UNIVERSITY OF CALIFORNIA, SAN DIEGOadavol/dissertation/dis_wp.pdf · university of california, san diego ... concrete cylinders under uniaxial compression ... figure 5-5 typical compression

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

Copyright

Andrew Davol, 1998

All rights reserved.

iii

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

iv

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.

v

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..................................................................................................1

2. RESEARCH REVIEW..........................................................................................7

3. MATERIAL CHARACTERIZATION..............................................................13 3.1 ADVANCED COMPOSITE SHELLS.......................................................................13

3.1.1 Fiber Reinforcement .................................................................................13 3.1.2 Matrix Materials .......................................................................................15 3.1.3 Manufacturing Processes ..........................................................................17 3.1.4 Typical Ply Properties...............................................................................17 3.1.5 Classical Lamination Theory ....................................................................18 3.1.6 Equivalent Plate Properties .......................................................................27 3.1.7 First Ply Failure Criteria ...........................................................................28 3.1.8 Thermal Expansion ...................................................................................28

3.2 CONCRETE........................................................................................................31 3.2.1 Uniaxial Compression...............................................................................32 3.2.2 Biaxial and Triaxial Stress States - Confinement Effects.........................32 3.2.3 Tension......................................................................................................35

4. ANALYTICAL MODELING OF CONCRETE FILLED FRP SHELLS......38 4.1 CIRCULAR SHELLS............................................................................................38

4.1.1 Compression .............................................................................................38 4.1.2 Tension......................................................................................................50 4.1.3 Shear .........................................................................................................50 4.1.4 Bending.....................................................................................................54

4.2 CONREC SHELLS...............................................................................................56 4.2.1 Compression .............................................................................................57 4.2.2 Bending.....................................................................................................68

vi

5. EXPERIMENTAL PROGRAM TO VALIDATE CONCRETE FILLED FRP TUBE BEHAVIOR.....................................................................................................71 5.1 SMALL SCALE SHELLS......................................................................................71

5.1.1 Concrete Characterization.........................................................................74 5.1.2 Compression .............................................................................................76 5.1.3 Bending.....................................................................................................89

5.2 FULL SCALE BENDING TESTS .........................................................................110 5.2.1 Concrete Properties.................................................................................118 5.2.2 Hollow Shell ...........................................................................................119 5.2.3 Concrete Filled Shells.............................................................................123 5.2.4 Concrete Filled Shell with integral Concrete Deck ................................137

6. CORRELATION OF ANALYTICAL MODELS TO EXPERIMENTAL DATA .........................................................................................................................145 6.1 CIRCULAR SHELLS..........................................................................................145

6.1.1 Small Scale Shells...................................................................................145 6.1.2 Full Scale Specimens ..............................................................................153

6.2 CONREC SHELLS.............................................................................................159 6.2.1 Compression ...........................................................................................159 6.2.2 Bending...................................................................................................162

7. PARAMETER STUDIES OF MATERIAL, LAY-UP, THICKNESS AND SHAPE VARIATIONS.............................................................................................168 7.1 CIRCULAR SHELLS..........................................................................................169

7.1.1 Compression Behavior............................................................................169 7.1.2 Bending Behavior ...................................................................................172

7.2 CONREC SHELLS.............................................................................................184 7.3 HYBRID SHELLS..............................................................................................187

7.3.1 Compression ...........................................................................................188 7.3.2 Bending...................................................................................................188

8. STRESS CONCENTRATIONS, TENSION STIFFENING AND THERMAL EXPANSION EFFECTS..........................................................................................189 8.1 STRESS CONCENTRATIONS....................................................................189

8.1.1 Closed Form Solution .............................................................................189 8.1.2 Parameter Study......................................................................................192 8.1.3 Case Study ..............................................................................................198

8.2 EFFECTS OF TENSION STIFFENING ......................................................201 8.3 THERMAL EFFECTS ..................................................................................204

8.3.1 Thermal Strains in Circular Sections ......................................................204 8.3.2 Thermal Testing......................................................................................205 8.3.3 Parameter Studies for Thermally Induced Strains ..................................207

8.4 LOCAL COMPRESSION BUCKLING OF CONCRETE FILLED FRP SHELLS...........209

9. CONCLUSIONS ................................................................................................211

vii

APPENDIX - MOMENT CURVATUR PROGRAM............................................220

REFERENCES..........................................................................................................236

viii

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

ix

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 4-10 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, 0% ±10O PLIES, FLAT TO

RADIUS RATIO .5 ..............................................................................................62

FIGURE 4-15 AREA STRAIN RATIO PROFILE, 0% ±10O FIBERS, FLAT TO

RADIUS RATIO OF 1.........................................................................................62


RADIUS RATIO OF 2.........................................................................................63


RADIUS RATIO OF 3.........................................................................................63

FIGURE 4-18 AREA STRAIN RATIO FOR 0% ±10O CONREC SHELL..........64

FIGURE 4-19 AREA STRAIN RATIO FOR 50% ±10O CONREC SHELL........64

FIGURE 4-20 AREA STRAIN RATIO FOR 80% ±10O CONREC SHELL........65

x

FIGURE 4-21 HOOP STRESS IN SHELLS WITH FLAT TO RADIUS RATIO

OF .5 ......................................................................................................................66


OF 1 .......................................................................................................................66


OF 2 .......................................................................................................................67


OF 3 .......................................................................................................................67

FIGURE 4-25 MOMENT CURVATURE OF TYPICAL CONREC

SECTION..............................................................................................................69

FIGURE 4-26 CONCRETE STRESS STRAIN RELATION.................................70

FIGURE 4-27 COMPARATIVE MOMENT CURVATURE FOR CONREC

SECTION WITH VARIOUS CONCRETE MODELS ....................................70

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

xi

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 5-10 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........90

FIGURE 5-19 FOUR POINT BENDING TEST ON SMALL SCALE

SPECIMEN...........................................................................................................90

FIGURE 5-20 STRAIN GAGE LAYOUT FOR CIRCULAR BENDING

SPECIMENS ........................................................................................................91

xii

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 5-30 LONGITUDINAL STRAIN VS. MOMENT FOR THICK

CIRCULAR SECTION .......................................................................................98

FIGURE 5-31 HOOP STRAIN VS. MOMENT FOR THICK CIRCULAR

SECTION..............................................................................................................99

xiii

FIGURE 5-32 SHEAR STRAIN VS. APPLIED SHEAR FOR THICK

CIRCULAR SPECIMEN ....................................................................................99

FIGURE 5-33 FAILURE OF THICK CIRCULAR SECTION ...........................100

FIGURE 5-34 STRAIN GAGE LAYOUT FOR CONREC BENDING

SPECIMENS ......................................................................................................101

FIGURE 5-35 LOAD - DISPLACEMENT CURVE FOR THIN CONREC

BENDING SPECIMEN .....................................................................................102

FIGURE 5-36 STRAIN PROFILE FOR THIN CONREC SECTION IN

CONSTANT MOMENT REGION ..................................................................102

FIGURE 5-37 STRAIN PROFILE FOR THIN CONREC SECTION IN SHEAR

AREA ..................................................................................................................103

FIGURE 5-38 LONGITUDINAL STRAIN VS. MOMENT FOR THIN CONREC

SPECIMEN.........................................................................................................103

FIGURE 5-39 HOOP STRAIN VS. MOMENT FOR THIN CONREC

SPECIMEN.........................................................................................................104

FIGURE 5-40 SHEAR STRAIN VS. APPLIED SHEAR FOR THIN CONREC

SPECIMEN.........................................................................................................104

FIGURE 5-41 FAILURE OF THIN CONREC SECTION ..................................105

FIGURE 5-42 LOAD-DISPLACEMENT RESPONSE FOR THICK CONREC

SPECIMEN.........................................................................................................106

FIGURE 5-43 STRAIN PROFILE FOR THICK CONREC SECTION IN

CONSTANT MOMENT REGION ..................................................................107

xiv

FIGURE 5-44 STRAIN PROFILE FOR THICK CONREC SECTION IN

SHEAR AREA....................................................................................................107

FIGURE 5-45 LONGITUDINAL STRAIN VS. MOMENT FOR THICK

CONREC ............................................................................................................108

FIGURE 5-46 HOOP STRAIN VS. MOMENT FOR THICK CONREC

SECTION............................................................................................................108

FIGURE 5-47 SHEAR STRAIN VS. APPLIED SHEAR FOR THICK CONREC

SPECIMEN.........................................................................................................109

FIGURE 5-48 FAILURE OF THICK CONREC SECTION ...............................109

FIGURE 5-49 FULL SCALE FOUR POINT BENDING TEST..........................110

FIGURE 5-50 SCHEMATIC OF FULL SCALE BENDING TESTS .................111

FIGURE 5-51 ENDBLOCK FOR SUPPORT OF BENDING TEST

SPECIMENS ......................................................................................................112

FIGURE 5-52 SHELL END DIAPHRAGM ..........................................................113

FIGURE 5-53 STEEL CONNECTION CAGE......................................................114

FIGURE 5-54 SHELL WITH CONNECTING CAGE .........................................114

FIGURE 5-55 ENDBLOCK LOWER SECTION .................................................115

FIGURE 5-56 PLACEMENT OF SHELL INTO ENDBLOCKS........................115

FIGURE 5-57 PVC PIPE FOR PUMPING SHELL .............................................116

FIGURE 5-58 COMPLETED END BLOCK FORM............................................116

FIGURE 5-59 INSTRUMENTATION LAYOUT FOR HOLLOW SHELL

TEST....................................................................................................................121

xv

FIGURE 5-60 LOAD DISPLACEMENT CURVE FOR HOLLOW SHELL

TESTS .................................................................................................................121

FIGURE 5-61 LONGITUDINAL AND HOOP STRAINS FOR HOLLOW

SHELLS ..............................................................................................................122

FIGURE 5-62 SHEAR STRAINS FOR HOLLOW SHELL TESTS ..................122

FIGURE 5-63 DISPLACEMENT INSTRUMENTATION FOR FILLED SHELL

TEST....................................................................................................................123

FIGURE 5-64 STRAIN GAGE LOCATIONS AND DESIGNATION FOR

FILLED TUBE TESTS......................................................................................124

FIGURE 5-65 LOAD DISPLACEMENT PLOT FOR FILLED SHELL

TEST #1...............................................................................................................126

FIGURE 5-66 STRAIN PROFILE IN CONSTANT MOMENT SECTION,

FILLED SHELL TEST #1 ................................................................................127

FIGURE 5-67 STRAIN PROFILE IN SHEAR SECTION, FILLED SHELL

TEST #1...............................................................................................................127

FIGURE 5-68 LONGITUDINAL STRAIN VS. MOMENT IN COMP. ZONE

FOR FILLED SHELL #1 ..................................................................................128

FIGURE 5-69 LONGITUDINAL STRAIN VS. MOMENT IN TENSION ZONE

FOR FILLED SHELL #1 ..................................................................................128

FIGURE 5-70 HOOP STRAIN VS. MOMENT IN COMPRESSION ZONE FOR

FILLED SHELL #1............................................................................................129

xvi

FIGURE 5-71 HOOP STRAIN VS. MOMENT IN TENSION ZONE FOR

FILLED SHELL #1............................................................................................129

FIGURE 5-72 SHEAR STRAIN VS. APPLIED SHEAR FOR FILLED

SHELL #1 ...........................................................................................................130

FIGURE 5-73 FAILURE OF FILLED SHELL #1................................................130

FIGURE 5-74 LOAD DISPLACEMENT PLOT FOR FILLED SHELL

TEST #2...............................................................................................................132

FIGURE 5-75 STRAIN PROFILE IN CONSTANT MOMENT SECTION,

FILLED SHELL TEST #2 ................................................................................133

FIGURE 5-76 STRAIN PROFILE IN SHEAR SECTION, FILLED SHELL

TEST #2...............................................................................................................133

FIGURE 5-77 LONGITUDINAL STRAIN VS. MOMENT IN COMP. ZONE

FOR FILLED SHELL #2 ..................................................................................134

FIGURE 5-78 LONGITUDINAL STRAIN VS. MOMENT IN TENSION ZONE

FOR FILLED SHELL #2 ..................................................................................134

FIGURE 5-79 HOOP STRAIN VS. MOMENT IN COMPRESSION ZONE FOR

FILLED SHELL #2............................................................................................135

FIGURE 5-80 HOOP STRAIN VS. MOMENT IN TENSION ZONE FOR

FILLED SHELL #2............................................................................................135

FIGURE 5-81 SHEAR STRAIN VS. APPLIED SHEAR FOR FILLED

SHELL #2 ...........................................................................................................136

FIGURE 5-82 FAILURE OF FILLED SHELL #2................................................136

xvii

FIGURE 5-83 DISPLACEMENT INSTRUMENTATION FOR FILLED SHELL

WITH SLAB TEST............................................................................................138

FIGURE 5-84 STRAIN GAGE LOCATIONS AND DESIGNATION FOR

SHELL ................................................................................................................138

FIGURE 5-85 STRAIN GAGES PLACED ON SHEAR CONNECTION

DOWELS ............................................................................................................139

FIGURE 5-86 INSTRUMENTATION LOCATIONS AND DESIGNATION FOR

TOP MAT OF STEEL REINFORCEMENT IN DECK................................139

FIGURE 5-87 STRESS CONCENTRATION STRAIN GAGE LOCATIONS

AND DESIGNATION........................................................................................140

FIGURE 5-88 LOAD DISPLACEMENT ENVELOPE FOR FILLED SHELL

WITH SLAB .......................................................................................................142

FIGURE 5-89 STRAIN PROFILE ACROSS SECTION FOR FILLED SHELL

WITH SLAB .......................................................................................................142

FIGURE 5-90 STRESS CONCENTRATION AROUND PENETRATION IN

SHEAR SPAN.....................................................................................................143

FIGURE 5-91 STRESS CONCENTRATION IN THE CONST. MOMENT

REGION..............................................................................................................143

FIGURE 6-1 CONCRETE STRESS VS. STRAIN FOR SMALL SCALE

COMPRESSION SPECIMENS........................................................................146

FIGURE 6-2 RADIAL VS. LONGITUDINAL STRAIN FOR SMALL SCALE

COMP. SPECIMENS ........................................................................................147

xviii

FIGURE 6-3 LOAD VS. DISPLACEMENT FOR SMALL SCALE CIRCULAR

SECTIONS..........................................................................................................150

FIGURE 6-4 STRAINS IN CONSTANT MOMENT REGION FOR THIN

CIRCULAR SHELL ..........................................................................................150

FIGURE 6-5 STRAINS IN CONSTANT MOMENT REGION FOR THICK

CIRCULAR SHELL ..........................................................................................151

FIGURE 6-6 STRAINS IN SHEAR AREA FOR THIN CIRCULAR

SHELL ................................................................................................................151

FIGURE 6-7 STRAINS IN SHEAR SPAN FOR THICK CIRCULAR

SHELL ................................................................................................................152

FIGURE 6-8 SHEAR STRAIN IN THIN CIRCULAR SHELL ..........................152

FIGURE 6-9 SHEAR STRAIN IN THICK CIRCULAR SHELL .......................153

FIGURE 6-10 LOAD DISPLACEMENT CURVES FOR FULL SCALE FILLED

SHELL TESTS ...................................................................................................155

FIGURE 6-11 EXTREME FIBER STRAINS IN CONSTANT MOMENT

REGION FOR SHELL #1.................................................................................156

FIGURE 6-12 EXTREME FIBER STRAINS IN SHEAR SPAN FOR

SHELL #1 ...........................................................................................................156

FIGURE 6-13 EXTREME FIBER STRAINS IN CONSTANT MOMENT

REGION FOR SHELL #2.................................................................................157

FIGURE 6-14 EXTREME FIBER STRAINS IN SHEAR SPAN FOR

SHELL #2 ...........................................................................................................157

xix

FIGURE 6-15 SHELL CENTERLINE SHEAR STRAINS FOR SHELL #1 .....158

FIGURE 6-16 SHELL CENTERLINE SHEAR STRAINS FOR SHELL #2 .....158

FIGURE 6-17 LOAD VS. LONGITUDINAL STRAIN FOR THICK CONREC

CYLINDERS ......................................................................................................161

FIGURE 6-18 HOOP STRAINS VS. LONGITUDINAL STRAIN IN THICK

CONREC SECTION .........................................................................................161

FIGURE 6-19 LOAD DISPLACEMENT FOR CONREC BENDING

SPECIMENS ......................................................................................................164

FIGURE 6-20 STRAINS IN CONSTANT MOMENT REGION FOR THIN

CONREC SHELL ..............................................................................................164

FIGURE 6-21 STRAINS IN CONSTANT MOMENT REGION FOR THICK

CONREC SHELL ..............................................................................................165

FIGURE 6-22 STRAINS IN SHEAR AREA FOR THIN CONREC SHELL ....165

FIGURE 6-23 STRAINS IN SHEAR AREA FOR THICK CONREC

SHELL ................................................................................................................166

FIGURE 6-24 SHEAR STRAIN IN THIN CONREC SHELL ............................166

FIGURE 6-25 SHEAR STRAIN IN THICK CONREC SHELL .........................167

FIGURE 7-1 COMPRESSION BEHAVIOR OF CARBON EPOXY

SHELLS ..............................................................................................................171

FIGURE 7-2 COMPRESSION BEHAVIOR OF E-GLASS SHELLS................171

FIGURE 7-3 CONFINEMENT EFFICIENCY OF E-GLASS VS. CARBON

SHELLS ..............................................................................................................172

xx

FIGURE 7-4 GEOMETRY FOR MOMENT CALCULATION .........................173

FIGURE 7-5 NORMALIZED MOMENT CURVATURE, R/T=10, CARBON

EPOXY SHELL .................................................................................................175


EPOXY SHELL .................................................................................................175


EPOXY SHELL .................................................................................................176


EPOXY SHELL .................................................................................................176

FIGURE 7-9 NORMALIZED MOMENT CURVATURE, R/T=10, E-GLASS

SHELL ................................................................................................................177


SHELL ................................................................................................................177


SHELL ................................................................................................................178


SHELL ................................................................................................................178

FIGURE 7-13 MOMENT CURVATURE WITH AXIAL LOAD, R/T=10, 10%

HELICAL FIBERS ............................................................................................180


HELICAL FIBERS ............................................................................................180

xxi


HELICAL FIBERS ............................................................................................181


HELICAL FIBERS ............................................................................................181


HELICAL FIBERS ............................................................................................182


HELICAL FIBERS ............................................................................................182

FIGURE 7-19 FLEXURAL STIFFNESS OF E-GLASS VS. CARBON

SHELLS ..............................................................................................................183

FIGURE 7-20 MOMENT CURVATURE RESPONSE FOR CONREC

SECTIONS, D/T=20...........................................................................................186

FIGURE 7-21 MOMENT CURVATURE RESPONSE FOR CONREC

SECTIONS, D/T=50...........................................................................................186

FIGURE 7-22 NORMALIZED MOMENT CURVATURE RESPONSE FOR

CONREC SECTIONS, D/T=20 ........................................................................187

FIGURE 8-1 INFINITE PLATE WITH A CIRCULAR INCLUSION...............191

FIGURE 8-2 LOAD CASES FOR STRESS CONCENTRATION STUDY .......194

FIGURE 8-3 STRESS CONCENTRATION FACTORS......................................197

FIGURE 8-4 TANGENTIAL STRESS CONCENTRATION VARIATION

AROUND HOLE FOR 80% HELICAL SHELL ...........................................197

FIGURE 8-5 SHEAR CONNECTION ...................................................................198

xxii

FIGURE 8-6 FINITE ELEMENT MODEL FOR STRESS CONCENTRATION

STUDIES.............................................................................................................201

FIGURE 8-7 AVERAGE STRESS VS. AVERAGE STRAIN FOR TENSION

STIFFENING .....................................................................................................203

FIGURE 8-8 LOAD IN CONCRETE FILLED CARBON SHELL IN PURE

TENSION WITH AND WITHOUT TENSION STIFFENING

EFFECTS............................................................................................................203

FIGURE 8-9 THERMALLY INDUCED MECHANICAL STRAINS PER

DEGREE CENTIGRADE.................................................................................206

FIGURE 8-10 COEFFICIENTS OF THERMAL EXPANSION FOR FIBER

REINFORCED SHELLS ..................................................................................208

FIGURE 8-11 HOOP STRESS IN SHELL 90O PLIES DUE TO A

TEMPERATURE RISE OF 55 OC...................................................................208

FIGURE 8-12 ULTIMATE BUCKLING STRAIN FOR ALL BENDING

SPECIMENS ......................................................................................................210

xxiii

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 .................................................................................................................117

TABLE 5-8 VENDOR SUPPLIED PLY PROPERTIES ......................................117

TABLE 5-9 EQUIVALENT PLATE PROPERTIES FOR LARGE SCALE

TESTS .................................................................................................................118

xxiv

TABLE 5-10 CONCRETE MIX USED FOR FILLED SHELLS ........................119

TABLE 5-11 CONCRETE PROPERTIES FOR FILLED SHELL TESTS .......119

TABLE 7-1 PLY PROPERTIES FOR PARAMETER STUDIES.......................169

TABLE 7-2 GEOMETRY OF CONREC SECTIONS FOR NORMALIZED

COMPARISON ..................................................................................................185

TABLE 8-1 COMPOSITE ARCHITECTURES FOR STRESS

CONCENTRATION STUDY ...........................................................................195

TABLE 8-2 STRESS CONCENTRATION FACTORS........................................196

TABLE 8-3 STRESS AROUND PENETRATION FOR BEAM AND SLAB

SHEAR CONNECTION ...................................................................................200

TABLE 8-4 PLY PROPERTIES FOR THERMAL TESTING ...........................206

xxv

LIST OF SYMBOLS

Scalars

Ac � cross sectional area of concrete core

E1 � ply modulus in fiber direction

E2 � ply modulus transverse to fiber direction

Ec � concrete tangent modulus

Eca � average tangent stiffness of concrete core

Eco � initial concrete modulus

EH � equivalent modulus in hoop direction for shell

EL � equivalent modulus in longitudinal direction for shell

Ex � modulus in x direction

Ey � modulus in y direction

f - flat length for conrec section

f�c � compression strength of unconfined concrete

fc � concrete stress

ft � tension strength of concrete

G12 � ply in-plane shear modulus

Gxy � shear modulus in x-y plane

hn � distance from mid-plane to near surface of ply n

M � moment in section

Mx � moment applied to laminate about y axis

MxT � equivalent thermal moment about y axis

xxvi

Mxy � twisting moment applied to laminate

MxyT � equivalent thermal twisting moment

My � moment applied to laminate about x axis

MyT � equivalent thermal moment about x axis

Nx � force applied to laminate in x direction

NxT � equivalent thermal force on laminate in x direction

Nxy � in-plane shear force applied to laminate

NxyT � equivalent thermal in-plane shear force applied to laminate

Ny � force applied to laminate in y direction

NyT � equivalent thermal force applied to laminate in y direction

Pc � load in concrete

Ps � load in shell

qc � shear flow between shell and core

qs � shear flow in shell

r - radius for conrec section

R � shell mean radius

t � thickness of laminated shell

teff � effective thickness of concrete used for shear calculation

u � displacement in x direction

uo � mid-plane displacement in x direction

v � displacement in y direction

V � shear load on section

xxvii

Vc � shear capacity of concrete core

vo � mid-plane displacement in y direction

w � displacement in z direction

wo � mid-plane displacement in z direction

αx � coefficient of thermal expansion in structural x direction

αxy � apparent shear coefficient of thermal expansion in structural x-y plane

αy � coefficient of thermal expansion in structural y direction

α1 − ply coefficient of thermal expansion in fiber direction

α2 − ply coefficient of thermal expansion normal to fiber direction

β1 − bond characteristic factor for tension stiffening

β2 − loading characteristic factor for tension stiffening

ε1 � ply strain in fiber direction or longitudinal concrete strain

εa � area strain

εcf � average strain for tension stiffening

εv � volume strain

εx � strain in structural x direction

εxo � mid-plane strain in structural x direction

εxT � thermal strain in structural x direction

εy � strain in structural y direction

εyo � mid-plane strain in structural y direction

εyT � thermal strain in structural y direction

xxviii

ε1Τ − ply thermal strain in fiber direction

ε2 − ply strain normal to fiber direction

ε2Τ − ply thermal strains normal to fiber direction

ε3 − ply strain normal to fiber direction

γxy � shear strain in structural x-y plane

γxyT � thermal shear strain in structural x-y plane

γ12 − ply shear strain 1-2 plane



κx � curvature about y axis

κxy � twist

κy � curvature about x axis

λ − concrete density factor

ν12 − ply in-plane Poisson�s ratio for loading in the fiber direction

ν21 − 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

σx � stress in structural x direction

σy �stress in structural y direction

σz � stress in structural z direction

σ1 − ply stress in fiber direction or longitudinal concrete stress

xxix

σ2 − ply stress normal to fiber direction

σ3 − ply stress normal to fiber direction

τxy � shear stress in x-y plane

τ12 − ply shear stress 1-2 plane



εL � longitudinal strain in shell

εH � hoop strain in shell

νLH � Poisson�s ratio in shell for loading in the longitudinal direction

νHL � Poisson�s ratio in shell for loading in the hoop direction

νc � concrete equivalent tangent Poisson�s ratio

νco � initial concrete Poisson�s ratio

σL � longitudinal stress in shell

σH � hoop stress in shell

εr � radial strain in concrete core

σr � radial stress in concrete core

µ − dilation rate

T − temperature

σTx - thermal stress in x direction

σTy - thermal strain in y direction

τTxy

- thermal shear stress in x-y plane

xxx

σhyd - hydrostatic pressure

νmax - maximum equivalent tangent Poisson�s ratio for concrete

µmax - maximum dilation rate

µu - ultimate dilation rate

aij - coeficients of deformation for stress concentration analysis

Eθ - modulus tangent to cutout

p - far field stress for stress concentration analysis

fcr - cracking stress for concrete

εco - strain at maximum stress for unconfined concrete

εMx - mechanical strain in x direction

εMy - mechanical strain in y direction

γMxy - mechanical shear strain in x-y plane

εTr - radial strain due to temperature change

εTH - hoop strain due to temperature change

γoxy - mid-plane shear strain

Vectors

{εo} - mid-plane strains

{κ} - curvatures

{σ1} - stresses in material coordinate system

{ε1}- strains in material coordinate system

{σx} - stresses in structure coordinate system

xxxi

{εx} - 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

xxxii

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

xxxiii

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.

