ABSTRACT. JIANG, GUOLIANG. Self-Monitoring Fiber Reinforced Polymer Retrofits. (Under the direction of Kara Peters.)

Transcription

1 ABSTRACT JIANG, GUOLIANG. Self-Monitoring Fiber Reinforced Polymer Retrofits. (Under the direction of Kara Peters.) Fiber reinforced polymer (FRP) composites have been widely applied to strengthen existing civil critical infrastructures due to their advantages such as light weight, high strength, corrosion resistance and easier installation in field applications. Since the strengthened structures are externally wrapped by non-transparent FRPs, it is difficult to monitor the structural integrity of the FRP strengthening or the underlying structure. One critical aspect of the strengthening to be monitored is the layers, interface condition between the FRP and the underlying structure. This dissertation presents the development of a FRP retrofit system with self-monitoring capabilities. Firstly, a FRP retrofit concept with sensing capability to monitor FRP debonding is designed. This FRP system is based on global and local fiber optic sensors embedded in between FRP layers or at the surface of the concrete structure. For global strain measurements, a newly developed oscillator interrogated interferometer is adapted, in combination with a prefabricated fiber-optic ribbon, to large-scale FRP strengthened structures. For local strain measurements, existing fiber Bragg grating (FBG) sensors are incorporated. Secondly, a simplified theoretical model for real-time data processing of sensor information is derived. In this model, the majority of the calculations for the sensor response or actuator excitations are performed only once for a given FRP system geometry. The computationally efficient three dimensional shear-lag model is developed through the extension of an optimal, two-dimensional theory previously reported in the literature.

2 Numerical predictions for the interfacial shear stresses and average normal stresses in a single-fiber composite and a unidirectional laminated composite are presented. Comparison to finite element analyses demonstrates that the new reduced model can be used to rapidly estimate the average normal stress distribution in the various constituents and therefore the embedded sensor responses. Finally the performance of the self-monitoring FRP retrofit system is demonstrated through two series of experiments. In the first series, a static loading test of a FRP strengthened reinforced concrete beam shows the feasibility and durability of the oscillator interrogated time-of-flight optical interferometer to provide global strain information. The measurable displacement range is determined by the oscillator frequency and therefore can be designed to be significantly larger than current global strain measurement systems. An optical path length change resolution of 9.5 mm and range of 1.39 m are demonstrated experimentally. To easily embed the sensor into the FRP retrofit and increase the fiber sensing length, a commercially available prefabricated fiber ribbon is used. In the second series, an example self-monitoring FRP system containing local sensors is fabricated incorporating embedded local fiber optic sensors. The ability of the self-monitoring FRP system to identify the initiation and development of crack-induced debondings in a FRP-steel splice joint are evaluated. Comparison of the predictions of the reduced three dimensional shear-lag model and the experimental results demonstrate the accuracy of the reduced model.

3 Self-Monitoring Fiber Reinforced Polymer Retrofits By Guoliang Jiang A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Mechanical Engineering Raleigh, North Carolina 2007 APPROVED BY: Dr. Jeffrey W. Eischen Dr. Gracious Ngaile Dr. Sami H. Rizkalla Dr. Kara Peters Chair of Advisory Committee

4 TABLE OF CONTENTS List of Tables...vii List of Figures...viii CHAPTER 1 INTRODUCTION FRP Retrofit Types and Processing Methods FRP Debonding Failures FRP Retrofits with Sensing/Controlling Capabilities Research Scope and Research Objectives Thesis Organization...10 CHAPTER 2 BACKGROUND Non-Destructive Evaluation (NDE) Methods for FRP Retrofits Self-Monitoring FRP Retrofits Fiber Optic Global Strain Measurement System Fiber Optic Bragg Grating Sensors Shear-Lag Analysis for Fiber Reinforced Composites...31 CHAPTER 3 SIMPLIFIED ANALYSIS MODEL FOR REAL-TIME DATA PROCESSING Theory Basic Shear-lag Equations Solution Methods Constant Boundary Conditions Piecewise Constant Boundary Conditions Arbitrary Boundary Conditions Numerical Results Unidirectional Single-Fiber Composite Unidirectional Laminated Composite Conclusions...75 CHAPTER 4 OSCILLATOR INTERROGATED TIME OF FLIGHT OPTICAL FIBER INTERFEROMETER Time-of-Flight Fiber Interferometer Time-of-Flight Measurements Oscillator Interrogated Interferometer Experimental Verification...85 v

5 4.2.1 Calibration of Interferometer Interrogation System Application of Displacement Sensor to FRP Retrofits Conclusions...98 CHAPTER 5 FEASIBILITY STUDY ON FRP RETROFITS WITH SELF-MONITORING CAPABILITY Experimental Methods Specimen Configurations Specimen Preparation Testing Procedure Simulations Experimental Results Comparison of Experimental Results and Theoretical Predictions Conclusions CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH Conclusions Recommendations for Future Research BIBLIOGRAPHY vi

6 LIST OF TABLES Table 3.1: Material properties of fibers, matrix and fiber reinforced lamina used for simulations...61 Table 5.1: Location and initial Bragg wavelengths for each specimen Table 5.2: Material properties used in the 2D FEA model vii

7 LIST OF FIGURES Figure 1.1: Photograph of wet lay-up FRP application to a bridge girder...4 Figure 2.2: Photograph of precured FRP strip prior to application...4 Figure 1.3: Common forms of debonding in FRP strengthened RC beams (from Teng et al., 2002)...6 Figure 1.4: Schematic diagram of example configuration of an intelligent FRP retrofit for strengthening of RC beam...7 Figure 2.1: Configuration of an ultrasonic pulse velocity interrogation system (from Mirmiran and Wei, 2001)...15 Figure 2.2: Configuration of electrochemical impedance spectroscopy for large scale structure with sensor locations (from Hong and Harichandran, 2004)...17 Figure 2.3: Schematic of near-field microwave measurement in FRP strengthened structures (from Akuthota et al., 2004)...19 Figure 2.4: Test configuration for PWAS with E/M impedance technique for FRP debonding monitoring (from Giurgiutiu et al., 2003)...22 Figure 2.5: Configuration of a low-coherence double Michelson interferometer (from Inaudi et al., 1994)...27 Figure 2.6: Schematic of Bragg grating sensor written into optical fiber core...28 Figure 2.7: Schematic of instrumentation for interrogation of fiber Bragg grating sensor in transmission...30 Figure 3.1: Schematic of discretization of multilayered structure cross-section into an n m layered composite...41 Figure 3.2: Configuration of unidirectional single fiber composite...62 Figure 3.3: Simulation results for the cantilevered single-fiber composite of Fig. 3.2 with E f / E m = 20 subjected to shear loading along entire length...65 Figure 3.4: Secondary loading condition for single fiber composite of Fig. 3.2 (a)...67 Figure 3.5: Average normal stress results of the single-fiber composite with E f / E m = 20 subject to outer surface shear loading over finite length...68 Figure 3.6: Average normal stress results of the single-fiber composite with E f / E m = 2 subject to outer surface shear loading over finite length...70 viii

8 Figure 3.7: Configuration of unidirectional laminated composite...72 Figure 3.8: Applied loadings for unidirectional laminated smart structure composite...73 Figure 3.9: Unit stress results for unidirectional laminated composite predicted using normal stress boundary conditions (FEA), shear stress boundary conditions (FEA) and shear stress boundary conditions (3D shear-lag)...74 Figure 4.1: Schematic of the interrogator for the time-of-flight interferometer...82 Figure 4.2: Photograph of laboratory set-up for the time-of-flight interrogation system...83 Figure 4.3: Example of sensor and reference signal acquired data points (post-filtering)...83 Figure 4.4: Photograph and dimensions of the 24-fiber prefabricated ribbon...85 Figure 4.5: Measured time-of-flight delay as a function of optical path length change achieved by splicing additional lengths of fiber to original section...87 Figure 4.6: Photograph of optical fiber ribbon tensile test using uniaxial tensile machine...88 Figure 4.7: Measured time-of-flight delay as a function of optical fiber length change...89 Figure 4.8: Photograph of C-channel concrete girders after sandblasting...92 Figure 4.9: Photograph of bonding epoxy application...92 Figure 4.10: Photographs of FRP bonding and fiber optic ribbon sensor embedding...93 Figure 4.11: Photograph of ribbon placement during installation of FRP reinforcement on lower surfaces on bridge girder...94 Figure 4.12: Load vs. deflection at supports for four-point beam bending tests...95 Figure 4.13: Photograph of testing on the full-scale girder which was loaded quasi-statically at the midspan until failure...96 Figure 4.14: Photograph of fiber splices for the twelve fiber optical fiber ribbon...97 Figure 4.15: Photograph of the midspan location along bridge girder post failure...97 Figure 4.16: Measured integral displacement and average strain as a function of mid-span load...98 Figure 5.1: Configuration of the double shear lap steel joint test specimen ix

9 Figure 5.2: Details of the double shear lap steel joint specimen Figure 5.3: Configuration of the three test specimens Figure 5.4: Photographs of wet lay-up procedure and test specimen preparation Figure 5.5: Photographs of FRP debonding simulations Figure 5.6: Configuration of the distributed sensing technique by use of distributed multiplexing FBG sensor network Figure 5.7: 2D FEA model created using ANSYS Figure 5.8: Structural geometries modeled in theoretical modeling Figure 5.9: Boundary shear loading applied to the FRP system under axial loading of 10 kn for model MOD A Figure 5.10: Self-monitoring FRP retrofit Figure 5.11: Average normal stress in optical fiber 4 for the four different modeled geometries Figure 5.12: FRP strengthened specimen FRP1 after specimen failure Figure 5.13: Specimen FRP 2 post-failure Figure 5.14: Strain measurements from specimen FRP Figure 5.15: Strain measurements from specimen FRP Figure 5.16: Strain measurements from specimen FRP Figure 5.17: Simulation results in specimen FRP 1 with comparisons to experimental results from FBG and strain gage sensors at location y = 18 mm Figure 5.18: Simulation results in specimen FRP 1 with comparisons to experimental results from FBG and strain gage sensors at location y = 25 mm Figure 5.19: Simulation results in specimen FRP 1 with comparisons to experimental results from FBG and strain gage sensors at location y = 75 mm Figure 5.20: Simulation results in specimen FRP 2 with comparisons to experimental results from FBG and strain gage sensors at location y = 0 mm Figure 5.21: Simulation results in specimen FRP 2 with comparisons to experimental results from FBG and strain gage sensors at location y = 100 mm x

10 Figure 5.22: Simulation results in specimen FRP 2 with comparisons to experimental results from FBG and strain gage sensors at location y = 175 mm Figure 5.23: Simulation results in specimen FRP 3 with comparisons to experimental results from FBG and strain gage sensors at location y = 0 mm Figure 5.24: Simulation results in specimen FRP 3 with comparisons to experimental results from FBG and strain gage sensors at location y = 25 mm Figure 5.25: Simulation results in specimen FRP 3 with comparisons to experimental results from FBG and strain gage sensors at location y = 50 mm Figure 5.26: Simulation results in specimen FRP 3 with comparisons to experimental results from FBG and strain gage sensors at location y = 100 mm Figure 5.27: Normalized strain profile for the CFRP in specimen FRP 3 from FBG sensor output at different load levels Figure 5.28: Strain profile for specimen FRP 3 from both simulation (MOD D) and FBG sensor output at different load levels Figure 5.29: Normalized strain profile for specimen FRP 3 from both simulation (MOD D) and FBG sensor output at different load levels Figure 5.30: Simulation results with sensor size effect considered for specimen FRP 1 with comparison to experimental results from FBG and strain gage sensors at y = 18 mm Figure 5.31: Simulation results with sensor size effect considered for specimen FRP 2 with comparison to experimental results from FBG and strain gage sensors at y = 0 mm Figure 5.32: Simulation results with sensor size effect considered for specimen FRP 3 with comparison to experimental results from FBG and strain gage sensors at y = 50 mm xi

11 CHAPTER 1 INTRODUCTION Fiber-reinforced polymer (FRP) composites have been widely used to strengthen existing structurally deficient civil infrastructures as an alternative to traditional heavy steel plate bonding techniques. FRP retrofits present several advantages such as low weight, high strength, corrosion resistance and easier installation in field applications (Rizkalla and Tadros, 2003). Researchers have applied and evaluated FRP composites as externally bonded supplements to various existing infrastructures, ranging from reinforced concrete (RC) (Teng et al., 2002), prestressed concrete (Rosenboom et al., 2007), steel (Schnerch et al., 2007) and wood structures (Dagher et al., 2002). The primary goal of FRP strengthening of structures is to improve either the structural flexural or shear strength of the existing structures, which will in turn extend their service life. Rosenboom et al. (2007) performed detailed experimental studies to evaluate the effectiveness and the structural performance of FRP strengthened, prestressed concrete infrastructures under static loading. Altogether eight C-Channel bridge girders were experimentally tested until failure in three-point loading. Six of the girders were strengthened with various CFRP composites including externally bonded wet lay-up systems, externally bonded pre-cured strips, and near surface mounted (NSM) FRPs. Rosenboom et al. (2007) concluded that the ultimate capacity of prestressed concrete bridge girders can be increased by as much as 73% using CFRP without sacrificing the ductility of the original member (Rosenboom et al., 2007). The testing results also demonstrated that through CFRP strengthening, the concrete crack spacing and width can be substantially reduced by as much 1

12 as 400% in comparison to unstrengthened girders. Rosenboom et al. (2007) thus concluded that amongst the existing strengthening methods, the application of externally bonded FRP is the most cost-efficient system that can be used for strengthening of existing deficient infrastructures. The fatigue behavior of FRP strengthened prestressed concrete infrastructures was further studied through experimental testing by Rosenboom and Rizkalla (2006). Seven bridge girders were strengthened with CFRP near-surface-mounted bars and strips, externally bonded precured strips and wet lay-up sheets. These experiments demonstrated that strengthening of existing structures using CFRP composites can reduce the stress ratio in the prestressing strands because of their capability to control crack widths and increase the overall stiffness. Similar strength improvements have also been reported for FRP strengthened steel and wood structures. Although these and other successful applications have been reported on the use of FRP to strengthen existing, structurally deficient infrastructures, experimental results also show that premature failures such as debonding of the FRP usually happen before the full structural strength is achieved. For example, Rosenboom et al. (2007) cautioned that proper field installation of the CFRP strengthening system is necessary to guarantee the full FRP strengthening performance. The authors further demonstrated that special attention should be paid to the debonding failure between the CFRP pre-cured laminates and the strengthened concrete structure. Additional strengthening methods such as the use of transverse FRP U-wraps were also suggested to delay debonding failures in externally bonded FRP infrastructures. This dissertation investigates the potential of self-monitoring FRP retrofits that could alter the user to localized debonding prior to failure of the strengthened structure. 2

13 1.1 FRP Retrofit Types and Processing Methods Currently three types of fibers and three types of resins are widely used for FRP retrofits, i.e. carbon, glass and aramid fibers and epoxy, polyester and vinylester resins (Teng et al., 2002). FRP retrofit composites are generally classified by their fiber type and referred to as: carbon-fiber-reinforced polymer (CFRP), glass-fiber-reinforced polymer (GFRP) or aramid-fiber-reinforced polymer (AFRP) composites. For structural strengthening in field applications, generally two forms of FRP composites are commercially available: wet lay-up sheets and precured strips (ACI Committee 440, 2006). Figures 1.1 and 1.2 show photographs of a typical wet lay-up and a typical precured FRP applied in field. As its name indicates, the application of wet lay-up FRP requires on-site saturation of unidirectional or multidirectional fiber sheets with resin. The saturated FRP layers are then bonded to the structural surface using the same resin. Finally both the saturated FRP and bonding epoxy are cured on-site, usually at ambient temperature. In comparison, precured FRP systems are resin saturated and cured prior to application in the field. A bonding adhesive is used to bond the precured FRP to the infrastructure. A third configuration exists in the form of a partially cured FRP prepreg system. Additional bonding resin may or may not be needed to bond the prepreg system to the structural surfaces depending on customer requirements (ACI Committee 440, 2006). Detailed descriptions of the available FRP forms under each of these three groups can be found in ACI Committee 440 (2006). 3

14 Figure 1.1: Photograph of wet lay-up FRP application to a bridge girder. Figure 1.2: Photograph of precured FRP strip prior to application 4

15 1.2 FRP Debonding Failures The successful extension of the structural service life through externally bonded FRP flexural or shear strengthening elements requires a condition of perfect bonding between the FRP and the structure. However, debonding of the FRP plates from the structure is a common failure mode for such reinforcements. Figure 1.3 shows the common debonding failure types for FRP strengthened reinforced concrete beams, dominated by either plate end debonding or intermediate interfacial debonding (Teng et al., 2002). As can be seen in the drawings, the debonding initiation and development are usually closely related to the concrete cracking development. Additionally, pre-existing defects such as air voids may be produced between the FRP and structural surfaces during the FRP application. These defects will also significantly impair the integrity of the strengthened structure. Since the FRP debonding prevents the full ultimate capacity of the strengthened structures from being utilized, it is crucial to be able to identify the initiation and development of plate debonding. 1.3 FRP Retrofits with Sensing/Controlling Capabilities Several non-destructive evaluation (NDE) techniques have been presented and analyzed by different researchers to identify FRP debonding. However, they are generally based on qualitative diagnostics and therefore can only be used to identify the existence of pre-existing debonding defects. These techniques will be reviewed in Chapter 2. Furthermore, it is critical to be able to identify the initiation and development of the FRP plate debonding. In fact, the prediction of FRP debonding still remains an open topic requiring more extensive research, both experimental and theoretical analysis (ACI 5

16 Committee 440, 2006). At the same time, once debonding has been identified, especially during inspection immediately after the FRP bonding completion, mitigation of the debonding is usually necessary. For example for wet lay-up FRP sheets, users typically inject resin or apply an overlapping FRP repair patch to cover the identified problematic zone. These repairs can be both time and labor consuming. (a) (b) (c) Figure 1.3: Common forms of debonding in FRP strengthened RC beams: (a) plate-end concrete cover separation; (b) plate-end interfacial debonding; (c) intermediate interfacial debonding (from Teng et al., 2002). 6

