BEHAVIOR OF CONCRETE COLUMNS UNDER VARIOUS CONFINEMENT EFFECTS
by
AHMED MOHSEN ABD EL FATTAH
B.S., Cairo University, 2000
M.S., Kansas State University, 2008
AN ABSTRACT OF A DISSERTATION
Submitted in partial fulfillment of the requirements for the degree
DOCTOR OF PHILOSOPHY
Department of Civil Engineering
College of Engineering
KANSAS STATE UNIVERSITY
Manhattan, Kansas
2012
Abstract
The analysis of concrete columns using unconfined concrete models is a well established
practice. On the other hand, prediction of the actual ultimate capacity of confined concrete
columns requires specialized nonlinear analysis. Modern codes and standards are introducing the
need to perform extreme event analysis. There has been a number of studies that focused on the
analysis and testing of concentric columns or cylinders. This case has the highest confinement
utilization since the entire section is under confined compression. On the other hand, the
augmentation of compressive strength and ductility due to full axial confinement is not
applicable to pure bending and combined bending and axial load cases simply because the area
of effective confined concrete in compression is reduced. The higher eccentricity causes smaller
confined concrete region in compression yielding smaller increase in strength and ductility of
concrete. Accordingly, the ultimate confined strength is gradually reduced from the fully
confined value f
cc
(at zero eccentricity) to the unconfined value f
c
(at infinite eccentricity) as a
function of the compression area to total area ratio. The higher the eccentricity the smaller the
confined concrete compression zone. This paradigm is used to implement adaptive eccentric
model utilizing the well known Mander Model and Lam and Teng Model.
Generalization of the moment of area approach is utilized based on proportional loading, finite
layer procedure and the secant stiffness approach, in an iterative incremental numerical model to
achieve equilibrium points of P
ε
and M
ϕ
response up to failure. This numerical analysis is
adaptod to asses the confining effect in circular cross sectional columns confined with FRP and
conventional lateral steel together; concrete filled steel tube (CFST) circular columns and
rectangular columns confined with conventional lateral steel. This model is validated against
experimental data found in literature. The comparison shows good correlation. Finally computer
software is developed based on the nonlinear numerical analysis. The software is equipped with
an elegant graphics interface that assimilates input data, detail drawings, capacity diagrams and
demand point mapping in a single sheet. Options for preliminary design, section and
reinforcement selection are seamlessly integrated as well. The software generates 2D interaction
diagrams for circular columns, 3D failure surface for rectangular columns and allows the user to
determine the 2D interaction diagrams for any angle α between the xaxis and the resultant
moment. Improvements to KDOT Bridge Design Manual using this software with reference to
AASHTO LRFD are made. This study is limited to stub columns.
BEHAVIOR OF CONCRETE COLUMNS UNDER VARIOUS CONFINEMENT EFFECTS
by
AHMED MOHSEN ABD EL FATTAH
B.S., Cairo University, 2000
M.S., Kansas State University, 2008
A DISSERTATION
Submitted in partial fulfillment of the requirements for the degree
DOCTOR OF PHILOSOPHY
Department of Civil Engineering
College of Engineering
KANSAS STATE UNIVERSITY
Manhattan, Kansas
2012
Approved by:
Major Professor
Hayder Rasheed
Copyright
AHMED MOHSEN ABD EL FATTAH
2012
Abstract
The analysis of concrete columns using unconfined concrete models is a well established
practice. On the other hand, prediction of the actual ultimate capacity of confined concrete
columns requires specialized nonlinear analysis. Modern codes and standards are introducing the
need to perform extreme event analysis. There has been a number of studies that focused on the
analysis and testing of concentric columns or cylinders. This case has the highest confinement
utilization since the entire section is under confined compression. On the other hand, the
augmentation of compressive strength and ductility due to full axial confinement is not
applicable to pure bending and combined bending and axial load cases simply because the area
of effective confined concrete in compression is reduced. The higher eccentricity causes smaller
confined concrete region in compression yielding smaller increase in strength and ductility of
concrete. Accordingly, the ultimate confined strength is gradually reduced from the fully
confined value f
cc
(at zero eccentricity) to the unconfined value f
c
(at infinite eccentricity) as a
function of the compression area to total area ratio. The higher the eccentricity the smaller the
confined concrete compression zone. This paradigm is used to implement adaptive eccentric
model utilizing the well known Mander Model and Lam and Teng Model.
Generalization of the moment of area approach is utilized based on proportional loading, finite
layer procedure and the secant stiffness approach, in an iterative incremental numerical model to
achieve equilibrium points of P
ε
and M
ϕ
response up to failure. This numerical analysis is
adaptod to asses the confining effect in circular cross sectional columns confined with FRP and
conventional lateral steel together, concrete filled steel tube (CFST) circular columns and
rectangular columns confined with conventional lateral steel. This model is validated against
experimental data found in literature. The comparison shows good correlation. Finally computer
software is developed based on the nonlinear numerical analysis. The software is equipped with
an elegant graphics interface that assimilates input data, detail drawings, capacity diagrams and
demand point mapping in a single sheet. Options for preliminary design, section and
reinforcement selection are seamlessly integrated as well. The software generates 2D interaction
diagrams for circular columns, 3D failure surface for rectangular columns and allows the user to
determine the 2D interaction diagrams for any angle α between the xaxis and the resultant
moment. Improvements to KDOT Bridge Design Manual using this software with reference to
AASHTO LRFD are made. This study is limited to stub columns
viii
Table of Contents
List of Figures...xiii
List of Tables.......xxv
Acknowledgements............xxvii
Dedication.........................xxviii
Chapter 1  Introduction .................................................................................................................. 1
11 Background ............................................................................................................................... 1
12 Objectives ................................................................................................................................. 1
13 Scope ........................................................................................................................................ 3
Chapter 2  Literature Review ......................................................................................................... 4
21 Steel Confinement Models ....................................................................................................... 4
211 Chronological Review of Models ......................................................................................... 4
2111 Notation.............................................................................................................................. 4
212 Discussion ........................................................................................................................... 50
22 Circular Columns Confined with FRP.................................................................................... 56
221 Past Work Review ............................................................................................................... 56
222 Discussion ......................................................................................................................... 115
23 Circular Concrete Filled Steel Tube (CFST) Columns ........................................................ 118
231 Past Work Review ............................................................................................................. 118
222 Discussion ......................................................................................................................... 134
24 Rectangular Columns subjected to biaxial bending and Axial Compression ....................... 135
241 Past Work Review ............................................................................................................. 135
242 Discussion ......................................................................................................................... 194
ix
Chapter 3  Circular Columns Confined with FRP and lateral Steel .......................................... 196
31 Introduction .......................................................................................................................... 196
32 Formulations ......................................................................................................................... 197
321 Finite Layer Approach (Fiber Model) ............................................................................... 197
322 Present Confinement Model for Concentric Columns ...................................................... 197
3221 Lam and Teng Model ..................................................................................................... 197
3222 Mander Model for transversely reinforced steel ............................................................ 199
323Present Confinement Model for Eccentric Columns.......................................................... 206
3231 Eccentric Model Based on Lam and Teng Equations .................................................... 209
3232 Eccentric Model based on Mander Equations ............................................................... 210
324 Moment of Area Theorem ................................................................................................. 212
33 Numerical Formulation......................................................................................................... 216
331 Model Formulation ............................................................................................................ 216
332 Numerical Analysis ........................................................................................................... 219
34 Results and Discussion ......................................................................................................... 226