1

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

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

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.

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.

5

Figure 1-1 Conventionally Reinforced Concrete Column

Figure 1-2 Carbon Fiber Shell for Full Scale Testing

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

7

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 [10]. The variable Poisson�s ratio was derived from compression

tests on unconfined concrete cylinders. The confining pressure in this approach is

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 ±45o 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

9

that the volumetric strain (εv), 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 80-100% of the strain reached at

maximum stress for an unconfined cylinder (εco). 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 µmax at an axial

strain close to the ultimate axial strain of the unconfined concrete and then decreases

until it stabilizes at an ultimate value µu. 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 ±75o from the longitudinal

direction.

10

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 1950�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 [20][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

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.

12

P

1

2 3

Undeformed Cylinder

Deformed Cylinder

Area Strainε ε εA = +2 3

Volume Strainε ε ε εv = + +1 2 3

Figure 2-1 Area and Volume Strain Definition

Longitudinal Strain, ε1

-0.010-0.008-0.006-0.004-0.002

Vo

lum

etri

c S

trai

n, ε

v

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002 Elastic (1-2ν)ε1

Unconfined ConcreteLinear Elastic ConfinementMild Steel Confinement

Figure 2-2 Expansion Behavior of Concrete

13

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.

3.1.1 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.

14

Glass fibers have been in use for engineering applications since the early

1940s. 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.

x10-6 oC-1

Young�s Modulus

GPa (msi)

Tensile Strengtha

GPa (ksi)

Poisson�s Ratio

E 2.54 5 72.4-76 (10.5-11)

3.6 (520)

.21

AR (Z) 2.68 7.5 70-80 (10.2-11.6)

3.6 (520)

.22

S 2.48 2.9-5.0 86 (12.5)

4.6 (667)

b

aVirgin Strength values. Actual strength values prior to incorporation into composite are ~2.1 GPa (305 ksi). bValue varies widely with changes in the manufacturing process.

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

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.

x10-6 oC-1 Longitudinal Transverse

Young�s Modulus

GPa (msi)

Tensile Strength

GPa (ksi)

TYPE I 1.95 -0.5 to -1.2 7-12

390 (56.6)

2.2 (319)

TYPE II 1.75 -0.1 to -0.5 7-12

250 (36.3)

2.7 (392)

3.1.2 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.

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 three-

dimensional 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.

x10-6 oC-1

Young�s Modulus

GPa (msi)

Tensile Strength

MPa (ksi)

Poisson�s Ratio

Epoxy 1.1-1.4 60 3-6 (.44-.87)

35-100 (5.1-14.5)

.38-.4

Polyester Vinylester

1.2-1.5 100-200 2-4.5 (.29-.65)

40-90 (5.80-13.1)

.37-.39

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.

17

3.1.3 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 10o [27].

3.1.4 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-

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.

3.1.5 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 uo, vo and wo in the x, y and z directions

respectively the deformation of any point is given by,

u u z wx

v v z wy

w w

oo

oo

o

= −

= −

=

∂∂

∂∂

. (3-1)

19

Table 3-4 Typical Ply Properties for Fiber-Reinforced Epoxy Resins [28]

Property E-Glass Carbon Fiber Volume Fraction 46 63 Specific Gravity 1.8 1.61 Tensile Strength, 0o MPa

(ksi)1104 (160)

1725 (250)

Tensile Modulus, 0o GPa(msi)

39 (5.66)

159 (23.1)

Tensile Strength, 90o MPa(ksi)

36 (5.22)

42 (6.09)

Tensile Modulus, 90o GPa(msi)

10 (1.45)

10.9 (1.58)

Compression Strength, 0o MPa(ksi)

600 (87.0)

1366 (198)

Compression Modulus, 0o GPa(msi)

32 (4.64)

138 (20.0)

Compression Strength, 90o MPa(ksi)

138 (20.0)

230 (33.4)

Compression Modulus, 90o GPa(msi)

8 (1.16)

11 (1.60)

In-Plane Shear Strength MPa(ksi)

95 (13.8)

In-Plane Shear Modulus GPa(msi)

6.4 (0.93)

Longitudinal Poisson�s Ratio (νLT) 0.25 0.38 Longitudinal Coef. of Thermal Expansion (10-6/oC)

5.4 0.045

Transverse Coef. of Thermal Expansion (10-6/oC)

20.2 36

20

The strains can be derived from the assumed displacement field as

ε ∂∂

∂∂

ε ∂∂

∂∂

γ ∂∂

∂∂

∂∂ ∂

xo o

yo o

xyo o o

ux

z wx

vy

z wy

uy

vx

z wx y

= −

= −

= + −

2

2

2

2

2

2

. (3-2)

Defining mid-plane strains as

{ }εεεγ

∂∂∂∂

∂∂

∂∂

oxo

yo

xyo

o

o

o o

uxvy

uy

vx

=�

��

��

�

��

��

=

+

�

�

��

�

��

�

�

��

�

��

, (3-3)

and mid-plane curvatures as

{ }κκκκ

∂∂

∂∂∂∂ ∂

=�

��

��

�

��

��

= −

�

�

��

�

��

�

�

��

�

��

x

y

xy

o

o

o

wxwy

wx y

2

2

2

2

2

2

, (3-4)

we can then write the general strains as

εεγ

εεγ

κκκ

x

y

xy

xo

yo

xyo

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

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 ε1 1= Q , (3-6)

with the stress vector given by

{ }σ

σσστττ

1

1

2

3

23

13

12

=

�

�

��

�

��

�

�

��

�

��

(3-7)

and the strain vector given by

{ }ε

εεεγγγ

1

1

2

3

23

13

12

=

�

�

��

�

��

�

�

��

�

��

. (3-8)

The nonzero terms of the 3x3 stiffness matrix for the in-plane behavior are given by

22

( )

( )

( )

QE

QE

QE

Q G

111 23 32

12 21 23 32 13 31 12 23 31

222 13 31

12 21 23 32 13 31 12 23 31

122 12 13 32

12 21 23 32 13 31 12 23 31

66 12

11 2

11 2

1 2

=−

− − − −

=−

− − − −

=+

− − − −=

ν νν ν ν ν ν ν ν ν ν

ν νν ν ν ν ν ν ν ν ν

ν ν νν ν ν ν ν ν ν ν ν

, (3-9)

with E1 the modulus of the composite lamina in the fiber direction, E2 the modulus

normal to the fiber direction, νij the Poisson�s for loading in the i direction and G12 the

in-plane shear modulus. This material stiffness matrix is valid for a coordinate system

aligned with the material directions. For a coordinate system oriented arbitrarily to the

material coordinate system it becomes necessary to transform the stiffness matrix,

from the orthogonal material coordinate system (1,2,3) to the orthogonal structure

coordinate system (x,y,z). The transformation between the two is given by

[ ]123

�

��

��

�

��

��

=�

��

��

�

��

��

Txyz

. (3-10)

If the 3 and z directions are parallel and the x,y coordinate system is rotated relative to

the 1,2 coordinate system by an angle θ (see Figure 3-1), [T] is the 3x3 transformation

matrix shown in equation 3-11.

23

1

2

3,z

xy

Figure 3-1 Material and Structural Coordinate Systems

[ ]T = −− −

�

�

��

�

�

��

cos sin sin cossin cos sin cos

sin cos sin cos cos sin

2 2

2 2

2 2

22

θ θ θ θθ θ θ θ

θ θ θ θ θ θ (3-11)

The stresses and strains are then transformed as

[ ]σστ

σστ

1

2

12

�

��

��

�

��

��

=�

��

��

�

��

��

Tx

y

xy

(3-12)

and

[ ]εε

γ

εε

γ

1

2

1212

12

�

�

��

�

��

�

�

��

�

��

=

�

�

��

�

��

�

�

��

�

��

Tx

y

xy

. (3-13)

The stress strain relation in the structure coordinate frame (x,y,z) is then expressed as

{ } [ ]{ }σ εx xQ= , (3-14)

where

θ

24

[ ] [ ] [ ][ ]Q T Q T= −1 (3-15)

and

{ }

{ }

σσστ

εεεγ

x

x

y

xy

x

x

y

xy

=�

��

��

�

��

��

=�

��

��

�

��

��

. (3-16)

The nonzero terms of the stiffness matrix in the structure coordinate frame are given

as

[ ]

[ ]

Q Q Q Q QQ Q Q Q Q

Q Q Q Q Q

Q Q Q Q Q

Q Q Q Q Q

Q Q

11 114

224

12 662 2

12 11 22 662 2

124 4

16 112

222

12 662 2

22 224

114

12 662 2

26 112

222

12 662 2

66 662 2 2

2 24

2

2 2

2

= + + +

= + − + +

= − − + −

= + + +

= − + + −

= − +

cos sin ( ) cos sin( ) cos sin (cos sin )

cos sin ( )(cos sin ) cos sin

cos sin ( ) cos sin

sin cos ( )(cos sin ) cos sin

(cos sin )


θ θ θ θ θ θ

θ θ θ θ

θ θ θ θ θ θ

θ θ ( ) cos sinQ Q Q11 22 122 22+ − θ θ

. (3-17)

The ply stress-strain relation shown in equation 3-14 can now be used to derive

the resultant forces and moments in the laminated plate by integrating through the

thickness of the laminate. The geometry of the plate is defined in Figure 3-2.

25

12

k

n

t

x

z

h0h1 h2

hkhn-1

hn

midplane

laminanumber

Figure 3-2 Geometry of Laminate

The section forces and moments can now be written as

{ }

{ }

NNNN

dz

MMMM

zdz

x

y

xy

x

y

xyt

t

x

y

xy

x

y

xyt

t

=�

��

��

�

��

��

=�

��

��

�

��

��

=�

��

��

�

��

��

=�

��

��

�

��

��

−

−

�

�

σσσ

σσσ

/

/

/

/

2

2

2

2

. (3-18)

These integrals can be handled as summations of the individual layers of the laminated

plate.

{ } [ ] { } [ ] { }

{ } [ ] { } [ ] { }

N Q dz Q k zdz

M Q zdz Q k z dz

ko

kh

h

h

h

k

n

ko

kh

h

h

h

k

n

k

k

k

k

k

k

k

k

= +��

��

��

��

= +��

��

��

��

−−

−−

��

��

=

=

ε

ε

11

11

1

2

1

(3-19)

Combining the stress and section force definitions the relation between the section

forces and deformations is derived.

26

NNNMMM

A BB D

x

y

xy

x

y

xy

xx

yy

xy

x

y

xy

�

�

��

�

��

�

�

��

�

��

=�

�

�

�

�

��

�

��

�

�

��

�

��

εεγκκκ

, (3-20)

with

( ) ( )

( ) ( )

( ) ( )

A Q h h

B Q h h

D Q h h

ij ij k k kk

n

ij ij k k kk

n

ij ij k k kk

n

= −

= −

= −

−=

−=

−=

�

�

�

11

21

2

1

31

3

1

1213

, (3-21)

where [A], [B] and [D] are 3x3 matrices. The A matrix represents the in-plane force

strain relations. The B matrix represents the coupling between in-plane and out-of-

plane behavior (extension - bending). The D matrix represents the out-of-plane

behavior.

The analysis presented for circular shells only considers the in-plane properties

of the shell. This means that the coupling and bending matrices, curvatures and

moments do not enter into the calculations. As soon as the section considered is no

longer circular the bending behavior must be considered. If only the in-plane behavior

is of interest equivalent orthotropic plate properties can be determined as described in

the following section.

27

3.1.6 Equivalent Plate Properties

Assuming that only the in-plane behavior of the laminate is of concern

equivalent orthotropic plate properties can be derived for a symmetric laminate. These

equivalent properties will be used later in the analytical relations derived for circular

sections. For a symmetric laminate there is no shear extension coupling which gives

A16=A26=0. The in-plane flexibility relation is given as

εεγ

x

y

xy

x

y

xy

A AA A

A

NNN

�

��

��

�

��

��

=�

�

�

�

��

��

�

��

��

11 12

21 22

66

00

0 0

* *

* *

*

, (3-22)

with

( )( )

( )

A A A A A

A A A A A

A A A A A

A A

11 22 11 22 122

22 11 11 22 122

12 12 11 22 122

66 661

*

*

*

*

/

/

/

/

= −

= −

= − −

=

. (3-23)

The same relation for an orthotropic plate assuming plane-stress can be written as

εεγ

ν

νx

y

xy

x

xy

x

xy

x y

xy

x

y

xy

tE tE

tE tE

tG

NNN

�

��

��

�

��

��

=

−

−

�

�

�

�

��

��

�

��

��

1 0

1 0

0 0 1

. (3-24)

Equating equations 3-22 and 3-24 we can derive the equivalent orthotropic plate

properties as

28

E tA A A A tAE tA A A A tA

A tE A A

E E A A

G tA A t

x

y

xy x

yx xy y x

xy

= = −

= = −

= − =

= =

= =

11

1

11 11 22 122

22

22 11 22 122

11

12 12 22

12 11

66 66

/ ( ) ( ) // ( ) ( ) /

/

( / ) /

/ ( ) /

*

*

*

*

ν

ν ν

. (3-25)

3.1.7 First Ply Failure Criteria

For the analyses discussed in this document the shell will be defined to have

failed when one of its plies has exceeded its allowable stress in the fiber direction. To

determine the ply stresses it is necessary to transform the strains in the global

coordinate system into the local ply coordinate system from which the ply stresses can

be determined. For the analysis presented in this document the strains in the structure

coordinate system are known. Equation 3-14 is used to transform these strains to

stresses in the structural frame. Equation 3-12 is then applied to transform the stresses

into the material coordinate system. Once the stresses in the material coordinate frame

are known they can be compared to the ply allowables.

3.1.8 Thermal Expansion

Concerns have been raised based on the potential difference in the coefficient

of thermal expansion for the composite shell and the concrete core. As can be seen in

Table 3-1 and Table 3-2 the coefficient of thermal expansion for reinforcing fibers in

the fiber longitudinal direction can be very low or even negative. This low thermal

expansion coefficient for the fibers can lead to very small thermal strains being

29

induced in the laminated shell by a temperature change depending on the composite

architecture chosen.

At the lamina level these materials behave orthotropically under temperature

changes due to the orientation of the fibers in one direction and the difference in the

coefficients of thermal expansion for the fibers and the matrix. The thermal strains in

the longitudinal and transverse directions are given by the following equations:

ε αε α

1 1

2 2

T

T

TT

==

∆∆

, (3-26)

with

ε1T - the thermal strain in the fiber direction

ε 2T - the thermal strain in the transverse direction

α 1 - the coefficient of thermal expansion in the fiber direction

α 2 - the coefficient of thermal expansion in the transverse direction

∆T - temperature change

Equivalent thermal expansion coefficients for a given lamina in the structure

coordinate frame may be determined from the transformation

αα

α


θ θ θ θ θ θ

αα

x

y

xy12

22

0

2 2

2 2

2 2

1

2

�

�

��

�

��

�

�

��

�

��

=−

− −

�

�

�

�

��

��

�

��

��

cos sin sin cossin cos sin cos

sin cos sin cos cos sin, (3-27)

30

where αxy is an apparent coefficient of thermal shear. With these coefficients the

thermal strains in the ply can be given by

εεγ

ααα

xT

yT

xyT

x

y

xy

TTT

�

��

��

�

��

��

=�

��

��

�

��

��

∆∆∆

. (3-28)

If free to expand no stresses would develop in the ply, however in a laminated

composite the plies are not free to expand due to the restraint offered from the adjacent

plies. The stresses generated from this restraint can be related to the mechanical strains

which are given as

εεγ

εεγ

ααα

xM

yM

xyM

x

y

xy

x

y

xy

TTT

�

��

��

�

��

��

=�

��

��

�

��

��

−�

��

��

�

��

��

∆∆∆

. (3-29)

The first strains on the right side of equation 3-29 are defined in Section 3.1.3. The

stress strain relation for a ply can now be expressed in terms of the mid-plane strains,

curvatures and thermal strains.

[ ]σστ

ε κ αε κ α

γ κ α

xT

yT

xyT

xo

x x

yo

y y

xyo

xy xy

Qz Tz Tz T

�

��

��

�

��

��

=+ −+ −+ −

�

��

��

�

��

��

∆∆∆

(3-30)

The laminate will deform under thermal loading with no external forces being applied.

Using equations 3-18 through 3.20 and 3-30 the following �thermal forces� are

obtained. These are the forces that if applied would give the same deformations as the

change in temperature.

31

[ ] ( )NNN

Tk

Q h hxT

yT

xyT

n

k

x

y

xy k

k k

�

��

��

�

��

��

= �=

�

��

��

�

��

��

− −∆1 1

ααα

(3-31)

[ ] ( )MMM

Tk

Q h hxT

yT

xyT

n

k

x

y

xy k

k k

�

��

��

�

��

��

= �=

�

��

��

�

��

��

− −12 1

21

2∆ααα

. (3-32)

Using these fictitious thermal forces and moments the mid-plane strains and

curvatures for the laminate can be calculated from equation 3-20. By carrying out this

analysis with a unit temperature change, equivalent coefficients of thermal expansion

can be determined for the laminate. This will result in not only expansional

coefficients but also shear and curvature coefficients if the lay-up is not symmetric.

For a symmetric lay-up there is no coupling between the in-plane strains and the

curvatures which allows the in-plane thermal expansion coefficients to be calculated

from the [A] matrix alone.

3.2 Concrete

Concrete is one of the most commonly used and commonly analyzed structural

civil engineering materials in use today and yet its response to applied loads is still not

fully understood. This difficulty in characterizing the mechanical behavior of concrete

is due to its highly nonhomogeneous structure and its nonlinear behavior due to crack

formation.

32

3.2.1 Uniaxial Compression

Although concrete is composed of materials that individually can be

considered to display brittle elastic behavior its stress strain curve is nonlinear. This

can be explained by the formation of microcracks as the loading progresses. The

volumetric strain defined as εv=ε1+ε2+ε3 (see Figure 2-1) initially decreases as would

be predicted by the theory of elasticity but almost immediately the volume strain

begins to deviate from this prediction and the volume begins to decrease at a slower

rate. When the concrete gets close to its ultimate strain the volume strain has actually

become positive (net volume expansion).

3.2.2 Biaxial and Triaxial Stress States - Confinement Effects

The response of concrete to bi- and triaxial stress states has been the subject of

many studies over the years. Kupfer, Hillsdorf and Rusch [30], Kupfer and Gerstle

[31], Gerstle et al. [32] and Vecchio and Collins [33] have studied concrete under

biaxial loading to determine the effects of an orthogonal load on the strength of

concrete. All researchers report an increase in the strength of concrete due to an

orthogonal compression stress and a decrease due to an orthogonal tension stress.

Kupfer and Gerstle have proposed a correlation between the strength increase and the

orthogonal confining stress. Veccio and Collins have proposed a relation based on the

orthogonal strain. This allows for the gradual reduction of strength observed based on

the orthogonal tensile strains.

33

Concrete under triaxial compression has been studied since early this century.

Hydraulic chambers and mechanical restraints have both been used to achieve the

desired stress state. Many models have been proposed to account for the increase in

strength and ductility observed in confined concrete. The main distinguishing point

between these models is the nature of the confining pressure. Some models assume a

constant radial pressure throughout the loading history. This simulates the case of steel

reinforcing spirals and hoops well since the steel yields fairly early on in the loading

and then produces a constant confining pressure proportional to the yield strength, area

and spacing of the transverse reinforcement. Mander has proposed such a model for

steel reinforced concrete [34][35]. Figure 3-3 shows the response of a typical concrete

cylinder under compression for various levels of confining pressure as predicted by

Mander�s model. The second type of model considers the variation in the confining

stress due to the actual dilation of the concrete core at any given point in the loading.

This second type of model works well for concrete confined by a linear elastic shell.

One popular method for formulating a confinement model dependent on the current

dilation state is to use a constant pressure model incrementally [36]. For each

increment of load the dilation of the system is calculated, knowing the dilation the

confining pressure is calculated and a constant pressure model can be used to predict

the current stress state. This is shown graphically in Figure 3-3 by the incremental

model which has increasing pressure as the axial strain increases. This causes the

incremental model to cross the constant pressure curves at points where the pressure in

the core equals that used for the constant pressure model.

34

The difficulty in pursuing this approach is the ability to accurately predict the

dilation of the confined concrete throughout the loading. Almost immediately upon

loading the concrete will begin to form microcracks and the dilation behavior will

deviate from that which would be predicted by Poisson�s effect in linear elasticity (see

Figure 2-2). The approach that is taken for the analysis described in Chapter 4 is

similar to the passive modeling approach described above except that the current

dilation state is used directly to estimate the tangent modulus of the confined concrete.

This approach is detailed in Section 4.1.1.2

Strain

0.000 0.005 0.010 0.015 0.020 0.025 0.030

Str

ess/

f'c

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5Mander's ModelIncremental Mander's Model

Unconfined

fl/f’c

0.05

0.10

0.20

0.40

Figure 3-3 Stress Strain Models for Confined Concrete

35

3.2.3 Tension

In general the tension strength of the concrete is ignored in the analytical

models used to predict the strength of the concrete filled FRP members studied in this

document. This is done to give a conservative estimate of the performance of this

system. The tension carrying capacity of the concrete may effect the stiffness of the

system even after the concrete is significantly cracked due to tension stiffening effects.

Studies will be presented to demonstrate the effects of including the tension carrying

contribution of the concrete on the system behavior. For the purpose of these studies

the following relations taken from Collins and Mitchell [37] will be used to estimate

the tension carrying capacity of the concrete. The cracking strength of the concrete for

stiffness evaluation, fcr, will be given by

f f MPa

f f psi

cr c

cr c

=

=

0 63

7 5

.

.

'

'

λ

λ . (3-26)

The λ factor is used to account for concrete density with

λ=1.00 for normal weight concrete

λ=0.85 for sand-lightweight concrete

λ=0.75 for all-lightweight concrete

The analytical models presented in Chapter 4 are based on a section analysis

approach that establishes a secant stiffness for the structural member at the current

loading state. This secant modulus is then used to predict member deformations under

load. As has been demonstrated in conventional reinforced concrete structures the

36

actual stiffness of a structural member will be greater than that predicted by a section

analysis approach due to the effect of tension stiffening. Tension stiffening is a

phenomenon that occurs due to the fact that the concrete cracks discretely on the

tension side of the member leaving sections of the member in between cracks intact or

uncracked. These uncracked sections will tend to stiffen the structure as load transfer

will still occur from the reinforcement into the uncracked concrete. To account for this

effect an average tension stress for the cracked concrete, fc, is used. This smeared

average concrete stress is approximated by

f fc

cr

cf

=+β β

ε1 2

1 500, (3-27)

where β1− factor accounting for bond characteristics of reinforcement

β1=1.0 for deformed reinforcing bars, or ribbed shells

β1=0.7 for plain bars, wire, strands or smooth shells

β1=0 for unbonded reinforcement

β2=factor accounting for sustained or repeated loading

β2=1.0 for short-term monotonic loading

β2=0.7 for sustained and/or repeated loading

εcf=average strain in section. The effect of this phenomenon will be studied in Chapter 8.

37

The material behavior presented in this section will be used in Chapter 4 to

develop the analytical models that describe the behavior of these concrete filled fiber

reinforced shells under various loading conditions.

38

4. ANALYTICAL MODELING OF CONCRETE FILLED FRP SHELLS 4.1 Circular Shells

The analysis presented in this section assumes that the shell is thin compared to

the radius of the section and that the strain gradient in the shell is negligible. Thus for

a circular section the bending stiffness of the laminate has no effect on the analysis

and the equivalent plate properties presented in Section 3.1.5 can be used. In this

section the compression and tension behavior are first derived. These relations are then

used to develop the bending behavior. An incremental elastic approach is taken for

this analysis. The fiber reinforced shell is assumed to remain linear elastic throughout

the loading. The tangent modulus and an equivalent tangent Poisson�s ratio of the

concrete are used in an incremental relation to introduce the nonlinear concrete

behavior.

4.1.1 Compression

4.1.1.1 Linear Elastic Relations

4.1.1.1.1 Shell Properties

For analysis purposes equivalent orthotropic properties for the laminated plate

are used to describe the shell behavior. Once a suitable lay-up is defined the lamina or

ply properties are used to derive equivalent plate properties through the use of

classical lamination theory as described in Section 3.1. This analysis results in

39

Young�s moduli for the longitudinal and hoop directions along with the associated

Poisson�s ratios. A biaxial stress state is assumed to exist in the shell. Thus the

longitudinal and hoop strains can be given respectively as

ε σ ν σL

L

LHL

H

HE E= − (4-1)

and

ε σ ν σH

H

HLH

L

LE E= − . (4-2)

4.1.1.1.2 Concrete Properties

The concrete model employed assumes that the compression response is

dependent on the radial expansion of the concrete. The longitudinal and radial strains

in the concrete are given respectively as

ε σ ν σ1 11 2= −

Ecc r( ) (4-3)

and

ε ν σ ν σrc

c r cE= − −1 1 1(( ) ) . (4-4)

4.1.1.1.3 System Behavior

The system is assumed to work together with no slip between the shell and the

concrete core. This is referred to as full composite action. Assuming full composite

action between concrete core and composite shell the following must hold.

40

ε εε ε

σ σ

L

H r

H rRt

==

= −

1

(4-5)

Utilizing equations 4-1 through 4-5 the radial pressure, radial strain and axial stress

can be determined.