17 One hypothetical solution to address the issues previously mentioned for monitoring of debonding in FRP strengthened critical structures and provide options for debonding or cracking mitigation is an intelligent FRP retrofit concept (Jiang and Peters, 2006). Figure 1.4 shows an example configuration of such an intelligent FRP retrofit applied to strengthening of a reinforced concrete (RC) beam. Also depicted in the figure is a typical failure mode of the FRP through end concrete cover separation, which is associated with the development of cracking in concrete. Figure 1.4: Schematic diagram of example configuration of an intelligent FRP retrofit for strengthening of RC beam. This intelligent retrofit system is based on an easy-to-apply configuration of FRP pre-preg tapes with multiple stacked layers of piezoelectric or shape memory alloy (SMA) actuators and integrated global and local optical fiber sensors. Both local and global actuation 7

18 can be provided, while detailed local sensor measurements, combined with global sensor information, would identify the location and the form of the damages. Simultaneously, a controlled loading could be applied to the FRP layer through the actuators for future integrity evaluation. In principle, such intelligent retrofits would be able to not only monitor the integrity of the FRP bond and opening of cracks in the concrete, but also minimize crack openings and retard the progression of further FRP debonding. Expected advantages to this approach are that the retrofit of Figure 1.4: is compatible with and minimizes perturbation to the original concept of FRP strengthening; is able to distinguish between different failure modes, i.e. whether failure occurs in the FRP, in the reinforced concrete structure or in the adhesive interface between the two; is durable and cost-efficient for application to large critical civil infrastructures. There are several steps required to develop and demonstrate this intelligent FRP system, including the selection of global and local strain measurement sensors and actuators; theoretical modeling of the FRP response to embedded actuation and the sensor response to the FRP deformations for both bonded and debonded cases; and the demonstration of the compatibility and durability of the sensing / actuation system. The work of this dissertation accomplishes several of these tasks, in particular those related to sensing and interpretation of the state of debonding. The specific research scope and objectives of this dissertation are listed in the following section. 8

19 1.4 Research Scope and Research Objectives Three main research areas are targeted in this dissertation. Firstly, a simplified theoretical model based on shear lag analysis is developed for real-time data processing. Secondly, a global strain measurement system using fiber optic sensors newly developed in our research lab is adapted and demonstrated for monitoring of FRP strengthened large-scale infrastructures. Finally, experimental studies are performed to demonstrate the sensing capabilities and robustness of the embedded sensor concepts. The following research objectives were accomplished: Design of an FRP retrofit concept with sensing capability to monitor FRP debonding. This FRP system is based on global and local fiber optic sensors embedded in between FRP layers or at the surface of the concrete structure. It is demonstrated that this concept will function as a self-monitoring method under externally applied loads and in future studies, could be integrated within the intelligent/smart FRP retrofit system. Derivation of a simplified theoretical model for real-time data processing of sensor information. In this model, the majority of the calculations for the sensor response or actuator excitations are performed only once for a given FRP system geometry. A new computationally efficient three-dimensional shear lag model was developed through the extension of an optimal two-dimensional theory previously reported in the literature. This model could also be applied for the design of intelligent retrofits and future modeling of the retrofit response to actuation. 9

20 Choice or development of appropriate sensing systems for self-monitoring FRP retrofit system. Both global and local measurements were performed based on embedded fiber optic sensors. For global strain measurements, a newly developed oscillator interrogated interferometer, previously developed in our research lab, was adapted, in combination with a prefabricated fiber-optic ribbon, to large-scale FRP strengthened structures. The system sensitivity and durability were assessed through ribbon tensile testing and a full-scale concrete beam test. For local strain measurements, existing fiber Bragg grating (FBG) sensors were incorporated. Experimental demonstration of a self- monitoring FRP system. An example self-monitoring FRP system was fabricated incorporating embedded fiber optic sensors. The ability of the self-monitoring FRP systemto identify the initiation and development of crack-induced debondings in a FRP-steel splice joint are evaluated. Comparison of the predictions of the reduced theoretical model and the experimental results were also used to evaluate the accuracy of the model. 1.5 Thesis Organization Chapter 1 presents the motivation and application of this research. Existing FRP retrofit types and common bonding methods for field applications are introduced. Different FRP debonding failure modes are briefly reviewed from the published literature, with an emphasis on FRP strengthened RC infrastructures. The research concept of the FRP system with sensing / controlling capabilities is presented and the research scope and specific objectives are introduced. 10

21 Chapter 2 summarizes background research applicable to this dissertation. Several non-destructive evaluation methods previously used to identify FRP debonding are introduced. Literature reviews on existing global strain measurement systems, fiber Bragg grating (FBG) sensors, and the application of fiber optic sensors in FRP strengthened structures are presented. The application of shear-lag analysis as a simplified method to stress/strain analysis in fiber reinforced composite structures is also reviewed. Chapter 3 presents the development of a simplified analytical model for the design and real-time processing of the intelligent FRP system. The derivation of the model is outlined and solution methods for various boundary conditions are discussed. In order to validate the new model, two numerical examples are analyzed with comparison to finite element analysis predictions. Chapter 4 describes the calibration and application of the newly developed global strain measurement system for use in FRP strengthened infrastructures. Its measurement principle is described and various calibrations performed. A full-scale beam test is also performed to show the feasibility of this technique for future intelligent FRP applications. Chapter 5 describes the experimental methods and results of the application of an example FRP system with self-monitoring capability to a spliced FRP-steel joint. Several specimens with differing sensor layouts were tested to identify crack initiation in the epoxy bonded FRP-steel joints. The experimental data was then compared to theoretical predictions using the reduced shear-lag method derived in Chapter 3. Finally, conclusions from this research are summarized and suggestions for future research directions are outlined in Chapter 6. It is expected the simplified theoretical model 11

22 could be easily modified to incorporate both sensing and actuation capabilities in future work. 12

23 CHAPTER 2 BACKGROUND This chapter presents a review of the literature relevant to this dissertation. Existing non-destructive evaluation (NDE) methods used to identify the integrity of FRP strengthened structures are introduced. Previous research on FRP retrofits with self-monitoring capabilities is also reviewed. The application of fiber optic sensors in FRP strengthened structures, both for global and local strain measurements, is summarized. Finally the application of shear lag analyses to the stress/strain evaluation in fiber reinforced composite structures and the estimation of axial stresses in embedded optical fiber sensors for smart structures are introduced. 2.1 Non-Destructive Evaluation (NDE) Methods for FRP Retrofits Since FRP strengthened infrastructures are partially or totally wrapped by non-transparent FRP plates, it is difficult to monitor the initiation and existence of failures such as FRP debonding or underlying concrete failure. To better understand the necessity for an intelligent FRP retrofit for critical infrastructures, we first review traditional non-destructive evaluation (NDE) methods used to identify FRP debonding. A more extensive review of the application of NDE methods in FRP rehabilitated concrete structures was presented by Kaiser and Karbhari (2004). Acoustic sounding, recommended by ACI Committee 440 (2006), from tap testing using a hammer or a coin is the simplest method currently applied in the field to identify the existence of air voids. Well trained technicians are required to make qualitative judgments 13

24 from the emitted sounds, which, at the same time, are only emitted by larger voids. This measurement is therefore a qualitative evaluation of the condition of bonding between the FRP and the underlying concrete structure which requires that the structure be accessible to technicians. Two non-destructive evaluation (NDE) techniques, namely acoustic emission (AE) and ultrasonic pulse velocity have been proposed for application to structural integrity monitoring of FRP strengthened structures by Mirmiran et al. (Mirmiran et al., 1999; Mirmiran and Wei, 2001). Mirmiran et al. (1999) applied acoustic emission (AE) to monitor the structural integrity of FRP strengthened columns. Different types of damages in the FRP strengthened structures such as FRP debonding and concrete cracking are related to different AE energy levels and amplitudes. These measurements can be used in turn to determine the condition of structural health. Mirmiran et al. tested specimens with different configurations and demonstrated that a strong relation exists between the rate of change of the cumulative AE counts and damage within the underlying concrete. Mirmiran and Wei (2001) further showed the application of ultrasonic pulse velocity to monitor concrete-filled FRP columns. The characteristics of an ultrasonic pulse signal depend on the structural condition as it propagates through the structure. A typical experimental setup for ultrasonic pulse measurements is shown in Fig The authors report that the sensitivity of the UPV technique applied to FRP strengthened structures is much higher than when applied to plain concrete structures. For better monitoring performance the UPV damage index could be combined with the normalized acoustic emission counts by the previous AE method (Mirmiran and Wei, 2001). One problem with the ultrasonic pulse velocity method is that sometimes the ultrasounic wave can not travel sufficiently deep through the FRP due to the 14

25 high attenuation properties in these materials. This technique is also sensitive to the appearance of aggregates in concrete (Feng et al., 2002). The authors did not specifically address the sensitivity of the two techniques to detect and evaluate FRP debonding. Figure 2.1: Configuration of an ultrasonic pulse velocity interrogation system: (a) sensors and actuators mounted on FRP wrapped concrete cylinder; (b) measurement instrumentation (from Mirmiran and Wei, 2001). 15

26 Hong and Harichandran (2005) analyzed the use of electrochemical impedance spectroscopy for FRP debonding identification. Different sensor configurations were tested as electrodes to measure an equivalent impedance of the FRP strengthened structures. These sensor configurations were combinations of surface bonded metallic copper tapes on the CFRP surface, embedded stainless steel wires in the concrete, and the existing steel reinforcing bars in the RC concrete. Existing debonding between the CFRP and concrete surface was simulated by use of Teflon insert and the development of the debonding was manually controlled by a wedge (sharpened saw blade) which was driven with a hammer. Measurements were taken from the impedance between various sensor combination pairs at different interfacial crack lengths. The authors applied an equivalent circuit analysis to analyze the impedance spectra from the measurements. This analysis demonstrated that the capacitance parameters in the equivalent circuit correlated strongly with the interfacial crack length and can be used to assess the global state of the bond between CFRP sheets and concrete (Hong and Harichandran, 2005). The authors further suggested that the impedance measurements taken between the embedded wire sensors can be used to identify the debonding locations. However, for the extension of this technique to large scale infrastructures, the necessity to embed wire sensors and connect the circuit to the steel rebar can make the installation complex, as shown in Fig 2.2. Further, it is not always possible to connect the rebars in existing structures to the sensor network. 16

27 Figure 2.2: Configuration of electrochemical impedance spectroscopy for large scale structure with sensor locations (from Hong and Harichandran, 2004). Akuthota et al. (2004) demonstrated the detection of FRP debondings by the use of the near-field microwave NDT technique. The measurement principle is shown in Fig 2.3. Comparison can be made between the magnitude and phase of the reflected signal by the structure and the incident signal transmitted by the probe. The comparison measurement is 17

28 then used to identify the existence of FRP debondings because the reflected signal is related to the dielectric constant difference between the different materials and the air-voids. The authors fabricated a small 50 cm 38 cm 7.6 cm mortar block with several simulated debondings of different size, locations and geometries by injecting air between the FRP and the mortar surfaces. They reported the success of this technique in identifying air voids as small as 2.0 cm 0.5 cm. They also demonstrated the potential to assess the quality of FRP debonding repairs by epoxy injecting. However, in order for this technique to be sensitive to the presence of FRP debonding, it is necessary to optimize the standoff distance between the probe and the structural surface, which may be affected by the structural surface roughness and unevenness (Akuthota et al., 2004). In practical concrete structures, the existence of large aggregates and steel rebars could affect the effectiveness of the microwave technique because signals can be scattered, although the authors recommended the use of higher frequency and lower incident microwave power. In comparison, Feng et al. (2002) exploited the application of microwaves using sinusoidal plane waves to FRP wrapped concrete columns. A special dielectric lens was designed to focus the plane wave on the bonding interface to solve the problem of the relatively small reflection signals. The authors fabricated concrete circular columns without rebars which were wrapped with a three-layer glass FRP jacket. They introduced various voids and debonding conditions between the FRP layers and at the FRP - concrete interface and reported the successful detection of both the FRP debonding location and extent. As mentioned before for the near-field microwave technique, the interference from the steel rebars in reinforced concrete structures was not evaluated. 18

29 Figure 2.3: Schematic of near-field microwave measurement in FRP strengthened structures (from Akuthota et al., 2004). Shih et al. (2003) analyzed the application of non-contact infrared thermography as an NDE method to locate air-void induced FRP debondings. For a review of this technique applied to FRP structures Ghosh and Karbhari (2006). Shih et al. artificially introduced air blisters by the use of plastic rings to control the size of the trapped air voids. At first double lap shear concrete specimens were made with circular air voids of 15 mm and 20 mm in diameter between the FRP and concrete. The remote infrared thermography successfully located the air-voids. It is difficult for this technique to detect the exact radial location and the depth of the air-voids, due to the anisotropic properties of the FRP composites (Feng et al., 2002). Furthermore, the impact of the air-voids on the structural behavior was evaluated by the use of FRP strengthened RC beams under four-point bending tests. Two sizes of air-void induced debondings comprising 2.04% and 5.29% of the total FRP area were analyzed. It was reported that a curtailment, although not that significant, appeared in the beam strength and ductility curves, which may be due to the presence of air-voids. As pointed out by Feng et al. (2002), infrared thermography is not very accurate due to the 19

30 limitations in the heat distribution and the amplitude-only measurement and its application distance is limited. In addition, further damage may be introduced to the partial or totally non-cured FRP composites by the heat used to build a thermographic profile (Feng et al., 2002). Although each of the above techniques has proved to be useful for air void detection, each of them is based on qualitative diagnostics and focuses on the identification of pre-existing defects (Kaiser and Karbhari, 2004). In other words, no information on debonding initiation and development is provided. Also, since these NDE techniques are applied by a user directly at the location of the reinforcement or within a short detection distance, in practice the structures must be accessible for inspection. As realized, structural evaluation at both global and local levels are necessary to provide successful monitoring of FRP strengthened infrastructures. Therefore, in this dissertation we will develop a self-monitoring FRP retrofits with both global and local fiber optic sensors. In the next sections, background related to this technique will be introduced. 2.2 Self-Monitoring FRP Retrofits Previous researchers have similarly developed FRP retrofits with self-monitoring capabilities, applying sensors such as conventional strain gages and piezoelectric ceramics. This section will briefly review advances in this field. Sarazin and Newhook (2007) applied surface bonded strain gages to the monitoring of FRP strengthened concrete beams based on strain index concepts. Debonding zones of 20% and 30% of the span length were simulated by embedding polyurethane sheets between 20

31 the FRP and concrete. The beams were then tested under four-point bending loading at different loading stages, i.e. service load condition, cracked beam and ultimate load testing. The strain profiles along the FRP length were derived from the strain gage measurements. It was determined that a flat zone in the profile was observed within the debonding region, which can be used as an index of both FRP debonding location and size. The application of strain profiles to identify FRP debonding can be a very successful method, however, conventional strain gages are not suitable for many practical applications. Surface bonded strain gages are easily damaged and have a limited lifespan in field applications. Also the spacing limitations of bonding strain gages may prevent the identification of the critical length for FRP debonding. As recommended by Sarazin and Newhook (2007), the application of sensing systems based on fiber optic sensors is a viable solution, and will be pursued in this dissertation research. Another technique used in self-monitoring FRP retrofits is based on application of electromechanical piezoelectric sensors. Both theoretical and experimental results have been reported for surface bonded sensor patches (Cheng and Taheri, 2005), surface mounted piezoelectric wafer active sensors (PWAS) (Giurgiutiu et al., 2003) and surface mounted and embedded SMART (Stanford Multi-Actuator-Receiver Transduction) (SMART) Layers (Lin et al., 2001; Qing et al., 2006; Wu et al., 2006). Giurgiutiu et al. (2003) applied PWAS to a concrete block retrofitted with GFRP with artificially created debondings, as shown in Fig The initiation and development of FRP debonding were induced through loading of the GFRP. As the PWAS uses high frequency standing waves to measure the change of the electromechanical (E/M) impedance in the structure, it is very sensitive to local FRP debondings. Afterwards, a fatigue test was 21

32 performed on a large scale RC beam strengthened by CFRP composites with eighteen surface mounted PWAS applied for monitoring of the beam. The authors demonstrated that FRP debonding initiation and location can be detected with the distributed PWAS due to the fact that local damage induces localized high-frequency vibration modes (Giurgiutiu and Zagrai, 2001). Figure 2.4: Test configuration for PWAS with E/M impedance technique for FRP debonding monitoring (from Giurgiutiu et al., 2003). Wu et al. (2006) packaged an integrated network of piezoelectric sensors and actuators into a SMART Layer for the monitoring of FRP retrofits (Lin et al., 2001). The layer can therefore generate and receive diagnostic signals. The whole SMART Layer is in the form of a single ply that can be either embedded in the FRP laminate or externally bonded on the structural surface. Wu et al. (2006) tested the SMART Layer system on a three-girder two bay bridge deck assembly. During the testing, initial cracking was introduced to the concrete slab and then both the slab and girders were strengthened by 22

33 CFRP composites. Six SMART Layers, each composed of eight piezoelectirc transducers were applied by either direct bonding to the FRP surface or embedded between FRP and concrete. An ultrasonic stress wave was generated by the actuator piezoelectric sensor and propagated through the structure. PZT sensors at other locations receive the signal, whose properties are coupled with the structural condition along the path traveled by the wave between the actuator and sensors. A damage index in terms of the wave scattering energy was used to derive damage images from which the authors showed that FRP debondings can be identified. Although both of the above techniques showed their feasibility in application of the structural health monitoring mostly in aerospace engineering field, their application to FRP retrofits in civil infrastructures was limited. Also in the sensor layout, only local strain measurement sensors were incorporated. For example, with consideration of the SMART Layers size, it is difficult to embed both global and local sensors. In comparison, sensor systems based on multiplexing fiber optic sensors (FOS) can solve these problems. We will briefly introduce research and application background on FOS in the following sections. As mentioned before, with their durability, multiplexing capabilities, small size, material compatibility and ability to measure over several length scales, fiber optic sensors (FOS) are ideal as embedded sensors in the FRP retrofits. Also, both global and local structural conditions can be monitored if appropriate sensors are multiplexed into one common interrogation fiber. Most importantly, these sensors can be easily integrated with active materials to produce local monitoring and self-repair capabilities for the retrofits. Although much research has been put on the applications of fiber optic sensors to structural 23

