F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

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Hillerborg's strip method 321indicate that the designer, using his intuition, has decided to carry all theload in the areas 1 by x-strips (i.e. strips spanning in the x-direction) and tocarry all the load in the areas 2 by y-strips. Suppose the uniformlydistributed load on the slab is of intensity q. Then a y-strip, such as A-A,will be loaded along its entire length, as shown in Fig. 8.6-2, so that thebending moment diagram is of parabolic shape with a maximum ordinateof qb 2 /8. They-strip B-B will be loaded only for a lengthy at each end andunloaded at the centre, because in the central length (b - 2y) the load iscarried by x-strips. Similarly, the x-strip C-C is loaded as shown. Thus,once the decision is made regarding the lines of discontinuity, the designerimmediately has (a) all the bending moment values with which to calculatethe required reinforcement and (b) the support reactions required fordesigning the supporting beams. But this is where design judgementcomes in. Referring to Fig. 8.6-2, the angle a defining the line of discontinuitybetween areas 1 and 2 is up to the designer to decide. If, forexample, a is made equal to 90°, then the result is a slab reinforced for onewaybending. By the lower-bound theorem, it is safe; but it is notserviceable, because excessive cracking will occur near the edges in they-direction. Hillerborg has suggested that for such a simply supported slab,the angle a should be 45°.In Fig. 8.6-2, the bending moment diagrams for typical strips are asshown. It is obviously impracticable to reinforce a slab to match thesemoments exactly. Hillerborg chose to reinforce the full length of eachstrip to withstand the maximum moment acting on it. However, even thesemaxima themselves vary with the position of the strip. For example, themaximum moment for a y-strip A-A is qb 2 /8 while that for B-B is qy 2 /2.Hillerborg decided to have strips of uniform reinforcement giving a slabyield moment equal to the average of the maximum moments found in thatstrip. Thus the slab in Fig. 8.6-3(a) might be divided into three x-stripsof widths bt. b2 and b1 respectively, and three y-strips of widths ft. 1 2 and 1 1' t ,',,'' ' ,,' ' ,,'' ,,'',4- >---------< _.,,'','~, ,'', ',, ', ' '(a)·I-X-r-~~------------~--~Bt+ ---,d' t r--~ '----,e' ,___ _!II8'3 - .. I -----------1 I -+ r---•e I---,82 :.......f.= --;•dI':'---~ .·~4s•(b)Fig. 8.6-3
322 Reinforced concrete slabs and yield-line analysisfa)®~No loadi!l!t·emStrip <D-<D10kNm~·f10kNmL§. ® ~~IJ!!l.fStrip@}@ 20kNt1m I Strip ~~20t;Ni~40kNm~mJ20kN~40kN Strip @-~ 40kN(b) Loads and momentsfor x-strips~~~40kN Strip ®-@ 40kN~r~f. 5m ~50kN50kNOne metre width ofstrip®-@(c) Loads and momentsfor y-strips50kN/m4m. :em(d) SUpport reaction edge beFig.-8.6-4(e) SUpport reactionedge ab
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Reinforced and Prestressed Concrete
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First edition 1975Reprinted three t
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viContents3 Axially loaded reinforc
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viii Contents9.5 The ultimate limit
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Preface to the Third EditionThe thi
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NotationThe symbols are essentially
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Notationxvhmax (hmin)IMMadd=larger
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Notation xvnVtminVtuVuwkwkXXtYtzZt
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Chapter 1Limit state design concept
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Statistical concepts 3occur in prac
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Statistical concepts 5results in th
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Statistical concepts 7an important
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Statistical concepts 9Example 1.3-1
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Statistical concepts 11these limits
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Partial safety factors 13proof stre
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Limit state design and the classica
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ReferencesReferences 171 Harris, Si
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Cement 19composition of Portland ce
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Aggregates 21Mortar cubes3-day comp
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Aggregates 23of concrete mainly ind
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2.5(a)Strength of concreteStrength
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Strength of concrete 21- Ordinary P
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Creep and its prediction 29..EE....
