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: RM Nimbus, IBM-PC/XT, IBM-PC/AT and IBM Compatibles (e.g. Amstrad-PC) Memory 512 K bytes are more than adequate Operating system MS-DOS Disk drive Single disk drive is adequate Monitor Monochrome is adequate Printer 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/mm 2 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 Method of Oimensions Total length code measurement tobe given in of bar bar schedule
20 1-- A----I straight A h--1 32 c A+h ~{: A--I A h"'" r----.h 33 c "") A+2h vf7 A5 A 34 n'),.!:Q==== A+n 'V'~ A---I A 35 I I A+ 2n n~LA=S'n A 37 I~r AL- ~B~ A+B +C 38 !Jr r~r~ AL-.J { I--B =:rT B - r - 20
A+B+C { if CIt ~45' 41 A+B +C -r-20 { if O( > 45 o
o = bar size
h} n ond r: see Fig. A2 -1
Fig. A2-2(a) Bending dimensions-BS 4466 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 A+S-1. r -0 ~I~- standard) A~ 2 S...... l
60 2(A+B)+20 0 E] lAs] J { Â+ C 62 \~>-B ~~ 1- C--I'v~ f -. (ifa< S 45°) ,-=A:='"1..i A 81 ~: ~8 2A+3S +220 t €::f?
A 83 A+2B+C+D '* ~ ~ - 2r - 40 Isometric~ ~ 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 4-0 % x/d = 0-3 ...... _~ 12 fcu 40 2-0 % - r -b- 2 / 1 / ~II I I J I I I I I ~ 1 I I I I I I I I I I Il lL O 0-5% 1-0% 1-5 % 2-0% 2-5% 3-0% 3'5% p (= As / bd) Fig. A2-3 Beam design chart-ultimate Urnit 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 1r oAro 1 -;-35 ~N>K I IIL .~.j E 30 E ~25 fcu 40 -.&:. .o 20 fy 460 "Z- d/h 0·85 15 10 ~ -1- K~~ 1 Jt-)-,--) ---1-)--jl----II--t-)-t --J---ii 5 o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 M/bh 2 (N/mm2 ) Fig. A2-4 Column design chart-BS 8110 Index Note: Bold numbers indicate main references Actual resistance moments 154 deftection calculation (BS 8110) Aggregate interlock 202 175,182 Aggregates 21 deftection control 168,172 characteristics 22, 27 design chart (BS 8110) 98, 498 coarse 21,23 design details (BS 8110) 142 fine 21,23 design and detailing example 147, grading 23,51 432 lightweight 21 effective span 103 sizes 21,23 effective width 127,148 strength 24 elastic theory 157, 335, 382 unit weight 24 fire resistance 146 Anchorage bond length 145,222 ftanged beam 127,234,360 Anchorage bond stress 221 ftexural strength 86,89,98,105, Anchorage of links 143, 221, 236 111,124 Anchorage of tendons 335 generalftexuraltheory 86 K' (= Mul/cu bd2) 105,110,123 Balanced beam section 91,96, 106 lever-arm factor zid 89, 103, 111, Balanced failure 124 beams 91,96,106 moment redistribution 120,133 columns 252, 280 preliminary analysis and sizing 402 Balanced steel ratio 91, 97, 106 rectangular stress block 96, 104, 119 Bar bending dimensions (BS 4466) shear strength 198,209,362 496,497 simplified stress block (BS 8110) 96, Bar bending schedule 452 104,119 Bar mark 82,214 slenderness limit 144 Bar size 73, 82, 494 span/depth ratio 169,200,402 Beams .85,156 stress blocks 87, 94, 96 axialthrust 103 torsion 224, 234, 368 bar spacing 144,174 ultimate moment of resistance 87, computer-aided design (BS 8110) 105,120 475,477 Beams, prestressed (see Prestressed computer application 475, 477 beams) concrete cover 39,146 Bending and axialload continuous beams 133,409,410 beams 103 crack control 173,326 columns 248, 265 crack width calculation (BS 8110) Bending moment tab les (BS 8110) 187,188 continuous beams 410 deep beams 218 continuous slabs 411 502 Index Bending moment envelope 139,434 interaction diagram 254,266,272 Bending schedule 452 magnification factor 288 Bends and hooks 222, 495 minimum eccentricity (BS 8110) Bent-up bars 206,213 265 Biaxial bending 267, 271 preliminary analysis and sizing 402 design method (BS 8110) 267,269, short 68,76 285 slender 273,278 