34 health monitoring, especially in fiber reinforced composites, their use in structural integrity assessment in FRP strengthened infrastructures is still limited. Based on low coherence interferometry, Zhao and Ansari (2002) applied long gage length optical fiber sensors to measure interface strains between the FRP and concrete in a FRP repaired beam. In this preliminary study it was found that the gage length of the long gage sensor plays a critical role in the FRP peeling identification, which was associated with a drop in strain measurement in the sensor near the peeled section. Lau et al. (2001) embedded fiber Bragg grating (FBG) sensors at the interface between the concrete surface and external bonded FRP composites to measure the strain variations in a concrete beam during three-point bending testing. It was demonstrated that the FBG sensors indicated high local strain values when debonding occurred at the concrete-frp interface. 2.3 Fiber Optic Global Strain Measurement System Fiber optic sensors (FOS) have been widely applied in structural health monitoring applications due to their durability, multiplexing capabilities, small size, ability to measure over several length scales and electro-magnetic immunity. For the measurement of global structural parameters such as displacements, interferometric sensors are often applied due to the simplicity of the sensor and data acquisition systems and their ability to integrate local effects along the structure, eliminating the need to know a-priori where important events will occur (Measures, 2001). Of all the available interrogation methods for FOS interferometers, the measurement of time-of-flight (or time of arrival) is one of the most straightforward to apply. 24

35 Time-of-flight measurements are absolute, insensitive to intensity fluctuations from the laser source and do not require a separate reference fiber (although a separate fiber could be used for temperature compensation if required), all of which are important for structural monitoring applications (Donati, 2004). In addition, since the lightwave propagating through the sensor fiber does not physically interfere with a second lightwave, polarization fading does not have the same adverse effect that can deteriorate the output signals of other fiber interferometers (Measures, 2001). The primary disadvantage to applying time-of-flight measurements, however, is the extremely small optical path length change and associated time delay to be measured, i.e., on the order of millimeters (or ns). As alternatives, multiple reflectometry and interferometry interrogation methods have been developed for the measurement of optical path length changes (Measures, 2001). Among them, optical time-domain reflectometry (OTDR) has been one of the most commercially successful methods (Barnoski and Jensen, 1976; Personick, 1977). In the conventional OTDR configuration, a laser pulse of short duration is launched into a sensing fiber and the Fresnel reflections are measured. The light propagation time can be directly determined from the reflected powers at the fiber near and far ends. The resolution is controlled by the both the input pulse duration and the detector bandwidth. Although pulses with less than one picosecond duration can be generated, the sampling resolution of the data receiver is usually relatively low. This in turn limits the optical path length resolution of the conventional OTDR to the order of meters (Sorin, 1992). Furthermore, the resolution is reduced by fluctuations in the laser intensity since the power from the Fresnel reflections is measured. The resolution of OTDR has since been increased through the application of optical frequency domain reflectometry (OFDR) (Uttam and Culshaw, 1985; Braun and 25

36 Leyde, 1989). In OFDR, the reflected power amplitude and phase at each swept modulation frequency of the continuous optical signal are recorded and the time domain information is then derived using an inverse Fourier transformation. An optical path length resolution of several tens of millimeters has thus been achieved (Uttam and Culshaw, 1985). Resolutions below sub-millimeters or even micrometers have been reported using coherent interferometry (Barfuss and Brinkmeyer, 1989) or low coherence interferometry (also called white light interferometry) (Takada et al., 1987; Youngquist et al., 1987; Inaudi et al., 1994) with Michelson or Mach-Zehnder interferometers as time-of-flight interrogators. Due to its higher resolution, low coherence interferometry has been more widely analyzed. However, the requirement of matching the sensing optical path length with a second reference path length (within the coherence length of the laser source) necessitates movable mirrors or other components. A typical low coherence interferometry is shown in Fig. 2.6 (Inaudi et al., 1994). As can be seen, this adds complexity to the data acquisition system and can severely limit the measurement range of the system. In addition to complex interferometric interrogation techniques, two methods have been applied to increase the displacement or strain sensitivity of time-of-flight measurements by effectively increasing the sensing fiber length. The first is to physically increase the sensor length by wrapping several parallel sensing fibers adjacent to one another. The embedment of several optical fibers in close proximity in a structure can be both labor and time intensive therefore specially designed prefabricated fiber panels (Fuhr et al., 1993) and multi-pass looped FOS patches (Lou et al., 1995) have been proposed to extend the optical path length. A second method to increase the effective optical path length is to circulate the lightwave through the sensing fiber multiple times through a reentrant loop (Lou et al. 1995; 26

37 Danielson, 1985). This technique does not require additional embedded fiber lengths; however it significantly increases the complexity of the data acquisition and is limited to a few recirculations due to the rapid signal decay. Figure 2.5: Configuration of a low-coherence double Michelson interferometer (from Inaudi et al., 1994). The above fiber optic interferometer interrogation methods have each been applied successfully to monitor global parameters of civil infrastructure systems. However, there is now a significant need to develop a small, low power, relatively inexpensive demodulation system that could be applied for field applications to civil infrastructures (Measures, 2001). Meggitt et al. (2000) replaced the translational stage with an optical fiber wrapped around a piezo-electrical cylinder to control the reference path length. This configuration removes the 27

38 need for externally moving components. On the other hand, distributed measurement systems based on stimulated Brillouin or Raman scattering measurements or multiplexed low coherence interferometry systems provide an additional benefit of strain localization along the optical fiber. 2.4 Fiber Optic Bragg Grating Sensors The fiber Bragg grating (FBG) sensor, shown in Fig. 2.6 is written into the core of an optical fiber in the form of a periodic modulation of the local index of refraction (Meltz et el., 1989). Due to the change of the core index of refraction, a specific wavelength from a broadband light source will be reflected, according to the criterion λ = 2Λ n (2.1) B eff where λ B is the peak (Bragg) wavelength of the reflected spectrum, Λ is the period of the core index modulation and n eff is the fiber core effective index of refraction. Figure 2.6: Schematic of Bragg grating sensor written into optical fiber core. 28

39 Figure 2.7 shows a typical interrogation used to measure the transmitted spectrum of a Bragg grating sensor. A similar approach could be applied to measure the reflected spectrum, if required. A narrow wavelength of light is emitted from the tunable laser, for which the center wavelength can be scanned over approximately 40 nm. As the wavelengths are scanned, the output transmission from the FBG sensor is measured by the photodetector. By controlling the scanning operation and measured transmitted intensity data acquisition through LabView, the transmitted spectrum of the Bragg grating sensor can be reconstructed. While producing an accurate measurement of the complete reflected spectrum, a spectral scan of nm can only be achieved at a rate less than 0.01 Hz. This is particularly limiting when several Bragg gratings are multiplexed in series and are to be monitored at the same time. For practical measurements, this data acquisition rate is too slow. Therefore, a Bragg grating sensor interrogation system recently developed by Micron Optics Inc. that measures only the peak wavelength in reflection, however can acquire the data at a rate of 1 KHz was also applied in the work of this dissertation. More details will be provided on these measurements in Chapter 5. The operation principle of the FBG as a strain or temperature sensor is through the induced changes in the FBG period and the index of refraction due to the applied strain or temperature. The change in index of refraction is due to either the strain-optic or thermo-optic effect. If only axial strain is applied to the FBG sensor, the relation between wavelength shift and strain is linear and can be derived as 29

40 Figure 2.7: Schematic of instrumentation for interrogation of fiber Bragg grating sensor in transmission. Δ λ = (1 p ) λε (2.2) B eff B 2 where Δλ B is the wavelength shift, ( /2) ( 1 ν) peff = ne p ν p is the is the effective strain-optic coefficient for the axial loading case, p 11 and p 12 are strain-optic coefficients of silica, n e is the effective index of refraction of the propagating mode (here we consider only the fundamental mode) and ε is the applied axial strain (Lawrence et al., 1997). When other applied strain components are applied to the FBG, the wavelength shift remains linear, however the effective strain-optic coefficient must be modified to include the other components. 30

41 FBG sensors have been widely applied as embedded sensors in fiber-reinforced composites to monitor structural integrity. A thorough review would be too extensive for this dissertation, such a review can be found in Measures (2001). In particular, FBGs can be easily embedded during the fabrication of laminated composites and demonstrate excellent durability over the lifetime of the compsite. Additionally, strains at different structural locations can be measured simultaneously by multiplexing multiple FBG strain sensors together into a single optical fiber to reduce the ingress/egress connections and instrumentation systems required. Different techniques can be used to acquire data from the different FBG sensors. The most common technique is called wavelength division multiplexing, in which the multiplexed FBGs are written at different wavelengths and therefore can be individually identified in the transmitted or reflected spectrum of the optical fiber. As mentioned previously, scanning the required wavelength range with a tunable laser cannot be done at suitable rate, therefore, a modified high-speed interrogation system was applied in the experimental study described in Chapter Shear-Lag Analysis for Fiber Reinforced Composites As a computationally efficient analytical method, the shear-lag method has been widely applied as a computationally efficient analytical method to analyze the stress distribution in fiber reinforced composite structures. Cox (1952) first introduced the shear-lag method for a single fiber embedded in a matrix material with axisymmetric geometry. Hedgepeth (1961) later derived the shear-lag method through the concept of influence functions and applied the solution to the calculation of static and dynamic stress 31

42 concentrations in neighboring fibers surrounding broken fiber in a unidirectional reinforced composite. Nairn and Mendels (2001) present an extensive review of the literature on shear-lag methods applied to axisymmetric and two dimensional (2D) planar geometries. Historically, many researchers have combined the shear-lag based calculation of load transfer from broken fibers to surrounding intact fibers with statistical fiber failure models for the prediction of composite strengths (Landis et al. 1999; Landis and McMeeking, 1999; Beyerlein and Landis, 1999; Okabe et al., 2001; Okabe and Takeda, 2002; Xia et al., 2002). More recently, the shear-lag method has been applied to estimate axial stresses in embedded optical fiber sensors for smart structures (Yuan and Zhou, 1998; Ansari and Yuan, 1998; Yuan et al., 2001; Okabe et al., 2002; Prabhugoud and Peters, 2003; Li et al., 2006), analyze fiber bundle pull-out tests (Brandstetter et al., 2004), calculate effects of residual stresses on stress fields in delaminated cross-ply laminates (Banks-Sills et al., 2003), and predict the electrical conductivity of a graphite reinforced composite (Xia et al., 2003). With recent advances in experimental methods such as micro-raman spectroscopy, it has been possible to evaluate the shear-lag method predictions for load transfer between neighboring fibers. In general, the shear-lag method predicts the load transfer in high volume polymer matrix composites well. However, this is not the case for composites with matrix to fiber modulii ratios near unity due to the increased role of the matrix axial stresses in the (Beyerlein and Landis, 1999). Stress concentrations in neighboring fibers due to fiber breakage have also been shown to depend on the matrix modulus (Wagner et al. 1996). Anagnostopoulous et al. (2005) induced local fiber discontinuities in high volume fraction aramid/epoxy composites and applied laser Raman spectroscopy from which they demonstrated that the quality of the interface was maintained, justifying the use of an elastic 32

43 transfer model. Furthermore, the authors demonstrated that at applied strains below the residual strain threshold in high volume composites, the local residual stresses strongly influence the local stress transfer. This effect can be accounted for in shear-lag analyses, however the local shear modulus of the as-processed composite must be known for the shear-lag method to yield good predicitions. Once the residual strains have been overcome by the applied strains the shear-lag method based on the shear modulus of the bulk matrix material works well. These experimental results emphasize the importance of the role of the matrix axial and residual stresses, however give strong support to the use of shear-lag methods for appropriate applications. Most of the early shear-lag models were based on the assumption that the matrix material shear behavior is controlled by the axial displacements of the surrounding fibers and the role of its axial stiffness can be neglected. The reduced role of the matrix is therefore merely to transfer axial loads in adjacent fibers through shear deformation. In other words, the matrix material is modeled as a shear spring to connect the embedded fibers. 1 As a result, these methods are only suitable for composites with a high modulus ratio between fiber and matrix and high fiber volume fraction (Nairn and Mendels, 2001). Xia et al. (2002) confirmed this limitation by comparison of shear-lag methods to finite element analyses. In addition, these early models cannot differentiate between a transverse crack stopping at the fiber-matrix interface or continuing through the matrix. This can result in significant errors for systems where the matrix strain to failure is lower or similar to that of the fiber such as for ceramic or metal matrix composites (Beyerlein and Landis, 1999). 1 An alternate perspective is that a matrix material with zero axial stiffness yet finite shear stiffness can be interpreted representing a matrix material that has failed in tension through either cracking or yielding at low stress magnitudes (Landis and McMeeking, 1999). 33

44 Recent efforts have been made to incorporate the role of the matrix stiffness into shear-lag analyses. Beyerlein and Landis (1999) and Landis and McMeeking (1999) derived a shear-lag method for which the governing equations for a periodic system are generated via the finite element method. The stiffness parameters for the model were calculated in a mechanically consistent manner through the principle of virtual work. The resulting finite element equations describing the displacement distributions were transformed into coupled ordinary differential equations (ODEs) by taking the limit as the longitudinal step size approaches zero and solved using Fourier transform techniques. By adding additional degrees of freedom into the matrix material elements of the original finite element model, Landis and McMeeking (1999) were then able to include the role of the matrix axial stiffness, random fiber spacing and non-perfect interface conditions. However, for any of these modifications the solution procedure becomes more complex as Fourier transform techniques can no longer be applied (Landis and McMeeking, 1999). Landis and McMeeking (1999) demonstrated that this shear-lag reduced finite element method produces excellent predictions for a variety of applications, however the modeling of a specific geometry through the development of a finite element model and then its transformation into differential equations can be computationally intensive (although significantly less computationally intensive than modeling the complete material system using a full three-dimensional finite element method). In a separate approach, Nairn (1997) and Nairn and Mendels (2001) revisited the shear-lag equations for axisymmetric and multilayered planar problems, starting from exact elasticity equations. Furthermore, they derived an optimal shear-lag theory based on the minimum number of non-unique assumptions required. The shear stress in each fiber or 34

45 matrix layer was interpolated through the use of shape functions. This optimal shear-lag theory yields a series of coupled ODEs with constant coefficients, similar to the previous finite element approach. However, the resulting system of equations are functions of the unknown interfacial shear stresses or layer average axial stresses directly and can be uncoupled using eigenvalue and eigenvector analysis. Nairn and Mendels (2001) also demonstrated that the appearance or propagation of a crack can be rapidly introduced through a change of the boundary conditions and the re-calculation of the necessary constants, without re-solving the entire system. As the derivation does not neglect the matrix axial stiffness, the optimal shear-lag theory of Nairn and Mendels (2001) works well for fiber/matrix or other multilayered structures independent of the modulus ratio. As for previous methods, the prediction of integrated quantities such as average axial stresses, displacements or total strain energy work better than those for shear stresses, transverse stresses and energy release rates (Nairn, 1997). However, the results of Nairn and Mendels (2001) do demonstrate that the prediction of interfacial shear stresses can be a reasonable approximation for many applications. Nairn (2004) also incorporated the role of imperfect or debonded interfaces through axial displacement discontinuity parameters (Nairn, 2004). Hedgepeth and Van Dyke (1967) first extended two-dimensional (2D) shear lag models to three-dimensional (3D) models for square and hexagonal fiber arrays based on their previously derived influence functions. 2 It was assumed, as for many later works, that the stress transfer to a given fiber is only influenced by its immediate neighbors. The efficiency 2 Here we use the terminology common in the literature that a 2D shear-lag model reduces a 2D material system to a 1D planar or axisymmetric array of fiber and matrix elements, whereas a 3D shear-lag model reduces a 3D material system to a 2D array of fiber and matrix elements. We do not imply that either system must be geometrically periodic. 35

46 of the model comes from the periodic geometry, limiting the application to transversely isotropic material systems. In a more recent example, Okabe et al. (2001) extended the 3D shear-lag model for square fiber arrays to predict fiber damage accumulation. As pointed out by Landis et al. (1999), one of the limitations of these 3D shear-lag models is in the simplification of the matrix as 1-D shear springs connecting one specific fiber only to its nearest immediate neighbors. Based on the principle of virtual work, Landis et al. (1998; 2000) expanded this 3D shear-lag model to connect each fiber to its nearest and near nearest fibers. In this method, the matrix is modeled as 3D finite elements with an infinite square or hexagonal array of fibers. Each of these examples provides rapid calculation of internal stresses, however their limitations are the same as the classical 2D shear-lag methods previously mentioned due to the neglect of the matrix axial stiffness. Landis and McMeeking (1999) thus incorporated the matrix stiffness into the 3D model of Landis et al. (1998) by adding extra degrees of freedom into the matrix elements. This model can only be applied to transversely isotropic material systems due to the model reduction through periodic fiber array geometries and near-neighbor influence assumptions. A second approach to modeling 3D laminated composites using a shear-lag analysis is to model the system as a multilayer planar structure and consider each lamina as homogeneous (Han et al., 1988; Dharani and Tang, 1990). For cross-ply laminates, the laminate model can be reduced through periodicity conditions, otherwise each layer must be modeled separately. While suitable for the prediction of transverse cracking or delaminations in the lamina (Han et al., 1988; Dharani and Tang, 1990; Banks-Sills et al, 2003), this approach cannot be applied to laminates with embedded sensors and actuators as it cannot 36

47 resolve stress distribution within each layer. Chapter 3 derives a 3D shear-lag model by extension of the 2D optimal shear-lag method of Nairn and Mendels (2001). This model will be applicable not only to analysis of unidirectional laminated composites with different material properties, but also to future analysis of smart structures with embedded sensors and actuators, such as seen in Fig. 1. It is important to point out that bending of the laminate cannot be modeled using the method of this article due to the fact that transverse shear deformations are neglected (Xia et al., 2002). Residual stresses could be included in this model using the application of the superposition principle, although this will not be explicitly addressed directly in this article (Landis and McMeeking, 1999). 37

