Luận văn Numerical analysis of externally prestressed concrete beams

shear force Q in continuous function which may be differentiated with respect to x and gives: -41- q GAdx dQ GAdx vd s ωω =−=2 2 (3.29) where dx dQq −= By summation of Eq.(3.27) and Eq.(3.29) gives a general equation to determine deflection with considering shearing force effect in Eq (3.30). ⎟⎠ ⎞⎜⎝ ⎛ += q GA EIM EIdx vd y ϖ12 2 (3.30) Fig.3.3 shows the effect of shear deformation on the total deflection of the beam. When beam is subjected to loading, the total displacement is equal to the sum of flexural displacement and shear displacement and expressed by an equation νy=νb+νs. However, when the span-to-depth ratio is big, the effect of shear deformation on the total displacement is extremely small, and in practice most analyses are usually taken by ignoring of shear deformation. But, the span-to-depth ratio is small, especially for the deep beams, it should be taken into account, because the shear deformation is approximately 10~15% of the total deflection at ultimate. 3.2.3 Shear stiffness model Based on Timoshenko’s theory, shear deformation as well as shear strain depends on value of the shear stiffness GA. However, in the elastic zone, effect of shear deformation (before formation of crack) on the total deformation is extremely small. In this case, the shear stiffness can be approximately calculated from the well-known relationship: )( AE GA c ν+ = 12 (3.31) Fig.3.3 Effect of shear deformation Lo ad Deflection Vb Vs GA is infinite GA is variable Total displacement Vy=Vb+Vs Vy Lo ad -42- where Ec is Young’s modulus of concrete;ν is Poisson coefficient; A is area of cross section. In the beams that are subjected to large shear forces and are web reinforced accordingly, diagonal cracks must be expected during the service condition. These cracks can increase the shear deformation of the beam, considerably. Shear distortions occurring in the web may be approximated by using the analogous truss model, in which vertical stirrup and 45o diagonal concrete struts are assumed to form web member (see Fig.3.4). The elongation of the stirrups is Δs, and the shortening of the compression strut is Δc. Applying Williot’s principle, the shear distortion can be found as: csRsv ΔΔΔΔΔ 2+=+= (3.32) where vs s s AE SV =Δ and wc s c bE V . 22 =Δ and substituting into Eq.(3.32), the shear distortion is given as: ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ +=+= c s v w ws s wc s vs s v E E A Sb bE V bE V AE SV 422 2Δ (3.33) where Es is the Young’s modulus of stirrup; S is the stirrup space and bw is the width of the web. Therefore, the shear distortion per unit length of the beam becomes: ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ +=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ += Δ = n dbE V E E A Sb dbE V d vws s c s v w ws sv 41 4 ρ γ (3.34) where ρv=Av/bwS is ratio of stirrup per unit of space of stirrup and n=Es/Ec is modulus ratio. Therefore, the shear stiffness of beam with 45o diagonal cracks, in accordance with truss action is the value of Vs when γ=1, and is thus given by: Fig.3.4 Truss model for shear stiffness model Δ S CΔ Δ R ΔV ΔC Δ S Δ R VS VS 45o d d s d -43- ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ + = v v ws n dbEGA ρ ρ 41 (3.35) Similar expression can be derived for other inclination of compression struts α and stirrupβ. It may be easy shown for the general case that the stirrup stress will be: ββαρ 2sin)cot(cot += v s s Vf (3.36) where the stirrup length is d/sinβ. Therefore, the shear stiffness can be defined as expression: dbE n GA ws v v βρα βαβαρ 44 244 sinsin )cot(cotsinsin + + = (3.37) where )sin( βρ SbA wvv = for the general case. 3.3 PROPOSED EQUATION OF CABLE STRAIN Despite the extensive literature on the topic of beam prestressed with external cables, the problem has been approached mainly in the elastic range. The solved case, for which a clearly stated analytical model is available both in linear and nonlinear range, is the case of the simply supported beams with rectilinear cables, for which the writing of compatibility equations is trivial and balance conditions are particularly simple. The treatment for more complex situations led to formulations referring to very specific cases and the solving method often adopted involved procedures. In particular, the coupling between an entire beam deformation and the cable local strain does not make it convenient to write the balance equations in terms of the equilibrium of beam cross section. Therefore, the global deformation compatibility between the concrete beam and the prestressing cable is necessary in order to establish the member-analysis procedure for the evaluation of externally prestressed concrete beams. In order to provide a satisfactory answer to these questions, a solution for computing