modeling of confined concrete by esneyder montoya
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MODELING OF CONFINED CONCRETE
Esneyder Montoya
A Thesis submitted in Confomiity with the reqllirements
for the Degm of M88ter of Applied Science
GradwPi, Department of CM1 Engineering
University of Toronto
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Modeling of Confined Concra8
Master of Agplied Sciem, 2000.
Graduata Deportment of Civil Engineering
University of Toronto
Abstract
Constitutive modelr for confineci conciete were invesügated using the
nonlineer finite dement program SPARCS, in which the influence of lateml
pressure in tiad reinforced cancrete (RC) columns subj8CSeâ to monotonie axial
compression is fonnulated in ternis of threedimensional stress states. A
combination of ascending and decrcending k a r t c h of axial stress- axial strein
relationships for confineci concrete proposeâ by various authors mre evatuated.
The influence of variable Poirwn's M o , cornpression sdening of concnite, and
concret8 -ver spalling in the ~88ponse of RC columns nnrs alro analyzed. Data
from cdumns testeâ by various remarchem was u W to establirh the validity of
the pro08dums implemented in SPARCS. In g m l , g m û agreement wîth the
Acknowledgements
The author is grateful for the constant support of Professors F.J. Vecchio
and S. A Sheikh during this project; and the financial support provided by the
Netunl Sciences and Enginming Research Council of Canada.
Contents
Ud of figures
List of tablas
1. Introduction
1.1. Overview of Concrete Response
1.2. Objectives
1.3. Summary
2. Literrture Review
2.1. Plasticity and Fradure Energy based models
2.1 .1 . Liu and Foster (1 998)
2.1 -2. Xi.; MacGregor; and E M (1 996)
2.1 -3. Karibanis and Kiousir (1 993)
2.1 -4. Chen and Mau (1 989)
2.2. Linear and Nmlineat Elastic ModeCs
2.2.1. Mau; EM; Md ïhou (1 998)
2.22. Bamgar cnd Maipudi (1997)
ii
iii
xiii
wxi
1
1
3
4
7
8
8
10
12
13
15
15
16
2.2.3. Bortolotti (1 994)
2.2.4. Sdby (1990)
2.2.5. Vlcehio (1992)
2.2.6. Selby and Vecchio (1 993)
2.2.7. Abdel-Halirn and Abu-Lebdeh (1 989)
3. SPARCS 24
3.1. Program Description 24
3.2. Finite Element Library 26
3.2.1. Hexahedron 26
3.2.2. Pentahedra (Wedge) 27
3.2.3. Tniss Bar 28
3.3. Corwtitutive Models for Concrete 29
3.3.1. Base StmssStrain Cuwes for Concret8 in Compmssion,
Pre-Peak Behaviour
3.3.1 .l. Hognestad parabole
3.3.1 -2, Thorenfeld et al.
3.3-1 .3, Hoshikuma el al-
3.3.2. Base Shbu-Strain CUIV~S for Concret8 in Compnmion,
Pos-Peak Behaviour 33
3.3*2.1. Moditied Kent and Park (Scott 1982) 34
3.3.2.2. Popovicr (1 973) 35
3.3.2.3. HoshiCaima ebt al. (1 996) 36
3.3.3. Base Stress-Strain Cuwe fw Concmte in Tension
3.3.3.1. V m i o (1982)
3.3.3.2. CollinbMitchell(1987)
3.3.3.3. larmo, Maekawa et. al.
3.3.4. Feilue Criteria for Conmete
3.3.4.1. Hsieh-Ting-Chen Criterkm (1 979)
3.3.4.2. Cracking Criteria
3.3.4.2.1 . Mohr-Coulomb Criterion (stress formulation)
3.3.4.2.2. Moht-Coulomb Criterion ( m i n formulation)
3.3.4.2.3. CEB-FIP Criterion
3.3.5. Confined Strength tp and Stmin at Peak Stress sp
3.3.5.1. Selby (1 993)
3.3.5.2. Veochio (1 992)
3.3.6. Compression Sdtening
3.3.6.1. V d i o and Collins (1982)
3.3.6.2. Vecchio and Collins (1986)
3.3.6.3. Vecchio 1992-A
3.3.8.4. Vecchio 1992-8
3.3.7. VafiabCe Poisson's Ratio
3.4. Constitutive Modd for Steel
3.5. SPARCS Stnidun
- - - - - - - - -
4.1, Introdudion
4.2. Selection of Parametric Variables
4.2.1. Pre-Peak Base Curves for Conccete
4.2.2. Post-Peak Base Curves for Concrete
4.2.3. Concmte Cracking
4.2.4. Confinement Enhancement
4.2.5. Variable Poisson's Ratio
4.2.6. Compression Softening
4.3. Sheikh and Uzumeri Tests (1980)
4.3.1. Column Geometry
4.3.2. Longitudinal Bar and Tie Setup
4.3.3. Test Instrumentation and Procedure
4.3.4. Selected Sheikh and Uzumeri Columns 60
4.4. Finite Elements MocCels
4.4.1. Geometry
4.4.2. Material Types
4.4.3. Parameter Combinations
4.4.4. Anelyris PrOCBCILIre
4.5. Analysir Rewlb
4.5.1 . Column 2A1-1
4.5.1 -1. Cmss Section and Profile Results
4.5.1.2. Lod-Defomiation Curves Combination 2
4.5.1.3. Summary of Load-ûehnWion Cuwes 74
4.5.2. Column 48349
4.5.3. Column 2C5-17
4.5.3.1. Cmss Section Results
4.5.3.2. Summary of Load-Deformation Cuwes
4.5.4. Column 406-24
4.6. Effedr of Mode1 Combination on the Response of the
Seleded Columns
4.6.1. Peak Load
4.6.2. Strain at Peak Load œ
4.6.3. Post-Peak Behavior
4.6.4. Effect of Compression Softening
4.7. Cornparison of Analytical Results with a Previous Version
of the Program
4.8. Study of Poisson's Ratio
4.8.1 Procedure to Obtain the Experimental Poisson's Ratio
4.8.2 €xperimental Variable Poisson's Ratio
S. ConObontion with ExpedmntJ Studios 122
5.1. Introduction 122
5.2. Lui, Forâr, and AttaICf Tests (1 998) 123
5.2.1. Column Gemetry 123
5.2.2. Longitudinal and Lateral Steel Anangement8 124
5.2.3. T m kidnmentsüon end P w 124
Con t8 nt
-
5.2+4+ Seleded Liu at al. Columns
5.2.5. Fhite Element ModeIr . 5.2.5.1 . Geometry
5.2.5.2. Material Types
5.2.5.3. Material ModeIr
5.2.6. Analytical and Experimontal Results of Liu el ai. Columns
5.2.6.1 . Column 2C64-10S50-15
5.2.6.2. Column 2C60-1 OS1 00-1 5
5.2.6.3. Column 2C6û-1 OS1 50-1 5
5.2.6.4. Column 2C8410S50-15
5.2.6.5. Column 2CMSSO-15
5.3. Mander et al. Specimens (1 984)
5.3.1 .Column Geometry
5.3.2. Longitudinal and Lateral Steel Arrangements
5.3.3. Test Instrumentation and Procedure
5.3.4. Selected Mander el al. Sp8dmens
5.3.5. Finite Element Modds
5.3.5.1. Gmetry
5.3.6. Analyticil and Experimental Reurlts of Mander et d. Specimens
5.3.6.1. Wall 11
5.3.6.2. Scott Column
5.4. Sheikh anâ U z u M Colwnns
5.4.1, Column 4-
Content
5.4.2. Column 484-20
5.4.3. Column 4D3-22
General Reaultr 153
Analysir of Short Column Sedias Oesigned According to
CSA23.3-94 and AC191 M 5 R Code Provitsions 155
Cornparisons with Sheikh-Uwmeri and Rasvi-Saatdoglu
Confinement Moâels 159
Finite Elernent Analysis of a Shear Wall 164
6. Conclusion
6.1 Conclusions
6.1.1. Summary
6.1.2. Specific Conclurions
6.2 Improvements, Limitations and Remmmendations for Future Work
8.2.1. Impmvemnts in SPARCS
6.2.2. Limitations of the Analysis
6.2.3. Recommendaüons for Future Work
Appndices 178
A ôesfgn of Short C d u m &dons .ceodîng to CSA23=3-81
8nd ACll.318-8UR 178
X
Mo&Iing of cenfinedCOlrcna&
Al. Section Pmperties
A2. Design using Canadian Standard CSA23.3-94
(Seismic Provisions Section 21)
A3. Design check American Standard ACIœ31&95R
(Seismic P ravisions Sedion 21 )
B. Shaikh and Uwmrri, and R u v i and Srritcioglu Modal Calculations
B. 1. Data for Wall 1 1
8.2. Sheikh and Uzumeri Model
8.2.1. Parameters
6.2.2. Stress-Strain Cuwes
8.2.3. M a l Load Versus Axial Strain Cuwe
8.2.4. Parameten d;r. Q, and ki for All Columns
8.3. Rasvi and Saatcioglu Moûel
8.3.1. Parameters
8.3.2. Stress-Strain Cuwes
6.3.3. Axial Load versus Axial Strain Curve
8.3.4. Rarvi and Saatcioglu Parameten for All Columns
8.4. Axial Shortening CUMS
8.4.1. Sheikh and Uzumeri Columns
8-4.2. Liu el al. Columns
8.4.3. Scott Colunn (Mander)
-
C. Vuirble P O ~ ~ ~ S Ratio in Liu et al. Columiis 206
List of Figures
3.1. Eightnoded brick (hexahedron)
3.2. Deformed brick
3.3. Pentahedra (Wdge)
3.4. Truss Bar
3.5. Hognestad Parabola
3.6. Generalized Popovicr Strsss-strain Cuwe
3.7. Ascending Branch of Hoshikuma et al. Mode1
3.8. Description of Laterd Prassure f i
3.9. Adapteâ Version of the Modified Kent and Park Post-Peak Model
3.1 O. Popovicr Port-f eak Relationrhip
3.1 1. Hoshikuma et al. PosbPeak Curve
3.12. Reinforcd Concrete in Tension
3.1 3. Mohr-Coulomb Criterion
3.14. Comprsuion Soflening Ef fd
3.1 5. Lateml Expansion
3.16. Poiuon's Ratio Relationrhip
3.17. Steel Model 48
3.18. Secant Modulus Definition 51
3.19. Solution Algorithi for SPARCS (Selby and Vecchio 1993) 53
4.1. Column Dimensions 58
4.2. Shdkh-and-Uzumeri Colurnn Sections 59
4.3. Section Datail for 2A1-1 62
4.5. Material Types
4.6. Concrete Axial Stress for Combination 2 (2A1-1)
4.7. Variation of sd /o, Ratio for Combination 2 (2441 -1 ) 71
4.8. Profile Deformation, Column 2A1-1 (Combination 2)
4.9. Concrete Lateral Stress States, 2A1-1, Combination 2
4.1 0. Rqxmse of 2Al-1, Combination. 2 73
4.1 1. Lateral Reinforcement Response of 2A1-1, Combination 2 74
4.12. Axial Response of 2A1-1, Combination 1 75
4.13. Axial Respnam of 2A14, Combination. 3 75
4.1 4. Axial Responre of 2A1-1, Combination. 4
4.15. Axial Rmponse d 2A1-1, Combination. 5
4.16. Axial Response of ?Al-1, Combination. 6
4.17. Axisl Response of 2/41 -1, All M e t Combinations
4.1 8. Lateml Reinbmmmt Respmse of 2A1-1, Ail Combinations 78
4.1 9. Axial Respome of 483-19, Combinatbn. 1 81
4.20. Axial Rerpocire of 483-19, Combinafion. 2 82
4.21. Axial Response of 463-1 9, Combination. 3
4.22. Axkl Responro of 483-1 9, Combination. 4
4.23. Axial Responw of 4B3=1@, Combination. 5 r
4.24. Axial Response of 483-1 9, Combination. 6
4.25. Axial Respnse of 483-1 9, All Model Combinations
4.26. Confinement of 2C5-17, Combination. 2
4.27. Sketch of Tie Strain Distribution, Column 2C5-17
4.28. Axial Repense of 2C5-17, Combination. 1
4.29. Axial Responw of 2C5-17, Combination. 2
4.30. Axial Response of2C2C5- Combination. 3
4.31 . Axial Responro of 2C5-17, Combination. 4
4.32. Axial Response of 2CS-17, Combination. 5
4.33. Axial Response of 2C5-17, Combination. 6
4.34. Axial Response of 2CS-17, Combination. 7
4.35. Axial Response of 2C5-17, All Model combinations
4.36. Lateral Reinfbmment Response of 2CS-17, All combinations
4.37. Axial Responw of 4W-24, All Madel Combinations
4.38. Compeiirons of Strength Increasa
4.39. E f f e of Portpak Cuwes (1: W i e d Kent 8 Parlc,
2: Popovicr, 5: Hoshikuma et ad.)
4.40. Effed of Compnssion Sdtening
4-41. R.uilt C O ~ ~ ~ O ~ S f ~ r 2Al-1
4.42. Rouit Comwms Ik 2CH7
-
4.43. Equilibrium of Forces in a Circuler Section
4.44. Load-Strain Curve of Column 2C90-1 OSlûû-25
4.45. toad-Poisson's Ratio Cunre of Column 2C9û-iOS100-25
4.46. Poisson's Ratio M a l Strain Curve of Column 2C90-10S100-25
4.47. Volumetric Strain Variation of 2C90-1 OS1 00-25
4.48. Load-Strain Curve of Column 2CûMS50-15
4.49. Load-Poisson's Ratio Curw, Colum 2C80bS50-25
4.50. Poisson's Ratio Axial Strain Cuwa of Column 2C8WS50-25
4.51. Volumetric Strain Variation of 2Cüû-6S50-25
4.52. Lateral Expansion Variation of Series I
4.53. Laterd Expansion Variation of Series II
4.54. Lateral Expansion Variation of Series III
5.1. Detail of Columns Tested by Liu et al.
5.2. Cross Section Model for Liu Columns
5.3. Material Types for Liu et al. Columns
5.4. AxiPl Shortening of 2C6û-lOSSO-15
S.S. Lataml Expansion of 2C60-1 OS5W 5
5.6. Aaal Shortening of 2C80-iOSI 00-1 5
5.7. Lateml Expansion of 2C60-1OS100-15
5.8. Mal Shortening d 2C80-1OS150-15
5.9. Lateml Expansion of 2C6û-1OS1 SO-15
5.10. Axial Shortming of 2CBO-1OS50-15
5.1t. Latemi mnsiond2C80-fOSSO-15
-- - -
5.12. Axial Shortening of 2CB06SSO-15
5.13. Lateral Expansion of 2C806SSO-15
5.14. Mander et al. Wall
5.15. Square Column of Scott SWes
5.1 6. Finite Elernent M W for WalI 1 1
5.1 7. Prototype of a Hollow 8ridge Pier
5.1 8. Axial Shortening of Wall 11
5.1 9. M a l Shortening of Scott Colwnn
5.20. Axial Response of 4C6-5
5.21. Axial Response of 4 M
5.22. Lateml Expansion of 4B4-20
5.23. Axial Response of 403-22
5.24. Maximum Anal ytical to Eqmimental Load Ratio (wo J Softening )
5.25. Maximum Analyücal to Experimmtal Load Ratio (W. I Softening)
5.26. Localization of Cr- Sections
5.27. Column Section and FE Mesh (CSA, ACI)
5.28. Axial Shortening of Section in Central Zone
5.29. Axial Shortming of Section In Plastic Hinge ZOM)
5.W. W l Compwiron for Column 483-1 9
5.31. Model Compwiwxi of Column 4û6-24
5.32. Modd Compatkon of Colunn 2 C 8 M H M S
5.33. Modd C o m p s r i for WalI 11
5.34. Ldsr et al. Wall SW16
Content
5.35. Finite Element Mesh for Wall SW16
5.36. Material Zone distribution for SW?6
5.37. Horizontal Responm of Wall SW16
5.38. Sketch of Horizontal Displacement of Wall SW16
A. 1. Design Sedion Using CSA and AC1 Codes
B.1 Cross Section of Wall 1 t
8.2. Strain Cuwes for Conwete, Wall 11, Sheikh and Uwmeri Model
8.3. Stmin Curve for Longitudinal Steel, Wall 1 1
8.4. Load-Strriin Curve of Wall 1 1, Sheikh and Utuneri Model
B.5. Stmin Cuwes for Concrete, Wall 1 1, Rasvi end Saatcioglu Model
8.6. Load-Strain Cuve of Wall 1 1, Rasvi and Saatcioglu Model
8.7. Column 2A1-1
8.8. Column 466-5
8.9. Column 2C54 7
B.10. Column 483-1 9
B. 1 1. Column 464-20
8.12. Column 4D3-22
6.1 3. CoCumn 4D6-24
B.14. Column 2-1OS50-15
B-15. Column2C60-1OSi~~5
8.16. Column 2C60-15û-15
8-17. Column 2CSO-1OS5û-15
B.f8, Column 2 C M 5 0 - 1 5
B. 1 9. Scott Column
C.1, Load-Strain Curve of 2CW-1 OSHM 5
C.2. LU-Poimon's Ratio C w e Of 2-1 0550-15
C.3. Posson's Ratio-Axial Strain Cunre of 2C60-t 0SW-15
C.4. VolumeûieAxial Strain Curve of 2C60..10S50-15
C.5. Load-Strain Cuwe of 2C60-3 OS1 00-1 5
C.6, Load-Poisson's Ratio Curve of 2C60-1 OS1 00-1 5
C.7. Poisson's Ratio-Axial Strain Curve of 2C-1 OS1 00-1 5
C.8. Volumetric-Axial Strain Curve of 2C-1 OS1 00-1 5
C.8- Load-Sîrain Curve of 2C60-1 OS1 50-1 5
C.10. Load-Poisson's Ratio Cuwe of 2C6û-1 OS1 50-1 5
C.11. Poisson's Ratio-Axial Strain Curve of 2C60-iOS150-15
C. 1 2. Volumetric-Axial Strain Curve of 2C-1 OS 1 50-1 5
C. 13. Load-Strain Curve of 2C80-1 OS504 5
C.14- Load-Poisson's Ratio Cuwe of 2C80-i OS5û-15
C.l S. Poisson's Ratio-Awal Strain Curve of 2C80-1 OSSO-1 5
C.16. Volumetri~~Axial Strain Curve of 2C8û-10S50-15
C.17. LoadSttain Curve of 2Cûû-6S100-15
C.18. Load-Poisson's Raüo Cuwe of 2C80-6Slûû-15
C.19- P o i m ' s RBtiOlAXial Strain Curve of 2CûMSlûO-15
C.20. V o l u r n ~ ~ a l Strain Curve of 2cBo-6S100-15
2 Load-Strain Cuwe d 2C906S100-25
C.22. Lod-Poisson's Ratio Cwe of 2CQWS100-25
C.23. Poisson's Ratio-Axial Stroin Cuve d 2C908S100-25
C.24. Volumetric-Axial Strain CUM of 2 C W S 1 Oe2S
C.25. LoadSWn Cuwe of 2C90-6S5ô-25
C.26. Lord-Poisson's Ratio Cunn d 2C906650-25
C.27. Poisson's Ratio-Axial Stnin Curw of 2C90-6S50-25
C.28. Volumetri~~Axial Stnin Curve of 2CgOSSiôU-25
C.29. LoadSWn Curve of 2 C S 1 OS1 00Q
C.30. Load-PoiuonJr Ratio Cuwe of 2C90-1 OS1 00-0
C.31. Poisson's Ratiodxial Strain Cuwe of 2C90-1 OS1 00-0
C.32. Volumetric-Axial Strain Cuwe of 2C90-1 OS1 000
List of Tables
4 1 Sheikh-and-Uzumei Sel8Cfed Column Properties
4.2. Finite Element Model Gemmetry
4.3. Element Material Types
4.4. Parametric Combinations for SPARCS Analyses
4.5. Analytical to Experimental Ratios for 2Ai -1
4.6. Analytical to Experimental Ratios for 483-1 9
4.7. Andytical to Expeimental Ratios for 2C517
4.8. Analytical to Experimental Ratios for 4û6-24
4.9, Analytical to Expewirnentel Peak L o d Ratios
4.10. Strength lncreare
4.1 1. Anelytical to Experimental Peak Strain Ratios
4.1 2. Selby'r anâ Cumnt Modela
4.1 3. Liu et al. Column Pmpwtim for the Study of Poisron'a Ratio
4.14. a and Y. Values fbr Liu et al. Columns
5.1, Liu et ai. Columns, Material PmpWes
5.2. Gmmby of Finite Element Modela
Con- -- - - - - - - - - -- - - - - - -
5.3. Liu et al. Material Types
5.4. Mander et al. Material Properties
5.5. Material Propwties of Additional Sheikh and Uwmeri Columns
5.6. Maximum Analyüaal to Gcprimental Load Ratio
5.7. Short Colurnn Sections in Seismic tones, Model Comparisons
5.8. Theordical to Gcperimentai Maximum Load Ratios for Square and
5.9. Theoretical to Experimental Maximum Load Ratios for Cirwlar Columns
5.1 0. Material Properties of Wall S W16
5.1 1. Material Zones for FE Model of SW16
k 1. Code Provisions for designing Compression Members
B. 1. Wall 1 1 Properties
6.2, Parameters for SheiWi-Uzumeti Model
8.3. Rasvi and Saatcioglu Parameters
Chapter 1
Introduction
Three-dimensional nonlinear finite element analysis of reinforced
cancrete has been investigated for almost four decades. Constitutive material
models that take into account the influence of triaxial states of strains and
stresses have been developed for both plain conmete and steel.
These models have b e n based on piinciples founded in theories such as
plasticity, fracture mechanics and elasticity (linear and nonlinear). In classical
plasticity theory, both materials (i.e., conmete and steel) behave elastically until
'yieldingn; then, materials behave plastically and follow associative or
nonassociative flow rules. The conœpt of Yfradure energy" is used in ftacture
mechsnicr to establish failum criteria that depnd on the state of stress to which
an element of concmte is subjected. Failuo sucfaces define the upper boundery
of conaute ttrwigth. Finally, in linear elasticity, the simplest mode1 follmvs a
Hdce's Iaw whefe stresses am dindly PIopOrtional to Wainr, without changes
in matrn0al pmpdesw In nonlinear elasticily, comate a d steel bahve
material properties are changed as load increaws (or deweases). Secwit and
tangentiel stiffness matri*œs have ken developed to account for the change in
material behaviour, and a number of mocklr have been implemented in finite
element programs mat use eithew formulation (Chen 1982).