48 CHAPTER 3 SIMPLIFIED ANALYSIS MODEL FOR REAL-TIME DATA PROCESSING Since the proposed self-monitoring FRP system is based on a multi-layer configuration of various materials, the stress/strain transfer between the FRP and sensor layers are expected to be complex. Although finite element analysis (FEA) methods could always be applied to predict the response of the sensors to various loading cases, a reduced model is needed to interpret sensor data. For the inverse problem of identified changes in loading or bonding condition, multiple iterations may be required to match the data to assumed debonding states. Full 3D FEA models are thus too computationally intensive to be applied for real-time processing of the sensing information. As an added benefit, the reduced model can also be used for design of self-monitoring FRP retrofits. Similar to finite element method, a structural model is created in matrix form which can be generated so that only changes to external loading or boundary conditions need to be incorporated for each iteration and the majority of the calculations are only performed once. The difference in this model is that the number of degrees of freedom is reduced by orders of magnitude through incorporating the shear-lag assumptions into each element. The stress state predictions are therefore not as accurate as full 3D finite element analyses, however it will be demonstrated that they are reasonably close for much less computational effort. Therefore, this chapter derives a new, computationally efficient 3D shear lag model through the extension of a 2D optimal shear-lag theory previously reported in the literature (Nairn and Mendels, 2001). The theoretical derivation of this new model will be briefly presented and two numerical examples will be analyzed. In addition, comparisons between 38

49 the results from this method and full 3D FEA analyses will be made. It is important to point out that bending of the laminate cannot be modeled using this method due to the fact that transverse shear deformations are neglected (Xia et al., 2002). In future studies, the role of embedded actuators could also be incorporated into the model through local boundary conditions on axial or interfacial shear stresses. Residual stresses could be included in this model using the application of the superposition principle, although this will not be explicitly addressed in this chapter (Landis and McMeeking, 1999). The derivation of the new 3D optimal shear-lag model is presented in Section 3.1. The characteristics and the solution algorithm of the modeling equation are discussed in Section 3.2. Numerical predictions of the model for two separate configurations are presented and compared to full 3D finite element analyses in Section 3.3. Finally possible applications and limitations of the new model are discussed. 3.1 Theory This section derives the 3D shear-lag model through extension of the 2D optimal shear-lag model of Nairn and Mendels (2001). The derivation procedure therefore follows that of Nairn and Mendels (2001) closely. To begin, the structure cross-section is discretized and the two relevant interfacial shear stress components are interpolated through shape functions. Afterwards, the fundamental shear-lag assumption is applied, leading to expressions for the relative average axial displacements of adjoining cells in terms of the average interfacial shear stresses. By enforcing equilibrium in the axial direction, the axial normal stresses are then related to the average axial displacements and average interfacial 39

50 shear stresses. Combining these relations yields a system of ordinary derivative equation (ODEs) from which the average interfacial shear stresses can be solved through application of boundary conditions. It will be shown that the solution method of the system of ODEs highly depends upon the form of the boundary conditions. Unlike the 2D shear-lag method of Nairn and Mendels (2001), this ODE system cannot be transformed into a system in terms of the average axial stresses due to the limited number of independent equations. While this places a restriction on the types of boundary conditions that can be applied, the method for appropriate problems will be shown to perform well in the numerical examples of Section Basic Shear-lag Equations We first idealize the cross-section of the unidirectional multilayered structure as a 2D array of n m material cells as shown in Fig Depending upon the scale of the problem and the computational resources available, the material cells can be individual fibers, embedded sensors and actuators, regions of matrix or reinforced material or an entire lamina. The coordinate system is oriented such that the x and z directions are in the plane of the cross-section and the y axis is the direction orthogonal to the cross-section. We also define the local displacements in the x, y and z directions as U, V and W respectively. The material cell labeled (i, j) is bounded by the domain x i-1 <x<x i and z j-1 <z<z j (i = 1 n, j = 1 m) with thicknesses t i * and t j in the x and z directions respectively. The material of each cell is considered to be linear thermoelastic and orthotropic with relevant material properties E (i,j) x, E (i,j) y, E (i,j) z, G (i,j) xy, and G (i,j) yz. Thermal loading is not included explicitly in this analysis, but will be discussed later in the description of future applications. 40

51 Figure 3.1. Schematic of discretization of multilayered structure cross-section into an n m layered composite. The x and z coordinates are in the plane of the cross section, while the y coordinate is in the axial direction. The cell (i, j) and its neighboring cells are indicated. The U, V and W displacements are also labeled. Inset shows interfacial shear stress distribution applied on the boundary of an infinitesimal section of the cell (i, j) as well as assumed averaged distribution. We define the nondimensional local coordinates for the cell (i, j), 41

52 X i x x z z i 1 j 1 = ; Z * j = (3.1) t t i j where 0 X < 1 and 0 Z < 1. The interfacial shear stresses will be interpolated < i < j between the values at the cell boundaries through shape functions applied in the same manner as in Nairn and Mendels (2001), except that the interpolation is expanded to the two independent shear stresses τ xy and τ yz. Accordingly, we first evaluate the interfacial shear stresses on each of the boundaries of the cell (i,j): τ ( x, y, z), xy x= x i 1 τ ( x, y, z), xy x= x i τ ( x, y, z) yz z= z j 1 and τ ( x, y, z) yz z= z ji, which are each indicated in the inset of Fig These four interfacial shear stresses will later be related to the average axial stress in cell (i, j) through equilibrium conditions. To simplify the notation, we further write τ ( x, y, z) = τ [ x, z ] xy x= xa, z= za xy a a where it is implied that τ x, z ] is a function of y. As we will be concerned with averaged stress xy[ a a values, we need only consider the variation of τ xy in the x direction through a cell and not within the plane over which it is applied to the cell (y-z plane). Therefore, we will interpolate τ xy in the x direction between its average value on the left and right hand boundaries of the cell. For further simplification of the notation, we label these average boundary stresses as τ 1 j [ ] τ [ ] z ave j xy xi 1, zj 1 < z< z j = τxy xi 1, z dz z xy xi 1 t j 1 j τ 1 j [ ] τ [ ] z ave j xy xi, zj 1 < z< z j = τxy xi, z dz z xy xi t j 1 j (3.2) 42

53 Similarly, we will interpolate τ yz in the z direction between its average values on the upper and lower boundaries of the cell and therefore write, τ 1 x ave i i yz xi 1 < x< xi, z j 1 = τ, * x yz x z j 1 dx yz zj t τ 1 i 1 i τ 1 x ave i i yz xi 1 < x< xi, z j = τ, * yz x z j dx yz zj t x τ i 1 i (3.3) The location of each of these averaged interfacial shear stresses on a material cell is also shown in Fig In terms of these averaged stresses, we now interpolate the shear stresses through the cell as τ ( xyz,, ) = τ [ x ] L + τ [ x] R ( ij) j ( i, j) j ( i, j) xy xy i 1 xy i τ ( x, yz, ) = τ [ z ] B + τ [ z] T ( ij) i ( i, j) i ( i, j) yz yz j 1 yz j (3.4) ( i, j) ( i, j) ( i, j) ( i, j) where L ( X i ), R ( X i ), T ( Z j ) and B ( Z j ) are shape functions active at the left, right, top and bottom interfaces of unit (i, j). The choice of shape functions is arbitrary and can be different for each individual cell, the only requirement being that each function satisfy the boundary conditions, (, i j) L (0) = 1 (, i j) L (1) = 0 (, i j) R (0) = 1 (, i j) R (1) = 0 (, i j) B (0) = 1 (, i j) B (1) = 0 (, i j) T (0) = 1 (, i j) T (1) = 0 (3.5) 43

54 The interpolation of shear stresses through linear shape functions in a single direction has been previously applied by Nairn and Mendels (2001) and McCartney (1992) with excellent results for 2D shear-lag models. Nairn and Mendels (2001) also investigated scaling the shape functions to match the strain energy of the interpolated stress field and the actual stress field for a particular multilayered structure. Their results demonstrate that such a modified shape function does improve the prediction results of the optimal shear-lag method. The choice to interpolate the two shear stress components in only a single direction through (3.4) is the simplest method of interpolation and will be shown in this article to predict averaged axial stress distributions well, even when linear functions are applied. Improvements to this interpolation for a given problem could made by either modifying the interpolation functions L (, i j), (, i j) (, ) R, T i j and (, i j) B (per Nairn and Mendels (2001)), increasing the order of the interpolation, e.g. ( ) ( ) i j ( ) + τ ( ) τ ( xyz,, ) = τ [ x, z ] L X, Z + τ [ x, z ] L X, Z ( ij) ( i, j) ( i, j) xy xy i 1 j 1 i j xy i j 1 i j τ (, i j) (, ) xy[ xi 1, zj ] L Xi, Z j xy[ xi, zj ] R Xi, Z j + (3.6) or increasing the number of cells. The relative merits of each of these strategies will be discussed in Section 3.2. Now that the shear stresses have been formulated in terms of boundary values, we apply the fundamental shear-lag assumption that U / y << V / x and W / y << V / z, 44

55 τ τ xy yz G G xy yz V x V z (3.7) Therefore, any loading applied to the multilayered structure must be slowly varying in the y direction as compared to the in-plane cell dimensions in order for the shear-lag approach to model the problem well. We now enforce displacement compatibility in the y-direction between adjacent cells along their common borders, and therefore derive expressions for average displacements in the y direction within a cell. Substituting Equation (3.1) and Equation (3.4) into Equation (3.7) and re-arranging terms yields t (, i j) (, i j) V V x i j (, i j) j (, i ) = = τ (, ) xy [ x i 1] L xy [ x i j + τ i ] R j Xi x Xi G xy (3.8) and t (, i j) (, i j) V V z j i (, i j) i (, ) = = τ z (, ) yz j 1 B τ yz z i j + i j j T Z j z Z j G yz (3.9) Following the transfer method (Nairn and Mendels, 2001; McCartney, 1992) we multiply both sides of Equations (3.8) and (3.9) by A X ) and A Z ) respectively, ( 1 i ( 2 j 45

56 V t ( A X ) ( A X ) L R (, i j) i j (, i j) j (, i j) 1 i = 1 i τ (, ) xy[ x i 1] τxy[ x i j + i] Xi G xy (3.10) V t ( A Z ) ( A Z ) B T (, i j) j i (, i j) i (, i j) 2 j = 2 j τ z (, ) yz j 1 τ yz z i j + j Z j G yz (3.11) where A 1 and A 2 are arbitrary constants. Integration of Equations (3.10) and (3.11) through * the thicknesses t and t respectively yields i j 1 (, i j) (, i j) (, i j) 1 x= xi 1 x= xi 1 0 ( A 1) V AV + V dx i = t G 1 { } [ ] { } (, 1 i j ) j ti (, i j ( ) ( ) ) j[ ] 0 1 i i xy x i 0 1 i i x A X R dx τ + A X L dx τ i j xy i 1 i (, i j) (, ) xy Gxy (3.12) 1 (, i j) (, i j) (, i j) 2 z= zj 2 z= zj 1 0 ( A 1) V AV + V dz j = G t 1 t 1 { } { } (, ) j ( A Z ) T dz ( ) (, ) 0 2 j τ z + A Z B dz 0 2 j i j τ 1 j i j i i j i z (, i j) j yz j (, ) j yz j yz G yz (3.13) Now we apply Equation (3.12) to two horizontally adjacent cells while choosing A 1 =1 for unit (i+1, j) 1 ( i+ 1, j) ( i+ 1, j) V x= x+ V dx i+ 1 i = 0 t G 1 { } [ ] { } ( 1 i+ 1, j ) j ti 1 ( i+ 1, j (1 ) j Xi 1) R dxi+ τxy x i 1 (1 Xi 1) L dxi+ 1 τxy[ x i] 0 0 i ( i+ 1, j) ( i+ 1, j) xy Gxy (3.14) 46

57 and A 1 =0 for unit (i, j), 1 (, i j) (, i j) V x= x+ V dx i i = 0 t G 1 { } [ ] { } (, 1 i j ) j ti (, i j ) j XR i dx i τxy x i i i xy[ x XL dx τ i 1] 0 i j 0 i (, i j) (, ) xy Gxy (3.15) The relative axial displacement between the two cells can then be solved by subtraction of Equation (3.14) from Equation (3.15) and simultaneously enforcing displacement continuity along their common border, ( i+ 1, j) ( i, j) V x= x = V i x= xi, { (1 1) } [ x i+ τ xy i+ 1] ( i+ 1, j) ( i, j) i 1 ( i+ 1, j) j V dx i+ 1 V dx i = + X R dx ( 1, ) i i+ j + G 0 xy i (, i j) 0 xy t t 1 1 i 1 ( i 1, j) t + + i ( i, j) j + (1 X 1) i 1 i ( 1, ) (, ) [ x i j ] 0 i L dx X i j 0 ir dx τ + + G xy G xy i (3.16) xy t 1 (, i j) j { XL i dx i} τ xy[ x i 1] G Similarly, applying the same procedure to two vertically adjacent cells (i, j+1) and (i, j) by evaluating Equation (3.13) with A 2 = 1 for unit (i, j+1) and A 2 = 0 for unit (i, j) and applying the displacement continuity condition ( i, j) ( i, j+ 1) V z= z = V j z= z j, 47

58 { (1 j ) } (, i j 1) (, i j) j (, i j 1) i V + + dz j+ 1 V dz j Z (, 1) 1 T + dz j 1 τ z 0 0 i j yz j+ 1 G yz j i (, i j) τ yz z j 0 1 yz t = tj 1 1 t 1 + (, i j+ 1) j (, i j) i + (1 Z (, 1) 1) j 1 z i j 0 j B dz Z (, i j) j 0 jt dz τ + yz j + G yz G yz t 1 (, i j) { ZB j j dz j G } (3.17) Finally, integration of Equation (3.16) with respect to x where x i-1 < x < x i yields ( i+ 1, j) ( i, j) < V > < V >= t G i+ 1 ( i+ 1, j) 0 xy 1 ( i+ 1, j) j { (1 1) 1} [ x Xi+ R dxi+ τ xy i+ 1] t + + G t 1 (, i j) j { XL i dxi} τ xy[ x i 1] G 1 1 i+ 1 ( i+ 1, j) i ( i, j) j (1 X ( 1, ) 0 i 1) L dx i 1 X i j ( i, j) 0 ir dx i τ xy xy G xy i (, i j) 0 xy t [ x ] i + (3.18) where we define the notation < > to indicate the value averaged over the surface area of the cell, < >= 1 xi z j dxdz. Equation (3.18) establishes the relationship between the x z t t i 1 j 1 i j axial displacements and the interfacial shear stresses τ xy at x i-1, x i and x i+1 in any two horizontally adjacent cells which will be related to axial stresses later. Integrating Equation (3.17) we derive a similar relationship between the average axial displacements in any two vertically adjacent cells, 48

59 (, i j+ 1) (, i j) < V > < V >= G t 1 { (1 Z ) T dz } τ j+ 1 (, i j+ 1) i (, 1) j 1 j 1 z i j yz j+ 1 yz tj 1 1 t 1 + (, i j+ 1) j (, i j) i + (1 Z j 1) B dz + j+ 1 + Z jt dz j τ z (, i j+ 1) 0 (, i j) 0 yz j G yz G + yz t 1 (, ) { Z B dz G } τ j i j i (, ) j j z i j 1 0 yz j yz (3.19) For an m x n cell configuration, there are a total of 2mn (m+n) independent relationships through Equations (3.18) and (3.19). Our next step is to convert the left hand side (LHS) of Equations (3.18) and (3.19) to average axial stresses in order to relate these to the interfacial shear stresses as well. First we consider stress equilibrium in the y direction of the cell (i, j), σ y τxy τ yz + + = 0 y x z (3.20) Integration of Equation (3.20) over area the surface area of the cell yields tt d < σ > (, i j) i j ( y ) j j i i + tj ( τxy [ x i ] τxy [ x i 1] ) + ti ( τ yz z j τ yz z j 1 dy ) = 0 (3.21) where < ( i, j) σ y > is the unit average normal stress in the cell. Next we consider Hooke s law in the y direction 49

60 ε y V σ y νxyσx ν yzσz = = y E E E y x z (3.22) Differentiating Equation (3.22) with respect to y, σ ν σ ν σ 2 ( i, j) (, i j) (, i j) (, i j) (, i j) (, i j) V y xy x yz z = 2 ( i, j) ( i, j) ( i, j) y Ey y Ex y Ez y (3.23) Assuming ν σ xy E y x x ν yz σ z σ y, << and integrating Equation (3.23) over the surface area E y E y z y of the cell, we find d 2 ( i, j) (, i j) ( V ) d( < σ y > ) 2 ( i, j) dy Ey dy < > = (3.24) Combining Equations (3.21) and (3.24) yields the relationship between the average axial displacement and the interfacial shear stresses, d ( < V > ) ( i, j) j j i i = ( 2 (, ) xy [ x i 1] xy [ x i ]) ( yz z j 1 yz z i j τ τ + τ τ j ) dy Ey ti t j (3.25) Differentiating Equations (3.18) and (3.19) with respect to y twice, 50

61 2 ( i+ 1, j) 2 ( i, j) d ( < V > ) d ( < V > ) 2 2 = dy dy G 1 ( i+ 1, j) { (1 0 Xi+ 1) R dxi+ 1} [ x ] d ( τ ) 2 j ti+ 1 xy i+ 1 ( i+ 1, j) 2 xy dy 2 j 1 1 ( 1, ) (, ) [ x i j t d τ + xy i ] i i j (1 X 0 i 1) L dx + i+ 1 X 0 ir dx i dy t + + G ( ) + i+ 1 ( i+ 1, j) ( i, j) 2 G xy Gxy 2 j t 1 d ( τ i ( i [ x 1] ), j) xy i XL (, i j) { 0 i dx i} 2 xy dy (3.26) 2 ( i, j+ 1) 2 ( i, j) d ( < V > ) d ( < V > ) = 2 2 dy dy G i j { (1 Z j+ 1) T dz j+ 1 0 } 2 i t 1 1 z j+ (, + 1) yz j+ 1 (, i j+ 1) 2 yz dy 2 i t ( z ) j t d τ + (, i j+ 1) j (, i j) yz j + (1 Z j 1) B dz j 1 Z jt dz j (, i j+ 1) 0 (, i j) 0 2 G yz G + yz dy G d { } 2 ( τ ) i t j 1 z (, i j) yz j 1 Z jb dz j ( i, j) 0 2 yz dy d ( τ ) (3.27) and substituting Equation (3.25) for d 2 < V ( i, j) > / dy 2 yields two ODEs, in terms of the interfacial shear stresses, coupled between horizontally and vertically adjacent cells respectively, 51