interaction surface 272, 273 Compacting factor 47 Biaxial stress state 40, 42 Compatibility torsion 224 Bond coefficient 221 Compression reinforcement 86,105, Bond length 222, 223 143 Bond stress 221,223 Computer application Bonded (unbonded) post-tensioned beams 475,477 beam 355 columns 475,481 Braced columns 264 mix design 473 Braced frame analysis 412 prestressed concrete 488 Buckling of deep beams 220 program listings 456,492 shear 479,489 Cantilever 146, 169 slabs 486 effective span 103 slender columns 483 Cantilever method 420 torsion 480 Cement, properties 18 Concordant tendon profile 388 types 19,32 Concrete unit weight 21 characteristic strength 12, 13 Characteristic breaking load 334 cover to reinforcement (durability) Characteristic loads 13 39 Characteristic strength 12 cover to reinforcement (fire) 78, Circular columns 287 146,328 CIRIA deep beam guide 144, 219 density 24 Coarse aggregate 23 design stress block (BS 8110) 70, Collapse mechanism 133,293 94,96 Column interaction diagram DoE mix design method 54 biaxial bending 272 durability 39 uniaxial bending 254,266, 267, 499 failure criteria 40 Columns grades 406 axially loaded 68 lightweight 21 biaxial bending 267, 269, 271 mix design and statistics 50,54,61 braced 264 modulus of elasticity 38, 180 circular 287 properties 18, 24 computer-aided design (BS 8110) Road Note No. 4 method 50 475,481 strength 25 computer application 475,481 stress block 70, 87, 94, 96 concrete cover 39, 78 ultimate strain 70,71 design charts (BS 8110) 266, 499 unit weight 24 design charts (I.Struct.E. Manual) workability 46, 49 267 Confident level, limits 10 design detailing example 78,451 Continuous beams 133,409,410 design details (BS 8110) 76,286 effective span 103 design minimum eccentricity 265 Cover to reinforcement eccentrically loaded 248,265 durability requirement 39 effective height 264, 265 fire resistance (beams) 146 fire resistance 78 fire resistance (columns) 78 footings 454 fire resistance (slabs) 328 initial crookedness 289 Crack control and width 173,187,326 Index 503 Cracked section 157 lever-arm factor zId 111.124 Creep coefficient 180 maximum bar spacing 174 Creep of concrete 28, 180 modulus of elasticity (concrete) 38 loss of prestress due to 351 neutral axis depth factor xld 111, Critical section 339 124 Cube strengh 25 partial safety factor Ye 14 Curing of concrete 28 partial safety factor Ym 15 Current margin in mix design 50,61 prestressed concrete beam 337, 347, Curtailment of bars 145,327 354 Curvature 160,164, 166, 181,372 reinforcement bar areas 494 Curvature-area theorem 175,195 shearforces (continuous beam) 410 Cylinder strength 25 shearforces (continuous slab) 411 shear links (Asv/sv) 213 I>eadload 14,402,405 shear stress Ve 210 I>eep beams span/depth ratio 169 buckling 220 torsional shear stress 235 CIRIA guide 219 ultimate anchorage and lap lengths instability 220 145 shear strength 218 I>etailing notation 82,214 slender deep beams 220 I>etailing structural idealization 220 beams 149,242,444 web openings 220 columns 81,451 I>eflection of prestressed beams 368 footings 454 long-term 371 slabs 431 short-term 369 stairs 453 I>eflection of reinforced concrete beams I>iagonal tension 198, 200 168,175 I>oE mix design method 54 long-term 175,182 I>oubly reinforced beams 98 short-term 175,182 I>owel action 202 I>esign charts (BS 8110) I>urability of concrete 38, 39 columns 266, 499 concrete cover requirement r.c. beams 98, 498 (Table 2.5-7) 39 I>esign details (BS 8110) beams 142, 144 Eccentrically loaded columns 248 columns 76,451 design chart 254, 266, 267 slabs 325,327,431 Effective depth 86 I>esign load 13 Effective height of column 264 