Appendix 1 How to order the program Iistings and the floppy disks

The complete listings of alI the computer programs in Chapter 12, together with the associated floppy disks, may be ordered from the Publishers:

Chapman and HalI Ltd 11 New Fetter Lane London EC4P 4EE

At.t Instructions 1. Study Section 12.1, before you place an order for any or alI of the

items below. 2. If you need only the printout of the program listings (with Commen­

tary and User Instructions), order Item (a) below. 3. If you need only the floppy disk to generate the printout of program

listings yourself, order Item (b1) or (b2) below. Items (b1) and (b2) are floppy disks containing alI the programs

stored as 'source files'-see explanation in Section 12.1(d). 4. If you need only the floppy disk to run the programs, order Item (cI)

and (c2) below. Items (el) and (c2) are floppy disks containing alI the programs

stored in machi ne code as 'executable files'-see explanation in Section 12.1(d).

ITEM (a): Wong, H. H. A. (in collaboration with Kong, F. K. and Evans, R. H.), Complete listings of the computer programs (with Commentary and User Instructions) in 'Kong and Evans: Reinforced and Prestressed Concrete, Van Nostrand Reinhold, 3rd Edition, 1987', Van Nostrand Reinhold, Wokingham, 1987.

ITEM (b1): Floppy Disk 1: Source files of the computer programs in 'Kong and Evans: Reinforced and Prestressed Concrete, Van Nostrand Reinhold, 3rd Edition, 1987', for RM. Nimbus and IBM Compatibles with 3.5 inch disk drive.

Appendix 1 How to order program listing and disks 493

ITEM (b2): Floppy Disk Ia and Ib: Source-files of the computer programs in 'Kong and Evans: Reinforced and Prestressed Concrete, Van Nostrand Reinhold, 3rd Edition, 1987', for lBM-PC/XT, IBM-PC/AT or IBM Compatibles with 5.25 inch disk drive.

ITEM (el): Floppy Disk II: Executable files of the computer programs in 'Kong and Evans: Reinforced and Prestressed Concrete, Van Nostrand Reinhold, 3rd Edition, '1987', for RM Nimbus and IBM Compatibles with 3.5 inch disk drive.

ITEM (c2): Floppy Disks Ha and IIb: Executable files of the computer programs in 'Kong and Evans: Reinforced and Prestressed Concrete, Van Nostrand Reinhold, 3rd Edition, 1987', for IBM-PC/XT, IBM-PC/AT or IBM Compatibles with 5.25 inch disk drive.

Al.2 System requirements The system requirements necessary to run the Authors' computer programs are summarized as follows.

Personal computers:

Memory Operating system Disk drive Monitor Printer

RM Nimbus, IBM-PC/XT, IBM-PC/AT and IBM Compatibles (e.g. Amstrad-PC) 512 K bytes are more than adequate MS-DOS Single disk drive is adequate Monochrome is adequate Optional

Appendix 2 Design tables and charts

TableA2-1 Areas of groups of reinforcement bars (mm2)

Bar Number of bars size

(mm) 1 2 3 4 5 6 7 8 9 10

8 50 101 151 201 251 302 352 402 452 503

10 79 157 236 314 393 471 550 628 707 785 12 113 226 339 452 565 679 792 905 1017 1131

16 201 402 603 804 1005 1206 1407 1608 1809 2011 20 314 628 942 1257 1571 1885 2199 2513 2827 3142

25 491 982 1473 1963 2454 2945 3436 3927 4418 4909 32 804 1608 2412 3216 4021 4825 5629 6433 7237 8042

40 1256 2513 3769 5026 6283 7539 8796 10050 11310 12570

TableA2-2 Reinforcement-bar areas (mm2) per metre width for various bar spacings

Bar Bar spacing (mm) size

(mm) 75 100 125 150 175 200 225 250 275 300

8 671 503 402 335 287 252 223 201 183 168

10 1047 785 628 523 449 393 349 314 286 262 12 1508 1131 905 754 646 566 503 452 411 377

16 2681 2011 1608 1340 1149 1005 894 804 731 670 20 4189 3142 2513 2094 1795 1571 1396 1257 1142 1047

25 6545 4909 3927 3272 2805 2454 2182 1963 1785 1636 32 8042 6434 5362 4596 4021 3574 3217 2925 2681

40 10050 8378 7181 6283 5585 5027 4570 4189

Appendix 2 Design tables and charts 495

Bar fy= 250 N/mm2 Jy = 460 N/mm2

siz.