62 2 j 2 j ( i 1, j) d ( τ [ x 1] ) + xy i+ ( i 1, j) ( i, j) + d ( τxy[ x i] ) t t t d + b + c G dy G G dy i+ 1 i+ 1 i ( i+ 1, j) x 2 ( i+ 1, j) x ( i, j) x 2 xy xy xy t ( ) a = τ + G dy E t E t E t τ [ ] τ + τ τ E 2 j (, ) d τ [ x 1] [ x ] τ [ x ] i i j xy i j j (, i j) x 2 ( i+ 1, j) xy i+ 1 + ( i+ 1, j) (, i j) xy i xy y i+ 1 y i+ 1 y i j i+ i i+ (, i j) xy x i 1 ( 1, ) yz z j z z (, ) yz j + i+ j i j ( i+ 1, j) yz j 1 y ti Ey tj Ey tj Ey tj 1 τ E t (, i j) y j i yz z j 1 + (3.28) 2 i 2 i t ( ) d ( τ z 1, 1 1 ) 1 (, 1 ) (, ) ( z ) j i j yz j tj i j t d τ + + j i j yz j d (, i j 1) z + b 2 (, i j 1) z c (, i j) z + 2 Gyz dy Gyz Gyz dy 2 i t d ( τ z (, ) 1 ) j i j yz j 1 i 1 1 a i z (, i j) z = τ 2 (, i j+ 1) yz j 1 (, i j 1) (, i j) Gyz dy Ey t τ yz z j+ 1 Ey tj+ 1 Ey t j j (3.29) 1 i 1 j+ 1 1 j 1 j+ 1 τ (, ) yz z j 1 (, 1) xy [ x i] (, ) xy[ x i] (, 1) xy [ x i j τ τ τ i j i j i j i 1] + + E t + + E t E t E t y j y i y i y i 1 τ E t (, i j) y i j xy [ x ] i 1 where we have simplified the notation by defining the constants, 1 1 ( i, j) ( i, j) ( i, j) ( ) ( i, j 1 ) x = i i x = i a X L dx b X L dx ( i, j) ( i, j) ( i, j) ( i, j) x = i i x = ( 1 i) c X R dx d X R dx ( i, j) ( i, j) ( i, j) ( i, j) z = j j z = ( 1 j) a Z B dz b Z B dz ( i, j) ( i, j) ( i, j) ( i, j) z = j j z = ( 1 j) c Z T dz d Z T dz 0 0 i i j j (3.30) 52

63 For the special case of linear interpolation functions L ( i, j ) = 1 X i, ( i, j ) R = X i, ( i, j ) B = 1 Z j, and ( i, j ) T = Z j, we find ( i, j) ( i, j) ( i, j) ( i, j) a = d = a = d = 1/6 and x x z z b ( i, j ) ( i, j ) ( i, j ) ( i, j ) x = c x = b z = c z = 1/ 3. Equations (3.28) and (3.29) can now be evaluated for all adjacent pairs of cells, yielding a system of 2mn (m+n) coupled ODEs, the solution of which will be discussed in the following section Solution Methods We now evaluate Equations (3.28) and (3.29) for all adjacent horizontal and vertical cells respectively and combine them to form a system of coupled ODEs in terms of the j unknown interfacial shear stresses [ ] i τ and [ ] xy x i τ, yz z j [ A] d 2 { τ} [ B ]{} τ {} dy = (3.31) 2 0 {} τ is then the vector of length 2mn + (m+n) composed of the interfacial shear stresses in j arbitrary order {} τ τ [ x ] i { } M{ τ [ z ] } =. The matrices [A] and [B] are constant and depend xy i yz j only on the geometry and material properties of the composite system and the chosen shape functions. As for the previous 2D model (Nairn and Mendels, 2001), the matrix [A] is tridiagonal. However, the matrix [B] is no longer tridiagonal, due to the multiple connectivity of the cells in the 3D model geometry. The system of Equation (3.31) has 2mn - (m+n) equations for 2mn + (m+n) unknowns, therefore we require 2(m+n) boundary conditions, for example the shear stresses on the outer surfaces of the laminate. 53

64 { } u We next reorder and partition the vector { τ } into subvectors of unknown values τ and known boundary conditions{ τbc} in order to write τ u τ u d M L M L = (3.32) dy τ bc τ bc [ A A ] [ B B ] {} where the submatrices [A 1 ] and [B 1 ] are square of dimension 2mn - (m+n). Rearranging Equation (3.32), we can write 2 d { τ u} A1 B 2 1{ τ u} = { τ } (3.33) dy where 2 d { τ bc} { τ } = A2 + B 2 2{ τ bc} (3.34) dy We have relaxed the constraint on the boundary conditions of Nairn and Mendels (2001), i.e. { } bc τ does not have to be constant or linear in y, therefore [ ] A does not necessarily equal zero. Such nonlinear boundary conditions appear frequently in bonded laminate problems. Continuing, we premultiply Equation (3.33) by [ A ] 1 ([ ] 2mn-(m+n)) to obtain, 1 2 A is always of rank 1 54

65 2 d { τ u} 2 [ M ]{ τ u} = { p} (3.35) dy with M 1 1 = A 1 B and { p } = A { τ }. Equation (3.35) is the same equation addressed in Nairn 1 1 and Mendels (2001) with two significant exceptions: (1) the matrix [ M ] is singular due to the fact that the original matrix [ B 1] is singular; (2) the vector { τ } is not necessarily a constant vector. Two methods can be used to solve Equation (3.35), i.e. state space transfer and eigenvalue and eigenvector decoupled methods (Boyce and Diprima, 1986), of which we will apply the latter in this dissertation. The matrix [B 1 ] is a square matrix of dimension 2mn - (m+n). However, the components of [B 1 ] are combinations of the relative average axial displacements between adjoining horizontal, < V ( i, j ) ( i 1, j > < V ) > ( ) ( ) > i, j i, j 1, and adjoining vertical, < V > < V, cells through Equations (3.18) and (3.19). For m n cells, there are mn-1 such independent relative average axial displacements. Therefore the rank of [B 1 ] is mn-1, leading to mn ( m + n) + 1 degenerate eigenvalues. At the same time, [B 1 ] has independent eigenvectors therefore [M] can be diagonalized. We can thus proceed with the eigenvalue, eigenvector decoupling of Equation (3.35). Writing [T] as the matrix of eigenvectors of [M] and [Q] the diagonalized matrix of eigenvalues, we diagonalize [M], 55

66 1 [ M ] [ T] [ Q][ T] = (3.36) with the j th column of [T] being the j th eigenvector corresponding to the j th eigenvalue λ j. Premultiplying Equation (3.35) by [T] and defining the vector{ r} [ T ]{ τ } =, we find u 2 d {} r 2 [ Q ]{ r } = { p } (3.37) dy 1 where { p} [ T ] { p} =. Equation (3.37) is then a system of uncoupled, second order ordinary differential equations which can be solved analytically. We now divide the solution of Equation (3.37) into three separate conditions, depending upon the form of the boundary condition dependent vector{ τ }: (1) { τ } is a constant vector; (2) {} τ is a piecewise constant vector; and (3) { τ } is a vector that varies arbitrarily in y. We consider each of these possibilities below Constant Boundary Conditions For the nonzero eigenvalues of [M], we find the solution in terms of the general solution and one specific solution, q r T a e b e ) (3.38) n1 λjy λjy j i = i, j( j + j 2 j= 1 λ j 56

67 where a j and b j are unknown constants. For values of λ = 0 which occurs for the repeated eigenvalues of [M], we find the solution, i 2 mn ( m+ n) λjy λjy j 2 ri = Ti, j( aje + bje y ) j= n q + (3.39) Ordering the eigenvalues such that the first mn-1 are non-zero and the rest are zero, we can write the complete solution to Equation (3.37) as, q n1 2 mn ( m+ n) λjy λjy j λjy λjy j 2 Ui = Ti, j ( aje + bje ) + 2 Ti, j ( aje + bje + y ) (3.40) j= 1 λ j j= n τ q To complete the solution, there are 2(2mn-m-n) unknown coefficients a j and b j, which must be solved from the shear stress boundary conditions. These will be determined in the later numerical examples on two fixed planes y = y1 and y = y2. Once the unknown shear stresses are solved, we can then calculate the average ( i, j) normal stresses < > in each unit by integrating Equation (3.21), σ y 1 1 < σ >= ( τ ( x ) τ ( x )) dy + ( τ ( z ) τ ( z )) dy + C 0 (3.41) (, i j) j j i i (, i j) y xy i 1 xy i yz j 1 yz j ti tj where the unknown coefficients conditions. C ( i, j) 0 are to be determined from the normal stress boundary 57

68 At this point, it is important to highlight a difference between the application of the optimal 2D shear-lag model and this extension to 3D configurations. Since there are considerably more unknown interfacial shear stresses than average axial stresses a transformation matrix relating the two stress systems (through Equation (3.21)) would not be square. Therefore, one cannot transform the linear system of (35) in terms of interfacial shear stresses into a similar system in terms of the average axial stresses with the same number of boundary conditions. Such a transformation was performed earlier for the 2D model by Nairn and Mendels (2001). From a modeling perspective, the important consequence is that all boundary conditions, other than C ( i, j) 0 for the solution to (41), must be in terms of shear stresses. Such a restriction can severely limit the application of this model to specific loading conditions (beyond the limitations of the shear-lag analysis itself). One strategy to extend the applicability of the model is to replace normal stress boundary conditions by equivalent shear stress boundary conditions. Although not an exact equivalence, good estimates can be obtained, as will be demonstrated later in one of the numerical examples Piecewise Constant Boundary Conditions If the loading { τ } can be represented as a piecewise constant vector, e.g. when external constant shear loadings are applied only over certain segments of the boundary, the vector {} τ can also be discretized along the loading length of the laminate. For h piecewise segments ( { τ} = { τ} 1 0 y y1,, { } { } yh 1 extended to h τ = τ y L ) the system can be 58

69 d τ u τ u τ M = M (3.42) h h τ τ u τ [ A] M [ B ] dy h u With the added continuity boundary conditions i i+ { τ } = { τ } 1 i = 1,, h 1 u u y= yi y= yi K (3.43) Thus, the number of degrees of freedom a j and b j increases h times. The solution to a specific discretized equation thus follows the procedure for constant boundary conditions previously described. An alternative solution procedure can be applied to exploit the asymptotic behavior of the shear-lag solution. As long as each segment length of the piecewise constant distribution is longer than the development length of the particular configuration the total interfacial shear stress (and therefore the average axial stress solutions) can be found by superposing the interfacial shear stress solutions to constant stress boundary conditions over the different spans, h h 1 { τ} = { τ} ( ) + { τ} { τ} ( ) + + { τ} { τ} ( ) H y H y y1 L H y y h 1 (3.44) where H is the Heaviside step function. As the individual solutions are asymptotic within a few cross-sectional widths (to be seen in the following section), the detailed interfacial stress 59

70 } i bc distribution for each solution due to { τ need only be considered in a small region near y i-1. An example of this solution method will be presented in section 3.2. This alternative solution method allows one to solve h linear systems of smaller dimensions Arbitrary Boundary Conditions The final case considered is that of arbitrary applied loading boundary conditions which can either be expressed as explicit mathematical functions in terms of the variable y or through discrete data points. Except for the rare explicit function where (37) can be solved directly, a numerical method such as the fourth order Runge-Kutta method must be applied. 3.2 Numerical Results To evaluate the predictions of the 3D shear lag model derived in the previous section, we consider two numerical examples. The first is a simple single embedded fiber specimen from which we will outline the solution process for different boundary conditions and evaluate the predictions of the interfacial shear stresses and axial stresses. The second is a configuration typical of those in both unidirectional laminated composites with different material properties and smart structures with embedded sensors and actuators. Then the stress transfer in a smart structure is evaluated, which is loaded axially on some of its edge units. For each case, comparisons are made to predictions from a full 3D finite element analysis of the same configuration. 60

71 Unidirectional Single-Fiber Composite To evaluate the 3D shear lag model, we consider the benchmark problem of a cantilevered composite beam with a single fiber embedded in surrounding matrix as shown in Fig. 3.2 (a). The dimensions of the cross section and its division into unit cells are shown in Fig. 3.2 (b). We divide the cross section into the minimum number of nine unit cells, resulting in twenty-four interfacial shear stresses. This beam has a total length of L = 60 mm in its axial direction (yielding a length to width ratio of 15). The fiber and matrix units are initially chosen to be glass and epoxy with properties given in Tab. 1 (i.e. a high modulus ratio composite, E f / E m = 20). The material properties of the GFRP are taken from Kachlakev (2002). Material Elastic modulus (GPa) Poisson's ratio Shear modulus (GPa) GFRP E x =E z =6.89 E y =20.7 ν xy =ν yz =0.26 ν xz =0.30 G xy =G yz =1.52 G xz =2.65 Glass fiber E = 70 ν = 0.29 G =27.13 Polymer fiber E = 7.0 ν = 0.29 G = 2.71 Epoxy matrix E = 3.5 ν = 0.33 G = 1.32 Table 3.1. Material properties of fibers, matrix and fiber reinforced lamina used for simulations. 61

72 (a) (b) Figure 3.2. Configuration of unidirectional single fiber composite: (a) Unidirectional single fiber composite with fixed boundary condition at y = L (not to scale); loading case shown in uniform shear stress along full length; (b) division of cross section into nine unit cells; number for each interfacial shear stress and surface over which it is applied is indicated. 62

73 Using the numbering scheme of Fig. 3.1 (b), the unknown and known interfacial shear stress vectors are { } T τ = { τ ; τ ; τ ; τ ; τ ; τ ; τ ; τ ; τ ; τ ; τ ; τ } u { } T τ = { τ ; τ ; τ ; τ ; τ ; τ ; τ ; τ ; τ ; τ ; τ ; τ } bc (3.45) Linear interpolation shape functions are assumed for the calculation of the structural matrices [A], [B] and [M]. Eigenvalue analysis of [M] yields four zero eigenvalues, as discussed previously in Section 3.1. To provide the comparison of this and all other simulations in this paper, we modeled the same specimen geometry using the finite element analysis (ANSYS). A density of one-hundred 3D eight-noded brick elements (solid45) elements were meshed along the length of the specimen (y-direction) with four hundred elements in the cross-section. This relatively fine density of elements was considered as a suitable benchmark to compare the predictions of the significantly reduced 3D shear-lag model of twenty-four interfacial shear stresses within the cross-section. The first loading case to be analyzed was a unit shear loading on its outer surfaces all across the span, τ = 1 MPa, as shown in Fig. 3.2 (a). This case is an example of the constant shear stress boundary condition discussed in section 3.2. The calculated average axial stress ( ) 2,2 distributions in the fiber unit, < >, and one of the matrix units, σ y σ y ( 1,2) < >, are plotted in Fig. 3.3 (a) as a function of axial position from both the 3D shear-lag model and the finite element analysis. For the finite element analysis, the average axial stress for a given unit cell was calculated as the average axial stress value for all nodal locations within the unit cell. As 63

74 can be seen in Fig. 3.3 (a), the difference between the predictions of axial stress from these two methods is small. For the same loading case, two of the calculated interfacial shear stresses, one within the matrix material, τ 2, and one at the fiber-matrix interface, τ 6, are plotted in Fig. 4(b). For both of these shear stresses, the 3D shear-lag method predicts the classical asymptotic form of shear stress, due to the inherent assumptions in the shear-lag theory. The discrepancy between the two methods is significantly more pronounced in the calculation of the interfacial shear stresses which can be clearly seen in Fig. 4(b). The shear-lag model underpredicts the rate of interfacial shear-stress increase from the free edge of the specimen, but overpredicts the steady-state shear-stress. This behavior is true for both the internal shear stress in the matrix, τ 2, and the shear stress at the fiber-matrix boundary, τ 6. Therefore, as will be confirmed by later simulations, the 3D shear lag model yields better predictions for average normal stresses than for interfacial shear stresses. This same behavior was observed by Nairn and Mendels (2001) for the 2D optimal shear-lag model. Care should thus be taken in applying the results of the 3D shear-lag simulations for the evaluation of shear stress components. 64

75 (a) Figure 3.3. Simulation results for the cantilevered single-fiber composite of Fig. 3.2 with E f / E m = 20 subjected to shear loading along entire length: (a) average normal stress in fiber and sample matrix units; (b) sample interfacial shear stresses (semi-span plotted only). Results plotted for both 3D shear-lag and finite element (FEA) analyses. (b) 65

76 The second loading condition applied to the single-fiber composite was a unit shear force applied only in the region 0 y L/ 10, as shown in Fig This case was analyzed using the method of superposition outlined in Section Fig. 3.5 (a) plots the average 2,2 normal stresses in the fiber unit, < >, and one of the matrix units, σ y ( ) σ y ( 1,2) < >, predicted using the 3D shear-lag and finite element analyses. Due to the local nature of the applied loading, the differences between the two methods is more pronounced than for the previous loading case, however the general comparison between the 3D shear-lag method and the FEA method is the same. The source of the differences can be seen in Fig. 3.5 (b) which plots two representative interfacial shear stresses, one at the interior of the matrix, τ 3, and one at the outer surface, τ 1. Once again, the 3D shear-lag analysis overpredicts the transfer of load from the matrix to the fiber. At y = 7 mm, just beyond the loaded region, the prediction of the average normal stresses are essentially the same. This approximately half the distance before the interfacial shear stresses are consistent, further reinforcing the point that the 3D shear-lag method should be used primarily for the prediction of average normal stresses. As mentioned in the background, the primary benefit of the 2D optimal shear-lag theory of Nairn and Mendels (2001) is its ability to predict stresses in low modulii ratio composites. Therefore for the third simulation of the single-fiber composite, we changed the material properties of the fiber to be that of a nylon fiber (see Tab. 3.1). For this example the modulii ratio is extremely low at E f /E m = 2. The same loading condition shown in Fig. 3.4 was applied. Fig. 3.6 plots the predicted average axial stresses in the fiber and matrix and sample interfacial shear stresses as before. For this case, two different interfacial shear stresses than those for the previous case were plotted. One can see that the comparison of results for both axial stress and interfacial shear stress is approximately of the same quality 66