σ ν ε εν νr

LH r H

HL LH

E tR

= − +−

( )( )

1

1 (4-6)

ε ε ν ν ν ν ν νν ν ν νr

c c HL LH LH H c c

H c c c HL LH

E R E tE t E R

= − + − −− − − − −

�

��

�

��1

2

2

1 1 21 2 1

( ) ( )( ) ( )

(4-7)

σ ε σ νL L L r

HL

H

EE

Rt

= −�

��

�

�� (4-8)

4.1.1.2 Nonlinear Concrete Response

The experimental data gathered from the confined cylinder tests described in

Section 5.1.2 was used to derive relations for the nonlinear concrete response. The

longitudinal and hoop strains are extracted directly from the strain gage and linear

potentiometer readings. The dilation rate µ is calculated from the strain data as shown

in equation 4-9.

µ εε

= ∆∆

r

1

(4-9)

The experimental dilation rate shows considerable scatter due to the fact that the

relation is defined incrementally. A curve fit of the form suggested by Mirmiran [8] as

41

shown in equation 4-10 is used to smooth the data (see Figure 4-1). In this relation

µ0 represents the initial dilation rate that is calculated from equation 4-7 with the

initial concrete Poisson�s ratio and modulus.

µµ ε

εεε

εε

εε

=+ +

�

��

�

��

+ +�

��

�

��

01 1

2

1 1

2

1

a b

c d

co co

co co

(4-10)

A second equation is derived from the test data relating the incremental change in the

concrete stress to the incremental axial strain as shown in equation 4-11.

σ

εε

εε

εε

1

1

1 12

1

=

+ +�

��

�

��

fa

b cc

co

co co

' (4-11)

The concrete stress is determined by subtracting the load in the shell from the total

applied load. Equations 4-10 and 4-11 are used in a program to simultaneously solve

equations 4-3 and 4-7 for the equivalent tangent Poisson�s ratio and tangent concrete

modulus as a function of the axial or radial strain. As discussed in Section 3.2.2 the

behavior of the confined concrete can be characterized by the confining pressure or by

the actual deformations. Models have been proposed that take the latter approach to

define a tangent or secant modulus for the concrete based on the current dilation or

�damage state� of the core [14]. For this analysis a curve fit of the form shown in

equation 4-12 was used to describe the tangent modulus as a function of the initial

modulus and the current radial strain.

42

E Ea

b c

c co

r

co

r

co

r

co

=

+

+ +�

��

�

��

1

12

εε

εε

εε

(4-12)

Longitudinal Strain

-0.04-0.03-0.02-0.010.00

Dila

tio

n R

ate

0.0

0.1

0.2

0.3

0.4

ExperimentalSmoothed

Figure 4-1 Experimental and Smoothed Dilation Rate

This evaluation was carried out for each of the compression cylinders tested. As

described in Chapter 5 two of the thick all hoop cylinders experienced considerable

bending and this data was considered compromised and was not utilized for the

analysis. The two helical cylinders had the hydrostone break away from the shell

during loading (see Chapter 5 for details) which made it impossible to accurately

determine the concrete stress at high radial strains. The analytical relation for the

concrete tangent modulus was based on the six remaining cylinders. The tangent

43

modulus vs. the radial strain for these six cylinders is shown in Figure 4-2 along with

the average tangent modulus. Table 4-1 gives the constants used for the analytical

model in equation 4-12 that represents the average of the test data.

Radial Strain

0.000 0.001 0.002 0.003 0.004

Co

ncr

ete

Tan

gen

t M

od

ulu

s (k

si)

0

500

1000

1500

2000

2500

3000

3500

Co

ncr

ete

Tan

gen

t M

od

ulu

s (M

Pa)

0

5000

10000

15000

20000

25000

AverageExperimental

Figure 4-2 Concrete Tangent Modulus vs. Radial Strain

Table 4-1 Constants for Tangent Modulus Relation

Constant a b c d Eco S.I. Eco U.S.

Value 1.0 -0.09967 5.802 7.061 20.9 (MPa) 3.03x106 (psi)

Figure 4-3 shows the equivalent tangent Poisson�s ratio vs. the radial strain for

the six cylinders analyzed. From this plot it can be seen that the maximum equivalent

tangent Poisson�s ratio for all cylinders occurs when the radial strain is approximately

44

0.2%. For the analytical model it has been assumed that the maximum equivalent

tangent Poisson�s ratio is a function of the ratio of the hydrostatic pressure (σhyd) in the

concrete when the radial strain reaches 0.2% to the unconfined concrete strength. A

cubic polynomial is used to approximate the relation between the equivalent tangent

Poisson�s ratio and the radial strain. The initial slope of these curves was observed to

be constant (~200) for all cylinders tested and will be represented by νci�. The initial

value at no radial strain is simply the initial concrete Poisson�s ratio. At a radial strain

of 0.002 the value of the polynomial must be νmax and the slope of the curve must be

0. These conditions are sufficient to establish the constants of the polynomial as

shown in equation 4-13. To simulate the observed behavior that the dilation rate seems

to flatten out the Poisson�s ratio is assumed to stay constant after a radial strain of

0.0025 has been reached. This value was based on the compression test data. Figure

4-4 compares the analytical model to the experimental curves.

ν ν ε ε εν

ν ν νν

c co r r r

ci

co ci

ci

b c dbcd c e

= + + +

=

= − −

= − −

2 3

750 000 100033333 8 333 4

'

max'

'

, ( ). .

(4-13)

Figure 4-5 shows the variation of the maximum Poisson�s ratio with

hydrostatic pressure. At this time this relation is based on scant data especially in the

region of low confinement. The maximum value seen in Figure 4-5 of 0.5 for the

maximum tangent Poisson�s ratio is used here for lack of any reliable data in this

range. From Figure 2-2 it can be seen that a value higher than 0.5 must be possible for

45

the slope of the volume strain curve to be positive as is the case for unconfined

concrete. More experimental investigation in this range is needed to accurately predict

the behavior for low levels of confinement. A relation of the form shown in equation

4-14 defines the analytical curve used for this analysis. This relation is the last piece

needed to formulate the compression behavior.

νσ σ

max ' '. . .= + −�

��

�

��0 488 0 047 0 035

2hyd

c

hyd

cf f (4-14)

The flow for the analytical model to predict the compression behavior is

depicted schematically in Figure 4-6. The determination of the maximum equivalent

tangent Poisson�s ratio is shown in Figure 4-7.

The above findings demonstrate the connection between the dilation and load

carrying capacity of the concrete core. Since the tangent modulus is based on the

current dilation state of the core, delaying this dilation will enhance the load carrying

capacity of the core. The critical variables for offering maximum dilation control are

the stiffness of the shell in the hoop direction and Poisson�s ratio of the shell for

loading in the longitudinal direction. A steel shell while offering high stiffness in its

linear range also has a high Poisson�s ratio meaning that in the initial stages of loading

the shell will expand faster than the concrete core. When the steel shell reaches its

yield strain, which for mild steel occurs at a radial strain of approximately 0.2%, the

hoop stiffness essentially becomes zero allowing the expansion of the core to increase

and eventually the net volume strain will become positive as described in Chapter 2.

46

Radial Strain

0.000 0.001 0.002 0.003 0.004 0.005

Eq

uiv

alen

t T

ang

ent

Po

isso

n's

Rat

io

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Thin Shells ExperimentalThick Shells Experimental

Figure 4-3 Equivalent Tangent Poisson's Ratio for Test Cylinders

Radial Strain

0.000 0.001 0.002 0.003

Eq

uiv

alen

t T

ang

ent

Po

isso

n's

Rat

io

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Thin Shells ExperimentalThick Shells ExperimentalThin Shells AnalyticalThick Shells Analytical

Figure 4-4 Analytical Equivalent Tangent Poisson's Ratio

47

σhyd/f'c1.0 1.5 2.0 2.5

Max

imu

m E

qu

ival

ent

Tan

gen

t P

ois

son

's R

atio

0.40

0.42

0.44

0.46

0.48

0.50

ExperimentalAnalytical

Figure 4-5 Maximum Tangent Poisson's Ratio vs. Hydrostatic Pressure

48

Increment axial strain

Calculate dεr from eqn (4-7)

εr=εr+dεr

Calculate tangent modulus Ec from eqn (4-2)

Calculate Poisson�s ratio from eqn (4-13)

Calculate orthotropic plate properties for shell

Calculate σL from eqn (4-8)

σ1=σ1+dε1 Ec Figure 4-6 Analytical Model For Compression Behavior

49

Increment axial strain

Guess max Poisson�s ratio vmax

Calculate dεεεεr from eqn (4-7)

εεεεr=εεεεr+dεεεεr

Define cubic relation for vc from eqn (4-13)

Calculate tangent modulus from eqn (4-12)

Calculate Poisson�s ratio from eqn (4-13)

Does εεεεr=.002?

Yes

Calculate σσσσhyd

Calc vmax_anal from eqn (4-14)

Does vmax =vmax_anal

No

Yes No

vmax =vmax_analContinue

Calculate orthotropic plate properties for shell

Figure 4-7 Determination of Maximum Equivalent Tangent Poisson's Ratio

50

4.1.2 Tension

In tension the longitudinal stress in the concrete core is taken to be zero. The

shell will tend to contract radially due to its Poisson�s ratio. This radial contraction is

resisted by the concrete core. The radial stress in the concrete is assumed to be

σ εr H coE= . (4-15)

Using this relation, equations 4-1, 4-2 and the last of equations 4-5 we can derive the

hoop strain as a function of the axial strain as

ε ε νν ν νH L

H LH

LH HL co H co LH

E tE R E t E R

=− −

�

��

�

�� . (4-16)

This relation is linear and mainly dependent on the Poisson�s ratio and the hoop

modulus of the shell as well as the initial Young�s modulus of the concrete. An

isotropic material such as steel would behave exactly the same as long as the material

remained linear.

4.1.3 Shear

Two cases will be discussed for determining the shear strains in the shell. The

first case applies to the condition of an inadequate or nonexistent shear transfer

mechanism between the shell and the encased concrete core. The second case applies

to a shell and concrete composite system that has an adequate shear transfer

mechanism. Figure 4-8 shows the forces in a section of shell and concrete necessary

to maintain equilibrium in a section of the member with shear forces present. The load

51

in the shell increases from P to P� over a defined distance. To maintain equilibrium the

shear in the shell at the section of interest, qs, plus the shear transferred between the

shell and the concrete core, qc, must equalize the difference between the shell loads. If

the shear mechanism provided between the shell and the concrete core is inadequate to

provide a shear transfer equal to qc then the core and shell will slip relative to each

other and the shear distribution between the shell and the concrete will be determined

by the relative shear stiffness of the members. If the shear transfer mechanism is

adequate to take the shear qc then the members will deform together and a plane-

sections-remain-plane assumption may be used.

Ps`

Ps

Pc

Pc` qs

qs

qc

qc

qc`

FRP Shell

Concrete Core

Figure 4-8 Shear Transfer Between Concrete Core and Composite Shell

52

For the first case the concrete and shell shear resistance must be considered

separately. The initial shear stiffness of the concrete core is much greater than the

initial shear stiffness of the shell. It is therefore assumed that the concrete takes the

entire shear until it has reached its shear carrying capacity. The shear capacity of the

concrete is estimated by equation 4-17 following the ACI recommendation for flexure

shear cracking of a reinforced concrete beam. The shell is assumed to take any

additional shear above this level.

V v A f

v MPa unitsv psi units

c k c

k

k

=

≤ ≤≤ ≤

'

. .. .

016 0 2919 35

(4-17)

For the second case the section is analyzed as monolithic, assuming no slip

occurs between the concrete core and the shell. A plane-sections-remain-plane

assumption is carried throughout thus assuming that the shear deformations are

insignificant compared to the bending deformations. Referring to Figure 4-9 at a

given angle θ the total load above the section of interest can be calculated. At a

distance dx along the member the moment has changed by an amount

M M Vdx' = + . (4-18)

For a small value of dx the neutral axis location is assumed constant and the total load

above the section of interest at a distance dx from the initial section is given by

P P M VdxM

'= + . (4-19)

53

The shear that must be transferred across the shear reference plane is thus

q P Pdx

PVM

= − =' . (4-20)

From the experimental data it can be seen that the shear strain in the shell increases

faster than the applied shear. This is attributed to the reduced shear carrying capability

of the concrete core. An effective thickness is used to calculate the shear strain from

the shear flow. The thickness assigned to the concrete is reduced by a ratio of the

mean concrete modulus above the shear reference plane to the modulus of the shell in

the longitudinal direction.

t t zEEeff

ca

L

= +2 , (4-21)

where the average concrete modulus is calculated from

EE dy zdy zcac= �

�. (4-22)

The concrete tangent modulus Ec is defined by equation (4-12).

θ

cy

z

N.A.

Figure 4-9 Geometric Properties for Determination of Shear Stress

54

4.1.4 Bending

A moment curvature program was written based on the above relations for

compression, tension and shear behavior. The maximum compression strain is

increased incrementally. The program iterates on the neutral axis location until

equilibrium is achieved. The plies in the shell are checked at each step until the first

fiber failure is encountered. This analysis can be entered with the constitutive fiber

and matrix properties that are then used to calculate lamina properties or the lamina

properties can be entered directly. Figure 4-10 maps out the analysis flow to

determine the bending capacity.

FIBER PROPERTIES

MATRIX PROPERTIESHAHN�S EQUATIONS

LAMINA PROPERTIES

CLASSICAL LAMINATION THEORY

EQUIVALENT HOMOGENEOUSORTHOTROPIC PLATE PROPERTIES.

CONCRETE PROPERTIES

CONCRETE 3-D COMPRESSIONCONSTITUTIVE EQUATIONS

LINEAR ELASTIC 2-DCONSTITUTIVE EQUATIONS

INCREMENTAL DEVELOPMENTOF COMPRESSION BEHAVIORFOR CYLINDRICAL SHELLSFILLED WITH CONCRETE

FIRST PLY FAILURE CRITERIA

TENSION BEHAVIORFOR CYLINDRICAL SHELLSFILLED WITH CONCRETE MOMENT CURVATURE ANALYSIS BENDING CAPACITY

Figure 4-10 Analysis Flow For Bending Behavior

55

The moment curvature program is based on conventional reinforced concrete

analysis with the main difference being the importance of accurately predicting the

hoop and shear strains in the shell throughout the loading so that the complete biaxial

stress state of the shell is known and can be checked against the ply allowables.

Classic lamination theory as described in Chapter 3 is used to determine the equivalent

orthotropic plate properties for the shell. To initialize the program an increment of

compression strain is applied to the extreme compression fiber of the member, a

neutral axis location is assumed for the composite system of shell and concrete core

and then the shell and core are integrated to determine the load and moment balance of

the section. The shell behavior can be determined from the biaxial linear elastic

equations presented in equations 4-1 and 4-2. Given the axial strain, the radial strain

can then be determined based on the appropriate compression or tension relation.

Utilizing the axial and radial strains the axial stress is then established. The total load

and moment in the shell are determined by integrating around the shell in 1o

increments from 00-180o and multiplying this result by 2. The concrete is divided into

horizontal slices or layers. The radial strain is determined based on the axial strain in a

given slice. The strain state is then used to estimate the axial stress in the slice. The

neutral axis location is iterated until equilibrium is achieved. For each increment of

compression strain applied the shell is checked for first ply failure. This is

accomplished by once again stepping around the shell in 1o increments calculating the

axial, radial and shear strains in the shell in the structure coordinate system,

transforming these strains to the ply coordinate system for each ply angle (since no

56

bending is assumed to exist in the shell each ply angle need only be checked once

regardless of the number of plies in the laminate at this angle), calculating the ply

stresses and comparing these ply stresses to the ply allowables. The analysis is

continued until a fiber or shear failure is encountered. The program output includes the

equivalent orthotropic plate properties calculated for the shell, the moment at first

matrix cracking and the moment and secant stiffness at first ply failure.

The concrete contribution in tension is generally considered to be negligible.

As described in Chapter 8 the effects of considering tension stiffening in the concrete

for a beam in bending are very small although if a beam and slab system is being

considered where the entire concrete filled fiber reinforced shell may be in tension

these effects could begin to be considerable.

4.2 Conrec Shells

A second cross section geometry considered in this document is a square

section with rounded corners that is referred to as a conrec section (see Figure 4-11).

The analysis for the conrec sections is based on the relations developed for the circular

sections. The area strain is used to calculate the concrete tangent modulus just as in the

case of the circular section. Finite element models have been used to investigate the

area strain patterns in the conrec sections as well as to quantify the effect of local shell

bending. Parameter studies were done for shells with 90o and ±10o plies. The ±10o

plies were increased from 0% of the lay-up to 80% of the lay-up. Studies were also

performed with various flat to radius ratios ranging from .5 to 3.

57

RadiusFlat

Figure 4-11 Conrec Cross Section

4.2.1 Compression

As mentioned in Section 4.1.1.2 the area strain in a circular section under

uniform compression is constant across the section. The models developed for the

circular cross section take advantage of this trait in that only one value of the

maximum Poisson�s ratio is necessary to characterize the behavior of the entire

section. When we get away from the circular cross section the area strain will vary

around the cross section and each point will experience different behavior based on

how much confinement the shell offers in that specific area of the section. An

incremental finite element approach has been developed that allows each element in

the model to respond differently under an applied uniform compression load. A small

increment of compression strain is applied to the models, the area strain in the

concrete core throughout the section, the hydrostatic pressure, as well as the hoop

strain in the shell are recorded. The area strain at the centroid of each element is used

to calculate a new Young�s modulus for that element. As explained in Section 4.1.1.2

the maximum equivalent tangent Poisson�s ratio occurs at a radial strain of 0.2% for

the circular sections. This is equivalent to an area strain of 0.4% (εA=2εr). The

58

hydrostatic pressure is used along with a linear relation (equation 4-27) to predict the

hydrostatic pressure that would be expected at an area strain of 0.4%.

σσ

εhyd mhyd

A_

.=

0 004 (4-27)

The predicted hydrostatic pressure is then used to estimate the maximum equivalent

tangent Poisson�s ratio from which the current equivalent tangent Poisson�s ratio is

calculated as described for the circular section. This process is repeated for subsequent

strain steps. The strains and stresses are accumulated for all steps until the final axial

strain of interest has been achieved.

To investigate the behavior of the conrec sections under compression finite

element modeling was used to determine the effect of varying the shell lay-up and the

flat to radius ratio. The models were constructed with one layer of solid elements to

model the concrete and composite shell elements around the perimeter (Figure 4-12)

to model the composite shell with proper bending stiffness (equivalent plate properties

described in Section 3.1.4 are for in-plane properties only). Elastic properties were

assumed for the concrete core. One quarter of the section was modeled with symmetry

boundary conditions being used. An equivalent circular section was also analyzed for

each lay-up with the same composite shell thickness and a diameter equal to the flat to

flat distance for the conrec. The area strains and hoop strains from the conrec sections

were ratioed to the values determined from the �equivalent circular sections�.

59

Figure 4-13 shows the 4 flat to radius ratios studied in this analysis. The flat to

flat dimension for all models was held constant at 381mm (15in.). A symmetric lay-up

of the form shown in Table 4-2 was assumed for all cases with the ply thicknesses

varied to give the desired percentage of helical plies. Three dimensional

representations of the area strain ratio (conrec/circular) for the shells with all hoop

plies are shown in Figure 4-14 through Figure 4-17 to demonstrate the effect of the

conrec geometry on the confinement of the concrete core. These plots represent one

quarter of the conrec section as shown in Figure 4-12 with the center at (0,0). These

plots look the same for all lay-ups studied with the exception of the scale of the strain

ratio axis. In Figure 4-18 through Figure 4-20 the values of the strain ratio for three

points in the section are given for the different shell geometries and lay-ups.

x y

z

y-symmetry x-symmetry

Figure 4-12 Finite Element Model Used for Evaluation of Conrec Sections

60

It can be seen from these plots that the shells with high hoop stiffness (high

percentage of 90o fibers) show a much greater loss of confinement efficiency (higher

area strains) than do the shells with low hoop stiffness (high percentage of ±10o

fibers). This is due to the fact that the circular sections with low hoop stiffness are not

very effective at confining the concrete core due to the high Poisson�s ratio of the shell

for loading in the longitudinal direction for these lay-ups. Bending applications for

which the shells with high axial stiffness are most advantageous do not suffer much

from the confinement losses as is demonstrated in the following section. For

compression behavior the shells with predominantly hoop fibers excel. The mean area

strain is an indicator of the response of the member in compression as this is a direct

indication of the load carrying capacity. The sections with large radii (F/R≤1) maintain

an area strain approximately 10% greater than that of the equivalent circular section

for an all hoop shell. As the flat to radius ratio increases the area strain as compared to

the equivalent circular section also increases. From this study a flat to radius ratio of 3

with a shell composed of all hoop fibers showed a mean area strain increase of

approximately 20% over the circular section.

61

Table 4-2 Composite Lay-Ups Used for Conrec Studies

% ±10o Plies Lay-up Ply Thickness mm (in.)

0 [90] [10.2]

([0.40])

50 [90,10,-10]sym [2.54,1.27,1.27]sym

([0.10,0.05,0.05]sym)

80 [90,10,-10]sym [1.02,2.03,2.03]sym

([0.04,0.08,0.08]sym)

a) 1/2 b) 1/1

c) 2/1 d) 3/1

x

y

x

y

x

y

x

yFlat/Radius

Figure 4-13 Conrec Geometries Used for This Analysis

62

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

01

23

45

67

1 2 3 4 56

7

Area S

train Ratio (C

onrec/Circ)

x

y

Profile of Section in x-y Plane

Figure 4-14 Area Strain Ratio Profile, 0% ±10o Plies, Flat to Radius Ratio .5

0.8

0.9

1.0

1.1

1.2

1.3

1.4

01

23

45

67

1 2 34 5

67

Area S

train Ratio (C

onrec/Circ)

x

y


Figure 4-15 Area Strain Ratio Profile, 0% ±10o Fibers, Flat to Radius Ratio of 1

63

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

01

23

45

67

1 2 3 45

67

Area S

train Ratio (C

onrec/Circ)

x

y



0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

01

23

45

67

1 2 34 5

67

Area S

train Ratio (C

onrec/Circ)

x

y



64

Flat/Radius Ratio

0.5 1.0 1.5 2.0 2.5 3.0

Are

a S

trai

n R

atio

(C

on

rec/

Cir

cula

r)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Figure 4-18 Area Strain Ratio for 0% ±10o Conrec Shell

Flat/Radius Ratio

0.5 1.0 1.5 2.0 2.5 3.0

Are

a S

trai

n R

atio

(C

on

rec/

Cir

cula

r)

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15


65

Flat/Radius Ratio

0.5 1.0 1.5 2.0 2.5 3.0

Are

a S

trai

n R

atio

(C

on

rec/

Cir

cula

r)

0.94

0.96

0.98

1.00

1.02

1.04


Unlike in the circular sections the hoop stresses in the conrec shells vary

around the section and also through the thickness due to bending. Variation of the

hoop stress is dependent on the shell lay-up and the geometry of the shell. Figure 4-21

through Figure 4-24 show the variation in the hoop stresses extracted from the finite

element models for the inside and outside surface of the shell ratioed to that of the

equivalent circular section at the same axial strain. The horizontal axis represents one

quarter of the perimeter of the shell starting from the center of a flat side as shown in

the figures. It can be seen from these plots that the maximum tension strain occurs in

the shell at the center of the flat on the outside surface for the geometries with a large

radius but can occur near the start of the radius due to bending if the radius is small.

66

Distance Around Section (mm)

0 50 100 150 200 250 300

Str

ess

Rat

io (

con

rec/

circ

)

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

Distance Around Section (in)

0 2 4 6 8 10 12

Outside 0% ±10o

Inside 0% ±100

Outside 50% ±10o

Inside 50% ±10o

Outside 80% ±10o

Inside 80% ±10o

Start of Radius

Area for Hoop Stress Plots

Figure 4-21 Hoop Stress in Shells With Flat To Radius Ratio of .5


0 50 100 150 200 250 300

Str

ess

Rat

io (

con

rec/

circ

)

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10


0 2 4 6 8 10 12

Outside 0% ±10o

Inside 0% ±10o

Outside 50% ±10o

Inside 50% ±10o

Outside 80% ±10o

Inside 80% ±10o


Start ofRadius

Figure 4-22 Hoop Stress in Shells With Flat To Radius Ratio of 1

67


0 50 100 150 200 250 300

Str

ess

Rat

io (

con

rec/

circ

)

0.4

0.5

0.6

0.7

0.8

0.9

1.0


0 2 4 6 8 10 12

Outside 0% ±10o

Inside 0% ±10o

Outside 50% ±10o

Inside 50% ±10o

Outside 80% ±10o

Inside 80% ±10o


Start ofRadius



0 50 100 150 200 250 300

Str

ess

Rat

io (

con

rec/

circ

)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0


0 2 4 6 8 10 12

Outside 0% ±10o

Inside 0% ±10o

Outside 50% ±10o

Inside 50% ±10o

Outside 80% ±10o

Inside 80% ±10o


Start ofRadius


68

4.2.2 Bending

The bending analysis for the conrec sections is carried out exactly as for the

circular sections. The concrete stress strain relation is developed using an equivalent

circular section which is a circular section with the same lay-up as the conrec section

and a radius equal to the flat to flat distance of the conrec. The following analysis is

presented to show that the use of the equivalent circular section for the bending

calculations does not effect the system response. The contribution to the bending

stiffness from the concrete is greatest for shells with predominantly hoop fibers. This

analysis was carried out for a conrec shell with a 127mm (5 in.) flat and a 127mm (5

in.) radius. The shell was carbon with the ply properties presented in Table 5-2, and

had a total thickness of 10.2mm (0.4 in.) with 90% of the fibers in the hoop direction

and 10% in the ±10o direction. The concrete strength assumed was 45.5 MPa (6.6 ksi).

Figure 4-25 shows the moment curvature response for this section and for the shell

alone. The difference between these two curves is the contribution to the behavior

provided by the concrete core. To show that the concrete model used has no great

effect on the overall behavior two concrete models were assumed, one based on the

equivalent circular section and one based on a perfectly plastic behavior cutting off the

concrete strength at f�c. This second model is a lower limit on the concrete strength.

