NUMERICAL AND ANALYTICAL MODELING OF CONCRETE CONFINED WITH FRP WRAPS.

Transcription

1 The Pennsylvania State University The Graduate School Civil & Environmental Engineering NUMERICAL AND ANALYTICAL MODELING OF CONCRETE CONFINED WITH FRP WRAPS. A Thesis in Civil Engineering by Omkar Pravin Tipnis 2015 Omkar Pravin Tipnis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2015

2 The thesis of Omkar P. Tipnis will be reviewed and approved* by the following: Maria Lopez De Murphy Associate Professor of Civil Engineering Thesis Adviser Ali Memari Professor of Civil Engineering Gordon P. Warn Associate Professor of Civil Engineering Peggy Johnson Professor of Civil Engineering Department head, Civil Engineering ii

3 Abstract This thesis is intended at studying and comparing empirical models that have been proposed for the modeling of the stress-strain response of a FRP confined concrete subjected to axial load. An attempt has been made to model the experimental set up for the compression test of a concrete cylinder confined with FRP sheet in AbaqusCAE. The results so obtained have been compared and analyzed against the experimental test results & the results obtained from a chosen mathematical model (Modified Lam & Teng). An attempt was made to create a new material model in Opensees that follows the chosen mathematical model. However, this was not achieved due to the reasons that will be explained in the later sections. Reinforced concrete confined with steel is typically designed by considering the Manders model (Mander et al., 1988), which assumes a constant confining pressure. This is true with the case of steel as it is a ductile material and one assumes the steel to be yielded. However with the case of FRP jackets, this is not true. FRP is a linear elastic and brittle material and does not yield, which makes the Manders model inaccurate for its analysis. Many models have been proposed which take into account the increasing confining pressure due to the FRP wrap. A comparative study of the constitutive models proposed for FRP confined reinforced concrete has been done in this study. Finally after a series of numerical interpretations of different specimens and their comparison with the experimental data, the utility and accuracy of the new modified Lam & Teng s model was validated. The validation process included comparison and analytical data obtained via finite element simulation in Abaqus, empirical model results and the experimental data. iii

4 Contents List of Figures Chapter 1 Introduction Introduction Scope of the study 2 Chapter 2 Literature Review Introduction Mechanism for Concrete Confinement by Transverse Reinforcement Modeling of Concrete in Compression Modified Hognestad Model: Kent and Park model Stress-Strain Response of FRP-Confined Concrete First Zone Transition Point Second Zone Failure Mode Post Failure Proposed models Samaan and Mirmiran Model (1998) Mander s Model (1984) Lam & Teng Model (2003) Modified Lam & Teng (Liu et al., 2013) Drucker-Prager Plasticity Model Conclusions 34 Chapter 3 Experimental Database & Preliminary results Introduction Preliminary study Test Database Abaqus modeling Opensees Modeling Results and Discussion 41 v

5 Chapter 4 AbaqusCAE Finite Element Modelling Introduction Concrete Elastic Properties Plastic Properties Fiber Reinforced Polymeric Jacket Abaqus model: Assembly Boundary Conditions & Analysis Step Interaction Meshing Conclusions 50 Chapter 5 Comparison Study and Analysis of Results Introduction Performance of the Drucker-Prager Model Performance of the Modified Lam & Teng model Regression Analysis Field Retrofitting cases 66 Chapter 6 Conclusions Summary Conclusions 69 References 71 Appendix A : Numerical and Analytical modeling results (Tabular) 76 Appendix B : Numerical and Analytical modeling results (Graphical) 79 Appendix C : Graphical comparison with Confinement ratios 106 Appendix D-1 : Bridge Retrofitted Data 107 Appendix D-2 : CalTrans retrofitting guidelines 109 iv

6 List of Figures Figure 2-1 : Confinement by Square hoops and Circular Spirals. 4 Figure 2-2 : Confinement of concrete by spiral reinforcement. (Park & Paulay,1875) 5 Figure 2-3 : The modified Hognestad model for compressive stress-strain curve for concrete (MacGregor, 2012) 6 Figure 2-4 : Proposed stress-strain model for confined & unconfined concrete Kent & Park (1971) model 8 Figure 2-5 : Typical Stress-Strain behavior of FRP confined concrete (Saafi & Toutanji, 1999) 9 Figure 2-6 : Failure modes of circular FRP-confined concrete cylinders (Micellli & Modarelli, 2012) 11 Figure 2-7 : Failure modes of sqauare FRP-confined concrete cylinders (Micellli & Modarelli, 2012) 11 Figure 2-8 : Tensile Coupon testing of CFRP, GFRP & Steel (Benziad et al (2010)) 12 Figure 2-9 : : Karabinis et al (2002). A2 & B2 are unconfined control specimens. 13 Figure 2-10 : Failure of concrete core ( Li et al (2006)) 14 Figure 2-11 : Parameters of Bilinear Confinement Model (Samaan & Mirmiran, 1998) 16 Figure 2-12 : Arching effect in Concrete. (Mander et al (1989)) 19 Figure 2-13 : Possible arching effect in non-uniformly confined concrete by FRP (Saadatmanesh et al (1994)) 20 Figure 2-14 : Free body diagram of FRP confinement (DeLorenzis et al (2003)) 21 Figure 2-15 : Lam & Teng s stress-strain model for FRP-confined concrete. (Lam & Teng, 2003) 24 Figure 2-16 : Envelope curve and hysterical rule of proposed FRP confined concrete. (Lui et al, 2013) 26 Figure 2-17 : Friction model (J-F. Jiang et al (2012)) 31 Figure 2-18 : Existing models for plastic dilation rate. (J-F. Jiang et al (2012)) 32 Figure 2-19 : Typical Dilation curve (J-F. Jiang et al (2012)) 33 Figure 4-1 : Creating a Concrete cylinder 3D extrude Part in Abaqus 46 Figure 4-2 : FRP jacket created in Abaqus 48 Figure 5-1 : Typical curves for confinement ratio Figure 5-2 : Typical curves for confinement ratio Figure 5-3 : Typical curves for confinement ratio Figure 5-4 : Typical curves for confinement ratio Figure 5-5 : Typical curves for confinement ratio Figure 5-6 : Typical curves for confinement ratio v

7 Figure 5-7 : Typical curves for confinement ratio Figure 5-8 : Typical curves for confinement ratio Figure 5-9 : Comparitive stress-strain curve for ID no Figure 5-10 : Comparative Stress-Strain curve for Specimen ID29 (Confinement Ratio = 3.9) 57 Figure 5-11 : Comparative Stress-Strain curve for Specimen ID29 (Confinement Ratio = ) 58 Figure 5-12 : Ultimate Strength comparison (Modified Lam & Teng vs Experimental) 59 Figure 5-13 : Ultimate Strain comparison (Modified Lam & Teng vs Experimental) 60 Figure 5-14 : Summary Report for Multivariable regression analysis 63 Figure 5-15 : Final Model performance report 64 Figure 5-16 : Ultimate Strain Comparison (New proposed equation vs Experimental) 65 Figure 5-17 : Histogram for retrofitted bridge columns based on confinement ratios. 67 Figure App-C-1 : Plot showing the variation of the ratio of the compressive strengths predicted to the experimental compressive strengths with respect to the confinement ratio 106 vi

8 Chapter 1 Introduction 1.1 Introduction Today, many reinforced concrete structures are in a bad condition. According to the ASCE report card 2013 for America s infrastructure, one in nine of the bridges in the United States is structurally deficient. (2013 Report card for America s Infrastructure,ASCE) The report also mentions that the average age of the bridges in the country is 42 years. Most of them need some rehabilitation and repair work to either restore them to their full capacity or to increase their design capacity in order to meet their growing demand. Causes of deterioration can range from corrosive environmental conditions, damage due to natural cause such as earthquakes & tornadoes or by human factors such as traffic accidents, use of substandard quality of construction material, faulty construction practices or increase in the load demand for the structure. Indication of a deteriorated reinforced concrete column is the spalling action of the concrete cover leading to exposure of the steel reinforcement in the column which leads to corrosion of the steel, eventually leading to reduced performance of that structure element. With respect to deteriorated reinforced concrete columns, one could conclude that the causes stated above result in deterioration because of lack of lateral confinement. The longitudinal reinforcement in the reinforced concrete columns provide very little lateral confinement effect, which is not adequate for most loading conditions. As a structural designer one always tries to design the reinforced concrete structures in a manner so that they exhibit ductile behavior. Lateral confinement in a reinforced concrete column provides the column with the required ductility. Under seismic loading, this additional confinement could ensure adequate strength for the column and increase its deformation capacity which improves its performance in an event like an earthquake. (Park et al., 1982; Mander et al., 1988; Shams & Saadeghvaziri, 1997) 1

