UNIVERSITY OF CALIFORNIA, SAN DIEGO

Transcription

1 UNIVERSITY OF CALIFORNIA, SAN DIEGO Structural Characterization of Concrete Filled Fiber Reinforced Shells A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Engineering Science (Structural Engineering) by Andrew Davol Committee in charge: Professor Frieder Seible Professor Gilbert Hegemier Professor Vistasp Karbhari Professor Donald Smith Professor Audrey Terras 1998

2 Copyright Andrew Davol, 1998 All rights reserved.

3 The dissertation of Andrew Davol is approved, and it is acceptable in quality and form for publication on microfilm: Chair University of California, San Diego 1998 iii

4 Dedication This work is dedicated to my first true companion Max, whose boundless energy was a constant inspiration to me, and to my life companion Michele, whose vision of our shared future has made the completion of this dissertation possible. iv

5 TABLE OF CONTENTS SIGNATURE PAGE... iii DEDICATION...iv TABLE OF CONTENTS...v LIST OF FIGURES... viii LIST OF TABLES... xxiii LIST OF SYMBOLS...xxv ABSTRACT...xxxii 1. INTRODUCTION RESEARCH REVIEW MATERIAL CHARACTERIZATION ADVANCED COMPOSITE SHELLS Fiber Reinforcement Matrix Materials Manufacturing Processes Typical Ply Properties Classical Lamination Theory Equivalent Plate Properties First Ply Failure Criteria Thermal Expansion CONCRETE Uniaxial Compression Biaxial and Triaxial Stress States - Confinement Effects Tension ANALYTICAL MODELING OF CONCRETE FILLED FRP SHELLS CIRCULAR SHELLS Compression Tension Shear Bending CONREC SHELLS Compression Bending...68 v

6 5. EXPERIMENTAL PROGRAM TO VALIDATE CONCRETE FILLED FRP TUBE BEHAVIOR SMALL SCALE SHELLS Concrete Characterization Compression Bending FULL SCALE BENDING TESTS Concrete Properties Hollow Shell Concrete Filled Shells Concrete Filled Shell with integral Concrete Deck CORRELATION OF ANALYTICAL MODELS TO EXPERIMENTAL DATA CIRCULAR SHELLS Small Scale Shells Full Scale Specimens CONREC SHELLS Compression Bending PARAMETER STUDIES OF MATERIAL, LAY-UP, THICKNESS AND SHAPE VARIATIONS CIRCULAR SHELLS Compression Behavior Bending Behavior CONREC SHELLS HYBRID SHELLS Compression Bending STRESS CONCENTRATIONS, TENSION STIFFENING AND THERMAL EXPANSION EFFECTS STRESS CONCENTRATIONS Closed Form Solution Parameter Study Case Study EFFECTS OF TENSION STIFFENING THERMAL EFFECTS Thermal Strains in Circular Sections Thermal Testing Parameter Studies for Thermally Induced Strains LOCAL COMPRESSION BUCKLING OF CONCRETE FILLED FRP SHELLS CONCLUSIONS vi

7 APPENDIX - MOMENT CURVATUR PROGRAM...22 REFERENCES vii

8 LIST OF FIGURES FIGURE 1-1 CONVENTIONALLY REINFORCED CONCRETE COLUMN...5 FIGURE 1-2 CARBON FIBER WRAP APPLIED TO BRIDGE COLUMN FOR SEISMIC RETROFIT...ERROR! BOOKMARK NOT DEFINED. FIGURE 1-3 CONCEPT FOR CONCRETE FILLED FIBER REINFORCED SHELL...6 FIGURE 1-4 PROTOTYPE BRIDGE STRUCTURE WITH CARBON SHELL GIRDERS AND A FIBER GLASS DECK SYSTEM...6 FIGURE 2-1 AREA AND VOLUME STRAIN DEFINITION...12 FIGURE 2-2 EXPANSION BEHAVIOR OF CONCRETE...12 FIGURE 3-1 MATERIAL AND STRUCTURAL COORDINATE SYSTEMS...23 FIGURE 3-2 GEOMETRY OF LAMINATE...25 FIGURE 3-3 STRESS STRAIN MODELS FOR CONFINED CONCRETE...34 FIGURE 4-1 EXPERIMENTAL AND SMOOTHED DILATION RATE...42 FIGURE 4-2 CONCRETE TANGENT MODULUS VS. RADIAL STRAIN...43 FIGURE 4-3 EQUIVALENT TANGENT POISSON'S RATIO FOR TEST CYLINDERS...46 FIGURE 4-4 ANALYTICAL EQUIVALENT TANGENT POISSON'S RATIO...46 FIGURE 4-5 MAXIMUM TANGENT POISSON'S RATIO VS. HYDROSTATIC PRESSURE...47 FIGURE 4-6 ANALYTICAL MODEL FOR COMPRESSION BEHAVIOR...48 viii

9 FIGURE 4-7 DETERMINATION OF MAXIMUM EQUIVALENT TANGENT POISSON'S RATIO...49 FIGURE 4-8 SHEAR TRANSFER BETWEEN CONCRETE CORE AND COMPOSITE SHELL...51 FIGURE 4-9 GEOMETRIC PROPERTIES FOR DETERMINATION OF SHEAR STRESS...53 FIGURE 4-1 ANALYSIS FLOW FOR BENDING BEHAVIOR...54 FIGURE 4-11 CONREC CROSS SECTION...57 FIGURE 4-12 FINITE ELEMENT MODEL USED FOR EVALUATION OF CONREC SECTIONS...59 FIGURE 4-13 CONREC GEOMETRIES USED FOR THIS ANALYSIS...61 FIGURE 4-14 AREA STRAIN RATIO PROFILE, % ±1 O PLIES, FLAT TO RADIUS RATIO FIGURE 4-15 AREA STRAIN RATIO PROFILE, % ±1 O FIBERS, FLAT TO RADIUS RATIO OF FIGURE 4-16 AREA STRAIN RATIO PROFILE, % ±1 O FIBERS, FLAT TO RADIUS RATIO OF FIGURE 4-17 AREA STRAIN RATIO PROFILE, % ±1 O FIBERS, FLAT TO RADIUS RATIO OF FIGURE 4-18 AREA STRAIN RATIO FOR % ±1 O CONREC SHELL...64 FIGURE 4-19 AREA STRAIN RATIO FOR 5% ±1 O CONREC SHELL...64 FIGURE 4-2 AREA STRAIN RATIO FOR 8% ±1 O CONREC SHELL...65 ix