Figure 4-26 shows the difference in the concrete stress strain behavior assumed.

When these models are used in a moment curvature analysis no discernible difference

is seen between the two (see Figure 4-27). This is due to the fact that the ultimate

strain for the shell is approximately 1.2% in the longitudinal direction which only puts

69

a small amount of concrete in a range where the choice of concrete model makes a

difference. For the parametric studies presented in Chapter 7 the equivalent circular

section is used to predict the behavior of the concrete in compression. If very low shell

thicknesses are being investigated it may become necessary to take into account the

variation in the area strain of the conrec section when performing a bending analysis.

Experimental investigations carried out for validation and calibration of the

models presented in this chapter will be discussed in the following sections.

Curvature (1/in)

0.0000 0.0004 0.0008 0.0012

Mo

men

t (k

ip-i

n)

0.0e+0

1.0e+6

2.0e+6

3.0e+6

Curvature (1/mm)

0e+0 1e-5 2e-5 3e-5 4e-5 5e-5

Mo

men

t (k

N-m

)

0.0e+0

1.0e+5

2.0e+5

3.0e+5

4.0e+5

Full SectionShell alone

Figure 4-25 Moment Curvature of Typical Conrec Section

70

Strain

-0.015-0.010-0.0050.000

Str

ess/

f'c

0.0

0.5

1.0

1.5

Equivalent Circular SectionPerfectly Plastic

Figure 4-26 Concrete Stress Strain Relation

Curvature (1/in)

0.0000 0.0004 0.0008 0.0012

Mo

men

t (k

ip-i

n)

0.0e+0

1.0e+6

2.0e+6

3.0e+6

4.0e+6

Curvature (1/mm)

0e+0 1e-5 2e-5 3e-5 4e-5 5e-5

Mo

men

t (k

N-m

)

0.0e+0

1.0e+5

2.0e+5

3.0e+5

4.0e+5

5.0e+5Cicular Relations for Concrete BehaviorPerfectly Plastic Relation for Concrete Behavior

Figure 4-27 Comparative Moment Curvature for Conrec Section With Various

Concrete Models

71

5. EXPERIMENTAL PROGRAM TO VALIDATE CONCRETE FILLED FRP TUBE BEHAVIOR

An experimental program designed to validate the concept of using concrete

filled FRP shells for bridge components has been under way at UCSD for the past

several years. Bridge column studies comparing concrete filled carbon reinforced

shells with various connection details to a standard reinforced concrete column have

been completed [38][39]. This document outlines: (1) compression tests done on small

scale concrete filled carbon fiber reinforced composite cylinders to investigate the

behavior of the concrete confined by a linear elastic shell, (2) small and full scale

bending tests to validate the models described in Chapter 4, and (3) a beam-and-slab

assembly with a cast in place concrete deck used here to investigate the stress

concentrations around the penetrations for the deck to beam connection and elsewhere

for evaluation of the system characterization [3][5]. Currently, a full scale bridge

section with concrete filled carbon girders and E-Glass/polyester deck panels is being

cyclically loaded to verify the fatigue response of such a proposed system (see Figure

1-3).

5.1 Small Scale Shells

The small scale test program consisted of bending, compression and thermal

expansion tests. Two shapes were investigated, circular cylinders with a diameter of

152mm (6 in.) and �conrec� cylinders. The conrec is a 152mm (6 in.) square with

72

51mm (2 in.) radii at the corners as shown in Figure 5-1. These shells were filament

wound with carbon epoxy by Alliant Techsystems Inc. A total of eight 2.44m (8 ft)

shells were manufactured. One set of shells was wound with purely hoop fibers. These

shells were utilized for compression tests. Another set of shells had approximately

85% of the fibers at ±10o from the longitudinal axis and 15% hoop fibers. A 2.13m (7

ft) section from these specimens was used for the bending tests and the remaining

305mm (1 ft) was tested in compression. The shells used for the small scale test

program are summarized in Table 5-1. The vendor supplied ply properties are listed in

Table 5-2. These properties were used to generate the equivalent plate properties for

the shells as listed in Table 5-3.

152mm6.0 in

152m

m

6.0

in

R51mm2.0 in

Figure 5-1 Nominal Geometry of Conrec Section

73

Table 5-1 Small Scale Test Shells NOMINAL

THICKNESS SHAPE LAY-UP S.I. U.S.

CIRC [90] 2.29mm .09 in.

CIRC [90] 4.57mm .18 in.

CIRC [90,±10,±10,90] 2.29mm .09 in.

CIRC [90,±10,±10,90]SYM 4.57mm .18 in.

CONREC [90] 2.29mm .09 in.

CONREC [90] 4.57mm .18 in.

CONREC [90,±10,±10,90] 2.29mm .09 in.

CONREC [90,±10,±10,90]SYM 4.57mm .18 in.

Table 5-2 Vendor Supplied Ply Properties

S.I. U.S.

E11 121 GPa 17.5 Msi

E22 6.90 GPa 1.0 Msi

G12 4.83 GPa 0.70 Msi

G23 2.62 GPa 0.38 Msi

ν12 0.30 0.30

ν23 0.40 0.40

74

Table 5-3 Equivalent Plate Properties

[90] [90,±10,±10,90]

S.I. U.S. S.I. U.S.

EL 6.90 GPa 1.00 msi 101 GPa 14.7 msi

EH 121 GPa 17.5 msi 19.8 GPa 2.87 msi

GLH 4.83 GPa 0.70 msi 7.52 GPa 1.09 msi

νLH 0.3 0.3 0.24 0.24

5.1.1 Concrete Characterization

Three 152mm (6 in.) diameter 305mm (1 ft) long concrete cylinders were cast

when the small scale shells were filled. These cylinders were tested in compression

with four longitudinal and four hoop strain gages so the stiffness, strength and

Poisson�s ratio of the concrete could be determined. The concrete mix used was a 35

MPa (5 ksi) nominal mix with light weight aggregate. Figure 5-2 shows the stress

strain curves for the three cylinders tested. The strain is the average of the four

longitudinal gages and the stress is derived from the applied load. The initial Young�s

modulus is determined from the average of these curves between 0 and 500

microstrain and was found to be 20.1 GPa (2.99 msi). This value correlates well with

the ACI recommended relation with weight (w) correction given by equation 5-1

which gives 19.9 GPa (2.88 ksi) using the nominal strength and 22.8 GPa (3.3 ksi)

using the measured strength.

E w fc c= 33 1 5( ). ' (psi units) (5-1)

75

The ultimate compression strength for the mix was calculated from the average

ultimate strength of the three cylinders and was found to be 45.5 MPa (6.6 ksi) at 28

days. Figure 5-3 shows the hoop strain vs. the longitudinal strain for all three

cylinders. Fitting this data leads to an initial Poisson�s ratio of 0.2. The concrete

properties used for the analysis are summarized in Table 5-4.

Longitudinal Strain (micro-strain)

-3500-3000-2500-2000-1500-1000-5000

Str

ess

(MP

a)

0

10

20

30

40

50

Str

ess

(ksi

)

0

1

2

3

4

5

6

7cyl #1cyl #2cyl #3

Figure 5-2 Concrete Compression Stress Strain Relation

76


-3500-3000-2500-2000-1500-1000-5000

Ho

op

Str

ain

(m

icro

-str

ain

)

0

500

1000

1500

2000

2500

3000

cyl #1cyl #2cyl #3

Figure 5-3 Hoop Strain vs. Longitudinal Strain for Concrete Cylinders Under

Uniaxial Compression

Table 5-4 Experimentally Derived Concrete Properties

S.I. U.S.f�c 45.5 MPa 6.6 ksiEco 20.1 GPa 2.99 msiεco 0.003 0.003νco 0.2 0.2

5.1.2 Compression

The purely hoop shells were cut into three 610mm (2 ft) and one 305mm (1 ft)

sections for compression testing. One 305mm (1 ft) section was also cut from the

tubes with both hoop and helical fibers to be tested in compression. The matrix of test

77

specimens used for compression behavior characterization is shown in Table 5-5. It

was felt that an aspect ratio of 2 to 1 may not be sufficient to get data unpolluted by

the end effects. Finite element modeling of the circular cylinders was used to

investigate these end effects. Models with the ends free to expand radially were

compared with identical models with the ends fixed radially. This investigation

showed that the aspect ratio of the 610mm (2 ft) specimens, which was 4 to 1, should

give a zone in the center of the cylinder which is representative of an unrestrained

cylinder. An example of this analysis is shown below in Figure 5-4 for unconfined

circular cylinders with a Poisson�s ratio of 0.45. The typical test setup for the

compression tests is shown in Figure 5-5.

y/L

0.00 0.25 0.50 0.75 1.00

Ho

op

Str

ain

/ H

oo

p S

trai

n U

nre

stra

ined

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

UnrestrainedL/D=2/1L/D=4/1

Ly

D

Figure 5-4 Comparison of Hoop Strains for Cylinders With Various Aspect Ratios

78

Figure 5-5 Typical Compression Test Setup

79

Table 5-5 Compression Specimens

SHAPE LAY-UP LENGTH

THICKNESS

SPECIMEN DESIGNATION

CIRC [90] 610mm (2 ft)

2.3mm (0.09 in.)

c1,c2,c3

CIRC [90] 305mm (1 ft)

2.3mm (0.09 in.)

c4

CIRC [90] 610mm (2 ft)

4.6mm (0.18 in.)

c5,c6,c7

CIRC [90] 305mm (1 ft)

4.6mm (0.18 in.)

c8

CIRC [90,±10,±10,90] 305mm (1 ft)

2.3mm (0.09 in.)

c17

CIRC [90,±10,±10,90]sym 305mm (1 ft)

4.6mm (0.18 in.)

c18

CONREC [90] 610mm (2 ft)

2.3mm (0.09 in.)

c9,c10,c11


2.3mm (0.09 in.)

c12


4.6mm (0.18 in.)

c13,c14,c15


4.6mm (0.18 in.)

c16

CONREC [90,±10,±10,90] 305mm (1 ft)

2.3mm (0.09 in.)

c19

CONREC [90,±10,±10,90]sym 305mm (1 ft)

4.6mm (0.18 in.)

c20

80

5.1.2.1 Circular Cylinders

The circular cylinders were instrumented with four longitudinal strain gages

and four hoop strain gages as shown in Figure 5-6. Three linear potentiometers were

positioned around the specimens to evaluate the longitudinal strain at specimen

deformations past the capability of the strain gages.

The critical information derived from these tests is the stress vs. strain behavior

of the concrete confined by the linear elastic shell. An understanding of how the

concrete expands radially is essential to model this behavior. The load vs. longitudinal

strain plots for all four shell types tested are shown in Figure 5-7. It can be seen from

these plots that the thick all hoop shell offers the highest load carrying capacity. The

shells with the helical fibers are capable of carrying much more load in the axial

direction than the all hoop shells as the modulus of the helical shells is almost 15 times

greater than the all hoop shells in the loading direction. This explains why the thick

helical shell response looks very similar to the thin all hoop shell response as the

capacity lost in the concrete core due to more rapid expansion is made up by the load

carrying capability of the shell. The models presented in Chapter 4 base the concrete

modulus on the current expansion or damage state of the concrete. Figure 5-8 shows

the hoop strain vs. the longitudinal strain for the four shell architectures tested. As

expected the all hoop shells do a much better job of restraining the concrete. A

problem was encountered with the helical cylinders that rendered the data collected

from these tests very difficult to use. The hydrostone end caps broke away from the

shell fairly early on in the loading due to the high stiffness of the shells in the

81

longitudinal direction mentioned above. Once the end caps were gone the shells were

then free to expand longitudinally. The longitudinal strain in the specimen as

measured by the linear potentiometers is plotted along with the longitudinal strain as

recorded by the strain gages on the shell in Figure 5-9. Examination of these plots

leads to the conclusion that the desired state of equal longitudinal strain in the shell

and core was lost and that slippage was occurring between the two. A free edge

condition was created at the top and bottom of the shell meaning that any load in the

shell had to be transferred from the concrete core. Insufficient information is available

to assertion how this load transfer took place so these tests were not used in the

development of the analytical models. The stress strain response of the concrete core

is shown in Figure 5-10 for the all hoop shells. These curves are derived by estimating

the axial load in the shells and subtracting it from the total applied load to obtain the

load carried by the concrete core. An elastic perfectly plastic assumption was made for

the all hoop shells in the longitudinal direction with a yield strain of 24.8 MPa (3.6

ksi). The failure of the all hoop cylinders was dramatic with audible fiber failure and

substantial discoloration of the matrix observed before final failure. The hoop strains

at failure were in the range of 0.7% to 0.8%. The thin and thick helical cylinders

demonstrated different failure modes with the thin shell splitting at a 10o angle from

the vertical and the thick shell splitting vertically. Typical shell failures are depicted in

Figure 5-11 and Figure 5-12 for the all hoop and helical cylinders respectively.

82

HYDRASTONE END CAPS

BIAXIAL GAGES FOR LONGITUDINAL ANDHOOP STRAINS

610mm(2�)

152mm(6�)

B

B

B

B

B - BIAXIAL-LONGITUDINAL, HOOPLinearPots

- LINEAR POTS

Figure 5-6 Strain Gage Layout for Circular Compression Specimens


-60000-50000-40000-30000-20000-100000

Lo

ad (

kN)

-3000

-2500

-2000

-1500

-1000

-500

0

Lo

ad (

kip

s)

-600

-500

-400

-300

-200

-100

0

Thin All Hoop ShellThick All Hoop ShellThin Shell With HelicalsThick Shell With Helicals

Figure 5-7 Load vs. Strain Curves for Circular Cylinders

83


-60000-50000-40000-30000-20000-100000

Ho

op

Str

ain

(m

icro

-str

ain

)

0

2000

4000

6000

8000

Thin All HoopThick All HoopThin With HelicalsThick With Helicals

Figure 5-8 Hoop vs. Longitudinal Strains for Circular Cylinders

Load (kN)

-2000-1600-1200-800-4000

Lo

ng

itu

din

al S

trai

n (

mic

ro-s

trai

n)

-20000

-15000

-10000

-5000

0

Load (kips)

-400-350-300-250-200-150-100-500

Thin Shell Linear PotsThin Shell Strain GagesThick Shell Linear PotsThick Shell Strain Gages

Figure 5-9 Longitudinal Strain in Helical Circular Cylinders

84


-40000-30000-20000-100000

Str

ess/

f'c

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Thin All Hoop ShellThick All Hoop Shell

Figure 5-10 Concrete Stress Strain Curves for All Hoop Circular Cylinders

Figure 5-11 Typical Failure of All Hoop Circular Shell

85

Figure 5-12 Failure of Helical Circular Shells

5.1.2.2 Conrec Cylinders

In order to capture the nonconstant hoop strains as described in Chapter 4 a

more complex strain gage layout was required for the conrec specimens as compared

to the circular specimens. Four biaxial gages were used to track the strains in the

center of the flat sides and uniaxial hoop gages were placed as shown in Figure 5-13

to track the varying hoop strains around the corner radius. The outside gages are

placed at the tangent points where the radius meets the flat and the center gage is

placed at the center of the radius.

86

For the conrec specimens the load vs. longitudinal strain behavior is depicted

in Figure 5-14. The pronounced kink in these curves for the conrec sections is

attributed to the rapid expansion of the concrete core that takes place due to the poor

confinement offered by the flat sides. This effect is mitigated as the flat bulges and

begins to better confine the section. The hoop strains in the shells are plotted vs. the

longitudinal strains in Figure 5-15 for the center of the flat side. This plot

demonstrates the rapid expansion in the initial stages of loading. The same problem

with the end-caps was encountered with the conrec helical cylinders as was described

above for the circular sections. Failure of the conrec cylinders was very similar to that

of the circular sections with hoop strains in the 0.4-1.0% range. Typical failures are

shown for the conrec sections in Figure 5-16 and Figure 5-17 for the all hoop and

helical cylinders respectively.

HYDRASTONE END CAPS

GAGES FOR HOOP STRAIN

BIAXIAL GAGES FOR LONGITUDINAL ANDHOOP STRAINS

610mm(2�)

152mm(6�)

B

B

B

B

HH

H

HH

H

B - BIAXIAL, LONGITUDINAL, HOOPH - HOOP

LinearPots

- LINEAR POTS

Figure 5-13 Strain Gage Layout for Conrec Compression Specimens

87


-60000-50000-40000-30000-20000-100000

Lo

ad (

kN)

-3000

-2500

-2000

-1500

-1000

-500

0

Lo

ad (

kip

s)

-600

-500

-400

-300

-200

-100

0


Figure 5-14 Load vs. Longitudinal Strain Conrec Cylinders


-60000-50000-40000-30000-20000-100000

Ho

op

Str

ain

(m

icro

-str

ain

)

0

2000

4000

6000

8000

10000

12000


Gage Location

Figure 5-15 Hoop vs. Longitudinal Strain for Conrec Cylinders

88

Figure 5-16 Typical Failure of All Hoop Conrec Shell

Figure 5-17 Failure of Conrec Helical Shells

89

5.1.3 Bending

Four point bending tests were performed on 2.13m (7 ft) long beams with the

shells listed in Table 5-6. The test setup is shown schematically in Figure 5-18 and

pictured in Figure 5-19. These tests were used to verify the bending behavior

postulated by the analytical models described in Chapter 4. Both circular and conrec

sections were tested. The four point bending test was chosen to give a constant

moment region in the center of the beam and a shear span on the two ends. The load

spacing was chosen to avoid a shear failure outside the constant moment region so the

bending capacity could be determined. The load was applied through 51mm (2 in.)

thick elastomeric pads. This load application method was chosen to distribute the load

and help to prevent a failure due to localized effects. The ends of the beam were

supported on pivots that could not introduce any axial load as they were free to slide in

the longitudinal direction.

Table 5-6 Shells for Small Scale Bending Tests

THICKNESS

SHAPE LAY-UP S.I. U.S.

CIRC [90,±10,±10,90] 2.29mm .09 in.

CIRC [90,±10,±10,90]SYM 4.57mm .18 in.

CONREC [90,±10,±10,90] 2.29mm .09 in.

CONREC [90,±10,±10,90]SYM 4.57mm .18 in.

90

1.83m(6�)

152mm(6�)

406mm (16�)

711mm (28�)

Shear SpanConstant Moment

Region

Figure 5-18 Schematic of Four Point Bending Test Setup

Figure 5-19 Four Point Bending Test on Small Scale Specimen

91

5.1.3.1 Circular Cylinders

The circular bending specimens were instrumented with strain gages to verify

the hoop to longitudinal strain relations and the shear behavior put forth in Chapter 4.

The strain gage layout used is presented in Figure 5-20.

A

A

T T

B

B

T

T

B

B

B

B

B

C

C

B

B

D

D

A-A B-B C-C D-D

12

45 6

7

8

3

L

L

BL

L

457mm18 in

559mm

22 in

51mm2 in

610mm24 in

T - Triaxial - Longitudinal, +45, -45B - Biaxial - Longitudinal, HoopL - Longitudinal

Figure 5-20 Strain Gage Layout for Circular Bending Specimens

The thin circular shell reached a maximum moment of 44.3 kN-m (392 kip-

in.) with a peak center displacement of 48.0mm (1.89 in.) relating to a maximum

compression strain of -0.54% and a maximum tension Strain of 0.99%. The load vs.

center displacement of the specimen is shown in Figure 5-21. The response is nearly

linear. Figure 5-22 and Figure 5-23 show the strain profile across the section in the

constant moment region and shear span respectively. It can be seen from these figures

that the section did not deform as predicted by the plane-sections-remain-plane

assumption. Some of the data presented below indicates that the shell was too thin to

92

distribute the applied load and that extensive local cracking of the concrete under the

load application points led to unpredictable behavior. The analytical models predict a

negative hoop strain in the shell in the tension zone of the specimen which was not

observed for this test. The positive hoop strains in the tension zone may be explained

by the formation of compression struts from the load application points to the lower

section of the shell. The longitudinal strains in the specimen are plotted vs. the applied

moment in Figure 5-24. In these and all following figures in this section the locations

in the shear span are designated with solid symbols and the locations in the constant

moment region are designated with open symbols. These plots show that the

longitudinal strains in the shear span of the specimen for a given moment are higher

than the corresponding strains in the constant moment region which should not be the

case if the specimen behaved as predicted. The next figure (Figure 5-25) plots the

hoop strains in the shell versus the applied moment. It is notable that the hoop strains

in the shear span are all positive and much higher than those seen in the constant

moment region of the specimen. This additional positive hoop strain is believed to be

from the more extensive cracking of the concrete due to shear. The shear strain in the

shell is plotted vs. the applied shear in Figure 5-26. This figure demonstrates the shear

behavior discussed in Section 4.1.3 with the shell initially showing little to no shear

strain and then suddenly increasing shear strains as the concrete reaches its cracking

strain. A pronounced drop in load was seen at a load of 108 kN (24 kips) after which

the specimen began to pick up load again. The specimen failed on the compression

side in the constant moment region.

93

Center Displacement (mm)

0 10 20 30 40

Lo

ad (

kN)

0

20

40

60

80

100

120

140

Center Displacement (in)

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

Lo

ad (

kip

s)

0

5

10

15

20

25

30

Figure 5-21 Load - Displacement Curve for Thin Circular Bending Specimen

Axial Strain (micro-strain)

-6000 -4000 -2000 0 2000 4000 6000 8000

Dis

tan

ce F

rom

Cen

ter

(mm

)

-60

-40

-20

0

20

40

60

Dis

tan

ce F

rom

Cen

ter

(in

)

-3

-2

-1

0

1

2

3

25 kN50 kN75 kN100 kN

Load

25kN=5.62 kips

Figure 5-22 Strain Profile for Thin Circular Section in Constant Moment region

94

Dis

tan

ce f

rom

Cen

terl

ine

(mm

)

-60

-40

-20

0

20

40

60


-2000 -1000 0 1000 2000 3000 4000 5000

Dis

tan

ce f

rom

Cen

terl

ine

(in

)

-3

-2

-1

0

1

2

3

25kN50kN75kN100kN

Load

25kN=5.62 kips

Figure 5-23 Strain Profile for Thin Circular Section in Shear Area

Moment (kN-m)

0 5 10 15 20 25 30 35 40 45 50 55

Lo

ng

itu

din

al S

trai

n (

mic

ro-s

trai

n)

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

12000

Moment (kip-in)

0 100 200 300 400 500

a1la2la3la5la6la7lc1lc2lc3lc5lc6lc7l

a c1 2

3

5

7

6

Figure 5-24 Longitudinal Strain vs. Moment For Thin Circular Specimen

95

Moment (kN-m)

0 5 10 15 20 25 30 35 40 45 50 55

Ho

op

Str

ain

(m

icro

-str

ain

)

-500

0

500

1000

1500

2000

2500

3000

Moment (kip-in)

0 50 100 150 200 250 300 350 400 450 500

a1ha2ha3ha5ha6ha7hc1hc2hc3hc5hc6hc7h

a c

1 2

3

5

7

6

Figure 5-25 Hoop Strain vs. Moment For Thin Circular Specimen

Shear (kN)

0 10 20 30 40 50 60 70

Sh

ear

Str

ain

(m

icro

-str

ain

)

-2000

0

2000

4000

6000

8000

10000

Shear (kips)

-4 -2 0 2 4 6 8 10 12 14

a2sa3sa6sa7s

a1 2

3

5

7

6

Figure 5-26 Shear Strain vs. Shear For Thin Circular Specimen

96

The thick circular section achieved a maximum moment of 102 kN-m (899

kip-in.) with the maximum tension and compression strains both in excess of 1.0%.

The load displacement behavior of the thick circular specimen is described in Figure

5-27. More nonlinear behavior is apparent in this specimen than in the thin shell. It can

be seen from Figure 5-28 and Figure 5-29 that for the thick shell the plane sections

remain plane assumption was justified. The longitudinal strains shown in Figure 5-30

behave as predicted in the shear span and in the constant moment region. The hoop

strains in the constant moment region shown in Figure 5-31 behaved as predicted by

the analytical models although the increased positive hoop strains seen in the shear

span are at this time unaccounted for in the modeling. The shear strains are shown in

Figure 5-32. The bilinear approximation does not match as well as for the thin shell

although it is still a fair prediction of the behavior (see Chapter 6). The failure of the

thick circular section is pictured in Figure 5-33.

97


0 10 20 30 40 50

Lo

ad (

kN)

0

50

100

150

200

250

300


0.0 0.5 1.0 1.5

Lo

ad (

kip

s)

0

10

20

30

40

50

60

Figure 5-27 Load - Displacement Curve for Thick Circular Bending Specimen


-10000 -5000 0 5000 10000

Dis

tan

ce F

rom

Cen

ter

(mm

)

-60

-40

-20

0

20

40

60

Dis

tan

ce F

rom

Cen

ter

(in

)

-3

-2

-1

0

1

2

3

25 kN50 kN75 kN100 kN125 kN150 kN 175 kN200 kN225 kN250 kN

Load

25kN=5.62 kips

Figure 5-28 Strain Profile for Thick Circular Section in Constant Moment region

98


-4000 -2000 0 2000 4000 6000

Dis

tan

ce f

rom

Cen

terl

ine

(mm

)

-60

-40

-20

0

20

40

60

Dis

tan

ce f

rom

Cen

terl

ine

(in

)

-3

-2

-1

0

1

2

3

25kN50kN75kN100kN125kN150kN175kN200kN225kN250kN

Load

25kN=5.62 kips

Figure 5-29 Strain Profile for Thick Circular Section in Shear Span

Moment (kN-m)

0 10 20 30 40 50 60 70 80 90 100 110 120 130

Lo

ng

itu

din

al S

trai

n (

mic

ro-s

trai

n)

-15000

-12500

-10000

-7500

-5000

-2500

0

2500

5000

7500

10000

12500

15000

Moment (kip-in)

0 200 400 600 800 1000 1200

a1la2la3la5la6la7lc1lc2lc3lc5lc6lc7l

a c1 2

3

5

7

6

Figure 5-30 Longitudinal Strain vs. Moment For Thick Circular Section

99

Moment (kN-m)

0 10 20 30 40 50 60 70 80 90 100 110 120 130

Ho

op

Str

ain

(m

icro

-str

ain

)

-2000

-1000

0

1000

2000

3000

4000

Moment (kip-in)

0 200 400 600 800 1000 1200

a1ha2ha3ha5ha7hc1hc2hc3hc5hc6hc7h

a c1 2

3

5

7

6

Figure 5-31 Hoop Strain vs. Moment For Thick Circular Section

Shear (kN)

0 20 40 60 80 100 120 140 160

Sh

ear

Str

ain

(m

icro

-str

ain

)

-2000

0

2000

4000

6000

8000

10000

Shear (kips)

0 5 10 15 20 25 30 35

a2sa3sa7s

a1 2

3

5

7

6

Figure 5-32 Shear Strain vs. Applied Shear For Thick Circular Specimen

100

Figure 5-33 Failure of Thick Circular Section

5.1.3.2 Conrec Cylinders

The cross section geometry of the conrec section is shown in Figure 5-1. A

strain gage layout similar to that used for the circular bending specimens was used for

the conrec sections. Hoop gages were placed at the tangent points of the radius and at

the center of the radius in the constant moment region to characterize the hoop strain

distribution around the section. The strain gage layout for the conrec bending

specimens is shown in Figure 5-34.