77 for the low modulii ratio composite as for the previous high modulii composite. Therefore, the differences between the two analysis methods is not due to the axial contribution of the matrix material. The 3D shear-lag method of this article is therefore applicable for structures with a wide range of modulii ratios. In the following section, we consider a laminated composite various material constituents. Figure 3.4. Secondary loading condition for single fiber composite of Fig. 3.2 (a). Unit shear loading is applied in the region 0 y 6 mm (beam remains fixed at y = 60 mm). 67

78 (a) Figure 3.5. Average normal stress results of the single-fiber composite with E f / E m = 20 subject to outer surface shear loading over finite length: (a) average normal stress in fiber and sample matrix units; (b) interfacial shear stress on sample outer and inner surfaces (semi-span plotted only). Results plotted for both 3D shear-lag and finite element (FEA) analyses. (b) 68

79 Naturally, the predictions of the 3D shear-lag method plotted in Figs. 3.3, 3.5 and 3.6 could be improved by increasing the number of elements in the cross-section. However, it should be emphasized here that the goal of this article is to derive a rapid calculation method to estimate the axial stresses in the various constituents. For example, Prabhugoud and Peters (2006) demonstrated that the use of the average normal stress value in an optical fiber is a good representation of the behavior of the optical sensor, even though the mechanical properties may vary significantly over the cross-section of the optical fiber. Therefore the prediction of the axial stresses will be considered sufficient in evaluating the success of the prediction method. As demonstrated by Nairn and Mendels (2001), the predictions of the shear-lag method could also be improved by choosing more appropriate shape functions than the linear ones of Equation In order to make the choice of shape functions appropriate for a large range of problems, they should be derived based on equivalent energy concepts, rather than arbitrarily altering the values to better correlate the finite element and 3D shear-lag predictions of Figs. 3.3, 3.5 and 3.6. This derivation will not be investigated in the article, but could be the subject of future investigations. 69

80 (a) (b) Figure 3.6. Average normal stress results of the single-fiber composite with E f / E m = 2 subject to outer surface shear loading over finite length: (a) average normal stress in fiber and sample matrix units; (b) sample interfacial shear stresses (semi-span plotted only). Results plotted for both 3D shear-lag and finite element (FEA) analyses. 70

81 Unidirectional Laminated Composite The numerical example presented in this article is the unidirectional laminated composite with multiple constituents shown in Fig. 3.7 (a). The laminate is composed of layers of glass fiber reinforced polymer (GFRP) sheets, epoxy matrix and glass fibers. The glass fibers could represent embedded optical fiber sensors (here approximated as square rather than circular cross-sections). The particular configuration was chosen, however, to demonstrate the ability of the 3D shear-lag method to consider materials with orthotropic material properties and multiple modulii ratios. The GFRP sheet layers were modeled as transversely isotropic materials, while the glass fibers and the epoxy matrix were modeled as isotropic materials, whose material properties are listed in Tab The dimensions of the laminate and discretization of the cross-section into forty-five units are shown in Fig. 3.7 (b). The discretization for the finite element model was the same as for the previous single-fiber composite. The length of the beam was kept consistent with the previous simulations at L = 60 mm, as well as the fixed end condition at y = L. For this unidirectional laminated composite, two separate loading conditions were applied to the finite element model. The first loading condition was a normal stress applied over a 1 mm length of the upper and lower GFRP layers as shown in Fig. 3.8 (a). The stress field was applied on both layers to maintain symmetric loading about the midplane and therefore prevent bending of the laminate. The second loading condition was an equivalent shear loading condition applied on the upper and lower surfaces of the laminate, over a distance to create the same total applied force as shown in Fig. 3.8 (b). While, these two loading conditions are not the same, the goal was to determine whether the 3D shear-lag 71

82 model could be used for a restricted group of normal stresses boundary conditions. Only the loading condition of Fig. 3.8 (b) was applied to the shear-lag model, using the superposition method described for the previous single-fiber composite example. (a) (b) Figure 3.7. Configuration of unidirectional laminated composite: (a) Unidirectional laminated composite with embedded optical fibers; (b) division of cross section into forty-five unit cells. 72

83 (a) (b) Figure 3.8. Applied loadings for unidirectional laminated smart structure composite: (a) unit normal stress, σ = 1 MPa, applied to top and bottom GFRP sheets along 1 mm width (area highlighted in blue); (b) equivalent local shear stress τ = 1 MPa applied on the top and bottom surfaces of laminate along same width. The Fig. 3.9 plots the predicted normal stresses in two of the glass fiber units and two of the CFRP units and two of the interfacial shear stresses at the CFRP glass boundaries. The results are plotted for the FEA solutions using both the normal stress boundary conditions of Fig. 3.8 (a) and the equivalent shear stress boundary conditions of Fig. 3.8 (b). As for the previous simulations, the 3D shear-lag prediction of the average normal stresses are better than those of the interfacial shear stresses, however the predictions are quite good for both sets of stresses. These results demonstrate the ability of the 3D shear-lag 73

84 model to incorporate a multiple materials with different modulii, rather than a single high fiber to matrix stiffness ratio. (a) (b) Figure 3.9. Unit stress results for unidirectional laminated composite predicted using normal stress boundary conditions (FEA), shear stress boundary conditions (FEA) and shear stress boundary conditions (3D shear-lag): (a) average normal stress in sample fiber and matrix units; location of fiber and matrix units highlighted in red on inset figure; (b) sample interfacial shear stresses. 74

85 In addition, the difference in average normal stresses due to the normal stress boundary conditions and the shear stress boundary conditions calculated using FEA is considerably less than the difference between the 3D shear-lag and FEA predictions for the average normal stress based on shear stress boundary conditions. These results indicate that the 3D shear-lag method could be applied to obtain approximate average normal stresses for either boundary condition using the concept of equivalent shear stresses. Naturally, this does not imply that the 3D shear-lag method could be used to model any normal stress boundary condition. In particular, when normal stresses are applied to unit cells within the inner domain of the cross-section it may be difficult to produce an equivalent shear stress condition, although the shear stress boundary conditions can be applied to interfacial shear stresses within the interior of the laminate as well. 3.3 Conclusions In this section, we derive a three-dimensional shear-lag model for unidirectional multilayered structures based on the extension of a previous two-dimensional optimal shear-lag model. Solution methods for a variety of shear stress boundary conditions are presented. The prediction of stress distribution in a single-fiber composite and unidirectional laminated composite demonstrate that the 3D shear-lag method can be used to rapidly estimate the average normal stress distribution in the various constituents. The method is also applicable for low stiffness ratio composites, including the laminate example with multiple material constituent modulii ratios. Such a capability is extremely useful for the real-time prediction of sensor responses when embedded in laminated structures. 75

86 The modeling of an example applied normal stress is demonstrated through an equivalent shear stress boundary condition. Future work would be required to specify when such a substitution is appropriate and how such a substitution would be made for normal stresses applied to elements within the interior of the cross-section. Future work could also derive more appropriate shape functions than the linear ones applied in this work, for example, functions suitable for constituents with other than rectangular cross-sections. 76

87 CHAPTER 4 OSCILLATOR INTERROGATED TIME OF FLIGHT OPTICAL FIBER INTERFEROMETER The objective of this chapter is to demonstrate an oscillator interrogated time of flight optical fiber interferometer that provides global strain information for the use in the self-monitoring FRP retrofit when applied to large-scale structures (Jiang et al., 2007). As pointed out by Sarazin and Newhook (2007), multi-scale measurements are the best approach to structural health monitoring. Combining measurements over different spatial scales can provide better interpretation of the damage form and extent. Among them, a global strain measurement system is usually the first indicator of whether or not structural property changes have occurred, for example debondings in FRP strengthened infrastructures. Applying global strain sensors can also significantly decrease the number of local sensors required for the same level of information. As reviewed in Chapter 2, although there are several commercially available global strain measurement techniques, their limitations prevent their applications in FRP strengthened large-scale civil infrastructures. An oscillator interrogated time of flight fiber interferometer newly developed in our research laboratory was adapted for the use of global strain measurements in intelligent FRP retrofit bonded structures, which provides sufficient displacement resolution for a full-scale structure, yet addresses some of the limitations of the other global strain measurement techniques. The time-of-flight interferometer is relatively inexpensive and durable as compared to previous methods. The measurable displacement range is determined by the oscillator frequency and can therefore be designed to be significantly larger than that of current 77

88 systems. An optical length change resolution of 9.5 mm and range of 1.39 m are demonstrated experimentally. To easily embed the FOS in structures and effectively increase the fiber sensing length, a commercially available prefabricated fiber ribbon is used. Results from static loading tests of a FRP strengthened reinforced concrete beam with embedded fiber ribbons show the feasibility of this technique for the self-monitoring FRP retrofit system presented in Chapter Time-of-Flight Fiber Interferometer This section presents the theory and implementation of the time-of-flight interferometer strain measurement technique. The phase of a sinusoidally modulated optical sensor signal is compared to the original electrical signal used to modulate the laser source. The constant frequency modulation combined with digital filtering of the signal provides a high resolution of time of flight measurement. The data acquisition system is constructed of standard telecom components and can be interrogated through an oscilloscope and is therefore relatively inexpensive as compared to the previous methods (for similar optical path length resolutions). Furthermore the system is portable, durable (since no moving parts are required) and has relatively low power requirements. The resolution of the interrogation system is determined by the resolution of the oscilloscope, the quality of the filter, and the noise level of the oscillator and is augmented by the use of pre-fabricated optical fiber ribbons that can be easily embedded into structural systems. The measurable displacement range, on the other hand, is determined by the oscillator modulation frequency and can be quite large compared to previous interrogation methods. 78

89 Before introducing the data acquisition system used to measure time of flight, the relationship between the change in time of flight of a lightwave propagating through an optical fiber and the change in length of the fiber due to applied strain along its length is reviewed. is Time-of-Flight Measurements The time-of-flight, t, for a lightwave propagating through a sensing fiber of length L L L λ t = n = β (4.1) c π e 2 0 c0 where n e is the effective index of refraction of the propagating mode (here we consider only the fundamental mode), c 0 is the velocity of light in a vacuum (c 0 = m/s), λ is the wavelength of the lightwave and β is the propagation constant of the propagating mode (Donati, 2004). Herein for the application of this system in FRP strengthened structures, only the case of axial loading is considered which is applied along an otherwise free fiber, although the measurement could be extended to any form of applied loading (Haslach and Sirkis, 1991). Once an axial force is applied to the fiber, the fiber experiences an axial strain, ε a (s) = ε(s) and transverse strain components, ε t (s) = -νε(s), where s is the coordinate along the fiber path (0 < s < L) and ν is the Poisson s ratio of silica (ν = 0.16). The change in time-of-flight for the lightwave can be calculated as 79

90 λ Δ t = ( Δ Lβ + LΔ β ) (4.2) 2π c 0 The total fiber length change ΔL due to the axial strain ε a (s) is found by integrating the axial strain along the path, Δ L= ε () s ds 0 L (4.3) The change in the propagation constant Δβ is due to both the strain-optic effect with the change of the fiber index of refraction n e and the geometrical effect due to the change of the diameter of the fiber core, a, which can be written as dβ dβ 2π L dβ L Δ β = Δ ne + Δ a = Δ ne() s ds + a() s ds dn da λ 0 da 0 e (4.4) The second term in Eq. (4.4) due to the change in core diameter is negligible as compared to the first term (Butter and Hocker, 1978). For the case of a pure axial applied force, Δn e (s) can be expressed as 1 3 Δ ne() s = ne[ (1 v) p12 vp11 ] ε () s = peneε () s (4.5) 2 where p 11 and p 12 are strain-optic coefficients of silica (p 11 = 0.17, p 12 = 0.36), and 80

91 p e 2 ( n 2) ( 1 ) [ ν p12 ν 11 = / p ] is the effective strain-optic coefficient for the axial loading e case (Butter and Hocker, 1978). Substituting these equations into Eq. (4.2) yields the relationship between the measured time-of-flight delay and the change in fiber length, ΔL, or the average axial strain along the fiber,ε ( 1 ) L ( 1 ) ( 1 ) pe ne pe ne pe ne Δ t = () s ds L L c ε = Δ = ε (4.6) c c Oscillator Interrogated Interferometer Figure 4.1 shows a schematic of the oscillator interrogated interferometer applied in this work for the measurement of time of flight for the sensing optical fiber. The interrogator is entirely constructed of standard telecom components and was applied in combination with a 1GHz oscilloscope and an optical fiber ribbon described later in this section. The sensing optical fiber is illuminated by an 850 nm laser diode source (Excelight) mounted to a TX/RX evaluation board. The electrical input to the laser source (and therefore the output intensity) is modulated sinusoidally using a 200MHz oscillator (Vectron). At the same time, the electrical input is sent directly to the oscilloscope (Agilent) to serve as the electrical reference signal, therefore no optical reference signal is required. The output optical signal from the sensing fiber is then converted to an electrical signal at the TX/RX evaluation board and transmitted to the oscilloscope. The output from the oscilloscope is acquired through a GPIB (highspeed USB) interface. Figure 4.2 shows a photograph of the complete data acquisition system used for the laboratory measurements. Finally, the sampled sensor and reference 81

92 signals are filtered using a discrete sinc interpolation approach (Schanze, 1995). Due to the narrow frequency bandwidth of the signals, the sinc interpolation reconstructs the signals (in the presence of noise and digital round-off errors) at a significantly higher resolution than the original sampling rate of the oscilloscope (Dooley and Nandi, 2000; Candocia and Principe, 1998). The signal processing including averaging, zeroing the initial optical delay and filtering is performed using LabView. Typical sensor and reference signals acquired over six cycles are plotted in Fig. 4.3, demonstrating the lack of distortion in the reconstructed signals. The amplitude difference between the signals is due to losses from the coupling at the ingress and egress to the optical fiber, however does not affect the time of flight measurement. To further eliminate errors due to vibrations of the sensing fiber (which are particularly prevalent for long-path length optical fiber sensors) the average of sixty cycles was used to calculate each time delay data point. Figure 4.1. Schematic of the interrogator for the time-of-flight interferometer. Electrical signals are indicated as dashed lines, optical signal as a solid line. 82

93 Figure 4.2. Photograph of laboratory set-up for the time-of-flight interrogation system. (1) digital oscilloscope, (2) power supply, (3) mounting board including oscillator and laser module, (4) PC. Figure 4.3. Example of sensor and reference signal acquired data points (post-filtering). 83

94 In designing the interrogator for the time-of-flight interferometer, the wavelength of the laser source does not affect the displacement range of the system (unlike for phase shift measurements). The modulation frequency of the oscillator, f, however, determines the upper bound of displacement that can be measured before the time of flight information is no longer absolute. The maximum time-of-flight change that can be measured is thus 1 Δ tmax = (4.7) f Converting the time-of-flight into displacement, the maximum change in fiber length that can be measured is c0 Δ Lmax = n (1 p ) f e e (4.8) Therefore lowering the frequency of the oscillator increases the displacement range that can be measured. For the system used in these experiments, p e = 0.26, n e = 1.46, so the maximum displacement is ΔL max = 1.39 m for the 200MHz oscillator. The lower displacement resolution is determined by the resolution of the oscilloscope and choice of filtering function and not the modulation frequency. In addition to the interrogation system shown above, a twenty-four single-mode optical fiber ribbon (Sumitomo Electric Lightwave) was used to increase the strain resolution of the time-of-flight interferometer by splicing the individual fibers in series to form one continuous optical path. The ribbon and its dimensions are shown in Fig The 84

95 surrounding UV-acrylate matrix material serves as an excellent protection for the glass fibers and the flat ribbon form allows relatively unobtrusive mounting of the sensor into the intelligent FRP retrofits as will be demonstrated in the next section. Figure 4.4. Photograph and dimensions of the 24-fiber prefabricated ribbon. Section A-A is not drawn to scale. Individual fiber diameter is 125 μm. 4.2 Experimental Verification Calibration tests were first performed to determine the displacement sensitivity and linearity of the interferometer interrogation system. Afterwards, the optical fiber ribbon was embedded in fiber-reinforced polymers (FRP) strips typically used in the reinforcement of reinforced concrete to quantify any detrimental effects on the FRP itself. Finally, displacement measurements were performed on a full-scale FRP strengthened reinforced concrete beam tested to failure to demonstrate the durability and practicality of the technique. Each of these experiments is described in the following sections. 85

96 Calibration of Interferometer Interrogation System The calibration of the oscillator interrogated time of flight interferometer was performed in two steps. First the optical path length was changed by physically splicing additional optical fiber segments to the original sensing length (calibration without the photoelastic effect); then the optical path length was changed by applying tension to the sensing fiber (calibration with the photoelastic effect). For the first test, a single optical fiber of original length mm was used as the sensing fiber. The measured time of flight of delay versus the total length of additional fiber added is plotted in Fig The maximum value of ΔL applied in Fig. 4.5 corresponds to the complete displacement measurement range for the current system. After processing the signal data, obtained from the maximum peak locations, the minimum time interval resolution that could be measured was ns, significantly smaller than the 1 ns resolution of the oscilloscope. This corresponds to a minimum displacement resolution of 9.5 mm, making the system resolution comparable with the OFDR technique mentioned in Chapter 2. As observed from Fig. 4.5, the obtained data were extremely linear throughout the measurement range, with a linear fit of Δt = ns/mm ΔL ± ns. The theoretical slope of the curve is predicted to be Δt / ΔL = ns/mm from Eq. (4.6), applying p e = 0. The additional scatter in the experimental data and 0.61% difference in slope between the predicted and measured displacement sensitivity is primarily due to the difficulty in measuring the length of the additional spliced fiber. 86

