AXIALLY LOADED FRP CONFINED REINFORCED CONCRETE CROSS-SECTIONS

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1 AXIALLY LOADED FRP CONFINED REINFORCED CONCRETE CROSS-SECTIONS Bernát Csuka Budapest University o Technology and Economics Department o Mechanics Materials and Structures Supervisor: László P. Kollár

2 1. INTRODUCTION Axial resistance o concrete and reinorced concrete columns can be signiicantly increased by using lateral coninement. Frequently used solutions are steel helices, jackets or tubes. In the last 2 years instead o steel jackets the use o FRP (iber reinorced polimer) as coninement has increased due to its high corrosion resistance, high ultimate stress and because it is easy to use or repair and/or reinorcement o damaged columns. FRP coninement can be applied to any type o cross-sections but most requently circular- and rectangular cross-sections (with rounded edges) are used. Due to the loads internal orces develop in the column, which may result in ailure. The internal orces are: - axial orce (due to concentric axial load Figure 1.1a), - axial orce and bending moment (due to eccentric axial or axial and horizontal load Figure 1.1b) and - shear orce (due to horizontal load Figure 1.1c horizontal load also causes bending moment). In this thesis only the irst two are investigated. (a) (b) (c) F e F P N = F M = N = F M = ef V = P Figure 1: Typical loads on a column: concentric compression (a), eccentric compression (b) and horizontal load (c). Behavior o conined materials: The concrete core o an axially loaded column laterally expands due to the Poisson-eect. The coninement hinders this expansion and hence the concrete is subjected to triaxial compression and its axial resistance increases. The conining stress σ l in case o circular cross-sections can be calculated as ollows (Figure 2): 2σ t σ l =, (1) d where σ is the hoop stress in the coninement; t is the thickness o the coninement and d is the diameter o the cross-section. 2

3 σ l d σ t σ t Figure 2: Conined circular cross-section. The behavior o conined materials has already been investigated or 1. Kármán [3] experimentally investigated rigid materials (marble and sandstone) in triaxial stress-state and ound that with proper coninement plastic or even hardening behavior can be achieved. 1.1 Concentrically loaded circular cross-sections There are several experimental results and models or concentrically loaded FRP conined circular columns. Models are based either on experimental data (design oriented models) or on triaxial concrete material models (analysis oriented models). According to the models the ailure strength o FRP conined column is not aected by the stiness o the coninement. We may observe, however, that or a very sot coninement the concrete might ail beore the development o the conining stresses, and or a very rigid FRP the coninement may ail beore the concrete reaches its plastic state. 1.2 Eccentrically loaded circular cross-sections Relatively ew documented experiments on eccentrically loaded FRP conined circular columns can be ound in the literature. There are two types o models: simpliied models [1,2], where the Bernaulli-Navier hypothesis (plane crosssections remain plane) is combined with an axial stress-strain curve o concrete or concentric loading [4] (which contains the eect o coninement); and numerical 3D models, using commercially available FE programs with built-in triaxial material model or concrete. Existing simpliied models ail to properly predict the behavior o eccentrically loaded conined columns (Figure 3) while in numerical models only qualitative comparisons with experiments were made. Authors also admit [1] that urther research in this area is needed. 3

4 1. N Eurocode Lam and Teng.5 M Figure 3: Comparison o experimental results and simpliied models. 1.3 Concentrically loaded rectangular cross-sections For concentrically loaded FRP conined rectangular columns (with rounded edges) numerous experimental results are available in the literature. There are simpliied models based on design-oriented models or concentrically loaded circular columns using dierent methods to ind an equivalent diameter and eective area; and numerical 3D models similar to those used or eccentrically loaded circular columns. For the numerical models again only quantitative comparisons with experiments were made; while no simpliied model is available in the literature which can predict reliably the experimental data. This can be observed very clearly in Figure 4, where the data is given as a unction o the relative corner radius cc / c 2 r / b r b Mirmiran et al. Al-Salloum Harajli et al. Yousse et al. ( cu / c ) Yan et al. Lam and Teng ACI 44 Experimental: Wang and Wu C5 2 plies set Figure 4: Eect o rounding o the edges on the ratio o conined strength ( cc ) to unconined strength ( c ) or square sections. 4