10 FIGURE 4-21 HOOP STRESS IN SHELLS WITH FLAT TO RADIUS RATIO OF FIGURE 4-22 HOOP STRESS IN SHELLS WITH FLAT TO RADIUS RATIO OF FIGURE 4-23 HOOP STRESS IN SHELLS WITH FLAT TO RADIUS RATIO OF FIGURE 4-24 HOOP STRESS IN SHELLS WITH FLAT TO RADIUS RATIO OF FIGURE 4-25 MOMENT CURVATURE OF TYPICAL CONREC SECTION...69 FIGURE 4-26 CONCRETE STRESS STRAIN RELATION...7 FIGURE 4-27 COMPARATIVE MOMENT CURVATURE FOR CONREC SECTION WITH VARIOUS CONCRETE MODELS...7 FIGURE 5-1 NOMINAL GEOMETRY OF CONREC SECTION...72 FIGURE 5-2 CONCRETE COMPRESSION STRESS STRAIN RELATION...75 FIGURE 5-3 HOOP STRAIN VS. LONGITUDINAL STRAIN FOR CONCRETE CYLINDERS UNDER UNIAXIAL COMPRESSION...76 FIGURE 5-4 COMPARISON OF HOOP STRAINS FOR CYLINDERS WITH VARIOUS ASPECT RATIOS...77 FIGURE 5-5 TYPICAL COMPRESSION TEST SETUP...78 FIGURE 5-6 STRAIN GAGE LAYOUT FOR CIRCULAR COMPRESSION SPECIMENS...82 x

11 FIGURE 5-7 LOAD VS. STRAIN CURVES FOR CIRCULAR CYLINDERS...82 FIGURE 5-8 HOOP VS. LONGITUDINAL STRAINS FOR CIRCULAR CYLINDERS...83 FIGURE 5-9 LONGITUDINAL STRAIN IN HELICAL CIRCULAR CYLINDERS...83 FIGURE 5-1 CONCRETE STRESS STRAIN CURVES FOR ALL HOOP CIRCULAR CYLINDERS...84 FIGURE 5-11 TYPICAL FAILURE OF ALL HOOP CIRCULAR SHELL...84 FIGURE 5-12 FAILURE OF HELICAL CIRCULAR SHELLS...85 FIGURE 5-13 STRAIN GAGE LAYOUT FOR CONREC COMPRESSION SPECIMENS...86 FIGURE 5-14 LOAD VS. LONGITUDINAL STRAIN CONREC CYLINDERS...87 FIGURE 5-15 HOOP VS. LONGITUDINAL STRAIN FOR CONREC CYLINDERS...87 FIGURE 5-16 TYPICAL FAILURE OF ALL HOOP CONREC SHELL...88 FIGURE 5-17 FAILURE OF CONREC HELICAL SHELLS...88 FIGURE 5-18 SCHEMATIC OF FOUR POINT BENDING TEST SETUP...9 FIGURE 5-19 FOUR POINT BENDING TEST ON SMALL SCALE SPECIMEN...9 FIGURE 5-2 STRAIN GAGE LAYOUT FOR CIRCULAR BENDING SPECIMENS...91 xi

12 FIGURE 5-21 LOAD - DISPLACEMENT CURVE FOR THIN CIRCULAR BENDING SPECIMEN...93 FIGURE 5-22 STRAIN PROFILE FOR THIN CIRCULAR SECTION IN CONSTANT MOMENT REGION...93 FIGURE 5-23 STRAIN PROFILE FOR THIN CIRCULAR SECTION IN SHEAR AREA...94 FIGURE 5-24 LONGITUDINAL STRAIN VS. MOMENT FOR THIN CIRCULAR SPECIMEN...94 FIGURE 5-25 HOOP STRAIN VS. MOMENT FOR THIN CIRCULAR SPECIMEN...95 FIGURE 5-26 SHEAR STRAIN VS. SHEAR FOR THIN CIRCULAR SPECIMEN...95 FIGURE 5-27 LOAD - DISPLACEMENT CURVE FOR THICK CIRCULAR BENDING SPECIMEN...97 FIGURE 5-28 STRAIN PROFILE FOR THICK CIRCULAR SECTION IN CONSTANT MOMENT REGION...97 FIGURE 5-29 STRAIN PROFILE FOR THICK CIRCULAR SECTION IN SHEAR SPAN...98 FIGURE 5-3 LONGITUDINAL STRAIN VS. MOMENT FOR THICK CIRCULAR SECTION...98 FIGURE 5-31 HOOP STRAIN VS. MOMENT FOR THICK CIRCULAR SECTION...99 xii

13 FIGURE 5-32 SHEAR STRAIN VS. APPLIED SHEAR FOR THICK CIRCULAR SPECIMEN...99 FIGURE 5-33 FAILURE OF THICK CIRCULAR SECTION...1 FIGURE 5-34 STRAIN GAGE LAYOUT FOR CONREC BENDING SPECIMENS...11 FIGURE 5-35 LOAD - DISPLACEMENT CURVE FOR THIN CONREC BENDING SPECIMEN...12 FIGURE 5-36 STRAIN PROFILE FOR THIN CONREC SECTION IN CONSTANT MOMENT REGION...12 FIGURE 5-37 STRAIN PROFILE FOR THIN CONREC SECTION IN SHEAR AREA...13 FIGURE 5-38 LONGITUDINAL STRAIN VS. MOMENT FOR THIN CONREC SPECIMEN...13 FIGURE 5-39 HOOP STRAIN VS. MOMENT FOR THIN CONREC SPECIMEN...14 FIGURE 5-4 SHEAR STRAIN VS. APPLIED SHEAR FOR THIN CONREC SPECIMEN...14 FIGURE 5-41 FAILURE OF THIN CONREC SECTION...15 FIGURE 5-42 LOAD-DISPLACEMENT RESPONSE FOR THICK CONREC SPECIMEN...16 FIGURE 5-43 STRAIN PROFILE FOR THICK CONREC SECTION IN CONSTANT MOMENT REGION...17 xiii

14 FIGURE 5-44 STRAIN PROFILE FOR THICK CONREC SECTION IN SHEAR AREA...17 FIGURE 5-45 LONGITUDINAL STRAIN VS. MOMENT FOR THICK CONREC...18 FIGURE 5-46 HOOP STRAIN VS. MOMENT FOR THICK CONREC SECTION...18 FIGURE 5-47 SHEAR STRAIN VS. APPLIED SHEAR FOR THICK CONREC SPECIMEN...19 FIGURE 5-48 FAILURE OF THICK CONREC SECTION...19 FIGURE 5-49 FULL SCALE FOUR POINT BENDING TEST...11 FIGURE 5-5 SCHEMATIC OF FULL SCALE BENDING TESTS FIGURE 5-51 ENDBLOCK FOR SUPPORT OF BENDING TEST SPECIMENS FIGURE 5-52 SHELL END DIAPHRAGM FIGURE 5-53 STEEL CONNECTION CAGE FIGURE 5-54 SHELL WITH CONNECTING CAGE FIGURE 5-55 ENDBLOCK LOWER SECTION FIGURE 5-56 PLACEMENT OF SHELL INTO ENDBLOCKS FIGURE 5-57 PVC PIPE FOR PUMPING SHELL FIGURE 5-58 COMPLETED END BLOCK FORM FIGURE 5-59 INSTRUMENTATION LAYOUT FOR HOLLOW SHELL TEST xiv