the increase of strain in an external cable has been developed to analyze accurately the response of the beams prestressed with external cables, and will be presented in this section. It is important to note that the initial development of the current methodology for the analysis of externally prestressed concrete beams was carried out at Concrete Structures Laboratory of Nagoya University45~48). Although this methodology was applied for the analysis of simply supported beams with external cables and symmetrical loading conditions, it is, however, very important and useful materials for the further -44- development. It is truly said that without the previous researcher’s successes, the current developed methodology may not be possible to be going on. 3.3.1 Previous development of equation for cable strain Numerous investigations of analytical models for the beams prestressed with unbonded cables or external cables were carried out in the past. Here, only the most relevant models to the present research are briefly presented. Before the formulation of cable strain for externally prestressed concrete beams, it is very important to note that we cannot review the computing method for the cable strain without evoking of the initial formulation for prestressed concrete beams with partially bonded cables. For the calculation of cable strain of the concrete beams prestressed with partially bonded cables, Tanabe, T., et al.49) were earlier authors, who proposed the following equation: cs l cscsss dxl k εΔεΔεΔεΔ +⎟⎠ ⎞⎜⎝ ⎛ −= ∫01 (3.38) where ks is the slipping coefficient; Δεs Δεcs are the increments of cable strain and concrete strain at the cable level, respectively; l is the total length of cable between the extreme ends. From Eq.(3.38), it can be seen that there are two extreme cases, namely, perfectly bonded and completely free slip. For the case of perfect bond, the slipping coefficient should be equal to zero (ks=0), i.e., the cable strain Δεs is equal to the concrete strain Δεcs at the cable level. While, for the case of completely free slip, the slipping coefficient should be equal to 1.0 (ks=1.0), i.e., the total deformation of cable is equal to the total deformation of concrete beam at the cable level, and the cable strain is constant over the whole its length between the extreme end anchorages. To apply Eq.(3.38), an intensive investigation of experimental and numerical studies on the behavior of prestressed concrete frame with partially bonded cables was carried out by Umehara, H., et al.50~52) at Concrete Laboratory of Nagoya Institute of Technology. It is shown that the behavior of partially bonded PC beams can be satisfactorily predicted, and analytical results are in good agreement with the experimental data. The accuracy of the proposed equation for the cable strain is again verified by comparing the predicted results with the experimental observations. Umehara also carried out the numerical investigations on the effect of the bond condition, and the predicted results of these investigations is plotted in Fig.3.5. -45- A more application of Eq.(3.38) for the analysis of partially bonded PC plate was carried out by Qutait, A.R., et al.53). The authors stated that strain energy of cable varies according to the extent of bondage between the concrete and the cable. When perfect bond and no sliding are occurred, the change in the cable strain due to the applied load would exactly follow the change of strain in the concrete fiber at the same level. On the other hand, when the bond between the concrete and the cable is artificially reduced to zero as in the case of perfectly unbonded members, the magnitude of cable strain will be the same throughout its entire length. In the case of externally prestressed concrete beams, when the beam is subjected to bending, the deflection of external cable does not follow the beam deflection except at the deviator points. As a result, the cable strain cannot be determined from the local strain compatibility between the concrete and the cable. For the calculation of cable strain, it is necessary to formulate the global deformation compatibility between end anchorages. This means that the stress change in a cable is member-dependent and is influenced by the initial cable profile, span-to-depth ratio, deflected shape of structure, friction at the deviators, etc. In many studies, while computing the cable strain, two extreme cases are usually considered, namely, free slip (no friction) and perfectly fixed (no movement) at the deviators. In the first case, the cables move freely throughout the deviators without any restraint, i.e., the frictional forces that could develop between the prestressing cables and the deviators under increasing load to failure are neglected, and the cables are treated as the internally unbonded cables. The cable