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Worked Examples to Eurocode 2

25 February 2010

Amended 18 January 2012

Revisions required to Worked Examples to Eurocode 2

due to Amendment 1 to NA to BS EN 1992-1-1:2004 dated Dec 2009.

Amended 18/1/2012to illustrate 40K limit on l/d according to Table NA.5 Note 6

Page Where Old text Revised text

37/38 3.1.6 Allowable l/d = N u K u F1 u F2 u F3 where N = 25.6 (U= 0.41%, fck= 30) K = 1.0 (simply supported) F1 = 1.0 (beff/bw= 1.0) F2 = 1.0 (span < 7.0 m) F3 = 310/Vs” 1.5 where Vs=Vsn(As,req/As,prov) 1/G where

Vsn| 242 MPa (From Concise Figure 15.3 and gk/qk= 1.79, \2= 0.3, Jg= 1.25) G = redistribution ratio = 1.0 ?Vs | 242 u 594/645 = 222 ? F3 = 310/222 = 1.40 ? Allowable l/d = 25.6 u 1.40 = 35.8 Actual l/d = 4800/144 = 33.3 ? OK Use H12 @ 175 B1 (645 mm2/m) Allowable l/d = N u K u F1 u F2 u F3 where N = 25.6 (U= 0.41%, fck= 30) K = 1.0 (simply supported) F1 = 1.0 (beff/bw= 1.0) F2 = 1.0 (span < 7.0 m) F3 =As,prov/As,req” 1.50 = 645/ 594 = 1.09 ? Allowable l/d = 25.6 u 1.09 = 27.9 Actual l/d = 4800/144 = 33.3 ? no good Try 33.3/27.9 x 654 = 784 mm2/m i.e. H12@140 (807 mm2/m) F3 = 807/ 594 = 1.36 ? Allowable l/d = 25.6 u 1.36 = 34.8

(which is < 40K = 40) [Table NA.5 Note 6] i.e. > 33.3? OK Use H12 @ 140 B1 (807 mm2/m)

38 3.1.7 By inspection, OK

However, if considered critical:

V = 29.5 kN/m as before

VEd = 29.5 ȭ 0.14 u 12.3 = 27.8 kN/m vEd = 27.8 u 103/144 u 103= 0.19 MPa vRd,c = 0.53 MPa

By inspection, OK

However, if considered critical:

V = 29.5 kN/m as before

VEd = 29.5 ȭ 0.14 u 12.3 = 27.8 kN/m vEd = 27.8 u 103/144 u 103= 0.19 MPa U = 804/1000 x 144 =0.56%

say 50% curtailed use 0.28% vRd,c = 0.54MPa

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Vs = (fyk/JS) (As,req/As,prov) (SLS loads/ULS loads) (1/G) = fydu (As,req/As,prov) u (gk+\2 qk)/(JGgk+JQqk) (1/G) = (500/1.15) u (639/645) u [(5.9 + 0.3 u 3.3)/12.3] u 1.0811 = 434.8 u 0.99 u 0.56 u 1.08 = 260 MPa F3 = 310/260 = 1.19 N

Note: As,prov/As,reqd 1.50 Allowable l/d = N u K u F1 u F2 u F3

= 23.5 u 1.3 u 1.0 u 1.19 = 36.4

Max. span = 36.4 u 144 = 5675 mm, i.e. < 5795 mm ? No good Try H12 @ 150 B1 (754 mm2/m) Vs= 434.8 u 639/754 u 0.56 u 1.08 = 223 F3 = 310/223 = 1.39 Allowable l/d = 23.5 u 1.3 u 1.0 u 1.39 = 42.5 Max. span = 42.5 u 144 = 6120 mm, i.e. > 5795 mm OK ? H12 @ 150 B1 (754 mm2/m) OK Allowable l/d = N u K u F1 u F2 u F3 = 23.5 u 1.3 u 1.0 u1.01 = 30.8 Max. span =30.8u 144 =4435mm, i.e. < 5795 mm ? No good Try H12 @ 125 B1 (904 mm2/m) F3 = 904/639 = 1.41 Allowable l/d = 23.5 u 1.3 u 1.0 u1.41 =43.1 Max. span =43.1 u 144 =6206mm, i.e. > 5795 mm OK ? H12 @125B1 (904mm2/m) OK 43 3.2.7 F3 = 310/Vs” 1.5 where Vs = fydu (As,req/As,prov) u (gk+\2qk)/(JGgk+ JQqk) (1/G) = (500/1.15) u (465/502) u [ (5.9 + 0.3 u 3.3)/12.3] u 1.03 = 434.8 u 0.93 u 0.56 u 1.03 = 233 MPa F3 = 310/233 = 1.33 Allowable l/d = N u K u F1 u F2 u F3 = 35.8 u 1.5 u 1.0 u 1.33 = 71.4

Max. span = 71.4 u 144 = 10280 mm i.e. > 5795 mm OK F3 =As,req/As,prov” 1.5 = 502/ 465 = 1.08 Allowable l/d = N u K u F1 u F2 u F3 = 35.8 u 1.5 u 1.0 u1.08 =58.0 (which is < 40K = 40 x 1.5 =60)

[Table NA.5 Note 6]

Max. span = 58.0u 144 =8352mm i.e. > 5795 mm OK 44 3.2.9 In Figure 3.4 H12@150 In Figure 3.4 H12@125

46 3.2.10a) Crack control

As slab < 200 mm, measures to control

Crack control

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However, as a check on end span: Loading is the main cause of cracking, ? use Table 7.2N or Table 7.3N for wmax= 0.4 mm and V= 241 MPa (see deflection check). Max. bar size = 20 mm

or max. spacing = 250 mm

? H12 @ 150 B1 OK.

However, as a check on end span: Loading is the main cause of cracking, ? use Table 7.2N or Table 7.3N for wmax= 0.4 mm and V= 241 MPa (see deflection check). Max. bar size = 20 mm

or max. spacing = 250 mm

? H12 @ 125B1 OK

End supports: effects of partial fixity Try H12 @ 450 (251 mm2/m) U-bars at supports

End supports: effects of partial fixity Try H12 @ 500 (226 mm2/m)U-bars at

supports

46 3.2.10b) End span, bottom reinforcement

Try H12 @ 300 (376 mm2/m) at supports

End span, bottom reinforcement

Try H12 @ 250 (452 mm2/m) at supports

49 3.2.10b)

iii

Supporrt B bottom steel at support For convenience use H12 @ 300 B1 (376 mm2/m)

Support B bottom steel at support For convenience use H12 @250B1 (452

mm2/m)

51 3.2.11 Figure 3.6 See A below

57/58 3.3.5b) Span A®®B: Deflection F3 = 310/Vs” 1.5 where Vs = (fyk/JS) (As,req/As,prov) (SLS loads/ULS loads) (1/G) = 434.8(523/628) [ (4.30 + 0.3 u 5.0)/13.38] (65.3/61.7$) = 434.8 u 0.83 u 0.43 u 1.06 = 164 MPa F3 = 310/Vs

= 310/164 = 1.89#but ” 1.50 therefore say 1.50

? Permissible l/d = 22.8 u 1.3 u 0.8 u 0.93 u 1.50 = 33.0

Actual l/d = 7500/257 = 29.2 ? OK

Use 2 no. H20/rib (628 mm2/rib)

Span A®®B: Deflection F3 = 310/Vs” 1.5 where Vs = (fyk/JS) (As,req/As,prov) (SLS loads/ULS loads) (1/G) = 434.8(523/628) [ (4.30+ 5.0)/13.38] (65.3/61.7$) = 434.8 u 0.83 u0.70u 1.06 =267MPa F3 = 310/Vs = 310/267 =1.16 ? Permissible l/d = 22.8 u 1.3 u 0.8 u 0.93 u 1.16= 25.6 Actual l/d = 7500/257 = 29.2 ?no good Try 1H25 + 1H20/rib (805 mm2/rib)

d = 254 mm; K = 0.028; z = 241 mm Asreq= 529 mm2 ?Vs = 434.8(529/805) [ (4.30 + 5.0)/13.38] (65.3/61.7$) = 434.8 u 0.66 u 0.70 u 1.06 = 213 MPa F3 = 310/213 = 1.45 Permissible l/d = 22.8 u 1.3 u 0.8 u 0.93 u 1.45 = 32.0 Actual l/d = 7500/254 = 29.5 ? OK Use 1H25 + 1H20/rib (805 mm2/rib)