9 Many confinement techniques have been developed over the years; designing the columns with steel hoops (stirrups) or by providing steel jacketing techniques. The steel jacketing technique has been proved quite useful in the field of retrofitting the columns. However, corrosion of the steel can be of concern. It also increases the self-weight of the structure to a great extent which is always a tradeoff. In situations where the concrete cover is very loose and weak one cannot use the steel jacketing techniques as it might damage the column even more due to the bolting of the jackets. During recent decades, many researchers have been trying to replace the conventional steel jacketing technique by usage of fiber reinforced polymer (FRP) wraps. FRP wraps used as confinement can increase the ultimate compressive strength and the ultimate strain of the concrete. (Samaan et al.,1998; Toutanji, 1999). A lot of research has been carried out on developing a retrofitting technique with these FRP wraps. The main advantages these FRP wraps possess over the steel jackets are very high strength to weight ratio & high resistivity to corrosion. 1.2 Scope of the study The objective is achieved and restricted within the following scope of study: 1) Literature review to identify and choose the most relevant models for modeling of concrete confined with fiber reinforced polymers in compression. 2) Modelling and finite element analysis of the confined concrete compression test in AbaqusCAE. 3) Developing the stress-strain curve from several proposed empirical model (Modified Lam & Teng). 4) Survey of experimental data on confined concrete with FRP in order to generate an experimental database. 5) Comparison of the analytical results in order to define the strengths and limitations of the empirical model chosen to study. 2

10 6) Validation of the model chosen and a study on its relevance for use in typical bridge columns retrofitted with FRP jackets. 7) Proposing & validating changes to the empirical model for more accurate results. 3

11 Chapter 2 Literature Review 2.1 Introduction A Literature review was conducted to determine the appropriate confinement model for concrete in compression with FRP as confinement reinforcement. In order to understand the mechanism of confinement by FRP the models of the typical steel confinement were reviewed and analyzed first. This review of steel confinement models facilitated the study of FRP confined concrete models. 2.2 Mechanism for Concrete Confinement by Transverse Reinforcement Steel spirals or hoops are quite commonly used as transverse reinforcement in concrete compression members. Research as demonstrated that circular hoops are more effective in providing confinement as compared to square hoops. Due to their shape circular hoops are able to provide continuous confining pressure around the circumference of the compression members. The square hoops have a tendency to bend the sides outwards due to the pressure of the concrete against the sides. This is more effectively demonstrated in the figure shown below. Figure 2-1 : Confinement by Square hoops and Circular Spirals. 4

12 Richard, Brandtzaeg and Brown (1928) proposed the relationship for calculating the strength of concrete cylinders loaded axially to failure subject to confining fluid pressure. f cc = f c + 4.1f l Eqn 2-1 where, f cc = axial compressive strength of confined specimen f c = uniaxial compressive strength of unconfined specimen f l = lateral confining pressure When the confining pressure is due to the steel circular spirals the free body diagram of half the spiral is as shown in figure 2-2. f cc = f c + 8.2A sp f l / d s.s Eqn 2-2 Figure 2-2 : Confinement of concrete by spiral reinforcement. (Park & Paulay,1875) 5

13 2.3 Modeling of Concrete in Compression Many models have been proposed to capture the non-linear behavior of concrete in compression by transverse reinforcement. For the scope of this study, the modified Hognestad model and the Kent & Park model were studied. Both the models are quite capable of capturing the behavior of the confined and well as unconfined concrete under compression Modified Hognestad Model: This model was studied from MacGregor (2012). This model is capable of presenting the behavior of concrete in compression (confined and unconfined). Figure 2-3 : The modified Hognestad model for compressive stress-strain curve for concrete (MacGregor, 2012) 6

14 The modulus of elasticity for concrete Ec may be calculated as follows E c = w (f c ) psi Eqn 2-3 Where w = density of concrete in pounds per cubic foot, f c is the compressive strength in psi. f c is the maximum stress reached in the concrete. The extent of falling branch behavior depends on the limit of useful concrete strain assumed. The slope of the line is affected by the amount of confinement, which terminates at a strain of Kent and Park model Kent and Park (1971) proposed a stress-strain equation that can present both unconfined and confined concrete behavior under compressive loading. In this model they generalized Hognestad model (1951) equation to more completely describe the post-peak behavior. The ascending curve is represented by : (Region : ԑ c 0.002) f c = f c [ 2ԑ c ( ԑ c ) 2 ] Eqn 2-4 This curve is obtained modifying Hognestad second degree parabola by replacing 0.85 f c by f c and ԑ co by The post-peak branch is assumed to be straight line which has a slope that is primarily defined as a function of the strength of concrete. (Region : ԑ c ԑ 20c f c = f c [1-Z(ԑ c - ԑ co )] Eqn where, Z = [ ] Eqn 2-6 ԑ 50u ԑ co

15 ԑ 50u = the strains at 50% of the maximum concrete strength for unconfined concrete. ԑ 50u = f c(in Psi) f c 1000 ԑ 50u = f c (in MPA) 145 f c 1000 Eqn 2-7 Figure 2-4 : Proposed stress-strain model for confined & unconfined concrete Kent & Park (1971) model 8

16 2.4 Stress-Strain Response of FRP-Confined Concrete The behavior of concrete confined by FRP differs with respect to its stress-strain response as compared to normal concrete or steel-stirrup confined concrete. Basically, the stress-strain response of FRP-confined concrete can be studied by dividing it in four parts First Zone Figure 2-5 : Typical Stress-Strain behavior of FRP confined concrete (Saafi & Toutanji, 1999) In this zone the behavior is the same as that of unconfined concrete. The concrete takes up all the axial load and the slope of the curve is the same as the slope of the stress-strain curve for unconfined concrete. One can say during this phase the concrete behaves in a way that the FRP-confinement is not present. One can also conclude by saying the bond between concrete and FRP-confinement is passive and the FRP-confinement jacket is not yet activated. 9

17 The relations between axial stress and lateral strain can be derived from the conventional tri-axial stress-strain state in case of this zone which is the elastic zone. ԑ r (Park Kihoon, 2004) σ z = ԑ re v[1 1 E (1 v)(2te frp )] d Eqn 2-8 Where, σ z = P/A or axial stress, ԑ r = radial strain in concrete, v = Poisson s Ratio of concrete E = Elastic Modulus of concrete, E frp = Elastic Modulus of FRP d = diameter of concrete cylinder (Consistent system of units) Transition Point At the maximum level of unconfinement, this transition point occurs that indicates that the concrete crack has taken place. This point can be termed as the first failure point of the concrete core. At this point the FRP-confinement jacket starts developing its confinement effect Second Zone The second region has the concrete core which has already started to fail. The FRP-confinement jacket is activated and it confines the concrete core. The FRP-confinement jacket applies a continuously increasing pressure on the concrete core until the jacket reaches its first point of failure. The amount of confining pressure that would be exerted by the jacket will depend on the amount of FRO material in it. The concrete is tri-axially stressed and the FRP-confinement jacket is uni-axially stressed Failure Mode FRP material is a very brittle material, which means the failure of this material is accompanied by a large release of energy. Failure usually starts at the middle of the specimen with a sudden or gradual development of the crack towards the end. Ideally the failure point is assumed as that axial strain in the specimen for which the lateral strain in the concrete reaches the strain at the fracture of the FRP confining 10