97 Figure 4.5. Measured time-of-flight delay as a function of optical path length change achieved by splicing additional lengths of fiber to original section. Each measurement is calculated from the average of 60 signal cycles. Linear fit to data is plotted as a solid line. Error value for linear fit is calculated for one standard deviation. A second test was performed through tensile loading of the optical fiber to calibrate the complete ribbon system to applied axial displacement. For this test the 24 optical fiber ribbon described in the previous section was wrapped between two mandrels mounted into the grips of a uniaxial testing machine, as shown in Fig A total of twelve loops of the flat fiber ribbons were bonded to the mandrel surfaces yielding a sensing fiber length of meters after the twenty-four fibers were spliced in series. The ribbon was loaded in displacement control until the optical fiber reached approximately 2% axial strain. The measured Δt versus total change in fiber length for this test is plotted in Fig As observed 87

98 during the loading of the fiber ribbon, there was some initial sliding of the ribbon on the mandrel to align the ribbon fully with the loading axis of the testing machine. The data points obtained during sliding are indicated on Fig Once slipping stopped, the measured Δt versus ΔL curve was linear, as seen in Fig A linear fit was performed for the data resulting in a sensitivity of Δt = ns/mm ΔL ± ns, as compared to the theoretical sensitivity of Δt = ns/mm ΔL, predicted from Eq. (4.6). Figure 4.6. Photograph of optical fiber ribbon tensile test using uniaxial tensile machine m optical path length was achieved with 12 fiber ribbon wraps on the mounting mandrels. The ribbons were bonded to the mandrel surfaces using a standard strain gage adhesive. 88

99 Figure 4.7. Measured time-of-flight delay as a function of optical fiber length change. Linear fit to data is plotted as a solid line. The first 17 data points demonstrate slippage of the fiber ribbon on the mandrels and are therefore not included in the linear fit. Each measurement is calculated from the average of 60 signal cycles. Error value for linear fit is calculated for one standard deviation. The 4.2% difference between the theoretical and experimental curves is most likely due to the difficulty in determining the fraction of each fiber wrap length that is actually strained. Additional sources of error are the values of n e and p e for the optical fiber provided by the manufacturer, the alignment of the ribbon on the mandrels, and the strain transfer between the optical fibers and the polymer matrix, however these are unlikely to contribute significantly. The noise in the data is significantly higher than the resolution of the interferometer measured in the previous calibration and is considered to be primarily due to the drop in load at each measurement point as the loading machine was temporarily stopped 89

100 for each measurement. This is potentially due to some yielding of the polymer matrix, however the noise did not increase as the applied load increased and therefore was probably due to a small amount of additional sliding of the ribbon on the mandrel Application of Displacement Sensor to FRP Retrofits Once the optical fiber ribbon sensor and interrogation system had been calibrated, its durability and performance in a full scale structural component was finally demonstrated. The particular example chosen was a 9.0 m span prestressed C-channel concrete girder that was to be strengthened with two layers of FRP strips and loaded quasi-statically at the midspan until failure at the Constructed Facilities Laboratory at NCSU (see Fig. 4.8). During strengthening of the girder, the twenty-four optical fiber ribbon was bonded between the two layers of FRP on the lower surface of the girder. A detailed description of the girders and the FRP strengthening design can be found in Rosenboom (2006). Fan et al (1998) and Zhao and Ansari (2002) applied long gage length optical fiber sensors based on low coherence interferometry in similar locations to measure hoop strains in FRP strengthening of columns and interface strains between the FRP and concrete on a strengthened rectangular section beam. As shown in Fig. 4.8, the girders were at first sandblasted in the field to make sure the interfacial surfaces were smooth such that good bonding was guaranteed between the FRP and the girders. Possible damages on the bottom surface of the concrete girder were also fixed before the application of FRP strengthening. The bonding epoxy was then mixed and applied to both the FRP and concrete surfaces as shows in Fig As can be seen from the photograph, epoxy was also applied on the concrete surfaces where U-wraps were to be later 90

101 applied. A first layer of FRP sheet was bonded to the concrete girder as shown in Fig (a) and the fiber optic ribbon sensor was then attached to the FRP using the same adhesive epoxy. Since the ribbon is relatively stiff, it can be easily bonded on the flat surfaces without bending or twisting. Finally a second layer of FRP was attached as in Fig (b). Photographs of the ribbon placement can be clearly seen in Fig. 4.11, in which the left hand section shows the exposed ribbon prior to layup of the second FRP layer and the right hand section shows the completed layup. The ingress and egress points for the ribbon were at the end points of the girder span. Special attention was necessary for the ingress and egress points since they can be easily damaged during the FRP bonding and girder transportation. The optical fiber ribbon therefore measured the integrated strain along the lower face of the beam along the entire span length. As can be seen in Fig. 4.11, the dimensions of the optical fiber ribbon cross-section are considerably smaller than the FRP reinforcement, therefore it was not expected that the ribbon would significantly reduce the load bearing capacity or failure strength of the FRP. This assumption was verified by four-point bending tests to failure of two-layer FRP strips with embedded ribbons. A total of seven 25.4 mm 3.5 mm specimens were fabricated using the same procedure as those applied to the bridge girder with zero, one, or two embedded fiber ribbons. The load vs. deflection at the supports is plotted for each specimen in Fig It can be seen from the curves of Fig. 4.9 that the embedment of the ribbons did not cause a measurable reduction in ultimate strength of the FRP plates. The variation in loading histories between specimens observable in Fig. 4.9 is due to thickness and width variations in specimens developed during the preparation of the test coupons and is indicative of variations observed in FRP strengthening applied to actual structures. 91

102 Figure 4.8. Photograph of C-channel concrete girders after sandblasting. Figure 4.9. Photograph of bonding epoxy application. 92

103 (a) (b) Figure Photographs of FRP bonding and fiber optic ribbon sensor embedding: (a) Wet lay-up of first layer FRP; (b) Embedding of fiber optic ribbon sensor between the first and second FRP layers. 93

104 Figure Photograph of ribbon placement during installation of FRP reinforcement on lower surfaces on bridge girder. Left hand section shows exposed ribbon prior to layup of second FRP layer, right hand section shows complete layup. After allowing the FPR to cure for two weeks, the FRP strengthened girder with the FOS ribbons embedded between the FRP layers was then transported to the Constructed Facilities Laboratory at NCSU. It was loaded quasi-statically at the midspan until failure as shown in Fig Prior to loading, a total of twelve fibers in one of the ribbons were spliced together in series to create an unloaded optical path length of m as shown in 94

105 Fig Fig shows a photograph of the center span section of the bridge girder after failure occurred at the maximum applied load. It is important to note that the optical fiber ribbon sensor survived this condition and was still functioning after failure of the structure, demonstrating the durability of the embedded ribbon system. Figure Load vs. deflection at supports for four-point beam bending tests. Solid line represents beams with no embedded ribbons, dashed line represents beam with one embedded ribbon and dotted line represents beam with two embedded, parallel ribbons. Fig plots the displacement along the total span measured from the embedded FOS (and the calculated global average strain), together with the mid-span applied loads. From the data of Fig. 4.16, one can observe the phases of the beam deformation including 95

106 elastic deformation until cracking (at approximately ΔL = 5 mm) and the plastic deformation beyond this point. It should be noted the average strain plotted in Fig.4.16 is the strain averaged along the entire length of the beam and therefore not necessarily the local strain at which cracking occurs in the concrete. Also, the measurement error indicated in Fig could be reduced by splicing additional fibers of the ribbon together to increase the optical length as well as embedding more ribbons into the FRP. Figure Photograph of testing on the full-scale girder which was loaded quasi-statically at the midspan until failure. 96

107 Figure Photograph of fiber splices for the twelve fiber optical fiber ribbon. Figure Photograph of the midspan location along bridger girder post failure. Cables for external LVDT displacement sensors can also be seen. 97

108 Figure Measured integral displacement and average strain as a function of mid-span load. The measurement error bands for ΔL are also plotted, calculated from the time interval resolution of the DAQ system. 4.3 Conclusions This chapter presents an oscillator interrogated optical fiber time-of-flight interferometer for the measurement of integrated strains, suitable for application to intelligent FRP retrofits used for large scale structures. An extended measurement length optical fiber sensor was achieved by embedding a commercially available, prefabricated multi-fiber ribbon. An optical path length change resolution of 9.5 mm and range of 1.39 m are demonstrated experimentally. Also the lack of moving components in the interrogation system also makes this interrogator suitable for future dynamic measurements. Finally, results from testing of a full-scale FRP strengthened beam with the time-of-flight sensor embedded between FRP layers demonstrate the ease of application and durability of the 98

109 sensing system in the global strain measurement in intelligent FRP systems. The following chapter will present the integration of local strain measurement sensors and compare their measured strain response to that predicted by the previous 3D shear-lag method. 99

110 CHAPTER 5 FEASIBILITY STUDY ON FRP RETROFITS WITH SELF-MONITORING CAPABILITY The objective of this chapter is to demonstrate the self-monitoring capability of the FRP technique with embedded fiber optic sensors and to evaluate the effectiveness of the simplified model developed in Chapter 3 for sensor response estimation. Three double shear lap steel spliced joint specimens were tested under tensile loading until failure, for which strain measurements from both embedded fiber optic sensors and surface-mounted electrical resistance strain gages are evaluated, namely the abilities in identifying abnormal structural conditions such as epoxy cracking and FRP debonding. The particular example of the double shear lap steel spliced joint was chosen strengthened because of the linear elastic behavior in steel material. Theoretical modeling is also performed and the results are compared to those from experimental studies to evaluate the effectiveness of the reduced 3D shear-lag model in Chapter Experimental Methods The test specimen design and fabrication methods, as well as the experimental testing procedures are described in this section Specimen Configurations The double shear lap steel joint specimens shown in Fig. 5.1 and Fig. 5.2 were designed to incorporate both global and local strain sensing fiber optic sensors for future 100

111 applications. For these experiments, measurements were not collected from the global system due to the small scale of the specimens and therefore the limited strain resolution in the global system, as described in the previous chapter. As can be seen, the double shear lap systems are composed of two steel plates, each 508 mm in length, which are strengthened on both the top and bottom by wet lay-up carbon fiber reinforced polymer (CFRP) sheets. The strengthening sheets were 400 mm in length with an epoxy bonding layer of thickness of approximately 0.5 mm. For each of the three specimens, only some of the embedded optical fibers contained FBG sensors at selected locations. However, to maintain the symmetry of the specimens, plain glass optical fibers of the same dimensions and properties were embedded at all other locations as shown in Fig All of the FBG sensors for a given specimen were embedded on the same side of the spliced steel plates due to the symmetry of the specimens. Figure 5.1. Configuration of the double shear lap steel joint test specimen: (a) Lay-up of the self-monitoring CFRP retrofits on a double shear lap steel joints; (b) Cross-section view of one quarter specimen with dimensions and layer compositions shown; (c) Side view of the test specimen along the length. Figures are not drawn to scale and all dimensions are in mm. 101

112 (a) (b) Figure 5.2. Details of the double shear lap steel joint specimen: (a) Cross-section view of one quarter specimen with dimensions and layer compositions shown; (b) Side view of the test specimen along the length. Figures are not drawn to scale and all dimensions are in mm. 102

113 All together three specimens were fabricated and tested, each with a different number and locations of FBG sensors. The objective of the first specimen was to evaluate the embedment of the FBG sensors in strengthening CFRP and to assess the joint self-monitoring capability with respect to epoxy crack initiation and development. The specific location of the FBG sensors in each specimen are listed in Tab. 5.1 and shown in Fig The first specimen (FRP 1) was fabricated without any pre-existing CFRP debonding as shown in Fig 5.3 (a). Three FBG sensors were located in the optical fibers labeled 2, 4 and 6 (see Fig. 5.3). The sensors were located at 18, 25 and 75 mm from the midspan of the spliced joint. The particular fiber in which each FBG is located should not affect the measured strains; however for these experiments we used only one FBG per optical fiber so as to not have to splice fibers close to the locations of the FBGs. For practical applications, the most efficient configuration would be to embedded a FBG array written into a single fiber. All FBGs had a gage length of 10 mm as specified by the manufacturer. The second double shear lap steel joint test was designed to evaluate the repeatability of the sensor responses as compared to the first specimen and to determine if the sensors can identify pre-existing FRP debondings. A second specimen (FRP 2) was fabricated including a pre-existing debonding zone of length 10 mm, centered at y = 175 mm (see Fig. 5.3 (b)). This length was chosen to simulate a debonding zone of 5 % of the total FRP length. To create the debonding zone, the steel plate was wrapped with Teflon tape prior to FRP bonding. Four FBG sensors were located in the optical fibers labeled 1, 3, 5 and 7, and located at 0, 100, 175 and 200 mm from the joint midspan. As will be seen in the later presentation of the test results, the small FRP debonding of specimen FRP 2 was too small to be detected by the small number of sensors. 103

114 (a) (b) (c) Figure 5.3. Configuration of the three test specimens: (a) FRP 1; (b) FRP 2; (c) FRP

115 Therefore, specimen three (FRP 3, see Fig. 5.3 (c)) was designed with a debonding length of 100 mm centered at the mid-span, corresponding to a debonded zone of 25% of the total FRP retrofit area. The objectives of this specimen were the same as for the previous specimens. For this specimen, a thin plastic film was used to wrap the surface of the steel plates instead of the previous Teflon tape. In this manner, the gap between the two steel plates could be maintained free of bonding epoxy which was observed to be a potential problem during the fabrication of specimen FRP 2. Five FBG sensors were introduced into specimen FRP 3 in the optical fibers labeled 1, 3, 4, 5 and 7 at locations of 0, 25, 50, 100 and 200 mm from the midspan. For all of the three specimens, conventional strain gage sensors were also bonded on the FRP outer surface at the same locations as the FBG sensors for an independent verification of the strain measurements. The electrical resistance strain gages also had a gage length of 10 mm, similar to the FBGs. The selection of the embedded FBG sensor initial Bragg wavelengths is important to ensure that there is enough separation between all the Bragg wavelengths throughout the loading regime, in order that the different sensors be identified from one another. For all of the specimens, the sensors had a wavelength spacing of at least 3 nm, which is sufficient to allow the maximum peak wavelength shift at which the FBG fractures. The initial, unloaded Bragg grating wavelengths corresponding to each sensor are also given in Tab

116 Specimen Preparation A wet lay-up process was used for the layup of the double shear lap steel joint specimens, which will be described in this section. At first the two steel plates were prepared through sand blasting, necessary to ensure the surfaces are free of damages (see Fig. 5.4 (a)). Then two layers of the CFRP were saturated by the bonding epoxy and bonded to the steel plates. The fiber ribbons and the plain optical fibers without the FBG sensors were then bonded on the epoxy layer on top of these two layers of CFRP (see Fig. 5.4 (b)). Specimen Number of FBG sensors FRP 1 3 FRP 2 4 FRP 3 5 FBG sensor Initial Bragg Optical fiber location, y wavelength number (mm) (nm) Table 5.1. Location and initial Bragg wavelengths for each specimen. One lesson learned from the fabrication of the first specimen was that the FBG sensors were very difficult to be positioned accurately. The manufacturer only marked the sensor location on the optical fiber within a region of 100 mm on the optical fiber, making the exact the sensor positioning unknown (the FBG sensors are not visible in the optical fiber). The optical fibers containing FBG sensors were therefore pre-stressed and glued them 106

117 between two clamped steel tubes with a UV-cured bonding agent, as shown in Fig. 5.4 (c). We then used a soldering iron to locate the position of the Bragg gratings along the fibers within 10 mm by monitoring the peak wavelength of the FBG while moving the iron along the fiber. This positioning error was one of the most significant sources of errors in the experiments, as will be discussed later; therefore more complex positioning methods such as optical low coherence reflectometry may be necessary for future experiments. (a) (b) (c) (d) Figure 5.4. Photographs of wet lay-up procedure and test specimen preparation: (a) Steel plates after sand blasting; (b) Lay-up of the sensing layer with embedded fiber sensors; (c) C-clamp to hold the FBG sensors and optical fiber in place prior to embedment; (d) Lay-up of the last two CFRP layers on top of the sensing layer. 107

118 After pre-tensioning and alignment of the optical fibers containing sensors, the FRP steel joint was positioned under the optical fibers. Finally, two more layers of saturated CFRP were put on the top of the sensing layer in seen in Fig 5.4 (d). Fig. 5.5 shows photographs of the Teflon tape and thin plastic film wrapped on the steel plates of specimens FRP 2 and FRP 3 to simulate pre-existing debondings. As can be seen clearly in Fig. 5.5 (c)-(d), the steel gap was free of bonding epoxy when the thin plastic film was applied at the mid-span in specimen FRP 3. (a) (b) (c) (d) Figure 5.5. Photographs of FRP debonding simulations: (a)-(b) Debonding simulation using Teflon tape wrapped around the steel plates; (c)-(d) Debonding simulation using plastic film wrapped around the steel plates. 108

119 Testing Procedure After the FRP strengthened double shear splice joint was allowed to cure for one week, each of the three specimens were tested in uniaxial tension using a MTS closed-loop universal testing machine in the Constructed Facilities Laboratory at NCSU. Displacement control mode was applied at a constant displacement rate for all tests. Data from both the electrical resistance strain gages and FBG sensors was acquired at a rate of 2 Hz. Loading of the specimen was stopped when complete failure of the specimen had occurred, indicated by a drop in loading capacity of the specimen. As previously mentioned in Chapter 2, all of the embedded FBG sensors for each specimen were connected in series. This was achieved by fusion splicing each of the sensing fibers outside of the specimen. The data acquisition system used is shown in Fig A 1 khz FBG interrogator (Micro Optics Inc.) was used to scan the FBG sensor wavelengths and monitor the peak wavelengths. As can be seen from the plot, only the Bragg wavelengths were recorded, rather than the entire spectrum. A laptop PC was used to acquire the measured data through a LabView program. Since the sensor measurements are encoded as wavelength rather than intensity information, light losses from splice joints does not affect the FBG sensor accuracy. Data acquisition from the FBG sensors was performed at 2 Hz to match the data acquisition rate of the electrical resistance strain gages. Strain measurements due to pre-tensioning was subtracted from FBG sensor measurements and strain due to clamping of the specimen was removed from FBG and strain gage measurements. 109