15 FIGURE 5-6 LOAD DISPLACEMENT CURVE FOR HOLLOW SHELL TESTS FIGURE 5-61 LONGITUDINAL AND HOOP STRAINS FOR HOLLOW SHELLS FIGURE 5-62 SHEAR STRAINS FOR HOLLOW SHELL TESTS FIGURE 5-63 DISPLACEMENT INSTRUMENTATION FOR FILLED SHELL TEST FIGURE 5-64 STRAIN GAGE LOCATIONS AND DESIGNATION FOR FILLED TUBE TESTS FIGURE 5-65 LOAD DISPLACEMENT PLOT FOR FILLED SHELL TEST # FIGURE 5-66 STRAIN PROFILE IN CONSTANT MOMENT SECTION, FILLED SHELL TEST # FIGURE 5-67 STRAIN PROFILE IN SHEAR SECTION, FILLED SHELL TEST # FIGURE 5-68 LONGITUDINAL STRAIN VS. MOMENT IN COMP. ZONE FOR FILLED SHELL # FIGURE 5-69 LONGITUDINAL STRAIN VS. MOMENT IN TENSION ZONE FOR FILLED SHELL # FIGURE 5-7 HOOP STRAIN VS. MOMENT IN COMPRESSION ZONE FOR FILLED SHELL # xv

16 FIGURE 5-71 HOOP STRAIN VS. MOMENT IN TENSION ZONE FOR FILLED SHELL # FIGURE 5-72 SHEAR STRAIN VS. APPLIED SHEAR FOR FILLED SHELL # FIGURE 5-73 FAILURE OF FILLED SHELL # FIGURE 5-74 LOAD DISPLACEMENT PLOT FOR FILLED SHELL TEST # FIGURE 5-75 STRAIN PROFILE IN CONSTANT MOMENT SECTION, FILLED SHELL TEST # FIGURE 5-76 STRAIN PROFILE IN SHEAR SECTION, FILLED SHELL TEST # FIGURE 5-77 LONGITUDINAL STRAIN VS. MOMENT IN COMP. ZONE FOR FILLED SHELL # FIGURE 5-78 LONGITUDINAL STRAIN VS. MOMENT IN TENSION ZONE FOR FILLED SHELL # FIGURE 5-79 HOOP STRAIN VS. MOMENT IN COMPRESSION ZONE FOR FILLED SHELL # FIGURE 5-8 HOOP STRAIN VS. MOMENT IN TENSION ZONE FOR FILLED SHELL # FIGURE 5-81 SHEAR STRAIN VS. APPLIED SHEAR FOR FILLED SHELL # FIGURE 5-82 FAILURE OF FILLED SHELL # xvi

17 FIGURE 5-83 DISPLACEMENT INSTRUMENTATION FOR FILLED SHELL WITH SLAB TEST FIGURE 5-84 STRAIN GAGE LOCATIONS AND DESIGNATION FOR SHELL FIGURE 5-85 STRAIN GAGES PLACED ON SHEAR CONNECTION DOWELS FIGURE 5-86 INSTRUMENTATION LOCATIONS AND DESIGNATION FOR TOP MAT OF STEEL REINFORCEMENT IN DECK FIGURE 5-87 STRESS CONCENTRATION STRAIN GAGE LOCATIONS AND DESIGNATION...14 FIGURE 5-88 LOAD DISPLACEMENT ENVELOPE FOR FILLED SHELL WITH SLAB FIGURE 5-89 STRAIN PROFILE ACROSS SECTION FOR FILLED SHELL WITH SLAB FIGURE 5-9 STRESS CONCENTRATION AROUND PENETRATION IN SHEAR SPAN FIGURE 5-91 STRESS CONCENTRATION IN THE CONST. MOMENT REGION FIGURE 6-1 CONCRETE STRESS VS. STRAIN FOR SMALL SCALE COMPRESSION SPECIMENS FIGURE 6-2 RADIAL VS. LONGITUDINAL STRAIN FOR SMALL SCALE COMP. SPECIMENS xvii

18 FIGURE 6-3 LOAD VS. DISPLACEMENT FOR SMALL SCALE CIRCULAR SECTIONS...15 FIGURE 6-4 STRAINS IN CONSTANT MOMENT REGION FOR THIN CIRCULAR SHELL...15 FIGURE 6-5 STRAINS IN CONSTANT MOMENT REGION FOR THICK CIRCULAR SHELL FIGURE 6-6 STRAINS IN SHEAR AREA FOR THIN CIRCULAR SHELL FIGURE 6-7 STRAINS IN SHEAR SPAN FOR THICK CIRCULAR SHELL FIGURE 6-8 SHEAR STRAIN IN THIN CIRCULAR SHELL FIGURE 6-9 SHEAR STRAIN IN THICK CIRCULAR SHELL FIGURE 6-1 LOAD DISPLACEMENT CURVES FOR FULL SCALE FILLED SHELL TESTS FIGURE 6-11 EXTREME FIBER STRAINS IN CONSTANT MOMENT REGION FOR SHELL # FIGURE 6-12 EXTREME FIBER STRAINS IN SHEAR SPAN FOR SHELL # FIGURE 6-13 EXTREME FIBER STRAINS IN CONSTANT MOMENT REGION FOR SHELL # FIGURE 6-14 EXTREME FIBER STRAINS IN SHEAR SPAN FOR SHELL # xviii

19 FIGURE 6-15 SHELL CENTERLINE SHEAR STRAINS FOR SHELL # FIGURE 6-16 SHELL CENTERLINE SHEAR STRAINS FOR SHELL # FIGURE 6-17 LOAD VS. LONGITUDINAL STRAIN FOR THICK CONREC CYLINDERS FIGURE 6-18 HOOP STRAINS VS. LONGITUDINAL STRAIN IN THICK CONREC SECTION FIGURE 6-19 LOAD DISPLACEMENT FOR CONREC BENDING SPECIMENS FIGURE 6-2 STRAINS IN CONSTANT MOMENT REGION FOR THIN CONREC SHELL FIGURE 6-21 STRAINS IN CONSTANT MOMENT REGION FOR THICK CONREC SHELL FIGURE 6-22 STRAINS IN SHEAR AREA FOR THIN CONREC SHELL FIGURE 6-23 STRAINS IN SHEAR AREA FOR THICK CONREC SHELL FIGURE 6-24 SHEAR STRAIN IN THIN CONREC SHELL FIGURE 6-25 SHEAR STRAIN IN THICK CONREC SHELL FIGURE 7-1 COMPRESSION BEHAVIOR OF CARBON EPOXY SHELLS FIGURE 7-2 COMPRESSION BEHAVIOR OF E-GLASS SHELLS FIGURE 7-3 CONFINEMENT EFFICIENCY OF E-GLASS VS. CARBON SHELLS xix