18 reinforcement. This means that the failure strength of the confined concrete is very closely related to the failure strength of the FRP strengthening material used. (Karabinis & Rousakis, 2002). However, experimental evidence shows that the value of the concrete hoop failure strain very much lower than ultimate failure strain of the FRP material. The predicted reasons for these are: (CEB- FIP) 1) The stress change in the jacket due to confinement pressure has a certain influence on the ultimate strength. 2) Due to inadequate surface preparation the specimens are of poor quality. 3) Size effects Figure 2-6 : Failure modes of circular FRP-confined concrete cylinders (Micellli & Modarelli, 2012) Figure 2-7 : Failure modes of sqauare FRP-confined concrete cylinders (Micellli & Modarelli, 2012) 11

19 One can observe from the figures above at the failure point of the FRP-confined concrete the concrete core has completely failed and the failure of the FRP-confinement starts from the middle Post Failure As mentioned earlier FRP is a very elastic-brittle material. The constitutive property of FRP tensile coupon testing can be represented in terms of the stress-strain curve as shown below. Figure 2-8 : Tensile Coupon testing of CFRP, GFRP & Steel (Benziad et al (2010)) One can infer from the graph shown above that the failure of FRP material is sudden and accompanied by a large amount of energy. The failure point of FRP confined concrete cylinders is defined as the point where the confining material, i.e. FRP fails in tension. Hence, the failure of the FRP confined concrete cylinder is also sudden and is accompanied by a large release of energy. Owing to this the, behavior of the confined cylinders post-failure would be expected to not be able to sustain any more loading pressure. This can be evident from the experimental results as shown in the research by Karabinis et al (2002) 12

20 Figure 2-9 : : Karabinis et al (2002). A2 & B2 are unconfined control specimens. 13

21 For Concrete type A (Unconfined concrete strength = 38.5 MPa) The three sets of specimens for this case (1 layer, 2 layers and 3 layers of CFRP) show that the confined specimen in the post failure regime failure to exhibit any resistance to the loading until as low as 5 MPa. This clearly shows, that yielding of this specimen does not take place. For Concrete type B (Unconfined concrete strength = 35.7 MPa) Similarly in this case, the three sets of specimens for this case (1 layer, 2 layers and 3 layers of CFRP) show that the confined specimen in the post failure regime failure to exhibit any resistance to the loading until as low as 15 MPa. The specimen with three layers shows some yielding behavior. The reason for this was however cited as different points of failure of multiple FRP fiber layers. Experimental data available is tested upto the the failure point in a uniaxial strain controlled compression test. Even though the concrete core is still intact and appear not to be crushed, the load carrying capacity of the same is negligible as one can see from the graphs above. Further more a closer representation of the concrete core would look as shown in the figure below. Figure 2-10 : Failure of concrete core ( Li et al (2006)) From the graphs and the figure shown above one can assume that the concrete looses its load carrying capacity during the loading regime 14

22 2.5 Proposed models In the early days the constitutive models for FRP-confined concrete were the same as those for the steel confined concrete. However, a number of research studies showed that this approach is not accurate and a significant difference exists between the behaviors of both these specimens. As the steel in the jacket of steel confined concrete is assumed to yield, one is safe to assume that the confining pressure exerted by steel jacket on the concrete is constant. (Mander et al., 1988b) However, this is not true in the case of FRP-confined concrete. FRP as a material does not yield and hence, the confining pressure exerted by the FRP jackets is not constant and keeps increasing until its failure point. Various FRP-confined concrete models have been proposed to date. All these models can be classified under two major categories; a) design-oriented models b) analysis-orientated models. Designoriented models are geared toward their use in the engineering design practices whereas analysis-oriented models can capture the detailed mechanical behavior exhibited by the specimen. According to a study (Ozbakkaloglu et al., 2013), it was concluded that the design-oriented models have a better capability to predict the ultimate strength and strain of the specimen. Thus this literature review study will explore design-oriented models. A few of these models were reviewed and studied thoroughly. Each model that was studied is presented in detail and the equations used are also shown. 15

23 2.5.1 Samaan and Mirmiran Model (1998) The Richard & Abott model (1975) was used and calibarated by Samaan & Mirmiran to represent the bilinear response of FRP-confined concrete. f c = (E 1 E 2 )ԑ c (1+( (E 1 E2 )ԑ c ) n 1 n ) fo + E 2 ԑ c Eqn 2-9 Figure 2-11 : Parameters of Bilinear Confinement Model (Samaan & Mirmiran, 1998) The confining pressure is given by f l = 2f lt j d Eqn 2-10 Where, f l = hoop strength of the tube; t j = Tube Thickness & d = core diameter. The strength of confined concrete can be linked to the confining pressure by FRP in the following way: f cu = f c f l 0.7 (Ksi) Eqn 2-11 To evaluate the first slope (E 1 ), this model adopts the formula proposed by Ahmad & Shah (1982) to predict the secant elastic modulus. E 1 = f c (Ksi) Eqn

24 The secant modulus changes at a point where the concrete reaches its unconfined strength. As shown in Fig 2-6, the second slope (E 2 ) is a function of the stiffness of the confining tube. Here, this slope depends more on the properties of the confining tube rather than the properties of unconfined concrete core. E 2 = f c E j t j D (Ksi) Eqn 2-13 Where, E j = effective modulus of elasticity of the tube in the hoop direction. The intercept in this model, f o is a function of the strength of the unconfined concrete & the confining pressure provided by the FRP-tube. This was estimated as: f o = 0.872f c f l (Ksi) Eqn 2-14 The ultimate strain ԑ cu is given as ԑ cu = (f cu f o )/E 2 Eqn 2-15 This model is however not very sensitive to the curve-shape parameter n, and has a constant value of 1.5 which was suggested by Samaan and Mirmiran (1998). This parameter is an important factor in understanding the ductility and the change in the behavior of FRP-confined concrete. 17

25 2.5.1 Mander s Model (1984) Mander et al (1984) proposed a unified stress-strain mathematical model for predicting the behavior of confined concrete. The basis for this model were the equations that were suggested by Popovics (1973). This approach was based on energy balance method. The stress equation suggested by this model was given by: f c = f cc xr r 1+x Eqn 2-16 Where: f cc = compressive strength of confined concrete. x = ԑ c ԑ cc Eqn 2-17 Where ԑ c = longitudinal compressive concrete strain ԑ cc = ԑ co (1 + 5 ( f cc f 1)) Eqn 2-18 co f co & ԑ co are the unconfined concrete strength and the corresponding strain respectively. r = E c E c E (se c) Eqn 2-19 E sec = f cc ԑ cc Eqn

26 Approach to compute the Effective lateral confining pressure The approach to compute the effective lateral confining pressure was based on the effective area which was confined between two steel stirrups. In the case of steel stirrups arching action takes place as shown in the figure. Figure 2-12 : Arching effect in Concrete. (Mander et al (1989)) The effective confined area of concrete reduces as we move towards the mid portion of from one stirrup and is least at the midpoint. Hence the relationship that was proposed to compute the lateral confining pressure reflected the fact that the effective lateral confining pressure exerted on the concrete will be a fraction of the confining pressure generated in the stirrup. Hence, f l = k e f l Eqn 2-21 Where, f l = lateral pressure generated in the transverse reinforcement. f' l = lateral pressure generated in the transverse reinforcement. Further, Mander et al (1989) defined k e as the ratio of the effective confined concrete area (A e ) to the area of concrete present between the center lines of two stirrups (A cc ) A cc = A c (1 ρ cc ) Eqn 2-22 Ρ cc = ratio of area of longitudinal reinforcement to area of core section A c = area of core enclosed between center lines of two stirrups 19

