Retangular Filament-Wound GFRP Tubes Filled with Conrete under Flexural and Axial Loading: Analytial Modeling Amir Fam 1, Siddhwartha Mandal 2, and Sami Rizkalla 3 ABSTRACT This paper presents an analytial model proposed to predit the behavior of onrete-filled retangular fiber reinfored polymer tubes (CFRFT) subjeted to bending, axial ompression loading or ombined loading. The model aounts for different laminate strutures of the flange and the web of the FRP tube through the Classial Lamination Theory (CLT). The gradual redution of stiffness resulting from the progressive failure of different FRP layers oriented at various angles is aounted for through the Ultimate Laminate Failure (ULF) approah. The model adopts the raked setion analysis using layer-by-layer approah and aounts for both totally filled tubes and tubes with inner voids. The model is apable of prediting the momenturvature responses of beams, axial load-strain responses of olumns and the omplete interation urves of beam-olumn members. The model is verified using experimental results, presented in a ompanion paper, and is utilized to evaluate the effets of laminate struture, hybrid laminates, varying the thikness of the tube and optimization of the inner void of partially-filled tubes. Comparisons of the behavior of CFRFT with onventional RC setions under different loading onditions showed that CFRFT ould provide axial load-bending moment interation urves omparable to those of RC setions of similar reinforement index. Also, providing a small fration of arbon fibers in the flanges ould substantially improve the flexural performane. The First Ply Failure (FPF) approah ould highly underestimate the strutural performane of CFRFT. Keywords: ultimate laminate failure (ULF), model, retangular, FRP, tube, interation urve, onrete-filled, lassial lamination theory (CLT) 1 Assistant Professor and Canada Researh Chair in Innovative and Retrofitted Strutures, Dept. of Civil Eng., Queen s University, Kingston, ON K7L 3N6 2 Master s student, Dept. of Civil Eng., Queen s University, Kingston, ON K7L 3N6 3 Distinguished Professor of Civil Engineering and Constrution, Diretor of the Construted Failities Laboratory, Civil Engineering Department, North Carolina State University, Raleigh, NC 27695-7533 1
INTRODUCTION The potential appliations of irular onrete-filled FRP tubes in new onstrution where the tubes provide permanent formwork, protetion for the onrete ore and reinforement, have been widely investigated by several researhers [Seible (1996), Burgueno et al. (1998), Mirmiran et al. (1998), Fam (2000)], inluding field appliations in bridges [Fam et al. (2003a)]. In onrete-filled irular tubes, the FRP tube provides a passive onfining pressure, leading to a signifiant inrease in the onrete s strength and dutility. Several analytial models have been developed to predit the behavior of suh hybrid systems under pure ompression [Samaan et al. (1998), Saafi et al.(1999), Spoelstra and Monti (1999) and Fam and Rizkalla (2001)], under bending [Fam and Rizkalla (2002)] and ombined bending and axial loading [Fam et al. (2003b)]. Very limited researh work has been done on square and retangular FRP tubes filled with onrete under various loading onditions and is summarized in a ompanion paper [Fam et al. (2003)]. Unlike irular FRP tubes, square and retangular FRP tubes filled with onrete show marginal improvement in strength and dutility under axial ompression [Mimiran et al. (1998), Fam et al. (2003)]. The effetiveness of onfinement of square and retangular FRP tubes depends on the orner radius and the aspet ratio of the tube. Davol (1998) has developed a finite element model for fiber-wrapped retangular setions with rounded orners and various laminate strutures and flat-to-radius ratios to study the transverse strain distribution along the perimeter under axial ompression. Mirmiran et al. (1999) has also developed a model to predit the beam-olumn behavior of square onrete-filled FRP tubes assuming a bilinear stress-strain urve for onrete and assuming a linear elasti behavior of FRP tube up to failure. No models, however, have been developed to aount for the stiffness degradation of FRP tubes based on progressive laminate failure and ULF approah, tubes with different laminate strutures in the 2
flange and web or tubes partially filled with onrete. This paper presents an analytial model to predit the behavior of CFRFT under pure bending, axial ompression loading or ombined bending and axial loading. The auray of the model is verified by experimental results provided in a ompanion paper [Fam et al. (2003)]. The model is based on equilibrium, strain ompatibility, raked setion analysis and utilizes the layer-by-layer approah taking into onsideration the gradual and progressive laminate failure of the GFRP tube. The model also aounts for different laminate strutures in the flange and web of the tube and an be applied to hybrid tubes with different types of fibers as well as tubes partially filled with onrete. The proposed model is used to evaluate the effets of FRP laminate struture, wall thikness, hybrid tubes and onrete strength. A omparison of the behavior of CFRFT to onventional RC setions under different loading onditions is also presented CONSTITUTIVE PROPERTIES OF MATERIALS The proposed analytial model is based on the following onstitutive properties of the FRP laminate of the tube and onrete. FRP Laminate The fiber orientation and staking sequene of layers in a multidiretional laminate ould lead to signifiant non-linear stress-strain response under axial tension or ompression. This is evident from the oupon test results [Fam et al. (2003)]. The non-linearity ould also arise from the progressive laminate failure of the FRP tube. In order to model the stress-strain response of multidiretional GFRP laminate, the Classial Lamination Theory (CLT) together with the progressive and Ultimate Laminate Failure (ULF) approah has been adopted. Typial FRP tubes, inluding the ones used in the present study, have a symmetri balaned laminate for both 3