20 FIGURE 7-4 GEOMETRY FOR MOMENT CALCULATION FIGURE 7-5 NORMALIZED MOMENT CURVATURE, R/T=1, CARBON EPOXY SHELL FIGURE 7-6 NORMALIZED MOMENT CURVATURE, R/T=15, CARBON EPOXY SHELL FIGURE 7-7 NORMALIZED MOMENT CURVATURE, R/T=2, CARBON EPOXY SHELL FIGURE 7-8 NORMALIZED MOMENT CURVATURE, R/T=25, CARBON EPOXY SHELL FIGURE 7-9 NORMALIZED MOMENT CURVATURE, R/T=1, E-GLASS SHELL FIGURE 7-1 NORMALIZED MOMENT CURVATURE, R/T=15, E-GLASS SHELL FIGURE 7-11 NORMALIZED MOMENT CURVATURE, R/T=2, E-GLASS SHELL FIGURE 7-12 NORMALIZED MOMENT CURVATURE, R/T=25, E-GLASS SHELL FIGURE 7-13 MOMENT CURVATURE WITH AXIAL LOAD, R/T=1, 1% HELICAL FIBERS...18 FIGURE 7-14 MOMENT CURVATURE WITH AXIAL LOAD, R/T=1, 5% HELICAL FIBERS...18 xx

21 FIGURE 7-15 MOMENT CURVATURE WITH AXIAL LOAD, R/T=1, 9% HELICAL FIBERS FIGURE 7-16 MOMENT CURVATURE WITH AXIAL LOAD, R/T=25, 1% HELICAL FIBERS FIGURE 7-17 MOMENT CURVATURE WITH AXIAL LOAD, R/T=25, 5% HELICAL FIBERS FIGURE 7-18 MOMENT CURVATURE WITH AXIAL LOAD, R/T=25, 9% HELICAL FIBERS FIGURE 7-19 FLEXURAL STIFFNESS OF E-GLASS VS. CARBON SHELLS FIGURE 7-2 MOMENT CURVATURE RESPONSE FOR CONREC SECTIONS, D/T= FIGURE 7-21 MOMENT CURVATURE RESPONSE FOR CONREC SECTIONS, D/T= FIGURE 7-22 NORMALIZED MOMENT CURVATURE RESPONSE FOR CONREC SECTIONS, D/T= FIGURE 8-1 INFINITE PLATE WITH A CIRCULAR INCLUSION FIGURE 8-2 LOAD CASES FOR STRESS CONCENTRATION STUDY FIGURE 8-3 STRESS CONCENTRATION FACTORS FIGURE 8-4 TANGENTIAL STRESS CONCENTRATION VARIATION AROUND HOLE FOR 8% HELICAL SHELL FIGURE 8-5 SHEAR CONNECTION xxi

22 FIGURE 8-6 FINITE ELEMENT MODEL FOR STRESS CONCENTRATION STUDIES...21 FIGURE 8-7 AVERAGE STRESS VS. AVERAGE STRAIN FOR TENSION STIFFENING...23 FIGURE 8-8 LOAD IN CONCRETE FILLED CARBON SHELL IN PURE TENSION WITH AND WITHOUT TENSION STIFFENING EFFECTS...23 FIGURE 8-9 THERMALLY INDUCED MECHANICAL STRAINS PER DEGREE CENTIGRADE...26 FIGURE 8-1 COEFFICIENTS OF THERMAL EXPANSION FOR FIBER REINFORCED SHELLS...28 FIGURE 8-11 HOOP STRESS IN SHELL 9 O PLIES DUE TO A TEMPERATURE RISE OF 55 O C...28 FIGURE 8-12 ULTIMATE BUCKLING STRAIN FOR ALL BENDING SPECIMENS...21 xxii

23 LIST OF TABLES TABLE 3-1 TYPICAL PROPERTIES OF COMMERCIAL GLASS FIBER REINFORCEMENTS...14 TABLE 3-2 MECHANICAL PROPERTIES FOR SELECT CARBON FIBERS...15 TABLE 3-3 MECHANICAL PROPERTIES FOR COMMON THERMOSETTING RESINS...16 TABLE 3-4 TYPICAL PLY PROPERTIES FOR FIBER-REINFORCED EPOXY RESINS...19 TABLE 4-1 CONSTANTS FOR TANGENT MODULUS RELATION...43 TABLE 4-2 COMPOSITE LAY-UPS USED FOR CONREC STUDIES...61 TABLE 5-1 SMALL SCALE TEST SHELLS...73 TABLE 5-2 VENDOR SUPPLIED PLY PROPERTIES...73 TABLE 5-3 EQUIVALENT PLATE PROPERTIES...74 TABLE 5-4 EXPERIMENTALLY DERIVED CONCRETE PROPERTIES...76 TABLE 5-5 COMPRESSION SPECIMENS...79 TABLE 5-6 SHELLS FOR SMALL SCALE BENDING TESTS...89 TABLE 5-7 COMPOSITE ARCHITECTURES FOR LARGE SCALE TESTS TABLE 5-8 VENDOR SUPPLIED PLY PROPERTIES TABLE 5-9 EQUIVALENT PLATE PROPERTIES FOR LARGE SCALE TESTS xxiii

24 TABLE 5-1 CONCRETE MIX USED FOR FILLED SHELLS TABLE 5-11 CONCRETE PROPERTIES FOR FILLED SHELL TESTS TABLE 7-1 PLY PROPERTIES FOR PARAMETER STUDIES TABLE 7-2 GEOMETRY OF CONREC SECTIONS FOR NORMALIZED COMPARISON TABLE 8-1 COMPOSITE ARCHITECTURES FOR STRESS CONCENTRATION STUDY TABLE 8-2 STRESS CONCENTRATION FACTORS TABLE 8-3 STRESS AROUND PENETRATION FOR BEAM AND SLAB SHEAR CONNECTION...2 TABLE 8-4 PLY PROPERTIES FOR THERMAL TESTING...26 xxiv

25 LIST OF SYMBOLS Scalars A c cross sectional area of concrete core E 1 ply modulus in fiber direction E 2 ply modulus transverse to fiber direction E c concrete tangent modulus E ca average tangent stiffness of concrete core E co initial concrete modulus E H equivalent modulus in hoop direction for shell E L equivalent modulus in longitudinal direction for shell E x modulus in x direction E y modulus in y direction f - flat length for conrec section f c compression strength of unconfined concrete f c concrete stress f t tension strength of concrete G 12 ply in-plane shear modulus G xy shear modulus in x-y plane h n distance from mid-plane to near surface of ply n M moment in section M x moment applied to laminate about y axis M T x equivalent thermal moment about y axis xxv