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F3 = 310/VS” 1.5 where

Vs= (fyk/Js) (As,req/As,prov) (SLS loads/ULS loads) (1/G) = 434.8 u (474/628) [(4.30 + 0.3 u 5.0)/13.38](61.1/55.9) = 434.8 u 0.75 u 0.43 u 1.09 = 153 MPa F3 = 310/Vs = 310/153 = 2.03, say = 1.5022 ? Permissible l/d = 26.8 u 1.5 u 0.8 u 0.77 u 1.50 = 37.1 Actual l/d = 9000/257 = 35 ? OK

?Use 2 H20/rib (628 mm2/rib)

F3 = 310/VS” 1.5 where

Vs= (fyk/Js) (As,req/As,prov) (SLS loads/ULS loads) (1/G) = 434.8 u (474/628) [(4.30 + 5.0) /13.38](61.1/55.9) = 434.8 u 0.75 u0.70u 1.09 =249MPa F3 = 310/Vs = 310/249= 1.24 ? Permissible l/d = 26.8 u 1.5 u 0.8 u 0.77 u1.24 = 30.7 Actual l/d = 9000/257 = 35 ? no good Try 1H25 + 1H20/rib (805 mm2/rib)

d = 254 mm; K = 0.025; z = 241 mm Asreq= 480 mm2 ?Vs = 434.8(480/805) [ (4.30 + 5.0)/13.38] (61.1/55.9) = 434.8 u 0.60 u 0.70 u 1.09 = 199 MPa F3 = 310/199 = 1.56 but ” 1.50 Permissible l/d = 26.8 u 1.5 u 0.8 u 0.77 u 1.50 = 37.1 Actual l/d = 9000/254 = 35.4 ? OK Use 1H25 + 1H20/rib (805 mm2/rib)

65 3.3.10 Figure 3.15 See B below

69 3.3.10b)

viii

Support B (and C): bottom steel curtailment B

BA and BC To suit prefabrication 2 no. H20/rib will be curtailed at solid/rib interface, 1000 mm from BA(B towards A) and BC.

Support B (and C): bottom steel curtailment B

BA and BC To suit prefabrication 1 H20 + 1H25 /rib will be curtailed at solid/rib interface, 1000 mm from BA(B towards A) and BC.

69 3.3.10c) Laps

At AB, check lap 1 no. H20 B to 2 no. H20 B in rib full tension lap:

l0=D1D6lb,rqd> l0,min where

D1 = 1.0 (cd= 45 mm, i.e. < 3M)

D6 = 1.5 (as > 50% being lapped) lb,rqd = (M/4) (Vsd/fbd) where M = 20 Vsd = 434.8 fbd = 3.0 MPa as before Laps

At AB, check lap 1 no. H20 B to 1 H20 B + 1H25 Bin rib full tension lap:

l0=D1D6lb,rqd> l0,min where

D1 = 1.0 (cd= 45 mm, i.e. < 3M)

D6 = 1.5 (as > 50% being lapped) lb,rqd = (M/4) (Vsd/fbd)

where

M = 20

Vsd = 434.8(bar assumed to be fully stressed)

fbd = 3.0 MPa as before

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Vs=Vsn(As,req/As,prov) 1/G where

Vsn= (500/1.15) u (8.5 + 0.3 u 4.0) /16.6 = 254 MPa

(or | 253 MPa (From Concise Figure 15.3 for Gk/Qk= 2.1, \2= 0.3 and Jg= 1.25) G= redistribution ratio = 1.03 ?Vs| 253 u (1324/1570)/1.03 = 207 ? F3 = 310/207 = 1.50 ? Allowable l/d = 20.3 u 1.2 u 1.50 = 36.5 Actual l/d = 9500/260 = 36.5 ? OK Use H20 @ 200 B1 (1570) Vs=Vsn(As,req/As,prov) 1/G where Vsn= (500/1.15) u (8.5 + 4.0) /16.6 = 327 MPa G=redistribution ratio = 1.03 ?Vs|327 u (1324/1570)/1.03 = 267 ? F3 = 310/267 = 1.16 ? Allowable l/d = 20.3 u 1.2 u1.16= 28.2 Actual l/d = 9500/260 = 36.5 ?no good Try H16 @100 (2010 mm2/m) and F3 = As,prov/As,req ” 1.50 = 2010/ 1340 = 1.50 ” 1.50 ? Allowable l/d = 20.3 u 1.2 u 1.50 = 36.5 Actual l/d = 36.5 ? OK Use H16@100B1 (2093 mm2/m) 78 3.4.7b) Allowable l/d = N u K u F1 u F2 u F3 where N = 26.2 (U= 0.40%, fck= 30) K = 1.2 (flat slab) F1 = 1.0 (beff/bw= 1.0)

F2 = 1.0 (no brittle partitions)

F3 = 310/Vs where

Vs=Vsn(As,req/As,prov) 1/G where

Vsn | 283 MPa (from Concise Figure 15.3 and Gk/Qk= 3.6, \2= 0.3, Jg= 1.25) G = redistribution ratio = 1.08 ?Vs| 283 u (959/1005)/1.08 = 250 ? F3 = 310/250 = 1.24 ?Allowable l/d = 26.2 u 1.2 u 1.24 = 39.0 Actual l/d = 5900/240 = 24.5 ? OK Use H16 @ 200 B2 (1005 mm2/m) Allowable l/d = N u K u F1 u F2 u F3 where N = 26.2 (U= 0.40%, fck= 30) K = 1.2 (flat slab) F1 = 1.0 (beff/bw= 1.0)

F2 = 1.0 (no brittle partitions)

F3 = As,prov/As,req ” 1.50 = 1005/ 959 = 1.05 ” 1.50 ?Allowable l/d = 26.2 u 1.2 u1.05 = 33.0 Actual l/d = 5900/240 = 24.5 ? OK Use H16 @ 200 B2 (1005 mm2/m) 90 3.4.13 Spans 1®2 and 2®3:

Column strip and middle strip: H20 @ 200 B

Spans 1®2 and 2®3:

Column strip and middle strip: H16 @ 100 B1

92 3.4.13 Figure 3.26 See D below

112 4.2.6 F3 = 310/Vs” 1.5

where

Vs in simple situations = (fyk/JS) (As,req/As,prov) (SLS loads/ULS loads) (1/G). However in this case separate analysis at SLS would be required to determine Vs.

Therefore as a simplification use the conservative assumption: 310/Vs= (500/fyk) (As,req/As,prov)

F3 = As,prov/As,req ” 1.50

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Vs=(fyk/JS)(As,req/As,prov)(SLS loads/ULS loads)(1/G) =434.8u(5835/5892) [(47.8+0.3u45.8)/(1.25 u 47.8 + 1.5 u 45.8)] u (1/0.945) = 434.8 u 0.99 u 0.48 u 1.06 = 219 MPa 124 4.3.5b) F3 = 310/Vs” 1.5 = 310/219 = 1.41

? Permissible l/d= 17.8u1.3 u0.90u0.95 u1.41 = 27.9

Actual l/d = 7500/252 = 29.8 ? no good Try 13 no. H25 B (6383 mm2) F3 = 310/Vs = 310/219 u 13/12 = 1.5333= say 1.50 ? Permissible l/d = 17.8u1.3u0.90u0.95u1.50 = 29.7 Actual leff/d = 7400/252 = 29.8 Say OK Use 13 no. H25 B (6383 mm2)

?Permissible l/d= 17.8u1.3 u0.90u0.95 u1.01 = 20.0

Actual l/d = 7500/252 = 29.8 ? no good Try 12 no. H32 B (9648 mm2) F3= As,prov/As,req ” 1.50 =9648 /5835 = 1.6544= say 1.50 ? Permissible l/d = 17.8u1.3u0.90u0.95u1.50 = 29.7 Actual leff/d = 7400/252 = 29.8 Say OK Use 13 no. H25 B (6383 mm2)

124 4.3.6 Figure 4.20 See E below

127 4.3.7d) F3 = 310/Vs” 1.5

where

Vs=(fyk/JS)(As,req/As,prov)(SLS loads/ULS loads)(1/G) =434.8u (3783/3828)[(47.8+0.3u 45.8)/(1.25 u 47.8 + 1.5 u 45.8)] u (1/0.908) = 434.8 u 0.99 u 0.48 u 1.10 = 227 MPa F3 = 310/Vs = 310/227 = 1.37 ? Permissible l/d = 44.7 u 1.37 u 0.88 u 0.95 u 1.37 = 70.1 Actual l/d = 7500/252 = 29.8 ? OK Use 8 no. H25 B (3928 mm2) F3 = As,prov/As,req ” 1.50 = 3828/3783= 1.01 ? Permissible l/d = 44.7 u 1.50u 0.88 u 0.95 u 1.01 = 56.6 (which is < 40K = 40 x 1.5 =60)