Behaviour of confined c o m t e is different ftom that of unconfined
conuete. Concret8 cm be considered mnfined when subjeded to triaxial
comprenions; the triaxial compression increases the conaete'r capacity to
sustain largef compressive sbmgVis and deformations. When a concrete
element is laterally rdnforced (e.g., by ties, hoops or rpirals) and subjected to
&al compression, lateral expansion of aie ekment in the plane perpendicular to
the axial compression activates the lateral steel, which confines th8 element by
exerting lateral pressure. Confined concrete generally fails in a ductile manner,
Werear unconfineâ comtete fails in a bfittle manner. As tensile strains develop
in unconfined concret8 rubjecteâ to compression, concret8 Mens and strength
demases. It is also knuwn that Poisson's ratio for concrete is not constant as
load increases; it inereares with m*al strain increments. This phenornenon is
beneficial in activating lateral steel.
The degm of confinement ir masund analytically by the increment in
compressive strength and compressive itrriin at peak stress with respect to the
unconljneâ compn#rive aC8ngVi and m i n at peak, respecüvely.
1.2 Objectivas
The general purpose of this work was to detemine the capabilities of
SPARCS to model confimd concrete. SPARCS is a nonlinear elastic finite
elemmt pmgnm developed a the University of Toronto (Sdby 1990, Sdby and
Vecchio 1993) for Vie analysis of minforced concret8 solids. The prognm user
constitutive models bas& on the Modifed Compression Field Theory (MCFT)
(Vwxhio and Collins 1986) and has incorpo,mted simple finite elements such as:
trws bars, &iodeci bricks and h o d d wsdges. Details of the program will be
given in Chapter 3.
The rpific objectives of this work can be wmmarized as follawcr:
r To extend the capabilities of SPARCS to analyze large models unâer
imposed displacements.
r To m a k a parametric study on the influence of various models on confined
concrete behavior; incîuding: barn stress-strain curve for c o m t e in
compression; compression softening; cracking criteria; strength
enhancement; and Poisson's ratio.
To impiement baae stress-$train m e s for pre-peak and post-peak behavior.
r To corrObOTate the analytical model for variable Poisson's ratio impl8mentd
in SPARCS wHh expwimentaI wlb.
O To corroborate the COCtfimment modrls i m p l ~ t e d in SPARCS with
cdumns testad under mOclOtmic i m s i n g axial campnuion.
such concealeâ colwms in wllk and
To comment on ACI41û-95 anâ CWCAN 4423.3094 code provisions for the
design of tie setups of short columns ôased on msults obtained fmm
m-ld sections that satirfy Wir requiriwnts.
1.3 Summuy
A set of ten columns testd by Liu et ai. (1998) was analyzed to detemine
the validity of the variable Poisson's ratio mode1 in SPARCS (Kupfer 1869). Plots
of the axial load venus Poisson's ratio, and axial stmin versus Poisson ratio
obtained from the expwimental msuk will be given in Chapter 4.
For the parametric study, a set of four cdumns tested by Sheikh and
Uzumefi (1978) wem modeled with SPARCS and analyzed to establirh the
senribiiity of each of the modela rnentioned above, in the ôehaviour of confined
conmete. Dwing mis study, the stress-$train m e for concrete proposed by
Hoshikums et al. (1996), and a tentative mode1 for strength enhancement
proposed by Vecchio (1992) wen implemented in SPARCS and used in the
parametric study. A tt\Orough mvision of the mwilts of this study will al«, k
given in Chapter 4.
Once the panmeûic study was completed and the influence of each
modd haâ be8n eaiiblished, sets of cdumnr testecl by Liu et al. (1 QM), Mander
et al. (1988), Scott et al. (1982)' and thme add i t i d Sheikh and Uarmwi
colurnns wsn modokd in SPARCS and cuinpad and d i w d with the
- - - - -
axial strain arrve, the experimental to analytical p a k l a d ratio, the stfength
gain in the concrete, and the lateral expansion (i.e., tie or spiral strain history).
Two short cdumn sections d-igned according to the AC1 31û-95, and
CSAiCAN 23.3994 provisions were modeled with SPARCS. The results are
discussed almg with the analytical cwobwation in Chapter 5.
The Sheikh-Uzumeri (1 982) and Rezvi-Seatdoglu (1 999) analytical
models for confinai cdumns were used to cornpute the m*al response of the
specimens modeled with SPARCS. Camparisons of these models with the finite
element solutions are given in Chapter 5 and Appendix B.
A genenl discussion and conclusion of aie results is pmsented in
Chapter 6, with reaimmendations for Mure work
Due to the amount of msub deriveû from SPARCS, an spcial-purpose
post-processor program with threedimensional graphical and analytical
capabilities w u developeâ for cheddng and plotüng: geometry, deformation
animation, sûess and strain statea in cross sections, key indiceton of damage,
secant moduli convergence. Some of the plots prssented in this wwk have b e n
taken directly fmm thir poa-pœuor.
In general, al1 the column models analyzed with SPARCS showed
excellent correlation with the experimental result~. The strength gain avetrago
was 3% higher than the actuel tests wiOi a standard deviation of 11 %. The port-
peak behavior of romo d the modeleâ Sbikh and Uawneri calumns was
significantly impved w t m compeâ with a pnvious version of the pmgram
(Selby anâ V a i o 1993). ExpwhentaI 1oaddal m i n wwm for Liu et al.
columns w r e adequately simulateâ. The ovmll behaviw of th Scott and
Mander specirn8ns was wll capturd by SPARCS.
Finally, the cumnt mdel iq tooh of SPARCS will help in the
understanding of cornplex elementr or sûudums whem thmedimensional
analysis is unavoidable.
Chapter 2
Literature Review
Confined mcmte can be defineci as ancrete that is restrained latemlly
by reinforcemant wnsisting of steel 8timps or spimls. This reinforcement exerts
lateral passive pressure against the concret8 a8 it expands due to the Poisson's
efbct when subjected to compressive load in one direction. The major effect of
the confinement is to enhance the stfength and dudility of reinforcd oonaete.
Concrete conlinernent ha8 been studied sin- early in the century
(Richart et al. 1928), and with smal interest in column behavior for the last
Wree decades (from Kent and Park (1972) to Rami and Saatcioglu (1 998)).
Constitutive material models for concrete in triaxial stress states have
been proposeci and adopted for use in numeical analyses, uring fradure
mechanicl, plaaticity or nonlinear elasüc analysis approaches. Stress-min
cuwes for confined conmete have bmn deriveci fmm @ally loaded cdumns
tested undet different load rates.
arrangements; tie setup; lateml steel spadng; and cover dimensions (mg., Ksnt
anâ Park 1972, Sheikh and Uuimeri 1978, Scott et al. 1982, Mander et al. 1988,
Rami and Saatcioglu 1992, Curwn and Paultre 1995). The analytical stress-
m i n wwes were fit with great accuracy to mts of colwnns teaed by their awn a authors, but they Iack general applicability.
The search for a general constiMive mode1 that can be applied not only
to reinforcd concrete columns, but to confined concrete in other structure
elements where threedimensional behavior can be expeded (i.e., concealeâ
columns in walls, beamcolumn joints), is undennay.
This chapter describes b M y sorne of the recent publisheâ workr in
rnodeling of confineci concrete, its applicability and success, as well as a brief
discussion of each one.
2.1 Plaatidty and Fracture Enrrgy Basad Moâels
2.1.1 Uu and Fostrr (1 888)
The authors revised the material mode1 proposed by Caml et al. (1992),
and calibrateâ the parameten of the mode1 for highstrength concmte (HSC).
The 'mkmprane modal"of Carol et al. is bared on the microplane conœpt
introduœd by Bazânt et al. (1984). At the micro Ievel, an arbitrary plane that
passe8 through a point maiin the concret8 is used to define nonnal and shear
rtnins acting on the plane. The fomicK Pcb perpmdiailarly, and the latter acts
parailel to the plam
The normal strain is divided into its volumetic and deviatoric
components, and rtressrtnain nlatiwhips for the volumetric, deviatoric, and
tangentiel stresses (i.e., shear stresses) are detemineâ using empirical
parametem. A mlationship ir thm established ôetween the microplane stresses
and macro stress tenrors using the principle d virtual work. The change in the
stress tensor dm, is given by
where D~~ is the tangential material stiiess matrix, and d e is the change in
the strain tansor. It was assumed that conesponding 8tresses and strains have
the same direction (i.e., shear stress and shear m i n , normal stress and normal
strain.).
The empirical parameters needed to calibrate the mode1 showeâ a wide
range of variation; as pointed out by various reseanhers. The authon
developed an wigsymmetric m&l, using the rnodel of Card et al. to analyze
confird and unconfined HSC, with sûengths up to 100 MPa.
A set of five parameters wvm calibrated for diffatent cancrete strengths
and different valwr of the aJas ratio, Wre trf ir the major principal stress, and
is the rninor principal stress.
confined cylinders with d-t confining pressures tested by Dahl (1992) were
used to match the expewimental data with the analyticd results, showing good
agreement in both pre- and post-peak behavior.
An analytical cirarlsr cdumn was alro modeleci using either four- or eight-
node cwisymmetric dementa, tnirs bars for the longitudinal steel, and a point-
type dement for the lateral confining steel. It was shown thot the analyses using
fow OF e1gM d e elemento are similar for concrete strength up to 60 MPa and
diier for 90 MPa comte. Radial expansion in the cirarlar section seems to be
concentrateci in a hinge region, after the peak stress is reached. The stimips did
not yield at peak load and the mode1 was able to represent convete softening
and cowr spalling, as well as dudile or brittle post-peak behavior.
2.1.2 Xie; MacGmgo~ and Elwi (19B6)
ln remnt yearr, various researchen have testeci high strength concrete
(HSC) columns, and have k n investigated finite element column models of
normal strength concrete (NSC).
Numerical analyris plays an important rob in investigating parameten af
column failum wid stretches #a study of column behaviour beyond expetimental
tests. The oôjective of thir paper was to evaluate analyücally, the behavior of
four columnr tested at the University of Alberta. Thom analyses were canjed out
using a finite e l m t d l in which th8 material mockl proposed by Pramono
and Willam (1 989) wrrt implemnteô.
The constitutive d l is a f n d u n 8nergy-bad plasticity fomiulatian
thrit use8 the same fomiulaüon for c o m ~ r # r h and tamion in CO(Nnefe and
considen rtroin Menin9 for triaxial states (tensile, compressive and triaxial
compmrive -8). At initial load stages, the modd bhaves elastically
within a yied s u m . As load increases, plastic flow ocairs within a failure
surface accofding to a nonassociative ruk. if plastic flow continues beyond
failure surface, the material will follow a suftening path.
The failure criterion u s d war proposd &y Leon (Romano 1969) and was
rnodified to account for the triaxial stress states. Parameters needed in the
material mode1 wem adjusteci for HSC and wem taken from data obtained from
cylinders testecl at the Univemity of Alberta. Specitically, a n w definition for
'cmck spaUg" as a function of principal stress ef accounts for l e u stMened
HSC cdumns with moderate confinement. Finally, the program ABAQUS was
ured to implement the new mockl.
Four ~olumns testeci by Ibrahim and MacGregof (1994) were chosen to
pmvo the analytical rnodel. These columnt had equal dimensions and steel
properties, but different hwk spacing and concrete strength, with the latter
ranging from 59.3 to 124.8 MPa. Eccentric loading was applied, cawing a zero
stress condition on one of the dges of each column.
One quarter of each d the colunnr was mudeled wing the ABAQUS 20-
node brick fw concret8 elements in which longitudinal and pamllel-to-eâge
reinforcement wre mbedded (i.e., meamd out into concret@). Oiamond-type
internai sümp8 were W i e d as ûuu bars, anâ -ab aswmptions wem
made for d i r p l ~ - c o n t r o I I e d loading and mrtnintr.
Litemmehiew 12
The msub agnw, well with 8xperimental values ancl the materiil mode1
has enough capscity to describe failum modes for columns ranging from poorly
confinai to well conflnd. Poorly confined columns prssented brittle behavior
after pek load. Alüiough triaxhal compressive stress sûates developed at tk
levels, local failure occurred at midlevel betwwn stimipr due to tenrile strains
in the c o m t e wre.
Cover spalling almg the compretssive fa- was typical of wellconfined
colmn failure. High moment and load capacities after cover spalling were the
result of highet lateral pressures due to yielding of stimipr.
ABAQUS a l l m the uwr to input material models and failure criteria. It
seems that some difficulties arose h m the lack of automation in load increments
within the program. Use of ûuss ban attachexi only to the ends could not
adequately mpresent the &ual confining behwior.
2.1.3 Karibanis and Kiousis (1893)
A Drucker-Prager plasticity-type moâel was developed to analyze aie
behaviout of columns confined with either tier or spiralt. The mode1 indudes a
nonassociative flow ruleI rtfain hatûening, and a limitai tensile strength for the
concmte. Steel plasticity was modeled with a Ramberg-ûsggood sûess-rtnin
wrve.
ïhe elastic behaviow of ancrate f d lws a simple H W s IawI and in h
plastic range follows a loaâing function which ir based on a min hardening
funcüon of pkticrtnins. (Le., imv(WQbie rtninr) .
ElartopMic stress incrementa mm, computeà wing e rather
complicated constitutive mabrix ttmt -8 waluated numerically. In this model,
different confining zoner betwwn ties or spirals weie defined. The least
confined zone was mid-way between lies. While plastic behavior was obsmed
in the wsakest zone, other zoner could experienœ elastic unloading during
large deformations (i.e., postpeak behavior)
The mode1 sucwssfUlly predided the behaviour of some of the columns
tested by Mander et al. (1998), and lyengar et al. (1970). H is to be noted that
the mode1 was developed only for circular columns, but can be u s d in columns
of different size because it is not bssed on Mistical regressions of a
determineci set of specimens.
2.1 A Chan and Mau (1 888)
The authors recalikated the mode1 proposed by Bezant and Kim (1979)
with ercperimental data from uniaxial biaxial and triaxial tests from other
nwearctiers. Regions of high axial compressive stress and low lateral stresses
in those tests were of sWal importance in Mdivvely simulating the behaviour
of concmte columns.
The original mdel of Bazânt and Kim is an incmmental stress-min
relationrhip of the type
where d# and d e am the incremental stress and incremental m i n matric~s,
nspedively; and the material maMx is mlwlated as
wtiere the right hand ride ternis stand for elastic, plastic and fredure material
matrioes, respedively. nie material rnatrix CI is mitten in tems of invariants that
make it suitabk for finite element anallyis (FEA).
The recalibrated mode1 was irnplemented in a FEA progrorn that uses
two-dimensional (2D) axisymmetric triangular ring elements for plain concrete,
and elasto-plastic springs for sted spinlr. A Newton-Raphson rdieme was
used for the loaddisplacemcmt history, and the cracking of conmete was
modeled using an iterative produre that accounted for stress nlease (i.e.,
stress softening ).
Circular cylinden with spird confinement testeâ by Ahmad and Shah
(1982) and by Mander et al. (1984) wbn geometrically madeleci in the numetical
analyses. Only slices of experirnental columns were modeled due to their
syrnmetry and the assumption that the behavior of aie fest of the column is
similar.
The analytical rmults showad vwy good agre«nent with those obbined
fm the tests. The nuwc8l mwlts prov8d 10 be an 8xcellrnt methOd for
analyzing stress dirtrikrüonr within the caicnite c m of columnr. The authon
compafed their recalibrated mode1 with cylinders with mdmte to hioh
volumetric nüos p @ r 1.57% - 3.1%) and wÏth a wdl-mfined cduM tested
by Mander et al. The moâel, houmver, was not compared with pooriy confiruad
columns to demonstrate its ability to pmdict a wide range of column khavion.
2.2 Linear and Nonlineu €lutlc Models
2.2.1 Mau; €hi; and Zhou (189û)
An analytical study of confinement in cirailar columnr is presented in thir
pawr. The geneml solution folIowa a Iinear elastic appmach, whereby the
radial, tangential and axial stresses are obtained from a solution of the
displacements of a cylinder subjected to a ring load (i.e., radial compmssion).
The Wten defined an '8- &ning stressN as aie average of the
tangential and d i a l Messes in the cylinder. The eMiw confinihg dsss ir a
fundion of the distance from the axis of the cylinder. Several cornparisons were
made for âiienmt spiral pitch-t~radius ratios, and for different transverse
sections dong the axh of the Sinder.
From these compafirons, an ua~rage confinement hctof" was detined for
sections at miblevel bekwn spimlr as s fundion of the spiral pitcMmdius
ratio. It was found that thir factor c m k u s d in the interpretation of nonlinear
tests as wa8 estrblirhed from analyses of ~ ~ r n m r testeâ by other authors.
Linear elastic analysam ~ l s ~~ in rectangular columns cordineci
with welded wire faMc (WWF), wbm shilar confinement fadm w m
obtained.
2 2 2 Buzegar .nd Maddipuâi (t887)
The authon developed a thmedimcmrional (30) nonlineat finite element
analysis (NLFEA) pmgmm for modeling of reinforcd concrete structurer. The
concret8 is modeled as an otthotropic matefial, with rmeared cracks in
nonorthogonal directions.
Concrete failure stresses are calailated wing the five-parameter ultimate
stiength criterion proposeci by Willam and Wamnke (1975). The h p l a s t i c
constitutive rnodel of Stankwki and Gentle (1985) was adopted with some
modifications. In mis model, increments of principal strains are tnindomeâ into
oûahedral normal and shear Wain increments. The latter a n multiplid by a
material rnatrix to calculate increments in octahedral normal and shear stresses,
The material matrix is a fundion of the tangential shear and bulk rnoduli, and
coupling rnoduli.
The post-peak behaviour of concrete was also taking into acewnt A
stress decrement L computed using a reduction factor that affects the tangent
stiflhess material ma te Smeared cracks in multiple directions are based on a
fradure energy mode1 developed by De Bont and Nauta (1985). Finally, the
authors also implementrd models for embeddeâ minforcement and bond slip. To
conoborate the poa-peaak model, the program wu uwd to analyze om of the
columnr tested by Scott et al. (1982); the rsurltr dmmâ good agreement with
2m2.3 ~ 0 l o t t l ( 1 8 @ 4 )
The author proposeâ a #lum criterion for concrete in ternis d the tensile
strengai of ancrete, and the intemal hidion angle. B a d on this criterion, a
constitutive law for c o m t e in axial tension was deriveci, The failure criterion
was developâ to be applied to mially loeded confined m e t e colums.
It was shown mat the confinement strength is a fundion of the tenrile
strain of the cover concret8 at failure. ln the case of cirarlar cdumnr, the cover
shell is idealized as a tube subjeded to intemal radial tensile pressure due to
expansion of the conmete m. The cover starts its tension softening behaviour
after it reaches its peak tensile strain.
A formula for the minimum transverse steel ratio was deduceâ from the
failum criterion. K the quantity of transverse steel exceeds the minimum, cover
concrete wuld mach the ultimate tensile strain,
A set of tolumns testeâ by 0th- was compred with the formulae
deducd for peak lord and peak strain. Analytical mults varied fmrn about Ml%
to 16û% of the experimental mults.
2m2.4 Sdby (1 980)
The author develop8â program SPARCS, the fint University of Toronto
FEA pmgm for nonlinear elastic anilysir d r e i n f ' ammte rdidr. Th.
constitutive nldionrhipr fiom th0 Modifid Cmpnrrion Field ï h m y (MCFT)
Conmte wa8 moâeleâ as en irotropic mmal bebm craddng, and with
orthotropic properties dter devdoping cracks. Steel wru srneamd out in the
comte. Principal stress directions coind*ded with principal rtrains. The
nonlinear elastic analyses wwe carried out updating the 8ecant rüffness
matefial matiœs for both m u e t e and steel. An iroparametric right-noded
brick, with thme degrees of freedom (DOF) per node and a cloced-fm stiffness
mate was implemented. Material behaviour included: a Hognestad parabols for
stress-strain u w e for concrete in compression; canprersion 80ftening due to
transverse tensile strsins; linear elastic khaviour of uncracked ancrete in
tension; tension stwening; and stress checks in cracked concret@. Steel could be
wienteâ in any diredion, and was modeled following an elastic-perfedlyglastic
curve, which also included $train hardrning. Pe f lH bond war also assumed.
First checks of the program wre made cornparino analytical to
exparimental mults of beams subjected to torsion and testeâ by other
researdier.
2.26 Vecchio (1 882)
In 19û6, Vecchio and Collins proposeci The MOdified Compression Field
Thtwy (MCFT). This theoiy ha8 been used to pprrdict and to analyze the
behviw of planedmss end plane-min elements wbjected to rhear and
n o m l stresses. T b original MCFT war extendeci b aanunt for the eîf8ds of
tow tnr ik stresses, ~~ mhr- due to confinam& nd variable
Poisson's ratio. ModMcationr to concret8 material matruc derived from the MCFT,
and to dnustrain cuwes wwo proposed.
In the finite elment fornulotion of the MCFT, orthobopic materials have
symmetrical material matrices. As compression in conaete increases, the secant
stiffness for different values of compressive stress varies. The Poisson's ratio is
expected to change as a result, thus making aie materiel matrlx for ancrete
The concept of a 'pre-in metM is in tmdud to account for non-
stress related strains in concrete elements. Prestrains are transfomed into
Yorcesn in this formulati~. The expansion eff8d is included in the apmSfrein
mat&" as
where e& is the prestrain vector in the principal directions 1 and 2, vu is the
Poisson's ratio in the direction i when subjecteâ to a stress in the direction j,
and €cf ir the recant stiffness of concmte in the direction ' To avoid numerical pmblems in the solution d the FE4 the Poisson%
ratio ve wer dividd into an "elatic portionu vr' and a msidual component vr * - v~ - ve .The dastic porüon vu %as d i d y induded into the material matrix,
and the msiducil portion v~ 'into the presûain matiix, keeping the symmetry and
the odhotmpical condition8 of the material ma-
- ---- . -
To account for sûtmgth enhancement, a formula proposed by Kupfer
(196Q) for biaxial stress-states is 8180 included in the formulation. ïhe increment
in the concret8 strength is
where K. is aie ~&8ngth enhanament fador (Kupfer), P. is the unconfined
strength of concrete and 1, is aie confind strength. And the main at peak stress
e, = Ke.e,
where g is the strain at peak unconfined stress.