341 StressStrain Curve Comparisons with Experimental Work ............................................. 226
342 Interaction Diagram Comparisons with Experimental Work ............................................ 236
Chapter 4  Circular Concrete Filled Steel Tube Columns (CFST) ............................................ 245
41 Introduction .......................................................................................................................... 245
42 Formulations ......................................................................................................................... 246
421 Finite Layer Approach (Fiber Model) ............................................................................... 246
422 Present Confinement Model for Concentric Columns ...................................................... 247
4221 Mander Model for transversely reinforced steel ............................................................ 247
x
4222 Lam and Teng Model ..................................................................................................... 254
423Present Confinement Model for Eccentric Columns.......................................................... 255
4231 Eccentric Model based on Mander Equations ............................................................... 258
4232 Eccentric Model Based on Lam and Teng Equations .................................................... 260
424 Moment of Area Theorem ................................................................................................. 262
43 Numerical Model Formulation ............................................................................................. 265
431 Model Formulation ............................................................................................................ 265
432 Numerical Analysis ........................................................................................................... 273
44 Results and Discussion ......................................................................................................... 279
441 Comparisons with Experimental Work ............................................................................. 279
Chapter 5  Rectangular Columns subjected to biaxial bending and Axial Compression .......... 293
51 Introduction .......................................................................................................................... 293
52 Unconfined Rectangular Columns Analysis......................................................................... 294
521 Formulations...................................................................................................................... 294
5211 Finite Layer Approach (Fiber Method).......................................................................... 294
5212 Concrete Model .............................................................................................................. 295
5213 Steel Model .................................................................................................................... 296
522 Analysis Approaches ......................................................................................................... 296
5221 Approach One: Adjusted Predefined Ultimate Strain Profile ........................................ 296
5222 Approach Two: Generalized Moment of Area Theorem ............................................... 300
5222a Moment of Area Theorem .......................................................................................... 300
5222b Method Two ............................................................................................................... 304
523 Results and Discussion ...................................................................................................... 313
xi
5231 Comparison between the two approaches ...................................................................... 313
5232 Comparison with Existing Commercial Software ......................................................... 315
53 Confined Rectangular Columns Analysis............................................................................. 318
531 Formulations...................................................................................................................... 318
5311 Finite Layer Approach (Fiber Method).......................................................................... 318
5312 Confinement Model for Concentric Columns................................................................ 319
5312a Mander Model for transversely reinforced steel ......................................................... 319
5313 Confinement Model for Eccentric Columns .................................................................. 327
5313a Eccentric Model based on Mander Equations ............................................................ 333
5314 Generalized Moment of Area Theorem ......................................................................... 337
532 Numerical Formulation ..................................................................................................... 341
533 Results and Discussion ...................................................................................................... 350
5331 Comparison with Experimental Work ........................................................................... 350
5332 Comparison between the surface meridians T & C used in Mander model and
Experimental Work .............................................................................................................. 358
Chapter 6  Software Development............................................................................................. 363
61 Introduction .......................................................................................................................... 363
62 Interface Design .................................................................................................................... 364
621 Circular Columns Interface ............................................................................................... 364
622 Rectangular Columns Interface ......................................................................................... 368
Chapter 7  Conclusions and Recommendations ........................................................................ 372
71 Conclusions .......................................................................................................................... 372
72 Recommendations ................................................................................................................ 374
xii
Appendix A  Ultimate Confined Strength Tables ..................................................................... 392
xiii
List of Figures
Figure 21: General StressStrain curve by Chan (1955)................................................................ 7
Figure 22: General StressStrain curve by Blume et al. (1961) .................................................... 8
Figure 23: General StressStrain curve by Soliman and Yu (1967) ............................................ 10
Figure 24: StressStrain curve by Kent and Park (1971). ............................................................ 12
Figure 25: StressStrain curve by Vallenas et al. (1977). ............................................................ 14
Figure 26: Proposed StressStrain curve by Wang et al (1978) .................................................. 15
Figure 27: Proposed StressStrain curve by Muguruma et al (1980) .......................................... 17
Figure 28: Proposed general StressStrain curve by Sheikh and Uzumeri (1982). ..................... 20
Figure 29: Proposed general StressStrain curve by Park et al (1982). ....................................... 23
Figure 210: Proposed general StressStrain curve by Yong et al. (1988) ................................... 26
Figure 211: Stress Strain Model proposed by Mander et al (1988) ........................................... 28
Figure 212: Proposed general StressStrain curve by Fujii et al. (1988) .................................... 30
Figure 213: Proposed StressStrain curve by Saatcioglu and Razvi (19921999). ..................... 32
Figure 214: Proposed StressStrain curve by Cusson and Paultre (1995). .................................. 38
Figure 215: Proposed StressStrain curve by Attard and Setunge (1996). .................................. 40
Figure 216: Mander et al (1988), Saatcioglu and Razvi (1992) and ElDash and Ahmad (1995)
models compared to Case 1. ................................................................................................. 54
Figure 217: Mander et al (1988), Saatcioglu and Razvi (1992) and ElDash and Ahmad (1995)
models compared to Case 2. ................................................................................................. 54
Figure 218: Mander et al (1988), Saatcioglu and Razvi (1992) and ElDash and Ahmad (1995)
models compared to Case 3. ................................................................................................. 55
xiv
Figure 219: Axial StressStrain Curve proposed by Miyauchi et al (1997) ................................ 65
Figure 220: Axial StressStrain Curve proposed by Samaan et al. (1998) .................................. 68
Figure 221: Axial Stress(axial & lateral) Strain Curve proposed by Toutanji (1999) ............... 74
Figure 222: variably confined concrete model proposed by Harries and Kharel (2002) ............ 80
Figure 223: Axial StressStrain Model proposed by Cheng et al (2002) ................................... 87
Figure 224: Axial StressStrain Model proposed by Campione and Miraglia (2003)................. 88
Figure 225: Axial StressStrain Model Proposed by Lam and Teng (2003) ............................... 90
Figure 226: Axial StressStrain Model proposed by Harajli (2006) ......................................... 103
Figure 227: Axial StressStrain Model proposed by Teng et al (2009) .................................... 112
Figure 228: Axial StressStrain Model proposed by Wei and Wu (2011) ................................ 115
Figure 229: Axial stressstrain model Proposed by Fujimoto et al (2004) ................................ 129
Figure 230: stress_strain curve for confined concrete in circular CFST columns, Liang and
Fragomeni (2010) ................................................................................................................ 131
Figure 231: relation between T and C by Andersen (1941) ...................................................... 138
Figure 232: relation between c and α by Bakhoum (1948) ....................................................... 141