[Table NA.5 Note 6]

Actual l/d = 7500/252 = 29.8 ? OK

Use 8 no. H25 B (3928 mm2

133 4.3.10 Figures 4.22 and 4.23 See F below

146 5.3 Figure 5.6

300 mm flat slabs All columns 400 mm sq 300 mm flat slabs All columns 500mm sq

153 5.3.8 2WP]VTa`TÓ³S

183

Refer-ences

1a National Annex to Eurocode 2- Part

1-1. BSI 2005

1a National Annex to Eurocode 2- Part

1-1.Incorporating Amendment No.1BSI

2009

183

Refer-ences

5 R S NARAYANAN & C H GOODCHILD.

The Concrete Centre. Concise Eurocode 2, CCIP-005. TCC 2006

5 R S NARAYANAN & C H GOODCHILD.

The Concrete Centre. Concise Eurocode

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2010

190 B1.1 Delete existing text. See G below New text:

According to Note 5 of Table NA.5 the NA to BS EN 1992-1-1[1a]the modification factor for service stress used with the l/d method of checking the SLS state of deformation (designated factor F3 in TCC publications) may be determined as either

x 310/Vsusing characteristic load combinations to determine the service stress, Vs

or

x (500/fyk)(As,prov/As,req)

In either case the modification factor is restricted to a maximum of 1.50. In the UK

fyk= 500 MPa.

Assuming Vsis proportional to Vu, and using characteristic load combinations to determine Vsproduces values of 310/Vs= 1.00 +3% - 6%. and so is not as attractive to using As,prov/As,req.

The use of F3= As,prov/As,req ” 1.5 is therefore advocated in checking deformation using the l/d method.

190 B1.2 See H below

191 B1.4 See J below

191 B1.5 See K below

201 C7 See L below

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B Amends to p 65 Figure 3.15

C Amends to p 70 Figure 3.17

300 250 150 125 300 250 2H20/rib 1H25 + 1H20/rib x 3

Note curtailed bat

2H20/rib

1H25 + 1H20/rib x 2

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D Amends to p 92 Figure 3.26

E Amends to p 124 Figure 4.20

H20@200 H16@100 25 bar 32 bar 20 bar 20/25 bar

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G Amends to p190 B1.1

13H25 12H32 x2 T25 in section x2 NewText

According to Note 5 of Table NA.5 the NA to BS EN 1992-1-1[1a]the modification factor for service stress

used with the l/d method of checking the SLS state of deformation (designated factor F3 in TCC

publications) may be determined as either

x 310/Vsusing characteristic load combinations

to determine the service stress, Vs

or

x (500/fyk)(As,prov/As,req)

In either case the modification factor is restricted to a maximum of 1.50. In the UK

fyk= 500 MPa.

Assuming Vsis simply proportional to the ultimate

stress, Vu, produces values of 310/Vs= 1.00 +/- 5%

and so for hand calculations is not as attractive as using As,prov/As,req.

The use of F3= As,prov/As,req ” 1.5 is therefore

advocated in checking deformation using the l/d method by hand.

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H Amends to p190 B1.2

J Amends to p191 B1.4

combination values of actions [1a] [1] were

Add new text

using the quasi permanent values of actions (see B.4)

New para

In the light of the above disparity and pending clarification, the UK NA was revised and published in December 2009 as detailed in B1.1 above.

NewText

an accurate

combination values of actions taking account of SLS moments, compression reinforcement, As,prov/As,req, etc.

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L Amends to p201 C7

New paras

Prior to the publication of the revised UK NA[1a] it was allowable to calculate the moderation factor using in l/d verifications of

deformation(F3) by using quasi permanent loads.

Assuming Vsdue to quasi

permanent actions is proportional

toVu, the method as outlined in C8

may be used to determine Vsin

verifications of crack widths, etc.

New text:

either

x 310/Vsusing characteristic load

combinations to determine the service stress, Vs

or

x (500/fyk)(As,prov/As,req)

In either case the modification factor is restricted to a maximum of 1.50.

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M Amends to p204 C8

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The introduction of European standards to UK construction is a significant event as, for the first time, all design and construction codes within the EU will be harmonised. The ten design standards, known as the Eurocodes, will affect all design and construction activities as all current British Standards for structural design are due to be withdrawn in 2010.

The cement and concrete industry recognised the need to enable UK design professionals to use Eurocode 2, Design of concrete structures, quickly, effectively, efficiently and with confidence. Supported by government, consultants and relevant industry bodies, the Concrete Industry Eurocode 2 Group (CIEG) was formed in 1999 and this Group has provided the guidance for a coordinated and collaborative approach to the introduction of Eurocode 2.

As a result, a range of resources are being delivered by the concrete sector (see www.eurocode2.info). The aim of this publication, Worked Examples to Eurocode 2: Volume 1 is to distil from Eurocode 2, other Eurocodes and other sources the material that is commonly used in the design of concrete framed buildings.

These worked examples are published in two parts. Volume 2 will include chapters on Foundations, Serviceability, Fire and Retaining walls.

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The original ideas for this publication emanates from the research project `Eurocode 2: Transition from UK to European concrete design standards’, which was led by the BCA and part funded by the DTI under their PII scheme and was overseen by a Steering Group and the CIEG. The work has been brought to fruition by The Concrete Centre from early initial drafts by various authors listed on the inside back cover. The concrete industry acknowledges and appreciates the support given by many individuals, companies and organisations in the preparation of this document. These are listed on the inside back cover.

We gratefully acknowledge the authors of the initial drafts and the help and advice given by Robin Whittle in checking the text. Thanks are also due to Gillian Bond, Kevin Smith, Sally Huish and the design team at Michael Burbridge Ltd for their work on the production.

The copyright of British Standards extracts reproduced in this document is held by the British Standards Institution (BSI). Permission to reproduce extracts from British Standards is granted by BSI under the terms of Licence No: 2009RM010. No other use of this material is permitted. This publication is not intended to be a replacement for the standard and may not reflect the most up-to-date status of the standard. British Standards can be obtained in PDF or hard copy formats from the BSI online shop: http://shop.bsigroup.com or by contacting BSI Customer Services for hard copies only:

Tel:+44 (0)20 8996 9001, Email: cservices@bsigroup.com.

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All advice or information from MPA - The Concrete Centre is intended only for use in the UK by those who will evaluate the significance and limitations of its contents and take responsibility for its use and application. No liability (including that for negligence) for any loss resulting from such advice or information is accepted by MPA - The Concrete Centre or its subcontractors, suppliers or advisors. Readers should note that the publications from MPA - The Concrete Centre are subject to revision from time to time and should therefore ensure that they are in possession of the latest version.

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 Lrf[hel bb * Bgmkh]n\mbhg * *'* :bf * *'+ >nkh\h]^3;Zlblh_lmkn\mnkZe]^lb`g , *', >nkh\h]^*3:\mbhglhglmkn\mnk^l -*'- >nkh\h]^+3=^lb`gh_\hg\k^m^lmkn\mnk^l -*'. GZmbhgZe:gg^q^l . *'/ ;Zlblh_ma^phkd^]^qZfie^lbgmablin[eb\Zmbhg . *'0 :llnfimbhgl / *'1 FZm^kbZeikhi^kmb^l / *'2 >q^\nmbhg / + :gZerlbl%Z\mbhglZg]ehZ]ZkkZg`^f^gml 0 +'* F^mah]lh_ZgZerlbl 0 +'+ :\mbhgl 0 +', <aZkZ\m^kblmb\oZen^lh_Z\mbhgl 0 +'- OZkbZ[e^Z\mbhgl3bfihl^]ehZ]l 1 +'. OZkbZ[e^Z\mbhgl3lghpehZ]l *+ +'/ OZkbZ[e^Z\mbhgl3pbg]ehZ]l *, +'0 OZkbZ[e^Z\mbhgl3hma^kl *0 +'1 I^kfZg^gmZ\mbhgl *1 +'2 =^lb`goZen^lh_Z\mbhgl +* +'*) EhZ]ZkkZg`^f^gmh_Z\mbhgl3bgmkh]n\mbhg +. +'** EhZ]ZkkZg`^f^gmlZ\\hk]bg`mhma^ NDGZmbhgZe:gg^qmh>nkh\h]^ +. +'*+ >qZfie^lh_ehZ]bg` +0 , LeZ[l ,. ,') @^g^kZe ,. ,'* Lbfierlniihkm^]hg^&pZrleZ[ ,/ ,'+ <hgmbgnhnlhg^&pZrlheb]leZ[ -) ,', <hgmbgnhnlkb[[^]leZ[ .+ ,'- ?eZmleZ[ 0* ,'. LmZbk_eb`am 2. - ;^Zfl 20 -') @^g^kZe 20 -'* <hgmbgnhnl[^Zfhgibglniihkml 21 -'+ A^ZoberehZ]^]E&[^Zf *)--', <hgmbgnhnlpb]^M&[^Zf **2 . <henfgl *,-.') @^g^kZe *,-.'* >]`^\henfg *,. .'+ I^kbf^m^k\henfg!bgm^kgZe^gobkhgf^gm" *,2 .', Bgm^kgZe\henfg *-/ .'- LfZeei^kbf^m^k\henfgln[c^\m mhmph&ahnk_bk^k^lblmZg\^ *.0 / PZeel *// /') @^g^kZe *// /'* La^ZkpZee */0 0 K^_^k^g\^lZg]_nkma^kk^Z]bg` *1, :ii^g]bq:3 =^kbo^]_hkfneZ^ *1. :ii^g]bq;3 L^kob\^Z[bebmrebfbmlmZm^ *2) :ii^g]bq<3 =^lb`gZb]l