In the case of ûWal sess states, the author suggested a tentative
formula to compute the strength enhancement, which is a combination of the
relationships proposeci by Richart (1 928) and Kupfer (1 969).
The stressatrain curve for confinecl ancrete is a liberal modification of
the Modified Kent and Park mode1 (Scott et al. 1982) which consirts of a
parabola for ththe ascending branch of the curve, and a straight line for the pst-
peak behavior.
A total of 1 1 pamlr tested by V-io a d Collin8 (1 986) and 13 sbar
4 1 s terted by Ldas et al. (1990) wsm a~lyzeâ using the FEA progrpm TRUC
The panels wem subj8deâ to combinations of bim*al and $Mar sûeases. The
set of panels chown wws those which apwiemœcâ awhing d Coclcrete, and
walls were subjected to monotonic inmments of lateml load. and constant
compression at the top. The walls were constmtd with conœaled columnr at
each end, thur providing an excellent test to check triaxial compressive stresses
under the load conditions menti- above.
The effect of concret@ expansion was well modeled in the finb ekment
approadi, the strength enhancement was also well captwed for all tests.
However;' the analytical nrwlts shwed a stiffer responw, and underedimated
the defledion of the walls.
The author extendeci the s q e of the MCFT to accwnt for strength
enhancement anâ variable expension d concret0 in twodimenrional strudural
elements.
2.2.6 8rlby and Vecchio (1883)
SPARCS was furüw updated and imptoved to include hno additional
finite elements: sixnoded isopacametric wedges (rsgular pentahedm), and ûuss
ban. The effect of expansion due to Poisson's ratio was implemented using the
concept of a @prestmin metnbC(Vecchio 1992).
New constitutive models were added to SFARCS. Base stress-$train
curves for concret8 in compression now indude Popovicr (1973) Wednrwtnin
curve for high Wmgth o0c)ciete (HSC). The eff- of confinernt dm to triaxial
rtnu states was also implmmted; a failum wwe propomd by Hsieh et al.
(1970) is used to ampute the ultimate coinpcwriue dnngüi. Finally, a mviwd
version d the modilkd lbnt wid Pwk p o o t m ampmssiva wnm (Scott et al.
1982), and the variable Poisson's ratio proposd by Kupfer et al. (1969), were
added to account for dudility and variable lateml expansion in conamte.
The first wthw modeleâ six of of columns tested by Sheikh and Uzumeri
(1978). Eightnoded kicks wrs used in two diffmnt types of meshw; one
coniidered boai are and cover m t e , and the other considered only the
conuete are. lmpooed loado were applied to a stiff plate at the top of each
model, and a set of springs was used to mode1 the postgeak behaviour.
Column peak strengths obtained with SPARCS agfeed fairly well with the
experimental mm~lts from Sheikh and Uzumeri. However, the post-peak
responses were not rstitfactory.
2.2.7 Abdel-Halim and Abulebdeh (1 988)
A set of 8 redangular cdumna, hno of them originally tested by Scott et al.
(1982) wws examined analytidly. The linear elastic FEA program SAP N was
useâ in a stepbyatep lord increment approach to account for nonlinear material
behaviour of reinfwd concrete.
At each loaâ s t a ~ , the finite elaments (B-noded bricks) were checked to
detrrmine if t h y aither cnckeâ or re8chad a previourly defined failun ruflaœ.
Al=, material plopcwti*e$ (i.e., mateda1 stiffnes8 matrices) were dionged
accarding to the cumt $train state. The columns w m brought to failus and
the analyücsl rewlts compared against üw experint#rtrl values, and against
The three-dimensional rtnss-strain nlationrhipr and ultimate strength
surface proposeâ by Celodin et al. (1977) wwe adopted in these analyses. Steel
was modeled using expecimental stress-min wwer with stnrin harâening.
Good agreement with the experimental nsultr d Scott et al. wru obtained.
This is one of the fint nonlimer elastic finite elment analyses of confineâ
concrete repoted in the Iiterature. Voiumetric ratio of confining steel,
arrangement of longitudinal steml, and the iwement in compressive strength
due to confinement were wme of the aspds investigated.
Chapter 3
SPARCS
In this chapter, a brief description of the pmgmm SPARCS is presenteâ.
The finite elements developed for the pmgram are de&bed, f o l l d by a
review of each of the constitutive modelo for concrete and steel that have been
implemented to date. Finally, the SPARCS stnicture and solution algorithm is
desctibed.
3.1 Pmgnm Description
Program SPARCS (i.e., Selby's Program for the Analysis of Reinforcecl
C o m t e Solids) is a nonlinear finite dement program that has been developeâ
at the University of Toronto (Selby 1990, Selby and Vecchio 1993 and1 997) for
the analysit of reinforced concret8 rdids. The üifeedimensional (3D) state of
stresses in reinforcBd m t e rdidr is taken into account by extrapdating the
stress-$train cwws derived from the Modified Compfession Field Theory
(MCFT) (Vécchio and Collins 1986) b its thfeedimensionrl fomiulrüon.
The 30 dnss -te is mlated to the 30 m i n state through a comtiMive
where (e) is the stmu vector, [Dl ir the material stifhess matfix, and {c) is the
sttain vector.
The material maio< [Dl is given in ternir of the secant stiffness moduli,
Poisson's ratio v, and Shear moduli G in three diredions (Le., local, global or
principal directions). Secant moduli Vary at each losd state as a fundion of the
stress atate. Stresses are cornputeci accordhg to base stress-$train curves
derived for both conmte and steel subj8ded to either compresrive or tensile
strains. Material behwior dations have been adopted from the MCFT and
theory of plardicity; inciuding: strength Menina due to tensile stmins, strength
enhancernent due to confinement, variable leteml expansion, cancrete cracking,
It is amurneâ in SPARCS that concret8 behaves isotropically befm
cracking, and orthotropically afterwards. Cracks are assumed to be smeareâ
within concretet thus allm*ng the user to maintain the same finite elment mesh
during the analysis proceu, and not having to change it due to llocsized cracks.
Although cracks am arrumed smewed, shss checks at crack ruHaces are
perfomd to reüw compatibility and equilikiun.
St-1 can k modeled u rmeared within the conamte elemmb, or
repmmnteâ as trum bprr attached to rdid elements. In any caset p w k t bond
A description of the finite dement library, and the material constituüve
laws and failure criteria followsi.
3.2 Fînk Elment Ukuy
SPARCS ha8 aime finite elements in itr libraw an &nodeci brick
(hexahedon), a 6-noded brick (pentahedra or wedge), and a tniss bar.
3.2.1 Hexahedron
The 811oded brick ir shomi in Flg. 3.1. It is an isoparametric elemnt with
orthogonal rides and 24 degrees of freedom (DOF), thme at each node.
The relative displacement between two adjacent nodes is assumeâ linsar,
so that Wges ma in stfaight, as ahanin in F~Q. 3.2. Infinitesiml rotations and
small defOnnationr are assumed in the cmputatim of the elment süffness
matrix k:
whem [BI is the m i n displaœment ma* mat depends on linear displaœment
fundions, anâ [a is as defined above. The dosed-îbm solution of Eq. (3.2) ir
obtained by direct integration, as illcorporated in SPARCS by Selby (1990).
Figura 3.2 Womnd Brick
It ir noted that nonorthogonal hexahedranr are not allowed in SPARCS;
the program has a subrouthe to check element geometry. Numkring must be
counterclese as s h m in Fig. 3.1.
3.2.2 Pentatwâtl (Wedge)
The 6-noded kick is rhown in Fig. 3.3. lt is also an iropammetric element
with 18 âegmes of freedom (d.0.f.); thme at emch nade. It must be prismatic (Le.,
mintaining th. same b9nsverse wdim throughout), and the bottom and top
fiaces (fsce 123 and face 456 in Fig.3.3) mu& k of wual %ma and must be
Iocated in paralid planes. Dista0111 am not a l lwd in its genenüon. Linear
displacements and infinitesimal d&ormationa are also assumed. The element
stiffness matrix k (Eq. 3.2) is obtaineâ from numerical gauss integration.
3.2.3 Trum Bar
This element has two nodes, and three d.0.f. at each end, as show in
Fig. 3.4. The element deformation ir computed as the relative displacement
between the two nodes divided by the Irngth of the element. A simple direct
amputation of the element stiffness rnaûix b given in the program (Selby and
Vecchio 1993).
Although the element formulation ir baseâ rolely on axial defomatians,
kidding is rot taken into eccount whm the bar is subjected to compression.
Bending ir alur ignomd in its stifhar mrtrix.
This section describes the modelr for conmete implementeâ in SPARCS
and used in the analysir of confined concrete. It begins with the base stress-
strain cunres for concnte in compression and tension, and continues with the
failure criteria for concrete under M a l state of stresses, and cracking. The
suggested models for compreuive strength enhancement are also ieviewed.
Finally, comprersion-softening models derived ftom the MCFT are presented.
The section ends with o description of the variable Poison's ratio.
3.3.1 Bru Stms-Stnki Cuwes for Concmk in Compnuion, Pn-peak
6ehaviour.
The followhg exprewionr are intendeci for the modeling of the arcending
branch of the 8tress-$train curve.
The parabola proposed by Hognestad is a widely useâ stress-strain cuwe
far the behuviwr of n o m l stnngth. anâ is calculated as:
whem C and tr are the compressive stress and strain in the principal Miredion
respectivdy, and f, and Q an, the peak stress and $train et peak stress.
rerpctively. Eq. (3.3) is depided in Fig. 3.5.
3.3.1.2 Thonnfeldt et al. (see Collin8 et al.)
This relationship is a generalized model of the baw shss-strain cuwe
pmposed by Popovics (1973). The mlationship represents well the stiffer
ascending branch and steqw falling branch of high strength concrete (HSC).
The stre#anin cwve ir computed as follawr:
and
k = 1 .O for the ascending bmnch, or for the destanding bmnch
The generelized Popovics arve is gmphed in Fig. 3.6. A variation in the
descendhg slop can be seen as the ancrete strengai imases.
C .
Figure 3.6 Gewalizrd Popovfcs Stwm-Straln Cuwe
3.3.1.3 Horhlkumr et ai.
Horhikuma et al. (1906) pre8ented a m a l for confinement dbcb on the
stress-strain relation of reinforeed conccete, suitaôle for pier design in seismic
zones. The base stress-strain cuwe of this mode1 was extractecl and
Pmplemented in SPARCS. The asœnding branch satidies the follmving
boundary conditions:
i) td =O at w 0
ii) The initial stiniress E. (tMgential) is cornputeci as the derivative of
the stmsr with respect to the atrain, at zero initial strain & 4:
iii) fd= f, at -= 4, and
iv) The curve is horizontal at pak stress (i.e., at = 6 )
The expression for the etcending bmnch is given as:
and
- - -
Figure 3.7 shows several asœnding branches obtained with the Eqs.
(3.9) to (3.11). It is noted that al1 wwes fiatten at pek, thus validating the
assumptions of the mdek
Figun 3.7 Ascending Brnch of HosNkumri et al. Modd
3.3.2 Basa S m - S b r i n Cuma for Concmte in Compmsaion, Post-peak
The post-peak behaviour of confinecl concrete (Le., after peak strass)
depends mainly on the trimial streu state. In wllconfined concrete, as lateral
pressure increates or nmains constant after the peak load has been reached,
the element will be able to sustain ked et large deformations, thus exhibiting
ductile behaviour. On the other hand, poorly confinecl concfete (Le., luw levelr of
lateral pressure) will fail shortly aftat the peak I d . This section describes the
desœnding stress-$train wnmv uuie in SPARCS for anfinad concret0 ber
peak sûe88er.
-
3.3.2.1 ModiRed Kent nd Puk (Scott 1882)
The current version of SPARCS contains a version loorely based on mis
rtres2)-strain cuwe, whidi w m i n t rodd originally by Selby and Vecchio
(1 993). The compressive rtnrs in the principal I direction, k is calarlated as:
and
H m , Il is Vie fimt stress invariant in ternis of the principal mcrete
stresses Gr, fd and fd (fd < fe < fer , compreuion negative ) , 6 is the strain
at peak uncontined stress P. . th is the assunnd lateml prersve and i8
caiailateâ as the algebraic sum of the principal sbsses in the plane
pependicular to the principal stress king calculatd (i.r., perpendicular to fd),
as 8hown in Fig. 3.8 for the puüarlar case fd = fd.
Figum 3.8 Description d Latenl Pn#un tL
Figure 3.9 shows the Post-peak h n c h of this stress-strain curve.
Figure 3.8 Adripted Version of the ModMeâ Kent and P u k Pest-Peak M d e l
3.3.2.2 Popovics (1973)
Popovics stres8-8tmin avrs wsr implemnteâ in SPARCS as anothr
alternative for representing the poot~peak behaviour of COCtfireed comete. This
version wsr also mI8ded by Mandar et a1 (1988) in the fmulation of hi8
Confind concret8 modd. Th. stress-Wain ann is computed as:
whem n ir the mm8 a8 in EqD (3.10) and the initial tangent stiffness is (in MP8.):
The relationship is sdiematically presented in Çig 3.1 0.
3.3.2.3 Hodiikum rt d. (1886)
The straight lim propomd by Hoshikunu et al. (1996) for the postpeak
behaviout of Confined concret0 was impiemented in SPARCS, with a slight
variation in the computaüon of desœnding 8lop & of the wrve. The stress-
stmin wnre is given as:
Hoshikuna et al. defined an ultimate stmin 8, as the strain comesponding
to an stress equal to half the p a k stress f, in the dercending brandr. This value
was detemined based on observations of crushing of cornete and budding of
longitudinal bars in theif sWmenr. Hcwver, this criterion was not taken into
account in the SPARCS fonnulation. The failun criteria for concrete will be
reviewed later in this mion.
The original formula for the descending slope Ec (see Fig. 3.1 1 .) was:
whem p is the volumetric ratio of lateml steel in a column, and G is the yieldinQ
stress of the latml steel. It is noteâ that the âenminator in the right hand r i a
of Eq. (3.20) is m p l d with the t m fml 2 in Eq. (3.19)
Figum LI 1 Horhikum at d. Po8t-pak c u m
3.3.3 Base Stmss-Stfain Cuwe for Concrete in Tension
Reinforced conaete follows a linear ascending branch up to the tensile
rtrength Pt, WhiCh is an input parameter for the program. The m i n at cracking
stress & b ghren as:
where & is aie initial tangential stiffness of concrete.
As aacûs are smeared out in minforad concrete, average tensile
stresses mer reaching Pt csn be cornputexi due to tendon stiffening (me Figue
3.12). 7hm tension stiffening models have been ad- to SPARCS:
3 . 3 . ~ V ~ C M O (1902)
Tmsik atmm (conriderd positive in SPARCS) are computed es:
wheie c* must be a principal tensile m i n in the diredion under consideration.
This tension-stiffmin~ madel w m d8rivd from the Modifieci Compmssion Field
Theory (MCFT) and obtaineâ from test msults of panels subjeded to biaxid
stresses, and extrapolated to the triaxial condition.
Figu, 3.12 Reinfoffiod Concmb In Tension
3.3.3.2 Collins-Y itchelî (1 987)
This is a variation of the Vecchio-1982 mode1 in which tension stiffening
is computed with the ielationship (me Collins and Mitchell 1997):
3.3.3.3 Inimo, Maekawa et. al.
Mer cracking, average tcmrile stress is kept constant (Le., f* Pt) up to a
stfain W*ce as large as e, The str~smr thereafter are given by:
3.3.4 Failun criteriri for concmta
3.3.4.1 Hdeh-TingChen CCrltrrlon (1 978)
Brittle failure of concrete ir expeded under tensile stresses, and ductile
failure can occur under trimeal compressive stresses. The failun Maface
proposed by Hsieh et al. (1979) (se8 Chen 1982) mpresents with good accumcy
the failure of c o m t e in: uniaxial compression, uniaxial tension, biaxial
compression, triaxial compression andor tension. It was c h e n (Selby and
Vecchio 1993) to be implemented in SPARCS, and is given belaw in ternis of
stress invariants.
where 4 is detennineâ from Eq. (3.1 5), and:
A Newton me- is used in the program to sdve Eq.(3.25) for the compressive
failum sûeas = fa
3A42 Cmcîcing Criteria
The Mohr-Coulomb criterion and some proposed variations w c ~ e alui
impiementecl in the m n t version of ihe program to account for cracking sûesr
f, under triaxial conditions. A review of hi8 two-parameter failum criterion is
given belaw.
3.3.4m2m1 Mohr-Coulomb Criterion (strass formulation)
if t, > fd 3 fa , the largest Mohr's cirde is definecl by the exlreme
principal stresms fd , and fd , as show in Fig. 3.1 3
- - - . . . . -
The strsight-line envelope cuwe is detennined as (sete Chan 1982):
whem + i8 the intemal Mdion angle d concret8 (asrumed 37@ in SPARCS), and
c is the cohesion . Definino Q a8:
from which the cohesion c can k readily detemined. The cracking stress fr is
determined Rom Eq. (3.27) to (3.29) as:
The ternir f!!, and P. in Eq. (3.30) am n p l a d by for and 6 ,
mspectiveiy:
3.3.4.2.3 CEB-RP Critarion
For this case:
and
3.3.6 Confinrd Stnngth f, and S W n rt Puk Stress q,
Once the base stress-strain cuww for concret8 in compression have
b e n defined, the confined stress (Le., peak stress) and its corresponding strain
are computed followBng either of the folkmbng aiteria.
3.3.6.1 Selby (1 893)
The author uses the failure criterion proposeâ by Hsieh et al. (1979)
(Sedion 3.3.4.1) to define the pmk sfress hdor & :
When compression roffening of c o m t e (i.a, strength reducüon due to high
tenrik sûains) ir not taken into ~ccount, f, and 6 am computed in SPARCS as:
3.3.6.2 Vacchlo (1992)
A Szliess enhattœmnt fBdor W. ir defineci by the author as:
K. = 1 L 0 + 0 9 2 ~ - 0 . 7 ( ~ ) ' ] ft= f'. +a& fC
where
and
Eqs. (3.37) to (3.39) am valid If O + fd > fa > f& , f& being the maximum
compressive stress, and K. cornputecl in the direction of fa . The expression in
brackets in the right hand terni of Eq. (3.37) is a relationship that approximates
that of Kupfer et ai. (196s) for biaxial compression, and the second temi
represents a slight modification of the mode1 pmposed by Richart et al. (1928)
(me Vecchio 1992).
Again, when compression raltening ir not a kdor in the analysir, and
q, are detmineâ as (mlling Eqs. (2.5) and (2.6)):
The MCFT (Vecchio and Collins l m ) recognized the effects of tensik
stmins in cnckeâ comrrte. As concret0 cracks, tansile stresses develop in the
concrete between cracks; frOm zero stress a a crack location to a maximum half
way between cracks. (see Figure 3.12) . The effect of tensile strains is to rduœ
the compressive strength in the direction parallel to the cracks.
To account for thir effect. several compression-softening expressions
have bwn proposed and implemented in SPARCS.
3.3.6.1 Vecchio and Collins (1982)
The reduCtEOn factor & (&s 1 .O) to account for compression SQftening is
cala lated as:
where t.r is the tensik strain in the diredion normal to the compressive strain
It is assumed that cortesponding principal strains and stresses have the
same direction in the MCFT. The peak compressive dnu, anâ the m i n at
peak drws, euwming no sûength enhancement (Le*, K, =1 .O or K. =1 .O) are:
3.3.6.2 VItdrio uK) Collins (19û6)
where ei, is the strain at f. for a uniaxially compressed cylinder.
in ail COIN the peak M(Ns fc and g am obtained from Eq. (3.43) and (3.44).
A g e m l sketch of the f , vamus k nlaüon~I~ip is rhown in Fig. 3.14
3.3.7 Vuirble Poisson's Ratio
Concrete subjeded to compression expends laterally. (see Figure 3.15).
As compressive strains incmase, the Poisson's ratio v also increases from its
initial value. An expression proposed by Vecchio (1992) and rlightly rnodified in
aie current version of SPARCS is presemted.
when ~ r ) is the compressive strain in Miredion, and is the strain at peak
unconfined stress f!* ,and va is the initial Poison's ratio. The original fornula
(VBCChio 1992) indudes p, instead of p, in Eq. (3.48). The Poisson's ratio is
depicted in Figure 3.16.
Figure 3.16 Polssonps ratio niatlonrhîp
Changes in Poison's ratio are expacted if the compressive strein > 0.- as
indicated in Fig. 3.16.This threshold value cwld be considered as the beginning
of nonlinear behavior of normal sbength ancrete. The maximum value allowed
in SPARCS is v, =O.S.
3.4 Constituüve Modal for Staal
The monotonie response of steel was modeled with a simple elasto-
plastic mode1 with $train hardening . (Figure.3.17.). Althoug h SPARCS has
modelr for hysteretic response of steel in cyclic toading, they will not be
presentd in this work
f
PARCS 49
whem c, L the yielding stmin, a is the m i n at hardening, E , ir oie initial
stiffneit, E i is the rtrein hardening moduktr, f, ir the yidding stress and f. is
the rupture stress. The mockl ir valid for either compressive or tensile stnwser.