Figure 233: geometric dimensions in Crevin analysis (1948) ................................................... 143
Figure 234: Concrete center of pressure Vs neutral axis location ,Mikhalkin 1952 ................. 144
Figure 235: Steel center of pressure Vs neutral axis location, Mikhalkin 1952 ........................ 144
Figure 236: bending with normal compressive force chart np = 0.03, Hu (1955) .................... 146
Figure 237: Linear relationship between axial load and moment for compression failure
Whitney and Cohen 1957 .................................................................................................... 150
Figure 238: section and design chart for case 1(rx/b = 0.005), Au (1958) ................................ 152
Figure 239: section and design chart for case 2, Au (1958) ...................................................... 153
xv
Figure 240: section and design chart for case 3(d
x
/b = 0.7, d
y
/t = 0. 7), Au (1958) .................. 153
Figure 241: Graphical representation of Method one by Bresler (1960) .................................. 158
Figure 242: Graphical representation of Method two by Bresler (1960) .................................. 158
Figure 243: Interaction curves generated from equating
α
and by Bresler
(1960)
.................. 159
Figure 244: five cases for the compression zone based on the neutral axis location Czerniak
(1962) .................................................................................................................................. 161
Figure 245: Values for N for unequal steel distribution by Pannell (1963) .............................. 166
Figure 246: design curve by Fleming et al (1961) .................................................................... 168
Figure 247: relation between
α
and
θ
by Ramamurthy (1966) ................................................. 169
Figure 248: biaxial moment relationship by Parme et al. (1966) .............................................. 170
Figure 249: Biaxial bending design constant (four bars arrangement) by Parme et al. (1966) . 171
Figure 250: Biaxial bending design constant (eight bars arrangement) by Parme et al. (1966) 171
Figure 251: Biaxial bending design constant (twelve bars arrangement) by Parme et al. (1966)
............................................................................................................................................. 172
Figure 252: Biaxial bending design constant (6810 bars arrangement) by Parme et al. (1966)
............................................................................................................................................. 172
Figure 253: Simplified interaction curve by Parme et al. (1966) ............................................. 173
Figure 254: Working stress interaction diagram for bending about xaxis by Mylonas (1967) 174
Figure 255: Comparison of steel stress variation for biaxial bending when ψ = 30 & q = 1.0
Brettle and Taylor (1968)...................................................................................................... .201
Figure 256: Non dimensional biaxial contour on quarter column by Taylor and Ho (1984). ... 180
Figure 257: P
u
/P
uo
to A relation for 4bars arrangement by Hartley (1985) (left) non dimensional
load contour (right) ............................................................................................................. 181
xvi
Figure 31: Using Finite Layer Approach in Analysis ............................................................... 197
Figure 32:Axial StressStrain Model proposed by Lam and Teng (2003). ............................... 198
Figure 33: Axial StressStrain Model proposed by Mander et al. (1988) for monotonic loading
............................................................................................................................................. 201
Figure 34: Effectively confined core for circular hoop and spiral reinforcement (Mander Model)
............................................................................................................................................. 202
Figure 35: Effective lateral confined core for hoop and spiral reinforcement (Mander Model) 203
Figure 36: Confinement forces on concrete from circular hoop reinforcement ........................ 204
Figure 37: Effect of compression zone depth on concrete strength .......................................... 206
Figure 38: Amount of confinement gets engaged in different cases ......................................... 207
Figure 39: Relation between the compression area ratio to the normalized eccentricity .......... 208
Figure 310: Eccentricity Based Confined Lam and Teng Model ........................................... 210
Figure 311: Eccentricity Based Confined Mander Model  ..................................................... 212
Figure 312: Transfering moment from centroid to the geometric centroid ............................... 215
Figure 313: Equilibrium between Lateral Confining Stress, LSR and FRP Forces .................. 216
Figure 314: FRP and LSR Model Implementation .................................................................... 219
Figure 315: Geometric properties of concrete layers and steel rebars ...................................... 220
Figure 316: Radial loading concept ........................................................................................... 221
Figure 317: Transfering Moment from geometric centroid to inelastic centroid ...................... 222
Figure 318: Flowchart of FRP wrapped columns analysis ........................................................ 225
Figure 319: Case 1 StressStrain Curve Compared to Experimental Ultimate Point ................ 227
Figure 320: Case 2 StressStrain Curve Compared to Experimental Ultimate Point ................ 228
Figure 321: Case 3 StressStrain Curve Compared to Experimental Ultimate Point ................ 229
xvii
Figure 322: Case 4 StressStrain Curve Compared to Experimental Ultimate Point ................ 230
Figure 323: Case 1 Proposed StressStrain Curve Compared to Experimental and Eid and
Paultre (2008) theoretical ones ........................................................................................... 232
Figure 324: Case 2 Proposed StressStrain Curve Compared to Experimental and Eid and
Paultre (2008) theoretical ones ........................................................................................... 232
Figure 325: Case 3 Proposed StressStrain Curve Compared to Experimental and Eid and
Paultre (2008) theoretical ones ........................................................................................... 233
Figure 326: Case 4 Proposed StressStrain Curve Compared to Experimental and Eid and
Paultre (2008) theoretical ones ........................................................................................... 233
Figure 327: Case 5 Proposed StressStrain Curve Compared to Experimental and Eid and
Paultre (2008) theoretical ones ........................................................................................... 234
Figure 328: Case 6 Proposed StressStrain Curve Compared to Experimental and Eid and
Paultre (2008) theoretical ones ........................................................................................... 234
Figure 329: Case 7 Proposed StressStrain Curve Compared to Experimental and Eid and
Paultre (2008) theoretical ones ........................................................................................... 235
Figure 330: Case 8 Proposed StressStrain Curve Compared to Experimental and Eid and
Paultre (2008) theoretical ones ........................................................................................... 235
Figure 331: Case 1 Proposed Interaction Diagram compared to Experimental point from Eid et
al (2006) .............................................................................................................................. 237
Figure 332: Case 2 Proposed Interaction Diagram compared to Experimental point from Eid et
al (2006) .............................................................................................................................. 238
Figure 333: Case 3 Proposed Interaction Diagram compared to Experimental point from Eid et
al (2006) .............................................................................................................................. 238
xviii
Figure 334: Case 4 Proposed Interaction Diagram compared to Experimental point from Eid et
al (2006) .............................................................................................................................. 239
Figure 335: Case 5 Proposed Interaction Diagram compared to Experimental point from Eid et
al (2006) .............................................................................................................................. 239
Figure 336: Case 6 Proposed Interaction Diagram compared to Experimental point from Eid et
al (2006) .............................................................................................................................. 240
Figure 337: Case 7 Proposed Interaction Diagram compared to Experimental point from Eid et
al (2006) .............................................................................................................................. 240
Figure 338: Case 8 Proposed Interaction Diagram compared to Experimental point from Eid et
al (2006) .............................................................................................................................. 241
Figure 339: Case 9 Proposed Interaction Diagram compared to Experimental point from Eid et
al (2006) .............................................................................................................................. 241
Figure 340: Case 10 Proposed Interaction Diagram compared to Experimental point from Eid et
al (2006) .............................................................................................................................. 242
Figure 341: Case 11 Proposed Interaction Diagram compared to Experimental point from
Saadatmanesh et al (1996) .................................................................................................. 242
Figure 342: Case 12 Proposed Interaction Diagram compared to Experimental point from
Sheikh and Yau (2002) ....................................................................................................... 243
Figure 343: Case 13 Proposed Interaction Diagram compared to Experimental point from
Sheikh and Yau (2002) ....................................................................................................... 243