26 M xy twisting moment applied to laminate M T xy equivalent thermal twisting moment M y moment applied to laminate about x axis M T y equivalent thermal moment about x axis N x force applied to laminate in x direction N T x equivalent thermal force on laminate in x direction N xy in-plane shear force applied to laminate N T xy equivalent thermal in-plane shear force applied to laminate N y force applied to laminate in y direction N T y equivalent thermal force applied to laminate in y direction P c load in concrete P s load in shell q c shear flow between shell and core q s shear flow in shell r - radius for conrec section R shell mean radius t thickness of laminated shell t eff effective thickness of concrete used for shear calculation u displacement in x direction u o mid-plane displacement in x direction v displacement in y direction V shear load on section xxvi

27 V c shear capacity of concrete core v o mid-plane displacement in y direction w displacement in z direction w o mid-plane displacement in z direction α x coefficient of thermal expansion in structural x direction α xy apparent shear coefficient of thermal expansion in structural x-y plane α y coefficient of thermal expansion in structural y direction α 1 ply coefficient of thermal expansion in fiber direction α 2 ply coefficient of thermal expansion normal to fiber direction β 1 bond characteristic factor for tension stiffening β 2 loading characteristic factor for tension stiffening ε 1 ply strain in fiber direction or longitudinal concrete strain ε a area strain ε cf average strain for tension stiffening ε v volume strain ε x strain in structural x direction ε o x mid-plane strain in structural x direction ε T x thermal strain in structural x direction ε y strain in structural y direction ε o y mid-plane strain in structural y direction ε T y thermal strain in structural y direction xxvii

28 ε Τ 1 ply thermal strain in fiber direction ε 2 ply strain normal to fiber direction ε Τ 2 ply thermal strains normal to fiber direction ε 3 ply strain normal to fiber direction γ xy shear strain in structural x-y plane γ T xy thermal shear strain in structural x-y plane γ 12 ply shear strain 1-2 plane γ 13 ply shear strain 1-3 plane γ 23 ply shear strain 2-3 plane κ x curvature about y axis κ xy twist κ y curvature about x axis λ concrete density factor ν 12 ply in-plane Poisson s ratio for loading in the fiber direction ν 21 ply in-plane Poisson s ratio for loading normal to the fiber direction θ rotation angle between ply and structure coordinate systems or angle around hole for stress concentration analysis σ x stress in structural x direction σ y stress in structural y direction σ z stress in structural z direction σ 1 ply stress in fiber direction or longitudinal concrete stress xxviii

29 σ 2 ply stress normal to fiber direction σ 3 ply stress normal to fiber direction τ xy shear stress in x-y plane τ 12 ply shear stress 1-2 plane τ 13 ply shear stress 1-3 plane τ 23 ply shear stress 2-3 plane ε L longitudinal strain in shell ε H hoop strain in shell ν LH Poisson s ratio in shell for loading in the longitudinal direction ν HL Poisson s ratio in shell for loading in the hoop direction ν c concrete equivalent tangent Poisson s ratio ν co initial concrete Poisson s ratio σ L longitudinal stress in shell σ H hoop stress in shell ε r radial strain in concrete core σ r radial stress in concrete core µ dilation rate T temperature σ T x - thermal stress in x direction σ T y - thermal strain in y direction τ T xy - thermal shear stress in x-y plane xxix

30 σ hyd - hydrostatic pressure ν max - maximum equivalent tangent Poisson s ratio for concrete µ max - maximum dilation rate µ u - ultimate dilation rate a ij - coeficients of deformation for stress concentration analysis E θ - modulus tangent to cutout p - far field stress for stress concentration analysis f cr - cracking stress for concrete ε co - strain at maximum stress for unconfined concrete ε M x - mechanical strain in x direction ε M y - mechanical strain in y direction γ M xy - mechanical shear strain in x-y plane ε T r - radial strain due to temperature change ε T H - hoop strain due to temperature change γ o xy - mid-plane shear strain Vectors {ε o } - mid-plane strains {κ} - curvatures {σ 1 } - stresses in material coordinate system {ε 1 }- strains in material coordinate system {σ x } - stresses in structure coordinate system xxx

31 {ε x } - strains in structure coordinate system {N} - section forces {M} - section moments Matrices [A] - in-plane stiffness matrix [A * ] - in-plane flexibility matrix = [A] -1 [B] - in-plane out-of-plane coupling matrix [D] - out-of-plane stiffness matrix [Q] - ply stiffness matrix in material coordinate system [ Q ]- ply stiffness matrix in structure coordinate system xxxi

32 ABSTRACT OF THE DISSERTATION Structural Characterization of Concrete Filled Fiber Reinforced Shells by Andrew Davol Doctor of Philosophy in Engineering Sciences (Structural Engineering) University of California, San Diego, 1998 Professor Frieder Seible, Chair Optimizing structures often leads engineers to combine several materials into a hybrid system which utilizes the advantages inherent in each of the constituents. Reinforced concrete is a classic example of such a system combining the superior tension carrying capability of steel with the compression capacity and low cost of concrete. A similar concept is being investigated which replaces the steel in a conventional reinforced concrete member with a fiber reinforced polymer (FRP) shell. These shells are manufactured with continuous relatively stiff fibers imbedded in a softer matrix material. The nature of these materials allows the properties in various xxxii

33 directions to be controlled by placing fibers with prescribed orientations permitting the engineer to tailor the material to a specific application. It is felt that such a system may lead to efficient construction techniques that could reduce erection times and construction costs due to the light weight of the FRP shells. This document examines the structural behavior of concrete filled FRP shells concentrating on compression and bending behavior, thermal response and stress concentration effects. Analytical models are proposed to predict the stress and deformation state of the fiber reinforced shell and concrete core under various loading conditions. The nonlinear response of concrete confined by a linear elastic shell under compressive loads is investigated. Experimental validation and calibration of these models has been carried out and is presented. Related documents associated with this project explore the joining of advanced composite components and the behavior of structural systems assembled with these components. xxxiii