5 2. PROBLEM STATEMENT Based on the inormation ound in the literature the ollowing questions arise or concentrically loaded FRP conined circular columns: - How does the stiness o FRP coninement aect the behavior o the conined concrete column? - Under what conditions can it be assumed, that the strength o the conined concrete is not aected by the stiness o the coninement? These questions have practical importance as the stiness o FRP may strongly vary and, in addition, in the uture new materials may also be applied as FRP coninement. For the case o eccentrically loaded FRP conined circular columns our aim is to develop a new model (Figure 1b) with the aid o which we wish to predict the experimental data and to explain the behavior o conined columns. For concentrically loaded FRP conined rectangular columns our aim is to develop a model and design expressions or the calculation o ailure loads. 3. METHOD OF SOLUTION As we stated earlier simpliied design-oriented models ail to properly predict the behavior o conined concrete. To ulill our aim we will use an analysisoriented model based on a sophisticated material law [5]. The ollowing steps are perormed: - a inite element model is developed; - the model is veriied with available experimental data; - several numerical calculations are perormed by changing the geometrical and material properties o the conined column; - empirical design expressions are developed by itting curves on the results o the FE calculations. 4. MODEL Material laws: The FRP coninement is modeled with the classical laminate plate theory and it is assumed that it behaves in a linearly elastic manner until ailure. For the steel reinorcement i present a simple elastic-plastic stressstrain curve is used. For the concrete a new, quite accurate (coninement-sensitive, non-associated) material law is used proposed by Papanikolaou and Kappos [5]. Typical stress-strain diagrams are shown in Figure 5. In Figure 5c or unconined concrete ater reaching the ailure state the concrete sotens and a decreasing path appears in the stress-strain diagram (solid line). This post-ailure behavior changes due to the FRP coninement (dotted lines) and hence the conined concrete strength ( cc ) is higher than the unconined concrete strength ( c ). (New results: 1.) This material law does not contain the time-dependent behavior o concrete, eects o creep and shrinkage are not investigated in this work. 5

6 σ (a) σ (b) σ (c) conined yd unconined c ε ε ε y Figure 5: Typical stress-strain diagrams or the materials used in the model: FRP coninement (a), steel (b) and concrete (c). ε ε c ε Numerical model: A 2D inite element model was developed or the calculation o the cross-section (Figure 6). Note that the inite element mesh is twodimensional, however the strains and stresses are three-dimensional: it is assumed that the axial strain varies linearly through the cross-section (and the nonlinearly varying axial stress is calculated by the FE code). concrete inite element (a) z FRP boundary element stirrup boundary element (b) z FRP boundary element concrete inite element y y rebar elements Figure 6: The inite element mesh o a circular cross-section with reinorcement (a) and rectangular cross-section (b). Due to the high nonlinearity o the concrete material law a strain-controlled incremental calculation was implemented, the block diagram is shown in Figure 7. (New results: 4., 5.) Input data materials, geometry Creating FEM mesh Increase strains yield criterion reached in some points Plastic strains added in points reaching yield criterion no FRP ailure? 6 FEM calculation materials in elastic state FEM calculation with considered plastic strains Figure 7: Block diagram o the calculation. yes Output data

7 5. VERIFICATION 5.1 Concentrically loaded circular cross-sections The axial concrete strengths at ailure were calculated numerically or all the experimental cases (154 cases) and the average absolute error was ound to be 9.95%. A ew experimental stress-strain diagrams were compared with the calculations recommended by dierent authors and also with our model (Figure 8). It can be observed that the proposed model can ollow the shape o the diagram, however the accuracy is poor because o the lack o material properties needed or the calibration o the concrete behavior. Axial stress Lateral strain Experimental data Saenz Popovics Lam and Teng Yousse Proposed Axial strain Figure 8: Comparison o the experimental results and the calculated σ(ε) diagrams. 5.2 Eccentrically loaded circular cross-sections The comparison o experimental results and numerical calculations is shown in Figure 9. N 6 Eurocode 4 Lam and Teng 2 Proposed model M Figure 9: Comparison o experimental results ound in the literature and models. 7