Figure 344: Case 14 Proposed Interaction Diagram compared to Experimental point from
Sheikh and Yau (2002) ....................................................................................................... 244
Figure 41: Using Finite Layer approach in analysis (CFST section) ........................................ 247
xix
Figure 42: Axial StressStrain Model proposed by Mander et al. (1988) for monotonic loading
............................................................................................................................................. 248
Figure 43: Effectively confined core for circular hoop and spiral reinforcement (Mander Model)
............................................................................................................................................. 249
Figure 44: Effective lateral confined core for hoop and spiral reinforcement (Mander Model) 250
Figure 45: Confinement forces on concrete from circular hoop reinforcement ........................ 251
Figure 46:Axial StressStrain Model proposed by Lam and Teng (2003). ............................... 254
Figure 47: Effect of compression zone depth on concrete strength .......................................... 256
Figure 48: Amount of confinement gets engaged in different cases ......................................... 256
Figure 49: Relation between the compression area ratio to the normalized eccentricity .......... 258
Figure 410: Eccentricity Based Confined Mander Model  ..................................................... 260
Figure 411: Eccentricity Based Confined Lam and Teng Model ........................................... 261
Figure 412: Transfering moment from centroid to the geometric centroid ............................... 264
Figure 413: 3D Sectional elevation and plan for CFST column. .............................................. 265
Figure 414: f
cc
vs f
c
for normal strength concrete ................................................................... 268
Figure 415: f
cc
vs f
c
for high strength concrete ........................................................................ 269
Figure 416: CFST Stressstrain Curve for different cases from Table 41 ............................... 270
Figure 417: Case 1 StressStrain curve using Lam and Teng equations compared to Experimetal
curve. ................................................................................................................................... 271
Figure 418: Case 2 StressStrain curve using Lam and Teng equations compared to Experimetal
curve. ................................................................................................................................... 271
Figure 419: Case 3 StressStrain curve using Lam and Teng equations compared to Experimetal
curve. ................................................................................................................................... 272
xx
Figure 420: CFST Model Flowchart ......................................................................................... 272
Figure 421: Geometric properties of concrete layers and steel tube ......................................... 274
Figure 422: Radial loading concept ........................................................................................... 274
Figure 423: Moment transferring from geometric centroid to inelastic centroid ...................... 275
Figure 424: Flowchart of CFST columns analysis .................................................................... 278
Figure 425: KDOT Column Expert Comparison with CFST case 1: ........................................ 281
Figure 426: KDOT Column Expert Comparison with CFST case 2 ......................................... 282
Figure 427: KDOT Column Expert Comparison with CFST case 3 ......................................... 282
Figure 428; KDOT Column Expert Comparison with CFST case 4 ......................................... 283
Figure 429: KDOT Column Expert Comparison with CFST case 5 ......................................... 283
Figure 430: KDOT Column Expert Comparison with CFST case 6 ......................................... 284
Figure 431: KDOT Column Expert Comparison with CFST case 7 ......................................... 284
Figure 432: KDOT Column Expert Comparison with CFST case 8 ......................................... 285
Figure 433: KDOT Column Expert Comparison with CFST case 9 ......................................... 285
Figure 434: KDOT Column Expert Comparison with CFST case 10 ....................................... 286
Figure 435: KDOT Column Expert Comparison with CFST case 11 ....................................... 286
Figure 436: KDOT Column Expert Comparison with CFST case 12 ....................................... 287
Figure 437: KDOT Column Expert Comparison with CFST case 13 ....................................... 287
Figure 438: KDOT Column Expert Comparison with CFST case 14 ....................................... 288
Figure 439: KDOT Column Expert Comparison with CFST case 15 ....................................... 288
Figure 440: KDOT Column Expert Comparison with CFST case 16 ....................................... 289
Figure 441: KDOT Column Expert Comparison with CFST case 17 ....................................... 289
Figure 442: KDOT Column Expert Comparison with CFST case 18 ....................................... 290
xxi
Figure 443: KDOT Column Expert Comparison with CFST case 19 ....................................... 290
Figure 51:a) Using finite filaments in analysis b)Trapezoidal shape of Compression zone . 294
Figure 52: a) Stress strain Model for concrete by Hognestad b) Steel stressstrain Model .. 295
Figure 53: Different strain profiles due to different neutral axis positions. .............................. 297
Figure 54: Defining strain for concrete filaments and steel rebars from strain profile ............. 298
Figure 55: Filaments and steel rebars geometric properties with respect to crushing strain point
and geometric centroid ........................................................................................................ 298
Figure 56: Method one Flowchart for the predefined ultimate strain profile method ............... 299
Figure 57: 2D Interaction Diagram from Approach One Before and After Correction ............ 300
Figure 58: Transfering moment from centroid to the geometric centroid ................................. 303
Figure 59: geometric properties of concrete filaments and steel rebars with respect to, geometric
centroid and inelastic centroid. ........................................................................................... 306
Figure 510: Radial loading concept ........................................................................................... 307
Figure 511 Moment transferring from geometric centroid to inelastic centroid ....................... 308
Figure 512: Flowchart of Generalized Moment of Area Method used for unconfined analysis 312
Figure 513: Comparison of approach one and two (
α
= 0) ....................................................... 313
Figure 514: Comparison of approach one and two (
α
= 4.27) .................................................. 314
Figure 515: Comparison of approach one and two (
α
= 10.8) .................................................. 314
Figure 516: Comparison of approach one and two (
α
= 52) ..................................................... 315
Figure 517: column geometry used in software comparison ..................................................... 316
Figure 518: Unconfined curve comparison between KDOT Column Expert and SP Column (
α
=
0) ......................................................................................................................................... 316
xxii
Figure 519: Design curve comparison between KDOT Column Expert and CSI Col 8 using ACI
Reduction Factors ............................................................................................................... 317
Figure 520: Design curve comparison between KDOT Column Expert and SP column using
ACI reduction factors .......................................................................................................... 318
Figure 521:a) Using finite filaments in analysis b)Trapezoidal shape of Compression zone 319
Figure 522: Axial StressStrain Model proposed by Mander et al. (1988) for monotonic loading
............................................................................................................................................. 320
Figure 523: Effectively confined core for rectangular hoop reinforcement (Mander Model) .. 321
Figure 524: Effective lateral confined core for rectangular cross section ................................. 322
Figure 525: Confined Strength Determination .......................................................................... 324
Figure 526: Effect of compression zone depth on concrete stress ............................................ 328
Figure 527: Amount of confinement engaged in different cases ............................................... 328
Figure 528: Normalized Eccentricity versus Compression Zone to total area ratio (Aspect ratio
1:1) ...................................................................................................................................... 330
Figure 529: Normalized Eccentricity versus Compression Zone to total area ratio (Aspect ratio
2:1) ...................................................................................................................................... 331
Figure 530: Normalized Eccentricity versus Compression Zone to total area ratio (Aspect ratio
3:1) ...................................................................................................................................... 331
Figure 531: Normalized Eccentricity versus Compression Zone to total area ratio (Aspect ratio
4:1) ...................................................................................................................................... 332
Figure 532: Cumulative chart for Normalized Eccentricity against Compression Zone Ratio (All
data points). ......................................................................................................................... 332
Figure 533: Eccentricity Based confined Mander Model ....................................................... 335
xxiii
Figure 534: Eccentric based StressStrain Curves using compression zone area to gross area
ratio ..................................................................................................................................... 336
Figure 535: Eccentric based StressStrain Curves using normalized eccentricity instead of