The thin conrec section failed at a moment of 54.4 kN-m (482 kip-in.) with a

peak center displacement of 31.0mm (1.22 in.) (see Figure 5-35) corresponding to a

maximum compression strain of -0.56% and a maximum tension strain of 0.76%. The

101

A

A

C

C

B

B

D

D

A-A B-B C-C D-D

12

45

6 7 8

3 L

L910

B

B

T

T T

T

B

B

HHHB

H HHB

L

L

457mm18 in

559mm22 in

51mm2 in

610mm24 in

T - Triaxial - Longitudinal, +45, -45B - Biaxial - Longitudinal, HoopL - LongitudinalH - Hoop

Figure 5-34 Strain Gage Layout for Conrec Bending Specimens

section behaved well and showed no signs of section warpage as can be seen in Figure

5-36 and Figure 5-37. The longitudinal strains responded the same in the shear and

constant moment spans as shown in Figure 5-38. As was seen in the circular sections

the hoop strains in the shear span are higher and much more erratic than the

corresponding hoop strains in the constant moment region (see Figure 5-39), again

this is attributed to the additional dilation due to shear cracking. The hoop strain in the

center of the flat on the compression side of the member is seen to decrease at the

higher load levels. The shear strains exhibit the bilinear behavior postulated in Chapter

4 and as expected the peak shear strains are found near the centerline of the shell as

shown in Figure 5-40. The thin conrec specimen failed on the compression side in the

constant moment region as seen in Figure 5-41.

102


0 5 10 15 20 25 30 35

Lo

ad (

kN)

0

20

40

60

80

100

120

140

160

180


0.0 0.5 1.0

Lo

ad (

kip

s)

0

5

10

15

20

25

30

35

40

Figure 5-35 Load - Displacement Curve for Thin Conrec Bending Specimen


-6000 -4000 -2000 0 2000 4000 6000 8000

Dis

tan

ce f

rom

Cen

terl

ine

(in

)

-3

-2

-1

0

1

2

3

Dis

tan

ce f

rom

Cen

terl

ine

(mm

)

-60

-40

-20

0

20

40

60

25kN50kN75kN100kN125kN150kN

Load

25kN=5.62

Figure 5-36 Strain Profile for Thin Conrec Section in Constant Moment region

103


-2000 -1000 0 1000 2000 3000 4000

Dis

tan

ce f

rom

Cen

terl

ine

(in

)

-3

-2

-1

0

1

2

3

Dis

tan

ce f

rom

Cen

terl

ine

(mm

)

-60

-40

-20

0

20

40

60

25kN50kN75kN100kN125kN150kN

Load

25kN=5.62 kips

Figure 5-37 Strain Profile For Thin Conrec Section in Shear Area

Moment (kN-m)

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Lo

ng

itu

din

al S

trai

n (

mic

ro-s

trai

n)

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

Moment (kip-in)

0 100 200 300 400 500 600

a1la3la5la6la8la10lc1lc5lc6lc10l

a c

1

5

6

10

234

98

7

Figure 5-38 Longitudinal Strain vs. Moment For Thin Conrec Specimen

104

Moment (kN-m)

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Ho

op

Str

ain

(m

icro

-str

ain

)

-400

-200

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Moment (kip-in)

0 100 200 300 400 500 600

a1ha3ha5ha6ha8ha10hc1hc2hc3hc4hc5h

a c1

5

6

10

234

98

7

Figure 5-39 Hoop Strain vs. Moment For Thin Conrec Specimen

Shear (kN)

0 20 40 60 80

Sh

ear

Str

ain

(m

icro

-str

ain

)

0

2000

4000

6000

8000

10000

12000

Shear (kips)

0 2 4 6 8 10 12 14 16 18 20

a3sa5sa8sa10s

a

1

5

6

10

234

98

7

Figure 5-40 Shear Strain vs. Applied Shear For Thin Conrec Specimen

105

Figure 5-41 Failure of Thin Conrec Section

The thick conrec section exhibited similar behavior to the thin conrec

specimen. The thick conrec section failed at a moment of 135 kN-m (1200 kip-in.)

corresponding to a maximum compression strain of -0.88% and a maximum tension

strain of 1.0%. A slight loss of load capacity was seen at 267 kN (60 kips) as shown in

Figure 5-42. The center displacement was greater than the capacity of the measuring

device which only had a clearance of approximately 38mm (1.5 in.). The strain

profiles presented in Figure 5-43 and Figure 5-44 show that the plane-sections-

remain-plane assumption held for this specimen. As with the previous tests the

longitudinal strains responded as expected (see Figure 5-45) and the hoop strains in

the shear span were much higher than those in the constant moment region (Figure

106

5-46). Again the bilinear nature of the shear response was observed as shown in

Figure 5-47. The thick conrec section as all of the other bending specimens failed in

the constant moment region on the compression side of the shell. The failed specimen

is pictured in Figure 5-48.


0 10 20 30 40

Lo

ad (

kN)

0

100

200

300


0.0 0.5 1.0 1.5

Lo

ad (

kip

s)

0

20

40

60

Figure 5-42 Load-Displacement Response for Thick Conrec Specimen

107


-10000 -5000 0 5000 10000 15000

Dis

tan

ce f

rom

Cen

ter

(mm

)

-60

-40

-20

0

20

40

60

Dis

tan

ce F

rom

Cen

ter

(in

)

-3

-2

-1

0

1

2

3

25kN50kN75kN100kN125kN150kN175kN200kN225kN250kN275kN300kN325kN350kN375kN

Load

25kN=5.62kips

Figure 5-43 Strain Profile for Thick Conrec Section in Constant Moment region


-4000 -2000 0 2000 4000 6000

Dis

tan

ce f

rom

Cen

ter

(in

)

-3

-2

-1

0

1

2

3

Dis

tan

ce f

rom

Cen

ter

(mm

)

-60

-40

-20

0

20

40

60 25kN50kN75kN100kN125kN150kN175kN200kN225kN250kN275kN300kN325kN350kN375kN

Load

25kN=5.62kips

Figure 5-44 Strain Profile For Thick Conrec Section in Shear Area

108

Moment (kN-m)

0 20 40 60 80 100 120 140 160

Lo

ng

itu

din

al S

trai

n (

mic

ro-s

trai

n)

-10000

-5000

0

5000

10000

15000

Moment (kip-in)

0 200 400 600 800 1000 1200 1400

a1la3la5la6la8la10lc1lc5lc6lc10l

a c

1

5

6

10

234

98

7

Figure 5-45 Longitudinal Strain vs. Moment for Thick Conrec

Moment (kN-m)

0 20 40 60 80 100 120 140 160

Ho

op

Str

ain

(m

icro

-str

ain

)

-2000

-1000

0

1000

2000

3000

4000

Moment (kip-in)

0 200 400 600 800 1000 1200 1400

a1ha3ha5ha6ha8ha10hc1hc2hc3hc4hc7hc10h

a c

1

5

6

10

234

98

7

Figure 5-46 Hoop Strain vs. Moment for Thick Conrec Section

109

Shear (kN)

0 50 100 150 200

Sh

ear

Str

ain

(m

icro

-str

ain

)

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

Shear (kips)

0 10 20 30 40

a3sa5sa8sa10s

a

1

5

6

10

234

98

7

Figure 5-47 Shear Strain vs. Applied Shear for Thick Conrec Specimen

Figure 5-48 Failure of Thick Conrec Section

110

5.2 Full Scale Bending Tests

Five four point bending tests were conducted on 345mm (14 in.) diameter

shells. Two shells were tested hollow to determine the accuracy of the stiffness

predicted utilizing classic lamination theory as described in Chapter 3. Two shells

were also tested filled with concrete and one shell was tested filled, with an overlaid

cast in place concrete deck. The test setup is pictured in Figure 5-49 and shown

schematically in Figure 5-50.

Figure 5-49 Full Scale Four Point Bending Test

111

7.92m

2.44m

FIXED SUPPORT PINNED SUPPORTCARBON SHELL

ENDBLOCK

2.74m(26�0�)

(8�0�) (9�0�)

Figure 5-50 Schematic of Full Scale Bending Tests

The tests were designed to put the specimen in four point bending with no

introduction of axial load. This was accomplished by casting a steel reinforced

concrete end block onto each end of the shell with an inset pipe as shown in Figure 5-

51. The endblock to tube connection was accomplished with steel bars that penetrated

914mm (36 in.) into the shell from the end of the endblock. These endblocks were

then supported on pin supports one of which was free to rotate at its base. The pivoting

pin support allows for axial deformation which eliminates the introduction of axial

load into the test specimen.

112

Figure 5-51 Endblock for Support of Bending Test Specimens

The construction of these test specimens was done in a way to develop a

construction technique that could be duplicated in the field. The idea was to pump the

tubes full with lightweight concrete in the horizontal position. To accomplish this a

diaphragm was required to seal the end of the shell through which the concrete could

be pumped (Figure 5-52). This diaphragm supported the connection cage as well as

the inlet for pumping the shell. The construction sequence used on the test specimens

was as follows. The steel cage for the endblock to tube connection was tied with the

diaphragm placed on the bars to seal the end of the shell (Figure 5-53). This cage was

then fitted into the shell so the diaphragm was approximately 51mm (2 in.) into the

shell. A pin was inserted through small holes in the end of the shell to hold the

connection cage in place (Figure 5-54). The endblock was designed so that the main

113

part of the endblock steel cage was tied leaving a gap in the front so the shell could be

dropped straight in from above. This part of the cage for the endblock was assembled

and placed inside the endblock casting forms (Figure 5-55). The shell with the

connection cages in place was then drooped into the casting forms from above (Figure

5-56). The pipe for filling the shell was then placed through the endblock cage (Figure

5-57). The top of the endblock cage and forms were then completed (Figure 5-58).

The endblocks were cast with conventional concrete leaving the pipe protruding from

the top. After the endblock had sufficiently cured the shell was pumped full with

lightweight concrete.

Figure 5-52 Shell End Diaphragm

114

Figure 5-53 Steel Connection Cage

Figure 5-54 Shell With Connecting Cage

115

Figure 5-55 Endblock Lower Section

Figure 5-56 Placement of Shell Into Endblocks

116

Figure 5-57 PVC Pipe for Pumping Shell

Figure 5-58 Completed End Block Form

117

Two composite architectures were used for the hollow and filled tube tests.

The lay-ups used are listed in Table 5-7, the vendor supplied ply properties are given

in

Table 5-8 and the equivalent plate properties derived from lamination theory are listed

in Table 5-9

Table 5-7 Composite Architectures for Large Scale Tests

THICKNESS

# LAY-UP S.I. U.S.

1 [90,±102,90,±102,90]sym 9.65mm .38 in.

2 [902,±102,902,±102,902,±10,902,±10,903] 8.89mm .35 in.

Table 5-8 Vendor Supplied Ply Properties

S.I. U.S.

E11 121 GPa 17.5 Msi

E22 6.90 GPa 1.0 Msi

G12 4.83 GPa 0.70 Msi

G23 2.62 GPa 0.38 Msi

ν12 0.30 0.30

ν23 0.40 0.40

118

Table 5-9 Equivalent Plate Properties for Large Scale Tests

Lay-Up #1 Lay-Up #2

S.I. U.S. S.I. U.S.

EL 97.2 GPa 14.1 msi 80.67 GPa 11.7 msi

EH 25.1 GPa 3.64 msi 42.9 GPa 6.22 msi

GLH 7.38 GPa 1.07 msi 6.89 GPa 1.00 msi

νLH 0.184 0.184 0.097 0.097

5.2.1 Concrete Properties

The shells used in these tests were pumped full with a lightweight concrete

pumpmix. A man-made lightweight aggregate with a maximum size of 9.5mm (3/8

in.) supplied by Pacific Custom Materials was used along with an expansive agent

(Interplast-N) supplied by Sika Corporation. Finally plastisizers were added before

pumping to ensure that the mix would flow easily through the shells. Two separate

concrete batches were used for the large scale test program. The first batch was used

for the first filled shell test and the beam and slab test. The second batch was used for

the second filled shell test. The first batch did not achieve the nominal design strength

of 27.6 MPa (4 ksi). This was due to the fact that insufficient plastisizer was added

and additional water was used to get the mix to flow. Table 5-10 shows the concrete

mix used and the proportions of the additives. The measured properties of both

batches are given in Table 5-11.

119

Table 5-10 Concrete Mix Used for Filled Shells

Material Loose Volume

Batch Weight

(lb.)

Absolute Volume

(ft3) Cement, Type 2 7.5 sacks 705 3.59

Washed Concrete Sand, SSD - 1586 9.52 Baypor F-60, SSD @ 55.0 lb/ft3 13.0 ft3 715 7.39 Active Water 39.0 Gals. 325 5.21 Absorbed Water, Maximum 3.0 Gals. - - Total Water Allowed 42.0 Gals. - - WRDA 79 @ 5 fl. oz./cwt. - 32.9 fl. oz. - Daravair @ 0.5 fl. oz./cwt. 4% air 3.3 fl. oz. 1.08 Interplast-N @ 1.0% per cwt. 0.8% air 6.6 0.21 Plastisizer (ADVA) 3 fl. oz./cwt.

- 23 oz. -

*Target slump 8-9 in.

Table 5-11 Concrete Properties for Filled Shell Tests

Batch 1 Batch 2 f�c 20.7 MPa

(3.0 ksi) 31.0 MPa (4.5 ksi)

Ec

o

15.4 GPa (2.23 msi)

18.8 GPa (2.73 msi)

εco .003 .003

5.2.2 Hollow Shell

The hollow shell test specimens were fitted with endblocks the same as the

filled specimens except the location of the diaphragm was moved 762mm (30 in.) into

the shell to provide a sufficient bond length for the steel connection cage.

120

Two hollow tests were performed with composite architectures as listed in

Table 5-7. The structural stiffness was compared to analytical predictions based on the

individual ply properties. Due to the susceptibility of the unfilled shell to local stress

concentrations and bending effects (ovalization) at the load introduction points it was

decided to do nondestructive tests with loads chosen to avoid local failure.

Displacement readings were taken at 5 points along the top surface of the shell

and one point on the bottom surface. Rotation of the specimen was monitored with

rotational potentiometers at the south end. The location and designation of these

devices is shown in Figure 5-59. Strain gages were also placed on the specimen to

monitor the bending and shear strains in the shell. The location and designation of

these gages is also shown in Figure 5-59.

The load displacement curves for the two shells is given in Figure 5-60.

Figure 5-61 and Figure 5-62 show the strains at the maximum compression and

tension fibers in the constant moment region as well as the shear strain on the neutral

axis for both tests. All values correlated well with the predictions based on classic

lamination theory and the vendor supplied ply properties. The second shell was

substantially softer as expected. It is notable that the shear stiffness does not change

much between the two lay-ups as can be seen in Figure 5-62.

121

A

A

A-AB-B

B

B

B1

B2

A

1.22m(4'-0")

914mm(3'-0")

DESIGNATIONSSTRAIN GAGE

B1LLOCATION ORIENTATION

L-LONGITUDINALH-HOOPP- +45M- -45

1.22m(4'-0")

2.13m(7'-0")

T 1 T 2 T 3 T 4 T 5

B 2

610mm(2'-0")

2.44m(8'-0")610mm

(2'-0")

ROT_L

ROT_T

SUP 1 SUP 2

1.22m(4'-0")

1.22m(4'-0")

914mm(3'-0")

*ALL LINEAR POTS HAVE A DISP PREFIX (I.E. DISPT2)

Figure 5-59 Instrumentation Layout for Hollow Shell Test

Displacement (mm)

0 10 20 30 40 50

Lo

ad (

kN)

0

10

20

30

40

50

Displacement (in)

0.0 0.5 1.0 1.5

Lo

ad (

kip

s)

0

2

4

6

8

10Lay-Up #1Lay-Up #2

Figure 5-60 Load Displacement Curve for Hollow Shell Tests

122

Moment (kN-m)

0 20 40 60 80 100 120

Str

ain

(m

icro

-str

ain

)

-1500

-1000

-500

0

500

1000

1500

Moment (kip-in)

0 200 400 600 800 1000b1l_#1b1h_#1b2l_#1b2h_#1b1l_#2b1h_#2b2l_#2b2h_#2

b

1

2

Figure 5-61 Longitudinal and Hoop Strains for Hollow Shells

Shear (kN)

0 10 20 30 40 50

Sh

ear

Str

ain

(m

icro

-str

ain

)

0

200

400

600

800

1000

1200

Shear (kips)

0 2 4 6 8 10

Lay-Up #1Lay-Up #2

Gage Location

Figure 5-62 Shear Strains for Hollow Shell Tests

123

5.2.3 Concrete Filled Shells

Two carbon shells were tested filled with a 27.5 MPa (4 ksi) lightweight

concrete pump mix. These tests were used to verify the analytical modeling done to

predict the stress and strain state in the shell due to the bending loads along with the

expansion of the enclosed concrete. Two different composite lay-ups were used as

listed in Table 5-7. The ply properties are listed in

Table 5-8 and the equivalent plate properties are listed in Table 5-9.

Displacement readings were taken at 5 points along the top surface of the

shells. The location and designation of these linear potentiometers is shown in Figure

5-63 for lay-up #1 and Figure 5-59 for lay-up #2. Strain gages were also placed on the

specimens to monitor the bending and shear strains in the shell. The location and

naming convention of these gages is shown in Figure 5-64.

D1 D2 D3 D4 D5

1.22m(4'-0") (2'-8")

784mm 1.32m(4'-4")

R1 R2

SD1 SD2

610mm(2'-0")

610mm(2'-0")

784mm(2'-8")

1.32m(4'-4")

Figure 5-63 Displacement Instrumentation For Filled Shell Test

124

Although the girder alone is not representative of a beam and slab system a

load was chosen that roughly represented a service bending demand on the filled shells

(51.2 kN (11.5 kips) per actuator). The specimens were cycled three times at this load

to see if any degradation was noted in the system stiffness. None was observed on

either test unit. The specimens were then loaded to failure.

A

A C

C D

D

A-A

12

3

4

567

8

C-CB-B

B

B

D-D

1

2

1.52m(5'-0")

1.37m(4'-6")

457mm(1'-6") 1.83m

(6'-0")

DESIGNATION

LONGITUDINALLOCATION

A4P

RADIAL LOCATION

ORIENTATIONL - LONGITUDINALH - HOOPP - +45M - -45

Figure 5-64 Strain Gage Locations and Designation for Filled Tube Tests

5.2.3.1 Shell #1

The lay-up used for the first bending test is shown in Table 5-7. This lay-up

yields a radius to thickness ratio of 18.2 with 84% of the fibers in the helical direction.

Failure occurred on the compression side of shell #1 in the constant moment region at

a longitudinal compression strain of ~0.55%. This value is approximately 46% of the

125

theoretical allowable strain based on a first ply failure model (1.2%). The tension

strain at failure was approximately 0.75%. It is felt that the single layer of hoop fibers

on the exterior of the shell was insufficient to keep the ±10o fibers from buckling

outward. It must be noted however that once the carbon girder is integrated into the

beam and slab system the shell sees no compression loading under service or factored

loadings (see filled tube with slab test).

The load vs. displacement curve for the first filled shell test is presented in

Figure 5-65. The nonlinear behavior due to the concrete behavior is more apparent

than for the small scale specimens. The analytical models used to predict the shell

strains are based on a plane sections remain plane assumption. From Figure 5-66 and

Figure 5-67, which show the strain profile across the section in the constant moment

and shear spans of the specimen respectively, it can be seen that this assumption is

justified. Figure 5-68 and Figure 5-69 show the longitudinal strains from the

compression and tensions sides of the shell plotted against the moment in the

specimen at the specific gage location. The behavior of the longitudinal strains in the

two spans is essentially the same as would be expected. The hoop strains are shown

versus the moment in Figure 5-70 and Figure 5-71. Here as in the small scale bending

tests the increased hoop strains in the shear span are seen especially on the tension side

of the shell where the cracking of the concrete core would be most pronounced. Shear

strains in the shear span of the specimen are shown versus the applied shear in Figure

5-72. The maximum shear strains were seen close to the centerline of the shell. The

gradual increase of the shear strains from the initiation of the loading is consistent

126

with the shear model presented in Chapter 4 for a shell with sufficient shear transfer

mechanisms to force the shell and core to deform as a unit. The failed shell is pictured

in Figure 5-73.


0 50 100 150 200 250

Lo

ad P

er A

ctu

ato

r (k

N)

0

50

100

150

200

250

300


0 2 4 6 8

Lo

ad P

er A

ctu

ato

r (k

ips)

0

10

20

30

40

50

60

Figure 5-65 Load Displacement Plot for Filled Shell Test #1

127

Strain (micro-strain)

-8000 -6000 -4000 -2000 0 2000 4000 6000 8000

Dis

tan

ce F

rom

Sh

ell C

ente

rlin

e (m

m)

-200

-150

-100

-50

0

50

100

150

200

Dis

tan

ce F

rom

Sh

ell C

ente

rlin

e (i

n)

-6

-4

-2

0

2

4

6

25 kN50 kN75 kN100 kN125 kN150 kN175 kN200 kN225 kN250 kN

Load

25kN=5.62 kips

Figure 5-66 Strain Profile in Constant Moment Section, Filled Shell Test #1


-6000 -4000 -2000 0 2000 4000 6000

Dis

tan

ce F

rom

Sec

tio

n C

ente

rlin

e (i

n)

-6

-4

-2

0

2

4

6

Dis

tan

ce F

rom

Sec

tio

n C

ente

rlin

e (m

m)

-200

-150

-100

-50

0

50

100

150

200


Load

25kN=5.62 kips

Figure 5-67 Strain Profile in Shear Section, Filled Shell Test #1

128

Moment (kN-m)

0 100 200 300 400 500 600 700

Lo

ng

itu

din

al S

trai

n (

mic

ro-s

trai

n)

-6000

-5000

-4000

-3000

-2000

-1000

0

Moment (kip-in)

0 1000 2000 3000 4000 5000 6000

a1a2a3c1c2c3

a c1 2

3

57

6

48

Figure 5-68 Longitudinal Strain vs. Moment in Comp. Zone For Filled Shell #1

Moment (kN-m)

0 100 200 300 400 500 600 700

Lo

ng

itu

din

al S

trai

n (

mic

ro-s

trai

n)

0

1000

2000

3000

4000

5000

6000

7000

Moment (kip-in)

0 1000 2000 3000 4000 5000 6000

a4a5a6a7a8c4c5c6c7c8

a c1 2

3

57

6

48

Figure 5-69 Longitudinal Strain vs. Moment in Tension Zone For Filled Shell #1

129

Moment (kN-m)

0 100 200 300 400 500 600 700

Ho

op

Str

ain

(m

icro

-str

ain

)

0

200

400

600

800

1000

1200

1400

Moment (kip-in)

0 1000 2000 3000 4000 5000 6000

a1a2a3c1c2c3

a c1 2

3

57

6

48

Figure 5-70 Hoop Strain vs. Moment in Compression Zone For Filled Shell #1

Moment (kN-m)

0 100 200 300 400 500 600 700

Ho

op

Str

ain

(m

icro

-str

ain

)

-600

-400

-200

0

200

400

600

Moment (kip-in)

0 1000 2000 3000 4000 5000 6000

a4a5a6a8c4c5c6c7c8

a c1 2

3

57

6

48

Figure 5-71 Hoop Strain vs. Moment in Tension Zone For Filled Shell #1

130

Shear (kN)

0 50 100 150 200 250

Sh

ear

Str

ain

(m

icro

-str

ain

)

0

500

1000

1500

2000

2500

3000

Shear (kips)

0 10 20 30 40 50

a2a3a4a6a8

a c1 2

3

57

6

48

Figure 5-72 Shear Strain vs. Applied Shear For Filled Shell #1

Figure 5-73 Failure of Filled Shell #1

131

5.2.3.2 Shell #2

The lay-up for the second shell is shown in Table 5-7. This lay-up yields a

radius to thickness ratio of 19.8 with 69% helical fibers. The lay-up used for the

second concrete filled shell bending test had more hoop fibers on the outer surface to

see if the buckling failure encountered in the first bending test could be mitigated. The

shell failed on the compression side in the constant moment region just inside one of

the actuators. The ultimate compression strain recorded was 0.83% which is a

substantial improvement over the 0.55% seen on the first shell. The load-displacement

curve for the second filled shell is shown in Figure 5-74. The ultimate center

displacement was 305mm (12 in.). The nonlinear response is more pronounced in this

specimen due to the increased contribution to the stiffness from the concrete core. The

two shells are compared along with the analytical predictions in Chapter 6. The strain

profiles across the section in both the constant moment and shear spans are not as

straight as for the shell in the first bending test as can be seen in Figure 5-75 and

Figure 5-76. The additional hoop fibers on the exterior of the shell produced a

relatively thick soft layer which may have contributed to the observed section

warpage. It also led to difficulties with the strain gages as is seen in the following

plots. The longitudinal strains on the compression and tension sides of the shell are

plotted vs. the moment in Figure 5-77 and Figure 5-78 respectively. The behavior

was as expected with the exception of the peak tension strain recorded in the constant

moment region which began to increase faster than expected in the later stages of

loading and eventually reached a strain of 1.4%. It is felt that this phenomena is

132

attributable to matrix cracking in the relatively soft layer on the outside of the shell in

the longitudinal direction (see Chapter 6). The hoop strain followed the trend of the

previous tests with the increased hoop strains once more observed in the shear span as

shown in Figure 5-79 and Figure 5-80 for the compression and tension sides of the

shell respectively. Figure 5-81 plots the shear strains vs. the applied shear load for the

second filled shell bending test. The maximum shear strains have moved away from

the centerline of the shell as compared to the first filled shell test. Again the response

is smooth and the behavior seems to correlate well with the second shear model

presented in Chapter 4. The compression failure of the shell is shown in Figure 5-82.