27 The lateral confining pressure would be found by considering half body of the stirrup. The assumption made in calculating the lateral confinement pressure is that the hoop tension is uniform and exerts a uniform pressure on the concrete. Also, the steel is assumed to have yielded which leads to a constant pressure exerted on the concrete. 2f yh A sp = f l sd s Eqn 2-23 f yh = yield strength of steel used as lateral reinforcement. Ρ s can be defined as the ratio of the volume of transverse confining steel to volume of confined concrete. Therefore, f l = 1 2 k eρ s f yh Eqn 2-24 In the case of FRP confined concrete arching does not take place and hence the effective confined area would be the same as the area between two boundaries of confinement. Hence it would be correct to assume that k e would be 1. However this is only true for completely wrapped concrete cylinders. One could compute k e in the same way as mentioned above for concrete column confined with spiral FRP. Figure 2-13 : Possible arching effect in non-uniformly confined concrete by FRP (Saadatmanesh et al (1994)) 20

28 Also the confinement pressure exerted by FRP would vary as the strength developed in FRP would depend on the level of hoop strain in FRP. Hence the equation for lateral confinement pressure would have to be modified Mander s model for FRP confined concrete Saadatmanesh et al (1994) extended the Mander s model to the case of FRP confinement by computing the lateral confinement pressure applied by the FRP jacket. By the Equilibrium of forces: Figure 2-14 : Free body diagram of FRP confinement (DeLorenzis et al (2003)) p = E l ԑ l = E l ԑ f Eqn 2-25 E l = Confinement/lateral modulus E l = 2E fnt D Eqn 2-26 Therefore, p u = 2E fntԑ f D = 2f funt D Eqn 2-27 As mentioned earlier k e would be 1. Hence f l = 1 2 ρ sf frp Eqn

29 Where, f frp = stress in FRP at a particular point of time. This depends on the hoop strain developed as FRP is an elastic brittle material and hence f frp would not be constant. For design purposes or to predict the ultimate confinement one could use the ultimate stress in FRP(f us ) in the above equation. f l = 1 2 ρ sf us Eqn Expression for Compressive strength of Confined concrete Mander s model suggests a nonlinear relationship between confined concrete strength and the confinement pressure applied based on the ultimate surface strength developed by Elwi & Murray (1979). The failure concepts applied in development of this model of concrete under tri-axial state of stress was based on the elastic perfectly plastic behavior of concrete in the compression regime. Also the lateral stress conditions assumed in this constitutive model are uniform. Saadatmanesh et al (1994) used the same model to predict the compressive strength of FRP confined concrete. f cc = f co ( f l f 2f l co f co ) Eqn 2-30 Where, f l is used from the equation specified above. This yields us the results at the ultimate state of FRP confined concrete. Hence, as shown by Imran & Pantazopoulou (1996) & Lan and Guo (1997), the Mander s model can be used to predict the ultimate condition of confined concrete. Further they concluded that this was true because the confined concrete strength was essentially independent of the shape of the loading path. Further they showed that this model can very accurately predict the ultimate condition provided the hoop strain at the failure is very close to the tensile strain of failure of FRP during the coupon testing. Research by Lam & Teng (2003) shows that, the two strains are not close to each other. They are related to each other by an FRP efficiency factor k ԑ which was approximated to a specific value of One could use the Mander s model at every increment of hoop strain in the FRP to compute the stress-strain curve which will basically be an envelope of family of Mander s model curves. 22

30 Research by Mirmiran et al. (1996) shows that the energy balance equation (Popovics (1973)), which considers concrete ductility to be proportional to the stored energy in the confining material cannot be applied in the case of FRP confinement. This was confirmed by Spoelstra et al. (1999) by comparing the results obtained by the Mander s model against the experimental data from the literature available. 23

31 2.5.3 Lam & Teng Model (2003) Lam & Teng proposed a design-oriented model which describes the stress-strain relationship of FRP-confined concrete. This model is only for uniformly confined concrete. The relationship is given by: f c = E cԑc (E c E 2 ) 2 4f co ԑ c 2 for 0 ԑ c ԑ t Eqn 2-31 f c = f co + E 2 ԑ c for ԑ t ԑ c ԑ cu Eqn 2-32 Where, f c & ԑ c are the axial stress and the axial strain of confined concrete respectively, ԑ t is the axial strain at the transition point & E 2 is the slope of the straight second portion. ԑ t = 2f co Eqn 2-33 E c E 2 E 2 = f cc f co Eqn 2-34 ԑ cu Figure 2-15 : Lam & Teng s stress-strain model for FRP-confined concrete. (Lam & Teng, 2003) 24

32 The compressive strength of FRP-confined concrete f cc is predicted using: f cc = f co + 3.3f l 0.7 Eqn 2-35 f l = 2σ jt d = 2E frpԑ j t d Eqn 2-36 ԑ j = ԑ h,rup ԑ h,rup = k ԑ ԑ frp Eqn 2-37 k ԑ is the FRP efficiency factor and has a value of Lam & Teng proposed that the ԑ j should be taken as the actual hoop rupture strain ԑ h,rup measured in the FRP jacket and not the ultimate FRP tensile strain ԑ frp as is assumed ideally. The ultimate concrete axial strain of uniformly confined concrete, ԑ cu is given by: ԑ cu ԑ co = f l f co ( ԑ hrup ԑ co ) 0.45 Eqn 2-38 Here, the axial strain (ԑ co ) at the compressive strength of unconfined concrete is taken as

33 2.5.4 Modified Lam & Teng (Liu et al., 2013) The constitutive model proposed by Lam & Teng (2003) is a very sophisticated model capable of capturing the behavior of FRP-confined concrete to a promising level. However, the model cannot represent the process of gradual development of confinement by the FRP-tube. He et al.,(2013) made an attempt to modify the parabola in a way which would be able to represent the transition more accurately. This model uses the slope and intercept proposed by Samaan et al. (1998) in Lam & Teng (2003) model. The first branch is parabolic which transitions into the second branch which is linear. The transition occurs at smoothly at a transition strain ԑ t ԑ t = 2f o (E c E 2 ) Eqn 2-39 E 2 = f cc f o Eqn 2-40 ԑ cu Figure 2-16 : Envelope curve and hysterical rule of proposed FRP confined concrete. (Lui et al, 2013) 26

34 This model requires the definition of three parameters based on the general shape of stress-strain curve described above. The three parameters include: the ultimate strength (f cc ), the ultimate strain (ԑ cu ) and the intercept stress( f o ). These are the three critical parameters critical to the constitutive model. Lam & Teng (2003) proposes that the intercept stress f o be taken equal to the compressive strength of the unconfined concrete. (f o = f * co ) The condition at ultimate state of FRP confined concrete is directly related to the ultimate transverse tension of FRP material which provides the confining pressure on the concrete. The amount of the maximum confining pressure that will be applied by the FRP tube on the concrete will be controlled by the ultimate strain (ԑ h,rup ) in the FRP. This strain is not equal to the ԑ FRP which is the ultimate tensile strain in the coupon testing. Hence a FRP-strain reduction factor (k e ) is defined to describe the relationship between ԑ h,rup & ԑ FRP. ԑ h,rup ԑ frp = k e Eqn 2-41 Now the actual confining pressure (f l,a ) at ultimate is calculated by using ԑ h,rup f l,a = k e f l = 2E frpt frp ԑ h,frp D Eqn 2-42 Where, D = Diameter, E frp = Youngs modulus of FRP & t frp is the thickness of FRP layer. FRP strain reduction factor( k e ) is very important in order to accurately describe the shape of the model. The values suggested by Lam & Teng (2003) for different types of confinement is not general and accurate enough. Lim & Ozbakkaloglu (2013) proposed an equation for the strain reduction factor which relates to the Youngs modulus of FRP and the unconfined concrete compressive strength. k e = f co x E f x 10 6 (MPa) for 10 5 MPa E f 6.4 x 10 5 MPa Eqn