34 1. INTRODUCTION Structural engineers have long known the value of combining materials into a composite structural system that takes advantage of the strengths inherent in each of its constituents. Steel reinforced concrete is a classic example of this type of structural system. These materials complement each other well due to the compression carrying capability of the concrete and the tension carrying capability of the steel reinforcement. Through the years this system has been improved upon as the understanding of the concrete, steel and the interaction between the two has increased. These improvements include the realization of the importance of transverse steel reinforcement to provide confinement for the concrete core and to increase the shear carrying capability of the structural member. It has also been well established that the load carrying behavior of concrete in one principal direction is greatly affected by the presence of stresses (or deformations) in the other principal directions. Specifically it has been shown that the strength and ductility capacity can be greatly increased by the presence of triaxial compression [1]. Such a triaxial state of compression is achieved by providing confinement for the concrete. Under uniaxial compression concrete will expand normal to the loading direction due to the Poisson s effect and microcracking. If reinforcement is placed to resist (confine) this expansion, the desired stress state of triaxial compression is achieved. In traditional reinforced concrete structures this confinement is attained by providing transverse reinforcement that takes the form of hoops, spirals or stirrups as shown in Figure 1-1. This enhanced strength 1

35 2 and ductility become very important when considering a structure's ability to withstand seismic loading. In many older structures insufficient transverse reinforcement was provided to achieve the strength and ductility required to resist seismic deformation demands. Brittle shear failures and ductile failures due to insufficient confinement of the concrete core have been documented in inadequately confined concrete members [2]. Retrofit measures have been developed to remedy these shortcomings [2]. One common retrofit measure consists of placing a jacket around a column to provide the lacking shear strength and or confinement. Steel jackets have been successfully implemented for this purpose [2]. The need to custom manufacture a steel shell for each column to be retrofitted and the time necessary to weld the jackets in place has led to the development of alternate advanced composite wraps such as fiber reinforced polymers (FRP) that are applied to the columns and avoid the need for custom manufactured jackets and can speed installation procedures [2]. These retrofit measures have led to the development of a new construction concept or system that replaces the steel in a conventionally reinforced concrete member with a premanufactured fiber reinforced shell as shown in Figure 1-. For this new system the shell takes over the tension carrying, shear and confining actions previously provided by the steel reinforcement and the concrete and shell combine for compression load transfer as shown in Figure 1-2. The concept being investigated in this project proposes using modular premanufactured fiber reinforced composite shells, set in place on site and then filled with concrete. This system offers the potential for substantial weight reduction as well as significantly reduced erection

36 3 times from those for current reinforced concrete structures as no removable forms or heavy lifting equipment are required. The premanufactured FRP shells are composed of relatively stiff fibers embedded into a softer matrix material. The fiber orientations in the composite are controlled to give the desired strength and stiffness in specified directions. The materials and technologies associated with these composites are not new to structural engineering. Aerospace structural engineers have long been taking advantage of the tailorable qualities of these light weight materials. Cost concerns and lack of design information consistent with civil engineering design practice have kept civil structural engineers from serious consideration of these advanced composites but new materials and manufacturing methods are under development that may change the cost equation enough for these composites to become practical alternatives to conventional structural systems. Furthermore, the constituent materials of the shell if chosen properly and used in a manufacturing process with quality control can offer good environmental resistance and longevity. The shells are filled with concrete that is used to carry compression loads and to stabilize the shell against buckling in compression as well as to aid in the joining of adjacent members. This system has the ability to take great advantage of the concrete core due to the confinement provided to the entire concrete core by the shell (no cover concrete) and the linear elastic nature of the shell which can control the dilation of the core much more than a ductile confining material such as steel.

37 4 This work represents one part of a program that was designed to prove the feasibility of this new concept and to establish some initial design guidelines for its implementation. The current document presents analytical models to characterize the structural behavior of concrete filled FRP shells. This characterization is mainly concerned with predicting the full stress state in the shell for all loading combinations including thermally induced stresses, establishing rational failure criteria on which design allowables can be based, investigating various material combinations and investigating stress concentrations. Experimental investigations along these lines are presented to verify the analytical modeling and to prove the viability of this concept. Other related documents from this program address the implementation of the advanced composite shell concept for new bridge structures [3] and joining concepts [4] [5]. A prototype structure under experimental evaluation is pictured in Figure 1-3. A review of the state of the art in areas pertinent to this analysis is presented in the following chapter.

38 5 Figure 1-1 Conventionally Reinforced Concrete Column Figure 1-2 Carbon Fiber Shell for Full Scale Testing

39 6 Figure 1-2 Concept for Concrete Filled Fiber Reinforced Shell Figure 1-3 Prototype Bridge Structure With Carbon Shell Girders and a Fiber Glass Deck System

40 2. RESEARCH REVIEW The use of advanced composite wraps for retrofit measures has led to considerable study in the area of concrete confined with linear elastic composite wraps. At the University of California, San Diego, design guidelines have been developed for advanced composite column retrofits [2]. The design guidelines are based on supplying the column with the lacking transverse reinforcement necessary to withstand seismic attack. These design equations were extended to a concrete filled filament wound carbon shell system by Seible, Burgueño, Abdallah and Nuismer [6]. The design models make no effort at predicting the actual radial strain in the shell throughout the loading. Compression tests on concrete filled advanced composite shells have shown that large strength and ductility enhancements are possible with these systems. Hoppel, Bogetti, Gillespie, Howie and Karbhari [7] investigated these effects and proposed a Hooke s law relation between the hoop strain in the shell, the confining pressure and the axial stress in the concrete. Mirmiran and Shahaway [8] proposed an incremental approach utilizing a cubic relation describing the change in radial strain as a function of the axial strain. The coefficients of this cubic relation were determined based on the unstressed and ultimate state (failure of the shell). This model incorporated a variable Poisson s ratio for the enclosed concrete based on a model proposed by Elwi and Murray [9] which has been used for finite element modeling of concrete [1]. The variable Poisson s ratio was derived from compression tests on unconfined concrete cylinders. The confining pressure in this approach is 7

41 8 calculated from the jacket hoop modulus, thickness, diameter and the radial expansion of the core. Once the confining pressure is known, a constant pressure confinement model is used to predict the concrete axial stress. This type of model works well for shells with predominantly hoop fibers, since this architecture leads to shells with low axial stiffness and a low Poisson s ratio for loading in the axial direction. A similar concept has been presented by Picher, Rochete and Lassiere [11] in the form of an effective confinement stiffness based on the jacket hoop modulus, thickness and diameter. These models do not work well for composite architectures including fibers oriented away from the hoop direction. This was demonstrated in the extreme case by tests performed by Howie and Karbhari [12]. In these tests various architectures were investigated including several with all ±45 o fibers. This lay-up leads to a very high Poisson s ratio for loading in the axial direction which was evident in the test results as no increase in the strength or ductility of these specimens was observed when compared to the unconfined concrete control specimens. The shell in this configuration simply expands faster than the confined concrete core and offers no resistance to the cracking in the concrete. Bending of members using a composite shell for transverse reinforcement and steel longitudinal reinforcement as in a conventional column was studied by Mirmiran, Kargahi, Samaan and Shahaway [13]. To be able to quantify the physical state of the composite shell under all loading conditions it is necessary to understand the deformation of the concrete core when put under load. In the case of an unconfined concrete cylinder it has been shown

