Structures, 18, 3, (198), [13] A.G. Cubus Software: Cedrus3 User s References Manual, Zürich, Switzerland, version 1.56, June, (1995). [14] J.T. Katsikadelis and C.B. Kandilas: A Flexibility Matrix Solution of the Vibration Problem of Plates Based on the Boundary Element Method, Acta Mechanica, 83, (199), 516. [15] J.T. Katsikadelis and F.T. Kokkinos: Static and Dynamic Analysis of Comosite Shear Walls by the Boundary Element Method, Acta Mechanica, 68, (1987),

17 ECCM 99, München, Germany [16] E.J. Saountzakis and J.T. Katsikadelis: Cree and Shrinkage Effect on Reinforced Concrete SlabandBeam Structures, Proc. of the International Congress Creating with Concrete, Dundee, Scotland, Setember 61, (1999). [17] H. Trost and J. Wolff: Zur Wirklichkeitsnahen Ermittlung der Beansruchungen in Abschnittsweise Hergestellten Sannbetontragwerken, Der Bauingenieur, 45, (197), [18] Eurocode No.: Design of Concrete Structures, Part 1 : General Rules and Rules for Buildings, Eurocode Editorial Grou, (1991). 17