3.5 SPARCS Sîrucîure
The material matrix d Eq.(3.1), [O] , comprises aie material matrices for
[&] is the sum of the steel material matrices, calwlated as:
where n is the number of steel components. Thus, the total material matrk can
be written as:
The orthotropic material st-s maûii for concnrte (Selby and Vecchio 1993)
in principal directions is:
whem E.r is the secant moâulur in the principal diredion Cl ,2,3. O* is the shear
mdulus given by
where vg is the Poisson's ratio component of strain in Vie Miredion due to a
The smeareâ steel material matru< in Mirection is given by:
where fd and e* are the stress and m i n in the Miredion, respdively, and pd
is the reinforcement ratio in that dimcüon. It is noted that the secant modulus for
Matri- [&'] and each [a mwt k transforrried from principal direcüons
to global coordinates befm summing them up.
At n y load sta~e the wmnt moduli depend on the stress-shin condition
in the direction king analyzed. h the case of concrete (se8 Fig. 3.18):
whem a is the stress-felated strdn in that direction. Other stnins due to
prestress, thermal loading or latml expansion are inttOdUCBd as Qpmstmins"
(V-io 1992) and conv8C1ed into quivaknt bms. As lateml expansion
changes at each load stage accorefing to aie stmin state, its related pmstnins
The prestmin vector due to lateral expansion, (5); can be written as:
where the rupemCpt indicfates 'pStrei#,
and:
PARCS 52
Lateral expansion is a non-stress nlated min, as it has no azwciated stress in
the expansion direction (Le., diagonal tems in Eq43.57) are null).
As mentioned in the introduction of this Chapter, SPARCS is a nonlinear
elastic pmgram that solver for displaœments as in a linear elastic program. The
global rtiflness mat& which ir the auembly of ebment stiffness matrices
computed with Eq.(3.2), is updatd a d useâ to wlve for new displaœments as
loed or imposeâ nodal displaœments are inmementeci.
A flow chart of the SPARCS solution algorithm adapted from Selby and
Vecchio (1 893) is shown in Figure 3.1 9.
Chapter 4 shows the parametric study of the models implemented in
SPARCS for the analysis of confined muete, and the analyses made to
detemine the variation of Poisson's ratio in coiumnr wbjeeted to monotonie
increadng compression.
Figure 3.18 Solution Algorithm foi SPARCS (Sulby and Vecchio 1983)
Chapter 4
Parametric Study and Poisson's ratio
Pmmetric Study
4.1 Introduction
In order to apply the materiel behavior models gresenteci in Chapter 3
and implementeâ in program SPARCS- to aie analysis of confined concrete, it
was neceuary to study the influence of @a& moâd on the three-dimensional
behavior of reinforcd concret8 when subjectd to tripual strass states.
The parameten analyzed were:
Prpesk compression base wwes for concret@.
Post-peak behavior of conflneâ cancrete.
C m t e crac&ing.
Confinement enhancement.
Variable Poisson's ratio, und
Compcw~ion softening of conamte.
-
Tha models for each parsmeter wwe useâ in the analysis of a set of fwr
columns ftom the expwimental wwk af Sheikh and Uuimwi (19ûO).
The cdumnr had dRwent tie arrangements, tie spacing, and longitudind
ber cages. They failed in diRemt man-, from Mffle to dudile behavior. The
set representeû a broad specûurn of possible configurationr of confineci
conerete. One limitation would be that ail the colurnns were made of normal
strength concrete, and had equal ske and rhape. A new $81 of rpecimens with
diff'erent size and material properüsr was analywd and pnrented in Chapter 5.
Each column was modeleci in SPARCS and analyzed using combinations
of Vie parameten mentioned above. The analytical resub wem cornparecl with
the experirnental results, and decitions wsre made on the mod effective models
that mpresented the tests accwately. The selected modelr were then ured in
modeling a n w set of columns, snd the rrwults presenteû in the next Chapter.
Later in thir Cheptsr, two of the cdumns of air pammetric study were
compareci with the mults obtained by Selby and Vecchio (1993) wing a
pfevious version of SPARCS. lmpmvement in the modeling capabilities of the
pmgram was achieved.
4.21 png..k 6 u m C w e s lkcomnb. Two of thme base curves fw pwpeak ôehavior of commte were chosan
for the parametric uielysir: The Hognahâ pambola (Eq.3.3) and the
Horhikuma et al. arve (Eq. 3.9). ïh gmmlized Popovio cuwe (Thomhldt
et al. Eq. 3.4) was not accwnbd for as it would adegenemalR into a parabola at
4.2.2 Poat-peak 6au Cu- for Comrete.
All am rnodels induded in SPARCS w r e useâ in this pwmetric study,
those wwe: A modified version of th Modifieci Kent and Park (MW) m a l (Eq.
3.12), The Popovics wwe (Eq.3.16), and the Hoshikuma et al. wwe (Eq.3.18). a
4
The first and third models are straight liner that depend directly on the
definition of lateml pressure. The Popovics cutve is indirectly related to the
lateral pressure through the sûength enhenœmnt factors diraisssâ in section
3.3.5.
4.2.3 Concrete Cracking
The *Stress" fmulatim of the Mohr-Coulomb meterion (Eq. 3.30) was
selected for the parametric study. T b criterion ha8 extensively been used to
piedid concrete failwe in biaxirl compression-tension, or uniaxial compression
(e.g., test cylindem) (see Cbn 1982). In columns, concrete cover fails at an
eariier stage than the confined corn dm to the cors ecp~sion, formation of
cracks, and co~xete awhing.
60th enhan- fadon implmnted in SPARCS wem useâ: Selby's
(Eq. 3.34), and Vecchio'r (Eq. 3.37). 60th stmngth edmœmmt CriMa have
dithrent Mnitimr of latml pressure (me Eqs 3.14, 3.38, and 3.39). In any
triaxial stress state, the lateral pressure defineci by Vecchio will a h y s be
smaller than Selby't . ît should be noted that the enhancement factor (K. or &) was alro wed
in calailoting both pwk rtnu anû p a k rûain. The prwriorw version of th
program had a factor I(, to cornpub th. peak rtnin. (Sdby and V e i o 1993)
#
4.M Variable Poisson8r Ratio
Aithough SPARCS has the option of r constant Poissm's ratio v., it wm
decidd to keep it variable. Ib variability was m b o m t e d wing the
experimental nsults of Liu et al. (1998) columnr, and is de8CljM at the end of
this Chapter. The suggested expression for v in Eq.(3.48) was then used.
The VBCCCIio-A mâeI(1992) (Eq. 3.46) was sebcted. As this parametric
study war the fint sttempt mide to ~ I u a t e the msibility of the tentative
models for conf id concrete, no pnkmnce was given to any pcvticular
softening model. if roftening is not considered, the œ s o B n i ~ kbf & is equal
to 1 *o.
4.3 Shalkh nd Uzumorl tests (lm)
This redion M b e s the set of colwnm testa by Sheh and Uanisn
(1 Qûô), anâ tb varhblw rtudîed.
4.3.1 Column Geonnty
A set of 24 short cdumns was tested under monotonic axial compression.
Each column war 3 ô h m 8quare and 1980-mm high; 368-mm tspeml ends
confined with welded steel plates were built bsfw casting to prevent frilurs
away from the testing zone. Details of the column geometry are show if Fig.4.1.
Figure 4 e 1 Column Dimensions
I L 2 Longitudinal Bar uid Th &tup
Defmeâ and plain ben wam uwd far the longitudinal and tie
minforcement, nrpectively. The centfa-to-canh spacing bebnmn outemat
ires was 267 mm in al1 rpedrnq providing a constant core ana quel to 77%
of the grou a m , butwith d ~ c o v ( ~ t h i d e n e s w s .
- - - - - - -- -
Four amngmentr were wlecteâ by th8 auttiom, and shown in Fig. 4.2.
Each amngement wer symmetric with respect to centre-lino axes, and the
number of lon~it~dinal bsm and spacing wa8 kept constant along section edges.
Longitudinal ban in a set had equil diameter. ïhe minimum nmbw of ban in a
column was 8.
The latemi ieinforœment was pmvidd by 2, 3, or 4 closed ües or
's813rn& hoop# (me AC131&95R, CSA23.344); each one s~ppart8d at Ieast 4
longitudinal ban, with the exception of the intemal tio in the D setup (see
Fig.4.2) that suppocted 8 ban. Tie diameter was also kept constant for both
extemal and intemal- ties in a section. In order to pievent undesired failwes, tie
hooks were extended 14 tiediameters into the concrete c m .
The specing between ties in the test mgbn (sac Fig. 4.1) um defined in
tems of the cmtm-tlcenlte vertical distance.
4.33 Test Instmnnntrtlon urd Pmmâum
T h 24 columnr wrwe instrumenteci with stmin gaugm to measure
defornations in tha longituâinal rt-1, ties and m t e . Each hoop in a set near
the column midhaight had four strain gauges. The load venus axial shortening
was m e a s u ~ wing hwo linear variable diiemntial tnn&mem ( L m placed
at opposite sides of the cdumn. Column tests iaded 3 to 6 houm.
The em*mental variables studieâ mm: longitudinal bar setup, tie
configuration and volumetric ratio, tie spacing , and tie steel properties.
4.3.4 Sdactsd Sheikh and Uzumri C o I u m
Four of the 24 columns tested by Shiekh and Uzumeri wm selected for
the parametric atudy. Each colmn haû one of the tie configurations showed in
Fige 4.2, and diffemd in volumtfic ratios, tie spacing and longitudinal bar setup.
Failure modes varid from brittle to ductile, thus giving o reasonable spectrum of
behaviors that might be modeled with SPARCS. The purpose of the parametric
study was to esblish a combination of the implemented material models in the
pmgram that bert npresented the adual column bhavior.
The matrriai properties of the selecteâ wlumns are shuwn in Table 4.1,
whem dh b the longitudinal bar diameter, A, ir the total longitudinal bar cross
sedionel are0 in the cokimn, p. ir the longitudinal steel ratio with mped to the
grou sedion, îv is the longitudinal steel yiddiw stress, & is the initial deml
rtiffiw)m, & ml ri* œe the stifflmm modulus and m i n at hadming,
rerp8ctively. Al=, dh and & are the tie diameter and a m , f i is the volumetric
ratio, f* is the tie yielding stress, s is the spadng behlY88n sets of ties.
Concnte
Column
(1)
2Al-1 483-19 2CS17 408.24
Speciman Column
- (1)
2A1-1 4BSlO 2C5-17 40624
Column (2) in Tabk 4.1 identifies the croor section type d Fig. 4.2, and the label
in column (1) wu given by the nwwcîws. The volumûic ratio b defined as
the volume of tir steal to volume of concnte ratio; f i vm*d fiwl highly confined
column (2.37%) to poorpoorconfined (0.843%). Tk rpadng varied from 0.125 b to
0.33 b, whem b is the column rize (305 mm). Fkidly, the average standard
cylinâer Wmgth was 35 MPa. (Le., nomual Wm@h anmite)
COM. W P
(2)
A 0 C O
6
anm] (3)
15.875 19.050 12.700 19.050
Tb Steel
&
m a (4)
1600 34û6 2065
4 Fm1 (10)
4.76 7.94 7.01 6.35
- PI
In1 (5)
1.72 3.67 2.22 3.67
&* tmrl (11)
17.8 4D.S 49.5 31.7
6
wp4 (6)
3ô7 392 167 392
p, 1 (1 2)
0.80 1 237 23û
El
w 4
2WûW 1- 1- 1-
8
1-1 (1 5)
57.1 101.6 101.6 38.1
fM Pnp4 (1 3)
540 48û 48û 4Uû
4 FnPiS (1 4)
200000 1W500 2QOWQ tBMW
PamwMcSyLdLand Poihon's Ratio 62
4.4 Finite Elsmnt Modal8
4.4.3 Geometry
Eight-noded hexahedrons and ûurr ban wem used to mode1 mctete
and tier, respectively. Due to th# srymmtry of the columnr, only one quartgr was
modeled. Lonoitudinal ban wem s r n a d out, and trurr ban were attacheci to
Iwo nodes of the same kick ar show in Fig. 4.3. The nunber of hexahedrons in
a section perpendiwlar to the colum longitudinal mis (hereaeter called layer),
and the number of laye# for each cdumn drrcribed in Tabb 4.1 varied from 13
to 17. Each layer contrined 36, 49 or 64 hexahedronr. The layer depth
depended on oie tie spacing for each column. An example of layer retup is given
in Fig. 4.4 for column 2A1-1.
A vertical view of the finite element confÏgufation is prerented in Fiq. 4.4a.
Imposed dirplacernents wefe applieâ to the top Iayer, which had stmnger
concrete properties and smeared rtwl in the thrm global dirediona. The top
Iayer was necessary to batter distribute the irnposeâ displaœments Vwough the
calumn depth. The disturbance caused by presem of the stiffer layer a the
top (ag., lateral rertraint) was O V B C C O ~ ~ with the mount of Iaym used in each
model. An effort wa8 made to keep hexahebon dimensions approximately equal
(Le., aspect ratio 1:1:1), and to avoid early failure of WaY bricks due to large
deformations; for thir remon, a limited number of layen were uted behween tie
setups (from 1 to 4 for the examined columns). Huwev, thir restriction did not
compromise the overall column behaviour, as w'll k explainad later.
P<uvrr~bic &t?y and Polsjon 's Ratio 64
Tho bottom layer wrs restraineâ against displacemonts, but allownd
to de fm Iatemlly. Latml resûaints along the axes of symmetry am also show
in Fig. 4.4b. Column mesher induded both cover and con comte. Table 4.2
gives the geometric propdes d rll the mdeled cdumns.
TABLE 4.2 Finit. Elamnt bdd Gwmatry
4.4.4 Matemial Typer
Longitudinal reinforcement was srneareci over a small area corresponding
to the location of each bar. The steel ratio p within a brick depends on the actuel
bar sizo and the brick dimensions. Figura 4.5 shows the different material types
rpecified in SPARCS, anâ Table 4.3 showr the longitudinal rtwl ratios and
concmte proprties for eah material type of the Sheikh-and-UZU~ columnr.
Column
(1)
2A1-1 4BS1 O 2CS-17 4D8-24
CIyrn
(2)
13 17 13 14
Lapn Inr*
(3)
2 4 4 1
layw Thkkrnu ml (s)
28.55 25.10 25.40 38.10
MlgM
Drim) (4)
971.15 431.W 330.20 533.4
Bricks
(O
637 612 832 soi
Truu h m par -P (fi)
18 24 U 17
Briclm par
m @ r .. (7)
49 3ô 64 W
Truu Bu+
(8)
126 120 176 255
Pmmekic Sb@ and Poiswn 's Ratio 65
computeâ wing:
Coîumn
(1)
Typo 1
61 (2)
Type2
tYi (3)
Type3
&l (4)
All t y p u 0,Wf' w'4
f8
(O w 4
f ', = 0 3 3 z
(4.3)
It ir widely acœpteâ that the plain concret0 column strength to standard cylinder
stmgth ratio is about 0.85 (@.o., Rami et al. (1996) showed msulb with a ratio
of 0.89 for high strength concrete columns). Instead of using the unconfined
concret8 sttmgth Pe in this study, aie values in dumn (5) of Table 4.3 were
assumed in the finite element analyses.
Srneaced steel and tmrs ban w m modeled uring the material properties
of Table 4.1, and followed the strain-hardening mdel depicted in Fig. 3.17.
From al1 the possible combinations of models saleded in section 4.2, only
thow s h m in Table 4.4 were chosen for running each column in SPARCS. The
set of modeIr that best rspresenteâ the actual behavior of el1 the columns was
also used in comborating the expen'mental rewilts of other specimens.
TABLE 4.4 Pmmebic Comblnrtlons for SPARCS Analyses
Modal
O MKP
Pogovies MKP
POP-
The Mening moâd spdfied in column (7) of Tabk 4.4 was the
Vecchi0-A (1 992) rnodel (Eq.(3.48)).
For each column model:
A panamter combination d i n g to Table 4.4 was selected in SPARCS
material spesifications.
The geometry and restreint conditions were dlecked.
An initial imposed displacement, and a number of load stage increments
were established.
The m8ctions at tha bottom layer were computed for eadi load ~tage, and
the load venur axial strain was grapheid.
If the pairs of axial losd-axial Wain points were too sparse, due to a large
imposai displacement, a n w run with a smller displacement was done,
until a smooth curve was oôtained.
A new combination of modela wu choeen and SPARCS was fun again with
the new pammtem.
The axial load venus axial m i n for each rock1 combination was plotted
andcompusd.
ïhe next ihowr the analysit naub for ttm Sheikh and Uzumeri
4.5 Anilyris Rosuits
Complete malysis msulb for column 2A1-1 am givm next. A summary of
the principd msultr for the rest of columns is presentd in this section
4.63 C O ~ U ~ 2Al-1
in &r to prorida an inride viw of the cumbersome finite eiement
results; only detaileâ resultr oMeined from combination 2 of Table 4.4 cdumn(1)
are presented. The axial load versur axial strain for the other combinations will
be presented in a rummary graph.
Recslling combination 2, the mdels studied WB:
Pre-peak cuwe for cmcmte: Hognestaâ
Post-peak auve for concrete: Popoviu
Confinement enhancement: Vecchio
Cracking criterion: Mohr-Coulomb
Poisson's ratio: Variable
C m t e Softening: Vecchio-A
4.SA .1 Cros8 Sadon uid Profile Resulb
The variation of axial 8ûem in cocIccBte for variws load stages in a
typical cross section ir rhainin in Fig. 4.6. The ascmâing portion of the concret8
arve is pnsmteâ in Fig. 4.6 (a) and (b). The figurer show a uniforni rtress
didrikrüon with maIl dnliwsncrt among alment8 (sea color =le). Thefe is
state when the mode1 had reached the maximum analytical load. (P,: 3370
KN). The difkrence be-n the concrete shell and the cors khavion becam
evident; the shell is now in the desœnding portion of the curve and cwshing.
The corn elements had machd sûesses almort wual to that of the standard
cylinder strength. It should lm noW that the pbin concrete stmngth Ibr the
SPARCÇ analysis was set to 0.85 Pc (P. = 37.5 MPa.), thus Iatenl reinfurcement
was activaW. Fig.4.6 (d) show the stress state befon failure, a conriderable
ana of the concmte core could not carry more load and the colum failed.
PmawHc S m @ and P o k m 's Ratio 70
The axial strain to drain at peak stress mtio; gd 1 O,, for the same loed
stages is grapheci in Fio. 4.7. Concret8 behavior at the fint load stage (Fig.
4.7a) was liiww dastiq cmcmte dnins were unifomily diWbuteâ and smaller
than 40n of peak stninr. As the load appmached 90% of the peak analytical
load (Fig. 4.7b), the straindio diabution remained unaltered between cover
and core. Noniinearity war evident in the ascmding bmnch, with the cover and
corn concret8 following diffsrent pa#s a8 load increaseâ to its peak value
(Fig.4.7~). The concrette m e r then failed due to sdkening of concret0 and
supporied no l a d aftemrdr. A! failum (Fig. 4.7d), some portions of the
conmete core reached a strain ratio of about 2.0 (i.e., a strain largef than
peak strain), and were no longer able to carry load. It cwld be condudeci that
the cdumn did not âevelop dudility.
A 'wire hame" viw of the vertical deformation is rhown in Fig. 4.8 for the
load stages mcmtioned above. Alto rhown Ir the average axial ttrsin a each
kad stage 8ri.. The peak m i n (i.e., drain at peak load) was 3.0 x IO= for this
model combination. The average strain a failum was about hnrice the peak
strain. As the Popwics post-peak curve used in this analysis can 'degenemte*
into a parabola, it will k seen later ümt this w s the case for this combination
The lest of tha columngrolik uid cmocmction gmphr rhawr the
variation af the concrete fatetal strain in one of the cross sedional diredion8
(Fig. 4.9). The ôehavioi in the 0 t h ~ direction wu symrnetric.
Figum 4.7 Variation of ml sp Ratio for Combi~tion 2 (ml-1)
At the Rnt load stage (Fig. 4.0a) mstraint due to lateml steel was not
perceptible; homnnt, it was pooribk to notice the 'amhi'$ variation of latenl
stresses khn#n longitudinal rupportsd ban. ûefbre paie load (Fig. 4.9b), the
Iabral steel mrs not adivatd, but cowr elemnb had cradced at stnins of
-1.524 '"'"lm. At P, confinement of the con occumd to a certain dogme, lateml
rtmwr m m high near longitudinal ban and decmaseâ with th8 distance fiom
the ruppofbâ barn. Near biture, some etmats were in Iabml compmsion
withh the diagonal band r h m in Fig.4.W. $ 1 .
pm:jym I(N R am 1 0 ~ ~,-e.130~10~
Figum 4.8 P d k Defonnrtion, Column ?Al-1 (Combination 2)
4.6.1.2 Load-tkfonnrtion Cuwa for Combination 2
The axial loed venus average axial strain wwe for the 2A1-1 modd is
presenteâ in Fig. 4.1 0, along with the experimntal cuwe. ln the saene manner,
F ~ Q u ~ 4.1 1 rhanr the w*al load v e m a average rtnin in outer tie. Cornparisons
among the modd combinaüm anâ the tqwhenbl msult8 will be discuswd
later for al1 colurnnr.
The average tie strain at peak load was 1.4 x 1 0 ~ in the analysis, which
was half the yidd strain for this colurnn (q, = 2.7 x 10' , I, = 540 MPa). The
lateral prerrure wm insdficient in devekping stfength enhancernent and
dudility. It should k noted that the volumetric raüo for this column was the
lowest of the set (p. = 0.80%), even though tie spacing was small(O.19b)
The load versus average axial m i n an»$ for the nmeining mode1
combinations of Table 4.4 am given in Figs. 4.12 to 4.1 6. Figure 4.17 and 4.1 8
show plots d aII the combinations, Md Table 4.5 shows the analyücal to
expet3mentQI tath forthe peaîc lord nd pek #min.
Pammetric Sb@ and P o h w 'SRcillo 75
O- o m om O- om 0 . m om om7
kdrl8brki
Figun 4.12 Axlal Responn of 2A1-1, Combination 1
LlrWdfdn
Figun 443 Mai Rmpmw d ?Al-1, Combination 3
Axlrl8tmln
Figum 4.14 Axial Rasponw d 2A1-1, Combination 4
1 SOO
Figun 4.16 Axial Rnponw of Ul-1, Combination 6
-1 +2 - t g 3 . * 0 - . . 4 5 +8 T E S T
0.0 0.5 1 .O 1 .S 2.0 2.5 3.0
am. rîmln ln oubr th (mmlm)
Figun 4.18 LaWmI Roinforcennnt Responw of 2A1-1, All Combinadion
TABLE 4.6 Anrlyaic~l to Exp~ilnnnfrl Rith for 2A1-1
Although concludons will be made later on the modd combination that
bast reprersnts the failun behavion studieâ, a kiM comment on the presented
curves follow8.