42 9 that the volumetric strain (ε v ), which is an invariant quantity defined as the sum of three orthogonal normal strains (see Figure 2-1), reaches an absolute minimum (maximum volume contraction) and then reverses until the net volume strain goes through zero at an axial strain of approximately 8-1% of the strain reached at maximum stress for an unconfined cylinder (ε co ). Beyond this level volume expansion seems to increase unrestrained (Pantazopoulou [14]). Work by Mirmiran and Shahaway [15] has shown that steel transverse reinforcement does delay this volume expansion but does not prevent it because at higher radial strain levels the steel reinforcement has yielded and no increase in confining pressure occurs with increasing dilation. In the case of a linear elastic shell the increase in pressure with expansion throughout the loading can prevent the volume expansion from occurring as seen in Figure 2-2. Based on compression tests done at the University of Central Florida Mirmiran shows that the dilation rate µ, defined as the incremental change in radial strain divided by the incremental change in longitudinal strain (tangent Poisson s ratio), increases from the initiation of loading to a maximum value µ max at an axial strain close to the ultimate axial strain of the unconfined concrete and then decreases until it stabilizes at an ultimate value µ u. The maximum and ultimate values are correlated empirically to the jacket hoop stiffness and the concrete strength. In these tests the composite shell thickness as well as the concrete strength was varied but the shell architecture was held constant with a wrap angle of ±75 o from the longitudinal direction.

43 1 To predict the behavior of noncircular sections under compression loading it becomes necessary to use approximate methods such as finite element analysis to account for the variation in the confinement of the concrete core across the section. Finite element modeling of concrete continues to be a much studied topic as many difficulties arise from the nonhomogeneous anisotropic behavior once cracking begins. Many models have been proposed through the years [16]. One of the earlier models used was based on the Drucker-Prager soil mechanics yield surface which expands the yield capacity of the material based on the current hydrostatic pressure [17]. Some success has been demonstrated with this model for square cylinders with rounded corners confined by linear elastic shells [18]. At the time of this writing no published material has been found on the bending behavior of concrete filled FRP shells, however much work has been published on concrete filled steel tubes (CFT). The use of the CFT system for building and bridge columns has been extensively investigated since the 195 s. Many of the design approaches put forth are conservative and ignore the contribution from the concrete fill. Furlong proposed a model in 1968 [19] which considered the concrete and steel separately and added the components to get the system behavior for axial and flexural loading. Tomii, Sakino, Watanabe and Xiao [2][21] investigated short columns utilizing steel shells locally for added shear reinforcement and confinement. Tomii and Sakino proposed a modification to a standard hoop reinforced concrete confinement model proposed by Park for the encased concrete. The steel shell was transformed into an equal number of spiral hoops. The modification was

44 11 used to slightly lower the predicted ultimate stress in the concrete to match experimental data. Lu and Kennedy [22] performed compression tests on CFTs and found that the addition of the contributions from the steel and unconfined concrete individually gives a good estimate of the behavior of the composite section. This makes sense as the shell will initially expand faster than the concrete core due to its higher Poisson s ratio leading to very little confinement of the concrete core. This document extends the previous works by including the effects of the Poisson s ratio of the shell, establishing relations for the nonlinear concrete response in compression and predicting the complete biaxial stress state in the shell under flexure including the shear strains. The constituent materials used in this system along with pertinent relations for their analysis are described in Chapter 3.

45 12 P Undeformed Cylinder Area Strain ε A = ε 2 + ε 3 Volume Strain ε = ε + v ε + ε Deformed Cylinder Figure 2-1 Area and Volume Strain Definition Volumetric Strain, ε v.2 Elastic (1-2ν)ε 1 Unconfined Concrete.1 Linear Elastic Confinement Mild Steel Confinement Longitudinal Strain, ε 1 Figure 2-2 Expansion Behavior of Concrete

46 3. MATERIAL CHARACTERIZATION 3.1 Advanced Composite Shells Advanced composite materials have been used extensively in the aerospace and defense sectors for over thirty-five years. The materials discussed in this document consist of stiff fibers embedded in a relatively soft matrix material. These materials are generally broken down into short fiber composites and continuous fiber composites. Only the latter will be considered as the former usually do not have sufficient stiffness for primary structural applications. This study is limited to composites that consist of structures assembled from individual plies or lamina. An individual lamina is composed of unidirectional fibers in a continuous matrix. These laminae are stacked with varying fiber orientations to obtain the desired properties of the assembled laminate. A brief description of the fibers and matrices used in this study is presented in the following sections Fiber Reinforcement There are several fiber reinforcement materials being investigated for use in civil applications. The most promising at this time are E-glass, popular due to its low cost and availability, and carbon useful for its excellent stiffness. Other popular fiber reinforcements include aramids (Kevlar) which have high strength and stiffness but experience problems with moisture absorption and polyester fibers that do not generally possess sufficiently high stiffness for structural applications. Although other fibers are available only carbon and E-glass will be considered here. 13

47 14 Glass fibers have been in use for engineering applications since the early 194s. Several different types of glass fiber are commercially available. The most common of these is designated E type. E-glass has low alkali content that attempts to ensure corrosion resistance and high electrical resistivity. The major drawback to E- glass for use in civil applications is that it has been reported to show poor chemical resistance in both acidic and alkaline solutions which makes it a poor choice where it is in contact with cement. A stiffer and stronger variant of E-glass was developed and given the designation S-glass. It however still shares the problems of E-glass. An alkali resistant fiber was developed and designated AR-glass or Z-glass but has had limited success in practice. Table 3-1 gives some pertinent mechanical properties of these fibers. Table 3-1 Typical Properties of Commercial Glass Fiber Reinforcements [23] Type of Fiber Specific Gravity Coef. Therm. Exp. x1-6 o C -1 Young s Modulus GPa (msi) E (1.5-11) AR (Z) ( ) S (12.5) Tensile Strength a GPa (ksi) 3.6 (52) 3.6 (52) 4.6 (667) a Virgin Strength values. Actual strength values prior to incorporation into composite are ~2.1 GPa (35 ksi). b Value varies widely with changes in the manufacturing process. Poisson s Ratio b Carbon fibers are attractive for civil structural applications due to their high stiffness and low density along with their resistance to chemical attack. A wide range of mechanical properties are available based on the precursor used to manufacture the

48 15 fibers and the manufacturing process itself. Table 3-2 shows some typical properties for commercially available carbon fibers, Type I produced for high stiffness and Type II for high strength. Note that the coefficient of thermal expansion is different in the longitudinal and transverse directions due to the nonisotropic nature of the carbon fibers. Table 3-2 Mechanical Properties For Select Carbon Fibers [24] Type of Fiber Specific Gravity Coef. Therm. Exp. x1-6 o C -1 Longitudinal Transverse TYPE I to TYPE II to Young s Modulus GPa (msi) 39 (56.6) 25 (36.3) Tensile Strength GPa (ksi) 2.2 (319) 2.7 (392) Matrix Materials The matrix material is used to bind the fibers and enable them to be combined to create a composite. One of the most prominent composite materials currently in use is the family known as polymeric composites or reinforced polymers. The matrix materials used in these composites fall into two main classes, thermosetting and thermoplastic resins. Thermoplastic resins are mainly used in short fiber applications and preimpregnated composite plies and will not be discussed further here. Thermosetting resins are predominant in continuous fiber applications and will be briefly described below.