compression zone ratio ....................................................................................................... 337
Figure 536: Transfering moment from centroid to the geometric centroid ............................... 340
Figure 537: geometric properties of concrete filaments and steel rebars with respect to crushing
strain point, geometric centroid and inelastic centroid. ...................................................... 343
Figure 538: Radial loading concept ........................................................................................... 344
Figure 539 Moment Transferring from geometric centroid to inelastic centroid ...................... 345
Figure 540: Flowchart of Generalized Moment of Area Method .............................................. 349
Figure 541:Hognestad column................................................................... 350
Figure 542: Comparison between KDOT Column Expert with Hognestad experiment (
α
= 0)
............................................................................................................................................. 351
Figure 543: Bresler Column ...................................................................................................... 351
Figure 544: Comparison between KDOT Column Expert with Bresler experiment (
α
= 90) .. 352
Figure 545: Comparison between KDOT Column Expert with Bresler experiment (
α
= 0) .... 352
Tie Diameter = 0.25 in. Figure 546 : Ramamurthy Column ................ 353
Figure 547: Comparison between KDOT Column Expert with Ramamurthy experiment (
α
=
26.5) .................................................................................................................................... 353
Figure 548 : Saatcioglu Column ................................................................................................ 354
Figure 549: Comparison between KDOT Column Expert with Saatcioglu et al experiment (
α
=
0) ......................................................................................................................................... 354
Figure 550 : Saatcioglu Column ................................................................................................ 355
xxiv
Figure 551: Comparison between KDOT Column Expert with Saatcioglu et al experiment 1 (
α
= 0) ...................................................................................................................................... 355
Figure 552 : Scott Column......................................................................................................... 356
Figure 553: Comparison between KDOT Column Expert with Scott et al experiment (
α
= 0) 356
Figure 554 : Scott Column................................................................. 357
Figure 555: Comparison between KDOT Column Expert with Scott et al experiment (
α
= 0) 357
Figure 556: T and C meridians using equations (5179) and (5180) used in Mander Model for
f
c
= 4.4 ksi .......................................................................................................................... 359
Figure 557: T and C meridians for f
c
= 3.34 ksi ....................................................................... 360
Figure 558: T and C meridians for f
c
= 3.9 ksi ......................................................................... 360
Figure 559: T and C meridians for f
c
= 5.2 ksi ......................................................................... 361
Figure 61: KDOT Column Expert classes ................................................................................. 363
Figure 62: KDOT Column Expert Initial form .......................................................................... 364
Figure 63: Circular Column GUI............................................................................................... 364
Figure 64: Circular Column Interface main sections ................................................................. 366
Figure 65: Different Interaction Diagrams plot in the Plotting areaCircular Section ............. 367
Figure 66: FRP formManufactured FRP................................................................................. 367
Figure 67: FRP formuser defined ........................................................................................... 368
Figure 68: Rectangular Column GUI ........................................................................................ 368
Figure 69: Rectangular Column Interface main sections .......................................................... 370
Figure 610: α angle form ........................................................................................................... 370
Figure 611: Different Interaction Diagrams plot in the Plotting area Rectangular Section ..... 371
Figure 612: 3D Interaction Diagram ......................................................................................... 371
xxv
List of Tables
Table 21: Lateral Steel Confinement Models Comparison ......................................................... 51
Table 22: Experimental cases properties ..................................................................................... 53
Table 31: Experimental data used to verify the ultimate strength and strain for the confined
model (Eid et al. 2006) ....................................................................................................... 226
Table 32: Experimental data used to verify the fully confined model ...................................... 231
Table 33: Experimental data used to verify the interaction diagrams. ...................................... 236
Table 41: CFST Experimental data ........................................................................................... 266
Table 42: Experimental data for CFST ...................................................................................... 279
Table 51: Data for constructing T and C meridian Curves for f
c
equal to 3.34 ksi .................. 362
Table 52: Data for constructing T and C meridian Curves for f
c
equal to 3.9 ksi .................... 362
Table 53: Data for constructing T and C meridian Curves for f
c
equal to 5.2 ksi .................... 362
Table A1: Ultimate confined strength to unconfined strength ratio for f
c
= 3.3 ksi ................. 393
Table A2: Ultimate confined strength to unconfined strength ratio for f
c
= 3.9 ksi ................. 394
Table A3: Ultimate confined strength to unconfined strength ratio for f
c
= 4.4 ksi (used by
Mander et al. (1988)) .......................................................................................................... 395
Table A4: Ultimate confined strength to unconfined strength ratio for f
c
= 5.2 ksi ................. 396
Table A5: Ultimate confined strength to unconfined strength ratio for f
c
= 3.3 ksi (using
Scickert and Winkler (1977)) .............................................................................................. 397
Table A6: Ultimate confined strength to unconfined strength ratio for f
c
= 3.9 ksi (using
Scickert and Winkler (1977)) .............................................................................................. 398
xxvi
Table A7: Ultimate confined strength to unconfined strength ratio for f
c
= 5.2 ksi (using
Scickert and Winkler (1977)) .............................................................................................. 399
xxvii
Acknowledgements
All praises to Allah the lord of mankind.
The author expresses his gratitude to his supervisor, Dr. Hayder Rasheed who was
abundantly helpful and offered invaluable assistance, support and guidance. Deepest gratitude
are also due to the members of the supervisory committee, Dr. Asad Esmaeily, Dr Hani Melhem,
Dr Sutton Stephens and Dr Brett DePaola,.
The author would also like to convey thanks to Kansas Department of transportation for
providing the financial means of this research
The author wishes to express his love and gratitude to his beloved family; for their understanding
& endless love, through the duration of his studies. Special thanks to his mother Dr Amany
Aboellil for her support
xxviii
Dedication
This work is dedicated to the memory of the ones who couldnt make it.
1
Chapter 1  Introduction
11 Background
Columns are considered the most critical elements in structures. The unconfined analysis
for columns is well established in the literature. Structural design codes dictate reduction factors
for safety. It wasnt until very recently that design specifi cations and codes of practice, like
AASHTO LRFD, started realizing the importance of introducing extreme event load cases that
necessitates accounting for advanced behavioral aspects like confinement. Confinement adds
another dimension to columns analysis as it increases the columns capacity and ductility.
Accordingly, confinement needs special non linear analysis to yield accurate predictions.
Nevertheless the literature is still lacking specialized analysis tools that take into account
confinement despite the availability of all kinds of confinement models. In addition the literature
has focused on axially loaded members with less attention to eccentric loading. Although the
latter is more likely to occur, at least with misalignement tolerances, the eccentricity effect is not
considered in any confinement model available in the literature.
It is widely known that code Specifications involve very detailed design procedures that
need to be checked for a number of limit states making the task of the designer very tedious.
Accordingly, it is important to develop software that guide through the design process and
facilitate the preparation of reliable analysis/design documents.
12 Objectives
This study is intended to determine the actual capacity of confined reinforced concrete
columns subjected to eccentric loading and to generate the failure envelope at three different
2
levels. First, the wellknown ultimate capacity analysis of unconfined concrete is developed
as a benchmarking step. Secondly, the unconfined ultimate interaction diagram is scaled
down based on the reduction factors of the AASHTO LRFD to the design interaction
diagram. Finally, the actual confined concrete ultimate analysis is developed based on a new
eccentricity model accounting for partial confinement effect under eccentric loading. The
analyses are conducted for three types of columns; circular columns confined with FRP and
conventional transverse steel, circular columns confined with steel tubes and rectangular
columns confined with conventional transverse steel. It is important to note that the present
analysis procedure will be benchmarked against a wide range of experimental and analytical
studies to establish its accuracy and reliability.