35 This expression is able to predict the strain reduction factor for FRP-confined concrete with concrete of unconfined compressive strength upto 120MPa. This expression is also valid for GFRP, CFRP and aramid FRP. With this, the maximum confining pressure can be accurately calculated. Lam & Teng (2003) model was verified based on the database specified in Teng et al.,(2003) which had concrete specimens of unconfined concrete specimens less than 43 MPa. With increase in the unconfined concrete strength f * co, the ultimate coefficients of stress and strain are needed to be modified. Hence the expression for the ultimate strain that has been proposed is: ԑ cu = c 2 ԑ co + 12 ( f l,a f ) ԑ 0.45 h,rup 0.55 ԑ c0 Eqn 2-44 co He et al (2013) used the database from Lim & Ozabakkaloglu (2013) which covered concrete specimens of unconfined concrete strength of 6.2MPa to MPa. With this database the normalized coefficients were statically determined: c 2 = (f co 20)& c 2 1 Eqn 2-45 The minimum threshold for FRP confined concrete to display complete strain-hardening is defined in terms of the FRP confinement stiffness K 1. With the help of this, we define the FRP stiffness threshold K lo. K 1 = 2E frpt frp D K lo = f co 1.65 Eqn & 2 If K 1 K lo f cc = c 1 f * co + 2 k 1 f l,a / k ԑ Eqn 2-47 c 1 = f l,a / (f * co ԑ h,rup ) Eqn

36 where k 1 =1 for wrapped FRP & k 1 = 0.9 for FRP tube If K 1 < K lo f cc = c 1 f * co + k 1 (f l,a - f lo ) Eqn 2-49 c 1 = (K 1 / f * co 1.6 ) 0.2 Eqn 2-50 ԑ lo = 24 (f * co / K ) 0.4 ԑ c0 Eqn 2-51 f lo = K 1 ԑ lo Eqn 2-52 where k 1 =3.18 for wrapped FRP & k 1 = 2.89 for FRP tube strength (f * co ). A specific relationship is proposed between the stress intercept (f o ) and the unconfined concrete f o = f * co Eqn

37 2.5.5 Drucker-Prager Plasticity Model Drucker & Prager proposed this model in There are three criteria that control the framework of this model. Every plasticity model is governed by a yielding and hardening criteria; the flow rule; path dependence; limited tensile strength and the pressure dependence critieria. In this model, the parameters that control the yielding and hardening criteria are the friction angle and the cohesion. The plastic dilation controls the flow rule of the plasticity model. A limited amount of study has been carried out regarding the plastic dilation rate for FRP-confined concrete. Mirmiran et al., (2000) & Karabinis et al., (2008) used a constant dilation rate which is not true. Yu et al., (2010) demonstrated with his research that the dilation rate varies with the plastic strains and the lateral stiffness. This was further verified by J- F. Jiang et al (2012). In order to implement this model in Abaqus we need to define three parameters : 1) Friction angle model, 2) hardening/softening function & 3) Plastic Dilation model Friction angle model theory as The friction angle φ can be related to the internal friction angle Φ defined in the Mohr-Coloumb tanφ = 6sinΦ 3 sinφ Eqn 2-54 The internal friction angle is the angle that the tangent line which is drawn to the Mohr s circle at the failure state makes with the X-axis or with the normal stress axis. In case of concrete the failure state is defined as the onset of peak strength. Mohr s circle is used to describe the behavior of brittle materials. While considering the plasticity model, we consider the softening behavior of concrete. Brittleness reduces under increasing deformation due to increasing softness. This causes the internal friction angle to reduce and the assumption of it being constant is not quite accurate. 30

38 Research by J-F. Jiang et al (2012) proposes an equation to determine the friction angle φ. φ = φ o + kԑ p Eqn 2-55 Where φ o = o, k = 226 ԑ p = Plastic Strain Cohesion Model Figure 2-17 : Friction model (J-F. Jiang et al (2012)) In the Drucker-Prager Plasticity Model, k is the hardening/softening which controls the development of the surface that yields. Most of the hardening-softening functions are based on the assumption that the internal friction angle φ is constant. Hence a different function is required which considers the variation in the internal friction angle φ. Based on the friction model described above the relationship between the normalized cohesion k/f c & ԑ p can be described as follows: k(ԑ p ρ) f C = k o + E p ( ԑ p 1+ ηԑ p ) + p 1 (ρ)ԑ p 2 + p 2 (ρ)ԑ p 2 Eqn 2-56 Where, k o = 1/8, E p = 2700 (initial slope of the cohesion curve), η = 6587, p 1 = ρ a 1 +a 2 ρ+a 3 ρ Eqn 2-57 p 2 = (b 1ρ+b 2 ) ρ+b 3 Eqn 2-58 Where, a 1 = 0.12, a 2 = , a 3 = , b 1 = -0.75, b 2 = , & b 3 =

39 Dilation Angle (Plastic Dilation Model) The FRP jacket induces a passive type of confinement on the concrete. Dilation is the measure of the volume change. In the case of FRP-confined concrete the plastic volumetric deformation/strain is critical as it controls the amount of confining pressure that will be generated by the FRP tube. Mirmiran et al.,(2000) pointed out that a zero plastic dilation rate, which is true for steel confined concrete can predict reasonably the behavior of FRP confined concrete but not accurately. Karabinis & Rousakis (2002) used an asymptotic function, where the plastic dilation angle β was assumed to decrease from o to o. Rousakis et al., (2008) later assumed a constant rate which depends on unconfined concrete strength. Figure 2-18 : Existing models for plastic dilation rate. (J-F. Jiang et al (2012)) J-F. Jiang et al (2012) proposed a mathematical model for the dilation angle. Test results from various research studies were used in this case and a model accurately satisfying these test results was arrived at. 32

40 Figure 2-19 : Typical Dilation curve (J-F. Jiang et al (2012)) β = β 0 + (M o + λ 1 β o )(ԑ c p ) + λ2 β u (ԑ c p ) λ 1 ԑ c p + λ 2 (ԑ c p ) 2 Eqn 2-59 The coefficients λ 1, λ 2 & β u are functions of β o, M o & ρ. β o = -37, M o = & ρ = 2E ft f Df c & λ 1 = 11.61ρ λ 2 = 5700ρ β u = exp( 0.06ρ) 37.5 Eqn 2-60 (a) (b) (c) 33

41 2.6 Conclusions As discussed in this chapter the stress-strain curve of a FRP confined concrete cylinder specimen can be divided into four portions: The first zone, transition point, second zone and the ultimate/failure point. The Samaan and Mirmiran, 1998 model provides the stress-strain curve and the ultimate stress and strain value. It provides two equations for the ultimate condition of the stress-strain curve. The stress equation is derived from experimental tests and the ultimate strain is derived from the geometric shape of the curve. The Manders model, 1984 is primarily derived for steel confinement, wherein the second zone is derived from the maximum confining pressure exerted by steel confinement. The Lam & Teng model and the Modified Lam & Teng s model provide several equations for each point on the stress strain curve. Accordingly, this study would be aiming at validating the modified Lam & Teng s model against the experimental stress-strain curves and those obtained via finite element simuation in AbaqusCAE by using the Drucker-Prager plasticity model. 34

42 Chapter 3 Experimental Database & Preliminary results. 3.1 Introduction In order to validate the reliability of the model chosen for this study it was necessary to gather an experimental database. The selection of the experimental database was subjected to the availability of the complete information related to the specimens, such as its geometric configuration, the material properties of concrete and FRP and the final results including the stress-strain curve. The experimental database chosen for this study includes 51 concrete cylinder specimens wrapped with Glass FRP and 54 wrapped with Carbon FRP. The database comprises of concrete cylinders having unconfined compression strength varying from 20 MPa to MPa. Source ID Type of FRP D (mm) H (mm) f' co (MPa) ԑ co (%) E frp (GPa) t (mm) ρ ԑ h,rup f' cc (MPa) ԑ cu 1 Carbon Carbon Carbon Carbon Carbon Lam & Teng (2004) 6 Carbon Carbon Carbon Carbon Glass Glass Glass Glass

43 Source Lam et al (2006) Teng et al (2007) I D Type of FRP D (m m) H (mm ) f' co (MPa) ԑ co (%) E frp (GPa ) t (mm ) ρ ԑ h,rup f' cc (MPa ) 14 Carbon Carbon Carbon Carbon Carbon Carbon Glass Glass Glass Glass Glass Glass ԑ cu Jiang et al (2007) 26 Glass Glass Glass Glass Glass Glass Glass Glass Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon

44 Source Harries & Kharel (2003) Shahway et al (2000) I D Type of FRP D (mm ) H (mm ) f' co (MPa ) ԑ co (%) E frp (GPa ) t (mm) ρ ԑ h,rup f' cc (MPa) 49 Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon ԑ cu Almu- Sallam (2007) 61 Glass Glass Glass Glass Glass Glass Glass Glass Glass Glass Glass Glass

45 Source Nanni & Bradford (1995) J.F. Berthet et al (2005) ID Type of FRP D (mm ) H (mm ) f' co (MPa ) ԑ co (%) E frp (GPa ) t (mm ) ρ ԑ h,rup f' cc (M Pa) 73 Glass Glass Glass Glass Glass Glass Glass Glass Glass Glass Glass Glass Glass Glass Glass Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Carbon Glass Glass Glass Carbon Carbon Carbon Glass Glass Glass Table : Experimental Database (ID 01 to ID 105) ԑ cu The literature review considered for this experimental database were from research studies that made an attempt to understand the behavior of FRP-confined concrete under uniaxial compression. 38

46 3.2 Preliminary study It was necessary that the procedure to be carried out for analytical and numerical modeling of the FRP confined concrete cylinders was feasible and valid. This was tested and verified by modeling the experimental setup of uniaxial compression test for unconfined concrete. The experimental stress-strain curve for unconfined concrete was chosen from the CSUF Test Report# SRRS-SCCI-OP1202., March This report was very detailed about the concrete properties and the findings from the uniaxial compression test setup. Hence the data from the CSUF Test Report# SRRS-SCCI-OP1202., March 2003 was used. 3.3 Test Database The cylinders under consideration were 6 x12 concrete cylinders. All the cylinders were allowed to cure for 28 days. The average compressive strength of the concrete used after 28 days of curing was 6200 psi (42.8MPa). Following was the database used: Specimen ID Compressive Strength, ksi (MPa) Axial Strain at Rupture CC (42.3) CC (41.41) CC (42.8) CC (43.92) CC (36.40) CC (43.40) Table 3-2 : Unconfined Concrete Specimen Data (CSUF Test Report 2003) 39

47 Following are the experimental stress-strain curves to be used: CC-6 CC-4 CC-3 CC-1 CC-5 CC-2 Fig 3.1: Experimental Stress Strain Plots (CSUF Test Report 2003) 3.4 Abaqus modeling The concrete was modeled in Abaqus as a 3D extrude cylindrical part with dimensions as per the the report under consideration. The concrete material was modeled as a Concrete damaged plasticity model. The concrete compression model was chosen to be Hognestad model as described in Section The Hognestad model used in Abaqus represents the behavior of confined concrete. The Kent-Park model used in Concrete02 material code in Opensees is derived by generalizing the Hognestad model. Hence it was believed that using Hognestad model in Abaqus would yield the most consistent results. 40

48 The cylinder was partitioned into a concentric cylinder for the purpose of uniform meshing. The mesh elements used were solid elements of wedge type and the interpolation was quadratic. A general static loading condition was created in which a displacement was applied in a equal time steps. The stressstrain data was exported from the output which was plotted with the experimental and Opensees stress strain curve. 3.5 Opensees Modeling The concrete cylinder was modeled in opensees using the pushover analysis code. Concrete02 model was used which follows the Kent-Park model. The analysis was performed on node to node element which is essentially a collection of 2D elements which replicates the behavior of a 3D object as in our case. The properties like unconfined compressive strength and ultimate strain were used from the report under consideration. 3.6 Results and Discussion Concrete specimen CC-1 was deemed to be an outlier pertaining to the extremely brittle failure. For the remaining specimens the results from Opensees and Experimental curve were very much in agreement with each other. The initial stiffness from the Abaqus stress strain plot was much higher than that obtained from the experimental and opensees stress strain plot. The Hognestad model used in Abaqus represents the behavior of confined concrete. It is expected that the stiffness of confined concrete be higher. However, the concrete cylinders tested are unconfined. Due to this difference in the modeling and actual scenario one could explain the difference between the stiffness obtained by Abaqus simulation and actual experimental testing. A rigorous study for mesh sensitivity would also help us to better understand the deviation in the stiffness in the results from Abaqus. The failure point in the Abaqus simulation is at a higher strain that in the experimental results and Opensees simulation. This can also be attributed to the Hognestad model exhibiting a confined concrete 41

49 behavior in which the confinement will cause the cylinder to rupture at a higher strain in comparison to the unconfined concrete cylinder. The ultimate stress values were consistent in all three results. However, the ultimate strain was lower in Abaqus simulation as compared to Opensees and Experimental results which yielded the same ultimate strain. Also, the rupture stress is less in Abaqus simulation as compared to Opensess and Abaqus. The shape of the stress strain curve is however similar in all three cases except for the point where the Hognestad model transitions from a linear behavior to a non-linear behavior. It is predicted that choosing a Todeschini model for Abaqus might prove more accurate. This could be verified in future studies. Figure 3-2 : Comparison of Stress Strain Plots of Abaqus, Opensees and Experimental Data CC2 42

50 Figure 3-3 : Comparison of Stress Strain Plots of Abaqus, Opensees and Experimental Data CC3 Figure 3-4 : Comparison of Stress Strain Plots of Abaqus, Opensees and Experimental Data CC4 43

51 Figure 3-5 : Comparison of Stress Strain Plots of Abaqus, Opensees and Experimental Data CC5 Figure 3-6 : Comparison of Stress Strain Plots of Abaqus, Opensees and Experimental Data CC6 44

52 Chapter 4 AbaqusCAE Finite Element Modelling 4.1 Introduction A lot of researchers have been exploring models that are accurate enough to portray the behavior of FRP-confined concrete in order to study its behavior. Based on the theory of plasticity, some constitutive models can capture the principle features of the highly complex nonlinear behavior of concrete. (Pekau et al.,1992). Karabinis et al., (2002) demonstrated that the behavior of confined concrete can be accurately demonstrated by the Drucker-Prager (DP) plasticity model. Therefore, for this study, the Drucker-Prager model was selected to model FRP-confined concrete in Abaqus. 4.2 Concrete The concrete cylinder is modelled as a full cylinder according to its geometric configurations as per the information obtained from the respective published literature which is the source of the experimental database as mentioned in Chapter 3.The cylinder is modeled as a 3D extrude part (see Figure 5-1). The cylinder is partition into another concentric cylinder. This assembly of 2 concentric cylinders is then partition into four equal parts. This procedure is carried out so that we can have a uniform mesh which will help us obtaining better results at a lower computational time. 45

53 Elastic Properties Figure 4-1 : Creating a Concrete cylinder 3D extrude Part in Abaqus The concrete material is modeled as a isotropic material. The ACI design equation was chosen to input the elastic modulus of concrete and a Poisson s ratio of 0.2 was adopted. E c = 4734 f c (in MPa) Eqn

54 Plastic Properties The Extended Drucker-Prager plasticity model will be used to assign plasticity to our model. We would be using the linear Plasticity model. The Drucker-Prager plasticity parameters based on the strain in concrete, was implemented using the subroutine option of SDFV in Abaqus CAE. The parameters were calculated using the equations mentioned in Section (Eqn 2-55 for friction angle, Eqn 2-56 for Cohesion & Eqn 2-59 for the Dilation angle) and were used in each FE input file. The parameters used in Abaqus for modelling the material Concrete are listed in Table 5-1 Material Property Notation Values used in this study Elastic Modulus E c 21171MPa to 49151MPa Concrete Poisson's Ratio ν c 0.2 Unconfined Compressive Strength f' co 20MPa to 107.8MPA Failure Strain Cohesion Dilation Angle Angle of Friction ԑco k β ϕ Table 4-1 : Concrete DP-model parameters 47