0 50 100 150 200 250 300 350

Lo

ad P

er A

ctu

ato

r (k

N)

0

50

100

150

200

250

300


0 2 4 6 8 10 12

Lo

ad P

er A

ctu

ato

r (k

ips)

0

10

20

30

40

50

60

Figure 5-74 Load Displacement Plot for Filled Shell Test #2

133


-10000 -5000 0 5000 10000 15000

Dis

tan

ce F

rom

Sec

tio

n C

ente

rlin

e (i

n)

-6

-4

-2

0

2

4

6

Dis

tan

ce F

rom

Sec

tio

n C

ente

rlin

e (m

m)

-200

-150

-100

-50

0

50

100

150

200


Load

25kN=5.62 kips

Figure 5-75 Strain Profile in Constant Moment Section, Filled Shell Test #2


-6000 -4000 -2000 0 2000 4000 6000 8000

Dis

tan

ce F

rom

Sh

ell C

ente

rlin

e (m

m)

-200

-150

-100

-50

0

50

100

150

200

Dis

tan

ce F

rom

Sh

ell C

ente

rlin

e (i

n)

-6

-4

-2

0

2

4


Load

25kN=5.62 kips

Figure 5-76 Strain Profile in Shear Section, Filled Shell Test #2

134

Moment (kN-m)

0 100 200 300 400 500 600 700

Lo

ng

itu

din

al S

trai

n (

mic

ro-s

trai

n)

-9000

-8000

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

Moment (kip-in)

0 1000 2000 3000 4000 5000 6000

a1a2a3c1c2c3

a c1 2

3

57

6

48

Figure 5-77 Longitudinal Strain vs. Moment in Comp. Zone For Filled Shell #2

Moment (kN-m)

0 100 200 300 400 500 600 700

Lo

ng

itu

din

al S

trai

n (

mic

ro-s

trai

n)

0

1500

3000

4500

6000

7500

9000

10500

12000

13500

15000

Moment (kip-in)

0 1000 2000 3000 4000 5000 6000


a c1 2

3

57

6

48

Figure 5-78 Longitudinal Strain vs. Moment in Tension Zone For Filled Shell #2

135

Moment (kN-m)

0 100 200 300 400 500 600 700

Ho

op

Str

ain

(m

icro

-str

ain

)

0

200

400

600

800

1000

1200

1400

1600

1800

Moment (kip-in)

0 1000 2000 3000 4000 5000 6000

a1a2a3c1c2c3

a c1 2

3

57

6

48

Figure 5-79 Hoop Strain vs. Moment in Compression Zone For Filled Shell #2

Moment (kN-m)

0 100 200 300 400 500 600 700

Ho

op

Str

ain

(m

icro

-str

ain

)

-800

-600

-400

-200

0

200

400

600

800

Moment (kip-in)

0 1000 2000 3000 4000 5000 6000


a c1 2

3

57

6

48

Figure 5-80 Hoop Strain vs. Moment in Tension Zone For Filled Shell #2

136

Shear (kN)

0 50 100 150 200 250 300

Sh

ear

Str

ain

(m

icro

-str

ain

)

0

500

1000

1500

2000

2500

3000

3500

Shear (kips)

0 10 20 30 40 50 60

a2a3a4a6a7a8

a c1 2

3

57

6

48

Figure 5-81 Shear Strain vs. Applied Shear For Filled Shell #2

Figure 5-82 Failure of Filled Shell #2

137

5.2.4 Concrete Filled Shell with integral Concrete Deck

The final test in this series consisted of a carbon girder filled with lightweight

concrete overlaid by a normal weight concrete slab connected through steel shear

dowels. This test verified the bending stiffness of the beam and slab system [3] as well

as demonstrated the efficacy of the shear connections [5]. The carbon shell was

prepared identical to the filled beam tests with the exception of the placement of steel

shear dowels to attach the concrete deck. The lay-up of the shell was the same as the

first bending test (lay-up #1) as shown in Table 5-7. A pair of 38mm (1.5 in.) holes

was drilled 102mm (4 in.) apart every 610mm (2 ft) along the shell. A pair of #6 bars

were placed into these holes prior to pumping the shell (see Figure 5-85).

Displacements were monitored at 5 points along the bottom surface of the shell

along with relative displacements between the deck and shell and endblock rotations.

The location of these potentiometers is shown in Figure 5-83. Strain gages were

placed on the shell as shown in Figure 5-84. One dowel of each pair was instrumented

with two strain gages, one on the north side and one on the south side of the bar at the

interface between the shell and the deck as shown in Figure 5-85. The top mat of steel

was gaged to study the strain profile across the section. These gage locations as well

as four displacement devices and one concrete gage are shown in Figure 5-86. Strain

gages were utilized to monitor the effective stress concentration around the

penetrations in the shell for the shear dowels. The gage layout and designation for

these gages is shown in Figure 5-87. Two locations on the top surface of the shell

were gaged, one in the shear span around the set of dowels 1.52m (5 ft) from the north

138

abutment (global location 1) and one in the constant moment region 3.35m (11 ft)

from the north abutment (global location 2).

SL1 SL2 SL3 SL4 SL5 SL6 SL7

D1 D2 D3 D4 D51.22m(4'-0")

1.32m(4'-4") 813mm

(2'-8")

305mm(1'-0")

610mm(2'-0") TYP 1.52m

(5'-0")

SUP-N SUP-S

ROT-N ROT-S

610mm(2'-0")

610mm(2'-0")

813mm(2'-8")

1.32m(4'-4")

Figure 5-83 Displacement Instrumentation for Filled Shell With Slab Test

A

A C

C D

D

A-A

12

34

567

8

C-C

5B-B

B

B

1

5D-D

1.83m(6'-0")

1.37m(4'-6")

1'-6" 1.22m(4'-0")

DESIGNATION

LONGITUDINALLOCATION

A4P

RADIAL LOCATION

ORIENTATIONL - LONGITUDINALH - HOOPP - +45M - -45

Figure 5-84 Strain Gage Locations and Designation for Shell

457mm

139

1 2 3 4 5 6 7 8 9 10 11

N

J3SJOINT

LOCATION

NORTHSOUTH

DESIGNATION

GAGE LOCATION305mm(1'-0") 610mm

(2'-0")TYP

Figure 5-85 Strain Gages Placed on Shear Connection Dowels

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

LOCATION

BA

R #

610mm(2'-0")TYP

457m

m(1

'-6")

TYP

-STEEL STRAIN GAGE LOCATION-CONCRETE STRAIN GAGE LOCATION

305mm(1'-0")

-VERTICAL DISPLACEMENT TRANSDUCERS* ALL TRANSDUCERS HAVE PREFIX DS (I.E. DSNE)

76.2mm(3") TYP

76m

m(3

") T

YP

SE

SW

NE

NW

SDC

DECK OUTLINE

Figure 5-86 Instrumentation Locations and Designation for Top Mat of Steel Reinforcement in Deck

140

8

9

10

11

12

13

14

15

5

6

7

2

3

4

1

51m

m2�

51m

m2�

76mm3�

76mm3�

76mm3�

SC1_3DESIGNATION

STRESS CONCENTRATIONGLOBAL LOCATION

GAGE LOCATION

Figure 5-87 Stress Concentration Strain Gage Locations and Designation

Initially all instrumentation was zeroed with the specimen supported by

scaffolding. The scaffolding was then removed and a scan was taken to evaluate the

dead load deflections and stresses. The specimen was cycled with actuator loads

chosen to match the shear demand on the connection dowels at peak service conditions

in a prototype bridge. This load was calculated to be approximately 67 kN (15 kips)

per actuator. Three cycles were applied at roughly 6 minutes per cycle to see if any

degradation in stiffness was evident. No degradation was seen after these initial cycles

so 1000 cycles were applied at a frequency of 1 Hz. The load was then increased in

steps of 22 kN (5 kips) per actuator until twice the service load had been applied (133

kN (30 kips) per actuator). At this level 100 cycles were applied to the specimen at 1

Hz. Still no degradation was evident. The load was then increased again in 22.25 kN

141

(5 kip) increments until three times the service load had been achieved (200 kN (45)

kips per actuator). Again at this load level 100 cycles were applied and a slight

softening was observed. The load was then stepped up to 223 kN (50 kips) per

actuator at which time the shear connection between the shell and the slab on the north

side of the specimen began to slip. Significant softening was observed. As the load

was increased to 245 kN (55 kips) per actuator the shear connection on the south side

began to slip and again softening was observed. The load was increased to an ultimate

value of 427 kN (95.9 kips) per actuator at which time the load plateaued due to

excessive deformations in the endblocks.

The load vs. deformation plot is shown in Figure 5-88. For this and the

following plots the cyclic loading has been removed (envelope is shown). Figure 5-89

shows the strain profile across the section for various load levels. Note that plane

sections remain essentially plane until the shear joint between the shell and deck slips

at which time the members begin to bend independently. The shear dowels showed

little to no strains until the slippage occurred at which time some of them yielded

immediately. This shows that the shear was being primarily carried by the friction

between the concrete and the carbon shell. When this friction joint failed the dowels

had insufficient strength to maintain the level of shear necessary to keep the system

acting monolithically. The results from the stress concentration arrays are presented in

Figure 5-90 and Figure 5-91 respectively for the shear and constant moment regions.

These plots show that there is a significant stress concentration around the

penetrations. A parametric study of this effect is presented in Chapter 8.

142


0 1 2 3 4 5 6

Lo

ad P

er A

ctu

ato

r (k

ips)

0

20

40

60

80

100


0 20 40 60 80 100 120 140 160

Lo

ad P

er A

ctu

ato

r (k

N)

0

100

200

300

400

500

Figure 5-88 Load Displacement Envelope for Filled Shell With Slab

Longitudinal Strain (microstrain)

-2000 -1000 0 1000 2000 3000 4000 5000 6000

Dis

tan

ce F

rom

Sec

tio

n B

ott

om

(m

m)

0

100

200

300

400

500

600

Dis

tan

ce F

rom

Sec

tio

n B

ott

om

(in

)

0

5

10

15

200 kN (0 kips)50 kN (11.25 kips)100 kN (22.5 kips)150 kN (33.7 kips)200 kN (44.9 kips)250 kN (56.2 kips)300 kN (67.4 kips)350 kN (78.7 kips)400 kN (89.9 kips)425 kN (95.5 kips)

Top of Shell

Load

25kN=5.62 kips

Figure 5-89 Strain Profile Across Section for Filled Shell With Slab

143

Actuator Load (kN)

0 25 50 75 100 125 150 175 200

Str

ain

(m

icro

-str

ain

)

-50

0

50

100

150

200

250

300

350

400

450

500

550

600

Actuator Load (kips)

0 10 20 30 40

Far Field LongitudinalHoop

Figure 5-90 Stress Concentration Around Penetration in Shear Span

Actuator Load (kN)

0 25 50 75 100 125 150 175 200

Str

ain

(m

icro

-str

ain

)

-50

0

50

100

150

200

250

300

350

400

450

500

550

600

Actuator Load (kips)

0 10 20 30 40

Far FieldLongitudinalHoop

Figure 5-91 Stress Concentration in the Const. Moment region

144

In the following chapter the experimental data presented above is used to

validate the analytical models presented in Chapter 4.

145

6. CORRELATION OF ANALYTICAL MODELS TO EXPERIMENTAL DATA

This chapter presents the correlations between the analytical models put forth

in Chapter 4 and the experimental investigations described in Chapter 5. The

experimental data from the small scale bending tests shows more scatter than from the

large scale tests. It is felt that the geometry of the small scale test specimens made it

difficult to get data unpolluted by the boundary effects as the gage locations were

fairly close to the points of load application and support. The correlations are

presented for the circular sections first, followed by comparisons for the conrec

sections.

6.1 Circular Shells

6.1.1 Small Scale Shells

6.1.1.1 Compression

The analytical model presented in Chapter 4 for the compression behavior of

concrete filled FRP shells is here correlated to the test data from the small scale

compression cylinders. The analytical model described in Section 4.1.1.2 is largely

based on this test data so good correlation is to be expected. Only the thin and thick all

hoop cylinders were able to be modeled due to the experimental difficulties with the

helical cylinders discussed in Chapter 5. The stress strain behavior of the cylinders is

compared in Figure 6-1. The thicker shell shows a significant increase in load

146

carrying and ultimate strain capacity. The expansion behavior is compared in Figure

6-2 where the radial strain is plotted vs. the axial strain for the six cylinders used in

this analysis. The increased strength cited above for the thick shells correlates with the

decreased radial expansion evident in Figure 6-2. It must be noted that the increase in

strength came at the cost of doubling the shell thickness. A point of diminishing

returns is reached with respect to increasing shell thickness where limited strength

gains are realized from the additional material.

Longitudinal Strain

-0.04-0.03-0.02-0.010.00

Str

ess

(ksi

)

-20

-15

-10

-5

0

Str

ess

(MP

a)

-120

-100

-80

-60

-40

-20

0


Thick Circular Shells

Thin Circular Shells

Figure 6-1 Concrete Stress Vs. Strain for Small Scale Compression Specimens

147

Longitudinal Strain

-0.04-0.03-0.02-0.010.00

Ho

op

Str

ain

0.000

0.002

0.004

0.006

0.008ExperimentalAnalytical

Thin Circular Shells

Thick Circular Shells

Figure 6-2 Radial Vs. Longitudinal Strain for Small Scale Comp. Specimens

6.1.1.2 Bending

Two small scale circular bending tests were performed as described in Section

5.1.3.1. The member stiffnesses are compared to the analytical predictions for both of

the circular shells in Figure 6-3. It can be seen that the member stiffnesses are over

predicted by the analytical model. The analytical stiffness of the shell alone (no

concrete fill) is also plotted. The low stiffness of the specimens seen in these plots is

not explained at this time. The author assumes that additional deflection was being

introduced into the system from the support structures. This additional deflection may

have come from the semicircular pivots designed to hold the circular specimens as no

such discrepancy was seen for the conrec specimens. The extreme fiber strains are

shown in Figure 6-4 and Figure 6-5 for the constant moment region and in Figure 6-

148

6 and Figure 6-7 for the shear span of the thin and thick shells respectively. The

correlation of theses strains also points to support deflection as the source of the

additional deflection since the curvature of the specimens was predicted fairly well by

the models especially for the thick shell. The analytical models developed to this point

can not accurately predict the hoop strain increase seen in the shear span. It is thought

that this increased strain is due to the increased cracking due to the shear loads.

Increased tension strains seen in the thin shell in the shear span (Figure 6-7) further

indicate the poor behavior of this specimen reported in Chapter 5. Shear strains were

predicted using the unbonded model described in Section 4.1.3 due to the fact that

these small scale shells had no ribs. A small amount of slippage between the concrete

core and carbon shell was observed during the tests which verifies the assumption that

the core and shell are not deforming as a unit.

Figure 6-8 and Figure 6-9 present the shear strains for the thin and thick shells

respectively. These plots show the analytical predictions with the minimum and

maximum values of vk presented in Section 4.1.3 (0.16-0.29 MPa units (1.9-3.5 psi

units)) used to show the effect of this parameter on the predicted shear strain. The thin

shell begins displaying shear strains at a shear load close to the minimum concrete

cracking load predicted by the analytical model. The rate at which the shear strain in

the shell increases after this point is lower than predicted by the analytical model. This

indicates that the concrete continues to increase the amount of total shear force it is

resisting beyond the shear at initial cracking. At approximately 56 kN (12.5 kips)

shear a small jump is seen in the shear strain in the shell. This indicates a loss of load

149

carried by the core instantaneously that was then shifted to the shell. After this point

the concrete core begins to pick up addition load again as the rate of increase in the

shell is still not as high as would be predicted by the analytical model which assumes

that all shear load past the initial cracking load is taken by the shell. The thick shell

also began to show shear strains in the shell at the minimum concrete cracking load

predicted. The concrete in this case however continued to carry significant additional

loads as the rate of the increase of shear strain in the shell was substantially less than

predicted by the analytical model. As loading progressed the rate of increase of the

shear strain came closer and closer to that predicted by the model. From these tests it

seems that using the proposed shear model with the lower limit for vk gives a

conservative estimate of the shear strain in the shell. The author would however

recommend that for bending applications a ribbed shell be used to ensure an adequate

shear transfer mechanism exists to force the system to deform as a unit as was done

for the full scale tests described later in this section.

150


0.0 0.5 1.0 1.5

Lo

ad (

kip

s)

0

10

20

30

40

50

60

70


0 10 20 30 40 50

Lo

ad (

kN)

0

50

100

150

200

250

300

350Thin Shell AnalyticalThin Shell ExperimentalThin Shell With No ConcreteThick Shell AnalyticalThick Shell ExperimentalThick Shell With No Concrete

Figure 6-3 Load vs. Displacement for Small Scale Circular Sections

Moment (kN-m)

0 10 20 30 40 50

Str

ain

(m

icro

-str

ain

)

-10000

-5000

0

5000

10000

Moment (kip-in)

0 100 200 300 400c1l_expc5l_expc1h_expc5h_expc1l_anac5l_anac1h_anac5h_ana

c1

5

Figure 6-4 Strains in Constant Moment region for Thin Circular Shell

151

Moment (kN-m)

0 10 20 30 40 50 60 70 80 90 100 110 120

Str

ain

(m

icro

-str

ain

)

-10000

-5000

0

5000

10000

Moment (kip-in)

0 100 200 300 400 500 600 700 800 900 1000c1l_expc5l_expc1h_expc5h_expc1l_anac5l_anac1h_anac5h_ana

c1

5

Figure 6-5 Strains in Constant Moment region for Thick Circular Shell

Moment (kN-m)

0 5 10 15 20

Str

ain

(m

icro

-str

ain

)

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

Moment (kip-in)

0 25 50 75 100 125 150 175a1l_expa5l_expa1h_expa5h_expa1l_anaa5l_anaa1h_anaa5h_ana

a1

5

Figure 6-6 Strains in Shear Area for Thin Circular Shell

152

Moment (kN-m)

0 10 20 30 40 50

Str

ain

(m

icro

-str

ain

)

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

6000

Moment (kip-in)

0 100 200 300 400a1l_expa5l_expa1h_expa5h_expa1l_anaa5l_anaa1h_anaa5h_ana

a1

5

Figure 6-7 Strains in Shear Span for Thick Circular Shell

Shear (kips)

0.0 2.5 5.0 7.5 10.0 12.5 15.0

Shear (kN)

0 10 20 30 40 50 60 70

Sh

ear

Str

ain

(m

icro

-str

ain

)

0

2000

4000

6000

8000

10000ExperimentalAnalytical vk=.17 S.I. (2 U.S.)

Analytical vk=.29 S.I. (3.5 U.S.)

Gage Locations

Figure 6-8 Shear Strain in Thin Circular Shell

153

Shear (kips)

0 5 10 15 20 25 30

Shear (kN)

0 20 40 60 80 100 120 140

Sh

ear

Str

ain

(m

icro

-str

ain

)

0

2000

4000

6000

8000

10000

12000ExperimentalAnalytical vk=.17 S.I. (2 U.S.)


Gage Locations

Figure 6-9 Shear Strain in Thick Circular Shell

6.1.2 Full Scale Specimens

Correlations for the full scale bending tests are presented below. The load vs.

displacement behavior is shown in Figure 6-10 for both of the shells tested. The

second shell is substantially softer than the first as would be expected due to the higher

thickness of the first shell and the higher percentage of helical fibers. The ultimate

displacement predicted for both shells is essentially the same since the model predicts

failure based on a ply stress allowable The neutral axis is shifted towards the

compression side of the section since concrete is assumed to take no tension. The

neutral axis location and the curvature determine the longitudinal strain in the extreme

fibers of the shell which in turn determines the peak ply stresses. The softer shell

154

forces the neutral axis more towards the compression side of the member causing a

slight decrease in the allowable curvature and a slight decrease in the ultimate

allowable deflection. Of course as mentioned in Chapter 5 the first ply failure model

does not take into account the compression buckling failure. This buckling failure is

addressed in Chapter 8. The extreme fiber longitudinal and hoop strains in the constant

moment region are shown in Figure 6-11 and in the shear span in Figure 6-12 for

shell #1. These plots are repeated for shell #2 in Figure 6-13 and Figure 6-14. Good

correlation is seen in all areas between the analytical model predictions and the

experimental results. Shell #2 showed increased tension strains at the higher load

levels in both the constant moment and shear spans as seen in Figure 6-13 and Figure

6-14. As stated in Chapter 5 it is believed that the relatively thick layer of pure hoop

fibers on the outside of the shell allowed for increased matrix cracking in the

longitudinal direction which led to the increased tension strains evident in these plots.

The analytical models predict the onset of microcracking for this shell at a moment of

approximately 293 kN-m (2600 kip-in.) which is below the moment at which the

increased strains are seen (600 kN-m (5000 kip-in.) in the constant moment region and

400 kN-m (3500 kip-in.) in the shear span). The shear strains are compared to the

analytical predictions for the two shells in Figure 6-15 and Figure 6-16. Shell #1

showed a gradual increase in the rate that the shell picked up shear strains which

matched the analytical prediction generated with the second model presented in

Section 4.1.3 for systems with an adequate shear transfer mechanism between the shell

155

and the concrete core. The results for shell #2 were very similar with slightly more of

a bilinear nature evident in the response.

Displacement (mm)

0 50 100 150 200 250 300 350 400

Lo

ad P

er A

ctu

ato

r (k

N)

0

50

100

150

200

250

300

350

400

450

500


0 2 4 6 8 10 12 14

Lo

ad P

er A

ctu

ato

r (k

ips)

0

20

40

60

80

100Shell #1 Exp.Shell #2 Exp.Shell #1 Ana.Shell #2 Ana.

Figure 6-10 Load Displacement Curves for Full Scale Filled Shell Tests

156

Moment (kN-m)

0 100 200 300 400 500 600 700 800

Str

ain

(m

icro

-str

ain

)

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

Moment (kip-in)

0 1000 2000 3000 4000 5000 6000 7000c1l_expc1h_expc5l_expc5h_expc1l_anac1h_anac5l_anac5h_ana

c1

5

Figure 6-11 Extreme Fiber Strains in Constant Moment region for Shell #1

Moment (kN-m)

0 100 200 300 400 500 600

Str

ain

(m

icro

-str

ain

)

-4000

-2000

0

2000

4000

6000

Moment (kip-in)

0 1000 2000 3000 4000 5000a1l_expa1h_expa5l_expa5h_expa1l_anaa1h_anaa5l_anaa5h_ana

a1

5

Figure 6-12 Extreme Fiber Strains in Shear Span for Shell #1

157

Moment (kN-m)

0 100 200 300 400 500 600 700 800

Str

ain

(m

icro

-str

ain

)

-10000

-7500

-5000

-2500

0

2500

5000

7500

10000

12500

15000

Moment (kip-in)

0 1000 2000 3000 4000 5000 6000 7000c1l_expc1h_expc5l_expc5h_expc1l_anac1h_anac5l_anac5h_ana

c1

5

Figure 6-13 Extreme Fiber Strains in Constant Moment region for Shell #2

Moment (kN-m)

0 100 200 300 400 500 600

Str

ain

(m

icro

-str

ain

)

-6000

-4000

-2000

0

2000

4000

6000

8000

Moment (kip-in)


a1

5

Figure 6-14 Extreme Fiber Strains in Shear Span for Shell #2

158

Shear (kN)

0 50 100 150 200 250 300

Sh

ear

Str

ain

(m

icro

-str

ain

)

0

500

1000

1500

2000

2500

3000

Shear (kips)

0 10 20 30 40 50 60


Gage Locations

Figure 6-15 Shell Centerline Shear Strains for Shell #1

Shear (kN)

0 50 100 150 200 250 300

Sh

ear

Str

ain

(m

icro

-str

ain

)

0

500

1000

1500

2000

2500

3000

Shear (kips)

0 10 20 30 40 50 60


Gage Locations

Figure 6-16 Shell Centerline Shear Strains for Shell #2

159

6.2 Conrec Shells

6.2.1 Compression

The conrec specimens tested in compression were modeled with finite element

analysis utilizing the concrete models developed for the circular sections as described

in Section 4.2.1. Figure 6-17 shows the load vs. axial strain relation as calculated

from the finite element analysis compared to the experimental results for the thick all

hoop cylinder compression tests described in Section 5.1.2.2. The experimental data

shows a definite kink in the stress strain response of the section that is not picked up

by the analytical approach. It is felt that this kink is due to the rapid expansion of the

concrete core after cracking that occurs due to the poor confinement of the concrete by

the flat sections of the conrec shell. As described in Section 4.1.1.2 the maximum

equivalent tangent Poisson�s ratio for little to no confinement is assumed to be 0.5. As

was described in Chapter 2 an unconfined cylinder or even a cylinder confined with

mild steel can experience volume expansion after the initial loading stages. This

volume expansion implies an equivalent tangent Poisson�s ratio above 0.5. Due to the

lack of data in this range it is not at this time possible to accurately predict the

behavior of the concrete at these low levels of confinement. The finite element

methods used here assume an isotropic elastic material for the concrete which does not

allow for a Poisson�s ratio equal to or greater than 0.5 due to the problems associated

with incompressible material models in finite element calculations. It may be possible

in the future as data becomes available for the lower confinement levels to extend this

method by using a nonisotropic material model in the finite element calculations that

160

would allow for different moduli and Poisson�s ratios to exist for the concrete in

different directions. Such a model could theoretically be introduced into a nonlinear

finite element code. The experimental radial strains in the shell for the center of the

flat , the tangent point of the radius, and the center of the radius are compared to the

analytical model predictions in Figure 6-18. The rapid increase in the hoop strain at

the center of the flat implies that the expansion postulated above is in fact taking

place. As the loading progresses the model and experimental results show the same

trend. It is felt that to accurately predict the behavior of noncircular sections not only a

nonlinear material model must be used but also nonlinear geometry must be

considered. For the conrec section being considered here as the concrete expands the

flat sides bulge out and the section will more closely resemble a circular section as the

loading progresses. This deformed section is better able to resist further expansion

than the original geometry.