It is also the objective of this study to furnish interactive software with a userfriendly
interface having analysis and design features that will facilitate the preliminary design of
circular columns based on the actual demand. The overall objectives behind this research are
summarized in the following points:
 Introduce the eccentricity effect in the stressstrain modeling
 Implement nonlinear analysis for considering the confinement effects on columns actual
capacity
 Test the analysis for three types of columns; circular columns confined with FRP and
conventional transverse steel, circular columns confined with steel tubes and rectangular
columns confined with conventional transverse steel.
 Generate computer software that helps in designing and analyzing confined concrete
columns through creating three levels of MomentForce envelopes; unconfined curve,
design curve based on AASHTOLRFD and confined curve.
3
13 Scope
This dissertation is composed of seven chapters covering the development of material models,
analysis procedures, benchmarking and practical applications.
 Chapter one introduces the objectives of the study and the content of the different
chapters.
 Chapter two reviews the literature through four independent sections:
1 Section 1: Reinforced concrete confinement models
2 Section 2: Circular Columns Confined with FRP
3 Section 3: Circular Concrete Filled Steel Tubes Columns (CFST)
4 Section 4: Rectangular Columns subjected to biaxial bending and Axial Compression
 Chapter three deals with Circular columns confined with FRP and lateral steel.
 Chapter four talks about concrete filled steel tube (CFST) circular columns
 Chapter five presents rectangular columns analysis for both the unconfined and confined
cases. Chapter three, four and five address the following subjects:
Finite Layer Approach (Fiber Model)
Present Confinement Model for Concentric Columns
Present Confinement Model for Eccentric Columns
Moment of Area Theorem
Numerical Formulation
Results and Discussion
 Chapter six introduces the software concepts and highlights the software forms and
components
 Chapter seven states the conclusions and recommendations.
4
Chapter 2  Literature Review
This chapter reviews four different topics; lateral steel confinement models,
Circular Concrete Columns Filled Steel Tubes (CFST) and Rectangular Columns
subjected to biaxial bending and Axial Compression.
21 Steel Confinement Models
A comprehensive review of confined models for concrete columns under concentric axial
compression that are available in the literature is conducted. The models reviewed are
chronologically presented then compared by a set of criteria that assess consideration of different
factors in developing the models such as effectively confined area, yielding strength and
ductility.
211 Chronological Review of Models
The confinement models available are presented chronologically regardless of their
comparative importance first. After that, discussion and categorization of the models are carried
out and conclusions are made. Common notation is used for all the equations for the sake of
consistency and comparison.
2111 Notation
A
s:
the cross sectional area of longitudinal steel reinforcement
A
st:
the cross sectional area of transverse steel reinforcement
A
e:
the area of effectively confined concrete
5
A
cc:
the area of core within centerlines of perimeter spirals or hoo ps excluding area of
longitudinal steel
b: the confined width (core) of the section
h: the confined height (core) of the section
c: centertocenter distance between longitudinal bars
d
s
: the diameter of longitudinal reinforcement
d
st
: the diameter of transverse reinforcement
D: the diameter of the column
d
s
the core diameter of the column
f
cc
: the maximum confined strength
f
c:
the peak unconfined strength
f
l:
the
lateral confined pressure
f
l:
the effective
lateral
confined pressure
f
yh:
the yield strength of the transverse steel
f
s:
the stress in the lateral confining steel
k
e:
the effective lateral confinement coefficient
q: the effectiveness of the transverse reinforcement
s: tie spacing
s
o:
the vertical spacing at which transverse reinforcement is not effective in concrete confinement
ε
co:
the strain corresponding to the peak unconfined strength f
c
ε
cc:
the strain corresponding to the peak confined strength f
cc
ε
y:
the strain at yielding for the transverse reinforcement
ε
cu:
the ultimate strain of confined concrete
6
ρ
s
: the volumetric ratio of lateral steel to concrete core
ρ
l:
the ratio of longitudinal steel to the gross sectional area
ρ: the volumetric ratio of lateral + longitudinal steel to concrete core
Richart, Brandtzaeg and Brown (1929)
Richart et als. (1929) model was the first to capture the proportional relationship
between the lateral confined pressure and the ultimate compres sive strength of confined
concrete.
lccc
fkff
1
'
+=
21
The average value for the coefficient k
1,
which was derived from a series of short column
specimen tests, came out to be (4.1). The strain corresponding to the peak strength
ε
cc
(see
Mander et al. 1988) is obtained using the following function:
+=
'
2
1
c
l
cocc
f
f
k
εε
12
5kk
=
22
where
ε
co
is the strain corresponding to
fc
,
k
2
is the strain coefficient of the effective lateral
confinement pressure. No stressstrain curve graph was proposed by Richart et al (1929).
Chan (1955)
A trilinear curve describing the stressstrain relationship was suggested by Chan (1955)
based on experimental work. The ratio of the volume of steel ties to concrete core volume and
concrete strength were the only variables in the experimental work done. Chan assumed that OA
approximates the elastic stage and ABC approximates the plastic stage, Figure (21). The
positions of A, B and C may vary with different concrete variables. Chan assumed three different
7
slopes E
c
,
λ
1
E
c
,
λ
2
E
c
for lines OA, AB and BC respectively. However no information about
λ
1
and
λ
2
was provided.
Blume, Newmark and Corning (1961)
Blume et al. (1961) were the first to impose the effect of the yield strength for the
transverse steel f
yh
in different equations defining the model. The model generated, Figure (22),
has ascending straight line with steep slope starting from the origin till the plain concrete peak
strength f
c
and the corresponding strain ε
co
, then a less slope straight line connect the latter point
and the confined concrete peak strength f
cc
and ε
cc
. Then the curve flatten till ε
cu
sh
fA
ff
yhst
ccc
1.485.0
'
+= for rectangular columns 23
psi
psif
c
co
6
'
10
40022.0 +
=
ε
24
ycc
εε
5=
25
sucu
ε
ε
5
=
26
Figure 21: General StressStrain curve by Chan (1955)
λ
2Ec
λ
1Ec
Strain
Stress
u
f
p
f
e
f
O
A
B
C
e
ε
p
ε
u
ε
γ
1Ec
γ
2Ec
8
Stress
Strain
0.85f'c
fcc
ε
co
ε
cc
ε
cu
Figure 22: General StressStrain curve by Blume et al. (1961)
where ε
y
is the strain at yielding for the transverse reinforcement, A
st
is the cross sectional area of
transverse steel reinforcement ,h is the confined cross sectional height,
ε
su
is the strain of
transverse spiral reinforcement at maximum stress and
ε
cu
is the ultimate concrete strain.
Roy and Sozen (1965)
Based on their experimental results, which were controlled by two variables; ties spacing
and amount of longitudinal reinforcement, Roy and Sozen (1965) concluded that there is no
enhancement in the concrete capacity by using rectilinear ties. On the other hand there was
significant increase in ductility. They proposed a bilinear ascendingdescending stress strain
curve that has a peak of the maximum strength of plain concrete f
c
and corresponding strain ε
co
with a value of 0.002. The second line goes through the point defined by
ε
50
till it intersects with
the strain axis. The strain ε
50
was suggested to be a function of the volumetric ratio of ties to
concrete core ρ
s
, tie spacing s and the shorter side dimension b (see Sheikh 1982).
9
s
b
s
4
'3
50
ρ
ε
= 27
Soliman and Yu (1967)
Soliman and Yu (1967) proposed another model that emerged from experimental results.
