2nd Int. PhD Symposium in Civil Engineering 998 Budapest IME-DEPENDEN ANAYSIS OF PARIAY PRESRESSED CONINUOUS COMPOSIE BEAMS M. Sar and Assoc. Prof. J. apos 2 Slova echnical University, Faculty of Civil Engineering Radlineseho, 83 68 Bratislava, Slova Republic SUMMARY In this paper a finite element model for the study of creep, shrinage, and concrete tensile stiffness between cracs effects in partially prestressed continuous composite beams with deformable shear connection is proposed. his model adopts an integral-creep law as constitutive relationship for concrete part under compression. So, step-by-step techniques are used to resolve this integral constitutive relation for any load history. Due to the nonlinearity of the problem, the stiffness of the element is calculated with an iterative technique every time-step using a layered model approach for concrete slab. Keywords: composite, creep, cracing, prestressed, continuous beams, flexible connection.. INRODUCION he design of Continuous composite beams for serviceability limit state is not straightforward, since the prediction of behaviour under sustained service loads is complicated by time-dependent deformations in the concrete slab due to creep and shrinage and additional non-linearity caused by cracing of the concrete over each interior support. he most powerful way that can prevent concrete cracing and improve the stiffness and strength of composite beam is using prestressed concrete. he prestressing can be obtained by relative movements of the internal supports or using internal tendons. Relatively little research has been published on the time-varying behaviour of continuous composite beams subected to sustained service loads. Dezi and arantino in 993 and 995 studied the viscoelastic analysis of non-prestressed or fully prestressed composite continuous beams with flexible shear connectors. hey consider a nonrealistic assumption that the slab everywhere through the beam is uncraced. Gilbert and Bradford in 995 described a simple analytical model of the behaviour of nonprestressed craced continuous composite beams under sustained service loads using the age-adusted effective modulus. Besides that they neglected the effects of slip between the concrete slab and the steel section as well as the tensile stiffness of concrete between cracs. PhD student 2 Assoc. Prof. in steel structures
q hc B Gc H h0 hs Gs node n Nn Mc n Nn Ms n Ks q (a) node n+ Nn+ Mc n+ Nn+ Ms n+ node n /2 /2 node n + (c) (d) Fig. (a)continuous composite beam; (b)cross section; (c)an elements connected only at nodes; (d) Nodal displacements of the element(c)[ is the length of the element]. In the present wor, the writers intend to develop previous studies by extending the analysis to partially prestressed continuous composite beams. Where the interest in these types of structures is raised because of the wide use of prestressed concrete in the construction of bridges as a way for preventing cracs and developing the durability. his paper describes dependent behaviour of partially prestressed continuous composite beams by a numerical model which is constructed using the finite element method. In this model, a numerical algorithm providing results of good accuracy, has been obtained by employing step-by-step procedures for creep analysis of the general method (Bazant 972). Incorporation with this, layered model approach is used in order to simplify the complication due to the non-linear behaviour because of cracing over the internal supports. his model maes it possible to evaluate the effects of concrete creep and shrinage, and stiffness of concrete between cracs on the deformations and stresses of the continuous composite beam at any time. In practical, this model maes it easy to investigate the actual distribution of stress though the composite section at any time for continuous composite beams. 2. FORMUAION OF HE FINIE EEMEN MODE Following the finite element method procedures, the continuous beam has to be replaced by a finite number of elements that are assumed to be connected only at their nodes Fig.c. he discretization of the beam to specified number of elements has to respect that the shear connectors in the middle of elements. If one considers some elements between the shear connectors the stiffness of connector in these elements will equal zero. he general steps of the finite element method are followed for the derivation of the proposed model. he finite element model proposed by the present study considers the composite beam of Fig.a as two parallel beam elements with the same deflection v at any cross section lined by means of shear connectors of linear rigidity K s. Also, this model assumes that no slips between the prestressing steel or the reinforcement steel and the concrete before cracing, and the total prestress losses due to relaxation are concentrated only at v2 u c u s v v /2 uc3 u s3 b (b) /2 u c u s v3 v3 v4 Xc Xs 2