0 r n h r n h

8 100 100 100 100

10 100 100 100 110

12 100 110 30 100 140

16 100 150 100 180 20

20 100 180 110 220

25 130 230 180 350

J2 160 290 40 230 450 40 200 360 280 560

Fig. A2-1 Minimum bend and hook allowances (mm)-BS 4466

496 Appendix 2 Design tab les and charts

Shape

code

20

32

33

34

35

37

38

41

Method of

measurement

1-- A----I h--1

~{: A--I

h"'" r----.h vf7 A5

n'),.!:Q==== 'V'~ A---I

n~LA=S'n I~r ~B~

!Jr r~r~ I--B =:rT

o = bar size

Oimensions

tobe given in

bar schedule

straight

c A

c "") A

A

I I

A

AL-

AL-.J B

h} n ond r: see Fig. A2 -1

Fig. A2-2(a) Bending dimensions-BS 4466

Total length

of bar

A

A+h

A+2h

A+n

A+ 2n

{ A+B +C - r - 20

{A+B+C

if CIt ~45'

{A+B +C -r-20

if O( > 45 o

Appendix 2 Design tables and charts 497

Shape Method of Oi mensions to Total length

code measurement be given in of bar bar schedule

..j A ,""", J\1 F-A-i {A+2S+C+E

43 ~ r r r:' r O ~ ( if O( S 45°)

-lCI- C

51 I~- A~ A+S-1. r -0 ~ standard) 2 S........l

60 E] lAs] 2(A+B)+20 0

J { Â+ C \~>-B ~~ 62

(ifa< S 45°) 1- C--I'v~ f -. ,-=A:='"1..i A

81 ~: ~8 €::f? 2A+3S +220 t

~ A 83 '* ~ ~

A+2B+C+D

Isometric ~ - 2r - 40 Isometric.

view O view

o = bar size

h, n and r: see Fig A2-1

Fig. A2-2(b) Bending dimensions-BS 4466

0-5

1

·0

1-5

2

-0

2·5

3-0

3'5

13

I -r

IT

III III II III IT

I

1 I

12

11

10 9 -NE

8 E

.........

7 Z

-

6 N

"'C

5 ~

"""-~

4

3 2 1

-b-<tl x

/d =

0-3

......... _~

fcu

4

0

L

-r

~" ;'~

x t-.~AS.!..:: T

fy 4

60

x/d =

0-4

------

~

~

1. .,

d d'/d

0-15 x/d =

0-5

----

~

~

J h

~

• As·

....1

~

I ~

~

.~

...... 1

-

~ ~

~ ~

/'

/ , /

/ ~II

I I

J I

I I

I I

~

1 I

I I

I I

I I

I I

I Il lL

4-0 % -

2-0 %

"'C

1-5 % ~

1-0

%

""'-....

CI)

~

II 0

·5%

--~

O

O

0-5

%

1-0

%

1-5

%

2-0

%

2-5

%

3-0

%

3'5

%

p (= A

s / bd

) F

ig. A2

-3

Beam

design chart-u

ltimate U

rnit state (BS 8110)

1 2

3 4

5 6

7 8

9 10

11 12

13 14

15 16

:: rn.:r'''' I~ '~~r ' , I ' .. , I ' , , , I ' , , , I ' .. , I ' , , , I ' , , , I ' , , , I ',' , , ~~ -1

40 r "'>oo"J: "'!-....

,'1 "2'1

,,""

1 1

1 1

1 1

1 1 r oA

ro 1

-;-35

~N>K I

II L .~. j E

30 E

~25

fcu 40

-.&:.

fy 460

.o 20 "Z-

d/h 0·85

15

10 ~ 1 Jt -)-,--) ---1-)--1--jl----II--t -)-t --J---ii K~~

5

o 1

2 3

4 5

6 7

8 9

10 11

12 13

14 15

16 M

/bh

2 (N/mm

2 )

Fig. A

2-4

C

olumn design ch

art-BS

8110

Index Note: Bold numbers indicate main references

Actual resistance moments 154 Aggregate interlock 202 Aggregates 21

characteristics 22, 27 coarse 21,23 fine 21,23 grading 23,51 lightweight 21 sizes 21,23 strength 24 unit weight 24

Anchorage bond length 145,222 Anchorage bond stress 221 Anchorage of links 143, 221, 236 Anchorage of tendons 335

Balanced beam section 91,96, 106 Balanced failure

beams 91,96,106 columns 252, 280

Balanced steel ratio 91, 97, 106 Bar bending dimensions (BS 4466)