strain is constant over its entire length regardless of friction resistance at the deviator, and the strain increment can be then expressed as: Fig.3.5 Effect of bond condition on ultimate strength of frame 0 5 10 15 200 200 400 600 800 A pp lie d lo ad [k N ] Displacement [mm] 3.0=sk 0.0=sk 0.1=sk Perfectly bonded Partially bonded UnbondedA pp lie d lo ad [k N ] -46- ∫Δ=Δ l css dxl 0 1 εε (3.39) where Δεs, Δεcs are the increments of cable strain and the concrete strain at the cable level, respectively; l is the total length of cable between the extreme ends. In the second case, the cables are considered to have a perfectly fixed at the deviators. This means that the cable strain of each segment is independent from that of the others. The strain variation in the cable depends only on the deformation of two successive deviators or anchorages. And the strain variation can be expressed as: i i si l lΔ =Δε (3.40) where Δli, li are the elongated and original lengths of a cable segment under consideration, respectively. For the former, if frictional resistance at the deviators is neglected, deflection and cracking may be overestimated at the service loading range, whereas for the latter, if perfectly fixed is assumed, the ultimate load capacity may be overestimated54). This phenomenon was verified by an intensive program of the analysis of rectangular-section beams prestressed with external cables (see Fig.3.6 for the model beam of the analysis), which was carried out by M’Rad, A.3). The analytical model was investigated with emphasis on the effect of bond condition of cable at the deviators for three cases: 1) free slip 2); possible slip with a friction of 0.2; and 3) perfectly fixed. The analytical results are plotted in Fig.3.7. Normally, there is frictional resistance between the cable and the deviator, and cable strain depends on the friction coefficient. When the friction at the deviators is considered, there is a slight difference in the cable deformation at the both sides of a deviator (see Fig.3.8). For this purpose, Hyoudo, T.45) and Terao, D.46) proposed that the difference in the cable strain could be formulated by an equation, which is expressed in terms of the friction coefficient kdi as: ∫ + + + + Δ + =Δ−Δ ii ll cs ii di sisi dxll k 1 01 1 εεε (3.41) In Eq.(3.41), the friction coefficient kdi is not coulomb friction, and does not know at present. This coefficient has only mathematical meaning, and is not familiar in the practice design for prestressed concrete structures. This coefficient is assumed to be a function of the inclination angle of cable, and has a value between 0 and 1.0. This value indicates the extent -47- fixity of cable at the deviators. It can be also seen that there are two extreme cases of bond condition of cable at the deviators, namely, free slip and perfectly fixed. For the case 1, when kdi = 0, the cable strains at the both sides of a deviator are equal. This means that there is no friction at the deviator, the cables freely move throughout deviator without any restraint. Fig.3.6 Model beam for the analysis by M’rad Fig.3.7 Effect of bond condition of cable at deviators Fig.3.8 Distribution of cable strain at a deviator 12.0 10.0 8.0 6.0 4.0 2.0 0 0.02 0.04 0.06 0.08 0.10 0.12 7.2 7.0 6.8 6.6 6.4 6.0 0 0.02 0.04 0.06 0.08 0.10 0.12 C ab le s tra in Deflection [m] 1. Free slip 2. Slip with friction of 0.2 3. Perfectly fixed 3 12 6.2 7.4 x10-3 Deflection [m] M om en t [ M N .m ] 3 2 1 1. Free slip 2. Slip with friction of 0.2 3. Perfectly fixed C ab le s tra in M om en t [ M N .m ] Segment (i+1) siεΔ 1+Δ siε il 1+il Cable 0=dik dik Segment (i) Deviator 10 0 200 2000 85 10 0 85 8585 10 0 85 10 0 85 8585 -48- Whereas, for the case 2, when kdi=1.0, the difference in the cable strain at the both sides of a deviator may have approximately the maximum value. By using Eq.(3.41), Terao, T., et al.46~48), Diep, B.K., et al.55~57) carried out numerical investigations of simply supported beam as well as continuous beams with external cables. Also the authors concluded that by the proper assign of friction coefficient kdi, the predicted increase of cable stress in the external cables is close to the measured value obtained from the experimental observations. Based on the intensive investigations of externally prestressed concrete beams, the relationship between the friction coefficient and the inclination angle of cable was proposed by Diep, B.K. et al.57), which is plotted in Fig.3.9. However, when the Fig.3.9 Relationship between the friction coefficient and cable angle Fig.3.10 Diagram of the typical change of strain in the external cables a) beam elevation; b) perfectly fixed; c) slips with friction; d) free slip Fr ic tio n co ef fic ie nt Cable angle [Rad] 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.1 0.12 0.14 0.16 0.18 0.2 0.22 