55 4.3 Fiber Reinforced Polymeric Jacket The FRP jacket is modeled as a shell in Abaqus. Abaqus allows the user to specify the shell thickness for the FRP jacket and hence the feature of varying thickness of FRP jackets can be taken into consideration with different specimens. The FRP sheet is modeled as elastic laminar with orthotropic elasticity in plane stress without bending and bending stiffness. The elastic modulus is only in the direction of the fibers which is along the circumferential direction. A Poisson s ratio of 0 was assigned. Modelling the FRP-tube as a shell-extrusion part and assigning it an elastic laminar behavior, following inputs are required. Material Property Notation Elastic Modulus E 1 FRP Elastic Modulus E 2 Poisson's Ratio ν c Shear Modulus G 12 Shear Modulus G 13 Shear Modulus G 23 Table 4-2 : Material Properties for FRP Figure 4-2 : FRP jacket created in Abaqus 48

56 4.4 Abaqus model: Assembly The 3D concrete cylinder and the FRP shell was assembled between two analytically rigid plates. These plates were included in the analysis in order to measure the strain and stress in accordance with the actual compression test. A reference point was created in the top plate (loading plate) in order to be able to record the history output of displacement along the height of the cylinder. Similarly a reference point was created in the bottom plate in order to record the reaction force Boundary Conditions & Analysis Step Boundary conditions were applied. An encastre (fixed) condition was applied at the reference point of bottom plate and a displacement was applied at the reference point on the top plate. A dynamic explicit step was created as convergence performance with the Drucker-prager plasticity algorithm is better with explicit analysis. Loading was applied by specifying the displacement as a constant rate which is in accordance with the uniaxial compression tests carried out in the experimental program from which we are using the experimental results Interaction There were two types of interfaces which needed to be defined. The interaction between FRP jacket and concrete surface was defined as a tied connection at edges and a no slip property was assigned along the surface. As for the interaction between the plates and concrete a tie connection property was assigned Meshing As mentioned earlier the concrete cylinder was partitioned for the purpose of uniform meshing. The central portion of the cylinder was assigned with tetrahedral type of elements and the outer portion was assigned with Hex-type elements. Results were obtained according to the history output requested at the two reference points. 49

57 meshing details: Total number of elements present was 900 and number of nodes were 976. Followinf are the Concrete Cylinder : Hexahedral Elements (C3D8R): 360 Concrete Cylinder : Tetrahedral Elements (C3D6) : 180 FRP Jacket : Quadrilateral Elements (S4R) : Conclusions During the course of modeling the uniaxial compression test of a concrete cylinder confined with FRP in Abaqus it was learnt that there was a need for a more robust model which would very accurately replicate the behavior of FRP confined concrete under uniaxial compression loading. The model used in this study had the following limitations. 1) Elastic Modulus of Concrete: The elastic modulus of concrete was not available as per the experimental testing of the concrete batch that was used in the study. Hence, the empirical relationship which is provided by ACI was used. This equation is found to be a good estimate for design purposes. But with respect to this study, where there was a need to have experimentally verified material properties in order to validate the chosen empirical model, the use of this equation was not desirable. 2) Poisson s Ratio: As discussed earlier, the behavior of FRP confined concrete can be explained in terms of the Drucker-Prager plasticity model. Also, at different stages of loading the confinement pressure exerted by the FRP would vary which would affect the Poisson s ratio. Providing a Poisson s ratio of 0.2 as a constant would be an approximation. A study needs to be carried out regarding the Poisson s ratio at different stages of loading in order to implement the exact material behavior in Abaqus. 50

58 3) Negative Dilation Rate: The equations used in the implementation of the Drucker-Prager plasticity model in Abaqus show that the dilation rate changes from a negative value to a regime of positive values depending on the FRP material and the grade of concrete used. (refer fig 2-19). Abaqus, by default does not consider negative dilation rate and assumes a dilation rate of 0 in this case. This would not represent the exact behavior of FRP confined concrete as verified by J-F Jiang et al.,(2012). 4) Mesh Convergence Studies: A systematic mesh convergence study was needed to be carried out. Finite element results are sensitive to the type and size of meshing that is used in the analysis. A mesh convergence study would help in determining the optimum meshing for the results to be accurate. 51

59 Chapter 5 Comparison Study and Analysis of Results 5.1 Introduction The ultimate stress and the ultimate strain obtained by the modified Lam & Teng s model and from the Abaqus simulation was compared with those obtained from the experimental results separately. The ultimate stress and strain values used in comparison study were the reported values in the literature that was used for the experimental database. The tabulated results have been listed in Appendix A. A comparison study was conducted between the results from the modified Lam & Teng s model, AbaqusCAE finite element simulation and the experimental database. From the comparison between these three results, the accuracy of the analytical models was examined. The comparison of the model was assessed based on the ultimate strength and ultimate strain. For design purposes it is of utmost importance that the empirical models used predict the ultimate condition accurately. In the case of FRP confined concrete, the failure point is accompanied by large release of energy which indicates a failure highly brittle in nature. With respect to civil structures such a failure can be catastrophic and hence an accurate assessment of this particular stage while design structures with FRP confined concrete is of paramount importance. Typical curves for various confinement ratios are shown below (from ρ = 3.9 to ρ = ). These are the representative stress-strain curves obtained for various confinement ratios. Confinement ratio is a quantitative measure of the amount of confinement available from the FRP jacket in comparison to the unconfined strength of concrete. It is given by the following equation. ρ = 2E ft f Df c Eqn 5-1 The complete set of stress-strain curves are listed in Appendix B for all the 105 specimens. 52

60 Abaqus Experimental Modified Lam & Teng Abaqus Experimental Modified Lam & Teng Figure 5-1 : Typical curves for confinement ratio 5-10 Figure 5-2 : Typical curves for confinement ratio Abaqus Experimental Modified Lam & Teng Abaqus Experimental Modified Lam & Teng Figure 5-3 : Typical curves for confinement ratio Figure 5-4 : Typical curves for confinement ratio Abaqus Experimental Modified Lam & Teng Abaqus Experimental Modified Lam & Teng Figure 5-5 : Typical curves for confinement ratio Figure 5-6 : Typical curves for confinement ratio

61 Experimental Modified Lam & Teng Abaqus Experimental Abaqus Modified Lam & Teng Figure 5-7 : Typical curves for confinement ratio 85 Figure 5-8 : Typical curves for confinement ratio Performance of the Drucker-Prager Model As mentioned earlier, the material constitutive model used for concrete in the Abaqus simulation is the Drucker-Prager hardening model. This model essentially exhibits the hardening behavior of the FRP confined concrete. However, in the case of FRP confined concrete the amount of hardening and its characteristics are governed by the amount of confinement provided. The Drucker-Prager model, which depends on parameters like friction angle and plastic dilation which are governed by the type and the amount of confinement present as discussed in section 5.2. It was observed in this study that this model fails to accurately represent the behavior of FRP confined concrete with lower (ρ < 5) and higher (ρ < 5) confinement ratios. Specimens with ID number 20, 21, 28, 29 have low confinement ratios (ρ = 3.9 to 4.52). The experimental stress-strain curve shows a curve shape similar to an unconfined concrete cylinder (Appendix B). A typical curve of Specimen with ID no 28 is shown below. 54

62 Abaqus Experimental Modified Lam & Teng Figure 5-9 : Comparitive stress-strain curve for ID no 28 This is expected as the amount of FRP confinement provided is not enough to impose any significant confinement pressure on the concrete cylinder and hence the actual behavior exhibited is similar to that of an unconfined concrete cylinder. However, the Drucker-Prager model still exhibits a certain amount of hardening behavior which causes the Abaqus simulations not to accurately replicate the experimental stress-strain curve. The equations governing the plastic dilation as mentioned in Section 5.2.5, show that for lower confinement ratios the plastic dilation does not change considerably throughout the loading regime which ensures a hardening branch of the stress strain curve with a positive slope. Specimens with ID number 36, 37, 38, 39, 54, 55 & 56 have very high confinement ratios (ρ = to ). There is a considerable deviation between the stress-strain curves obtained from the actual experimental study and Abaqus simulations. (Figure 6-7 & 6-8) In the case of higher confinement ratios, the complete capacity of confinement is not used. There can be two cases of higher confinement ratios 1) High thickness of the FRP jacket: In this case the outermost fibers are not stretched to their full capacity before the concrete core disintegrates completely. Hence, the amount of 55