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Creep and its prediction 31humidity
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Shrinkage and its prediction 33read
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Shrinkage and its prediction 35The
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2.5(d) Elasticity and Poisson's rat
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Durability of concrete 39(a) an upp
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Failure criteria for concrete 41e.g
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Failure criteria for concrete 43v,0
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Non-destructive testing of concrete
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Assessment of workability 47fSlump
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Principles of concrete mix design 4
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Traditional mix design method 51req
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w/c ratio by weight: 0.40 3.7 3.3 2
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DoE mix design method 55(a) The mix
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DoE mix design method 57always happ
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DoE mix design method 59For a given
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Step2Statistics and target mean str
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Statistics and target mean strength
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References 65(where 1.64a is the cu
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References 6732 Shacklock, B. W. Co
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Stress/strain characteristics of st
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Real behaviour of columns 71with a
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Real behaviour of columns 73settles
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Real behaviour of columns 75ultimat
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Design details 77horizontally cast
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Design and detailing-illustrative e
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References 83(a) Bar mark[ffJ IbJBa
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Chapter4Reinforced concrete beamsth
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A general theory for ultimate flexu
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Beams with reinforcement having a d
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Beams with reinforcement having a d
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Characteristics of some proposed st
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Characteristics of some proposed st
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BS 8110 design charts-their constru
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Derivation of design formulae 105co
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Derivation of design formulae 1010
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Step(b)Find lever arm z. From Fig.
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Design procedure for rectangular be
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Design formulae and procedure-BS 81
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so thatDesign formulae and procedur
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Flanged beantS 127d' 60x = (0.45)(7
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Flanged beams 129(4.8-2)When using
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Flanged beams 131(b) If Kr of eqn (
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Moment redistribution-the fundament
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Design details (BS 81 10) 143(d) Tr
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Design details (BS 81 10) 145(a) Wh
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Problems 151effects of torsion have
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Problems 153where z values are give
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References 155theory of structures.
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Elastic theory: cracked, uncracked
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-(I)Elastic theory: cracked, uncrac
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Deflection control in design (BS 81
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Deflection control in design (BS 81
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The actual span/depth ratio = (11 m
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Calculation of short-term and long-
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StepSCalculation of short-term and
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Calculation of crack widths (BS 811
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Calculation of crack widths (BS 81
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Problems 195moment-area method, whi
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References 19719 ACI Committee 224.
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Shear failure of beams without shea
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(c)Shear failure of beams without s
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VI-iShear failure of beams without
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Effects of shear reinforcement 205F
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Effects of shear reinforcement 207A
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Shear resistance in design calculat
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Shear resistance in design ca/cu/at
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Table 6.4-2 Values of A.vl Sv (mm)f
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Shear resistance in design calculat
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From Table 6.4-2, useSize 12 links
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Shear strength of deep beams 219ver
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Bond and anchorage (BS 8110) 221lat
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Bond and anchorage (BS 8110) 223Tab
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Torsion in plain concrete beams 225
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Torsioll ill plaill cOllcrete beal1
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Effects of tors ion reinforcement 2
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Design practice (BS 8110) 231of bea
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Structural behaviour 233CUrve II(Un
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Table 6.11-1 Torsional shear stress
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Torsional resistance in design calc
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(b) From Table 6.11-1,Viu = 5 N/mm
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References 2452TVt = 2hmin[hmax - (
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References 247beams with web openin
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Principles of column interaction di
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rr-b--j_ld'• A~ •• TPrinciple
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ThereforeN = afcubh = 0.219 X 40 X
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(c)e = 200 mm. Thereforee/h = 200/5
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BS 81IO design procedure 265Table 7
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BS 8110 design procedure 267(b) In
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BS 8110 design procedure 269yIT ·l
Page 288 and 289: Biaxial bending-the technical backg
Page 290 and 291: Additional moment due to slender co
Page 296 and 297: Slender columns (BS 8110) 279height
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Page 300 and 301: M 1 = Nemin where emin = (0.05)(400
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Page 304 and 305: Problems 2877.8 Computer programs(i
Page 306 and 307: Problems 289refers to a particular
Page 308 and 309: References 29111 Beeby, A. W. The D
Page 310 and 311: Yield-line analysis 2938.2 Yield-li
Page 312 and 313: Johansen's stepped yield criterion
Page 318 and 319: 8.4 Energy dissipation in a yield l
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Page 322 and 323: Energy dissipation in a yield line
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Page 326 and 327: Energy dissipation for a rigid regi
Page 334 and 335: m 1 [projection of eh on m1 moment
Page 336 and 337: Hil/erborg's strip method 319can be
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Page 342 and 343: Design of slabs (BS 8110) 325( . .