flange and web, therefore, the relationship assoiated with symmetri balaned laminate are redued from the general expressions in the following setion. The various assumptions involved and the detailed derivation an be found elsewhere [Daniel and Ishai (1994)]. The properties of a single lamina are first alulated from the onstitutive fiber and matrix properties using the rule of mixture. The Classial Lamination Theory (CLT) an then be used to determine the overall behavior of the multidiretional laminate, whih is a funtion of the properties and staking sequene of individual layers. By defining a loal oordinate system (1, 2) for the unidiretional lamina with the diretions 1 and 2 are parallel and normal to the fiber diretion, respetively, as shown in Fig. 1(a), the in-plane stress-strain relation for a given lamina an be written as: [ ] 1, 2 [ Q] [ ε ] 1, 2 σ = (1) 1, 2 Where[ Q] 1, 2 is the stiffness matrix with respet to loal axes (1, 2). As the axes (1, 2) do not oinide with the global loading axes (x, y), it is neessary to transform the stress and strain omponents referred to (1, 2) axes to the (x, y) axes through the transformation matrix[ T ]: [ σ ], 2 = [ T ][ σ ] x, y and [] ε, 2 = [][] T ε x, y The transformation matrix[ T ] is given in terms of [ ] 1 (2) 1 (3) m = osθ and n = sinθ by: 2 2 m n 2mn 2 2 T = n m 2mn (4) 2 2 mn mn m n Where the angle θ is measured positive ounterlokwise from the x-axis to the 1- axis. Thus the stress-strain relation in the x-y oordinate system beomes: 4
Where [ Q] x, y [ ] x,y [ Q] [ ε ] x, y σ = (5) x,y is the transformed stiffness matrix of the lamina and is given by: 1 [ Q] [ T ][ Q] [ T ] x,y = (6) 1,2 If we onsider an n-ply multidiretional laminate in the x-y plane as shown in Fig.1 (b), the strain at any layer within the laminate[ ε ] k x, y an be related to the strain at the referene plane (x-y) 0 [ ε ] x, y, whih is loated at mid-thikness of the laminate, as follows: k o [] x,y = [ ε ] x,y + z[] k x, y ε (7) Where z is the distane from the referene plane to the layer k and [ k ] x, y is the urvature vetor. Now, the stress-strain relation of a single layer k, in a laminate an be written as: k k [ ] [ ] [ ] k x, y σ = (8) x,y Q x,y ε Using the expression for the strains from Eq. (7), Eq. (8) an be re-written as: k k o k [ ] x,y = [ Q ] [ ε ] x,y + z[ Q] [ k] x, y σ (9) x,y For an n-ply laminate the total fores an be obtained by summing the stress integrals as n h n h h x,y k= 1 k= 1 k k k k k o k o [ N] x,y = [ σ ] x,ydz = [ Q] [ ε ] x,ydz + [ Q] x,y [ k] x,y zdz [ A ] x,y [ ] x,y + [ B] x,y [ k] x, y hk 1 hk 1 hk 1 x,y = ε (10) Where [ A ] represents in-plane fore-strain relationship, and [ B] is oupling between in-plane and out-of-plane behavior. For a symmetri balaned laminate, [ B ] beomes zero, thus the foredeformation relation for a symmetri laminate under uniaxial loading along the x-axis beomes: N 0 0 x A = A 0 xx yx A A xy yy 0 0 0 A SS ε x ε y 0 o o (11) 5
Young s modulus and Poisson s ratio of the laminate an be alulated as: E x N x = o hε x 1 = A h xx 2 A A xy yy and Axy ν xy = (12) A yy k Where h is the total thikness of the laminate and A Q ( h h ) the transformed stiffness of layer k. Failure riteria ij n = with i, j =x, y, s. k= 1 ij k k 1 k Q ij is Sine the FPF approah does not neessarily indiate omplete failure of the laminate and may highly underestimate the strength of the laminate, the ULF approah has been adopted to establish the stress-strain relation of the FRP laminate. To determine the lamina stresses, the strains in the x-y oordinate system are transformed into the loal lamina oordinate system by Eq. (3). The lamina stresses are then determined using Eq. (1). One the laminae stresses are obtained, the laminae safety fators and unidiretional lamina strengths are alulated using Tsai-Wu failure riteria [Daniel and Ishai (1994)]. The minimum safety fator of all laminae is onsidered as the laminate safety fator. One the load required to produe FPF is determined, the stiffness of the failed lamina or laminae are disounted through ply disount method and a new laminate stiffness[ A ] is alulated. The lamina stresses are realulated and heked against the seleted failure riteria to ensure that failure of the undamaged laminae does not our under the inreased share of stress following FPF. The load is inreased until failure of the next lamina ours. Ultimate failure of the laminate ours when the remaining intat laminae, at any stage of progressive laminate failure, annot sustain the stresses. Stress-strain response of GFRP laminate Fig. 2(a) shows the predited stress-strain responses using the CLT-ULF approah for GFRP oupons, from the flange and web, tested in axial tension. The predited stress-strain urve 6
shows good agreement with the oupon test results for the flange of the GFRP tube but it underestimates the ultimate strength of the web. Unlike the flange, where the ultimate strength is dominated by the longitudinal [0] layers, the web is mainly omposed of fibers with [±45] layers. In order to assess the sensitivity of the model to the fiber angle, the strength of a hypothetial laminate omposed of fibers with [±40] layers was predited. It was found that the 5 degrees hange in angle ould inrease the strength by 52 perent. Thus the higher ultimate strength measured for the web, ompared to the predited strength, ould be attributed to a slight hange in the winding angle during fabriation. Fig. 2(a) also shows that the FPF approah highly underestimates the strength. Fig. 2(b) shows the predited stress-strain responses of the GFRP laminate under axial ompression. While the ultimate strengths of both the flange and web are well predited, the ultimate strains are underestimated due to the fat that under ompression, the resin matrix, whih is a non-linear material, dominates behavior of the laminate. The CLT, however, assumes that the resin is a linear-elasti material until failure for both fibers and resins. Conrete Under Compression For irular onrete-filled FRP tubes (CFFT), Fam and Rizkalla (2002) have shown that, in pure bending, onrete in ompression exhibits extended strain softening but no inrease in strength, essentially insignifiant onfinement. Aordingly, onrete ompressive behavior an be adequately desribed by Popovis model (1973). This model is also adopted here for CFRFT. As shown in Fig. 1(), The ompressive stress f orresponding to a strainε is given by: f f ( ε / ε ) r + ( ε / ε ) r = (13) r 1 Where f is the unonfined onrete strength, ε is the orresponding strain, and r ( E E ) = Eo / o se. The tangent and seant moduli, o E and E se, are determined as 7