In t e m of peak load and peak strain, ôest msub wem obtaineû with
combinations 1,2 and 5; the dWemnœ among these mockls mis the postpeak
curve. Combination 5 had the Hoahikuma et al. modei for post~peak behavior,
the Popovics curve war useâ in combination 2, and the modifiecl Kent and Park
model (MW) in combination 1. The Hoshikurna et al. mode1 was developed for
conmete piers of bridges, with low volumetric ratios (les$ than 0.50%) as was the
case of this colurnn.
Combinations 3,4 and 6 overestirnated the pesk l a d and underestimated
the peak etraki. In this case, combinations 3 and 4 used the Selby model for
confinement, and MW and Popovio for post-peak behavior, respectively. No
differenœ was apparent in th8 cbscending branches.
In general, the behavior of column 2A1-1 was captured with gooci
acairacy for al1 mode1 combinations of SPARCS. Although the specimen had
poor Confjnement and shawed no ductility, it was not tested to failure. Concrete
mcked et longitudinal compressive sûainr of betwwn -1.5 and -2.0 """lm, and
the Iateml stwl did not yield at paak (me Sheikh et al. 1980). The average lie
m i n pmâicteâ by SPARCS was smallar than the actual tie stmin measured at
P, in the test (h (anal.): 1.4 x 103, cr, (test): 1.9 xl O=).
4.5.2 Column 483-19
The m*al response of column 483-19 is sum-Z8d in RQS. 4.19 to 4.25,
for al1 the mockl combinations. This spechen swtaimd M peak load over s
slightly lager range of axial m i n than dumn 2Al-1, thus cuihibiting some
ductility. Hwuever, due to the laqet tie spacing, which was 0.336, its capacity
dropped S ~ Q ~ Y .
F m the analysis, al1 combinations yielded 8omehow the rame general
axial response, with one peak at the beginning of aie plateau of the emrimental
am, follawed by a descending bmnch, which extendetâ approximately up to the
same longitudinal sûain as in the em*mental curve.
The plateau in the test ocwrred at an axial m i n of 3 x loJ to -8 x IO?
The researchen reporteâ that at an strain of 4.1 x IO= the colum reached the
maximum capacity and the average tie strain was 2.3 x lo4 . It was found in the
analysis that at the analytical peak load the avmge tie strain was between 0.8 x
1 o3 and 1.3 x 1 o3 ; and that in the descending branch; at an axial sbrrin of 6.0 x
1 O= , the average outer üe strain was between 2.3 x 1 o3 and 2.6 x 1 O* .
The cover cornete aackeâ at an axial $train of between -1.2 x 1 o5 and
-1.8 x loJ , a d by the time the load reacheâ the peak load, the cuver had
rpalled oîf.
A gain in axial capac*ty aïter p d c war oôserved for moâel combinations 3
and 4. (Figr 4.21,4.22). This was m m pronounad fw mâel combination 3.
Mer the cdumn loot ib cover, the conorste con um able to assuma the I d
shed by the cowr and m c h d the initial peak load IeveI. Bath combinations
used the Selby mode1 for confinement.
Combination 6 ir similr to combination 2 but without compression
softening considered. T b dedine in the desœnding kanch was lem apparent
for the former and exhibited mors dudility than combination 2 wrve.
Finally, Table 4.6 pmsents aie ~elytical b experimental peak $train and
peak load ratios for thir column.
Fîgun 4.20 M a l Response of 4B3-19, Combinrtbn 2
Flgun 4.22 M d Rosponsa of 4B3-19, Combination 4
TABLE 4.0 Anaiyticd to Exprrimntrl Ritlos for 483-18
From Table 4.6 combinations 1, 2, and 5 give the best predidionr for the
peak load, and combinations 3, and 4 give good predidions of peak strain. It
should be noteâ that the experimental peak m i n was taken as the strain at the
beginning of the plateau in the test arwe.
Aîl combinations overestimated the strength of the colwnn, but
combination 2 gave the Ieast dilference with respect to the mearured capacity
(43 KN). The confinement rnodel proposed by Selby pmduœd the highest
values for lateral pressure in the are.
Athough the volumeric ratio of this rpecimn wa8 relatively high (pv:
I.ûô%), the tie spWng wrr large urd th. concret8 cor8 at mid-level be-n
ties behaved uHconfined dkr pak load haâ been reactmd. Furthet
deteriorrition of the cor\cre€e corn irnw the lateml Meel from developing
MÏcient pressure, snd the âesœnding port- mpnw stacted jurt after the
The program wem to captura the efkt of tk spadng in the global
4.M Column 2C6-17
A brief isview of the c m sedon msub is presented, followsd by the
axial and üe resonses.
4.k3.1 CM W o n Re8ulb
Colum 2C5-17 had the highest v o l u ~ c raîio of the set (p,, = 2.37%) ,
and a L spacnig of 101.8 mm. (or 0.33 b). The degm of confinemnt at the tie
Ievel and at mid-height bebmen ties is s h m in Fig. 4.26 at the fint peak load
for mode1 combination 2.
Figui, 4.26 C o ~ ~ of 2C6-17, Combinibion 2
This ligure sham the m*a1 conmte $train to peak rtrsin ratio 1 q, . Alaiough the cdor wk is dHlbmt bt each kwl, the concmb corn at tie kvel
exhibitecf iowr ratios than that of th. con midway behmen ües.
P ~ M c S m ~ a n d P o i l r n o n ' s R d i o 88
4.8.3.2 Su- of Load-WWmaiictn CUMS
The dumn 2CS-17 hed a m p k x tie arrangementI with the lergest
number of longitudinal ban (Le., the rmallest centre to centre di- behmen
longitudinal reinbrœment). It al80 had the lagest volumetric ratio and tie
spacing. Dudik nsponse was expected ftom the analyses. The a*a l response
is depicted in Figs. 4.28 to 4.35; tie response b graphed in Fig. 4.36, and the
analytical to axperimental peak ratios are given in Table 4.7. A--
In general, the axial loaddcil shortening awsr presented two peakr.
The fimt peak ocairred as cover spalled aff; a slight m a s e in capacity then
ocarned due to the cover lou, folluweâ by a gain in strength due to the
activation of the lateml steel up to a second pesk Mer the last peak, the
column mode1 was able to carry the load without losing resistance during large
defomationr. Due to the large th spcidng, cross sections al midlevel betwean
ties collapseâ and the mode1 carried no further load.
Peak values varied for each combination. The fint peak was smaller than
the second peak foi mode1 combinations 1,3, and 4. On the other hand, the fint
peak wa8 slightly Iager for combination 2, and both peak wem quel for
combination 7. Combination 5, which umâ the Horhikuma et al. mode1 for port-
paak bdiwior only captured one peak lord and d o m steeply.
Initial sWlneu for al1 mockls wrr v«y rimilar to thrt of the qmcimen, but
dl shmd a Wbr re8pome at fird peak Although moda combinations 3 Md 6
preserrted the highest p d c load nüoo (1.25), t h y exhibited the large&
ducülities. In ths case of combination8 2 and 6, which urne rimilat exœpt for üm
cornpfes$ion-softening mode1 rctivated in combination 2, H was apparent that
some cbgrse d comte rdtming rhwld hava occurred behmn ties whem
confinement is not lotally dident, 80th airves could be judged as the lawer and
upper limits when the of tensik rtninr ir teksn into eccount.
2CS-17 Comb. 1
0.000 0.008 0.01 0 0.010 0.020 0.025
&hl 8ttrln
Figure 4.28 Axial Responw of ZC6-17, Comblnation 1.
2CS-17 Comb. 2
I
T U T
a
Figure 4.29 Axlal Remponw of 2C6-17, Combinatéon 2.
2C6-17 Comb. 3 SOOO
4SOO
4000
3500
3000
g as00
ZOO0
1 SOO
1000
800
PanamMc &bit# andPoissDn 's Ratio 91
2C8-17 Comb. 4 4WO
Figure 4.31 Mai Response of 2CS-17, Combination 4.
2CS-17 Comb. 5 4000 ,
2C5-17 Comb. 0
Flgun 4.33 Axial Reaponse of 2C6-17, Combination 6
2CS-17 Corn b, ?
Figum 4.36 )oh1 R#ponw of 2C6-17, All M a l Combi~tlons
TABLE 4.7 Anilyticrl to Exp.rinnntal Ratio6 for -17
Load venus average tie stnin cuwes am p l o W in Fig. 4.36. Model
combination8 3,4,6, and 7 reached the second peak just after yielding of Iatenl
SM (3 : 2.4 "'Tm), and combinations 1 and 2 reachd their peak at an average
O sûain ~~n 1.8 and 2.4 -lm. As mentiond abow, the & m i n 1 peak
Ibr the specitnen wu 1.8 -lm. It mis appamit from Fig. 4.36 that a change in
stihess occurred st a tie sûain of about 0.3 -/,,,, which ~ccurred at an axial
stmin of -1.6 Tm. 1 his correspondecl to the initiation of cracks on the conmte
cover.
4.Sm4 Column 406-24
This dumn had a rlighüy kwsr volumaiSc nüo than 2CS-17 :
2.3û%), and a smalkt th rprcing (8 : 0.1 36).
The rsaponse to momtonic ~Ctn!pnwdon of the spedmen was dudile due
to the dose-spawd steel cage. Them was one layer of bricks be-n tie setups
for the finite dement modal, thw avoiding Vat? demanta (i.e*,mrhrirl in8tability)
and the aspect ratio wm kept dom 10 1 :1:1.
All mode1 combinations but combination 6 showed W a k curves, as
couid be the case for wsii-confined dumns. The program was able to predid
cover spalling (at an axial stren of about -1.6 -lm, the maximum peak load in a
range d 96% to 109% of the actual pak load, and to a certain extent the
obsewed ductility. The lateral steel yielded after the fint peak but uval1 Wom the
second peak in the majority of cases. For instance, taking combination 2, the
average tie strain was 1.5 "'"''/,,, for the fint pak and 5.2 """lm for the second
peak. It should be noted that the experhental evemge tie strain was 4.5 "'"'/,,, at
peak
A plot with al1 the combinations is shown in Fig. 4.37. It is apparent that
combination 6 is the best fit of the experimental m e . Recalling f i m fable 4.4,
this was the only combination whem compression softening of conuete was not
taken into acaunt. As longitudinal and lateral steel encased the concrete corn
very dorely, triaxi*al compression stresse8 âeveloped for large deformations,
and the capacr*ty wu rurtlined.
Table 4.8 gives th8 wulytical to qmrimenfal peak load anû peak straln
ratios for thb c o l m . Alüiwgh an mkl m i n of -11.7 -lm was reporta at peak
load in the experimmt, a plateau stmtching from about 9.5 to -25.0 -lm wu
obrewed, and cdurnn (7) affable 4.8 may 1- d BCCWBC~.
An analysis of the unuribility of all the perameûic variables in the^ rn-l
combinations ir presented in the next section, and conciusions will be dmwn for
the modal cambination(r) that bet fit al1 of aie ailumi ôehavioun studied.
4.6 Effacte of M&el Combination on the Responu of the Selected
Columns
Table 4.4 ir repmduced in thir W i o n to facilitate the description of the
parerneters involved in the rtudy of confined c~clcrete behavior.
TABLE 4.4 Pamntric Combinations for SPARCS Analyses
MKP ~ O ~ O V ~ C S
MKP Pope-
Hoshikumr pope-
YES YES YES E S VES YES YES
YES E S E S E S YES NO YES
All the combinstionr have the mme pm-peak cuwe fw comte, and
diffemnt post-peak c u m and conlimnt rnodels. Combinatirnt 2 and 8 dmr
only in the comate wftmkig dtrrkn. ïhe purpose was to iidenüfy which group
of moâeb might npioduœ the goml behawi'or COllfined concmte. Peak
loads, peak strainr, port-peak behavior, and compression ratening am to k
compand in this section. As tie m i n data from the expmimentr war only
available for columnr 2A1-1 and 2CS-17 in this stuûy, sepamte cornparisons for
lateral expanrion will be made for these Iwo cdumnr aoainst msultr obtained
wMh a prwious version of the program.
4.6.1 Peak Load
The results of analytical to experimental peak load ratios for each wlumn
wen, ieorganized in Table 4.9-
TABLE 4.9 Anûyücrl to Ezpelmrnbl P a k Load ntios
Column l M a l Combination
Combinations 1,2, a d 5 useâ the tentative confinement mode1 proposed
2Al-1 4 B 1 9 2CS-17 4M-24 m a n
std.dav(%)
by Vecchio. The pndidion of the peak load wing this ml agmd well with
higher peak loads (eombinatiionr 3 anû 4) due to its definition of IaGnl pnrmum.
0.W 1 .O1 1-06 0-99 1-02 4.44
0.98 1 .O1 1 .O3 0.M f.00 M O
f -OU 1 -06 125 1 -O# 1.12 a66
1 .OU 1 .O6 1.17 1 .O7 t 0 9 646
0.99 1.01 1 .O3 O.= 1-00 299
1 .O7 1 .O4 1.24 1-03 1.09 @,56
Combination 6 desmes romo attention. Well-confined columns (i-r., with
high volumetric ratios and rmall tie rpadng u was the case of 463-19 and 406-
24) are lesr susceptible to develop concret8 wftening in the concret8 cors than
those nat so w l l confind (eg., column 2CI17, very high tie spacing), or poorly
confined (0-g., ZN-1). As tie rpedng inmases, Iateml pressure weakens and
softening might ckvelop within the CoClcTete are. SPARCS cannot account for
softening if it had not been set up at the beginning of the analysir.
For discu8siwi puqmses, the 'Sfrengtth imriease'' ir defined as:
where P, is the^ maximum umnfined l a d in the concret8 corn:
and A. is the area of the corn ftom centre to centre of aie perimeter hoop.
These expre88ions wem useâ to cornpute the rtrmgth inuease for each
combination and am prssenteâ in Table 4.10.
TABLE 4.10. Stnngth Inciuu
the- resulb are plotteâ in Fig 4.38 sgainrt the volurnetric ratk f i
Figum dD3û Comprrkons of Stmngth Incnaae
Combination 2 f i W wll al1 the experitnental peak loads. Combinations 3,
4 and 6 showad scattered stretngth increases for columns with high volumetric
ratios.
The unconfined plain concrute sûength (taken as 0.85Pc) ww, a good
approxitnaüon of the actual value for the columns analyzd.
The anrlytical to expeimntal p d c strain rdationships am summarizd
in Table 4.1 1. For ductile behavior, the qmcimens sustaineâ lord -out loring
ciprci(y br fa- de1Wmitions; thus the would k r range of axial rhorlsning
-
to which t h post-peak dudik mgim occum. ïhe analyücal v a l ~ taken in
Table 4.11 wem üK#o at which the second peak loadr occucted in dudile
column behavior. ïhe re8ults show more scatter man th. pndicted p a k loads,
and the program undemtimated these values in general.
TABLE 4.1 1 Anrtyticai Q ExprrlmanW Peak Stniwn Ratios
Column 1 Modd Cambinrtion
For combinations 3 and 4 where the Selby's confinement mode1 was
used, the predicted p a k sûains were Iarger than those predicted for the other
combinationr, in thro of the fou colunns. A better result was obtaineâ for the
poorlyçonfineâ 2A1-1 column with Vlcchio'r moâel. Although combination 3
presented the Ieast scatter in the peak m i n calculation, it ovendimated
cdumn capacity well rbove the othm.
Combinations 1 anâ 2 shuwmd rimilar man anâ rtPnderd deviation for
both the peak loaû and the peaic m i n ; howmver, OOmbination 2 dionnd
4.6.3 Post-Peak Behrvior
thm post-peak wrves for co~nete am plotted for Bach column in
Fig. 4.39, Mile keeping the othw combination panmaters constant.
Figure 4.38 Efhct d PObf-Peak Cu- (1: ModHied Kent & Puk, 2:
Popovics, 6: Hoshlkuma at 1)
No difhmce was appamnt in cdumn 2A1-1 in the post-pak mime.
Howevw the Hoshikuma et al. modd rhawbd a steep decsnding krnch for the
nmuining thme columns. The Modified Kant and Parû wid tfm Papovicr cuwes
sustained load sirnilsrly, but the Popovicr wwe producmi larger defonnationr.
Thur, il tmœd the exprimntil cuww with slightly better rcarraq for highly
Infiumœ on the dudility wrr noted if compfessiun softening was not
considweâ in the analysis. As the cmcrete stmgth did not waken due to
tensile mins, lateral expansion increased quiûûy at eady Ioad StBQeS, rerulting
in early activation of lateml steel. By the t i m the column M reactied ib peak,
the canfining pressure haâ eventually maclmd its maximum value, and ties had
The hivo combinations could be considered as bounds for the a*al
shoftening curves of the anal- columnr. They were useâ for comborating
the confinement behavior of the specimens presented in Chapter 5.
Selby (1993) mporleâ the analysis of six of the Sheikh-Uzumeri columnr
using SPARCS. Two of VHU. columnr were alro analyzeâ in this study, 2A1-1
and 2C5-17. In Selby'r early attempt to tnodel rtienglh enhancement, program
SPARCS haâ limited optionr in both constitutive modeling and analysis
capabilitier. Some of these limitations have b e n ovBcicome thraigh recent
improvernents. Differencer in rnodeling and analytical msults are discussed in
this sedion.
Table 4.12 shows a compwkon of th g m û y , steel anâ material
Porclllwbtc Slw& and Pohum's Raljo 105
stated"th8-art compter proœssom and romo modifications to vedor
dhensionr in the SPARCS wlvsr, it wu possible to incremmt the number of
elemmb subrtantially, and to =Ive conriô8rably lagw modela h a fraction of
the previous m i n g times (that Iasted up to ocn, week!)
S o m analysis arurmpths mnained the rame. Pefed bond was
assumeû behnreen steel ban and c o m t q bus8 &an wws ured to mode1 the
ties, and buckling of longitudinal bars w m not canridrred.
TABLE 4.12 Solby'r and Cunnt ModeIr
Tatol number of Ehkks
tha difhmcm k(inmn th. applied load and the load taken by tha springr was
the load th column rupported. It uns .Ir0 necessary to modd a s , l M m s h
(i.e., a mode1 without c o m t e mver) once peak loads had been m a W . The
current version of the program overcomw thir difficulty, and allowr the user to
Figures 4.41 and 4.42 show the axial rhortening and averaoe tie stnin for
2A1-1 and 2C5-17, mpectively. lïe rtrains followeâ the same load path in the
initial loading phau, rhowing a stiier msponse than the swmens. No
differenw in the pre-peak msponse war noted. Hwever, stmtching of ties
ocarrnd to some extend in th pst-peak branch for both wlunns in the cumnt
analyses.
The use of spalld mesher was not succssdul in capturing the post-peak
response of 4 of the 6 columns analyzed by Sdby. As the author comrnented,
pmatum aurhing of th concret8 core antributed to th la& of ductility. This
could tm ettributecl in part to the aspect ratio d the finite dement8 used.
moâel implemnteâ in SPARSC for variable Poisson's raüo was
compemd with tan d the wlumns t t d d by Liu et al (1998). Tha mxt section
Rgun 4.41 Resuit Compvlso~ for Ul-1
4.8 S W y of Pdasonms Rnio
Tm 250inm ciradar columnr tmted by Liu et al. (1998) wem cholcwi to
invertigate the variation of the Poism8s ratio with the incnaro in mial
cumprewion. Colum prop.rtie8 am shawn in Table 4.13; detailed description of
these columns will k given in Chapter 5. These wlumns hsd spinls or hoops
as lateral steel, cylinder strengüw for conwete rsnging from 60 to 90 MPa,
volumetric ratios frornj, .2 to 6.0%, and conuete covm of O, 15 and 25 mm.
The objedive of this study was to corroborate, by dired observation of the
plots given hemin, that lateral expanaion varies as axial load increme8, and to
detemine if the tentative ocpension mode1 proposed by Vecchio (see Veschio
1992) r8asonably reproducer Mir phanornefton.
TA6LE 4.13 Uu et ai. CoIum Propartlas for the Stwly of Poiuongs n o
+8m1 Procoâun to Obbin th. Exp.nmhî Poisson's ntlo
The Poisson's ratio variation was computeâ using the proœâure
a Ffom the experimcmtrl wkl loid venus lateral rteel min aime, a load
stage was wlected at which the Poiuon's ratio v was to be computexâ (Le.,
tie strain a),
a The lateral pressure f i on the concret8 corn was detemined from the
equilikium of the section as shown in Figure 4.43.
(4-6)
u b m f. is th actual lie 8&eSS1 r is the hoop -ng (or spiral pitch), d is
the œntte-toamûm tie diameter, A, i8 the ama dam hoop kg, and d is tha
column diameter.
- - -
Equation 4.6 a n ils0 be wfittm as:
(4.8)
p. is the volumetric lateral steel ratio in the concret8 corn. It should be noted
mat fu was assumed unifomly distributecl dong the tie spacing; however,
thk was not the case for confinec! columns.
As the hoop spacing varied f i 50 to 150 mm for the Liu et al. columns, the
letml pressure on the conuete con, was no longer considerd unifomly
distributecl. Thus, in order to obtain an equivaknt pmssum (Le., unifm
lateml pressure), a factor to r e d w fW war applied. me confinement
elkcîiwess --nt k. proposed by Mander et al. (1 998) wos used in this
study. The factor k. ir computed for dmlar hoops as:
and for circula? rpiralr r
---- -
whem 8' ir the dear spadng between hoop8 (or dear spiral pitch), and pr is
the longitudinal steel ratio in tha comte cors.