the initial time. Finally, the compacted steel sections which have an adequate stiffness against instability are used. 2. Constitutive equations of materials Generally, the concrete slab may contain three different materials (concrete, reinforcement steel, and prestress steel). Because of the a wide difference among the properties of these three materials as well as the difference between the behaviour of concrete in tension and in compression, the layered model approach will be useful for the analysis of such complex element Fig.c and Fig.2. he hypothesis that concrete can not resist tension is assumed here. (a) (b) (c) (d) (e) Fig. (2) General layered concrete slab: (a) mean total strain; (b) shrinage strain; (c) creep strain; (d) elastic strain; and (e) stress distribution. he concrete layers under compression are assumed to be a linear viscoelastic material governed by the well-nown integral-type relationship between a prescribed uniaxial stress history and corresponding strain (Bazant 972). t (, 0) n( ) c( 0) (, 0) (, ) d ( ) ε c t t ε t = σ t J t t + J t τ σ τ (2.) where ε c ( t t ) shrinage strain at time t; c( t ) t 0, 0 = the total strain at time t for the applied stress at time 0 ε n t = the σ 0 = the stress applied at time t 0 ; J( t, t 0 ) = the creep or compliance function (strain in time t caused by a constant unite stress applied in time t 0 defined by the CEB Model Code). Solution of structural analysis problems according to the creep law Eq. leads to Voltera s integral equations which can be solved only by a step by step numerical method suggested by Bazant 975 based on the replacement of the Stieltes heredity integral in Eq. by a finite sum. For this purpose time t may be t ; ( ) subdivided by discrete times t 0, t, t 2..., t N into subintervals t = t t = 2,,... N which will be considered as unequal. Applying the trapezoidal rule Eq. yields ( t )[ ] compressed concrete layers i> n σ c = E c ε c ε v ε n (2.2) where ε c, ε v, ε n are the incremental of total, viscous(creep), and shrinage strains respectively; σ ci ε v = [ J ( t, t I) + J ( t, t I ) J ( t, t I) J ( t, t I ) ]; I =.. 2 and E c( t ) = 2 [ J( t, t ) + J( t, t ) ]. herefore Eq.2 is considered the constitutive relationship of those layers under sustained compressive stress( yi y '). y i is y coordinate of the layer number i (,2,3... n (number of concrete layers)). y ' is the coordinate at which ε e equals zero; in time t, y '= ε e 0 α, α = ε e u ε e 0 H / 2. 0 ε e εe u y tensiled concrete layers i > n 3
When a crac is formed, the stress in the concrete immediately adacent to the crac drops to zero and the steel stress increases to that corresponding to the fully craced state. With increasing distance from the crac the stress in the concrete increases as force is transferred from the steel to the concrete by bond stress,, until at some distance, l t, from the crac the concrete again attains the tensile strength the concrete Fig.3. It should be noted that crac formation is an inherently semi-random process. he empirical formula suggested by CEB[967-978], for crac spacing, l (in mm), that will be recommended here is: ( ) ( )( ) l = 2 c + s 0 + 2 f τ A u (2.3) ct b ct s where c = the concrete cover (in mm); s = spacing of bars (in mm); A ct = area of concrete in tension; u s = the sum of the perimeters of the reinforcing bars; and 2 = a coefficient allowing for the effect of the distribution of tensile stress within the section, equals.0 for pure tension and 0.5 for pure bending. M l M l l M lt lt M M M concrete under compression ε r ε cr εr2 εcr εr2 ε cr σ r 2 σ r2 σ r (a) (b) (c) Fig 3 he different stages of the strain and stress Distributions between the cracs for reinforcement in partially craced concrete slab. [M < M < M ] It is assumed that the bond stress is independent of the displacement of the steel bar relative to the outer surrounding concrete. So, the distance t l is calculated as follows: ( 2 ) ( ) lt = As E r ε r ε cr us (2.4) where f E ( t) ε cr = ct c ; ε r 2 = the reinforcement strain at the crac position; and E r = reinforcement steel Young s modulus. With regard to the stress of the reinforcement at the crac position three stages can be distinguished see Fig 3. he incremental form of the stress at the crac position σ r 2 - mean strain ε rm relationship can be written as: [ ] σ = E ε + ε r 2 r rm r where ε rs = ε r ( t ) ε r ( t ), ε r ( t ) ε r ( t ) = 0. 0 for l t 0. 0, ε r ( t ) = ( l t )[ ε r ε cr ] [ us ] [ 4 E s As] for lt l / 2 ; and l t l (2.5) is memorised through the calculation, 2 for 0. 0 lt l / 2, ε r t =, are the values of l t, l in time t. In the same way, the behaviour of prestressed steel in concrete slab of composite continuous < < ( ) 4