I>esign (nominal) shear stress 199, Effective length of cantilever 103 210 Effective modulus of elasticity 72, 180 I>esign strength 13, 69 Effective prestressing force 335 I>esign tables (BS 8110) Effective span 103 areas of bars 494 continuous beams 103 bar spacing (crack control) 174 simple beams 103 bending moments (continuous beam) Effective width 127 410 Elastic theory bending moments (continuous slab) columns 71 411 prestressed beams 335,382 concrete cover (durability) 39 reinforced beams 157 concrete cover (fire) 78,146,328 shear in prestressed beams 364 effective column height 265 slabs 325 fire resistance (beams) 146 Equilibrium torsion 224 fire resistance (columns) 78 Equivalent section 72, 158 fire resistance (slabs) 328 Evaporable water 27 K' (= Mulfeubd2) 123 Exposure condition (BS8110) 40 504 Index Factor of safety 13, 14, 15 Johansen's stepped yield criterion Failure criteria, concrete 40 294,296 Fan mechanism 318 Final setting time 19 Laplength Fine aggregate 21 minimum lap length 144 Fineness of cement 19 required lap length 144,145 Fire resistance Level of significance 10 beams 146 Lever arm, factor 89,103 columns 78 BS 8110 simplified block 104,111, slabs 328 118,124 Flanged section 127 BS 8110's limit on zId 103 deftection control 169 design tables (BS 8110) 111,124 effective width 127 Lightweight aggregates 21 transverse (secondary) steel 128, Lightweight concrete 21 143 Limit design 136 Flexural stiffness (rigidity) EI 179, Limit state design 2, 15 234,369,371,407 Limits for main steel Flexure-shear cracks 200 beams 142,143 Floor load, reduction of 405 columns 76 Flowcharts (see Computer application) slabs 326, 327 Footings 454 Line of pressure (thrust) 381, 391 Frame analysis Linear transformation 387 braced 412 Links 77,143,206,211 unbraced 419 Links, anchorage of 143 Free water 50 Links, spacing 143,211,236 Full anchorage bond length 145,222 Loading 14,401 dead 14,402,405 Gel/space ratio 27 imposed 14,403,405 Grade of concrete 406 wind 14,419 Loading arrangement (BS 8110) Hardening of concrete 19 continuous beams 409 Heat of hydration 19 continuous slabs 410 Hillerborg's strip method 319 Local bond 223 Hognestad's stress block 92 Long-term deftection Hooks or bends 222,495 prestressed beams 371 reinforced beams 175,182 Ideal tendon profile 345 Loss of prestress 348 Imposed load 14,403 creep 351 reduction for ftoors 405 elastic deformation 350 Indirect tensile strength 25 multi-Ievel strands 354 Initial setting time 19 relaxation 349 Instability 144, 286 shrinkage 350 columns 275 Lower-bound theorem 319 deep beams 220 Interaction Maturity of concrete 28 bending and axialload 103 Maximum barspacing 144,173,174, biaxial bending and axialload 271 326 diagrams for columns 254, 266, 272 Maximum cement content 40 torsion and bending 231,232 Maximum moments diagram 139, 141, torsion and shear 232, 233 434 torsion, bending and shear 231,233 Maximum prestressing force 339 Interface shear transfer 202 Member sizing 402 Isotropically reinforced slabs 298 Membrane analogy for torsion 226 Index 505 Microcomputer (see Computer) Plastic design, plastic hinge 133, 136 Minimum bar spacing 144 Plastic moment of resistance 133 Minimum cement content 39 Plastic theory of structures 303,319 Minimum design dead load 14 Poisson's ratio of concrete 38 Minimum eccentricity (columns) 265 Portal method 422 Minimum lap length 144 Post-tensioning 335 Minimum links 143,210 Preliminary analysis and sizing 402 Minimum prestressing force 339 Prestressed concrete beams 333 Mix design 49 class 1, 2, 3 members 333 basic principles 50 computer application 488 DoE method 54 deflections, long-term 371 Road Note No. 4 