*2-Ohenf^+

!in[ebla^]l^iZkZm^er" ?hng]Zmbhgl Li^\bZe]^mZbel3lmknmmb^  L^kob\^Z[bebmr  ?bk^  K^mZbgbg`pZee  K^_^k^g\^lZg]_nkma^kk^Z]bg`  :ii^g]bq:3=^kbo^]_hkfneZ^  :ii^g]bq;3=^lb`gZb]l 

(18)

ii

Symbols and abbreviations used in this publication

Symbol Definition : <khll&l^\mbhgZeZk^Z4:\\b]^gmZeZ\mbhg : Lbm^Zembmn]^%f!lghp" : :embmn]^h_ma^lbm^bgf^mk^lZ[ho^f^Zgl^Ze^o^e!pbg]" :%;%< OZkbZ[e^lnl^]bgma^]^m^kfbgZmbhgh_lebf :\ <khll&l^\mbhgZeZk^Zh_\hg\k^m^ :] =^lb`goZen^h_ZgZ\\b]^gmZeZ\mbhg :>] =^lb`goZen^h_Zl^blfb\Z\mbhg :k^_ K^_^k^g\^Zk^Zh_ma^lmkn\mnk^hklmkn\mnkZe^e^f^gm!pbg]" :l <khll&l^\mbhgZeZk^Zh_k^bg_hk\^f^gm :l%fbg Fbgbfnf\khll&l^\mbhgZeZk^Zh_k^bg_hk\^f^gm :l%ikho :k^Zh_lm^^eikhob]^] :l%k^j :k^Zh_lm^^ek^jnbk^] :l* :k^Zh_k^bg_hk\bg`lm^^ebgeZr^k* :l+ :k^Zh_\hfik^llbhglm^^e!bgeZr^k+" :le :k^Zh_ma^m^glbe^k^bg_hk\^f^gm^qm^g]bg`Zme^Zlme[]$][^rhg]ma^l^\mbhg\hglb]^k^] :lF!:lG" MhmZeZk^Zh_k^bg_hk\^f^gmk^jnbk^]bglrff^mkb\Ze%k^\mZg`neZk\henfglmhk^lblmfhf^gm!ZqbZeehZ]"nlbg` lbfieb_b^]\Ze\neZmbhgf^mah] :lp <khll&l^\mbhgZeZk^Zh_la^Zkk^bg_hk\^f^gm4:k^Zh_ing\abg`la^Zkk^bg_hk\^f^gmbghg^i^kbf^m^kZkhng]ma^ \henfg :lp%fbg Fbgbfnf\khll&l^\mbhgZeZk^Zh_la^Zkk^bg_hk\^f^gm4FbgbfnfZk^Zh_ing\abg`la^Zkk^bg_hk\^f^gmbghg^ i^kbf^m^kZkhng]ma^\henfg :m :k^Zh_m^glbe^k^bg_hk\^f^gmbg_eZmleZ[\henfglmkbil Z =blmZg\^%ZeehpZg\^Zmlniihkml Z :qbl]blmZg\^_khfma^\hg\k^m^lnk_Z\^mhma^\^gmk^h_ma^[Zk!_bk^" Z :g^qihg^gm!bg\hglb]^kbg`[bZqbZe[^g]bg`h_\henfgl" Z Ikhc^\mbhgh_ma^_hhmbg`_khfma^_Z\^h_ma^\henfghkpZee Ze =blmZg\^[rpab\ama^eh\Zmbhgpa^k^Z[Zkblghehg`^kk^jnbk^]_hk[^g]bg`fhf^gmbl]blieZ\^]mhZeehp_hkma^ _hk\^l_khfma^mknllfh]^e_hkla^Zk'!ÂLab_mÃ]blmZg\^_hk\nkmZbef^gm" Z*%Z+% =blmZg\^_khf^]`^h_lniihkmmh\^gmk^h_lniihkm Z*%[* =bf^glbhglh_ma^\hgmkhei^kbf^m^kZkhng]Zg^ehg`Zm^]lniihkm!ing\abg`la^Zk" Z'f'l'e' :embmn]^Z[ho^f^Zgl^Ze^o^e [ Ho^kZeepb]mah_Z\khll&l^\mbhg%hk_eZg`^pb]mabgZM&hkE&[^Zf [ ;k^Z]mah_[nbe]bg`!pbg]" [^ >__^\mbo^pb]mah_Z_eZmleZ[!Z]cZ\^gmmhi^kbf^m^k\henfg" [^__ >__^\mbo^pb]mah_Z_eZg`^ [^j!a^j" >jnboZe^gmpb]ma!a^b`am"h_\henfg6[!a"_hkk^\mZg`neZkl^\mbhgl [fbg Fbgbfnfpb]mah_p^[hgM&%B&hkE&[^Zfl [m F^Zgpb]mah_ma^m^glbhgshg^'?hkZM&[^Zfpbmama^_eZg`^bg\hfik^llbhg%hgerma^pb]mah_ma^p^[blmZd^g bgmhZ\\hngm [p Pb]mah_ma^p^[hgM&%B&hkE&[^Zfl'Fbgbfnfpb]ma[^mp^^gm^glbhgZg]\hfik^llbhg\ahk]l [* AZe_h_]blmZg\^[^mp^^gZ]cZ\^gmp^[lh_]hpglmZg][^Zfl <^ >qihlnk^\h^__b\b^gm!lghp" <m Ma^kfZe\h^__b\b^gm!lghp" <p La^Zk\^gmk^

(19)