8 Our experiments: Fiteen circular column specimens were prepared and tested in our experimental program running in 26 at the Budapest University o Technology and Economics. Six columns were conined with carbon ibers conined polymer (CFRP) and six were conined using glass ibers (GFRP). The remaining columns were used as control specimens. The instrumentation or a CFRP conined specimen can be seen in Figure 1. (a) (b) Figure 1: Instrumentation o specimen 11 (a) and the specimen curved due to the load (b). Comparison o the load paths and the calculated capacity diagrams or GFRP conined columns are shown in Figure 11. For the calculation o the diagrams the measured maximal hoop strain was used instead o the rupture strain o the conining FRP. Ater the experiments the columns were cut and no delamination between the concrete core and the conining FRP was ound. Unortunately, the experimental results o CFRP conined columns showed a very bad correlation with the calculated values. The reason can be the extremely poor quality o the manuactured concrete columns N Column 2 N Column 3 Calculated capacity Load path (experiment) 1 M Figure 11: Comparison o the calculated load paths and the calculated capacity diagrams or GFRP conined columns. 5 Calculated capacity Load path ( experiment) M

9 5.3 Concentrically loaded rectangular cross-sections For veriications irst the experimental data o Wang and Wu [6] were compared with our numerical calculations (Figure 12). Our experimental results are quite accurate, especially i the case o the unconined circular cross-section was used or the determination o the concrete material properties (identiied as calibrated results). We emphasize that our model is the only one (see Figure 4) which can predict the concave shape o the experimental data (New results: 5.1.). We also calculated without any calibration the cc values or all available experiments, the average absolute error is 16.33%. 12 cc Experimental results Proposed numerical Proposed numerical (calibrated) 2 r / b 6. ACHIEVEMENTS Figure 12: Comparison o experimental results with our models. 6.1 Concentrically loaded circular cross-sections With the aid o the developed model we investigated the eect o the stiness o the coninement. An example can be seen in Figure 13 or a C3 concrete with unidirectional coninement. Each solid line belongs to a given stiness ratio (ρ s ) deined as (New results: 1.): 2E t ρ s =, (2) de where E c is the elastic modulus o concrete. (New results: 1.) We may observe (Figure 13) that or higher stiness the stress-strain curve is monotonic (curve c), while or lower stiness the diagram has one local maximum point and one local minimum point (curve d). The rupture o coninement (and hence the concrete strength at ailure) depends on the coninement ratio (ρ c ) [4]. c c ρ = /, (3) l c 9

10 In Figure 13 it can be seen that the stiness aects the strength o the conined concrete (New results: 1.). Eight stress-strain diagrams are shown, which belong to dierent stinesses, however to the same strength (coninement ratio). The dots show the end o the stress-strain curves. When the stiness is very high the FRP ruptures beore the concrete can reach the ailure state (curve a, E = 765 kn/mm 2 ). This is deined as overstiened concrete, and must be avoided (New results: 1.2.). σ [N/mm ] c 75 (a) E = 765 kn/mm2 (b) E = 153 cc,max (c) E = 51 cc,min 5 c 25 2 E = 76 E = 38 E = 25 E ε u = 382,91 N/mm Constant! E = 19 E = (d) 2 Low-stiness coninement High stiness coninement Overconined concrete Figure 13: Eect o the stiness o coninement on the stress-strain diagram o C3 concrete. An optimal stiness (ρ s,opt ) or the coninement can be ound where the ailure strength is maximum (curve b in Figure 13). This is also the limit between the overstiened concrete and the high-stiness coninement. The ollowing curve was itted on the numerically calculated results (New results: 2.1.): ρ s,opt =.1+.22ρ 2.2 c c.3 ε c. (4) I the stiness is high (but lower than ρ s,opt ) higher ailure strength can be achieved than or lower stiness. The limit between high-stiness and lowstiness coninement was investigated numerically. The data can be approximated by the ollowing expression (New results: 2.1.): ρ s,limit = c , i c (5) c , i c >