E o = 5000 f (MPa) and E f ε se =, respetively. It has also been established that in totallyfilled CFFT under bending, ompression failure of onrete would only happen as a onsequene of ompressive failure of the tube [Fam et al. (2003 b)] and that the ultimate ompressive strain of onrete an be assumed equal to the ultimate ompressive strain of the FRP flange. For tubes with a void, however, the ultimate strain is limited to 0.0035 as evident from the behavior of beam B2, reported in the ompanion paper [Fam et al. (2003 )]. Based on a study on fiber wrapped irular, square and retangular olumns, Rohette (1996) onluded that onfinement effetiveness depends on the orner radius of the setion. Using these results, Mirmiran et al. (1998) proposed a modified onfinement ratio (MCR) to define the onfinement effetiveness in terms of orner radius of the setion and showed that with MCR less than 15 perent, the FRP jaket is ineffetive in enhaning the onrete strength in axial members. In the present study, MCR is 8.5 perent, whih indiates insignifiant onfinement effet. Aordingly, the unonfined onrete model is used to predit the axial loadbending moment interation diagrams and the axial load-strain behavior of CFRFT. Conrete Under Tension The model proposed by Vehio and Collins [Collins and Mithell (1997)] has been adopted for onrete in tension. The tensile stress in onrete f orresponding to a strain ε is given by: for ε < ε r, f = E o ε (14) for ε > ε, r f α α f 1 2 r = (15) 1+ 500 ε Where f r is the raking strength of onrete, taken as f r = 0.6 f (MPa) and is related to the raking strain ε r as f r = E o ε r. α 1 and α 2 are fators aounting for the bond harateristis 8
and nature of loading, respetively. For GFRP tubes a value of 0.3 is used for α 1 as suggested by Fam (2000). For short-term monotoni loading the value of α 2 is normally taken as 1.0. ANALYTICAL MODEL FOR DIFFERENT LOADING CONDITIONS To predit the moment-urvature response and axial load-bending moment interation diagram of CFRFT, a strain ompatibility/equilibrium model has been developed. A layer-by-layer approah is adopted to integrate the stresses over the ross-setional areas of onrete and FRP. The setion is idealized as a perfet retangle and divided into n number of strips of equal thikness hi, as shown in Fig. 1(d). Exept for the top and bottom flanges of the tube, eah strip i, onsists of retangular portions of onrete and FRP web. The entroid of the strip is loated at its mid thikness and the depth to its enter from top level is h(i). It is assumed that perfet bond exists between the FRP tube and onrete and strains are linearly distributed along the depth of the setion. The stresses in the FRP tube are based on the stress-strain urves obtained from oupon tests or predited using CLT-ULF, aounting for the different properties of the flange and web. The stresses in the onrete are based on the onrete model desribed earlier. The stresses at the entroid of a strip are assumed to be onstant throughout its thikness. To obtain the moment-urvature response for beams, the proedure an be summarized as follows: 1. Assume a value of the extreme ompressive strain ε in onrete and a neutral axis depth. 2. Calulate the ompressive and tensile strains, ε t and ε b, in the top and bottom flanges of the tube and ompare them to the ultimate strains to hek for failure of the flanges. 3. Calulate the ompressive and tensile stresses, f fft and f ffb, in the top and bottom flanges of the FRP tubes, respetively. 9
4. For eah layer i, ompute the strain ε(i) and the orresponding ompressive or tensile stresses in the web of FRP tube, f fwt (i) or f fwb (i), and onrete f (i) or f t (i). If the tensile or ompressive strain ε(i) in the web exeeds the ultimate strain of the laminate, the ontribution of web from that layer is ignored. The model, therefore, aounts for gradual failure of the tube. 5. Calulate the ompressive fores in the flange, web of FRP tube and onrete as CF f, CF(i) and CC(i), respetively. Similarly, alulate tensile fores in the flange, web of FRP tube and onrete as TF f, TF(i) and TC(i), respetively. 6. Chek equilibrium by satisfying that the differene between the sum of the total ompressive and tension fores is less than the presribed tolerane. If equilibrium is not satisfied, go to step 1 and assume another value of and repeat the proess until equilibrium is reahed. 7. One equilibrium is satisfied, alulate the total moment M, by summing the internal moments of onrete and FRP in ompression and tension for all layers. The orresponding urvature ψ is the slope of the strain profile, i.e. ψ = ε /( t f ). 8. Enter a higher value of strain ε in step 1 and repeat the proess from step 2 to 7 until the omplete moment-urvature response is onstruted. 9. Calulate the defletion using the moment-area method [Collins and Mithell (1997)]. In order to generate the axial load-moment interation diagram, a similar approah is used first to ompute the moment-thrust-urvature response and is summarized as follows: 1. For a given eentriity e alulate the resultant of internal fores N and orresponding moment M using steps 1 to 5 (desribed earlier to ompute the moment-urvature response). 2. Compute the eentriity e = M/N and ompare it with that assumed initially. If the two values are different, assume a new value of neutral axis depth and realulate M and N 10