la^Zk" \Zem :embmn]^_Z\mhk!pbg]" \] =rgZfb\_Z\mhk!pbg]" \]bk =bk^\mbhgZe_Z\mhk!pbg]" \^%_eZm >qihlnk^_Z\mhk!pbg]" \_ ?hk\^\h^__b\b^gm!pbg]" \fbg Fbgbfnf\ho^k%!]n^mhma^k^jnbk^f^gml_hk[hg]%\fbg%[hk]nkZ[bebmr\fbg%]nk" \ghf GhfbgZe\ho^k'GhfbgZe\ho^klahne]lZmbl_rma^fbgbfnfk^jnbk^f^gmlh_[hg]%]nkZ[bebmrZg]_bk^ \i^ !>qm^kgZe"ik^llnk^\h^__b\b^gm!pbg]" \i^%*) !>qm^kgZe"ik^llnk^\h^__b\b^gm_hkZk^Zl7*f+!pbg]" \ib Bgm^kgZeik^llnk^\h^__b\b^gm!pbg]" \ikh[ Ikh[Z[bebmr_Z\mhk!pbg]" \l^Zlhg L^Zlhg_Z\mhk!pbg]" \l Lbs^_Z\mhk!pbg]" \r%\q <henfg]bf^glbhglbgieZg D\]^o :eehpZg\^fZ]^bg]^lb`g_hk]^obZmbhg = =bZf^m^kh_Z\bk\neZk\henfg4=bZf^m^k ] >__^\mbo^]^imamhm^glbhglm^^e ]+ >__^\mbo^]^imamh\hfik^llbhglm^^e ]\ >__^\mbo^]^imah_\hg\k^m^bg\hfik^llbhg > >__^\mh_Z\mbhg4Bgm^`kbmr!bg_bk^"4>eZlmb\fh]nenl >\] =^lb`goZen^h_fh]nenlh_^eZlmb\bmrh_\hg\k^m^ >\f L^\Zgmfh]nenlh_^eZlmb\bmrh_\hg\k^m^ >I ;^g]bg`lmb__g^ll >l =^lb`goZen^h_fh]nenlh_^eZlmb\bmrh_k^bg_hk\bg`lm^^e >qi' >qik^llbhg >JN LmZmb\^jnbeb[kbnf ^ >\\^gmkb\bmr ^) Fbgbfnf^\\^gmkb\bmrbg\henfgl ^+ =^_e^\mbhg!nl^]bgZll^llbg`F+bgle^g]^k\henfgl" ^b >\\^gmkb\bmr]n^mhbfi^k_^\mbhgl ^r%^s >\\^gmkb\bmr%F>](O>]Zehg`rZg]sZq^lk^li^\mbo^er!ing\abg`la^Zk" ?* ?Z\mhkmhZ\\hngm_hk_eZg`^]l^\mbhgl!]^_e^\mbhg" ?+ ?Z\mhkmhZ\\hngm_hk[kbmme^iZkmbmbhglbgZllh\bZmbhgpbmaehg`liZgl!]^_e^\mbhg" ?, ?Z\mhkmhZ\\hngm_hkl^kob\^lmk^llbgm^glbe^k^bg_hk\^f^gm!]^_e^\mbhg" ? :\mbhg ?>F ?bq^]^g]fhf^gm ?\!?l" ?hk\^bg\hg\k^m^!lm^^e" ?] =^lb`goZen^h_ZgZ\mbhg ?> M^glbe^_hk\^bgk^bg_hk\^f^gmmh[^Zg\ahk^] ?d <aZkZ\m^kblmb\oZen^h_ZgZ\mbhg

?k^i K^ik^l^gmZmbo^Z\mbhg!6r?dpa^k^r6_Z\mhkmh\hgo^km\aZkZ\m^kblmb\mhk^ik^l^gmZmbo^Z\mbhg" ?l M^glbe^_hk\^bgk^bg_hk\^f^gm

?m] =^lb`goZen^h_ma^m^glbe^_hk\^bgehg`bmn]bgZek^bg_hk\^f^gm

(20)

Symbol Definition ?O%>] MhmZeo^kmb\ZeehZ]!hg[kZ\^]Zg][kZ\bg`f^f[^kl" ?p K^lnemZgm\aZkZ\m^kblmb\_hk\^]n^mhpbg]'!L^^l^\mbhg+'/" _[] NembfZm^[hg]lmk^ll _\] =^lb`goZen^h_\hg\k^m^\hfik^llbo^lmk^g`ma _\d <aZkZ\m^kblmb\\hfik^llbo^\rebg]^klmk^g`mah_\hg\k^m^Zm+1]Zrl _\m%] =^lb`gm^glbe^lmk^g`mah_\hg\k^m^!a\m_\m%d(g<" _\m%d <aZkZ\m^kblmb\ZqbZem^glbe^lmk^g`mah_\hg\k^m^ _\mf F^ZgoZen^h_ZqbZem^glbe^lmk^g`mah_\hg\k^m^ _l\ <hfik^llbo^lmk^llbg\hfik^llbhgk^bg_hk\^f^gmZmNEL _r] =^lb`grb^e]lmk^g`mah_ehg`bmn]bgZek^bg_hk\^f^gm%:le _rd <aZkZ\m^kblmb\rb^e]lmk^g`mah_k^bg_hk\^f^gm _rp] =^lb`grb^e]lmk^g`mah_ma^la^Zkk^bg_hk\^f^gm _rp]%^_ >__^\mbo^]^lb`glmk^g`mah_ing\abg`la^Zkk^bg_hk\^f^gm _rpd <aZkZ\m^kblmb\rb^e]lmk^g`mah_la^Zkk^bg_hk\^f^gm @d <aZkZ\m^kblmb\oZen^h_Zi^kfZg^gmZ\mbhg @d%lni Nii^k\aZkZ\m^kblmb\oZen^h_Zi^kfZg^gmZ\mbhg @d%bg_ Ehp^k\aZkZ\m^kblmb\oZen^h_Zi^kfZg^gmZ\mbhg `d <aZkZ\m^kblmb\oZen^h_Zi^kfZg^gmZ\mbhgi^kngbme^g`mahkZk^Z Ab AhkbshgmZeZ\mbhgZiieb^]ZmZe^o^e A A^b`amh_[nbe]bg`!pbg]" a Ho^kZee]^imah_Z\khll&l^\mbhg4A^b`am aZo^ H[lmkn\mbhga^b`am!pbg]" a]bl =blieZ\^f^gma^b`am!pbg]" a_ =^imah__hhmbg`4Mab\dg^llh__eZg`^ al =^imah_leZ[ I L^\hg]fhf^gmh_Zk^Zh_\hg\k^m^l^\mbhg4Bg^kmbZ I BglneZmbhg!bg_bk^" b KZ]bnlh_`rkZmbhg D F>]([]+_ \d':f^Zlnk^h_ma^k^eZmbo^\hfik^llbo^lmk^llbgZf^f[^kbg_e^qnk^ D ?Z\mhkmhZ\\hngm_hklmkn\mnkZelrlm^f!]^_e^\mbhg" D  OZen^h_DZ[ho^pab\a\hfik^llbhgk^bg_hk\^f^gmblk^jnbk^] Do :\hkk^\mbhg_Z\mhk_hkZqbZeehZ] Df :\hkk^\mbhg_Z\mhk_hk\k^^i d <h^__b\b^gmhk_Z\mhk d K^eZmbo^_e^qb[bebmrhkk^eZmbo^lmb__g^ll e <e^Zka^b`amh_\henfg[^mp^^g^g]k^lmkZbgml e A^b`amh_ma^lmkn\mnk^bgf^mk^l e!hkE" E^g`ma4LiZg e) >__^\mbo^e^g`ma!h_\henfgl" e) =blmZg\^[^mp^^gihbgmlh_s^khfhf^gm e) =^lb`geZie^g`ma e)%_b >__^\mbo^e^g`mang]^k_bk^\hg]bmbhgl e[ ;Zlb\Zg\ahkZ`^e^g`ma e[] =^lb`gZg\ahkZ`^e^g`ma e[%^j >jnboZe^gmZg\ahkZ`^e^g`ma e[%fbg FbgbfnfZg\ahkZ`^e^g`ma iv

(21)