11 A simpliied model is introduced where or low-stiness coninement the ollowing expression is derived rom the model (New results: 2.3.): cc = max cc,min c, (6) or high-stiness coninement the ollowing approximation can be used (New results: 2.3.): = + ( ) s s,limit cc cc,min cc,max cc,min ρ ρ ρ s,opt ρ s,limit, (7) where the limit values derived rom the concrete material law are (New results: 2.2.): = , (8a) cc,min l l c 2 cc,max l 1.16 l c c = + +. (8b) The simpliied method described above was compared with the experimental results ound in the literature and 11.72% average absolute error was ound. 6.2 Eccentrically loaded circular cross-sections A typical calculated capacity diagram is shown in Figure 14 where the average axial stresses at ailure in certain points are also indicated. The average axial stress decreases rapidly as we move away rom the most compressed part o the crosssection, much aster than it is predicted by the models [1,2] based on the (linear) stress-strain diagram o Lam and Teng [4] recommended or concentrically loaded circular columns (New results: 4.1.). N M Stresses at ailure state: Figure 14: Average stress curves at ailure. 3 z d σ 11

12 Based on the average axial stresses simpliied stress-strain diagrams (P1 or P2) are recommended as shown in Figure 15 (New results: 4.2.). σ c cc Eurocode 2 cc,ec Lam and Teng P2 c P1 ε c ε t ε cu ε c,p1 ε c2,c ε cc ε cc,ec Figure 15: Stress-strain diagrams or simpliied design. We have calculated the capacity curves o experimental data available in the literature, and it was ound that the Eurocode 2 [2] and the Lam and Teng [4] curves overestimate the ailure load or eccentric loading, while both P1 and P2 seems reasonable (Figure 16) N Eurocode 2 Lam and Teng ε c P1 P2 4 Proposed model κ ε =.49 M Figure 16: Capacity diagrams or simpliied design (Bisby and Ranger ). 12

13 6.3 Concentrically loaded rectangular cross-sections According to the FE model two typical stress results are shown in Figure 16. The axial stresses are highest at the rounded edges and lowest (even zero) at the sides o the cross-section (New results: 5.1.). The stresses within the cross-section are more complex than it is indicated by the simple division o the section into a conined and an unconined area (as recommended by all simpliied models). Thereore we decided to use the cc calculated or a circular conined column (with d = b, denoted by cc ) as a basis o the simpliied method. Only square cross sections (b = h) with rounded edges are considered where the value o cc can be calculated using Equations (6 and 7) by assuming that d = b. Axial orce Sections: σ σ σ 2. σ Axial strain 2. Figure 2: Axial stresses in rectangular cross-sections ( c is indicated by dashed lines). The average axial strength ( cc ) o square sections with rounded edges are approximated as ollows: For low-stiness coninement (ρ s ρ s,limit ): 2r 7.23 ( ρ ρ ) l cc = ccο s,limit sο b l c 2 c , (9) (According to the results o the numerical calculations with high relative corner radius (1 > 2r / b >.9) a sudden drop appears in the value o cc. Due to this phenomenon the above presented equation underestimates the value o cc in case o circular cross-sections.) For high-stiness coninement (ρ s > ρ s,limit ): 13