The main parameters involved in the work done were tie spacing s, a new term represents the
effectiveness of ties s
o
, the area of ties A
st
, and finally section geometry, which has three different
variables; A
cc
the area of confined concrete under compression, A
c
the area of concrete under
compression and b. The model has three different portions as shown in Figure (23). The
ascending portion which is represented by a curve till the peak point (f
c
, ε
ce
). The flat straight
line portion with its length varying depending on the degree of confinement. The last portion is a
descending straight line passing through (0.8 f
c
, ε
cf
) then extending down till an ultimate strain.
(
)
2
0028.0
45.04.1
BssA
ssA
A
A
q
st
ost
c
cc
+
−
−=
28
(
)
qff
ccc
05.019.0
'
+= 29
6
10*55.0
−
=
ccce
f
ε
210
)1(0025.0 q
cs
+
=
ε
211
)85.01(0045.0 q
cf
+=
ε
212
where q refers to the effectiveness of the transverse reinforcement , s
o
is the vertical spacing at
which transverse reinforcement is not effective in concrete confinement and B is the greater of b
and 0.7 h.
10
Stress
Strain
'
c
f
'
8.0
c
f
ce
ε
cs
ε
cf
ε
Sargin (1971)
Sargin conducted experimental work on low and medium strength concrete with no longitudinal
reinforcement. The transverse steel that was used had different size and different yield and
ultimate strength. The main variables affecting the results were the volumetric ratio of lateral
reinforcement to concrete core ρ
s
, the strength of plain concrete f
c
, the ratio of tie spacing to the
width of the concrete core and the yield strength of the transverse steel f
yh
.
(
)
+−+
−+
=
2
2
'
3
)2(1
1
mxxA
xmAx
fkf
cc
213
where m is a constant controlling the slope of the descending branch:
'
05.08.0
c
fm −= 214
cc
c
x
ε
ε
=
215
'
3 c
ccc
fk
E
A
ε
=
216
'
3
245.010146.01
c
yhs
f
f
b
s
k
ρ
−+=
217
Figure 23: General StressStrain curve by Soliman and Yu (1967)
11
'
734.0
10374.00024.0
c
yhs
cc
f
f
b
s
ρ
ε
−+= 218
'
3 ccc
fkf =
219
where k
3
is concentric loading maximum stress ratio.
Kent and Park (1971)
As Roy and Sozen (1965) did, Kent and Park (1971) assumed that the maximum strength
for confined and plain concrete is the same f
c
. The suggested curve, Figure (24), starts from the
origin then increases parabolically (Hognestads Parabola) till the peak at f
c
and the
corresponding strain
ε
co
at 0.002. Then it descends with one of two different straight lines. For
the confined concrete, which is more ductile, it descends till the point (0.5 fc, ε
50c
) and continues
descending to 0.2fc followed by a flat plateau. For the plain concrete it descends till the point
(0.5 fc, ε
50u
) and continue descending to 0.2f
c
as well without a flat plateau. Kent and Park
assumed that confined concrete could sustain strain to infinity at a constant stress of 0.2 fc
−=
2
'
2
co
c
co
c
cc
ff
ε
ε
ε
ε
for ascending branch
(
)
[
]
coccc
Zff
εε
−−= 1
'
for descending branch 220
1000
002.03
'
'
50
−
+
=
c
c
u
f
f
ε
221
(
)
hbs
Abh
st
s
+
=
2
ρ
222
uch 505050
ε
ε
ε
−
=
223
s
b
sh
ρε
4
3
50
=
224
12
couh
Z
εεε
−+
=
5050
5.0
225
where ρ
s
is the ratio of lateral steel to the concrete core, Z is a constant controlling the slope of
descending portion.
Popovics (1973)
Popovics pointed out that the stressstrain diagram is influenced by testing conditions and
concrete age. The stress equation is:
n
cc
c
cc
c
ccc
n
n
ff
+−
=
ε
ε
ε
ε
1
226
0.110*4.0
3
+=
−
cc
fn 227
4
4
10*7.2
cccc
f
−
=
ε
228
Vallenas, Bertero and Popov (1977)
Stress
Strain
'
c
f
'
2.0
c
f
'
5.0
c
f
Stress
c
ε
u50
ε
c50
ε
c20
ε
Figure 24: StressStrain curve by Kent and Park (1971).
13
The variables utilized in the experimental work conducted by Vallenas et al. (1977) were
the volumetric ratio of lateral steel to concrete core ρ
s
, ratio of longitudinal steel to the gross area
of the section ρ
l
, ties spacing s, effective width size, strength of ties and size of longitudinal bars.
The model generated was similar to Kent and Park model with improvement in the peak strength
for confined concrete, Figure (25). For the ascending branch:
[ ])1(1
'
−−= xZk
f
f
cc
c
c
ε
kccc 3.0
ε
ε
ε
≤
≤
229
k
f
f
c
c
3.0
'
=
ck
ε
ε
≤
3.0
230
cc
c
x
ε
ε
=
231
'
ccc
kff =
232
x
kf
E
kxx
f
E
f
f
c
ccc
c
ccc
c
c
−+
−
=
21
'
2
'
'
ε
ε
ccc
ε
ε
≤
233
For the descending branch:
'
'
'
245.010091.01
c
yhl
s
st
f
f
d
d
h
s
k
+
−+=
ρρ
234
'
734.0
1005.00024.0
c
yh
cc
f
f
h
s
ρ
ε
−+=
235
002.0
1000
002.03
4
3
5.0
'
'
−
−
+
+
=
c
c
s
f
f
s
h
Z
ρ
236
14
where k is coefficient of confined strength ratio, Z is the slope of descending portion, d
s
and d
st
are the diameter of longitudinal and transverse reinforcement respectively.
Axial Stress
Axial Strain
ε
cc
f cc
ε
0.3k
0.3kf'c
Figure 25: StressStrain curve by Vallenas et al. (1977).
Wang, Shah and Naaman (1978)
Wang et al. (1978) obtained experimentally another stressstrain curve describing the
behavior of confined reinforced concrete under compression; Figure (26). The concrete tested
was normal weight concrete ranging in strength from 3000 to 11000 psi (20.7 to 75.8 MPa) and
light weight concrete with strength of 30008000 psi (20.7 to 55 MPa). Wang et al. utilized an
equation, with four constants, similar to that of Sargin et al.