following the proedure of step 1. The proess is repeated until the two values of eentriity are equal. This orresponds to point 1 in Fig. 1(e). 3. For different ompressive strains ε, repeat the proess until the maximum values of M and N are reahed as shown in Fig. 1(e) as point 2. This point orresponds to reahing the ultimate strength of the material or the peak moment value of the moment-thrust-urvature response (if the urve shows a desending portion). 4. Construt the full interation urve following the proedure desribed in steps 1 to 3 for different ombination of maximum M and N at different eentriities. To obtain the load-axial strain behavior under pure axial load, different levels of uniform strain are assumed aross the setion and the orresponding stresses in FRP and onrete are integrated to obtain the resultant ompressive fore. VERIFICATION OF THE MODEL Flexural Behavior The model is applied to the beam speimens to predit the moment-urvature and load-defletion behavior. Fig. 3 shows the predited moment-urvature responses of beams B1, B2 and B3 using both the oupon test properties and the CLT-ULF predited properties of the FRP tube. Fig. 3 shows that the model provides good predition for the flexural responses. Fig. 3 also shows that the tension stiffening (T.S.) has marginal effet on the moment-urvature response as evident in beam B1, and therefore, an be ignored. The predited responses using the stress-strain relations of the GFRP tube derived using CLT-ULF, indiate that the flexural apaities of beams B1 and B3 were slightly underestimated. This is due to the fat that CLT underestimated the ultimate 11
strength and strain of the web as disussed earlier. Point on the predited urves in Fig. 3 indiates initiation of failure of the web of the tubes. Fig. 4 shows the predited load-defletion responses of the three beams. The model slightly underestimated the defletion of beams B1 and B2. This differene is possibly due to the additional shear deformations, whih ontributed to the total defletion of B1 and B2. It should be noted that both beams B1 and B2 had lower shear span-to-depth ratios ompared to beam B3. Beam-Column Behavior The predited interation urve I o along with the test results of the beam-olumn speimens is shown in Fig. 5. Also shown on the same graph are additional urves I 1, I 2, and I 3, whih will be disussed later. The model, whih is based on unonfined onrete strength, shows good agreement with test results of CFRFT. This is ompletely different from irular onrete-filled FRP tubes, where the unonfined onrete model underestimated the interation urve [Fam et al. (2003b)]. Fig. 5 also shows that the pure axial strength is slightly overestimated. Rihart and Brown (1934) indiated that under axial loading the maximum ompressive strength of onrete should be taken between 0.80 and 0.85 f. Hognestad (1951) also suggested a fator of 0.85. This fator is attributed to the differene in size and shape of the olumn and onrete ylinder and the rate of loading. The predited axial strength orresponding to 0.85 f is 1854 kn whih is very lose to the average experimental results of 1896 kn. It should be mentioned that the predited interation urve I o is largely dominated by ompression failure exept for a small tension failure region at very low axial loads. Column Behavior The model has been used to predit the axial load-strain behavior of the olumns C1 and C2 as shown in Fig. 6. Also shown on the same figure are the individual ontributions of FRP tube and 12
onrete ore. The 85 perent strength redution fator is also shown. The model based on the unonfined strength of onrete, ombined with the ontribution of FRP tube predits well the axial strength, however, it does not provide an aurate post-peak response. COMPARISON WITH CONVENTIONAL RC SECTIONS In order to have a better understanding of the behavior of CFRFT, its behavior is ompared with onventional reinfored onrete (RC) setions. The omparison is provided for beams and beam-olumn members. The riteria used to design the RC setions are also disussed. Flexural Behavior An attempt is made to ompare the flexural behavior of beam B1 to RC beams with the same ross-setional dimensions and onrete strength as B1 and reinfored with Grade 60 steel rebar. In order to selet the steel reinforement ratio, four different riteria have been adopted: 1. Setion RC1 designed aording to ACI 318-99 to provide same moment apaity as B1. 2. Setion RC2 with the same reinforement ratio as B1 (ρ s =ρ f ). 3. Setion RC3 with reinforement of equal axial stiffness as B1 (E s A s =E f A f ). 4. Setion RC4 with the same reinforement index as B1 (ω s =ω f ). Fig. 7 shows a omparison of the four RC beams with CFRFT beam B1 in terms of the moment-urvature responses. The figure also shows the reinforement of the four RC beams. Beams RC2, RC3 and RC4 are all doubly-symmetrially reinfored. Beam RC1 designed aording to ACI 318-99, has similar moment apaity to beam B1 with a total reinforement ratio of 3.7 perent. When attempting to design RC1 as singly reinfored setion, the Code limitation of maximum reinforement ratio of 0.75 of the balaned reinforement ratio has resulted in lower moment apaity than B1. Therefore, ompression steel was added. The 13