14 14 ρs,limit+.161 ρsο (1) b l c 2 2r 7.23 cc = ccο (Note that the lower limit or both expressions is c.) The simpliied expression was veriied against the experimental results or square sections with rounded edges ound in the literature % average absolute error was ound (New results: 6.). 7. NEW RESULTS 1. A new model is developed based on a coninement-sensitive plasticity constitutive concrete material law or the calculation o concentrically loaded FRP conined circular cross-sections. (Models ound in the literature are built on less accurate concrete material laws.) The model was veriied by comparing the results o the new numerical method with the experimental data ound in the literature [7]. Based on the model the ollowing statements can be made: 1.1. Contrary to the statements ound in the literature the axial strength o the conined column depends on the stiness o the conining FRP but this eect is negligible in case o low-stiness coninement [7] It was ound that overstiening o the coninement is possible which leads to signiicantly lower axial strength than in case o low- or highstiness coninement [7] It was ound that an optimal stiness or the coninement can be deined which (or a given coninement ratio) leads to the maximal axial strength [7]. 2. New expressions are recommended or concentrically loaded FRP conined circular concrete cross-sections [7]: 2.1. Based on the results o numerical calculations simpliied expressions are recommended or the calculation o low-, high-stiness and optimal coninement and or the calculation o overstiening [7] Based on the model new analytical expressions were derived to determine the strength o conined concrete in case o low-stiness and optimal coninement (average error < 15%) [7] Based on the model simpliied expressions are recommended or the calculation o axial strength in case o high-stiness coninement (average error < 15%) [9].

15 3. By investigating the material properties o commonly used materials, it was ound that i glass ibers are used the coninement is low-stiness; while with carbon ibers high-stiness coninement can be achieved [7]. 4. A new model is presented based on the Bernoulli-Navier hypothesis or the calculation o eccentrically loaded FRP conined circular cross-sections. (Models based on concrete material laws or triaxial compression that give qualitative results can not be ound in the literature.) The model was veriied by comparing the results o the new numerical method with the experimental data ound in the literature [8,9]. Based on the model the ollowing statements can be made: 4.1. It was ound that the axial resistance in case o high eccentricities can be signiicantly lower than predicted by the models ound in the literature [8]. (Experimental results also veriy this statement.) 4.2. New stress-strain curves were recommended or a simple designoriented model or the calculation o the capacity diagram o eccentrically loaded FRP conined circular cross-sections [8]. 5. The model presented above was applied or the calculation o concentrically loaded FRP conined rectangular cross-sections. (Models ound in literature are less sophisticated or built on less accurate concrete material laws.) The model was veriied by comparing the results o the new numerical method with the experimental data ound in the literature [1]. Based on the model the ollowing statements can be made: 5.1. The exact axial stress-distribution in the cross-section was calculated [1]. (Only approximate methods were ound in the literature.) 5.2. The eect o the rounding o the edges on the axial strength was calculated [1]. (Models ound in the literature did not give reliable results.) 5.3. A new, simpliied expression was recommended or the calculation o concentrically loaded FRP conined rectangular cross-sections. The accuracy o the new model is acceptable (<15%) [1]. 15

16 REFERENCES Reerred articles [1] Bisby, L. and Ranger, M., Axial-lexural interaction in circular FRP-conined reinorced concrete columns, Construction and Building Materials, 24, (21). [2] Eurocode 2: Design o Concrete Structures Part 1-1: General rules and rules or buildings, ENV , (23). [3] Kármán, T., Mitıl ügg az anyag igénybevétele?, Magyar Mérnök- és Építész Egylet Közlönye, 44(1), (191). [4] Lam, L. and Teng, J. G., Design-oriented stress-strain model or FRP-conined concrete, Construction and Building Materials, 17, (23). [5] Papanikolaou, V. K. and Kappos, A. J., Coninement-sensitive plasticity constitutive model or concrete in triaxial compression, International Journal o Solids and Structures, 44, (27). [6] Wang, L-M. and Wu, Y-F., Eect o corner radius on the perormance o CFRP-conined square concrete columns: Test, Engineering Structures, 3, (28). Publications [7] Csuka, B. and Kollár, L. P., FRP conined circular columns subjected to concentric loading, Journal o Reinorced Plastics and Composites, 29(23), (211). [8] Csuka, B. and Kollár, L. P., FRP conined circular columns subjected to eccentric loading, Journal o Reinorced Plastics and Composites, (article in press 211). [9] Csuka, B. and Kollár, L.P., Analysis o FRP conined columns under eccentric loading, Composite Structures, (accepted or publication, 211). [1] Csuka, B. and Kollár, L. P., FRP conined rectangular columns subjected to concentric loading, (Manuscript to be published at Journal o Reinorced Plastics and Composites). 16