496,497 Bar bending schedule 452 Bar mark 82,214 Bar size 73, 82, 494 Beams .85,156

axialthrust 103 bar spacing 144,174 computer-aided design (BS 8110)

475,477 computer application 475, 477 concrete cover 39,146 continuous beams 133,409,410 crack control 173,326 crack width calculation (BS 8110)

187,188 deep beams 218

deftection calculation (BS 8110) 175,182

deftection control 168,172 design chart (BS 8110) 98, 498 design details (BS 8110) 142 design and detailing example 147,

432 effective span 103 effective width 127,148 elastic theory 157, 335, 382 fire resistance 146 ftanged beam 127,234,360 ftexural strength 86,89,98,105,

111,124 generalftexuraltheory 86 K' (= Mul/cubd2) 105,110,123 lever-arm factor zid 89, 103, 111,

124 moment redistribution 120,133 preliminary analysis and sizing 402 rectangular stress block 96, 104, 119 shear strength 198,209,362 simplified stress block (BS 8110) 96,

104,119 slenderness limit 144 span/depth ratio 169,200,402 stress blocks 87, 94, 96 torsion 224, 234, 368 ultimate moment of resistance 87,

105,120 Beams, prestressed (see Prestressed

beams) Bending and axialload

beams 103 columns 248, 265

Bending moment tab les (BS 8110) continuous beams 410 continuous slabs 411

502 Index

Bending moment envelope 139,434 Bending schedule 452 Bends and hooks 222, 495 Bent-up bars 206,213 Biaxial bending 267, 271

design method (BS 8110) 267,269, 285

interaction surface 272, 273 Biaxial stress state 40, 42 Bond coefficient 221 Bond length 222, 223 Bond stress 221,223 Bonded (unbonded) post-tensioned

beam 355 Braced columns 264 Braced frame analysis 412 Buckling of deep beams 220

Cantilever 146, 169 effective span 103

Cantilever method 420 Cement, properties 18

types 19,32 unit weight 21

Characteristic breaking load 334 Characteristic loads 13 Characteristic strength 12 Circular columns 287 CIRIA deep beam guide 144, 219 Coarse aggregate 23 Collapse mechanism 133,293 Column interaction diagram

biaxial bending 272 uniaxial bending 254,266, 267, 499

Columns axially loaded 68 biaxial bending 267, 269, 271 braced 264 circular 287 computer-aided design (BS 8110)

475,481 computer application 475,481 concrete cover 39, 78 design charts (BS 8110) 266, 499 design charts (I.Struct.E. Manual)

267 design detailing example 78,451 design details (BS 8110) 76,286 design minimum eccentricity 265 eccentrically loaded 248,265 effective height 264, 265 fire resistance 78 footings 454 initial crookedness 289

interaction diagram 254,266,272 magnification factor 288 minimum eccentricity (BS 8110)

265 preliminary analysis and sizing 402 short 68,76 slender 273,278

Compacting factor 47 Compatibility torsion 224 Compression reinforcement 86,105,

143 Computer application

beams 475,477 columns 475,481 mix design 473 prestressed concrete 488 program listings 456,492 shear 479,489 slabs 486 slender columns 483 torsion 480

Concordant tendon profile 388 Concrete

characteristic strength 12, 13 cover to reinforcement (durability)

39 cover to reinforcement (fire) 78,

146,328 density 24 design stress block (BS 8110) 70,

94,96 DoE mix design method 54 durability 39 failure criteria 40 grades 406 lightweight 21 mix design and statistics 50,54,61 modulus of elasticity 38, 180 properties 18, 24 Road Note No. 4 method 50 strength 25 stress block 70, 87, 94, 96 ultimate strain 70,71 unit weight 24 workability 46, 49

Confident level, limits 10 Continuous beams 133,409,410

effective span 103 Cover to reinforcement

durability requirement 39 fire resistance (beams) 146 fire resistance (columns) 78 fire resistance (slabs) 328

Crack control and width 173,187,326

Cracked section 157 Creep coefficient 180 Creep of concrete 28, 180

loss of prestress due to 351 Critical section 339 Cube strengh 25 Curing of concrete 28 Current margin in mix design 50,61 Curtailment of bars 145,327 Curvature 160,164, 166, 181,372 Curvature-area theorem 175,195 Cylinder strength 25

I>eadload 14,402,405 I>eep beams

buckling 220 CIRIA guide 219 instability 220 shear strength 218 slender deep beams 220 structural idealization 220 web openings 220