moment urvature response of RC1 is quite stiffer than beam B1 due to the higher Young s modulus of steel ompared to GFRP. The response of beam B1 is also ompared with beams RC2 and RC3 whih have similar reinforement ratio (ρ s = ρ f ) and similar axial stiffness (E s A s = E f A f ), respetively. It is lear that RC3 provides similar stiffness to that of B1, before yielding of steel, however, as the beam inluded only 0.8 perent reinforement ratio it showed signifiantly lower strength than B1. Beam RC2 showed a signifiantly higher strength than B1, however, when ompared with B1, the behavior of RC2 and RC3 ould be misleading as the ultimate strength fu of the GFRP tube is onsiderably different from the yield strength f y of the Grade 60 steel. Therefore, it would be more appropriate to ompare B1 with RC4 of a similar reinforement index, ω, defined as ω = ρ f / f = ρ f / f, where ρ f and f u s y ' ρ s are the reinforement ratios of CFRFT and RC beam, respetively. The reinforement ratio ρ of beam B1 was 9.6 perent. Using the similar reinforement index onept, the equivalent reinforement ratio in RC4 is 5.6 perent. The reinforement is symmetrially distributed along the perimeter of RC4. The flexural strength of beam RC4 is found to be almost similar to beam B1. Therefore, the flexural strength of CFRFT is quite omparable to onventional RC beam with similar reinforement index. Although beams RC1, RC2 and RC4 were all under-reinfored and failed in tension, they all showed very limited dutility due to the relatively high reinforement ratio in omparison to RC3. Interation Curves The interation urve of the CFRFT setion is ompared to those of RC setions of the same dimensions (152 x 254 mm) and reinfored with Grade 60 steel. Three RC setions were seleted based on the following riteria: f 14
1. Setion designed to provide the same axial ompressive strength of the CFRFT setion, (Interation urve I 1 in Fig. 5). This riteria resulted in ω s = 0.8 ω f. 2. Setion designed to provide the same bending strength as the CFRFT setion, (Interation urve I 2 in Fig. 5). This riteria resulted in ω s = 1.45 ω f. 3. Setion designed with the same reinforement index as the CFRFT setion (ω s =ω f ), (Interation urve I 3 in Fig. 5). All three setions are doubly-symmetrially reinfored. The three interation urves I 1, I 2, and I 3 are ompared with interation urve I o of the CFRFT setion in Fig. 5. Curve I 1 representing a RC setion with very small flexural apaity under pure bending while urve I 2 represents a RC setion with very high axial ompressive strength. Unlike urve I 1, urve I 2 is entirely governed by ompression failure. Curve I 3 shows moderate flexural apaity in bending but higher axial apaity than the CFRFT. The full interation urve I 3 is also governed by ompression failure. PARAMETRIC STUDY The analytial model has been used in a parametri study to examine a wider range of parameters beyond the limitation of the experimental program. The parameters studied inlude hybrid FRP laminate with glass and arbon fibers, laminates with different proportions of fibers in the axial and transverse diretions, tubes with different wall thikness, onrete of different ompressive strengths, and beams with different inner void sizes. Tube with Hybrid Laminate More than one type of fibers an be inorporated to optimize the FRP tube. The ombination of different types of fiber ould substantially hange the performane of the member. The behavior 15
of the CFRFT with hybrid tube inluding arbon fibers as replaement for the E-glass longitudinal [0] layers in the flanges has been investigated for both the large and small beams B1 and B3, respetively. The arbon fibers used in this analysis has a tensile strength of 3500 MPa and Young s modulus of 227GPa. The moment-urvature responses of the hybrid beams are shown in Fig. 8. For both small and large beams with hybrid tubes, the flexural strength was found to inrease by 56 and 62 perent, respetively. It is therefore lear that a small amount of arbon fibers plaed longitudinally in the flanges ould inrease the flexural strength substantially without muh inrease of the overall ost. Optimization of Inner Void in the Retangular Setion Fig. 9 shows the effet of the thikness of onrete flange on the moment apaity of partiallyfilled tube B2. It an be seen that the moment apaity inreases as the flange thikness-to-depth ratio (h t /D) inreases up to a ratio of 0.15, after whih, almost no inrease in flexural strength is observed. This is mainly beause extending the onrete flange below the neutral axis level is ineffetive. The setion an, therefore, be optimized with relatively small h t /D ratio. Voided setions provide a substantially redued self-weight and inreased effiieny in bending. Effet of Laminate Struture and Wall Thikness on Flexural Behavior The retangular GFRP tube used in this part of the parametri study has inner dimensions of 152 254 mm (similar to beam B3) and [0/90] s symmetri ross-ply E-glass/epoxy laminate. Five different laminate strutures are used in the analysis by varying the proportions of fibers in the axial [0] and transverse [90] diretions, inluding 9:1, 3:1, 1:1, 1:3 and 1:9 ratios. A 9:1 laminate indiates that 90 perent of its fibers are oriented in the axial diretion. For eah laminate struture, five different wall thiknesses have been hosen, inluding 1, 2, 4, 8 and 16 mm, equivalent to reinforement ratios of 2.1, 4.2, 8.6, 17.5 and 36.3 perent, respetively. The CLT- 16
ULF approah has been used to determine the onstitutive properties of the laminates. Different laminate strutures resulted in effetive elasti moduli and tensile strengths in the range of 41 to 13 GPa and 927 to 103 MPa, respetively, in the axial diretion. The range of onrete strengths examined is 20 to 80 MPa. The variation of the flexural strength with the reinforement ratio ρ, whih is defined as the ratio of ross setional area of GFRP tube to that of onrete, is shown in Fig. 10 for the 50 MPa onrete and for different laminate strutures. The urves are normalized with respet to the ross-setional dimensions, onrete ompressive strength f, and the ultimate tensile strength f u of the tubes. For a partiular laminate struture, inreasing the wall thikness results in inreasing the flexural apaity and ould lead to hange in the failure mode from tension to ompression. Fig. 10 also presents the lous of the balaned reinforement ratio, whih results in failure of the tube