Analytical values Proposed friction coefficient 8797.01734.44694.23 2 +−= θθdik Fr ic tio n co ef fic ie nt l2 l1 l3 b) c) d) 1 1 l lΔ 3 3 l lΔ2 2 l lΔ a) ∫ Δl csdxl 01 ε -49- beam is subjected to bending, according to the deflected shape of beam, the inclination angle of cable will change. As a result, the friction coefficient kdi at the deviators will change as well. Therefore, the value of friction coefficient is not constant during the loading step, and it should be changed and depending on the deformed shape of beam. For the beams prestressed with external cables undulated by many deviators or multiple span continuous beams, the value and sign of friction coefficient are often arisen in the calculation, and the computing process should be repeated until obtaining the desirable results. Fig.3.10 shows the diagram of the typical change of strain in the external cables due to the applied load for different cases of bondage of cable at deviators. 3.3.2 Deformation compatibility of beam Behavior of prestressed concrete beam depends on the bondage of prestressing cable with the concrete. When the perfect bond exists, the prestressing cable can be considered as a beam element. In this case, the cable strain and the concrete strain at the cable level must be the same. On the other hand, when the bond does not artificially exist, it becomes necessary to consider it as a tied arch. For externally prestressed concrete beams, there is no any bond between the prestressing cables and the concrete beam, and the cables are attached to the beam at some deviator point along the beam. Therefore, an analytical model for externally prestressed concrete beams cannot be developed without considering the total compatibility requirement that the total elongation of a cable must be equal to the integrated value of concrete deformation at the cable level between end anchorages. This assumption is referred in this study as”deformation compatibility of beam”, and this can be expressed in the following equation: ∑ ∫ = Δ=Δ n i l cssii dxl 1 0 εε (3.42) where Δεsi is the strain increase of (i) cable element; Δεcs is the strain increase of concrete element at the cable level; li is the length of (i) cable element; l is the total length of cable. 3.3.3 Force equilibrium at deviators Fig.3.11 shows that Fi, Fi+1 are tensile forces in cable segments (i) and (i+1) at the deviator. Correspondingly, θi, θi+1 are cable angles, respectively. Thus, the force equilibrium at the deviators in the X direction can be expressed as: -50- 1111 cos)sinsin()1(cos ++++ =+−+ iiiiiii k ii FFFF i θθθμθ (3.43) where coefficient ki depends on the slipping direction and has a value ki = 1 if Ficosθi > Fi+1cosθi+1 and ki = 2 if Ficosθi < Fi+1cosθi+1; μi is the friction coefficient at the deviator, and is assumed to be known at each deviator. Eq.(3.43) can be rewritten in terms of the increments of tensile force as: 1111 cos)sinsin()1(cos ++++ Δ=Δ+Δ−+Δ iiiiiii k ii FFFF i θθθμθ (3.44) where ΔFi, ΔFi+1 are the increments of tensile force in the cable at the either side of deviator. Most of the previous experiments showed that the unbonded cables did not develop the inelastic strain, even at the failure of the beam. It is fact that the cable stress is fairly distributed all through the entire length of cable. As a result, it is improbable that the cable develops the inelastic strain. Since the stress of the unbonded cables in general or the stress of the external cables in particular usually remains below the elastic limit even at the failure of beam, it is possible to rewrite the force equilibrium at the deviator in terms of increments of cable strain by dividing both sides of Eq.(3.44) by EpsAps and result in Eq.(3.45): 1111 cos)sinsin()1(cos ++++ Δ=Δ+Δ−+Δ isiisiisii k isi i θεθεθεμθε or [ ] [ ] 0sin)1(cossin)1(cos 111 =Δ−+−+Δ−+ +++ siiikisiiiki ii εθμθεθμθ (3.45) where Eps, Aps are the elastic modulus and area of prestressing cable, respectively. 3.3.4 Proposed equation for cable strain Combining Eq.(3.42) with the force equilibrium at the deviator, which is expressed in Eq.(3.45), one can analytically obtain the increment of cable strain of each segment at certain Fig.3.11 Force equilibrium at a deviator X Y ki=2 ki=1 μPi Pi Fi+1 Fi Θi+1Θi -51- loading stage, and it can be expressed as the following: ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢ ⎣ ⎡ −+−−+ −+−−+ ......000 ......000 ::::: ::::: ......)1()1(0 ......0)1()1( ...... 323222 212111 321 22 11 scsc scsc lll kk kk μμ μμ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦ ⎤ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎣ ⎡ = ⎪⎪ ⎪⎪ ⎭ ⎪⎪ ⎪⎪ ⎬ ⎫ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎨ ⎧ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥ ⎦ ⎤ −+−−+ −+− ∫ − −−−− −−− − −− − 0 0 : : 0 0 : : )1()1(...... 0)1(...... :::: :::: 00...... 