The lateral pressure on the concret8 corn was then computed as:
1 fi, =+%*A
O As the lateral iMin in concnte was axpected to ôe srrll, the conmete
behavior in the direction pependicular to the applied load wuld be
w n s i d ~ linear elastic. It was assumeci that if the lateral sûess k was
smallw than 40% of P. (cylinder sbengai for concnte), the strain in the
concret8 ~o due to confinement could k computed as:
when E. ir the stiiess modulus for the concret8 (shown in Table 4.13).
The Poisson's ratio v i8 given as:
wbm 6 is the tie M n , and k the average axid rtnM in the column,
obtaind frwn the experimental nwb.
O The volumetric dnin was compukd as:
eV =e, +2(6 +e,)
The procedure war mpeated for v v i o w load s t a ~ ~ s mtil the axial shortening
a d tie m i n CWM, for each dumn wwe completely detemined.
Axial loedPoi8son's ratio wwes wem normalizeâ with respect to th8 initial
Pois8on'r ratio vo and the m i n at paak cylinâw stress E, As the columns
w m maûe d high strmgth cormete, the values for 4 wre ohined as
foiiws:
where the stiinne8s modulus E. was obtoined from the experimental resub.
Table 4.14 shows these initial values for al1 the colurnns.
TABLE 4.14 uwl vo vJu# for Uu et ai. Columns
These nonnalized plots were compare with the variable Poisson's ratio mode1
(see Figure 3.16):
(Eq. 3.48)
It should k noted that two assumptions w r e made: concrete behavior was
considered elastic up to a stress of 0.4f. in the direction of expansion, which
was deemed reasonable; and the use d a factor to obtain an equivalent Ilitem1
pressure. Although variour factors have been proposed in the literature (e.g.,
Rami and Saatcioglu (1999)) it was not an objective hem to investi~ate them,
but to obtain a value for the lateral pmrure that Mected the adual
confinement behavior.
Axial load-oaial rtmin, axial loaâ-tie m i n , axial load-Poissonrs ratio, axial
stfain-Poisson's ntk cuwes, anâ the variation of the volumettic ratio am
presenteâ in Figr. 4.44 to 4.51 for two of the Liu et al. Colums. The reuilb for
the remrining cdumnt am shown in Appndix C.
Figure 4.44 Lord-Stmin Cuwe of Column 2CW-10S1W-25
Figure 4.48 Pokoon's Ratio Axlrl m i n Cuwe of Colum 2CW4OS100-26
Flgun 4.4û Load-Stmin C u m of Column 2CMEoSSO-16
Figum 4.48 Lad-Poiuon8s H o Cum of Cdumn 2C8-26
a00 0.20 440 am QI0 1.00 1 1.10 1.a 1.r) 200 - Flgun 4 . a Poissonsr Ratio k<kl Stmin Cuwe of Column 2C804SM)-25
Figures 4.52 to 4.54 show plots of the variation of Poisson's ratio with the
axial stnin Ibr each of the column series described in Tabk 4.14. Alro shown is
the analytiil expnuion for the Iabnl expansion mode1 of Eq. 3.48.
Figun 4.62 LaWml Exglmkn Variation of &rios I
The lbllowing can k wnduded from the obsewed khavior of these
columns:
The incmase in the Poisson's ratio began rt about 0.5 to 0.6 P , whem
P, is the mswimum load sustaineci by the column.
Lateml rteel stnins are pmctically negligibb Ibr m-al compms8ive lords
stnaller than 0.5 b 0.6 P, . lt is apparent that as the Poisson's ratio began to change, the voluntric
stnin sr incmamd (Le.. the cokimn secüon changeâ from being contractd b
king expandeâ).
The Poisson's ratio incmase ftm the initial value va m a n at axial m i n 8 rt
about 0.2 to 0.6 6
In general, the analytical cunn tfaceâ the adual variation in lateral
expansion as load increased.
It 8hould be noted mat this analysir wer limited to circular high-strength
columns. Furthw analysis of the eduPl variation of lateral expansion in othw
sections and concret8 types may be madd to validate the pmposeâ formula for
Vie Poisson's ratio variation implemented in program SPARCS.
lt teemeci Via Equation 3.48 wsr adquate in modeling the Poisson's
Mect on confinement, and it was decided to keep v variable during al1 of the
analyses made with SPARCS for al1 the rpecimens studied.
Chapter 5
Corroboration with Experimental Studies
6.1 introduction
In order to proviâe a doser v i w of the capabilities of the modek
implemented in the program SPARCS, a series of specimens tested by other
researchen were moddd and analyzed. This series induded five high strength
conmete (HSC) circular wlumns testeci by Liu et al. (1998), hnro nonnal strength
concrete (NSC) specimens testeâ by Mander et al. (1984), three additional
Sheikh and Uzumeri columns, and a shear wall testeâ by Lefas et al. (1990).
The objective was to corroborate the effediveness of the confinement
mode1 thmugh compafisonr of analyücal and experimental axial and lateml
deformation wrves. A description of the mmens and terüng procmdures am
pmsented, followaâ by the finite dement mdel and analysis results.
0.2 Uu, Fos-, and Atbrd Tosb (198û)
Cirarlar HSC columns with hoop8 or spirats w m t a e â and d e l d uring
the umkmplane mdf reported by the authon.
0.21 Colum Gwmtfy
A set of 18 short HSC drcular colums divided into 3 series (Le., i, il and
III), were subjeded to conmtric monotonie axial compression. The colunns had
a 2Semrn diameter and wre 1-m high. Top and bottom ends wete
heurshed to pmvent failure. Each haundi war 400 x 400 x 300 mm. The middle
Viird of each d u n n war gauged with m i n gauges. Details of the column
geometty and longitudinal and lateral steel distribution for %Il the series are
shown in Fig.S.1.
. . . . .. .... . . .. . :... J. ..'. . . i.. i. ... . . .. . . . . .
. ....
Figum 6.1 Dotdl of Cdums Tostad by Uu at rl.
- -- - -- -- --
6.2.2 LongîtudM uid Labrai Stod -nt8
Hot-rolled barn wwe useâ for the longitudinal reinforcement, and
deformecl ban for the circulu hoops and spiral tim. Th. cirailar hoop wwe
welded at the ends*
Hoop mng or spiral pitch wm 50,100, or lx) mm, end concrete cover
vatid from O mm to 25 mm (sec Fig. 5.1~). Eight 12mm ben wem ured fw the
longitudinal steel in al1 specimens, and 6-mm or 1Omm ban were useâ for hoop
or spiralr. Since the cover thickness varid, the centre401~8ntre diameter of the
hoops or spinls was not constant
6.2.3 T a t Instrumentation and Procadun
Strain gauges were mwnted on bo# lateml deel and longitudinal steel at
the midheight of the brting zone (800 mm). For ~ d e s 1, thme gaugas at 120"
wem used in each of two aâjaœnt hoops, and one gauge in each of two
opposite longitudinal ban. For rerim II and III, four $train gauges were used
along one pitch on the central spiral, and fou gauges on every second
longitudinal bar.
LWTs r p ~ n i n g 400 mm wem installed to measum the vertical
dirpl8œment d each oolunn. LoPd œllr wwe u d to mearum the applid
W.
authon; large ecœntricities mm mcordeâ duing th6 testing of m e d the
elements.
The experimental variables studied inciuded spacing and volumetric ratio
of rpirals, concret8 aver and concret. stfength.
6.24 Saiocted Uu at ri. Coiumns
Five of the 18 colurnns wwe moddeâ in SPARCS and compared with the
e>cperimental msults. ln mis seledion, the amcrete cover was kept constant (15
mm). Thme of the cdumns had a nominal concmte strength of 80 MPa,
correspondhg to Series I (8- Fig. S.la), and hnro were of 80 MPa,
corresponding to Series II (Fig. 5.lb). Th@ variables accwnted for in the
modeling were the hoop or spiral spocing, the conwete stfength, and the effect
of lateml steel yielding stress.
The material properties of the seiecteâ columns an, shown in Table 5.1.
The volumetric ratio p vafied from 2*0% to 64%, which could b
considerd high values. However, du8 to the kittle nature of HSC, large
amounts of laterai steel wrr mdad for these d u m m to undego dudik
ôehavior. lÏe sp.dng varied fram 20% to 60% of the cdumn diamrkr.
6.2.6 Finik Eiomont Modds
û.2.6.1 Gmmrtry
A8 *th the columns in Chapter 4, only one-quarter of the colwnn aorr
sedion war rodded due to symmetry. It should be noted that SPARCS has no
a%isymec elements in its library to mode1 bodies of rsvoluüon (ag., circular
colwnns). Instead, 6+1oded wedwr and truss ban mm, ured for conmete,
hmps and spinls, ntrpedively. Longitudinal ban were smeamd out, and trwr
ban wem attacha to adjacent wdge noda as show in Fig. 5.2. A summary of
the ~eomeûy for each column ir given in Table 5.2. The vertical profile of the
finite alement mode1 for column 2C80-1OSSO-15 was timiler to the Sheikh and
Uuimeri columns.
l'hem wem no differenm in the georneûy of al1 modeIr, but the numbr
of tniu ban used and the mechanical propetiea of the materials varied.
TABLE 6.2 Oeomoby of Finlte Hanmit Mockla
S.2.6.2 Materhl Types
The longitudinal minforcement was modeled as smeamâ within Vie
conciete. The steel ratios p used in the rnodelr are shown in Table 5.3 along
with the plain conctete strength and the tensik strength for concrete. A sketch of
the matefial distribution in aie cross section of the cireular columns is shown in
F~Q. 5.3.
Flgun 6.3 Materlai Typos for Uu a al. Colums.
The tensile
cornputecl uring:
stmngth for concria8 (see dumn (6) of Table 5.3) nnw
6.26.3 Material Models
For the corroboration study, the matwial behavior models that best fit the
experïmental data in Chapter 4 were ured. Those models wem:
Pre-peak cuve for conmete: Hognestad Parabda
Port-peak wwe œ . Popovics.
C o n f i ~ m o d s l . Vecchio
CmCkhIg Critm*~ Mohr-C~~l~mb ( W ~ S S fomiulation)
Compniuion softeniog : Vecchio-02A (if conridaad)
The Wainhardening modal for steel wu alro used in all wbsequmt
analyses. Although the Popovics equation tor HSC could alro k uwd in the
pre-pe~k mgime, it wa8 fornd that the simple pasbds equaüon provided good
approximations of the gcwieral axial-sMening behavior.
These wlumn modela wem al= subj@ded to imposeci dirplacemnts at
the top stiff Iayer. The reurlts are dividd into axial rtrein and üe m i n artver;
the effed of lateral expansion is d i r a i r d dong with the cornparisons with t e 3 experimental data. A summsry of the p a k lads and respective strains is
presented Iater in this sadion.
The FE rock1 behavior was compand with the experimental msult for
each column. Values given in parentheses in the following sedion are the axial
compressive strain in -lm et Mich an event occumed.
&2.6.1 Column 2CM4 OSSO4 6
The failum sequenœ in the analyücal m d d was as followt: cracking d
cover shell (-1.624); yielding of longitudinal steel (-2.170); fkrt peak l a d of
275ûKN and cover spalling (-2.81 1); yielding of hwpr (4.61 1 );and second peak
l a d of 2û33 KN (-8.187). At faikm the column capacity war mduced to l e u
than 50% of the maximum load sehieved. Ducülity of the mode1 was half that of
the specimm. The expansion o f k t wu ciphrnd masamble wsll (me Figue
5.5).
l t m d u m i 8pachn rhowrd rimilar khevior (me Liu et al. 1998) with
slight d i i in the wirl min. The f ~ i t u d i n r l bur yidâeâ bdon the fimt
peak As the fint peak of 2880 KN war mached, the cwer spalleâ off at an
average axial m i n of 3.0 -lm. The rtrain hardening of lies began immediately
Mer the yidd drain, and the lateral rtmr increaseâ aftw the fint p a k until a
second peak of 2970 KN wu teacbâ. The column wu heavily confined (6.0%)
and faild due to tie weld fndum. Sllght buâûing of longitudinal ben w s
apparent. Th8 SPARCS and experimental ames am plottacl in Figure 5.4.
)
S.2.6.2 Column 2C60-1 OS1 ô04 6
The model reached a -mum load of 2830 KN a an axial strain of-
2.831 """lm, which was larger Vlan that of the expdmnt (2460 KN a about -1 .S
"'"'/,,,). Due to this circumstance, the longitudinal reinforcement yielded More the
peak load in the model. As can be seen in Figure 5.6, the initial stifhess of the
modal was smaller than the record4 duinig testing. Hawmver, for boai the model
and the specimen, yielding of hoopr did not ocar at peak
During the testing of the column, a large accentriCity was detectad as the
l a d wsr applieâ. Tho3 column did not mach the 'SQUBS~ loed" (for this case,
calculated to be 2850 KN). which could k considerd as the minimum capacity
for concwWic laaded column. The partital spalling at p a k l a d obsewed in the
emmmt could alw be pmd d the accidental ~cc~c~tricity, nd could explain
the difbmme in tha axial shortming cuw#.
The hoop m i n i8 rhami in Figure 5.7. The R moâel faild befim
yielding of the Iatml steel.
O 6 1 O 1 a 20 rrhl rtnln (ma)
Rgu, S.? Latmnl Expw#ion of 2C6&lOSlûû-l6
6.26.3 Column 2Cûû-IOSI 00-18
The mode1 trecéd with good accuracy the a-ding bnnch of the axial
strain wwe. Cracking of the shell m n at an rtmin of -1.623 -lm; at mis
load stage, the lateml steel drain was pnicticdly negligibk (0.3 """lm). The peak
load (2880 KN, compression sdtening not considard) was reacheâ at an axial
strain of -2.813 -lm, Mer the longitudinal ban yielded. At peak load the wver
spalld M. Due to the large spacing betwsen hoops, the triaxial compression in
the concrete was hardly developed and the column mode1 fai ld in a brittle
manner.
For the rpea'men, dmilar failure conditions prevailed as for rwmen
2C6O-î OS100-15. A large ecœntricity wu recorâed (9 mm) in the strain gauges
instolled in the longitudinal reinforcement. Houvaver, the shape of the axial
shortening arrvo suggesteâ that the adual behevior could have been similar to
that dercribeâ for the moûel. The analytical peak load was the minimum
e-ed strength (Le., the squash load). Axial shortcming and tie drain graphs
are plotted in Figures 5.8 and 5.9, respectiveiy.
6.2.6.4 COIU~W~ 2C80-10SW-16
The modal wiüwut compreuion lOQtening M8d mat- vwy WII the
experimntal cuwe for axial shortening. In the modd with compression
8oftening, th post-peak mime wa8 'pwrlkIa to that of the expefïmental one,
butthefncingwasata lowc~load.
ssoo -,
Figun 6.8 Axial Shortedng of 2C60-1 OS1 60-1 6
For the Maiytici~ model, the arcsnding bmnch clorely mdcbd the
exparimental one until 70% of the fimt peak, evm though the Hognestad
panbols was u a in th8 modeling. Fimt cracks in the covw appared al about
-2.0 "'"'lm. The fimt prak load of 3590 KN ocairred 1 an axial m i n d -3.242
et this point, the longituâinal bars had alreaây yielded. Tho tie steel sa in
wes low (1.4 '""'&,). A8 lateral preuwe inweased aftw the fint pak, a second
higher peak of 3850 KN was obtaineâ in the model. The tie steel yielded before
the second peak was mached (me Fig. 5.10).
The test specimen reached the fnrt peak (3520 KN) at about -2.4 '""lm.
The vertical reinforcement yielded just More this peak, and significant cover
spalling was obsewed. The spiral stresws incre~died Mer the firot peak, until a
second peak (3880 KN) was recorded. The column ïailed after the second p8ak
due to ftadure of the spiral reinfomment in the gauged region.
6.2.6.6 Column 2C804SoO4S
The diffemnce with the pnwiouo column was the diameter of the spiral
(Le., diffmt volumetrigc ratio). The mode1 (without Wening Mect) cmcked
initially at -2.0 -l, axial strein; at thir load stage the average spiral strain was
only 0.8 mlm. The load-defomiation mspon8e paked (3700 KN) at 9.6 -lm,
and the lie m i n et thir baâ stage was 1.6 -lm*
0 .O 2.0 4 ,O 8.0 8 .O 10.0 12.0 14.0
8xlrl rtrrln (inmlm)
-- -
In the analytical model, the column cnisheâ at about 6.4 "'"''lm. The
mode1 wiai compression roftening fo l lwd the asamding h n c h but the covw
spalld off et a l m axial rtrsin. Due to the concret8 sdtening, the poa-pak
branch obtained in the finite rlemcmt analysis war l a w c ~ than the adual
behavior. No considerable cracking within the con, w l d have taen expedd
due, in part, to the tigM laterel reinfiorcement
nie failure sequeme for the rpedmen was: yielding of longitudinal ban;
peak load of 3850 KN reachd at about -3.0 mlm with cover spdling; desœnding
post-peak branch up to 6.0 -lm; and failure due to fhture of the spiral
reinforcernent in seven locations follawed by longitudinal bar budding. The
actual hoop strain plot apparently did net rdect the reporteâ spiral fracture.
Plots of &al shortening and tie strain am gnphed in Figs. 5.1 2 and 5.1 3.
0.0 1.0 2.0 3.0 4.0 5.0 8.0 7.0 1.0 r x k l atrriin (m mlm)
Figure 6.1 2 M a l Shortenlng of 2C8OdSb0-16
0.0 0.5 1.0 13 2.0 2.S 3.0 3.5 4.0 aua. 8 p h l dnh (niinh)
6.3 Mander et ai. Speclm«i. (1884)
Mander testeâ a series of cimlar, square, and redangulai columns. The
main objective d mis investigation war ta emluate column behavior under high
axial strain rates, thus simulating the m p n - of bridge cdumns under mirmic
load. A stress-strain mode1 for the concrete that considered the M8Ct of
confinement and the m i n rate was airo deveioped. The ultimate stmin was
basd on the rupture of the fimt hoop when a column was subjecteâ to either
monotonic or dynamic loading.
0.3.1 Column G o m b y
A set of 17 short circuler cdumns, 5 square columnr, and 16 walls
divided into two batches wero subjeded to mcentric monotonic or dynamic
load at both lw and high strain rates. The square colunn series was pmpaed
for testing by Scott (1982) to establish the influence of age in testing concrete.
Near-full-sùe column dimensions wefe chosen to avoid scale eff8ds. Scotrs
columns wefe 450mm square and 1200mm high, anâ the walls wem 700 x 150
mm redangular (scald to about 0.31 ratio of the rctwl dimensions of the
hollow sections, which wem 1900 x 300) and 12003mi high. In al1 carsr, tic or
spiml - n ~ WM mduω at both ends from 80% to 50% of the 8 W n g in the
test mgion. DehiIr d the column geomeûy, longitudinal and lateral steel
dirtribuüon for the walk and rqu~rs spedmenr are rhauni Figs.5.14 anâ 5.1 5
Wall Setup
Figure 6.14 Mander et al. Wall 11 . . . . . . . . . , .
Wall 11 had SO-mm hoop spadng. In orâer to avoid sudden kss of
capaciety during test in^, tha concret8 cover was kept to a minimum of 25 mm.
Tha SuNt mlumn rhown in Fig. 5.15 had 12 longituâinal deformed ban of
20 mm and Grade 3û0, Iatml steel wa8 plain round ban of 10 mm and Grade
275, with a cover to th hoopo of 20 mm. The th wtup war similar to the 'Type
D" arrangement of the Sheikh and Uzurneri cdumns (sa3 Chapter 4). 1
6.3.3 Tent InsûumenWion and Pmcodum
For the dmlar columnr, four linear potentiorneters wen, installed in pain
on the East-West and North-South faces of the colum, spanning over a length
equal to the cor9 diameter (450 mm). These potentiorneters wen, used to
meawre the axial displacernent in the column. It war asrumeâ thrt longitudinal
steel and cancrete mm, peHectly bonded, thus Wain gauges wre not mounted
on the longitudinal ban. The lateral steel strains were measufeâ using electrical
strain gauges.
For the walls, linear potentiometen were installed in pain symmetrically
placed dong the 7OO-mrn ride of the wall, and rpanning 400 mm. Eight strain
gaugas, four on ons of the inteml hoops and the 0th~ four on the oppsite
extemal hoop, wwe uwd to measure the lateml expansion.
For the Scott colum, a8 for the dreular column, fov pot8ntiometem wero
instaîleâ; one ab anch fim of the colum, spanning 400 mm. The author did not
r e m atmin muges on the tie steel forthi8 odum.
In al1 casesg a DARTEC tesüng machine was used. The losd and the
overall displacement behlV88n the machine heaâ anâ bottom plate m#e also
recorded.
Wall 11 m s t e d at a high sûain rate (0.01J/r), which was a
longitudinal displacement of about 20 mmlr. Cameras were mounted and
synchmnizod with the machine operation. EÎch single test lasteâ less than 5
seconds.
8.3.4 Selacteci Muidr et d. Spimens
The spedmens wlec!eâ for mode lin^ with SPARCS are shown in Figs.
5.14 and 5.15. lt should k noted that the general dimensions of these
specimens wn larger than the pmvious anal- columns. (Le., Liu et al. and
Sheikh and Uzumeri columns). The specimens were made of nomal strength
c o m t e (25 and 41 MPa).
The purpose was to establish the capacities of the program to mode1 the
confinement efled on different speci*men gminetry and steel arrangements.
Material properties for the specimens are given in Table 5.4.
The same pattern useâ in modding Sheikh and Uzumeri columns was
implemented for the Mander et al. 8pdment. The wall was modeled using 8-
noded bricks for concmte deme-, the longitudinal steel was smeared into the
conuete bricks, and the hoop sets wiere modeled with tniss ban as shomi in
Fi~ure 5.16 (dimensions in mm).
- - -
The material behavior modrls useâ for Liu et al. columns wre useâ in
modeling Mander's speci*mens. Eq. 4.4 was uwâ in calculating the tensik
strength for concrete, f
6.3.6 Anilytïcal rnd Expertmenhi Resulb of Mander et al. Spdnwns.
6,3,6.1 Wall 11
This wall was a scab mode1 (0.31 ratio) of a wall in a dudile hollow
sedion of a prototype bridge pier, as s h m in Figure 5.17.