e^__ >__^\mbo^liZg eg <e^ZkliZg er%es LiZglh_Zmph&pZrleZ[bgma^rZg]s]bk^\mbhgl F ;^g]bg`fhf^gm'Fhf^gm_khf_bklmhk]^kZgZerlbl F  Fhf^gm\ZiZ\bmrh_Zlbg`erk^bg_hk\^]l^\mbhg!Z[ho^pab\a\hfik^llbhgk^bg_hk\^f^gmblk^jnbk^]" F)%>ji ?bklmhk]^k[^g]bg`fhf^gmbgjnZlbi^kfZg^gmehZ]\hf[bgZmbhg!LEL" F)*%F)+ ?bklmhk]^k^g]fhf^gmlZmNELbg\en]bg`ZeehpZg\^l_hkbfi^k_^\mbhgl F)>] >jnboZe^gm_bklmhk]^kfhf^gmbg\en]bg`ma^^__^\mh_bfi^k_^\mbhgl!ZmZ[hnmfb]a^b`am" F)>]%_b ?bklmhk]^kfhf^gmng]^k_bk^\hg]bmbhgl F+ GhfbgZel^\hg]hk]^kfhf^gmbgle^g]^k\henfgl F>] =^lb`goZen^h_ma^Ziieb^]bgm^kgZe[^g]bg`fhf^gm F>]r%F>]s =^lb`gfhf^gmbgma^k^li^\mbo^]bk^\mbhg FK]r%FK]s Fhf^gmk^lblmZg\^bgma^k^li^\mbo^]bk^\mbhg Fm =^lb`gmkZgl_^kfhf^gmmh\henfg_khfZ_eZmleZ[ f Gnf[^kh_o^kmb\Zef^f[^kl\hgmkb[nmbg`mhZg^__^\m f FZll G :qbZe_hk\^ G ;Zlb\liZg&mh&^__^\mbo^&]^imakZmbh%e(]%_hkD6*') G)>]%_b :qbZeehZ]ng]^k_bk^\hg]bmbhgl G: GZmbhgZe:gg^q GZ%G[ Ehg`bmn]bgZe_hk\^l\hgmkb[nmbg`mhAb G>] =^lb`goZen^h_ma^Ziieb^]ZqbZe_hk\^!m^glbhghk\hfik^llbhg"ZmNEL G=I GZmbhgZeer=^m^kfbg^]IZkZf^m^k!l"Zlin[ebla^]bgZ\hngmkrÃlGZmbhgZe:gg^q g EhZ]e^o^eZmghkfZem^fi^kZmnk^l'<hgl^koZmbo^erg6)'0!_bk^" g :qbZelmk^llZmNEL g NembfZm^Z\mbhg!ehZ]"i^kngbme^g`ma!hkZk^Z" g K^eZmbo^ZqbZe_hk\^G>](!:\_\]" g[Ze Ma^oZen^h_gZmfZqbfnffhf^gmk^lblmZg\^ g)%gl Gnf[^kh_lmhk^rl Jd <aZkZ\m^kblmb\oZen^h_ZoZkbZ[e^Z\mbhg Jd*!Jdb" <aZkZ\m^kblmb\oZen^h_Ze^Z]bg`oZkbZ[e^Z\mbhg!<aZkZ\m^kblmb\oZen^h_ZgZ\\hfiZgrbg`oZkbZ[e^Z\mbhg" jd <aZkZ\m^kblmb\oZen^h_ZoZkbZ[e^Z\mbhgi^kngbme^g`mahkZk^Z j[ ;Zlb\pbg]ik^llnk^ ji I^Zdpbg]ik^llnk^ ji!s^" I^Zdo^eh\bmrik^llnk^Zmk^_^k^g\^a^b`ams^%!pbg]" K K^lblmZg\^4F^\aZgb\Zek^lblmZg\^!bg_bk^" K: K^Z\mbhgZmlniihkm: K; K^Z\mbhgZmlniihkm; K] =^lb`goZen^h_ma^k^lblmZg\^mhZgZ\mbhg k KZ]bnl kf KZmbhh__bklmhk]^k^g]fhf^gmlbg\henfglZmNEL LEL L^kob\^Z[bebmrebfbmlmZm^!l"¾\hkk^lihg]bg`mh\hg]bmbhgl[^rhg]pab\ali^\b_b^]l^kob\^k^jnbk^f^gmlZk^gh ehg`^kf^m l LiZ\bg` l LghpehZ]hgZkhh_

(22)

Symbol Definition ld <aZkZ\m^kblmb\`khng]lghpehZ] lk KZ]bZeliZ\bg`h_i^kbf^m^klh_la^Zkk^bg_hk\^f^gm lm MZg`^gmbZeliZ\bg`la^Zkk^bg_hk\^f^gmZehg`i^kbf^m^klh_la^Zkk^bg_hk\^f^gm m Mab\dg^ll4Mbf^[^bg`\hglb]^k^]4;k^Z]mah_lniihkm m) Ma^Z`^h_\hg\k^m^Zmma^mbf^h_ehZ]bg` NEL NembfZm^ebfbmlmZm^!l"¾Zllh\bZm^]pbma\heeZil^hkhma^k_hkflh_lmkn\mnkZe_Zbenk^ n I^kbf^m^kh_\hg\k^m^\khll&l^\mbhg%aZobg`Zk^Z:\ n I^kbf^m^kh_maZmiZkmpab\abl^qihl^]mh]krbg` n <bk\nf_^k^g\^h_hnm^k^]`^h_^__^\mbo^\khll&l^\mbhg!mhklbhg" n) I^kbf^m^kZ]cZ\^gmmh\henfgl!ing\abg`la^Zk" n* ;Zlb\\hgmkhei^kbf^m^k!Zm+]_khf_Z\^h_ehZ]"!ing\abg`la^Zk" n*# K^]n\^]\hgmkhei^kbf^m^kZmi^kbf^m^k\henfgl!Zm+]_khf_Z\^h_ehZ]"!ing\abg`la^Zk" nb E^g`mah_ma^\hgmkhei^kbf^m^kng]^k\hglb]^kZmbhg!ing\abg`la^Zk" nhnm I^kbf^m^kZmpab\ala^Zkk^bg_hk\^f^gmblghehg`^kk^jnbk^] O La^Zk_hk\^ O>] =^lb`goZen^h_ma^Ziieb^]la^Zk_hk\^ OK]%\ La^Zkk^lblmZg\^h_Zf^f[^kpbmahnmla^Zkk^bg_hk\^f^gm OK]%fZq La^Zkk^lblmZg\^h_Zf^f[^kebfbm^][rma^\knlabg`h_\hfik^llbhglmknml OK]%\fbg Fbgbfnfla^Zkk^lblmZg\^h_f^f[^k\hglb]^kbg`\hg\k^m^Zehg^ OK]%l La^Zkk^lblmZg\^h_Zf^f[^k`ho^kg^][rma^rb^e]bg`h_la^Zkk^bg_hk\^f^gm o[ ;Zlb\pbg]o^eh\bmr o[%) Ma^_ng]Zf^gmZe[Zlb\pbg]o^eh\bmr[^bg`ma^\aZkZ\m^kblmb\*)fbgnm^pbg]o^eh\bmrZm*)fZ[ho^`khng]e^o^ebg hi^g\hngmkr o[%fZi ?ng]Zf^gmZe[Zlb\pbg]o^eh\bmr_khf?b`nk^G:'*f(l o>] Ing\abg`la^Zklmk^ll o>] La^Zklmk^ll_hkl^\mbhglpbmahnmla^Zkk^bg_hk\^f^gm!6O>]([p]" o>]%s La^Zklmk^ll_hkl^\mbhglpbmala^Zkk^bg_hk\^f^gm!6O>]([ps6O>]([p)'2]" oK]%\ =^lb`gla^Zkk^lblmZg\^h_\hg\k^m^pbmahnmla^Zkk^bg_hk\^f^gm^qik^ll^]ZlZlmk^ll oK]%fZq <ZiZ\bmrh_\hg\k^m^lmknml^qik^ll^]ZlZlmk^ll P* ?Z\mhk\hkk^lihg]bg`mhZ]blmkb[nmbhgh_la^Zk!ing\abg`la^Zk" P^ I^Zd^qm^kgZepbg]ehZ] Pd <aZkZ\m^kblmb\oZen^h_pbg]Z\mbhg!G;ghmbgma^>nkh\h]^lZg]lahne][^k^`Zk]^]ZlZ_hkfh_Jd% \aZkZ\m^kblmb\oZen^h_ZoZkbZ[e^Z\mbhg" pd <aZkZ\m^kblmb\ngbmpbg]ehZ]' pd <kZ\dpb]ma pfZq Ebfbmbg`\Ze\neZm^]\kZ\dpb]ma Q)%Q:%Q< <hg\k^m^^qihlnk^\eZll^l Q=%Q?%QL q G^nmkZeZqbl]^ima q =blmZg\^[^mp^^g[nbe]bg`l!pbg]" q =blmZg\^h_ma^l^\mbhg[^bg`\hglb]^k^]_khfma^\^gmk^ebg^h_ma^lniihkm q%r%s <h&hk]bgZm^l4IeZg^lng]^k\hglb]^kZmbhg qn =^imah_ma^g^nmkZeZqblZmma^nembfZm^ebfbmlmZm^Z_m^kk^]blmkb[nmbhg S Shg^gnf[^kh[mZbg^]_khffZi!lghp" s E^o^kZkfh_bgm^kgZe_hk\^l vi