method 50 deflections, short-term 369 Modular ratio 72, 180 design procedure 374, 393 Modulus of elasticity 38, 72 design procedure (simplified) 376 concrete 38, 180 design tab les (BS 8110) 337, 347, prestressing tendons 361 354 reinforcement bars 69,72 generalflexuraltheory 355 Modulus of rupture 26 ideal tendon profile 345 Moment-area theorem 175 loss of prestress 348 Moment-axis notation 294 prestress loss ratio 336 Moment redistribution 120, 133 strength, flexural (BS 8110) 354 redistribution ratio 120, 137 strength, shear (BS 8110) 362,365 redistribution percentage 121 strength, torsion (BS 8110) 368 Moment vector notation 296 Prestressed continuous beams 380 Prestressing tendons Neutral axis depth, factor 86,89,159 characteristic breaking load 334 BS 8110 simplified block 96,105, characteristic strength 13 111,124 modulus of elasticity 361 design tab les (BS 8110) 111,124 Pre-tensioning 334 Nominal are a 76 Primary moment 380 Nominal (concrete) cover 39,40 Primary tension (compression) Nominallinks (see Minimum links) failure 92 Nominal shear stress 199 Primary torsion 224 Non-destructive testing 44 Probability 3 Non-evaporable water 26 Program listings (see Computer Normal moment in slab 295 application) Normal probability distribution 7,8 Rapid-hardening Portland cement 19, Openings in deep beams 220 20 Ordinary Portland cement 19 Rebound hardness test 45 Orthotropically reinforced slab 298 Rectangular stress block (BS Over-reinforced beams 89,96 8110) 95,96 Redistribution of moments 120, 133 Parabola, properties of 151,152 moment redistribution ratio, Pb Partial prestressing 333 120, 127 Partially cracked section (BS 8110) moment redistribution %, P% 121 164 Reinforcement Partial safety factors 13, 14, 15 anchorage bond length 45, 222 Permeability of concrete 39 areas, tables of 494 Permissible pressure (thrust) zone 391 bar mark 82,214 Permissible shear stress 210 bending dimensions (BS 4466) 496, Permissible tendon zone 343 497 Perry-Robertson formula 289 bending schedule 452 506 Index bends 222, 495 slabs 324 bond,anchorage 145,221,222 transfer mechanism 202 bond, local 223 Shear centre 238 characteristic strength 12, 97 Shear compression (tension) curtailment and anchorage 145, failure 201 146,327 Shear force envelope 215,435 density 405,406 Shear force tables (BS 8110) design strength 69, 97 continuous beams 410 design stress/strain curve 69,97 continuous slabs 411 design yield strain 69 Shear-span/depth ratio 200 detailing notation 82,214 Shear reinforcement 204, 211 distance between bars (max.) 144, Shear stress Ve 210 174,326 Shear stress (max.) 210 distance between bars (min.) 144 Short columns 68, 76 hooks 222, 495 Shrinkage curvature 181 lap length 144, 145 Shrinkage of concrete 33, 180 maximum steel ratio, beams 143 loss of prestress due to 350 maximum steel ratio, columns 76 Sieve analysis 23 minimum steel ratio, beams 142, Sign convention 143 bending moment (sagging +ve) 333 minimum steel ratio, columns 76 eccentricity of tendon 334 minimum steel ratio, slabs 326,327 moment vectors 296 modulus of elasticity 69,72 prestressed concrete 333 secondary reinforcement 326, 327, rotation vectors 301 429 shear force (Fig. 9.2-5(b» 345 service stress fs 171 yield line 293 shear 204, 211 Simplified stress block (BS 8110) 96, size of bars 73, 82, 406, 494 104,119 spacing (max.) 144,174,326 Size of reinforcement bars 73,82,406, spacing (min.) 144 494 torsion 228, 235 Sizing, preliminary 402 transverse (flanged beams) 128,143 Slabs unit weight 405,406 computer-aided design (BS 8110) Resulting moment 380 486 Rigid region 294, 308 computer application 486 Road Note No. 4 method 50 design 325 Robustness 