in tension and ompression simultaneously. The balaned reinforement ratio redues gradually with inreasing the stiffness of the tube in the axial diretion. Fig. 11 shows the variation of flexural strength with the perentage of fibers in the axial diretion. It an be seen that for a given wall thikness, the flexural apaity inreases as the perentage of fibers in the axial diretion [0] inreases. The rate of inrease is muh higher for thik tubes than for thin tubes. Varying the onrete strength has resulted in behavior very similar to that of irular speimens reported by Fam and Rizkalla (2002), where the onrete strength has insignifiant effet on the flexural strength. The effet of reinforement ratio on moment apaity is ompared for both CFRFT and onventional RC setions. It was found that a (1:1.5) GFRP laminate results in tensile strength f u similar to that of the yield strength f y of Grade 60 steel. Therefore, a CFRFT with (1:1.5) 17
laminate is used in this analysis. The omparison is provided in Fig. 12 for the CFRFT as well as two RC setions inluding singly and doubly reinfored setions. For the singly reinfored beam, the flexural apaity inreases with inreasing reinforement ratio in the tension failure region with a rate relatively higher than the CFRFT beam. Beyond the balaned reinforement ratio (in the ompression failure zone) the rate of inrease of flexural strength of singly reinfored RC setion is negligible ompared to the CFRFT setion, whih ontinues to inrease with almost a onstant rate. The doubly-symmetrially reinfored RC setion shows very similar trend to CFRFT. Therefore, flexural CFRFT members an provide a reliable replaement of onventional doubly reinfored RC setions and an also eliminate the problem of reinforement ongestions of RC beams at higher reinforement ratios, whih an ease the fabriation proess. Effet of Wall Thikness and Laminate Struture on Interation Curves In order to study the effet of laminate struture and reinforement ratio, whih is governed by the thikness of tube, a retangular tube with E-glass/Epoxy laminate, similar to that used in the parametri study in flexure has been hosen. Three different laminate strutures with various ratios of fibers in the axial and transverse diretions of 1:9, 1:1 and 9:1 are used. For eah laminate struture, two different wall thikness has been onsidered, inluding 2 and 16 mm. Fig. 13 shows the interation diagrams. The three urves in eah graph represent the three different laminate strutures for a given wall thikness. For a given laminate struture, the bending moment apaity is inreased with inreasing the wall thikness. Fig. 13 also indiates that for a given wall thikness, inreasing the perentage of fibers in the axial diretion results in signifiant inrease in the flexural apaity for both thin and thik tubes. However, its effet on the axial strength is only pronouned in thik tubes. 18
Fig. 13 also indiates that CFRFT with thin tubes, under small and intermediate eentriities, have a similar urve for all range of laminate strutures. However, at large eentriities, when the urve is dominated by tension failure, the bending apaities vary signifiantly. In this ase, the most effiient laminate is the one with maximum amount of fibers oriented in the axial diretion suh as the 9:1. In CFRFT with thik tubes, the most effiient design is governed by tubes with maximum amount of fibers oriented in the axial diretion (9:1), for both small and large eentriities. CONCLUSIONS A theoretial model has been developed to analyze the behavior of onrete-filled retangular FRP tubes (CFRFT) under different loadings. The model utilizes the Classial Lamination Theory inluding the Ultimate Laminate Failure approah (CLT-ULF) and employs a raked setion analysis using a layer-by-layer approah in order to apply the priniples of strain ompatibility and equilibrium. The model also aounts for different laminate strutures in the flange and web of the FRP tube and an be applied to tubes with hybrid fibers as well as tubes partially filled with onrete. The model has been verified using experimental results and used in a parametri study to evaluate the effets of thikness of the tube (reinforement ratio), laminate struture and onrete strength on the behavior of CFRFT under different loading onditions. The behavior of CFRFT was also ompared to that of onventional RC setions. The following onlusions are drawn: 1. The CLT-ULF model is apable of apturing the non-linear harateristis of FRP laminate resulting from progressive failure of the layers but annot apture the non-linear effet of resin, whih is highly pronouned under ompression or in [±45] laminates. 19
2. First Ply Failure approah (CLT-FPF) ould highly underestimate the strength of FRP laminate with layers in different diretions. 3. Using either the experimental oupon test results or the CLT-ULF results predits well the behavior of CFRFT. Coupon test results are, however, slightly more aurate. 4. Conventional RC setions of the same moment apaity as CFRFT have higher axial ompressive strength, while RC setions with the same axial strength as CFRFT have substantially lower flexural strength. 5. RC setions with similar flexural strength, reinforement ratio, or reinforement index to CFRFT are signifiantly stiffer, but have very low dutility. RC setions with reinforement ratio based on equivalent axial stiffness (EA) of reinforement, are quite dutile but substantially lower in flexural apaity than CFRFT. 6. A small fration of arbon fibers plaed longitudinally in the FRP flange of the tube (hybrid laminate), ould result in signifiant gain in flexural strength and stiffness. 7. In CFRFT flexural members, the void size an be optimized for maximum strength-toself weight ratio. 8. Inreasing the wall thikness for a given laminate or inreasing the perentage of fibers in the axial diretion of the laminate results in inreasing the flexural strength and stiffness of CFRFTs. However, failure mode ould hange from tension to ompression. Conrete strength, on the other hand has marginal effet on the flexural response of the CFRFTs. 9. The balaned reinforement ratio of CFRFT under pure bending depends on the laminate struture. It redues gradually with inreasing axial stiffness of the laminate of the tube. 10. Variation of flexural strength with reinforement ratio in CFRFT is quite different from singly reinfored RC setions but very similar to that of doubly reinfored RC setion. 20