120 Figure 5.6. Configuration of the distributed sensing technique by use of distributed multiplexing FBG sensor network. 5.2 Simulations Theoretical simulations were also performed of the double shear lap joint using the simplified shear-lag model derived in Chapter 3. These simulations will be used to compare the measured sensor response to that predicted by the reduced model and therefore evaluate the utility of the model. As mentioned in Chapter 3, the force boundary conditions to the laminate problem should ideally be in the form of shear stresses for the 3D shear-lag model. In this dissertation, a 2D finite element analysis of the spliced joint was used to derive the boundary shear loading applied to the FRP as axial stresses are applied to the end of the steel plates. Using the finite element method to calculate the applied boundary conditions means 110

121 that the steel plates do not have to be included in the 3D shear-lag model. The particular model, solved with ANSYS, is shown in Fig In the finite element model, the whole FRP strengthening composed of layers 1-3 (see Fig. 5.2) were modeled as a single transversely isotropic layer. The equivalent properties of the layer were calculated using the rule-of-mixtures (Jones, 1999). The properties of the FRP, bonding epoxy and the steel plate layers (see Fig. 5.2) are listed in Table 5.2. Due to the structural symmetry, it was necessary to model only one quarter of the splice joint. 2D 8-node elements (PLANE183), were used in the model. As shown in Fig. 5.8, four separate structural geometries were simulated, i.e. the perfect steel joint (MOD A), the steel joint with total cracking at the mid-point (MOD B), the steel joint with half cracking at the mid-point (MOD C) and the steel joint with 100 mm debonding centered at the mid-point (MOD D). These four different geometries were chosen to match conditions observed during the experimental tests. In particular, Models MOD B and MOD C were used to simulate two different cracking conditions at the mid-point. As the internal cracking was not visible during the tension tests, it is impossible to know how much the cracking developed into the splicing epoxy. Therefore these were chosen as typical cases. As will be shown later in the experimental results, the small FRP debonding simulated by the Teflon tape was not detected by the sensors. Therefore the second test (FRP 2) was treated analytically the same as the first one (FRP 1) and no theoretical simulation was performed for this FRP debonding condition. 111

122 Layers E 1 (GPa) E 2 (GPa) ν 12 G 12 (GPa) CFRP Steel Epoxy Table 5.2. Material properties used in the 2D FEA model. Figure D FEA model created using ANSYS. Symmetry boundary conditions were applied, therefore only one quarter of the joint was modeled. 112

123 (a) (b) (c) (d) Figure 5.8. Structural geometries modeled in theoretical modeling: (a) Perfect steel joint (MOD A); (b) Steel joint with total cracking at mid-point (MOD B); (c) Steel joint with half cracking at mid-point (MOD C); (d) Steel joint with 100 mm debonding centered at mid-point (MOD D). The plots are not drawn to scale and all units are in mm. 113

124 Fig. 5.9 shows a typical result from FEA for the perfect bonding simulation case MOD A under a tensile loading of 10 kn. A polynomial curve was fit to the interfacial shear stress so that the shear stress boundary conditions could be applied as explicit mathematical functions (see Section ). Figure 5.9. Boundary shear loading applied to the FRP system under axial loading of 10 kn for model MOD A. Polynomial curve fit also shown. Once the interfacial shear stress boundary conditions were available for all of the four structural geometries, the 3D shear-lag analysis of Chapter 3 was applied to determine the average normal stress in each of the FBG sensors. The same materials properties given in Table 3.1 were applied for this model. As for the finite element model, it was necessary to model only one quarter of the FRP strengthening applied to the splice joint. Fig shows the numbering of the FRP layers and their division into unit cells for the 3D shear-lag model. As can be seen, the cross-section is divided into 19 x 3 unit cells resulting in a total of

125 units. All together 136 interfacial shear stresses are available, among which 44 are known and 92 are unknown. The shear stress boundary conditions were applied to the self-monitoring FRP on the interfacial surface bonded to the steel plates. The FRP retrofit was modeled as a cantilever beam fixed at the midspan splice point. As a typical example, Fig plots the calculated average normal stress in optical fiber 4 (see Fig. 5.3) as a function of y for each of the four geometries for the case of a 10 kn axial load applied to the steel plates. (a) (b) Figure Self-monitoring FRP retrofit: (a) geometry; (b) unit divisions and stress numbering used in the shear-lag model. The plot is not drawn to scale and all units are in mm. Cross-section is doubly symmetric. 115

126 Figure Average normal stress in optical fiber 4 for the four different modeled geometries. The steel plates were loaded in tension by 10 kn. 5.3 Experimental Results Fig shows a photograph of specimen FRP 1 after complete failure of the specimen. As can be seen, FRP debonding dominated the specimen failure. During the tensile tests, it was observed that cracking first occurred in the epoxy filled gap between the two steel plates. Then the cracking further developed into the interface between the FRP and the 116

127 steel plates. Finally it was observed that FRP plate end debonding initiated at approximately the same time that the entire FRP plates debonded almost instantaneously. The photograph of Fig is also an excellent opportunity to observe the difference in cabling required for the same number of FBG and electrical resistance strain gage sensors. Figure FRP strengthened specimen FRP1 after specimen failure. 117

128 Fig shows a photograph of the small CFRP debonding region simulated by wrapping of Teflon tape in specimen FRP 2. This photograph was taken after failure of the specimen and shows the surfaces of the upper and lower CFRP after their separation. As can be seen, although the application of Teflon tape created debonding between the CFRP and the steel plates in some of the region, there are still some areas in which bonding epoxy leaked around the tape. Hence in the theoretical simulations, this specimen was not considered as one with CFRP debonding. Rather, it will be treated as the same as the first one (FRP 1). Figure Specimen FRP 2 post-failure. Upper and lower CFRP surfaces are shown after separation. 118

129 Figures 5.14 to 5.16 plot the strain measurements obtained from the three specimens, FRP 1, FRP 2 and FRP 3 respectively. Both the strain measurements from the FBG and conventional strain gage sensors are plotted for comparison. As can be seen from Fig. 5.15, in the second specimen, FRP 2, strain outputs from only three strain gages are plotted. It was observed that the strain gage bonded at the y = 200 mm (S4) was not well bonded due to the fact that excessive epoxy built up on the top of the FRP retrofits at this point. The specimens FRP 1 and FRP 2 had the same initial configuration, although the sensors were at different locations. As can be seen from Figures 5.14 and 5.15, the strain outputs from the FBG and strain gage sensors near the mid-span suddenly jumped at the point where cracking occurred at the splicing point in the steel joints. For specimen FRP 1, two sensors (S1 and S2) were located near the mid-span, whereas for specimen FRP 2 only one sensor (S1) was located near the mid-span. However, this strain output jump was not observed in the sensors far away from where cracking occurred, as expected. After initial cracking occurred in both specimens, the strain-load curve continued to change. This is an indication that more cracking occurred in the bonding epoxy, which continued to change the structural load behavior. Some discrepancies can be observed in Fig between the strain measurements of the FBG and strain gage sensors. Several possible experimental details could explain these differences. For example, there could be a strain differences at the surface and interior of the FRP laminate. However this was not supported by the shear-lag analysis. The most plausible explanation was that the FBG sensor positioning was poor in this first specimen. The positioning is most important in the region near the mid-span where the strain gradients are highest as seen in Fig After the better sensor positioning method was used, as 119

130 described in Section 5.1.1, more consistent strain measurements were clearly observed in the second and third tests (see Figures 5.15 and 5.16). Observation of the sensor outputs also shows that non-zero strain measurements exist in both the FBG and strain gage sensors at the free end of the laminate (y = 200 mm), where theoretically the axial stress should be zero in the FRP. This non-zero stress may be due to the pre-tensioning in the FBG sensors during the positioning procedure resulting in residual stresses or inter-lamina stresses developing in the laminate. A second contribution may be from the finite gage length of the sensors. The third specimen FRP 3 was designed to evaluate the capability of the sensors to identify FRP debonding. For this specimen a pre-existing debond of 25% of the laminate length was present from the beginning of loading, therefore no sudden jump in strain due to debonding initiation was observed. Only two strain gages were bonded within the 50 mm debonding region on one side, i.e. at y 1 = 0 mm and y 2 = 50 mm, due to the length of the gages. As can be seen in the strain- load relation in Fig. 5.16, the FBG sensors embedded within the debonding region have a much higher strain response than the sensors at y = 100 mm and the end-point y = 200 mm. To be better able to compare the effect of debonding at different load levels, Fig shows the strain profile under different loading levels along the structural length constructed from the strain output of the five FBG sensors. In the plot, the strain measurements are normalized with respect to the sensor output at location y = 100 mm. As can be seen, a relatively flat zone is observed in the strain profile, located within the debonding zone. A similar result was reported by Ghosh, et al. (2006). This is reasonable because no shear loading is transferred to the FRP retrofits in the debonded area. From the flat zone in the strain profile, the debonding location and size can theoretically be identified 120

131 by the embedded fiber optic distributed sensing system if reasonable sensor spacing is employed. Considering the FRP debonding length of 100 mm centered at the mid-point, the axial stress field is not constant at y = 100 mm. This can be seen from the theoretical model results in Fig For this example this strain value was used to normalize the strain profile for each load level. This value was chosen because the strain value at y = 200 mm is more sensitive to any residual inter-laminate stresses. However, the fact that the strain profile is not constant at this location, and will become more non-constant as the debonding progresses, does not prevent one from interpreting the normalized strain profile. From the theoretical results, the stress at the original far-field located at y = 100 mm for the FRP retrofit without debonding (MOD A) is less than the one for the FRP with debonding (MOD D). In other words, for the half length specimen as shown in Fig. 5.27, if the pre-existing FRP debonding develops beyond y = 50 mm into the interface on the right hand side, the strain at y = 100 mm will increase because the far-field stress region will shift towards the right hand side. However, the maximum stress level within the debonding zone will still remain constant. Correspondingly, the strain profiles normalized with respect to the strain at y = 100 mm are expected to decrease. As can be seen in Fig. 5.27, the strain profile in the debonding region decreased with increasing the load levels, which indicates that the pre-existing FRP debonding grew into the interface between the FRP and steel plates beyond the initial debonding length. 121

132 Figure Strain measurements from specimen FRP 1. Both FBG and conventional strain gage sensor measurements are plotted. 122

133 Figure Strain measurements from specimen FRP 2. Both FBG and conventional strain gage sensor measurements are plotted. 123

134 Figure Strain measurements from specimen FRP 3. Both FBG and conventional strain gage sensor measurements are plotted. 5.4 Comparison of Experimental Results and Theoretical Predictions Figures 5.17 to 5.26 plot the strain measurements for each sensor (both FBG and associated strain gage) superimposed on the predicted sensor strain response estimation based on the appropriate 3D shear-lag model (MOD A-D). 124

135 For the specimens FRP 1 and FRP 2, the pre-cracking geometry was modeled as MOD A. Cracking was then assumed to occur at the load level where the strain was discontinuous for each specimen. Once cracking occurred however, the extent of the cracking in the epoxy between the steel plates was not known. Therefore, two possible cracking conditions, complete cracking through the thickness of the epoxy (MOD B) and cracking halfway through the thickness of the epoxy (MOD C) were modeled analytically. Both simulations are plotted in Figs For the sensors of specimen FRP 1, the initial model prediction (MOD A) agreed reasonably well with the sensor measurements. The complete cracking through the epoxy thickness appears to better capture the strain gage results, however the strain jump in the FBG results are closer to the partial cracking model (MOD C). As mentioned earlier, however, the position of the FBGs was not well controlled for this experiment, therefore MOD B appears to be the better choice for this specimen. For the specimen FRP 2, the predictions for the initial specimen geometry through MOD A match well with the experimental results for the two sensors S2 and S3, however a considerable difference can be seen for the sensor S1 and the midspan of the specimen (see in Figs to 5.22). For this case, the partial cracking model (MOD C) better predicted the strain jump at the point of cracking. The difference between the theoretical modeling and the experiments may be due to the specimen dimension differences, which are difficult to control in hand prepared specimens. An additional source of error may be in the finite gage length effect. This effect will be addressed later in this section. As mentioned above, the simulation with complete cracking (MOD B) agrees better for the first test (FRP 1), whereas the simulation with half cracking (MOD C) fits better for the second test (FRP 2). Although cracking always occurred in the bonding epoxy at the 125

136 weakest point, different crack extents may exist in practical tests. In practice, it is expected that usually one major crack will occur and then the cracking will further propagate into the epoxy. This is confirmed from the change in the experimental curve slopes with increase of loading for both specimens. The difference in slopes from simulations MOD B and MOD C shows the same trend. One limitation of the 3D shear-lag model is that it is a linear model and therefore cannot predict the softening of the epoxy layer as multiple cracking occurs, apparent in the experimental results. For the specimen FRP 3, no sudden increase in strain occurred, therefore only predictions from the pre-existing FRP debonding (MOD D) were compared to the test results, as plotted Figs to The theoretical model consistently underpredicted the strain response of the all sensors. This may indicate that the specimen stiffness was higher than modeled. For additional comparison, the strain profiles predicted from the simulations and experimental measurements are plotted in Fig As mentioned previously, the strain measurement at the end-point gave non-zero information, so only four sensor measurements were compared to the theoretical profile. As can be seen in Fig. 5.28, the theoretical modeling yields consistent a strain profile with the experimental results. A maximum of 15% difference occurs in the strain values, however the profiles extremely close. For this reason, the normalized strain profiles are plotted in Fig Normalizing the profile thus removes errors such as that due to the specimen stiffness. The theoretical strain profile does not change as a function of applied load, however the measured strain profiles do change due to the nonlinearity mentioned previously or possible FRP debonding propagation. 126

137 A final source of error may be effect of the sensor the gage length. This effect would be more important near the specimen midspan where the strain gradients are the highest. This gage length was added into the theoretical predictions by taking the average of the strain along the 10 mm sensor length. As examples, the strain-load graphs of Figs. 5.17, 5.20 and 5.25 were re-evaluated including the gage length. These results are plotted in Figs to 5.32 respectively. As can be seen, the performance of the simplified model was improved after the sensor gage length was considered. 5.5 Conclusions The experimental study of this chapter demonstrates the feasibility of the self-monitoring FRP retrofits with embedded fiber optic sensors. It was shown that abnormal structural behaviors such as cracking in the epoxy can be identified from the sensor strain output. The location of the epoxy crack can be detected from the sudden change in the strain-load profile. Furthermore, the development of epoxy cracking may be monitored from the change of the strain-load curve slopes or the strain profiles. The position and size of the FRP debonding could theoretically be calculated from the constant zone in the reconstructed strain profile. The validity of the simplified shear-lag model was also evaluated with respect to the different relevant structural geometries. Consistent results were observed through comparison to experimental measurements, although some differences remain. In particular the 3D shear-lag model cannot predict the non-linear response of the laminate after significant cracking has occurred. Finally the gage length effect on the stress estimation in 127

138 embedded FBG sensors was analyzed. It was shown that better agreements were achieved if this effect was taken into consideration. 128

139 Figure Simulation results in specimen FRP 1 with comparisons to experimental results from FBG and strain gage sensors at location y = 18 mm. Three structural conditions were modeled: perfect joint (MOD A), epoxy complete cracking (MOD B) and epoxy half cracking (MOD C) at the splicing point. 129

140 Figure Simulation results in specimen FRP 1 with comparisons to experimental results from FBG and strain gage sensors at location y = 25 mm. Three structural conditions were modeled: perfect joint (MOD A), epoxy complete cracking (MOD B) and epoxy half cracking (MOD C) at the splicing point. 130

141 Figure Simulation results in specimen FRP 1 with comparisons to experimental results from FBG and strain gage sensors at location y = 75 mm. Three structural conditions were modeled: perfect joint (MOD A), epoxy complete cracking (MOD B) and epoxy half cracking (MOD C) at the splicing point. 131

142 Figure Simulation results in specimen FRP 2 with comparisons to experimental results from FBG and strain gage sensors at location y = 0 mm. Three structural conditions were modeled: perfect joint (MOD A), epoxy complete cracking (MOD B) and epoxy half cracking (MOD C) at the splicing point. 132

143 Figure Simulation results in specimen FRP 2 with comparisons to experimental results from FBG and strain gage sensors at location y = 100 mm. Three structural conditions were modeled: perfect joint (MOD A), epoxy complete cracking (MOD B) and epoxy half cracking (MOD C) at the splicing point. 133

144 Figure Simulation results in specimen FRP 2 with comparisons to experimental results from FBG and strain gage sensors at location y = 175 mm. Three structural conditions were modeled: perfect joint (MOD A), epoxy complete cracking (MOD B) and epoxy half cracking (MOD C) at the splicing point. 134

145 Figure Simulation results in specimen FRP 3 with comparisons to experimental results from FBG and strain gage sensors at location y = 0 mm. The FRP debonding at mid-span with a length of 100 mm was simulated (MOD D). 135

146 Figure Simulation results in specimen FRP 3 with comparisons to experimental results from FBG and strain gage sensors at location y = 25 mm. The FRP debonding at mid-span with a length of 100 mm was simulated (MOD D). 136

147 Figure Simulation results in specimen FRP 3 with comparisons to experimental results from FBG and strain gage sensors at location y = 50 mm. The FRP debonding at mid-span with a length of 100 mm was simulated (MOD D). 137

148 Figure Simulation results in specimen FRP 3 with comparisons to experimental results from FBG and strain gage sensors at location y = 100 mm. The FRP debonding at mid-span with a length of 100 mm was simulated (MOD D). 138

149 Figure Normalized strain profile for the CFRP in specimen FRP 3 from FBG sensor output at different load levels. 139

150 Figure Strain profile for specimen FRP 3 from both simulation (MOD D) and FBG sensor output at different load levels. For each load level, the simulation is plotted as outline symbol and the FBG output as solid symbol. 140

151 Figure Normalized strain profile for specimen FRP 3 from both simulation (MOD D) and FBG sensor output at different load levels. 141

152 Figure Simulation results with sensor size effect considered for specimen FRP 1 with comparison to experimental results from FBG and strain gage sensors at y = 18 mm. 142

153 Figure Simulation results with sensor size effect considered for specimen FRP 2 with comparison to experimental results from FBG and strain gage sensors at y = 0 mm. 143