412,429,442 design and detailing example 427, Rotation vector notation 301 431 design details (BS 8110) 326,328 St Venant's torsion constant 225 elastic analysis 325 Sand-heap analogy 226 flexural strength (BS 8110) 292 Schmit rebound hammer 45 Hillerborg's strip method 319 Second moment of area 161,163,407 serviceabi!ity 325, 326 Secondary moment 380 shear (BS 8110) 324 Secondary reinforcement 326, 327, yield-line analysis 293 429 Siender beams 144 Secondary torsion 224 Siender columns Serviceability !imit state 2, 156 design procedure (BS 8110) 278 Service stress f. 171 instability failure 274 Setting time 19 material failure 275,286 Shear Siender deep beams 220 beams, prestressed 345,392 Siump test 46, 47 beams, reinforced 198,209 Soundness of cement 20 deep beams 218 Index 507 Space truss analogy 228 Transformation profile 387 Span/depth ratio 169,200,402 Transformed section 72, 158,407 Splitting tensile strength 25 Transmission length 335 Standard deviation 6 Transverse reinforcement 128, 143 Statistics 3, 61 fianged beams 128,143 Steel ratio 77, 88 Triaxial stress state 43 Stirrups (see Links) Truss analogy 206 Strength of concrete 25, 61 Twisting moment in slab 295 biaxial strength 41,42 cube strength 25 Ultimate anchorage bond length 145, cylinder compressive strength 25 222 cylinder splitting strength 25 Ultimate anchorage bond stress 221 fiexural strength 25 Ultimate fiexural strength 87, 105, 120 indirect tensile strength 25,26 deep beams 220 modulus of rupture 26 prestressed beams 354,355 splitting tensile strength 25 Ultimate limit state 2, 85 tensile strength 25 Ultimate strain of concrete 71,94,96 triaxial strength 43 Ultrasonic pulse method 44 uniaxial strength 25,40 Unbraced frame analysis 419 Stress block 70, 87, 94, 96 Uncracked section 163 Stress/strain curves Under-reinforced beam 89,96 concrete 37, 70 Uniaxial compressive strength 25,40 reinforcement 69,97 Uniaxial stress state 40 Strip method, Hillerborg's 319 Upper-bound theorem 303 Structural idealization Ut tensio sic vis 85 deep beams 220 Sub-frames 408 VB consistometer test 47 VBtime 47 Target mean strength 50, 61 Verulam letters 155,245 T-beams (see Flanged section) Voids in concrete 27 Tendons, eccentricity 334,336 Tension reinforcement 86 Water 24 Theoretical cut-off point 145 free 26,50,57 Ties for robustness 412,429,442 evaporable 27 Torsion non-evaporable 26 box and hollow sections 234 Water/cement ratio 26,27,29 design details (BS 8110) 236 Web crushing 207 design method (BS 8110) 234 Web openings in deep beams 220 fianged sections 234 Web reinforcement 204,211,219 hollow sections 234 Web shear cracks 199 membrane analogy 226 Whitney's stress block 93 plain concrete 224 Wind loading 419 prestressed concrete 368 Workability 46,49 reinforced concrete reinforcement 228,235 Yield line 293, 301 sand-heap analogy 226 positive, negative 293 small sections 235 Yie\d-line analysis 293 space truss analogy 228 component vector method 308 torsion - bending interaction 231 concentrated load 316,317 torsion-shear interaction 232 equilibrium method 319 Torsional rigidity (stiffness) 234 fan mechanism 318 Torsional shear stress 225,234,235 interaction, top and bottom Torsion function 225 steel 307, 329 Transfer 335,346 isotropically reinforced slab 298 508 Index Johansen's stepped yield criterion skew reinforcement 299,316 294,296 upper-bound theorem 303 many variables 312 work method 303 nodal force method 319 Young's modulus orthotropical reinforcement 298 concrete 38, 180 sign convention, moments 296 Iightweight concrete 21 sign convention, rotations 301 prestressing tendon 361 sign convention, yield Iines 293 reinforcement 69,72