I>eflection of prestressed beams 368 long-term 371 short-term 369

I>eflection of reinforced concrete beams 168,175

long-term 175,182 short-term 175,182

I>esign charts (BS 8110) columns 266, 499 r.c. beams 98, 498

I>esign details (BS 8110) beams 142, 144 columns 76,451 slabs 325,327,431

I>esign load 13 I>esign (nominal) shear stress 199,

210 I>esign strength 13, 69 I>esign tables (BS 8110)

areas of bars 494 bar spacing (crack control) 174 bending moments (continuous beam)

410 bending moments (continuous slab)

411 concrete cover (durability) 39 concrete cover (fire) 78,146,328 effective column height 265 fire resistance (beams) 146 fire resistance (columns) 78 fire resistance (slabs) 328 K' (= Mulfeubd2) 123

Index 503

lever-arm factor zId 111.124 maximum bar spacing 174 modulus of elasticity (concrete) 38 neutral axis depth factor xld 111,

124 partial safety factor Ye 14 partial safety factor Ym 15 prestressed concrete beam 337, 347,

354 reinforcement bar areas 494 shearforces (continuous beam) 410 shearforces (continuous slab) 411 shear links (Asv/sv) 213 shear stress Ve 210 span/depth ratio 169 torsional shear stress 235 ultimate anchorage and lap lengths

145 I>etailing notation 82,214 I>etailing

beams 149,242,444 columns 81,451 footings 454 slabs 431 stairs 453

I>iagonal tension 198, 200 I>oE mix design method 54 I>oubly reinforced beams 98 I>owel action 202 I>urability of concrete 38, 39

concrete cover requirement (Table 2.5-7) 39

Eccentrically loaded columns 248 design chart 254, 266, 267

Effective depth 86 Effective height of column 264 Effective length of cantilever 103 Effective modulus of elasticity 72, 180 Effective prestressing force 335 Effective span 103

continuous beams 103 simple beams 103

Effective width 127 Elastic theory

columns 71 prestressed beams 335,382 reinforced beams 157 shear in prestressed beams 364 slabs 325

Equilibrium torsion 224 Equivalent section 72, 158 Evaporable water 27 Exposure condition (BS8110) 40

504 Index

Factor of safety 13, 14, 15 Failure criteria, concrete 40 Fan mechanism 318 Final setting time 19 Fine aggregate 21 Fineness of cement 19 Fire resistance

beams 146 columns 78 slabs 328

Flanged section 127 deftection control 169 effective width 127 transverse (secondary) steel 128,

143 Flexural stiffness (rigidity) EI 179,

234,369,371,407 Flexure-shear cracks 200 Floor load, reduction of 405 Flowcharts (see Computer application) Footings 454 Frame analysis

braced 412 unbraced 419

Free water 50 Full anchorage bond length 145,222

Gel/space ratio 27 Grade of concrete 406

Hardening of concrete 19 Heat of hydration 19 Hillerborg's strip method 319 Hognestad's stress block 92 Hooks or bends 222,495

Ideal tendon profile 345 Imposed load 14,403

reduction for ftoors 405 Indirect tensile strength 25 Initial setting time 19 Instability 144, 286

columns 275 deep beams 220

Interaction bending and axialload 103 biaxial bending and axialload 271 diagrams for columns 254, 266, 272 torsion and bending 231,232 torsion and shear 232, 233 torsion, bending and shear 231,233

Interface shear transfer 202 Isotropically reinforced slabs 298

Johansen's stepped yield criterion 294,296

Laplength minimum lap length 144 required lap length 144,145

Level of significance 10 Lever arm, factor 89,103

BS 8110 simplified block 104,111, 118,124

BS 8110's limit on zId 103 design tables (BS 8110) 111,124

Lightweight aggregates 21 Lightweight concrete 21 Limit design 136 Limit state design 2, 15 Limits for main steel

beams 142,143 columns 76 slabs 326, 327

Line of pressure (thrust) 381, 391 Linear transformation 387 Links 77,143,206,211 Links, anchorage of 143 Links, spacing 143,211,236 Loading 14,401

dead 14,402,405 imposed 14,403,405 wind 14,419

Loading arrangement (BS 8110) continuous beams 409 continuous slabs 410

Local bond 223 Long-term deftection

prestressed beams 371 reinforced beams 175,182

Loss of prestress 348 creep 351 elastic deformation 350 multi-Ievel strands 354 relaxation 349 shrinkage 350