Rgun 5.17 Pmtotypa of r Hollow Bildge Pkr
It wu testeci in rnonotmic compmmiun kit under high strain rates, thus
simulating the dynamic dlbct that wuld be producd by an earthquake. The
smmen was restmined against out4plane budding duting the temg. The
finite em8nt mode1 wrr rertnineâ a8 for the s q m ~~ of ChPpbr 4, but
Cmdmtion 143
any possible restraining dV&ct due to the perpendiwlar walls joining et boüi
ends war not taken into account, mither for Uie rpecimen nor for üie SPARCS
model.
The analytical and 8xp8rimenl miel shortening an, plotteâ in Figure
5.1 8. A, the wall tier mm, dosely spaced and the volumaric steel ratio uns high
(p,, = 2.3350, littie cracking in the concr8te coro wouid have been expected.
Thus, compression sofbning war not msidered in this analysis.
The sequeme of events in the analytical modd wm: c~clcrete cndu in
the cover at an axial compressive $train of about -1.8 ""''lm; yietding of
longitudinel ban; Nxnnum & a l load reached a -3.6 * I m wioi mver qalling.
Finally, signifiant l o u of capacRy dccurred, but the ductility was kept
compamble to that of the specimen up to a strain of -1 5 '""lm.
Mander reporteci (1984) vertical cndo developing at a sûain of about -2
"""lm, large pieces of concret8 covw spalling off, c o n s i ~ b l e shngth gain due
to the tie arrangement, and a steep pootpeak derœnding branch. This migM be
attributed to the large concrete cover to gros$ arma ratio (43%)
SPARCS captured ml1 the ovmll bhavior of the wall; however, soma
dillerence can be noted in the post-peak mgime. The finite element model tracad
the adual descending part of the wrw but at lower lords. The tie yield strain in
the analytical mode1 oaxrred after the peak load, whereoe the test specimen
tias yielded practically at the onmt d the post-peak behavior.
3 . 2 Scott Column
Scott (1982) pmpared for testing five square columns. Manâer (1W)
then testeci the rpecimens at high sûain rates in order to ertablish the influence
of age in ancrete. One of those -mens (specimen 16, me Mander 1984)
shown in Fig. 5.15, was testd at an age of 942 days and at a Wain rate of
0.0167h. Msnder found a Wength gain of the specimen in the post-peak region.
In order to avoid any auumptbn regarding shrinksge and u w p of concmte, the
finite dement mockl wr8 comparecl with the original column testeâ by Scott at
an age d 67 âays and at the ronn rtrsin rate (-men 13, Mander 1984).
The mial shortming curves afe given in Figure 5.19. ïha analytW
~ingkenchrrprstedfromthee~*mite la t rbwt5O%dthepeak
clwmbdion 147
load. Although the analytical modd mached concrete corn stresses of 37 MPa
(1.75 times the plain concfete stmgth, 0.85Pe), the peak load was only 81% of
the sdual maximum. Dynamic Mects of the impod load in the test were
appamtly more pronouncd in this column than in wall 11. The cover began
cracking at about -1.5 "Tm, and rpalled off et the peak load. the longitudinal
bar0 yielded belore the maximum load wa8 raached; Vie ties yieidd rAet the
peak load, and the mode1 sustained the load without losing capacity until the
concrete cor8 crushed.
Scott Column 10000
- - -
6.4 Shdh and Uainmi Columns
Thme additional wlumns wim, mdomly seledecl from the 24 temted by
the nsearchers in order to OOnfirm uiitability of aie material khavior models
that had been choosn in the parameMc dudy. The column descriptions, finite
element models, test setup and inafLImentation were descriW in Chapter 4;
hence, oniy the resulb d these columns am presented in this mcüon. Tho
material properties are shown in fable 5. Each column was analyzed two times,
one considering concrete Mening and the other without..
TABLE 6.6 Matanil Prop.rllm of Additionil Shelkh and Uzumwi Colums
Specimrn Longitudinal Steel ~inr,
(1)
-5 4û4-20 403-22
COM. m (2)
C 6 O
4
ml (3)
15.875 1B.OSû 19.05û
€8
IMpa (I)
200000 196100 1-
A,
mS1 (4)
32ûû W 34W
Eih
tMm (8)
9220 (5200
8200
A
FI (O)
3.U 3.67 3.87
8 i
àn-1 (S)
7.70 7.M 7.W
G
WPIS (6)
367 392 392
S.4.1 Column OC64
This column had a large volumetric ratio and a small tie rpacing,
combined with a tight cage and a large nurnber of longitudinal ban. The axial
responw is plotteci in Figure 5.20.
Figure 6.20 Axial Rasponse of 4C6-6
60th SPARCS analyses followed the expimental cuwe with good
acairacy. When SOffening wrr consiâered, the trace of the post-peak behavior
closely f o l l d the expimental curw. Two p.ab, me at the spelling of the
coMxas cover and one O(fw the tim yielded, wme apparent in the analysir with
concmte softening. Th8 s e a d p e k (4840 kN) was slightly Iagw than the fint
peak (4450 kN) indicating mne gain of c~lpacl'ly efkr Io= of cover. Yklding of
longitudinal bars and initiai mcking d the covw occurrsd at aie same load
stage at about -2.0 -lmD
The model behavior was equal for both analyses up to the fint p k of
the sdtened cww; bihimtion of the port-pak mime war apparent aftemrâs.
5.4.2 Column 4-20
This specimen had a maller volmetric ratio than the pmvious cdumn,
with an wual tie spacing but different tie arrangement. ln ais case, the
analytical response without conuete roftening traced the spcimen ductile
responso vwy well (see Figures 5.21 and 5.22). Cover cracking started et about
-1.9 Tm; the longitudinal ban yielded befon the p a k load, and the tie steel
yielded after aie onset of the post-peak &ranch.
It wer üier8fore apparent that the tie configuration influenced the overall
response of the finite dement model. Compering this column to 4C6-5, bath had
similar properties for the ancrete and longitudinal steel. Although there wem
differsnces in the amount of lon~it~dind steel and volwnetric ratio, th8 main
difference was the tie amngement. The Type C amngement (me de=-ption in
Chapter 4) producd lamer lateral &errer and larger strength gain than did
Type B.
-- -
8.43 Column 403-22
The tie rpcicing of this swmn w m the Iafge8t among the VKee
columu. O m the maximum l a d ws mached, the spchen wrtsined no load
at large skainr; instead, the capacity droppeâ after the peak. As for the pvious
models, this column wsr analyzed with and without softening. Due to the large
spacing between ties, credang within the concret0 core wwld have been
expected leading to rom softening of conaete. This behavior wrir captured by
SPARCS as rhown in Figure 5.23.
The failure sequence for the softened responm war as follows: yielding
of the longitudinal ban and initiation of verücd cracks in the cover. Fint peak
load (4330 kN) and cowr rpalling; second slightly smaller pak l a d (4300 kN);
and decrease of capacity until failun, due to m e t e crurhing.
4500.
4000-
3WIO-
mm- ~~, 3 ,.
6 . Gononl Rarults
The maximum loads obtaineâ from the tests am compared to the
analyücal peak loaâr obtaineâ from the SPARCS analyses in Table 5.6. The
analyticd values wem fwnd to k, betwwn 85% to 1W% of the experimental
values, with a standard deviation d 11 %. The Scott column (tested by Mander)
was not accounted for in the computation.
A graphical reprementation of Tabk 5.6 is show in Figures 5.24 and
5.25. Then, was no significant difference in the cornputed values with respect to
the experimental mes. Thur, it ir apparent that the tentative model for strength
enhancement proposd by Vecchio (1992) can be used in the analysis of
confineci conuete regardless of the dement shape (e.g., circular, rectangular,
and square columns), and concrete type (e.g., normal or high strength concrete).
TABLE 6.8 Maximum AnalyticJ to ExperlmentaI Load Ratio
~ 1 0 8 2 ~ 1 5 2c6cbms1SO,1s ZCU&lûW&15 2caoaS&15
Wall 11 sc&t
Maximum Load
1000 2000 3000 4000 5000 6000 7000 8000 9000
Experimonîml (LN)
Maximum Lord 9000
sooo - .--Ca-
7000 - SUratiL
O H l M
Figum 6.26 Mudmum kiily(icil to Exprrlnmitil Lord Ritk (wJWbting)
Two aou sections of a rhort square column were ckvlgned according to
the Canadien and Am#ican design provi8ionr for concret8 (i.e., CSA23.3-94
and AC1-31845. mspecüvely). Specifically, they wwe designed for the minimum
requirements for laterai steel in seirmic zones. The sections mm, auumed to
be one in th plastic hinge zone (Le., Mar the top and bottom of the column)
and one in the central mgion of the column sway Rom the plastic zone (see
Figure 5.26).
The column war amumeâ to ba constfW8d of m a l stmngth concnte
of 30 MPa, with 400 MPa aeel used for both tho longitudinal and letml steel.
Eight longitudinal ban mib rupported by the outecmost hoop and two cross
tics, as shuwn in Figure 5.27a. In the SPARCS wirîysest both mcüons w#s
Ciudomtion 156
subjected to comtric compression. As thir is not the ciuo for seismic actions,
a close look a the expeded dudility and rtrengtf~ enhamament due to
confinement w l d k seen from them analyses. The finite element mesh used
in the SPARCS is plotted in Figure 5.27b. k m i l d design calculationr and a
summary of the code provisions for the design of short columnr am given in
Appendix A
The axial load vamus axial shortening wrves for both redions are
graphed in Figures 5.28 and 5.29, along with the cuves predided by the
Sheikh-Uturneri and RazviSaatcioglu models (which are ecplained in Appendix
6). In the SPARCS analyses, the mateiel models c h o n in Chapter 4 wre
used with and without compression 60flening M8d,
CSA-AC1 (Central)
Figun 6.28 Wd Shortening of Sedon in Central Zone
CSA-AC1 (Hingr)
For the SPARCS analy8isI il is apparent that after the p a k load had been
reached, the column rustaind b load carrying capacity for up to about
the peak Wain. From this point onwards, e rudden lost of capacity ocairred.
The Sheikh-Uzumeri and Razvi-Saatcioglu modela followed paralbl post-peak
behevion but a difkrent load cspacitias, the bmer k i n g more conmative.
However, it should be notecl that the finite eiement solution without compression
softening lay between the Iwo anfinement mockls.
A cornparison of the maximum l a d and the sWin et peak load for both
sections is given in Table 5.7. Also shown is the m i n at 85% of the peak load
in the post-peak region, sw; anâ the pak m i n to su m i n ratio. Although the
w l m n in the potentiel hinge zone appermtly achieved rom8 ductility, aie axial
column capacity is reduœd between 5 and 17% (depending on the rnodel
considered) due to the higher tie spacing in the central zone, and the dudility
dropped more drastically. This drawbadc could k of m e importance in
columns with high axial load and low moments c a u d by laterd loads (e.~.,
ground motions), such us intmal columns in buildings.
TABLE 5.7 S h e Column Socths In Seismic tonos, Modo1 Compulsons
The axial thortening results obtained for the ret of 14 specimens
analyzed with SPARCS am compmd with the analytical m a I r for confined
concret8 columns subjected to concentric compression developeâ by Sheikh et
ai. (1982, 1992), and Rami and Saatcioglu (1099). Description of bobh models
and an example of how to cornpub the axial load versus axial stmin using them
are prewnted in Appndix B. Alro ahanin in Appendix B is the cornplete set of
axial ihortening graphs for aII of the spedmens analyzeâ in this work
The Sheikh and Uzumm* mode1 was originally developad for columns of
normal rtrength concrette; the rock1 was modified to allow its use with any
concrete strength. The Rami and Ssetcioglu mode1 was intended for general
application; it con k applied to reâangular, square or circular sections, end
concrete may be normal or high strength.
It should be noted that aie meterial behavior relationships used in the
finite element analyses wwe (me Chapter 3): the Hognestad panbola for the
pre-peak respnse; the Popovics c w e for poat-k regime; the Vecchio's
mode1 for strength enhancement; the modified Kupfer m a l for Iateel
expansion (Le., Poiuon'r ratio); and the Mohr-Coulomb criterion for conmete
cracking.
The n8ub for son18 of the columnr are plotted in Figr. 5.30 to 5.33.
Each gmph indudes tha axpWmmtal cwva, the FE cuwm fiom SPARCS (with
uid w t t w mirig), rid th. m~yllcr~ mode18.
' i f 1-
nQI-
aw- Sm- --
fm-
0 . m 0,006 O M O 0.010 0-
-w
Figun 1.30 Modal Compwlson for Column 483-19
O 2 4 6 0 rrtkl itnh (mmlm)
Figure 6.32 Modal Comprison for Column 2C80-6860-15
Figurm 6.33 Modal ~ d s o n for WdI 11
'The analytical moâels pmdict the compressive behavior but do not give
an ultimate axial strain. The poa-psak nsponse of the Sheikh-üzumeri and
Rami-Saatcioglu mod.18 wem cut off W1 the plots, as they continue beyond a
residual stress, wMch is 0.3f:, and 0.2 f' mspctively.
The finL ekment modeling followed the pre-peak and post-peak
branches in a manner similar to these wellkrown anaiyticai modeis. Tabies 5.8
and 5.9 give the theoretical to experimental ratios for the maximum load for al1
the modelr. SPARCS presented an average ratio be-n of 95% to 103% of
the actuel peak Ioaâ for rectangular culumns, and between 97% to 10996 for the
cirarlar cdumns. Ail models had rimilw standard deviationr. This prover the
efficiency of SPARCS to trace the actwl response of columns in concentric
compreuion.
TABLE 5 J Thont lc~l to Exprrlnwntil Maximum Lord Ratios for Squam
Column k a t
(Mt) ml (2)
3418 4710 3524 JOOI 4370 4300 4725 ';Mn YM
, Pm. @PARCS) I
a u %a#
, (wholt) Diiul (3)
3370 4836 3622 4137 4233 4333 4ûSû 42aO 8125
ta3
Sh0ikh4t~rnii
%a
(woJmft)
(4)
3WO 5102 4355 4277 13t0 4484 4858 M 8970
em (8)
0.98 0.W 1.03 1.01 0.91 O 036 0 0.71
- O
3903
3996 440s 4 7 a 48Q3 4m0 8070
mi(trrt_
(rm . ( 1
1.07 1 . 1 4 1-04 1.00 1.04 1.03 0.91 0.81
mm (8)
1.14 1.03 1.13 1 1 1.07 1.01 0 .a 0,Ut
W (8)
3UQ1 Si14 4 4SOI 4821 St3Q
mm 716t
mm) (1 O)
1.14 1.09 1.17 1.10 1-10 1.20
S l ~ l . 1 0 1.02 0.63
TABLE 6.8 Thoontlcd to Expedmontrl Muimum Lord Ratloo for Cimulu Columns
lt should k notd from fable 5.8 column (7), in the Sheikh and Uzumeri
rnodel; that the maximum analyticiil load for their columnr war slightly diffemnt
ftom the expimntal mes. In some of the expeiments, the peak load occuned
when the contribution of the comte mer vanished due to spalling, which was
not the case whm uring the analytical model.
One of the s h ~ a r wallr tested by Lefa8 et al. (1990) (see Vecchio 1992)
was chosen to examine the effact of confinement on concealecl columns. This
wall was analyted by V a i o (1992) uring the twdimensional finite element
program TRK, and by Selby and V-io (1993) uring a previous version of
The wall (SW16, see Vecchio 1992) had a height-to-uri*dth ratio of 1 .O; it
was a r t integnlly with a top rpreader beam and a heavily reinforcd beam at
Ir base. Properties for concret8 and steel for this wall are given in Table 5.10
and plotted in Figure 5.34.
TABLE 6.10 Materid Pmptkr of Waîl W16
The wall was subjecteâ to constant axial load and monotonically increamd
lateral load, with both kadr applied at the top spreader bearn. T'ha comprestive
The finite el«r#nt modd fW will SW16 is rhanni in Figure 5.35. A total of
285 hexahdms wre uird to modd minforced wncmte; vWd, hofkonhl,
and tie steel were smeared within the concret8 bricks. Table 5.1 1 shows the
material z o m for the finite element model.
r*
Figura 6.34 L d n J d. Wdl W 1 6
TABLE S.11 Matadai Zan# for FE Mak) of SW16
Figun 6.36 Finit. Elamnt Mnh (bt Wall SW16
The v8rtical lord of 460 I<N was applied at the top 8pnsckt barn as joint
forces, and the lateml lopd was applied as imposed dirplaœrnats at the
Ielùnost nodes of the top beam as shown in Fig 5.35. The enelytical moûel
considerd compression softening, the confinement mode1 proposeâ by Vlcchio,
and the variable Iateral expansion.
The material proparties for î h finits dement cwulysir wem: plain
conaek, stmgth 0.û5P. (37.4 MPa), the elaatic modulur 33150 MPa, and the
initial Poisson's ratio v = 0.15. Al=, for the steml: the 8Whrem rnodulus was
210000 MPa, the strain and stiffness rnodulus et hardening wem assumâ as
2.5 -lm and 1 OOOO MPa, respectively.
The expwimntd and analytical shar-latc~al displaœment cunres are
shami in Figure 5.37, and a plot of the ddomed wall near hilum is shown in
Figure 5.38 (horUontal load applieâ fmm Idt to right). The analytical response
was stifter than the expfimental, but the analytical shear rtrength was 351 kN,
which was only 1% smallrr than the actual (the wall shear mistance was 355
kN). SPARCS underestimatecl the Iateral dirplaœmnt. However, the analytical
cuve t r a d well the overall msponse of th. wall. Fkwral and flexural shear
cnido developd at the top ban and the conwaled column on the tension
side. Shear cmcks spfeaâeâ out into the wall web and the bottom d the
coc1c88leâ cdumn on the compremion riôe wsr wibjected to trimgal
cornpnrrive sûmmes machin9 a maximum principal compnuive stress of 43
O 1 2 3 4 5 6 7 8
h m n l #qkœmnt (mm)
Figure 6.37 Horirontal'Responae of Wall SW16
Chapter 6
Conclusion Y
6.1 Conclusions
The capabilities of finite element program SPARCS used in the modeling of
confineci conuete were examined. Constituüve material behavior rnodels for
strength enhancement, latenl expansion, conuete softening, and post-peak
ductility wwe combinai in a panmetrio rbidy of fov columns subjested to
monotonic concentiic compression. columns repmmnted diffenmt sted
configurations and failurs modes. The combination of pre- and post-peak
compression curves for conmete, strength enhancement and variable Poisson's
ratio modelr that best fit the experimental rerults was chosen from the set of
parameters analyzd. ïhe current version of the finite element pmgmm wm
u d to compare hno of the fou dumm menüoned above with the analytical
results obtained from a previaur vemion. SignMcant impmvment in the dudib
behaviœ of the models was noted.
The Poisson's ratio war shown to be variable in the test data fiom ten
dm la t high-Wength columnr. Ttw variable tateral expansion model of SPARCS
accurately trciced the khwior of the circuler columnr.
The wleded mode1 combination h m the paismetric study was used to
cornborate experimmtal msultr of another set of specimens testeâ by various
researchers. The fkst attempt to mode1 the confinement bshevior of high-
drength circuler cdumnr wer made in thir pmject. Elements of diffwmt size,
steel arrangement, and concrete type wrm, mockled and analyzed using
SPARCS. The modal combination accurately predided the strength and port-
peak behavior of the specimrnr. H was noted that compri#iion softening of
concmte influenced the axial shortening response of the columns.
The finite element msponw of the cdumns was cornpered with the
analytical strem-strain moâels for confinement propos4 by Sheikh and Uzumeri
and Rasvi and Saatcioglu. lt wai fwnd mat the capacity and post-peak behavior
predided uring SPARCS compared mawnably well.
Two sections of a theoretical coklmn were designed accordhg to the
minimum mquirements for wirmic zones of the Canadian CSNCAN A23.3-94
and Amwican ACI-31û-QSR coder. The sections wem located at the potential
plastic hinge and at th middk of th cdumn, mrpsdively. The SPARCS
analyses dmmâ l e u ductility and undler strength in the central zone than in
A squat War wall with conœaleâ columns wu ruûjected to constant
axial compmsion and monotonically i m a i n g laterai load was mdmied uring
the matecial models chosen in th paramettic study. It wu tound that triaxial
compmuim sûesses did occulted at the base of one of the conceoleû columns.
The stfength was pndicted with goad acairecy; huwever the^ analytical response
was somwhat stiif.
6.1.2 Sp i f i c Conclurions
The pmpak axial rerponu was not rignificantly aff8ded by the choren
base curve; the Hognestad parabola fit the adual responw with sufficient
accuracy.
The Popovics cunm reproduœd wll the post-peak behavior of axially loaded
reinfbfω concrete qmdmens.
The strength enhpncernent moâd prpropod by Vecchio predicted the
element capacity within a mean of 3% and standard deviation of 11 % for al1
the spedmens analyzed in this study.
The variable Poisson's ratio model dbdively dmulated the inueased lateral
pre88ure on aie axially loadeâ columnr.
The cament version of the program reasonably moâeled covw spalling and
tie strain variation of the column modeIr.
The use d imposed displacements in SPARCS eliminated the nwd for rüff
@rings to analytic~lly npmduœ th8 poat-pmk ngim of minbmd concnite
su- to comprsssion, a signincant improvement over th prsviow
version.
- -
The plain concret8 sûength of all the columns wer taken as 85% of the
standard cylinder Wength for concmte. This value was in accordance to the
observeci capsdty of the plain concret0 columns that have been tested.
6.2 Impvaments, Umihtlons uid R ~ ~ n d r t i o r i s for Futum Worû
6.2.1 implovammnb in SPARCS
The SPARCS structure wii, enhancd in the folluwing aspects:
In the solver, that hardes large morkls in any global direction.
In the geometry chedcs of bricks and wedges.
A more efficient method of nurnbering elements was suggested.
In the calculation of the peak s t m s 6 and $train at peak stress s,
In the variable Poisson's ratio.
Improvements in the fmulation am required as followr:
a In the post-peak stress-sûain ductility relation.
a In the transition betwleen the uMXMfjned and confined elemnts
In the Wtened-uMned transition.