Page 344 and 345: Design of slabs (BS 8110) 327(a) 0.
Page 346 and 347: Problems 329Problems8.1 In yield-li
Page 348 and 349: References 331(a)(b)Problem8.5reinf
Page 350 and 351: Chapter9Prestressed concrete simple
Page 352 and 353: Stresses in service: elastic theory
Page 364 and 365: Stresses at transfer 347(9.3-6)wher
Page 366 and 367: Loss of prestress 349therefore wher
Page 368 and 369: Loss of prestress 351The equilibriu
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Page 372 and 373: The ultimate limit state: flexure (
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Page 380 and 381: The ultimate limit state: shear (BS
Page 382 and 383: The ultimate limit state: shear (BS
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Short-term and long-term deflection
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Short-term and long-term deflection
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Problems 375Step3Determine the perm
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Problem 377Example 9.8-2 and compar
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References 37914 Guyon, Y. Limit-st
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Primary and secondary moments 381A~
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Analysis of prestressed continuous
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Analysis of prestressed continuous
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Linear transformation and tendon co
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Rule2Linear transformation and tend
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Applying the concept of the line of
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Summary of design procedure 393dePT
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(k:,~:J100100kN 100kN 100kN~ ~ ~ ~S
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Summary of design procedure 397(b)(
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Problems 399modulus is 78 x 10 6 mm
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Chapter 11Practical design and deta
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7000f = 460 N/mm 2yLoads-including
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Materials and practical considerati
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Analysis of framed structure (BS 81
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Braced frame analysis 41336 kN/m an
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---"'s::....I:>;:s"The symmetry of
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Table 11.4-4 Moment distributionBra
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11.4(c) Unbraced frame analysisUnbr
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Unbraced frame analysis 421IOkN J 5
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Unbraced frame analysis 423loading
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0iDesign and detailing-illustrative
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Design load F = 1.4gk + 1.6qkStep 3
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CASE IDesign and detailing-illustra
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Design and· detailing-illustrative
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leyfb = 4000/380 = 10.5 < 15Design
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Job No.: 1959/65/67Trinity and Newn
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Typical reinforcement details 453Co
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References 45512 Mayfield, B., Kong
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Program layout 457the efforts to ma
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Program layout 4596566676869707l727
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Program layout 46119819920020120220
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Program layout 46333133233333633533
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Program layout 465&65&66&67&68&69no
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599600601'602603 1000 FORMAT6046056
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How to run the programs 469numbers
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Worked example 471(2) External docu
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Program NMDDOE 473The rectanqular s
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12.3 Computer program for Chapter 3
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Program BDFLCK 477materials and the
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Program BSHEAR 479characteristic st
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Program RCIDSR 481The input data fo
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Program CTDMUB 483CommentThe comple
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12. 7(e)Program SRCRPR 485Program S
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Program SSH EAR 487123' 56789101112
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Program PBSUSH 489123'5678910111213
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References 491The input data for th
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Appendix 1 How to order program lis
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Appendix 2 Design tables and charts
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Appendix 2 Design tables and charts
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:: rn.:r'''' ',I~ '~~r I ' .., I ',
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502 IndexBending moment envelope 13
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504 IndexFactor of safety 13, 14, 1
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506 Indexbends 222, 495bond,anchora
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508 IndexJohansen's stepped yield c
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F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

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