Lower-bound theorem 319

Maturity of concrete 28 Maximum barspacing 144,173,174,

326 Maximum cement content 40 Maximum moments diagram 139, 141,

434 Maximum prestressing force 339 Member sizing 402 Membrane analogy for torsion 226

Microcomputer (see Computer) Minimum bar spacing 144 Minimum cement content 39 Minimum design dead load 14 Minimum eccentricity (columns) 265 Minimum lap length 144 Minimum links 143,210 Minimum prestressing force 339 Mix design 49

basic principles 50 DoE method 54 Road Note No. 4 method 50

Modular ratio 72, 180 Modulus of elasticity 38, 72

concrete 38, 180 prestressing tendons 361 reinforcement bars 69,72

Modulus of rupture 26 Moment-area theorem 175 Moment-axis notation 294 Moment redistribution 120, 133

redistribution ratio 120, 137 redistribution percentage 121

Moment vector notation 296

Neutral axis depth, factor 86,89,159 BS 8110 simplified block 96,105,

111,124 design tab les (BS 8110) 111,124

Nominal are a 76 Nominal (concrete) cover 39,40 Nominallinks (see Minimum links) Nominal shear stress 199 Non-destructive testing 44 Non-evaporable water 26 Normal moment in slab 295 Normal probability distribution 7,8

Openings in deep beams 220 Ordinary Portland cement 19 Orthotropically reinforced slab 298 Over-reinforced beams 89,96

Parabola, properties of 151,152 Partial prestressing 333 Partially cracked section (BS 8110)

164 Partial safety factors 13, 14, 15 Permeability of concrete 39 Permissible pressure (thrust) zone 391 Permissible shear stress 210 Permissible tendon zone 343 Perry-Robertson formula 289

Index 505

Plastic design, plastic hinge 133, 136 Plastic moment of resistance 133 Plastic theory of structures 303,319 Poisson's ratio of concrete 38 Portal method 422 Post-tensioning 335 Preliminary analysis and sizing 402 Prestressed concrete beams 333

class 1, 2, 3 members 333 computer application 488 deflections, long-term 371 deflections, short-term 369 design procedure 374, 393 design procedure (simplified) 376 design tab les (BS 8110) 337, 347,

354 generalflexuraltheory 355 ideal tendon profile 345 loss of prestress 348 prestress loss ratio 336 strength, flexural (BS 8110) 354 strength, shear (BS 8110) 362,365 strength, torsion (BS 8110) 368

Prestressed continuous beams 380 Prestressing tendons

characteristic breaking load 334 characteristic strength 13 modulus of elasticity 361

Pre-tensioning 334 Primary moment 380 Primary tension (compression)

failure 92 Primary torsion 224 Probability 3 Program listings (see Computer

application)

Rapid-hardening Portland cement 19, 20

Rebound hardness test 45 Rectangular stress block (BS

8110) 95,96 Redistribution of moments 120, 133

moment redistribution ratio, Pb 120, 127

moment redistribution %, P% 121 Reinforcement

anchorage bond length 45, 222 areas, tables of 494 bar mark 82,214 bending dimensions (BS 4466) 496,

497 bending schedule 452

506 Index

bends 222, 495 bond,anchorage 145,221,222 bond, local 223 characteristic strength 12, 97 curtailment and anchorage 145,

146,327 density 405,406 design strength 69, 97 design stress/strain curve 69,97 design yield strain 69 detailing notation 82,214 distance between bars (max.) 144,

174,326 distance between bars (min.) 144 hooks 222, 495 lap length 144, 145 maximum steel ratio, beams 143 maximum steel ratio, columns 76 minimum steel ratio, beams 142,

143 minimum steel ratio, columns 76 minimum steel ratio, slabs 326,327 modulus of elasticity 69,72 secondary reinforcement 326, 327,

429 service stress fs 171 shear 204, 211 size of bars 73, 82, 406, 494 spacing (max.) 144,174,326 spacing (min.) 144 torsion 228, 235 transverse (flanged beams) 128,143 unit weight 405,406

Resulting moment 380 Rigid region 294, 308 Road Note No. 4 method 50 Robustness 412,429,442 Rotation vector notation 301

St Venant's torsion constant 225 Sand-heap analogy 226 Schmit rebound hammer 45 Second moment of area 161,163,407 Secondary moment 380 Secondary reinforcement 326, 327,