Although an effort was maâe to mode11 the confinement eff" through a
wide spctrurn d alternatives (8-g., m e t e types, stwl arrangementa, rize,
failum modes), mme limitationr am 810teâ:
O The lateml pressure as W~iwid in Vwchio and Selby enhancement models
might overestimrh the compressive cipacity of triaxially stfesseâ memben
0th- than columns since the use of 0.85P' for plain cancrete strength is
cornmon only for ailumns.
r The analytical respnser have an uppr bound if softening of concret8 is not
included in the program options. If safienhg ir induclad, a ioww bound could
ocarr as both dudility and rtrength decrease.
r Buûûing of reinforcememt in compression ha8 not yet been implemented.
6.2.3 Recommondrtions for Futun Work
The fol lowino rec~mmemdations am m a :
In order to mode1 bodies of rwoluüon in SPARCS, such as circuler columns,
an axisymmetric demnt w l d facilitate the mesh generation.
The confinement mode1 should be testeci in rpecimens subjected to moment
and axial l a d (Le., eccentric loads).
O lmprovement In the Poiuon'r ratio to mode1 conditions beyond the 0.5 limit.
a Enhancement of the po8t-peak mockl for COltfined concret8 in termr of
Mudal sûe8m8, to mode\ ductility and failure conditions.
O Improvemmts to th tension behavior mode1 br concrete to bettw mode1
highly confined ~pecimena with pndically no cracking of the are.
O Improvomenfs to the definition of Iatml p w m in the confinement -8
to maider umvm triw~l o o m ~ i v e rtnu -es, such as at the borckr
References
Abdel-Halim, M.AH., and Abu-LeWeh, T.M. (1 989). 'Analytical study for
concrete confinement in tied columnsR. J. Shud. E h g a ASCE, v115, nt 1,
pp2810-2828.
Benegar, F o l and Maddipudi. S. (1997). "Threedimensional modeling of
concrete structures. 1: Plain comtea. J. Stwct. Engrg., ASCE, v123, n10,
~~1339-1346.
Barzegar, F*, and MaMipudi, S. (1097). 'ThreedHnensional modeling of
concrste sûuctures. II: Reinforcd con~ret0~. J. = Engip., ASCE, v123, n10,
~~1347-1358.
Bortoloti, L. (1994). 'InRuence d concret8 tenrilo duûility on compressive
sûength of confimd columns8. J. MahrSJs in Civ. Ehgrg., ASCE, 6, n4, ppYl2-
563.
-
Chen, B., and Mau, S.T. (1989). 'Remlibration of a plastic-fraduring madel for
conaete confinement". Cement and Concrefe Res., v19, pp143-154.
Chen, W.F. (1982). HasiMy in Wntbriced Commte. McGrew-Hill, New York,
NY.
Collins, M.P, and Mitchell, D. (1997). Presfriessed Commfe Structures.
Response Publications, Toronto, Canada.
Cuswn, D., and Paultre P. (1995). 'Stress-stmin mode1 for confineci high-
strength cornete". J. Strucf. Engtg. , ASCE, v121, n3, pp468477.
HoshikumP, J., Ksaihiko, K, Kazuhiko, Nol and Taylor, AW. (1996). 'A model
for confinement M8d on stress-Wain maltion of reinforcd concret8 columns for
seismic design". Paper No. 825 Pmœedihgs fp W o d Conibmm on
Ewfhquake Engrg.., Elreviw Science, London, UK
Liu, J.. and Fostw, S.J. (1998). 'Finitsalernent madel for confined concrete
columns*. J. Saud. Ehgrp., ASCE, v124, nQ, pp1 011-1 017.
Liu, J., Foster, S., and Attard, M. (1998). 'Behaviow of tied high stmngth
comate cdumnr Ioaded in mœnûic compressionm. UnW Rem No. U-372,
The University d South Wak, Sydney, Audnlia.
Khrabinis, Al., and Kiousir, P.D. (1994). 'Effècts of confinement on conmte
columns: Plastidty approecK. J. Sn& Ehgtg., ASCE, v120, n9, pp2747-2767.
J
Mander, B. (1984). 'Seismic design of bridge pierb. Thesis pmsented to the
University of Canterbury, at Christchurch, New Zealand, in partial fulfillment of
aie nquirements for the degree of Doc2or of Philooophy in Civil Engineering.
#
Mander, B., Priestley, M.J.N., and Park, R. (1988). 'Oboenred stress-strain
behavior of carfned conuetea. J. Strucf. €m., ASCE, VI 14, n8, pp1827-1849.
Mau, S.T., Elwi, A€., and Zhou, S. (1998). 'Analyticsl study of apacing of lateral
steel and column confimmienC J. StlUCt. Engrg., ASCE, v124, n3, pp262-269.
Popovim, S. (1973). 'A numerical approach to the wmplete stremstrain curve
of concretea. Cement end Commk Res., ~ 3 , n5, pp553-500.
Rami, S., and Saatcioglu, M. (1999). 'Confinement mdel for highaength
wnaaten. J. StW. Engrp., ASCE, v125, n3, pp281-289.
Richart, FE., Bnndzaeg, A., and Bmwn, RL. (1928). =A study of the failure of
conamte under combimd compmmive stremesn. Bulletrcn No. 185, University of
liiinois Enginmdng E x p r i n b l Station, Urbuu, Illinois.
- - -- - - --- - -
Scott, B.D., Park, R., anâ Priestley, M.J.N. (1982). 'Stress-strain behaviwr of
conaeb confineâ by overhpping hoops at low anâ high strain ratsr'. J. of the
Am8-n Commte Inditute, v79, ni, pp13-27.
Selby, R.G. (1990). 'Nonlineu finite elment analysb d reinlord ancrete
solidsa. Thesis ptesented to the University of Toronto, at Toronto, Canada, in
partial fulfillment for the mquinments of aie degrm of Master of Applied
Sciem.
Sdby, R.G., and Vecchio, F. J. (1 9D3). aThreedimmrional constitutive relations
for reinforcBd concntem. PubIhtbn No. 9342, Department of Civil Engineering,
University of Toronto, Tomnto, Canada.
Selby, R.G., and Vecchio, F.J. (1997). 'A constitutive model for analysis of
reinforcd concreten. Canadan J. dCiv. n24, pp 460-470.
Sheikh, SA, end Uzumri, S.M. (1 980). "Strength and dudility of tied concret8
columnsn. J. Wuct. W., ASCE, vlO6, nSTS, pp1070-1102.
Sbikh, S.A. mâ Umm*, S.M. (1882). 'Amlytical d e l far co~nete
confinement in t i d cdumns". J. Slnrd D k , ASCE, vlOB, nST12. ~2703-2722.
Sheikh, S.AI Md Yeh, C.C. (1992). uAnelyücal momentumatum relations for
tieâ concmte columnsm. J. @Nd. Engro., ASCE, vll8, n2, ~528544 .
Vecchio, F.J. (1992). 'Finite dement modeling d c o m t e expansion and
confinemeW. J. Smd. €''tg., ASCE, v118, n9, pp2390-2408.
Vecchio, F.J., and Collins, M.P. (1Sû2). The rssponse of reinforced conmete to
inplane shear and normal stressesm. PubIhtriOn No. 82-03, Department of Civil
Engineering, University of Toronto, Toronto, Canada.
Vecchio, F.J., and Collins, M.P. (1986). 'The modifiecl compression field theory
for reinfwC8d cornete elements wbjected to shear". 3. of the A m e n
Commte Institute, va , n2, pp219231.
Xie, J., MacGmgor, J.O., and Elwi, AE. (1998). 'Numerical investigation of
eccentrically Iaaded high-8trengai comte tied columnr". AC1 Stmct J., v93,
n4, pP44M1-
Appendix A
Design of Short Column Sections According to CSA23.3-94 and ACMI 8-95R
Table A l shows a summary of the provisions for ttw design of spirals and
ties in short columns aaording to aie Canadian and Ammwcan code provisions.
The table coven the requinwnents for seismic and 'non-seismie zones, and
notes are inciuded in W b where there are di irenœs betW88n the codes
nie section is shown again in Figure A.1.
Gmetry
Squan, cdumn, with 6 = H = 490 mm, Tic diamter dh = 11.3 mm, Longitudinal
Bar diameter db = 19.5 mm.
a MatedaIr
Unconfimi -te shngth P. = 3ôMPa, Welding stress of deel f, = 400 MPa
A2 h i g n urfng Canadian Standud CSAZ3.3-04 (Srismic Pmvîrionr
Section 21)
j21.4.3.11 Longitudinal steel ratio p :
a [21.4.4.2] Minimum lateral steel area AI, :
whem ho = 410 mm (outer-tolouter extemal hoop distance), Ad = h: A, is the
grors area section = 2401 00 mm2, and s is the tie -ng. Thw, for this section
the f m w equation govems, A,,, à 3.951s- As the section ha8 3 legs in each
diredion, 2 V(lW mm2) = 3ûô mm2, and
a 121 -4.4.4 Minimum tie spacing in potential plastic hinge zones
Since the calwlateâ spacing ir rmalkr than the values from mis dawe, s =
0 [21.4.4.6] Tie spacing in zones away from potential plastic hinoes:
Thus, s = 6(19.5mm) = 1 17 mm, s z 1 15 mm
A3 b i a n check Amerkm Staâlud ACI-318-86R (Seismic Provisions
Section 21)
a pl -4.1.11 8, H r 1Z (305 mm) o.k.
[21.4.1.2]BIH=1.0~0.4 o k
0 pl .4.3.1] Longitudinal steel ratio p = 1 % o.k.
a (21.4.4.11 s = 75 mm o.k.
a pl .4.4.2] Minimum spadng in plastic hinge zones
a [21.4.4.3] Maximum distance b W w n supported ban = 165 mm c 355 mm
o.k.
[21 A.4.61 Tia spacing outride hinge region
Thur r = 1 15 mm as for the CSAn.3-94 design.
Appendix B
Sheikh and Uzumeri, and Razvi and Saatcioglu Model Calculations
The axial load vems axial strain curve of an elernent subjected to &al
conœntric compression using the two moâelr ir given in thir appendix
Computations for al1 the rpecimens analyzed in thir project followed the same
procedure. Wall 11 (testeci by Mander) was chosen as example.
Bel WalIliData
The section of this wall k reproduced in Figwe 8.1 and its propwties are
oiven in Table B. 1.
TABLE 6.1 WdI11 Pm(ni(res
l
4 & 6 & y f8 6 ml mrl J pnp4 (Mk] (MW &J c-a lMgl
(1) (12) (W (14) (14 (1W (17) (98) (la) . (20) Wall 11 120 3217 3.m 2W 191000 3900 24.W 41 31000
8.2 Shdkh and Uzumwi Moâsl
ïb 'anelyticil moâel for concrete confinement in tied columnsn was
reported by the authors in 1982, and modifieâ by Sheikh and Yeh (1992). A
detaited description crin k found dsewhere (Sheikh at al. 1982, 1992). The
modd is applicable to columns of n o m l conmete rtrength. Notation follows that
of the papem. The Wei was applieâ to wall 1 1 in the following mennec
8.21 PammatwS
Geometrv:
Gmss h a of the section: 700 x 500 mm = 105000 mm2
Centre-to-cmtm distame d outmost tie: B = 944 mm, H = 94 mm.
Tie -ng: s = 50 mm
Coverto cenbs linedtie:25mm + d a =25 mm +3mm =28 mm
Volumetrioc ratio p. = 0.0233
Efledive canfined corn area to core are8 ratio at üe level Â
where Cl ir the wntmtoantre distance behneen longitudinal bars, for this
case: Cf= 66 mm (long direction), Ci = 88 mm (short direction)
a = 5.5 (stsüstically deteminecl factor)
n = number of arcs covemd b8tween longitudinal ban = 16
Thur,
Were 8 is the arc angle, taken a8 Mo.. Thur,
a Ratio of a m dbdivdy confined at critical section to the th. of the core kg:
whemP.=41 MPa,thwP,=1098M.
Gain in concret0 strength &
whee : 0.W71, y :OS an, statistic parameten given, and P. is the tie
stress, considerd as the yielding stress. Thur for this exampk:
K. ~1 .28
Detemination of minimum m i n consrponding to maxi*rnum concrete stress
H m , et = 2.816 ""lm
0 Determination of maximum m i n mesponding to maximum concrete stress
(8-7)
wbm C ir the spdng beîwwn longitudinal ban (rveraoeû out in thir
calculation to 85.3 mm). Thur w = f 1.5 """lm
Detemination of m i n at 85% of the maximum c~clcrete s t m s in the
d s m i r i g aMCh (port-peak~vior~
The Imat value for u was obtained in the short direction of the section (i-e.,
along ~ K J 1SOnim si*), thur = 16.7 """lm
Thus, f, = 44.6 MPa
B.2.2 Stmsa=Stm&n Cuwas
Confined Concrete
For the concfete core, the stress-strain arve kgins with a paraboh until the
maximum confined stress; f, , is reached at the m i n a,, followed by a plateau
up to tho $train ~ . r , anâ descmding with a dope determined by the $train et
0.85f,. The curvo continues from this point until a stress of 0.30fm. has been
reached. The mode1 wggcntr a horizontal lin0 from the latter point.
Unconfineci Concret8
Sheikh and U Z U ~ masurd strains in the coclctet8 cover up to about 3 to
3.5 -lm in thair colums (me Sheikh et al. 1980). It wsr ruggerted that the
unconfined comate m e u8ed in the calwletims of the total load vernus
m i n r pwrbolc with a mmimun dnu of 0.85Te anâ with a li~wr
desœnding kuich up to a meXimm dnin between the values mentionrd
above and a atm88 of 0.425P0. In thir work, the dercending bfanch war
calculated until a m i n equal to 1 .& , about 3 -lm for normal stiain concrete.
The rtrsu-str8in wnm f6r both coMneâ and unconfined anamte ir graphed
in Figure 8.2. Alro rhown in Figura 8.3 is the rrtresestmin cunn, for the
longitudinal st-l of the wll.
Figu, 6.2 SWn CUM~ for Concret., Wall 11, Sheikh and U w m d Modal
Fiaun B.3 SWn Cuwe for Long. S W , Wdl11
Tha axial loaâ b cornputecl uring the following formula:
where the fint tenn on the right hend ride refen to the axial load canied by the
concrete core (f, is the concrete con stress at a given exial strain), the second
tenn is the wkl load carried by the concret8 mer, and the lest t m is the load
in the longitudinal steel. Figure 8.4 shows the Load-Strain aiwe for Wall 11.
The contributions to the total load of each of the concret0 components and the
steel are also shown.
The parameters for the square mû redangular columns of this study are
given in Table 8.2. Values for Sheikh and Uturneri column8 wwe obtaineâ in
by the miter uving th. methoâ outlinml in the preceding sections.
TABLE 8.2 Pmmabn for Shaikh-Uzumeri Modei
The confinement mothl pioporCid by the aufhm (Rami and Saatcioglu
1999) was M d l y rummuizeâ in #a litsntun review in Chepter 2. The mode1 is
applicebk to nomal and hi@-strength concrete, and to circulalw a d redangular
sections (induding square sections). The tie mangement may be hoops, crou
ties, spimls or wslckd wire fabric. This modal war derived through an extensive
statisticsl study of swmens testeâ by aie researchen and other gioups. The
follwing calailations were mrids for Wall 1 1.
8.3.1 Prmmtws
Geometry (uring Rami and Saatcioglu's notation)
Taking x-x axis in the long direction (Le., dong 700-mm ride) and y-y axis in the
short diredion (i.r., along 1 50n,m side):
Centre-to-Centre distance of outermost hoop: b, = 644 mm, b, = 94 mm
fie spaci-ng: s = 50 mm
h a of ûansvem s t d : in x-x: A, = 283 mm2, in y-y A, = 56.6 mn2
Spacing betwamn supportaci long. bars: & = 88.9, sb = 72 mm
Tranrvema steel ratio pr:
For wall Il p = 0.0092.
a Rducüon ~c ientofawage lateral pressure ka:
Thur, for x-x dimcüon ka, = 1 .O, in y-y diredion kW = 0.24
Tie stms8 f, at maximum cantinement prsssum:
where & ir the tio yielding stress, and f, = 0.85$, In thir case f. = = 310 MPa
Average lateral pressure fi
where q nurnber of tie legs that crou t h w h the direction being analyzed (i.e.,
x-x or y-y), A. is the erea of a tie leg, a ir the angle of inclination of the leg with
respect to th crossing direction (ag., if the k leg is prpendicular to the x-x
For wall 1 1, A = 28.3 mm2 in both dimctions, a = SV, q p 10, qv = 2, thw fk =
2.72 MPa, f* = 3.73 MPa.
and a weigMed average may be umd whm f* ir d M m t in both difecüons:
Thur, fm = 2.72 MPa, Iw = 0.88 MPa, and the averaga fb = 2.49 MPa.
Thus, kc = 5.74 for wrill 1 1.
Confinement Strength Pa :
For the example, Pr = 49.1 MPa.
9 Factor that accwnts for type of wnaete (i.e., n o m l or high strength
(B. 1 O)
Thur, kt = 1 .O (Po = 0.85(41) = 34.9 MPa)
Fsdor that accounts f# lateral steel stmngth &
Thur, kr = 1 .O (fr = 31 O MPa)
Parameters for umnfined concret8 curve:
Strsin at perik unconineâ rtreu f, m:
If was pmviâed from tests of plain concmte cdumnr, it rhould not exceeâ the
value cornMd with th aôove equation. For wall11, gf = 2.0 """lm.
Strein at 0.85 Pm in the descending bmnch, &r
This equation is valid only if
Thur, as p. = 0.0092 * 0.02 o.k., then = 3.8 "Tm.
Parameten for confined muete wrve:
Strain at maximum peak confinexi stress P , et:
el = e,,(i +5k$)
- - -- -
6.3.2 Stms-Stnin Cuwm
Confined and Unconfined C o m t e
The remarchem pmpooed the use of similar shape8 for both typer of conuete.
For the asœnding branch, the Popovicr wwe in the fom:
and
For the confined cote and unconfined cover, the equations are valid up to the
strain et and m, rsspedivdy.
The post-pmk hanches follaw a 8bsight line with a dope defird by the strain
follawr a horizontal l i n aftewards (only for confined concnte).
The Scresr-rtrsin wnm for longitudinal steel it as for the Sheikh and Ummeri
m&l; dastolplastic with m i n haidening. Figure 8.5 shows the strass-strain
0.0 10B 2ofl 30B 40.0 SOB 60.0 70.0
Aiilal 8t rh @mhn)
Figun 6.5 (Itnln Cu- for Conont., WdI 11, Rwl and Saatcioglu Modal
8.3.3 Axial Lord vernus Axiai Stnin Curvs
The Equation (B.10) is ured to cornpute the contributions of ancrete and
longitudinal steel to the total axial load. Figure B.6 shuws the axial response of
wall 1 1 pmdicted by the Rami and Saatcioglu modd.
BA4 Rami and Srcitdoglu Parmaten for All Columns
Table 8.3 shows the parameters nemssary to cornpute the stress-sûain
ames for aII of the cdumnr analyzed in this work.
TABLE B.3 Ruvl uid SIatciogIu Pamnetam
B.4 Mai ShOCkig Cunn,
Th. ~ * m e n t u I cuwm for th8 columnr of Chapter 4 and 5 are
comparsd with the SPARCS analyses anâ the pmdideâ load versua deformation
3,000 0.005 0.010 0.015 0.020 0.025 0.030 &hi 8mln (mmlm)
Figure 6.8 Coiumn ?CS-17
Rgun 8.1 1 Cdumn 4B44û
Figun BA2 Cdumn 403-22
0.000 0.01 0 0.020 0,030 0.040
Axial m i n (mmlm)
Figun A13 Column 4W.U
6.4.2 Liu a ri Columna
O O 10 16 20 irlrl m i n (ma)
Figure 6.1 6 Column 2C6ô-108100-16
6.4.3 Scott Column (Mmâer)
Scott
Figun 8.18 Scott Column
Appendix C
Variable Poirson9s Ratio in Liu et al. Columns
The Poison's ratio; v, was calwlated for ton of the cokmns tested by Liu
et al. (1998). Plots for eight d these spedmens are shawn in this appendix, the
remaining two wcwe shown in Chepter 4. The gmpht inciude loa&shin wrves,
and normalized cuw8s of the variable Poisson's ratio versus axial and
volumetric strain as w l l as th8 variation of the lateral expansion with respect to
the applied axial load.
It was caiduâd that the mackl for variable Poisson's ratio proposed by
Vecchio (1992) and implemented in SPARCS predided reasonsbly well this
8fi8d on confined cmcfete.
Figun C.1 Lord-Sfnin Cuwe of 2CW-1 OSSO-1 5
Figum C.4 Volunirblc-Mal M n Cum d 2C6ô-lôSoO-16
Figura C.7 Poisson's R.tiobxid S W n C u m d 2CW-1OS100-16
40012 -
aooio
i a m .
am# -
Qom-
Qom2 *
Figure Cm@ Lord4tnin C u m of 2C60-4OS16O-l6
Flgw, C.13 Lord-SWn C u m of 2C8MOSSû-16
Figura C.16 Pokrongr Ratio-Axial Strain Cunn of 2Cûû-1 OS6O-16
Ffgum C.H Vdunntricurl (Itnln C u m of ZCûû-10S60-15
FTgum C.17 Lord-Stnin Cuwe of 2C8OdSlW-1 S
Figure C.19 P oissonDr Ratio-M.1 SWn C u m of 2C804S100-16
Figura C.23 Pduonms Ratio-Axial Stdn C u m of 2C00aS100-26
Figura C.24 VduHt-Axirl8biln Cuwe d 2CW-ûSfa0-26
Figun C.26 Lord4brin Cunn of 2C9016SM)-26
2C9011SSO-26
1.10 - 1.m 4
0.90
0.m
1 ::: as0 - 0.4 - 0.34 - 0.a 4 I r 1 1 1
Figun C.27 Poisson's Ratîo-A%Irl Shrin Cunm of 2CW4Sb0-26
Figura C.28 Load-SWn Cum of 2CW-10S100-û
0.30 . 0.20.
aio
Figum C32 Volumcldxirl Wii. Cum d 2C00,lOS~ûû-û