(23)

s^ K^_^k^g\^a^b`am_hkpbg]pZk]pZeelh_k^\mZg`neZk[nbe]bg`l!pbg]" a :g`e^4:g`e^h_la^Zkebgdlmhma^ehg`bmn]bgZeZqbl4KZmbh

a: :\h^__b\b^gm_hknl^pbmaZk^ik^l^gmZmbo^oZkbZ[e^Z\mbhgmZdbg`bgmhZ\\hngmZk^Zlniihkm^] a*%a+%a, ?Z\mhkl]^Zebg`pbmaZg\ahkZ`^Zg]eZilh_[Zkl

a-%a.%a/ a\\!a\m" :\h^__b\b^gmmZdbg`bgmhZ\\hngmehg`m^kf^__^\mlh_\hfik^llbo^!m^glbe^"ehZ]Zg]ma^pZrehZ]blZiieb^] a^ Fh]neZkkZmbh6>l(>\] ag :\h^__b\b^gm_hknl^pbmaZk^ik^l^gmZmbo^oZkbZ[e^Z\mbhgmZdbg`bgmhZ\\hngmgnf[^kh_lmhk^rllniihkm^] b :g`e^4KZmbh4<h^__b\b^gm b ?Z\mhk]^Zebg`pbma^\\^gmkb\bmr!ing\abg`la^Zk" g IZkmbZe_Z\mhk g< IZkmbZe_Z\mhk_hk\hg\k^m^ g? IZkmbZe_Z\mhk_hkZ\mbhgl%? g@ IZkmbZe_Z\mhk_hki^kfZg^gmZ\mbhgl%@ g@d%lni IZkmbZe_Z\mhkmh[^Ziieb^]mh@d%bg_ g@d%bg_ IZkmbZe_Z\mhkmh[^Ziieb^]mh@d%lni gJ IZkmbZe_Z\mhk_hkoZkbZ[e^Z\mbhgl%J gF IZkmbZe_Z\mhk_hkfZm^kbZe!nlnZeerg<hkgL" gL IZkmbZe_Z\mhk_hkk^bg_hk\bg`lm^^e d K^]blmkb[nmbhgkZmbh^jnZelkZmbhh_ma^k^]blmkb[nm^]fhf^gmmhma^^eZlmb\[^g]bg`fhf^gm!*¾k^]blmkb[nmbhg" e\n NembfZm^\hfik^llbo^lmkZbgbgma^\hg\k^m^ e\n+ NembfZm^\hfik^llbo^lmkZbgebfbmbg\hg\k^m^pab\ablghm_neerbgink^ZqbZe\hfik^llbhgZllnfbg`nl^h_ma^ iZkZ[heb\¾k^\mZg`neZklmk^ll¾lmkZbgk^eZmbhglabi!gnf^kb\Zeere\n+6e\n," e\n, NembfZm^\hfik^llbo^lmkZbgebfbmbg\hg\k^m^pab\ablghm_neerbgink^ZqbZe\hfik^llbhgZllnfbg`nl^h_ma^ [bebg^Zklmk^ll¾lmkZbgk^eZmbhglabi el\ <hfik^llbo^lmkZbgbgk^bg_hk\^f^gm elm M^glbe^lmkZbgbgk^bg_hk\^f^gm n ?Z\mhk]^_bgbg`^__^\mbo^lmk^g`ma!6*_hk©<.)(/)" n* <h^__b\b^gm_hk[hg]\hg]bmbhgl n+ <h^__b\b^gm_hk[Zk]bZf^m^k y :g`e^4:g`e^h_\hfik^llbhglmknml!la^Zk" yb Bg\ebgZmbhgnl^]mhk^ik^l^gmbfi^k_^\mbhgl l Le^g]^kg^llkZmbh l ?Z\mhk]^_bgbg`ma^a^b`amh_ma^\hfik^llbhgshg^!6)'1_hk©<.)(/)" l_b Le^g]^kg^llbg_bk^ lebf Ebfbmbg`le^g]^kg^llkZmbh!h_\henfgl" mb%m*%m+ LghpehZ]laZi^_Z\mhkl m_b KZmbhh_ma^]^lb`gZqbZeehZ]ng]^k_bk^\hg]bmbhglmhma^]^lb`gk^lblmZg\^h_ma^\henfgZmghkfZem^fi^kZmnk^ [nmpbmaZg^\\^gmkb\bmrZiieb\Z[e^mh_bk^\hg]bmbhgl v Lmk^g`mak^]n\mbhg_Z\mhk_hk\hg\k^m^\kZ\d^]bgla^Zk j K^]n\mbhg_Z\mhk(]blmkb[nmbhg\h^__b\b^gm'?Z\mhkZiieb^]mh@dbg;L>G*22)>qi'!/'*)[" r K^jnbk^]m^glbhgk^bg_hk\^f^gmkZmbh r =^glbmrh_Zbk!pbg]" r  K^bg_hk\^f^gmkZmbh_hkk^jnbk^]\hfik^llbhgk^bg_hk\^f^gm%:l+([]

(24)

viii Symbol Definition r* I^k\^gmZ`^h_k^bg_hk\^f^gmeZii^]pbmabg)'/.e)_khfma^\^gmk^ebg^h_ma^eZi[^bg`\hglb]^k^] re K^bg_hk\^f^gmkZmbh_hkehg`bmn]bgZek^bg_hk\^f^gm rer%res K^bg_hk\^f^gmkZmbhh_[hg]^]lm^^ebgma^rZg]s]bk^\mbhg r) K^_^k^g\^k^bg_hk\^f^gmkZmbh_\d)'.£*)¾, s`] =^lb`goZen^h_ma^`khng]ik^llnk^ sl Lmk^llbgk^bg_hk\^f^gmZmLEL sl :[lhenm^oZen^h_ma^fZqbfnflmk^lli^kfbmm^]bgma^k^bg_hk\^f^gmbff^]bZm^erZ_m^kma^_hkfZmbhgh_ma^\kZ\d sl\!slm" Lmk^llbg\hfik^llbhg!Zg]m^glbhg"k^bg_hk\^f^gm sl] =^lb`glmk^llbgma^[ZkZmma^nembfZm^ebfbmlmZm^ sln Ngfh]b_b^]l^kob\^lmk^llbgk^bg_hk\^f^gm]^m^kfbg^]_khfNELehZ]l!L^^?b`nk^<," h!e%m)" ?bgZeoZen^h_\k^^i\h^__b\b^gm h^_ >__^\mbo^\k^^i_Z\mhk f ;Zk]bZf^m^k c ?Z\mhkl]^_bgbg`k^ik^l^gmZmbo^oZen^lh_oZkbZ[e^Z\mbhgl c) <hf[bgZmbhgoZen^h_ZoZkbZ[e^Z\mbhg!^'`'nl^]pa^g\hglb]^kbg`NEL" c* ?k^jn^gmoZen^h_ZoZkbZ[e^Z\mbhg!^'`'nl^]pa^g\hglb]^kbg`pa^ma^kl^\mbhgpbeeaZo^\kZ\d^]hkghm" c+ JnZlb&i^kfZg^gmoZen^h_ZoZkbZ[e^Z\mbhg!^'`'nl^]pa^g\hglb]^kbg`]^_hkfZmbhg" w F^\aZgb\Zek^bg_hk\^f^gmkZmbh6:l_r](:\_\]©*

(25)

Introduction

Aim

Ma^Zbfh_mablin[eb\ZmbhgblmhbeenlmkZm^makhn`aphkd^]^qZfie^lahp;L>G*22+¾*¾*T*V !>nkh\h]^+"fZr[^nl^]bgikZ\mb\^mh]^lb`gbg&lbmn\hg\k^m^[nbe]bg`lmkn\mnk^l'Bmblbgm^g]^] maZm ma^l^ phkd^] ^qZfie^l pbee ^qieZbg ahp \Ze\neZmbhgl mh >nkh\h]^ + fZr [^ i^k_hkf^]' >nkh\h]^+lmkb\mer\hglblmlh__hnkiZkml!IZkml*¾*%*¾+%+Zg],"T*&-V[nm_hkma^inkihl^lh_ mablin[eb\Zmbhg%>nkh\h]^+k^_^klmhiZkm*¾*hger%nge^lljnZebË^]'Ma^phkd^]^qZfie^lpbee [^\Zkkb^]hnmpbmabgma^^gobkhgf^gmh_hma^kk^e^oZgmin[eb\Zmbhgleblm^][^ehp%Zg]beenlmkZm^] bg?b`nk^*'*3 Ma^hma^kmak^^iZkmlh_>nkh\h]^+'  N Hma^k>nkh\h]^l'  N FZm^kbZeZg]^q^\nmbhglmZg]Zk]l'  N In[eb\Zmbhgl[rma^\hg\k^m^bg]nlmkrZg]hma^kl'  N