161


-40000-30000-20000-100000

Lo

ad (

kN)

-3000

-2500

-2000

-1500

-1000

-500

0

Lo

ad (

kip

s)

-600

-500

-400

-300

-200

-100

0

ExperimentalFinite Element

Figure 6-17 Load vs. Longitudinal Strain for Thick Conrec Cylinders


-20000-15000-10000-50000

Ho

op

Str

ain

(m

icro

-str

ain

)

0

1000

2000

3000

4000

5000

6000


Figure 6-18 Hoop Strains vs. Longitudinal Strain in Thick Conrec Section

162

6.2.2 Bending

The small scale conrec bending specimens described in Section 5.1.3.2 are

here used to demonstrate the ability of the models presented in Chapter 4 to predict the

behavior of a conrec section in bending. The load vs. displacement curves for both the

thin and thick conrec shells are shown in Figure 6-19 along with the predicted

stiffness of the shells alone with no concrete fill. Good correlation was seen between

the experimental and analytical results with none of the additional displacement seen

for the circular sections evident for the conrecs. The extreme fiber strains in the

longitudinal and hoop directions in the constant moment region for the thin and thick

shells are shown in Figure 6-20 and Figure 6-21 respectively. The same data is shown

for the shear span in Figure 6-22 and Figure 6-23. For the thin shell a lag is seen in

the tension strain at the initiation of loading in both the constant moment and shear

spans and the extreme fiber strains are slightly overpredicted by the model. For the

thick shell good correlation was seen between the experimental and analytical values

of the longitudinal strains. The hoop strains predicted utilizing the �equivalent circular

section� described in Section 4.2.2 are reasonable estimates of the strains seen in the

tests for the constant moment region especially for the thick shell although in the shear

span the increased hoop strains seen in the circular sections are also evident for the

conrecs. Shear strains on the centerline of the shell are shown along with the

predictions generated using the shear model presented in Section 4.1.3 assuming no

shear transfer mechanism is present between the shell and the concrete core in Figure

6-24 and Figure 6-25 for the thin and thick shells respectively. The thin shell begins

163

to show significant shear strains at a shear force level equal to that predicted by the

shear cracking relation presented with the lower limit for vk. The rate that the shell

shear strain increases is lower than the model prediction indicating the continuing

increase in shear carried by the concrete core. At a shear load of approximately 52 kN

(12 kips) the rate of increase of the shear strains increases to a level equivalent to that

predicted which indicates no additional shear is being taken by the concrete core. The

thick conrec section also began to show shear strains in the shell in the range predicted

by the shear cracking model. The shear strains increased at a rate again slower than

predicted up to a shear load of approximately 130 kN (30 kips) at which point a

sudden increase of the shear strain in the shell was seen. This sudden increase

indicates a loss of shear load being carried in the concrete core which could indicate a

loss of aggregate interlock in the shear span common for lightweight concrete as the

shear strength of the aggregate is low compared to standard aggregate. The shear

strains in the shell after this point continued to increase at a rate which indicates no

additional shear being taken by the concrete core.

In the following chapter the analytical models will be used to investigate the

effects of varying the parameters that effect the response of the system.

164


0 5 10 15 20 25 30 35

Lo

ad (

kN)

0

50

100

150

200

250

300

350


0.00 0.25 0.50 0.75 1.00 1.25

Lo

ad (

kip

s)

0

10

20

30

40

50

60

70Thin Shell ExperimentalThick Shell ExperimentalThin Shell AnalyticalThick Shell AnalyticalThin Shell With No ConcreteThick Shell With No Concrete

Figure 6-19 Load Displacement for Conrec Bending Specimens

Moment (kN-m)

0 10 20 30 40 50 60

Str

ain

(m

icro

-str

ain

)

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

Moment (kip-in)

0 100 200 300 400 500c1l_expc1h_expc6l_expc6h_expc1l_anac1h_anac6l_anac6h_ana

1

6

c

Figure 6-20 Strains in Constant Moment region for Thin Conrec Shell

165

Moment (kN-m)

0 25 50 75 100 125 150

Str

ain

(m

icro

-str

ain

)

-10000

-5000

0

5000

10000

15000

Moment (kip-in)

0 200 400 600 800 1000 1200c1l_expc1h_expc6l_expc6h_expc1l_anac1h_anac6l_anac6h_ana

1

6

c

Figure 6-21 Strains in Constant Moment region for Thick Conrec Shell

Moment (kN-m)

0 5 10 15 20 25

Str

ain

(m

icro

-str

ain

)

-3000

-2000

-1000

0

1000

2000

3000

4000

Moment (kip-in)

0 50 100 150 200a1l_expa1h_expa6l_expa6h_expa1l_anaa1h_anaa6l_anaa6h_ana

1

6

c

Figure 6-22 Strains in Shear Area for Thin Conrec Shell

166

Moment (kN-m)

0 10 20 30 40 50 60

Str

ain

(m

icro

-str

ain

)

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

6000

Moment (kip-in)


1

6

c

Figure 6-23 Strains in Shear Area for Thick Conrec Shell

Shear (kN)

0 20 40 60 80

Sh

ear

Str

ain

(m

icro

-str

ain

)

0

2500

5000

7500

10000

12500

15000

Shear (kips)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

ExperimentalAnalytical vk=.17 S.I. (2 U.S.)


Gage Locations

Figure 6-24 Shear Strain in Thin Conrec Shell

167

Shear (kN)

0 25 50 75 100 125 150 175 200

Sh

ear

Str

ain

(m

icro

-str

ain

)

0

2500

5000

7500

10000

12500

15000

17500

20000

Shear (kips)

0 5 10 15 20 25 30 35 40

ExperimentalAnalytical vk=.17 S.I. (2 U.S.)


Gage Locations

Figure 6-25 Shear Strain in Thick Conrec Shell

168

7. PARAMETER STUDIES OF MATERIAL, LAY-UP, THICKNESS AND SHAPE VARIATIONS

In the following sections the behavior of concrete filled fiber reinforced shells

will be investigated varying a range of parameters. The parameters investigated are

radius to thickness ratios, composite architectures materials and shape variations. Four

radius to thickness ratios are studied varying from 10 to 25 in increments of 5. The

higher the radius to thickness ratio the greater the influence of the concrete core on the

behavior of the system. The composite shell architectures are composed of ±10o plies

and 90o plies. The percentage of helical plies (±10o) is varied from 10% to 90% of the

total lay-up thickness. Carbon and E-Glass composites are used with the pertinent ply

properties listed in Table 7-1. The concrete properties assumed for this analysis are

those of the compression cylinders tested which were used to develop the analytical

model presented in Chapter 4. This is a 45.5 MPa (6.6 ksi) lightweight concrete with a

modulus of 20.7GPa (3.0 msi).

The cost per unit weight of the finished FRP shell is highly dependent on the

cost of the constituent materials and the manufacturing method employed. These

values are constantly in flux. For the materials being investigated in this section at the

time of publication of this dissertation a reasonable estimate for a finished shell

produced with E-glass is $8.00/lb. and for carbon is $25.00/lb. This makes a carbon

shell 3.12 times more expensive per pound than an E-glass shell. The efficacy of these

169

materials in compression and flexure is compared on a weight basis in the following

sections.

Table 7-1 Ply Properties for Parameter Studies

E-Glass Carbon

E1 39 GPa

(5.66 msi)

121 GPa

(17.5 msi)

E2 6.90 GPa

(1.00 msi)

6.90 GPa

(1.00 msi)

ν12 .25 .30

Specific

Gravity

1.8 1.61

7.1 Circular Shells

7.1.1 Compression Behavior

In this section the compression behavior of circular shells will be investigated.

The stress is reported as a normalized ratio of the concrete stress at the allowable

longitudinal strain of the shell, which is taken as 1.2% for carbon and 1.4% for E-

glass, to the ultimate compression stress of the unconfined concrete, f�c. The plots

presented below compare the concrete response for a given radius to thickness ratio

and material for all composite architectures. Figure 7-1 is for carbon and Figure 7-2

is for E-glass.

170

The trends follow intuition with the highest confinement offered by smaller

radius to thickness ratios, more 90o plies and stiffer materials. The most confinement

and thus highest concrete strength is achieved with the predominantly hoop carbon

fiber shells giving a ratio of maximum concrete stress to f�c of 1.85 for a shell with a

radius to flat ratio of 10.

A comparison of the confinement efficiency of the two materials studied in this

section is presented in Figure 7-3. This plot shows the ratio of the weight of an E-

glass shell to the weight of a carbon shell required to give the same peak stress with

the same percentage of hoop and helical plies. This plot demonstrates that at the

current cost estimates for the two material systems carbon is more efficient at

confining the concrete core for all but the shells with a high percentage of helical

fibers (over ~75%).

171

% Helical Plies

10 20 30 40 50 60 70 80 90

σ 1/f' c

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

R/t=10R/t=15R/t=20R/t=25

Figure 7-1 Compression Behavior of Carbon Epoxy Shells

% Helical Plies

10 20 30 40 50 60 70 80 90

σ 1/f' c

1.24

1.26

1.28

1.30

1.32

1.34

1.36

1.38

1.40

1.42

1.44

R/t=10R/t=15R/t=20R/t=25

Figure 7-2 Compression Behavior of E-Glass Shells

172

% Helicals

10 20 30 40 50 60 70 80 90

Wei

gh

t E

-Gla

ss /

Wei

gh

t C

arb

on

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

R/t=10R/t=15R/t=20R/t=25

Equal Cost At Current Cost Estimates

Figure 7-3 Confinement Efficiency of E-Glass vs. Carbon Shells

7.1.2 Bending Behavior

The moment in the section can be written as described in equation 7-1. The

first term on the right side of the equation is the contribution from the shell and the

second term is the contribution from the concrete core. The geometric properties for

this relation are shown in Figure 7-4.

{ } { }[ ]M tR R R c R R R c dLo= − − + − −�2 2 2

1

180σ θ θ σ θ θcos( ) ( ) sin ( ) cos( ) ( ) (7-1)

With the radius to thickness ratio held constant and the lay-up unchanged the neutral

axis position (c) is proportional to the radius. Thus the thickness and neutral axis

173

position can both be expressed as a constant function of the radius as shown in

equation 7-2.

t C Rc C R

==

1

2

(7-2)

Combining equations 7-1 and 7-2 it can be seen that the moment is proportional to the

cube of the radius as shown in equation 7-3.

{ } { }[ ]M R C C C dLo= − − + − −�2 1 13

1 22

1 2

180σ θ θ σ θ θcos( ) ( ) sin ( ) cos( ) ( ) (7-3)

θ

c

N.A.

Rt

Figure 7-4 Geometry for Moment Calculation

The figures below depict the moment curvature response for the range of

radius to thickness ratios, composite architectures and materials described above.

Figure 7-5 through Figure 7-8 are for concrete filled carbon shells and Figure 7-9

through Figure 7-12 are for concrete filled E-glass shells. The moment is normalized

by the inverse of the cube of the radius as explained above and the curvature is

normalized by the radius to generalize the results. For most structural applications

174

these composite members tend to be stiffness critical. To achieve the highest stiffness

it is desirable to place as many fibers close to the longitudinal axis (0o direction) as

possible. With ±10o helical fibers the Poisson�s ratio for loading in the longitudinal

direction of the shell is greater than the initial Poisson�s ratio of the concrete core if

the percentage of helical plies exceeds approximately 85% for carbon epoxy and 80%

for E-glass. This situation should be avoided so the shell does not expand faster than

the concrete core in the initial stages of loading and offer no resistance to cracking.

This analysis does reflect the greater displacement capacity of the fiberglass shells due

to the higher ultimate strain allowable. The peak moment and curvature reported here

correspond to the first ply failure described in Chapter 3. As described earlier these

values are used as theoretical maximum values since they do not take into account

local buckling of the shell.

175

φr

0.000 0.002 0.004 0.006 0.008 0.010 0.012

M/r

3 (M

N-m

/m3 )

0.0e+0

1.0e+5

2.0e+5

3.0e+5

4.0e+5

5.0e+5

M/r

3 (ki

p-i

n/in

3 )

0.0e+0

1.0e+4

2.0e+4

3.0e+4

4.0e+4

5.0e+4

6.0e+4

7.0e+4

10%

90%

% Helicals

Figure 7-5 Normalized Moment Curvature, R/t=10, Carbon Epoxy Shell

φr

0.000 0.002 0.004 0.006 0.008 0.010 0.012

M/r

3 (M

N-m

/m3 )

0.0e+0

5.0e+4

1.0e+5

1.5e+5

2.0e+5

2.5e+5

3.0e+5

3.5e+5

M/r

3 (k

ip-i

n/in

3 )

0.0e+0

1.0e+4

2.0e+4

3.0e+4

4.0e+4

5.0e+4

10%

90%

% Helicals


176

φr

0.000 0.002 0.004 0.006 0.008 0.010 0.012

M/r

3 (MN

-m/m

3 )

0.0e+0

5.0e+4

1.0e+5

1.5e+5

2.0e+5

2.5e+5

M/r

3 (k

ip-i

n/in

3 )

0.0e+0

5.0e+3

1.0e+4

1.5e+4

2.0e+4

2.5e+4

3.0e+4

3.5e+4

10%

90%

% Helicals


φr

0.000 0.002 0.004 0.006 0.008 0.010 0.012

M/r

3 (M

N-m

/m3 )

0.0e+0

5.0e+4

1.0e+5

1.5e+5

2.0e+5

M/r

3 (ki

p-i

n/in

3 )

0.0e+0

5.0e+3

1.0e+4

1.5e+4

2.0e+4

2.5e+4

10%

90%

% Helicals


177

φr

0.000 0.005 0.010 0.015

M/r

3 (M

N-m

/m3 )

0.0e+0

5.0e+4

1.0e+5

1.5e+5

2.0e+5

2.5e+5

M/r

3 (k

ip-i

n/in

3 )

0.0e+0

1.0e+4

2.0e+4

3.0e+4

10%

90%

% Helicals

Figure 7-9 Normalized Moment Curvature, R/t=10, E-Glass Shell

φr

0.000 0.005 0.010 0.015

M/r

3 (M

N-m

/m3 )

0.0e+0

5.0e+4

1.0e+5

1.5e+5

M/r

3 (k

ip-i

n/in

3 )

0.0e+0

5.0e+3

1.0e+4

1.5e+4

2.0e+4

10%

90%% Helicals


178

φr

0.000 0.005 0.010 0.015

M/r

3 (M

N-m

/m3 )

0.0e+0

2.0e+4

4.0e+4

6.0e+4

8.0e+4

1.0e+5

1.2e+5

M/r

3 (ki

p-i

n/in

3 )

0.0e+0

5.0e+3

1.0e+4

1.5e+4

10%

90%

% Helicals


φr

0.000 0.005 0.010 0.015

M/r

3 (M

N-m

/m3 )

0.0e+0

2.0e+4

4.0e+4

6.0e+4

8.0e+4

1.0e+5

M/r

3 (ki

p-i

n/in

3 )

0.0e+0

2.5e+3

5.0e+3

7.5e+3

1.0e+4

1.3e+4

10%

90%

% Helicals


179

The addition of axial load to the section alters the moment curvature response

because the neutral axis is shifted and more of the concrete core is mobilized. This

effect is most notable for large radius to shell thickness ratios where the concrete

makes the greatest contribution to the moment resistance of the section. To

demonstrate this effect moment curvature plots for carbon shells with 10, 50 and 80

percent helical fibers for radius to thickness ratios of 10 and 25 are plotted in Figure

7-13 through Figure 7-18 for axial loads equal to 20, 40 and 60 percent of the capacity

of the unconfined concrete (f�c). These plots demonstrate the increased stiffness

possible with the addition of axial load into the system especially for shells with

predominantly hoop fibers and systems with relatively thin shells where the concrete

offers a greater proportion of the moment resistance. It is envisioned that post

tensioning could be used with this system for certain structural applications to increase

the system stiffness[40].

180

φr

0.000 0.002 0.004 0.006 0.008 0.010

M/r

3 (kN

-m/m

3 )

0e+0

1e+8

2e+8

3e+8

4e+8

5e+8

6e+8

7e+8

8e+8

9e+8

M/r

3 (ki

p-i

n/in

3 )

0.0e+0

2.0e+4

4.0e+4

6.0e+4

8.0e+4

1.0e+5

1.2e+520% f'c40% f'c60% f'c

Axial Load

Figure 7-13 Moment Curvature With Axial Load, R/t=10, 10% Helical Fibers

φr

0.000 0.002 0.004 0.006 0.008 0.010 0.012

M/r

3 (kN

-m/m

3 )

0.0e+0

2.0e+8

4.0e+8

6.0e+8

8.0e+8

1.0e+9

1.2e+9

1.4e+9

1.6e+9

1.8e+9

2.0e+9

M/r

3 (ki

p-i

n/in

3 )

0.0e+0

5.0e+4

1.0e+5

1.5e+5

2.0e+5

2.5e+520% f'c40% f'c60% f'c

Axial Load


181

φr

0.000 0.002 0.004 0.006 0.008 0.010 0.012

M/r

3 (kN

-m/m

3 )

0.0e+0

5.0e+8

1.0e+9

1.5e+9

2.0e+9

2.5e+9

3.0e+9

3.5e+9

M/r

3 (ki

p-i

n/in

3 )

0e+0

1e+5

2e+5

3e+5

4e+5

5e+5

20% f'c40% f'c60% f'c

Axial Load


φr

0.000 0.002 0.004 0.006 0.008 0.010

M/r

3 (

kN-m

/m3 )

0e+0

1e+7

2e+7

3e+7

4e+7

5e+7

6e+7

7e+7

M/r

3 (ki

p-i

n/in

3 )

0

2000

4000

6000

8000

10000

20% f'c40% f'c60% f'c

Axial Load


182

φr

0.000 0.002 0.004 0.006 0.008 0.010

M/r

3 (

kN-m

/m3 )

0.0e+0

2.0e+7

4.0e+7

6.0e+7

8.0e+7

1.0e+8

1.2e+8

M/r

3 (ki

p-i

n/in

3 )

0

4000

8000

12000

1600020% f'c40% f'c60% f'c

Axial Load


φr

0.000 0.002 0.004 0.006 0.008 0.010 0.012

M/r

3 (

kN-m

/m3 )

0e+0

5e+7

1e+8

2e+8

2e+8

M/r

3 (ki

p-i

n/in

3 )

0

5000

10000

15000

20000

2500020% f'c40% f'c60% f'c

Axial Load


183

As with compression the flexural efficiency of the two material systems is

compared on a weight basis. For flexure the comparison is made based on the secant

stiffness of the system. The weight of an E-Glass shell necessary to give the same

secant stiffness of a carbon shell with the same percentage of hoop and helical plies is

shown in Figure 7-19. The secant stiffness for both material systems is evaluated at

the ultimate curvature of the carbon shell. The comparison shown in Figure 7-19

indicates that the E-Glass shell is more efficient for flexure with the current cost

estimates. This does not take into account that E-glass must be isolated from standard

concrete infill (see Section 3.1.1) which could increase the cost of the E-glass system.

% Helicals

10 20 30 40 50 60 70 80 90

Wei

gh

t E

-Gla

ss /

Wei

gh

t C

arb

on

1.5

2.0

2.5

3.0

3.5

R/t=10R/t=15R/t=20R/t=25

Equal Cost At Current Cost Estimates

Figure 7-19 Flexural Stiffness of E-Glass vs. Carbon Shells

184

7.2 Conrec Shells

The motivation to study the conrec sections grew out of the concern that the

circular section is not a very efficient shape for bending applications such as girders or

beams and that connections to a circular section can be difficult due to the line contact

that results if an adjacent member, such as a deck in a beam-and-slab system, is placed

directly on the shell. In this section the moment curvature response of conrec sections

is compared to that of circular sections with the same composite shell architecture.

The comparison is first made based on holding the overall height of the section

constant. As expected a substantial stiffness improvement is realized from reducing

the radius on a conrec section. It must be realized however that the conrec section with

a small radius has a substantial amount more composite material in the shell than does

a circular section of the same overall height. The second comparison in this section

shows the moment curvature response for various shapes if the amount of composite

material is held constant.

For the remainder of this section the section depth (d) is taken to mean the

section total height and the corner radius (r) is taken to mean the radius in the corner

of the conrec section. Figure 7-20 and Figure 7-21 depict the moment curvature

response of three different conrec geometries as compared to a circular section with

the same composite lay-up for two different section depth to thickness ratios (20 and

50). The flat to corner radius ratios are varied from 1/2 to 3 for the conrec sections.

185

The same analysis results are presented in Figure 7-22 for a nominal section

depth to thickness ratio of 20. A nominal ratio is used here since the total height of the

conrec sections is varied to force the total amount of composite material in the shell to

be held constant. Table 7-2 shows the corner radius and flat dimensions used along

with the actual height of the section. All values in the plot are normalized to the same

nominal section depth which is equal to twice the radius of the circular section.

Table 7-2 Geometry of Conrec Sections For Normalized Comparison

Flat/Corner Radius Flat/d Corner Radius/d Height/d

Circle - 0.5 1

1/2 0.1896 0.3792 0.998

1/1 0.3055 0.3055 0.9665

3/1 0.5155 0.1718 0.9095

186

φr

0.000 0.002 0.004 0.006 0.008 0.010 0.012

M/r

3 (ki

p-i

n/in

3 )

0.0e+0

1.0e+4

2.0e+4

3.0e+4

4.0e+4

5.0e+4

6.0e+4

7.0e+4

8.0e+4

9.0e+4

M/r

3 (kN

-m/m

3 )

0.0e+0

1.0e+8

2.0e+8

3.0e+8

4.0e+8

5.0e+8

6.0e+8Circular SectionFlat/Rad=1/2Flat/Rad=1Flat/Rad=3

10%

50%

90%

%Helicals

Figure 7-20 Moment Curvature Response for Conrec Sections, d/t=20

φr

0.000 0.002 0.004 0.006 0.008 0.010 0.012

M/r

3 (ki

p-i

n/in

3 )

0

5000

10000

15000

20000

25000

30000

35000

M/r

3 (kN

-m/m

3 )

0.0e+0

5.0e+7

1.0e+8

1.5e+8

2.0e+8

Circular SectionFlat/Rad=1/2Flat/Rad=1Flat/Rad=3

10%

50%

90%

%Helicals

Figure 7-21 Moment Curvature Response for Conrec Sections, d/t=50

187

φr

0.000 0.002 0.004 0.006 0.008 0.010 0.012

M/r

3 (k

ip-i

n/in

3 )

0

10000

20000

30000

40000

50000

60000

70000

M/r

3 (kN

-m/m

3 )

0.0e+0

5.0e+7

1.0e+8

1.5e+8

2.0e+8

2.5e+8

3.0e+8

3.5e+8

4.0e+8

4.5e+8 Circular SectionFlat/Rad=1/2Flat/Rad=1Flat/Rad=3

10%

50%

90%

%Helicals

Figure 7-22 Normalized Moment Curvature Response for Conrec Sections,

d/t=20

7.3 Hybrid Shells

More than one material may be used to form the fiber reinforced shell. The

ever changing cost benefit equation must be taken into account when selecting the

fiber reinforced composite most suitable for a particular application. As can be seen

from the plots in Section 7.1.1 the response of the confined concrete does not change

much for shells with predominantly helical fibers when we consider carbon or glass

shells. In this section the behavior of a hybrid system with carbon used for the helical

plies and E-glass used for the hoop plies will be investigated.

188

7.3.1 Compression

The compression behavior of a concrete filled fiber reinforced shell is

dominated by the properties of the material used for the hoop direction fibers. There is

seen a slight decrease in the concrete stress due to the fact that the carbon fibers in the

helical plies make the Poisson�s ratio for loading in the longitudinal direction of the

shell slightly higher than for the all glass system thus slightly decreasing the

confinement efficiency of the section. This effect is not more than 3.5% for the

parameter studies presented above.

7.3.2 Bending

Unlike the compression behavior, the bending behavior of the shell is strongly

dominated by the material chosen for the helical plies. As shown in Chapter 6 the

contribution of the concrete core to the total bending stiffness is not great and thus a

change in the confinement of the core does not have a significant effect on the bending

behavior of the system. The moment curvature response of such a hybrid system is

essentially identical to that of a shell wound exclusively with the material used for the

helical plies.

189

8. STRESS CONCENTRATIONS, TENSION STIFFENING AND THERMAL EXPANSION EFFECTS 8.1 STRESS CONCENTRATIONS To use fiber reinforced composite shells as structural members for civil

applications it is necessary to understand the effects of penetrations in the shell that

may be necessary for joining members or attaching nonstructural members. One such

case studied in this document is the penetration used to attach a deck to a composite

shell girder through steel shear dowels. Anisotropic materials can have much higher

stress concentration factors than would be found in an isotropic material. This section

will look at the stress concentration around a circular hole in an orthotropic plate with

and without an elastic inclusion. Finite element models will be used to investigate the

effect of shell curvature on the stress concentrations as well as explore individual ply

stresses. A parametric study of the influence of composite architecture on the stress

concentration factors is performed. In the last section of this chapter a case study is

performed to investigate the stress concentration around the penetrations in the girder

necessary to attach the deck in a beam and slab system.