may be governed by Eq. 5. It should be noted that the value of ε p must be calculated with the bond stress between the prestressed steel and the surrounding concrete. he current study considers the connection between the concrete slab and steel section only at the shear connector s position. With the assumption that the concrete slab and the steel beam deform with the same curvature, one gets the following relation: [ ] ( cn sn) γ = h0 dv dx u u x= xn (2.6) whereγ = the slip between the concrete slab and steel section at the shear connector s location x= xn ; [ dv dx ] x= xn = the rotation of the composite section at the shear connector s location; and ucn, usn = the horizontal displacements of the points on the middle surfaces of the concrete slab and the steel beam at the shear connector s location, respectively. he shear force resisted by the connector Q can be estimated in an incremental form as: Q = K γ s (2.7) he stress-strain relationship of steel section can be assumed linear where the serviceability stresses are fare less than the ultimate strength. 2.2 Element formulation With reference to the parameters of the nodal displacements of the element proposed in Fig.d and according to usual technique for finite elements it is possible to suppose: ( ) = N uc ( x) = N c Uc v ( x) u x Us s s where Uc = { uc, uc2, uc3} ; Nc = { Nc( x) Nc2( x) Nc3( x) } vectors Us, V and for matrices of shape functions N horizontal displacement of the shear connector can be written as: = N V (2.8) v,, ;and analogously for the s, Nv. he incremental γ = A Uc + A2 Us + h0 N' vγ V (2.9) where A = { 0, 0, } ; 2 { 0 0 } = /2. he incremental mean strains cm ( x) A =,, ; and N'vγ = the matrix of first derivative of Nv at x ε of the concrete layer(i) can be expressed as a function of incremental nodal displacements ( ) ε cm x = N' c Uc y N'' v V (2.0) i where N'c = the first derivatives of the matrices Nc ; N''v = the second derivatives of the matrix Nv, and y i = the distance from the middle of concrete slab to the centroid of the concrete layer. Similarly, one gets the incremental mean strains of reinforcement steel and prestressed steel, and steel section. Applying the principle of virtual wor to a generic instant t with one element, considering the creep, shrinage, relaxation, change of element stiffness, and load applied in the time interval ( t, t ) for the iteration number one gets ( cm) c c ( sm) s s ( ) ( ) δ ε σ dv + δ ε σ dv + δ γ Q = δ v q dx (2.) Vc Vs 5
where Vc, Vs represent the volume of concrete and steel respectively, q the possible increment of distributed load applied to the element in the time interval( t, t ). Considering the constitutive relations and the strain-displacement relations in the previous section incorporated with the layered system assumed for concrete slab Fig., one obtains: n [ δ ( c ) N' c iδ ( V ) N'' v ] c( )[ N' c c ic N'' v V ε v ε n] B hi U y E t U y dx + n i [ δ ( c ) N' c iδ( V ) N'' v ] c( )[ ε n] [ δ ( c ) N' c rδ( V ) N'' v ] r[ N' c c r N'' v V ε r ] [ ( c ) ( ) ] [ ] δ N' c pδ V N'' v p N' c c pn'' v V + ε p B h U y E t dx + n Bh r U y E U y + dx + Bh p U y E U y dx + Vs [ δ ( s) s iδ ( ) ] [ ] v s s N' V N'' N' s si N'' v V U y E U y dx + [ δ ( ) ( ) ( ) ] δ δ c s [ ] γ v s c s v γ U A + U A2 + h0 V N' K A U + A2 U + h0n' V ( V ) = δ Nv q dx x (2.2) where B = the width of the concrete slab; h i, h r, h p are the thicness of the concrete, reinforcement steel, prestressing steel layers; E s, E p are steel and prestressed steel Young s modulus and n is the numbers of concrete layers under compression. As the equation of the virtual wor Eq.2 must be respected for every congruent δ ( Uc ) δ ( ) Us, δ ( V ) the system of three equations immediately can be obtained: [ s ] e c EAc N' N' c dx + K s A A Uc K A A Us + + 2, e c v s ESc N' N'' dx + K h0 A V = EAc N' c ε v dx + EAc N' c ε n dx + + EAr N' c ε r dx + EAp N' c ε p dx (2.3) [ s ] c K A U E s As s dx K s Us 2 A + N' s N' + A2 A2 + [ s A ] [ s vγ ] ESce N'' v N' c dx + K s h0n' vγ A Uc + K h0n' A2 Uc K h0 2 0 0 V =. (2.4) 2 + + EI e N'' v N'' v dx K s h0 N' vγ N' vγ V = Nv qx dx + ESc N'' v ε v dx + ES r N'' v ε r dx + ES p N'' v ε p dx (2.5) n( ) n where EAc = B hi E ( t ) ; ESc = h y E ( t ); EAc = Bh i E ( t ) EA ES r r r c n( ) B i ci c = E B h ; ESr = Er B hr yr ; EAp = E p Bhp, = E A y ; EAce = B h i E ( t ), + EA + EA, p p p p n c r p c ; 6