11. There is insignifiant onfinement effet on enhaning the strength of CFRFT. Unlike irular members, the unonfined onrete stress-strain urve used in the model predits well the behavior of CFRFTs under ombined bending and axial loading. 12. Inreasing the wall thikness or perentage of fibers in axial diretion of the tube, results in large gain in flexural strength. In thin tubes, inreasing the perentage of fibers in axial diretion has insignifiant effet on axial strength. ACKNOWLEDGEMENT The authors wish to aknowledge finanial support provided by the Network of Centres of Exellene on Intelligent Sensing for Innovative Strutures (ISIS Canada), Queen s University and North Carolina State University. NOTATIONS B = width of retangular ross-setion = neutral axis depth CF f = ompressive fore in the flange of FRP tube CF(i) = ompressive fore in web of FRP tube within strip i CC(i) = ompressive fore in onrete within strip i D = depth of the retangular ross-setion E o = initial tangent elasti modulus of onrete E f = effetive elasti modulus of FRP tube in axial diretion E s = Young s modulus of steel E se = seant modulus of onrete at f 21
e = eentriity of axial load f = stress in onrete at strain ε f = ompressive strength of unonfined onrete f (i) = ompressive stress in onrete in general strip i f r = raking strength of onrete in tension f fft = stress in the top flange of FRP tube f ffb = stress in the bottom flange of FRP tube f fwt (i) = stress in the ompression web of FRP tube within strip i f fwb (i) = stress in the tension web of FRP tube within strip i f t (i) = tensile stress in onrete in general strip i f u = ultimate tensile strength of tube in axial diretion f y = yield strength of steel hi = thikness of eah strip h(i) = distane form top of the tube to the entroid of layer i h n = distane from mid-plane to near surfae of layer n h t = thikness of onrete in ompression flange M = bending moment M u = ultimate moment apaity of beam N = axial ompressive fore N x = fore applied to laminate in x diretion n = number of plies within a laminate or number of strips within a ross-setion of FRP tube t f = thikness of flange of the FRP tube 22
t w = thikness of web of the FRP tube TC(i) = tension fore in onrete in general strip i TF f = tension fore in the flange of FRP tube TF(i) = tension fore in the web of FRP tube in general strip i z = distane between a general layer k and mid-thikness plane of a omposite laminate α 1 = fator for bond harateristis of reinforement α 2 = fator for nature of loading ε b = average strain in the bottom flange of the FRP tube ε = axial ompressive strain orresponding to f ε r = strain orresponding to f r ε t = average strain in the top flange of the FRP tube ε(i) = strain orresponding to strip i ε o = in-plane strain in a lamina at the level of referene plane loated at the mid-thikness θ = angle between referene axis x of a laminate and loal axis 1 of an individual lamina κ = urvature of a FRP lamina ρ = reinforement ratio ψ = urvature of a setion of a beam under bending ω s = reinforement index of RC setion ω f = reinforement index of FRP tube setion 23
[A] = in-plane stiffness matrix [B] = in-plane and out-of-plane oupling matrix [Q] = laminate stiffness matrix [T] = transformation matrix REFERENCES 1. ACI Committee 318 (1999) Building ode requirements for reinfored onrete and ommentary, ACI 318M-99/ACI 318RM-99, Amerian Conrete Institute, Detroit. 2. Burgueno, R., Davol, A., and Seible, F. (1998) The arbon shell system for modular bridge omponents. Pro., 2nd Int. Conf. on Composites in Infrastruture (ICCI 98), pp.341 354. 3. Collins, M. P., and Mithell, D. (1997). Prestressed onrete strutures, Response Publiations, Canada. 4. Daniel, I. M., and Ishai, O. (1994). Engineering mehanis of omposite materials, Oxford Univ. Press, New York. 5. Davol, A. (1998) Strutural haraterization of onrete filled fiber reinfored shells. PhD thesis, University of California, San Diego, USA. 6. Fam, A. Z. (2000) Conrete-filled fiber reinfored polymer tubes for axial and flexural strutural members. PhD thesis, The Univ. of Manitoba, Winnipeg, Canada. 7. Fam, A. Z., and Rizkalla, S. H. (2001) Confinement model for axially loaded onrete onfined by irular fiber-reinfored polymer tubes. ACI Strutural Journal, V. 98, No. 4, July-August, pp. 451-461. 24
8. Fam, A. Z., and Rizkalla, S. H. (2002) Flexural behavior of onrete-filled fiber reinfored irular polymer tubes. Journal of Composites for Constrution, ASCE, V.6, No.2, May, pp. 123-132. 9. Fam, A., Pando, M., Filz, G., and Rizkalla, S.(2003a) Preast piles for Route 40 bridge in Virginia using onrete filled FRP tubes. PCI Journal, V48, No. 3, May-June, PP. 32-45. 10. Fam, A., Flisak, B., and Rizkalla, S. (2003b) Experimental and analytial modeling of onrete-filled fiber-reinfored polymer tubes subjeted to ombined bending and axial loads. ACI Struural Journal, V.100, No. 4, July-August, pp.1-11. 11. Fam, A., Shnerh, D., and Rizkalla, S. (2003) Retangular filament-wound GFRP tubes filled with onrete under flexural and axial loading: Experimental Investigation. Submitted to ASCE, Journal of Strutural Engineering, September. 12. Hognestad, E. (1951) A study of ombined bending and axial load in reinfored onrete members. University of Illinois Engineering Experimental Station, Bulletin No. 399, June, 128 pp. 13. Mirmiran, A., Shahawy, M., Samaan, M., Ehary, H. E., Mastrapa, J. C., and Pio, O. (1998) Effet of olumn parameter on FRP-onfined onrete. Journal of Composites for Constrution, ASCE, V.2, No.4, Nov., pp.175-185. 14. Mirmiran, A., Shahawy, M., and Samaan, M. (1999) Strength and dutility of hybrid FRPonrete beam-olumns. Journal of Strutural Engineering, ASCE, V.125, No.10, Marh, pp.1085-1093. 15. Popovis, S. (1973) A numerial approah to the omplete stress-strain urve of onrete. Cement and Conrete Researh, V. 3, No.5, pp.583-599. 25