429 Secondary torsion 224 Serviceability !imit state 2, 156 Service stress f. 171 Setting time 19 Shear

beams, prestressed 345,392 beams, reinforced 198,209 deep beams 218

slabs 324 transfer mechanism 202

Shear centre 238 Shear compression (tension)

failure 201 Shear force envelope 215,435 Shear force tables (BS 8110)

continuous beams 410 continuous slabs 411

Shear-span/depth ratio 200 Shear reinforcement 204, 211 Shear stress Ve 210 Shear stress (max.) 210 Short columns 68, 76 Shrinkage curvature 181 Shrinkage of concrete 33, 180

loss of prestress due to 350 Sieve analysis 23 Sign convention

bending moment (sagging +ve) 333 eccentricity of tendon 334 moment vectors 296 prestressed concrete 333 rotation vectors 301 shear force (Fig. 9.2-5(b» 345 yield line 293

Simplified stress block (BS 8110) 96, 104,119

Size of reinforcement bars 73,82,406, 494

Sizing, preliminary 402 Slabs

computer-aided design (BS 8110) 486

computer application 486 design 325 design and detailing example 427,

431 design details (BS 8110) 326,328 elastic analysis 325 flexural strength (BS 8110) 292 Hillerborg's strip method 319 serviceabi!ity 325, 326 shear (BS 8110) 324 yield-line analysis 293

Siender beams 144 Siender columns

design procedure (BS 8110) 278 instability failure 274 material failure 275,286

Siender deep beams 220 Siump test 46, 47 Soundness of cement 20

Space truss analogy 228 Span/depth ratio 169,200,402 Splitting tensile strength 25 Standard deviation 6 Statistics 3, 61 Steel ratio 77, 88 Stirrups (see Links) Strength of concrete 25, 61

biaxial strength 41,42 cube strength 25 cylinder compressive strength 25 cylinder splitting strength 25 fiexural strength 25 indirect tensile strength 25,26 modulus of rupture 26 splitting tensile strength 25 tensile strength 25 triaxial strength 43 uniaxial strength 25,40

Stress block 70, 87, 94, 96 Stress/strain curves

concrete 37, 70 reinforcement 69,97

Strip method, Hillerborg's 319 Structural idealization

deep beams 220 Sub-frames 408

Target mean strength 50, 61 T-beams (see Flanged section) Tendons, eccentricity 334,336 Tension reinforcement 86 Theoretical cut-off point 145 Ties for robustness 412,429,442 Torsion

box and hollow sections 234 design details (BS 8110) 236 design method (BS 8110) 234 fianged sections 234 hollow sections 234 membrane analogy 226 plain concrete 224 prestressed concrete 368 reinforced concrete reinforcement

228,235 sand-heap analogy 226 small sections 235 space truss analogy 228 torsion - bending interaction 231 torsion-shear interaction 232

Torsional rigidity (stiffness) 234 Torsional shear stress 225,234,235 Torsion function 225 Transfer 335,346

Index 507

Transformation profile 387 Transformed section 72, 158,407 Transmission length 335 Transverse reinforcement 128, 143

fianged beams 128,143 Triaxial stress state 43 Truss analogy 206 Twisting moment in slab 295

Ultimate anchorage bond length 145, 222

Ultimate anchorage bond stress 221 Ultimate fiexural strength 87, 105, 120

deep beams 220 prestressed beams 354,355

Ultimate limit state 2, 85 Ultimate strain of concrete 71,94,96 Ultrasonic pulse method 44 Unbraced frame analysis 419 Uncracked section 163 Under-reinforced beam 89,96 Uniaxial compressive strength 25,40 Uniaxial stress state 40 Upper-bound theorem 303 Ut tensio sic vis 85

VB consistometer test 47 VBtime 47 Verulam letters 155,245 Voids in concrete 27

Water 24 free 26,50,57 evaporable 27 non-evaporable 26

Water/cement ratio 26,27,29 Web crushing 207 Web openings in deep beams 220 Web reinforcement 204,211,219 Web shear cracks 199 Whitney's stress block 93 Wind loading 419 Workability 46,49

Yield line 293, 301 positive, negative 293

Yie\d-line analysis 293 component vector method 308 concentrated load 316,317 equilibrium method 319 fan mechanism 318 interaction, top and bottom

steel 307, 329 isotropically reinforced slab 298

508 Index

Johansen's stepped yield criterion 294,296

many variables 312 nodal force method 319 orthotropical reinforcement 298 sign convention, moments 296 sign convention, rotations 301 sign convention, yield Iines 293

skew reinforcement 299,316 upper-bound theorem 303 work method 303

Young's modulus concrete 38, 180 Iightweight concrete 21 prestressing tendon 361 reinforcement 69,72