e n B i ( ) ESc = h y E t, + ES + ES, and e n B i ci c r p ( ) 2 2 2 ci c r r r pp p p s s EI = h y E t + E Bh y + E Bh y + I E. Eqs., 2 and 3 can put in the following form: + + = + + + (2.6) + + =. (2.7) K U + K U + K V = + + (2.8) c s v n r p K U K2 U K3 V c s K2 U K22 U K23 V 0 0 c s v q r p 3 32 33 3 3 3 3 K, K2,... are sub-matrices. One can rearrange the equations and carry out a static condensation procedure of the equations of the internal nodal parameters uc 3 and us 3 to obtain a reduced system of the type: K U = e e e ( 8x8) ( x8) ( x8) (2.9) e where the matrices K = condensed stiffness matrix of the element, and e U = u c, us, v, v2, uc2, us2, v3, v4. he values of the load sub-vectors { } n q and 3 can be easy obtained by integrating the shape functions = EAc N' ε dx = EAc ε n Z 3 q = N q dx = q Z2 nr c n where { 0 0} v x x Z =,,. and 2 2 2 { 2 2 2-2} Z = /, /, /, /. (2.20) Four values of the concrete elastic strains for any element will be calculated and memorised at the end of each time increment. hese values are the elastic strains in the 0 0 middle ( ε e, ε e2) and upper ( ε e u, ε e u 2) fibres of concrete slab at nodes and 2 of the element considering the creep for the whole section. hese strains are calculated using the fact that the total strain can be divided to elastic, shrinage and creep strains. Using these four values the stains at nodes and 2 for any layer, the elastic strain for any layer which is a linear function through the element can be estimated. hen, the load subvector can be written in the following v form: ( )[ ] v * = EAc N' c ε v dx = K R( t, t I ) E c t I EAc Π I + ESc Α I I = * where K = 0. 5 N' c Bdx; B = {( x) x } {( ) ( ) ( ) ( ) } e u u e 0 0 ε ε H 2 ε e2 ε e2 H 2 analogous way one obtains: (2.2),, ; 0 0 Π = ε e, ε e 2,, and Α = 0.0 { } v / /, / /,. For the sub-vector 3, in an ( )[ 3 3 ] v ** 3 = ESc N'' v v dx = K R( t, t I ) E c t I ESc Π I + EIc Α I I = ** where K = 05. N'' v B3dx B3 = ε (2.22) n( ) 2, EIc = B h i yci Ec( t ), { 0 0 Π = ε e ε e } 2, {,,, 0 {( x), x, 0.0, 0.0 }, and Α = ( ε e u ε e) / ( H / 2), ( ε e u 2 ε 0 e2) 7
/( / ),, }. rf p p H 2 Also, in a similar way, the load sub-vectors,, 3 and 3 can be written in the following form: r * r ** p * p ** = r r 3 = r 3r = p p 3 = p 3 p EA K S EA K S EA K S where r { } S = ε r, ε r 2,, { } 3r S = ε r ε r 2 ε p 2, }, and p { } 3 S = ε p, ε p 2,. r EA K S,,,, p { S = ε p,, For a generic beam made up of several elements, using the usual technique of the finite element method for assembling and applying the boundary conditions, the system of linear algebraic equations can be obtained. K U = F (2.23) where vector; K represents the stiffness matrix, U = the unnown nodal displacement F is the load vector including the nodal force and prestressing force vectors of the beam in the time increment () for iteration (J). Obviously, the overall matrices and F are assembled using the element s matrices in Eq.9. Once in any time increment the nodal displacement vector U K is obtained using Eq.23 the overall stiffness matrix and load vectors will be modified and give a new nodal displacement vector U. hese procedures will be repeated until the difference between U and U is in neglected order. hen, one can get the real displacements, strains, and stress's distribution through the sections at time t. 3. CONCUSIONS A method for the time-dependent analysis of partially prestressed continuous composite beam with flexible shear connectors was presented taing into account the creep and shrinage effects, the effect of progression of cracs and the stiffness of concrete between cracs. his method permits evaluating the time development of the displacements, strains, and stress's distribution due to the application of geometrical and static load for any load history. 4. REFERENCES Bazant, Z. P. (972). Prediction of concrete creep effects using age adusted effective modulus methods. American Concrete Institution Journal, 69(4), 22-27. Dezi,., and arantino, A. M. (993). Creep in composite continuous beams. I: heoretical treatment. Journal of the Structural Engineering, Vol. 9, No. 7, 2095-2. Dezi,., eoni, G., and arantino, A. M. (995). ime dependent analysis of prestressed composite beams. Journal of Structural Engineering, Vol. 2, No. 4, 62-633. Gilbert, R. I., and Bradford, A. (995). ime-dependent behaviour of continuous composite beams at service loads. Journal of Structural Engineering, Vol. 2, No. 2, 39-327. 8