8.1.1 Closed Form Solution The equations for a circular inclusion in a flat orthotropic infinite plate with

loading along one of the principal directions of the plate were used to parametrically

study the influence of lay-up geometry on the stress concentration factor. The

equivalent stiffness properties for the laminate were determined using classic

190

lamination theory assuming constant strain through the thickness as described in

Chapter 3. This method has been shown to be effective for determining stress

concentrations [41]. The equations presented below are extracted from Lekhnitskii,

Anisotropic Plates [42].

For an orthotropic material the Hooke�s law relations are written as

ε σ σε σ σγ τ

x x y

y x y

xy xy

a aa aa

= += +

=

11 12

12 22

66

(8-1)

The coefficients of deformation can be expressed in terms of the elastic constants as

shown in equations 10-2.

aE

aE E

aE

aG

111

1221

2

12

1

222

6612

1

1

1

=

= − = −

=

=

ν ν

(8-2)

The following relations will be expressed in terms of the coefficients of deformation

for both the infinite plate and the elastic plug. The constants referring to the plug will

be differentiated with a prime. The following constants will be used in this section.

191

k aa

EE

n EE

EG

EE

a aa

aa

= =

= −�

��

�

�� +

= ++

+

22

11

1

2

1

212

1

12

1 4 12 66

11

2 2 22

11

4

2

2

ν

θ θ θ θθ

sin sin cos cos

(8-3)

With

θ − angle around the inclusion measured from the x axis

E1 - plate modulus in the x direction

E2 - plate modulus in the y direction

For the plate and loading as shown in Figure 8-1 the tangential stress around the

inclusion is given by equation 8-4.

x

y

pp θ

Figure 8-1 Infinite Plate With a Circular Inclusion

( )

( )[ ][ ]

σ θ

θ θ

θ θ θ

θθ= −

+ − + + + + − + +

+ − + + + + + + −

p EE

a

n k k n a k a k n a

k k k n a k k a k n a k a

∆∆

∆

∆

11

6

21 2 3

4 2

22 3 4

2 4 24

6

2 1 2 2 1

1 2 2 1

{ sin

( ) ( ) ( )( ) sin cos

( )( ) ( ) ( ) sin cos cos }

(8-4)

192

with

( )[ ]a a a a n k a a a a a kn a a a

a a a a a a a k na a a a n a k a a a a a ka a a a a a a a a k n a a

a a

1 11 11 222

11 11 22 12 122

22 12 12

2 11 11 22 11 12 12

3 11 11 22 22 22 12 12 12 122

4 11 11 22 12 12 12 66 11 12 122

11 22

1

2

= − − + − + − − −

= − + − +

= − + + − + −

= − − + − + + + − −

=

( ) ( ) ( ) ( )

( ) ( ) ( )( )( ) ( ) ( )

( ) ( )[ ( )] ( )(

' ' ' ' '

' '

' ' ' '

' ' ' '

∆ + + + + + − −a a k a a a a a k a a n a a k11 22 22 66 12 11 22 22 11 12 1222' ' ' ' ' ') ( ) ( ) ( )

(8-5)

8.1.2 Parameter Study The equations presented in the previous section have been used to perform a

parametric study of the influence of composite shell architecture on the stress

concentration factor for loading in the longitudinal and hoop directions for both

compression and tension loads Figure 8-2 defines the cases that were studied. Cases 1

and 2 represent loading along the longitudinal direction in tension and compression

respectively, cases 3 and 4 represent loading in the hoop direction in tension and

compression respectively and cases 5 and 6 are for tension in the longitudinal and

hoop direction with no inclusion (open hole). For this analysis the inclusion is

assumed to be the concrete used to fill the shell. Since concrete has very limited

tension capacity an orthotropic plug will be assumed with a very low modulus in the

tension direction. The shell architectures used in this study consist of helical plies at

±100 and hoop plies at 90o and are listed in Table 8-1. The ply properties were

assumed to be those given in Table 5-2.

The results of the parametric analysis are presented in tabular form in Table 8-

2 and graphically in Figure 8-3. The peak stress concentration occurs at 90o from the

193

applied load unless otherwise specified. It can be seen from these results that tension

loading gives stress concentration factors almost identical to the open hole stress

concentration factors in all cases studied. This is due to the fact that the orthotropic

plug was given very low stiffness in the direction of the load for these cases. Further

study also showed that the compression values were almost identical to those

calculated assuming an isotropic plug. This leads to the conclusion that for the values

used in this study the stiffness of the plug transverse to the loading direction does not

have a significant effect on the tangential stress in the composite shell around the hole.

The variation of the stress concentration factor with θ is shown in Figure 8-4 for a

composite lay-up with 80% helical fibers for the first four cases.

194

Case 1 Case 2

Case 3 Case 4

Case 5 Case 6

Figure 8-2 Load Cases for Stress Concentration Study

195

Table 8-1 Composite Architectures for Stress Concentration Study

Lay-up Designation

% Helicals (±10o)

E1 MPa (msi)

E2 MPa (msi)

G12 MPa (msi)

ν12 ν21

1 100 111 (16.1)

7.10 (1.03)

7.86 (1.14)

.703 .0450

2 90 103 (14.9)

18.5 (2.68)

7.58 (1.10)

.258 .0464

3 80 92.4 (13.4)

29.9 (4.33)

7.31 (1.06)

.150 .0485

4 70 82.1 (11.9)

41.3 (5.99)

6.96 (1.01)

.102 .0513

5 60 71.0 (10.3)

52.7 (7.64)

6.67 (.967)

.074 .0549

6 50 60.7 (8.80)

64.1 (9.29)

6.36 (.922)

.056 .0591

7 40 49.9 (7.24)

75.2 (10.9)

6.05 (.878)

.044 .0662

8 30 39.2 (5.68)

86.9 (12.6)

5.74 (.833)

.034 .0754

9 20 28.4 (4.12)

97.9 (14.2)

5.44 (.789)

.027 .0931

10 10 17.7 (2.56)

110 (15.9)

5.13 (.744)

.022 .137

11 0 6.89 (1.0)

121 (17.5)

4.83 (.700)

.017 .300

196

Table 8-2 Stress Concentration Factors

Lay-up

Designation

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

1 5.73 3.00 -3.6-0o

2.2�55O

1.7�0O

.495

5.52 -3.8�0O

2.2�55O

2 5.24 2.82 2.77 .929 5.20 2.77

3 4.98 2.63 3.25 1.27 4.97 3.25

4 4.79 2.45 3.68 1.57 4.78 3.68

5 4.57 2.24 4.07 1.85 4.57 4.07

6 4.36 2.03 4.45 2.11 4.36 4.45

7 4.11 1.80 4.83 2.35 4.11 4.82

8 3.83 1.54 5.23 2.61 3.83 5.22

9 3.48 1.24 5.64 2.85 3.48 5.62

10 3.03 .896 6.13 3.11 3.03 6.09

11 -3.7� - 0O

2.34

1.6� 0O

.464

6.91 3.39 -4.1� 0O

2.34

6.70

197

% Helical Fibers

0 20 40 60 80 100

Str

ess

Co

nce

ntr

atio

n F

acto

r

0

1

2

3

4

5

6

7Case 1Case 2Case 3Case 4

Figure 8-3 Stress Concentration Factors

Angle From x Axis (Degrees)

0 10 20 30 40 50 60 70 80 90

Str

ess

Co

nce

ntr

atio

n F

acto

r

-3

-2

-1

0

1

2

3

4

5

6

Case 1Case 2Case 3Case 4

Figure 8-4 Tangential Stress Concentration Variation Around Hole for 80%

Helical Shell

198

8.1.3 Case Study A case study was performed on the shear connection used to connect the

concrete deck to the carbon girder in the beam and slab test described in Section 5.2.3

A similar connection using a fiberglass deck system was also investigated. For both of

these structural systems the shear transfer between the deck and the girder is

accomplished through friction, bonding and steel shear dowels. Figure 8-5 shows the

configuration of the connection for the composite deck system which is identical to the

concrete deck system as far as the shear connection to the girder is concerned. The

neutral axis of the concrete deck system is above the top of the girder leading to a state

of biaxial tension in the shell around the penetration. For the system with the

composite deck the neutral axis falls below the top of the girder giving compression in

the longitudinal direction and tension in the hoop direction. Both of these cases will be

evaluated with a finite element model and also using the concentration factors

calculated above.

Polymer Concrete Fiber Glass Deck Panel

Polymer ConcreteSaddle

Light Weight ConcreteSteel Shear Dowel Figure 8-5 Shear Connection

199

A finite element model was utilized to determine the ability of the orthotropic

plate assumption used above to predict the actual stress concentration in an individual

ply of a laminated composite. One quarter of the shell was modeled with symmetry

boundary conditions being used. Quadratic composite plate elements were used so that

the individual ply stresses could be monitored. Orthotropic properties were assumed

for the concrete plug with minimal stiffness assigned to the tension direction to

simulate the inability of the concrete to take tension. The analytical models presented

in Chapter 4 were used to predict the longitudinal and hoop strains present in the top

flange of the shell under maximum service conditions. The longitudinal strain was

applied to the model with imposed nodal displacements along the far field edge. The

reaction of the concrete was replaced with an internal pressure calculated from the

longitudinal and hoop strains. The model used is shown in Figure 8-6. For the case of

the equivalent orthotropic plate superposition is used for the longitudinal and hoop

stresses at the critical location.

The longitudinal strain in the girder at maximum service conditions was found

to be �0.01% for the composite deck system. From the methods described in Chapter 4

the radial strain is determined to be 0.002% and finally the radial pressure is found

from equation 4-6 to be 2.08 kPa (0.30 psi). Applying these loads to the finite element

model the maximum stress around the hole in the ±10o plies was found to be 25.6 MPa

(-3.66 ksi). For the orthotropic plate assumption the far field stresses are obtained from

equations 4-5 and 4-8 giving 9.72 MPa (-1.41 ksi) in the longitudinal Direction and

37.9 kPa (5.5 psi) in the hoop direction. From Figure 8-4 the concentration factor for

200

loading in the longitudinal direction for a shell with 85% ±10o plies is found to be ~2.7

at 90o from the loading direction for case 2 loading. For the hoop loading the

concentration is found to be �1.8 at 00 from the direction of the loading for case 3

(Figure 8-4). Using super-position the total stress around the penetration is found to

be 26.6 MPa (3.85 ksi).

The concrete deck system is evaluated similarly with a longitudinal strain of

0.01%. The pertinent values for both systems are presented in Table 8-3. It can be

seen that the stress concentration predicted in this analysis is substantially greater for

the system with the concrete deck than was recorded in the test. This may be due to

the fact that the gages on the test specimen are not precisely on the edge of the hole

and the concentration dies off quickly. Also the tension strains were so low on the top

surface of the shell that the concrete cracking strain was never reached. This may

allow load to transfer through the concrete and lower the observed stress

concentration.

Table 8-3 Stress Around Penetration for Beam and Slab Shear Connection

Composite Deck Concrete Deck εL -0.01% 0.01% εH 0.002% -0.0005% 8.1.3.1 Radial Pressure -2.08 kPa (-.30 psi) -18.6 kPa (-2.70 psi) F.E.M. Maximum Stress ±100 Plies

-25.6 MPa (-3.71 ksi) 44.5 MPa (6.45 ksi)

Orthotropic Plate Longitudinal Stress

-9.72 MPa (-1.41 ksi) 9.78 MPa (1.42 ksi)

Orthotropic Plate Hoop Stress

37.9 kPa (5.5 psi) 339 kPa (49.2 psi)

Orthotropic Plate Maximum Stress

-26.6 MPa (-3.86 ksi) 48.3 MPa (7.00 ksi)

201

Figure 8-6 Finite Element Model for Stress Concentration Studies

8.2 EFFECTS OF TENSION STIFFENING The relations presented in Section 3.2.2.1 will be used in this section to study

the significance of the tension stiffening effect on the stiffness of a concrete filled

fiber reinforced shell. The first portion of the analysis presents the load versus

deformation curves for a concrete filled fiber reinforced shell in tension. The tension

stiffening relations were then integrated into a moment curvature analysis to

investigate the effect on the bending stiffness. Figure 8-7 shows the stress versus

strain curves for various concrete strengths based on equation (3-27). The light-weight

reduction factor of 0.75 is used to reduce the rupture stress as described in Section

3.2.2.1. This plot demonstrates that the effect of the concrete compressive strength on

the average tension stress is fairly small. For the remainder of this analysis a concrete

202

strength of 27.6 MPa (4 ksi) will be assumed. Figure 8-8 shows the effect of tension

stiffening on a concrete filled carbon shell in pure tension. The geometry of the shell

was held constant with a radius of 171.5mm (6.75 in.) and a shell thickness of

9.65mm (0.38 in.). The composite architecture was assumed to be composed of helical

fibers at ±10o from the longitudinal axis and fibers at 90o from the longitudinal axis

(hoop direction). The percentage of helical fibers was varied from 0% to 100%. In

reinforced concrete analysis the effect of tension stiffening is usually limited to an

effective area of the concrete that is judged to be close enough to a reinforcing bar to

have a stiffening effect. For this analysis the entire concrete section was considered to

give an upper bound to the tension stiffening effect. The curves in Figure 8-8 indicate

that at service conditions the effect of tension stiffening is small. If the ultimate strain

in the composite shell is assumed to be close to 1% then the service strain should not

be over 0.3% which is the maximum strain plotted in Figure 8-8. When analyzed in

bending no significant difference is discernible in the member secant stiffness.

203

Average Strain

0.000 0.002 0.004 0.006 0.008

Ave

rag

e S

tres

s (M

Pa)

0.0

0.5

1.0

1.5

2.0

2.5

Ave

rag

e S

tres

s (k

si)

0

50

100

150

200

250

300

350

f'c=27.6 MPa (4 ksi)f'c=34.5 MPa (5 ksi)f'c=41.4 MPa (6 ksi)

Figure 8-7 Average Stress vs. Average Strain For Tension Stiffening

Axial Strain

0.000 0.001 0.002 0.003 0.004

Lo

ad (

kip

s)

0

100

200

300

400

500

600

700

800

900

Lo

ad (

kN)

0

500

1000

1500

2000

2500

3000

3500

4000

Without Stiffening EffectsWith Stffening Effects

%Helicals

10%

90%

Figure 8-8 Load in Concrete Filled Carbon Shell in Pure Tension With and

Without Tension Stiffening Effects

204

8.3 THERMAL EFFECTS

8.3.1 Thermal Strains in Circular Sections

As shown in Section 3.1.6 coefficients of thermal expansion for the shell in the

longitudinal and hoop direction can be calculated from the composite architecture and

individual ply properties. Once these coefficients are known the stresses induced in the

shell, due to the difference in thermal expansion of the concrete and the composite,

can be determined.

Assuming that the longitudinal stresses are negligible the radial strain in the

concrete is given by

ε ε σ νr rT r

ccE

= − −( )1 . (8-6)

For the composite shell the hoop strain is obtained from

ε ε σH H

T H

HE= + . (8-7)

Using the kinematic and equilibrium relations of equations 4-5 along with 8-6 and 8-7

the radial stress in the concrete can be determined.

σ α ανr

c H

c

c H

T

ER

tE

=−

−+

�

��

�

��

∆ ( )1

(8-8)

The total hoop strain in the shell is then given as

205

( )εα α

ναH

c H

H c

c

H

TtE

RE

T=−

−+

�

��

�

��

+∆

∆( )1

1. (8-9)

The first term on the right hand side of equation 4-26 is the mechanical strain or the

strain that produces stress in the shell and core.

8.3.2 Thermal Testing

To quantify the effects of thermal loading on the concrete filled fiber

reinforced shells a series of experimental investigations was undertaken. These

experiments involved the thermal loading of concrete cylinders wrapped with carbon

fiber. The cylinders were tested in an environmental chamber that allowed free

expansion of the specimens. Three concrete cylinders were wrapped with all hoop

carbon shells by a hand lay-up process. Table 8-4 gives the pertinent ply properties

obtained for this process. The cylinders had one, two and three layers of carbon

respectively. A hollow cylinder for each lay-up was manufactured to be used as a

reference during testing. The hoop strains in the concrete filled shell are compared to

the hoop strains recorded in the reference hollow shell to account for strain variations

in the gage due to temperature changes. Figure 8-9 plots the mechanical hoop strain in

the concrete filled shell for the three different lay-ups tested along with the theoretical

predictions from equation 8-9. It can be seen from this figure that the mechanical

strains are overpredicted by the theoretical models. This could be due to the

coefficients of thermal expansion for the composite or the concrete as these values

were not experimentally determined but nominal values were used.

206

Table 8-4 Ply Properties for Thermal Testing

Lay-Up Property. S.I. U.S.

1-Layer E1 67.4 GPa 9.78 msi

t 0.57mm 0.0225 in.

2-Layer Ε1 75.5 GPa 10.95 msi

t 0.99mm 0.039 in.

3-Layer Ε1 81.2 GPa 11.78 msi

t 1.40mm 0.055 in.

E*t (kN/mm)30 40 50 60 70 80 90 100 110 120

Mec

hani

cal S

train

in H

oop

Dire

ctio

n (m

icro

-stra

in)

0

1

2

3

4

5

6

7

8

9

10

11

E*t (lb*106/in)0.2 0.3 0.4 0.5 0.6


Figure 8-9 Thermally Induced Mechanical Strains Per Degree Centigrade

207

8.3.3 Parameter Studies for Thermally Induced Strains

The stresses induced in the fiber reinforced shell due to the difference in the

coefficient of thermal expansion of the shell and the concrete core will vary depending

on the lay-up chosen, the radius to thickness ratio, and the materials in the shell. The

analytical methods outlined in Sections 3.1.6 and 8.3.1 are here used to investigate

these effects. For this analysis the ply coefficients of thermal expansion are taken to be

those given in Table 3-4 for E-glass-epoxy and carbon-epoxy unidirectional lamina.

The coefficients of thermal expansion for the shell in the hoop and longitudinal

direction are given in graphical form in Figure 8-10. The induced stresses in the shell

are reported for a 55o C (100 oF) temperature increase in Figure 8-11. These figures

indicate that for an E-glass shell with greater than 60% of the fibers in the helical

direction the shell will expand faster than the concrete core. For carbon this does not

occur until the shell has 86% helical fibers. The lower stresses generated in the E-glass

shell occur because the coefficient of thermal expansion is closer to that of the

concrete and the hoop modulus is lower than is the case for the carbon shell.

208

%Helical Fibers

20 40 60 80

Co

effi

cien

t o

f T

her

mal

Exp

ansi

on

(x1

0-6/o

C)

0

2

4

6

8

10

12

14

16

18Carbon LongitudinalCarbon HoopE-Glass LongitudinalE-Glass HoopConcrete

Figure 8-10 Coefficients of Thermal Expansion for Fiber Reinforced Shells

% Helical Plys

20 40 60 80

% o

f A

llow

able

Str

ess

in H

oo

p P

lys

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0Carbon r/t=10Carbon r/t=15Carbon r/t=20Carbon r/t=25E-Glass r/t=10E-Glass r/t=15E-Glass r/t=20E-Glass r/t=25

Figure 8-11 Hoop Stress in Shell 90o Plies Due to a Temperature Rise of 55 oC

209

8.4 Local Compression Buckling of Concrete Filled FRP Shells

The compression buckling problem for concrete filled FRP shells is very

complex due to the anisotropic nature of the materials, the variation in the bending

stiffness as a function of the shell architecture and the geometric stiffness effects

introduced from the expanding concrete core which develops hoop tension in the shell.

Whitney presents approximate solution techniques for thin simply supported

cylindrical shells under compression and internal pressure loading [43]. These

techniques are limited to lay-ups with no extension twist coupling

(A16=A26=B16=B26=0). The radius to thickness ratios considered by Whitney are

several orders of magnitude greater than those considered in this dissertation. The

solution techniques for thin shells neglect the transverse shear deformations of the

shell. For the materials being investigated here the transverse shear stiffness is

generally very low compared to the axial stiffness which can make the shear

deformations much more of an issue in these advanced composite shells than is the

case for metallic shells. Higher order theories which consider the shear deformation of

the shell have been proposed for various boundary conditions [44][45].

The ultimate compression strain seen in the six bending test specimens is

plotted in Figure 8-12 vs. the ratio of the diameter to the thickness. A general increase

in the ultimate strain is seen with increasing thickness with the exception of the first

full scale specimen tested which failed at an ultimate strain level similar to the thin

small scale specimens.

210

D/t

30 35 40 45 50 55 60 65 70

Ult

imat

e C

om

pre

ssio

n S

trai

n (

%)

-1.1

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

Thin Small ScaleThick Small ScaleFull Scale #1Full Scale #2Thin ConrecThick Conrec

Figure 8-12 Ultimate Buckling Strain for All Bending Specimens

211

9. CONCLUSIONS

The experimental and analytical program outlined in this document have

answered some important questions regarding the use of concrete filled FRP shells for

structural members in civil engineering applications and have opened several new

questions that need to be addressed further.

The analytical modeling of the compression behavior is based on fairly scant

data at this point but the procedure outlined is capable of incorporating new data as it

becomes available for different levels of confinement and for different concrete mixes.

The problems encountered in the compression testing of the cylinders with 85%

helical fibers could be avoided if no end capping was done and instead the cylinders

were machined with two flat parallel ends and tested directly on the steel platens of the

testing machine. An aspect ratio of at least three between the height and diameter of

the cylinders is suggested to minimize the effects of the constraint on the ends. In this

way a database of compression behavior could be built and the relations described in

Chapter 4 for the concrete tangent modulus and equivalent tangent Poisson�s ratio

could be better determined as a function of the concrete strength and weight and

composite architecture.

The bending behavior of these members was modeled fairly well with the

analytical models presented. It was found that the stiffness of the shell in the axial

direction is far and away the controlling factor on the behavior of the system. The

hoop strains in a section under constant moment were predicted well by the models

212

and were found to be small in comparison to the longitudinal strains. The interaction

between the stresses was found to have little effect on the ultimate strength of the

members and an allowable strain in the principal loading direction based on the

composite architecture used is felt to be a justifiable criteria for design purposes. All

specimens tested in bending failed on the compression side of the shell at an ultimate

strain lower than that predicted by a first-ply-failure model even though the highest

strains were on the tension side. This is not surprising as this model does not consider

the stability of the shell. The second large scale filled shell bending test was

successfully designed with more hoop plies on the outside to mitigate this buckling

failure. No attempt was made in the current work to correlate the buckling of the shell

to the thickness and composite architecture but this is felt to be an important area of

study that should be undertaken.

The shear behavior of these members was observed and predictions based on

simple mechanics models were proposed to estimate the shear strain that could be

expected in the fiber reinforced shell. By design no shear failures were observed in the

testing that was completed thus far as the tests were planned to induce bending failure

before shear failure could occur. The ability of the models presented to predict the

shear strain in the shell was fairly good but the additional hoop strain seen especially

along the centerline of the shell is not at present accounted for. A comprehensive

program to quantify the behavior of various composite architectures under shear

loading should be undertaken along with more rigorous modeling to understand the

expansion of the cracked concrete which is driving the hoop strains in the shell up.

213

Concerns over possible induced stresses from thermal expansion effects were

found to be unwarranted for the lay-ups and radius to thickness ratios investigated in

this document. The temperature ranges encountered by a civil structure are not

excessive and the coefficients of thermal expansion for a laminated composite are

generally not as extreme as they can be for an individual ply due to the fact that the

fibers are not all aligned in the same direction. The study here concentrated on the

induced stress in the hoop direction due to a temperature change. The induced stresses

and deformations of a particular structural system assembled with these members must

be studied carefully to ensure that the structure does not encounter any adverse effects

from differential expansion or contraction. Such an analysis is being carried out for the

beam-and-slab assembly currently being tested.

Stress concentration effects will become an important consideration as the

potential uses for these and other composite members in civil applications increase.

The work presented here was intended to give designers a feel for how these

concentration factors change with composite architecture so that if penetrations are

needed appropriate consideration will be given.

The choice of materials for this system must incorporate many different

considerations. Research is underway at many universities and in industry to

characterize the long range environmental durability of composite materials for use in

civil infrastructure applications. From the work presented here it can be seen that with

the current cost estimates the carbon shells are more efficient for providing

confinement and E-glass shells are more efficient for providing bending stiffness. If

214

the cost ratio for carbon/E-glass is reduced to 2.5 carbon becomes the more

economical choice for both applications. The use of hybrid shells to optimize cost is

possible. It has been demonstrated in this dissertation that the compression behavior is

strongly dominated by the hoop fibers and the bending behavior is strongly dominated

by the helical fibers. The fluctuating cost ratio of the various materials must be

considered along with the potential complications in the manufacturing processes

when considering a hybrid shell.

The two shapes investigated in this document show strengths and weaknesses

for different applications. The circular shell is excellent for confining the concrete

core. It was also shown to be a good choice for bending applications if the total height

of the section is not a consideration as it offers superior stiffness for the same amount

of material as compared to the conrec section. If the total height of the section is a

critical consideration substantial stiffness increases are possible with a conrec section.

The loss of confinement efficiency in the conrec section was shown to have little

effect on the overall performance of the system in bending applications. Connections

to adjacent members may be simplified with the use of a conrec section as it offers flat

areas avoiding line contacts.

The construction procedure used to build the test specimens was extended to a

full scale bridge section currently undergoing fatigue testing in the structures lab at

UCSD. The construction procedure was found to be straight forward and the potential

for reduced erection times is felt to be substantial.

236

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UNIVERSITY OF CALIFORNIA, SAN DIEGOadavol/dissertation/dis_wp.pdf · university of california, san diego ... concrete cylinders under uniaxial compression ... figure 5-5 typical compression