49 16 Common thermosetting resins used for composite material applications are epoxy and polyester resins. The final product is produced by converting the liquid resin into a solid through chemical cross-linking which leads to a tightly bound threedimensional network of polymer chains. Curing can be achieved at room or ambient temperature but elevated temperature cure cycles can also be used. Thermosetting resins are usually isotropic and do not melt on reheating. Table 3-3 presents some typical properties for epoxy and polyester resins. Table 3-3 Mechanical Properties For Common Thermosetting Resins [24] [25] Type of Fiber Specific Gravity Coef. Therm. Exp. x1-6 o C -1 Young s Modulus GPa (msi) Epoxy ( ) Polyester Vinylester ( ) Tensile Strength MPa (ksi) 35-1 ( ) 4-9 ( ) Poisson s Ratio Epoxies are used extensively in structural applications due to the broad range of mechanical properties that can be achieved. Elevated temperature curing cycles are commonly used but are not required. Polyesters are the most widely used thermoset resin system accounting for about 75% of the total resin used. They are used extensively in marine applications and cure relatively quick at ambient temperature through the addition of a catalyst. Vinylester are often considered to be part of the polyester family. They were developed to combine the advantages of epoxies with the faster cure of polyesters.

50 Manufacturing Processes Many manufacturing methods have been developed to combine the fibers and matrices into a finished composite part [26]. In the aerospace industry preimpregnated plies (fibers preimpregnated with the uncured resin and then stored at low temperatures to retard curing) are combined and then put under elevated temperatures and pressures to achieve high fiber volume fractions (percentage of total volume of the composite occupied by the fibers). This method is expensive and does not hold great promise for civil applications. Methods more suited to civil applications combine the resin and fibers as the part is being manufactured. Those commonly used with continuous fibers include: (1) hand-lay-up where the dry fibers are placed at the desired orientation and the resin is applied by hand and squeegeed into the fibers to get complete coverage or wetting of the fibers, (2) filament winding where the dry fibers are taken through a resin bath and wound onto a mandrel with the desired orientation and (3) pultrusion which is similar to an extrusion process for metals with the exception of having the part pulled through a die rather than pushed. These processes all have limitations on the fiber volume fraction that can be achieved as well as the fiber orientations possible. The shells used for the tests in this program were manufactured with the filament winding process which limited the allowable angle from the longitudinal axis for the fibers to greater than 1 o [27] Typical Ply Properties Table 3-4 lists typical properties of unidirectional-fiber-reinforced epoxy resins. These properties are strongly influenced by the fiber volume fraction. The E-

51 18 Glass properties listed in Table 3-4 represent a composite with a fairly low fiber volume fraction most representative of a hand lay-up process whereas the carbon properties are more representative of an aerospace quality preimpregnated ply Classical Lamination Theory Classical lamination theory is the name given to the analytical methods used to predict the behavior of a laminated composite material. The analysis assumes a continuous displacement field through the thickness of the laminate which implies a perfect bond between adjacent plies (no slip). A thin plate assumption is also used which ignores the shear deformation of the composite. Thus a line initially perpendicular to the mid-plane of the plate remains straight and perpendicular to the mid-plane after deformation. It is also assumed that the through-the-thickness strains are negligible. A description of the pertinent relations used in this work are included here. Any text on mechanics of composite materials [28] [29] will give a full derivation of the following relations. The plate is assumed to lie in the x-y plane. If a point on the mid-plane of the undeformed section is displaced by u o, v o and w o in the x, y and z directions respectively the deformation of any point is given by, u u z w o = o x v v z w o = o y w= w o. (3-1)

52 19 Table 3-4 Typical Ply Properties for Fiber-Reinforced Epoxy Resins [28] Property E-Glass Carbon Fiber Volume Fraction Specific Gravity Tensile Strength, o MPa (ksi) 114 (16) 1725 (25) Tensile Modulus, o GPa (msi) 39 (5.66) 159 (23.1) Tensile Strength, 9 o MPa (ksi) 36 (5.22) 42 (6.9) Tensile Modulus, 9 o GPa (msi) 1 (1.45) 1.9 (1.58) Compression Strength, o MPa (ksi) 6 (87.) 1366 (198) Compression Modulus, o GPa (msi) 32 (4.64) 138 (2.) Compression Strength, 9 o MPa (ksi) 138 (2.) 23 (33.4) Compression Modulus, 9 o GPa (msi) 8 (1.16) 11 (1.6) In-Plane Shear Strength MPa (ksi) 95 (13.8) In-Plane Shear Modulus GPa (msi) 6.4 (.93) Longitudinal Poisson s Ratio (ν LT ) Longitudinal Coef. of Thermal Expansion (1-6 / o C) Transverse Coef. of Thermal Expansion (1-6 / o C)

53 2 The strains can be derived from the assumed displacement field as ε ε γ x o o y o o xy o o o u x z w x v y z w y u y v x z w xy = = = (3-2) Defining mid-plane strains as { } ε ε ε γ o x o y o xy o o o o o u x v y u y v x = = +, (3-3) and mid-plane curvatures as { } κ κ κ κ = = x y xy o o o w x w y w xy , (3-4) we can then write the general strains as ε ε γ ε ε γ κ κ κ x y xy x o y o xy o x y xy z = +. (3-5) Each individual lamina being composed of unidirectional fibers embedded in a matrix can be modeled as a transversely isotropic material, with the plane of isotropy

54 21 being normal to the fiber direction. If we define a local coordinate system (1,2,3) for the lamina with the 1 direction parallel to the fiber direction and the 2 and 3 directions normal to the fiber direction we can then write the in-plane stress strain relations for a given lamina as { σ } [ ]{ ε } with the stress vector given by = Q, (3-6) 1 1 { σ } 1 σ 1 σ 2 σ 3 = τ 23 τ 13 τ 12 (3-7) and the strain vector given by { ε } 1 ε1 ε 2 ε 3 =. (3-8) γ 23 γ 13 γ 12 The nonzero terms of the 3x3 stiffness matrix for the in-plane behavior are given by