16. Rihart, F.E., and Brown, R. L. (1934) An investigation of reinfored onrete olumns. University of Illinois Engineering Experimental Station, Bulletin No. 267, June, 128 pp. 17. Rohette, P. (1996) Confinement of short square and retangular olumns with omposite materials. MS thesis, The Univ. of Sherbrooke, Quebe, Canada. 18. Saafi, M., Toutanji H. A., and Li Z.(1999) Behavior of onrete olumns onfined with fiber reinfored polymer tubes. ACI Material Journal, V.96, No.4, July-Aug., pp.500-509. 19. Samaan, M., Mirmiran, A., and Shahawy, M.(1998) Model of onrete onfined by fiber omposites. Journal of Strutural Engineering, ASCE, V.124, No.8, Sept., pp.1025-1031. 20. Seible, F. (1996) Advaned omposite materials for bridges in the 21 st entury. Pro., 1st Int. Conf. on Composites in Infrastruture (ICCI 96), pp.17 30. 21. Spoelstra, M. R., and Monti, G. (1999) FRP-onfined onrete model. Journal of Composites for Constrution, ASCE, V.3, No.3, August, pp.143-150. 26
LIST OF FIGURES Fig. 1 Summary of the analytial model Fig. 2 Stress-strain behavior of the FRP tube Fig. 3 Moment-urvature response based on CLT-ULF and oupon test results Fig. 4 Load-defletion behavior of CFRFTs Fig. 5 Interation diagrams of CFRFT and RC setions Fig. 6 Axial load-strain behavior of olumns Fig. 7 Moment-urvature response of CFRFT beam B1 versus onventional RC beams Fig. 8 Moment-urvature response of hybrid CFRFT ompared to all-gfrp CFRFT Fig. 9 Effet of the thikness of onrete flange on moment apaity of CFRFT Fig. 10 Variation of the flexural strength with the reinforement ratio of CFRFT Fig. 11 Variation of the flexural strength with perentage of fibers in the axial diretion Fig. 12 Variation of moment apaity with reinforement ratio for CFRFT and RC beams Fig. 13 Interation urves of CFRFTs with different wall thikness and laminate struture 27
2 y 1 θ x (a) Unidiretional lamina Stress E o h n hn-1 h k-1 hk n k 3 2 1 z z k h x (b) Multidiretional laminate E se N f f Comp. 1 e 2 (M, N) max ε ε r 1 (M, N ) Ten. f f r ε ε ε u Strain M () Stress-strain urve of onrete (e) Interation diagram D hi t w B General layer i h(i) N.A. ε t ε ε(i) ψ CF f CF(i) f fft CC(i) f fwt (i) f (i) M N f fwb (i) TF(i) f t (i) TC(i) t f ε b Strains f ffb TF f FRP Stresses Conrete Stresses (d) Craked setion analysis Fig. 1 Summary of the analytial model
350 300 250 Failure of [±45] layers Failure of [0] layers Flange 350 300 250 Flange Flange Stress (MPa) 200 150 100 50 Failure of [90] layers Predited Experimental Web Stress (MPa) 200 150 100 50 Web Web Predited Experimental 0 0 5 10 15 20 25 30 35 40 Strain x (10-3 ) 0 0 5 10 15 20 25 30 35 40 Strain x (10-3 ) (a) Behavior under axial tension (b) Behavior under axial ompression Fig. 2 Stress-strain behavior of FRP tube
Moment (kn.m) 300 250 200 150 100 50 0 No T.S. T.S. 0 20 40 60 80 100 120 140 B1 B3 Predition (CLT-ULF) Predition (Coupon Tests) Experimental Moment (kn.m) 300 250 200 150 100 50 0 B2 Predition (CLT-ULF) Predition (Coupon Tests) Experimental 0 20 40 60 80 100 120 140 Curvature (1/m) x (10-3 ) Curvature (1/m) x (10-3 ) (a) Totally filled tubes (b) Partially filled tube Fig. 3 Moment-urvature response based on CLT-ULF and Coupon test results
700 600 B1 Load (kn) 500 400 300 200 100 B2 Predited Experimental B3 0 0 10 20 30 40 50 60 70 Defletion (mm) Fig. 4 Load-defletion behavior of CFRFTs
Axial Load (kn) 4000 3000 2000 1000 0 Experimental f = 52.6 MPa 152 x 254 setion 0.85 P o I 1 I 2 ω s =1.45ω f ω s =ω f 0 25 50 75 100 125 150 I o I 3 ω s =0.8ω f Moment (kn.m) As/2 RC Setions I 1, I 2, I 3 CFRFT I o Fig. 5 Interation diagrams of CFRFT and RC setions
Axial Load (kn) 2000 1600 1200 800 C2 Predited 0.85 P o C1 P Total = P f +P 400 0 GFRP (P f ) Conrete (P ) 0 2 4 6 8 10 12 14 16 18 20 Axial strain (x10-3 ) Fig. 6 Axial load-strain behaviour of olumns
Moment (kn.m) 400 300 200 100 ρ s =5.6 % RC1 #5 #7 #8 ρ s =3.7 % RC4 #5 #7 B1 #9 ρ f =9.6 % ρ s =9.6 % RC2 #6 #10 ρ s =0.8 % RC3 #4 0 0 20 40 60 80 100 120 Curvature (1/m) x (10-3 ) Fig. 7 Moment-urvature response of CFRFT beam B1 versus onventional RC beams
500 400 Modefied B1, CFRP [0] layers in flanges (Theoretial) Moment (kn.m) 300 200 100 0 0 20 40 60 80 100 120 140 Curvature (1/m) x (10-3 ) All-GFRP B1 (Experimental) Modefied B3, CFRP [0] layers in flanges (Theoretial) All-GFRP B3 (Experimental) Fig. 8 Moment-urvature response of hybrid CFRFT ompared to All-GFRP CFRFT
250 200 h t M (knm) 150 D = 374 100 265 50 0 0.2 0.4 0.6 0.8 1 h t / D Fig. 9 Effet of the thikness of onrete flange on moment apaity of CFRFT
M u f u / ( f ' bh 2 ) MPa 1000 800 600 400 200 Tension failure Compression failure f = 50 MPa 152 x 254 beam Lous of balaned failure 9:1 3:1 1:1 1:3 0 0 10 20 30 40 Reinforement ratio ρ (%) 1:9 Fig. 10 Variation of the flexural strength with the reinforement ratio of CFRFT
M u f u / ( f bh 2 ) MPa 1000 800 600 400 200 1:9 f = 50 MPa 152 x 254 beam 1:1 9:1 16mm 8 mm 4 mm 2 mm 1 mm 0 0 25 50 75 100 % age of axial fibers [0] Fig. 11 Variation of the flexural strength with perentage of fibers in the axial diretion
Normalized moment M / ( f bh 2 ) 0.4 0.3 0.2 0.1 0 f = 50 MPa 152 x 254 beam A s /2 (1:1.5) laminate 0 5 10 15 20 Reinforement ratio ρ (%) A s Fig. 12 Variation of moment apaity with reinforement ratio for CFRFT and RC beams
N/ ( f bh ) 3 2.5 2 1.5 1 t = 2 mm 1:1 1:1 9:1 f f = = 50 50 MPa MPa 152 152 x x 254 254 setion setion t = 16 mm 0.5 0 1:9 1:9 9:1 0 0.2 0. 0 0.2 0.4 0.6 0.8 1 M / ( f bh 2 ) Fig 13. Interation urves of CFRFTs with different wall thikness and laminate struture