2
2
1
DX
CX
BXAX
Y
+
+
+
=
237
Where
cc
c
f
f
Y =
238
cc
c
X
ε
ε
=
239
15
The four constant A, B, C, D were evaluated for the ascending part independently of the
descending one. The four conditions used to evaluate the constants for the ascending part were
dY/dX = E
0.45
/E
sec
at X=0 E
sec
= f
cc
/
ε
cc
Y = 0.45 for X = 0.45/(E
0.45
/E
sec
)
Y=1 for X=1
dY/dX = 0 at X=1
whereas for the descending branch:
Y=1 for X=1
dY/dX = 0 at X=1
Y = f
i
/f
cc
for X =
ε
i
/
ε
cc
Figure 26: Proposed StressStrain curve by Wang et al (1978)
where f
i
and
ε
i
are the stress and strain at the inflection point, f
2i
and
ε
2
i
refer to a point such
that
cciii
ε
ε
ε
ε
−
=
−
2
and E
0.45
represents the secant modulus of elasticity at 0.45 f
cc
Y = f
2i
/f
cc
for X =
ε
2
i
/
ε
cc
Muguruma , Watanabe , Katsuta and Tanaka (1980)
Strain
Stress
cc
f
cc
f45.0
cc
ε
i
ε
i2
ε
i
f
i
f
2
16
Muguruma et al. (1980) obtained their stressstrain model based on experimental work
conducted by the model authors, Figure (27). The stressstrain model is defined by three zones;
Zone 1 from 0A:
2
2
'
c
co
coic
cic
Ef
Ef
ε
ε
ε
ε
−
+=
(kgf/cm
2
)
coc
ε
ε
≤
≤
0
240
Zone 2 from AD
(
)
( )
( )
ccc
ccco
ccc
ccc
ffff −
−
−
+=
'
2
2
εε
εε
(kgf/cm
2
)
cccco
ε
ε
ε
≤
<
241
Zone 3 from DE
( )
ccc
cccu
ccu
ccc
ff
ff
εε
εε
−
−
−
+=
(kgf/cm
2
)
cuccc
ε
ε
ε
≤
<
242
(
)
cucc
cccc
u
fS
f
εε
ε
+
−
=
2
(kgf/cm
2
) 243
(
)
2000/100413.0
'
cu
f−=
ε
(kgf/cm
2
) 244
−=
W
s
f
f
Cc
c
yh
s
5.01
'
ρ
245
where
S is the area surrounded by the idealized stressstrain curve up to the peak stress and W is
the minimum side length or diameter of confined concrete
For circular columns confined with circular hoops:
17
(
)
'
1501
ccc
fCcf +=
(kgf/cm
2
) 246
(
)
cocc
Cc
ε
ε
14601
+
=
247
(
)
ucu
Cc
ε
ε
9901
+
=
248
Whereas for square columns confined with square hoops:
(
)
'
501
ccc
fCcf +=
(kgf/cm
2
) 249
(
)
cocc
Cc
ε
ε
4501
+
=
250
(
)
ucu
Cc
ε
ε
4501
+
=
251
Axial Stress
f cc
f'c
f'u
ε
cc
ε
cu
0
Α
D
Ε
Axial Strain
ε
u
f u
ε
co
Figure 27: Proposed StressStrain curve by Muguruma et al (1980)
18
Scott, Park, Priestly (1982)
Scott et al. (1982) examined specimens by loading at high strain rate to correlate with the
seismic loading. They presented the results including the effect of eccentric loading, strain
rate, amount and distribution of longitudinal steel and amount and distribution of transverse
steel. For low strain rate Kent and Park equations were modified to fit the experimental data
−=
2
'
002.0002.0
2
kk
kff
cc
cc
εε
k
c
002.0
≤
ε
252
[
]
)002.0(1
'
kZkff
cmcc
−−=
ε
k
c
002.0
>
ε
253
where
'
1
c
yhs
f
f
k
ρ
+=
254
k
s
b
f
f
Z
s
c
c
m
002.0
"
4
3
1000145
29.03
5.0
'
'
−+
−
+
=
ρ
f
c
is in MPa 255
where b is the width of concrete core measured to outside of the hoops. For the high strain
rate, the k and Z
m
were adapted to
)1(25.1
'
c
yhs
f
f
k
ρ
+=
256
k
s
b
f
f
Z
s
c
c
m
002.0
4
3
1000145
29.03
625.0
'
'
−+
−
+
=
ρ
f
c
is in MPa 257
and the maximum strain was suggested to be:
19
+=
300
9.0004.0
yh
scu
f
ρε
258
It was concluded that increasing the spacing while maintaining the same ratio of lateral
reinforcement by increasing the diameter of spirals, reduce the efficiency of concrete
confinement. In addition, increasing the number of longitudinal bars will improve the concrete
confinement due to decreasing the spacing between the longitudinal bars.
Sheikh and Uzumeri (1982)
Sheikh and Uzumeri (1982) introduced the effectively confined area as a new term in
determining the maximum confined strength (Soliman and Yu (1967) had trial in effective area
introduction). In addition to that they, in their experimental work, utilized the volumetric ratio of
lateral steel to concrete core, longitudinal steel distribution, strength of plain concrete, and ties
strength, configuration and spacing. The stressstrain curve, Figure (28), was presented
parabolically up to (f
cc
, ε
cc
), then it flattens horizontally till ε
cs,
and finally it drops linearly
passing by (0.85f
cc
, ε
85
) till 0.3 f
cc
, In that sense, it is conceptually similar to the earlier model of
Soliman and Yu (1967).
f
cc
and ε
cc
can be determined from the following equations:
cpscc
fkf =
'
cpcp
fkf =
85.0=
p
k 259
'
2
2
22
2
1
5.5
1
73.2
1
sts
occ
s
f
b
s
b
nc
P
b
k
ρ
−
−+=
260
6'
10*55.0
−
=
cscc
fk
ε
261
20
−+=
'
'
2
51
81.0
1
c
sts
cocs
f
f
b
s
c
ρ
εε
262
css
s
b
ερε
+= 225.0
85
263
s
b
Z
s
ρ
4
3
5.0
=
264
where b is the confined width of the cross section, f
st
is the stress in the lateral confining bar, c is
centertocenter distance between longitudinal bars,
ε
s85
is the value of strain corresponding to
85% of the maximum stress on the unloading branch, n is the number of laterally supported
longitudinal bars, Z is the slope for the unloading part, f
cp
is the equivalent strength of
unconfined concrete in the column, and P
occ
= K
p
f'
c
(A
cc
 A
s
)
Figure 28: Proposed general StressStrain curve by Sheikh and Uzumeri (1982).
Stress
Strain
cc
f
cc
ε
cs
ε
85
ε
21
Ahmad and Shah (1982)
Ahmad and Shah (1982) developed a model based on the properties of hoop
reinforcement and the constitutive relationship of plain concrete. Normal weight concrete and
lightweight concrete were used in tests that were conducted with one rate of loading. No
longitudinal reinforcement was provided and the main two parameters varied were spacing and
yield strength of transverse reinforcement. Ahmed and Shah observed that the spirals become
ineffective when the spacing exceeds 1.25 the diameter of the confined concrete column. They
concluded also that the effectiveness of the spiral is inversely proportional with compressive
strength of unconfined concrete.
Ahmad and Shah adapted Sargin model counting on the octahedral failure theory, the
three stress invariants and the experimental results:
2
2
)2(1
)1(
XDXA
XDXA
Y
ii
ii
+−+
−+
=
265
pcn
pcs
f
f
Y =
266
ip
i
X
ε
ε
=
267
where f
pcs
is the most principal compressive stress, f
pcn
is the most principal compressive strength,
ε
i
is the strain in the ith principal direction and
ε
ip
is the strain at the peak in the ith direction.
ip
i
i
E
E
A =
ip
pcn
ip
f
E
ε
=
E
i
is the initial slope of the stress strain curve, D
i
is a parameter that governs the descending
branch. When the axial compression is considered to be the main loading, which is typically the
case in concentric confined concrete columns, Equations (265), (266) and (267) become:
22
2
2
)2(1
)1(
DXXA
XDAX
Y
+−+
−+
= 268
cc
c
f
f
Y =
269
cc
c
X
ε
ε
=
270
sec
E
E
A
c
=
271
Park, Priestly and Gill (1982)
Park et al. (1982) modified Kent and Park (1971) equations to account for the strength
improvement due to confinement based on experimental work conducted for four square full
scaled columns (21.7 in
2
(14 000 mm
2
) cross sectional area and 10.8 ft (3292 mm) high), Figure
(29). The proposed equations are as follow:
−=
2
'
002.0002.0
2
kk
kff
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