00...... ...... 0 )1( 3 2 1 1111 121 1 11 2 l cs sn ns s s s nn k nnn k n nn k n nn dx scsc sc ll nn n εΔ εΔ εΔ εΔ εΔ εΔ μμ μ or [ ]{ } [ ]{ }dNM s =εΔ (3.46) where ci and si are denoted as cosine and sine of cable angle, subscripts under these letters indicate the cable angle number. Finally, the increment of cable strain is defined by using the inverse matrix operation as: { } [ ] [ ]{ } [ ]{ }dCdNMs == −1εΔ (3.47) where [N] is denoted as the right-hand side of Eq.(3.46) and {d} is the increment of nodal displacement vector. It can be seen from Eq.(3.46) that the strain variation in an external cable depends mainly on the overall deformation of beam, friction at the deviators and cable angles. The increasing beam deformation under the applied load is in the relative change of cable elongation. The adequate evaluation of cable strain depends on the accuracy in the calculation of concrete strain at the cable level. That is the strain variation in a cable depends on the displacement of every points of beam. Therefore, the beam should be necessarily divided into a large number of short elements by using the finite element method. Since the elongation of a cable depends on the concrete strain at the cable level, the portion under integral of Eq.(3.46) should be formulated in terms of the coordinate of the cable -52- elements. Hence, within one prestressing element, the coordinate of cable can be determined by the equation of straight line as y=px+q, and the following equation can be derived for the component under the integral of Eq.(3.46) as: { }d l pKpl Ts q Ts p Tsl pKq Ts p dxvqpxudxyvudx i i i l b l b l cs iii ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ +−−+−−= +−=−=Δ ∫∫∫ 61,,1,6,,1 ])([)( 0 ''' 0 ''' 0 ε (3.48) 3.3.5 Stiffness matrix of prestressing cable Strain energy of a prestressing cable can be expressed in the following equation: 2 2 spsps s lAE U ε = (3.49) where Us is the strain energy of prestressing cable; l is the cable length; εs is the cable strain corresponding to the applied load; Eps is the modulus of elasticity of cable; Aps is the area of cable element. The increment of strain energy of cable element can be then derived from Eq.(3.49), and results in: ∫∫∫ ⎟⎠⎞⎜⎝⎛= dV spspss dVAEU 22 1 εΔΔ (3.50) By substituting Eq.(3.47) into Eq.(3.50), and arrives at: { } [ ]{ }∫∫∫ ⎟⎠⎞⎜⎝⎛= dV Tpspss dVdCdlAEU '2 1Δ (3.51) where [C’]=[C]T[C] Based on the principle of minimum potential energy, the stiffness matrix of a prestressing cable can be written as: [ ] [ ]{ }( )dVdClAE d U K dV psps s s ∫∫∫== '∂Δ∂ (3.52) The total stiffness of the beam can be obtained by combining the stiffness matrix for the concrete element expressed in Eq.(3.23) with the stiffness matrix for the prestressing element expressed in Eq.(3.52). The well-known load displacement relationship can be expressed as: [ ]{ } { }extsc FuKK Δ=Δ+ (3.53) where {Δu} is the nodal displacement vector; {ΔFext} is the vector of the applied load. -53- 3.4 EVALUATION OF CABLE SLIP AT DEVIATORS 3.4.1 Review of computing method for cable slip Although a large number of analytical studies on the behavior of externally prestressed concrete beams were carried out in the past, most of them did, however, not consider slip at the deviators. Even though dedicated efforts for the calculation of cable slip in externally prestressed concrete beams have been made in limited research works, it is still questionable if any proposed methods are consistent at all. Virloguex, M.2~3) was one of the earlier authors, who took slip into account by using Cooley formulation for the calculation of friction losses in the cable having continuous contact with the concrete along its length. For the author’s model, the main restriction for the cable slip is that: )( 1 )( iiii x ii x i eFFeF ΔφαΔμΔφαΔμ + + +− ≤≤ (3.54) where Fi, Fi+1 are the cable forces in the segments (i) and (i+1), respectively; Δαi is the angular variation; Δxi is the length of deviation block in between; μ and φ are the friction coefficient and the wobble coefficient per unit length of cable, respectively. If these relations are satisfactory at any deviator, there is no slip at this deviator. Otherwise, if these relations are not respected for a single deviator, the cable will slip on this deviator from the left to the right-hand side, if this condition is satisfied: )( 1 ii x ii eFF ΔφαΔμ + + > (3.55) and from the right to the left hand-side, if this condition is satisfied: )( 1 ii x ii eFF ΔφαΔμ + + < (3.56) If denote gi, ki as amount of cable slip and coefficient of slipping direction at deviator (i), respectively. Depending on the slipping direction, coefficient ki may have a value (–1) or (+1). After the slipping, the equilibrium condition should be satisfied: )( 1 )()( iii xk iiii egFgF ΔφαΔμ + + = (3.57) where Fi(gi), Fi+1(gi) are the tensile forces in th