63 hardening computed by the equations specified in Section 5.2 is much more than the actual confinement pressure experienced by the concrete cylinder. 2) High E f /f c ratio : This is a case of having a low strength concrete confined by a high strength FRP jacket. In this case the concrete core tends to significantly deteriorate before the complete development of the FRP jacket can occur. However, the hardening criterion does not take this into account which leads to an estimation of higher confined strength of concrete. From the model chosen (mentioned in Section ) to represent the changing plastic dilation angle it was evident that specimens having higher confinement ratios had a negative dilation angles (see Figure 2-19). Abaqus does not take into consideration a negative dilation angle and assumes it to 0. This is also believed to be a reason for the Abaqus simulation results for higher confinement ratios not complying with the actual experimental results. 56

64 5.3 Performance of the Modified Lam & Teng model The modified Lam & Teng model is a design-oriented model. The model is expected to give accurate ultimate state conditions. It was observed that the model failed to predict the ultimate state in cases of heavily confined concrete or specimens with high confinement ratios. It was observed that the accuracy of the modified Lam & Teng model significantly depended on the amount of confinement that was provided by the FRP jackets. The model performed well and could replicate the ultimate states for confinement ratios up to ρ = 85. Specimens with ID number 36, 37, 38, 39, 54, 55 & 56 (ρ = to ) showed deviation as compared to the experiments. However, the model tends to work considerably well in terms of specimens having low confinement ratios. The graphical results are shown in Appendix B for all the specimens. Typical figures are shown for specimens with lower confinement ratio and higher confinement ratios. Abaqus Modified Lam & Teng Experimental Figure 5-10 : Comparative Stress-Strain curve for Specimen ID29 (Confinement Ratio = 3.9) 57

65 Experimental Abaqus Modified Lam & Teng Figure 5-11 : Comparative Stress-Strain curve for Specimen ID29 (Confinement Ratio = ) As observed, Eqn 2-44 fails to predict the ultimate condition for higher confinement ratio (ρ > 85). The reason for this was believed that the equation fails to take into consideration that at higher confinement ratios the complete FRP confinement pressure that is available is not utilized because the concrete core disintegrates much before the complete FRP confinement is used. This leads to the Eqn 2-44 predicting a higher ultimate strengths for highly confined specimens. There is a need for more sophisticated equations that take into consideration the effect of higher confinement ratios. 58

66 Modified Modified Lam & Lam Teng & Teng f'cc (MPa) ԑ cc (%) Following is a plot showing the ultimate strength predicted by modified Lam & Teng s model in comparison to the experimental results used. 250 Ultimate Ultimate Strain Strength Strain Comparison (New (New Proposed (Modified Proposed Equation) Lam Equation) & Teng) (ρ > 85) ρ < 5 5 < ρ < 85 ρ > Experimental Experimental f' ԑ cc (%) cc (MPa) Figure 5-12 : Ultimate Strength comparison (Modified Lam & Teng vs Experimental) The modified Lam & Teng model gives us an over prediction of strength which is not conservative. It was observed that more than 85% of the specimens showed an accuracy up to a difference of 15% in prediction of the ultimate strength as compared to the experimental results. Hence according to this study the performance of modified Lam & Teng s model in prediction of ultimate strength was acceptable. 59

67 Modified Lam Teng cc (%) Modified Lam & Teng ԑ cc (%) A similar plot was constructed for comparing the ultimate strain prediction by the modified Lam & Teng s model. 12 Ultimate Ultimate Ultimate Strain Strain Comparison Strain Comparison (Modified (New (New (Modified Proposed Lam Proposed & Teng Lam Equation) Equation) vs & Teng) Experimental) (ρ > 85) ρ < 5 5 < ρ < 85 ρ > Experimental ԑ cc cc (%) Figure 5-13 : Ultimate Strain comparison (Modified Lam & Teng vs Experimental) From the above graph it is evident that Eqn 2-44 proposed in the modified Lam & Teng s model yielded non conservative values of ultimate strain. It was observed that less than 65% of the specimens showed an accuracy upto a difference of 15% in prediction of the ultimate strain as compared to the experimental results (Appendix A). Hence a need to modify the equation for the prediction of ultimate strain condition was needed. Following this, a regression analysis and optimization was carried out in order to introduce a new equation for the ultimate strain for FRP confined concrete which was in better accordance with the experimental findings. 60

68 5.4 Regression Analysis Regression analysis is a statistical process for estimating the relationships among variables. As seen earlier the equation provided by Modified Lam & Teng s model (Eqn 2-44) was not able to predict the ultimate strain condition accurately. Hence there was a need to perform a regression analysis in order to establish a more robust relationship between the material properties of the FRP confined concrete and the ultimate strain which agrees with the experimentally observed values for the same. Minitab17 software was used to carry out the regression analysis. In the case of this study, it is expected that the ultimate strain depends on many variables. Hence, this was a case of multivariable regression analysis. The same experimental database (105 data points) used earlier was used in this case. The response for our analysis was set as the experimentally observed values of ultimate strain. As mentioned in the literature review, Lam & Teng.,(2003) proposed the ultimate strain equation as mentioned in Eqn 2-38 This equation was modified by Lui et al.,2013 by introducing a coefficient c 2 in place of the constant value of Hence the ultimate strain equation was modified as mentioned in Eqn 2-44 It was determined that a further modification, with respect to this coefficient, c 2 is needed in order to improve the performance of the modified Lam & Teng s model. Hence it was decided to carry out a multi-regression analysis for c 2. A correlation analysis was carried out with respect to several parameters which included unconfined concrete strength (f c ), unconfined ultimate concrete strain (ԑ c ), modulus of elasticity of FRP (E frp ), thickness of the FRP jacket (t frp ), ultimate strain of FRP (ԑ frp ), diameter of the specimen (D) & the ultimate hoop strain for FRP confined concrete (ԑ h,rup ). A correlation analysis measures the extent to which variables tend to change the response. The correlation analysis provides us with two values; 1) Pearson Coefficient & 2) P-Value. Pearson Coefficient is a type of correlation coefficient that represents the relationship between two variables that are measured on the same interval or ratio scale. The value of 61

69 the pearson coefficient will vary between -1 to +1. A value of 0 denotes no correlation between the variables. One can determine the effect a particular variable on the value of response (ultimate strain, ԑ cu ) based on this value. Higher the absolute value of Pearson coefficient higher is the correlation between the two variables. The p-value is defined as the probability, under the assumption of a hypothesis, of obtaining a result equal to or more extreme than what is actually supposed to be observed. The smaller the p-value, the larger is the significance because it tells us that the hypothesis under consideration may not adequately explains the observation. It is also referred to as the level of significance of a particular variable with respect to the response variable. Variables for the regression analysis were chosen based on these two values obtained via a correlation analysis. The variables chosen were; unconfined concrete strength (f c ), unconfined ultimate concrete strain (ԑ c ), modulus of elasticity of FRP (E frp ), thickness of the FRP jacket (t frp ), ultimate hoop strain for FRP confined concrete (ԑ h,rup ). A multiple regression analysis was carried out with respect to the 5 variables stated above. This analysis considered terms upto a degree of 2 (quadratic). A diagnostic was carried out on the regression model that was fitted by the analysis procedure. The regression model was able to explain 88.64% of the variation in the response quantity (ultimate strain). Also the P-value was less than which indicates all the parameters are significant to the model. 62

70 Figure 5-14 : Summary Report for Multivariable regression analysis The regression model is as follows c 2 = f co ( E frp 59.63) + ԑ co ( ԑ co 46.7E frp 5062t frp ԑ hrup ) + ԑ hrup ( ԑ hrup ) + t frp ( t frp 10.08ԑ hrup ) + E frp ( E frp ) Eqn

71 Figure 5-15 : Final Model performance report Based on this equation the ultimate strain was computed and compared against the experimental values. It was found out that more than 75% of the specimens have the predicted ultimate strain with an error of +-15% as compared to the earlier equation which showed only less than 65% having predicted ultimate strain with the same margin of error. One can also observe the data values on the graph below are less scattered as compared to the ones calculated by the equation proposed in the modified Lam & Teng s model. (Eqn 2-44) 64