Ma^k^ Zk^% ma^k^_hk^% fZgr k^_^k^g\^l mh hma^k ]h\nf^gml Zg] pabe^ bm bl bgm^g]^] maZm mabl in[eb\Zmbhg%k^_^kk^]mhZlPhkd^]^qZfie^l%\ZglmZg]Zehg^%bmblZgmb\biZm^]maZmnl^klfZr k^jnbk^ l^o^kZe h_ ma^ hma^k k^_^k^g\^l mh aZg]% bg iZkmb\neZk% <hg\bl^ >nkh\h]^ +T.V% pab\a lnffZkbl^l ma^ kne^l Zg] ikbg\bie^l maZm pbee [^ \hffhger nl^] bg ma^ ]^lb`g h_ k^bg_hk\^] \hg\k^m^_kZf^][nbe]bg`lmh>nkh\h]^+' BS EN 1991–1–1 BS EN 1992–1–1 DESIGN OF CONCRETE STRUCTURES BS EN 1991–2 ACTIONS NA NA NA NA NA NA NA NA NA NA NA –2 –3 –4 –6 PD6687 1–2 –3 –2 WORKED EXAMPLES TO EUROCODE 2 VOL 1 CONCRETE INDUSTRY PUBLICATIONS WORKED

EXAMPLES PUBLICATIONS BY OTHERS

Fire Bridges Liquid retaining Fire Snow Wind Execution Densities and imposed loads VOL 2 General STANDARDS PRECAST WORKED EXAMPLES BS EN 1990 BASIS OF DESIGN CONCISE EUROCODE 2 HOW TO DESIGN CONCRETE STRUCTURES www. Eurocode2 .info RC SPREAD SHEETS PRECAST DESIGN MANUAL MANUALS DETAILERS HANDBOOK DESIGN GUIDES BS EN 13670 EXECUTION OF CONCRETE STRUCTURES Ghm^ Ma^m^kfLmZg]Zk]l^g\hfiZll^l;kbmblaLmZg]Zk]l%>nkh\h]^l%GZmbhgZe:gg^q^l!G:"Zg]In[ebla^]=h\nf^gml ?b`nk^*'* Phkd^]^qZfie^lbg\hgm^qm Ma^]^lb`glZk^bgZ\\hk]Zg\^pbma;L>G*22+¾*¾*T*V%Zlfh]bË^][rma^NDGZmbhgZe:gg^qT*ZV

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2

@^g^kZeer% ma^ \Ze\neZmbhgl Zk^ \khll&k^_^k^g\^] mh ma^ k^e^oZgm \eZnl^l bg Zee _hnk iZkml h_ >nkh\h]^ +T*¾-V Zg]% pa^k^ ZiikhikbZm^% mh hma^k ]h\nf^gml' L^^ ?b`nk^ *'+ _hk Z `nb]^ mh ik^l^gmZmbhg'K^_^k^g\^lmh;L1**)T0Vk^_^kmhIZkm*nge^llhma^kpbl^lmZm^]' @^g^kZeer%ma^Âlbfie^Ã^qZfie^l]^i^g]hg^jnZmbhglZg]]^lb`gZb]l]^kbo^]_khf>nkh\h]^+' Ma^]^kbo^]^jnZmbhglZk^`bo^gbg:ii^g]bq:Zg]ma^]^lb`gZb]l_khfL^\mbhg*.h_<hg\bl^ >nkh\h]^+T.VZk^k^i^Zm^]bg:ii^g]bq;' Ma^^qZfie^lZk^bgm^g]^]mh[^ZiikhikbZm^_hkma^bkinkihl^%pab\ablmhbeenlmkZm^ma^nl^h_ >nkh\h]^+_hkbg&lbmn\hg\k^m^[nbe]bg`lmkn\mnk^l'Ma^k^Zk^lbfie^^qZfie^lmhbeenlmkZm^ahp mrib\ZeaZg]\Ze\neZmbhglfb`am[^]hg^nlbg`ZoZbeZ[e^\aZkmlZg]mZ[e^l]^kbo^]_khfma^<h]^' Ma^l^Zk^_heehp^][rfhk^]^mZbe^]^qZfie^lbeenlmkZmbg`ma^]^mZbe^]phkdbg`lh_ma^<h]^l' Bghk]^kmh^qieZbgma^nl^h_>nkh\h]^+%l^o^kZeh_ma^\Ze\neZmbhglZk^ik^l^gm^]bg]^mZbe_Zk bg^q\^llh_maZmg^\^llZkrbg]^lb`g\Ze\neZmbhglhg\^nl^klZk^_ZfbebZkpbmama^<h]^'MhZg ^qm^gm%ma^]^lb`glZk^\hgmkbo^]mhlahpoZeb]f^mah]lh_]^lb`gbg`^e^f^gml%mh`bo^bglb`am Zg]mha^eibgoZeb]Zmbg`\hfinm^kf^mah]l'Ma^rZk^ghmg^\^llZkberma^fhlmZiikhikbZm^%ma^ fhlm^\hghfb\hkma^hgerf^mah]lh_]^lb`gbg`ma^f^f[^klbeenlmkZm^]'

Sections 1 and 2 Worked examples

Cl. 6.4.4 K^e^oZgm\eZnl^lhk_b`nk^gnf[^kl_khf;L>G*22+¾*¾*!b_ma^k^_^k^g\^blmhhma^kiZkml%hma^k

>nkh\h]^lhkhma^k]h\nf^gmlmablpbee[^bg]b\Zm^]" Cl. 6.4.4

NA ?khfma^k^e^oZgmNDGZmbhgZe:gg^q!`^g^kZeermh

;L>G*22+¾*¾*" NA

Cl. 6.4.4 & NA ?khf[hma;L>G*22+¾*¾*Zg]NDGZmbhgZe:gg^q Cl. 6.4.4 & NA

Fig. 2.1 Section 5.2 K^e^oZgmiZkmlh_mablin[eb\Zmbhg Fig. 2.1 Section 5.2 EC1-1-1: 6.4.3 ?khfhma^k>nkh\h]^l3;L>G*22)%;L>G*22*% ;L>G*22+¾*¾+%^m\ EC1-1-1: 6.4.3 PD 6687[6] ;Z\d`khng]iZi^kmhNDGZmbhgZe:gg^q^l ;L>G*22+¾* PD 6687[6] Concise <hg\bl^>nkh\h]^+T.V Concise

How to: Floors [8] Ahpmh]^lb`g\hg\k^m^lmkn\mnk^lnlbg`

>nkh\h]^+T1V3?ehhkl How to: Floors[8]

Grey shaded tables Bg:ii^g]b\^l%]^kbo^]\hgm^gmbgmZ[e^lghm_khf >nkh\h]^+

?b`nk^*'+

@nb]^mhik^l^gmZmbhg

:llhf^h_ma^]^mZbebg`kne^lbg>nkh\h]^+Zk^`^g^kZeerfhk^bgoheo^]maZgmahl^mh;L1**)% lhf^ h_ ma^ ]^lb`gl ik^l^gm^] bg mabl in[eb\Zmbhg aZo^ [^^g ^qm^g]^] bgmh Zk^Zl maZm aZo^ mkZ]bmbhgZeer[^^gma^k^lihglb[bebmrh_]^mZbe^kl'Ma^l^^qm^g]^]\Ze\neZmbhglZk^ghmg^\^llZkber iZkm h_ÂghkfZeà ]^lb`g [nm Zk^ bg\en]^] Zm ma^ ^g] h_ lhf^ \Ze\neZmbhgl' Bm bl Zllnf^] maZm ma^]^lb`g^kpbee]bl\nllZg]Z`k^^pbmama^]^mZbe^kZk^Zlh_k^lihglb[bebmrZg]ma^]^`k^^h_ kZmbhgZeblZmbhg% ma^ ^qm^gm h_ ]^lb`gbg` ]^mZbel% Zll^llf^gm h_ \nkmZbef^gm Zg] hma^k Zli^\ml maZmma^]^mZbe^klahne]ng]^kmZd^'Bmblk^\h`gbl^]maZmbgma^oZlmfZchkbmrh_\Zl^l%ma^kne^l `bo^gbg]^mZbebg`fZgnZelT1%2Vpbee[^nl^]'Ahp^o^k%ma^^qZfie^lZk^bgm^g]^]mha^eipa^g \nkmZbef^gm%Zg\ahkZ`^Zg]eZie^g`malg^^]mh[^]^m^kfbg^]'

(27)

Eurocode: Basis of structural design

Bg ma^ >nkh\h]^ lrlm^f ;L >G *22)% >nkh\h]^3 ;Zlbl h_ lmkn\mnkZe ]^lb`gT*)V ho^kZk\a^l Zee ma^ hma^k>nkh\h]^l%;L>G*22*mh;L>G*222';L>G*22)]^Ëg^lma^^__^\mlh_Z\mbhgl%bg\en]bg` `^hm^\agb\Ze Zg] l^blfb\ Z\mbhgl% Zg] Ziieb^l mh Zee lmkn\mnk^l bkk^li^\mbo^ h_ ma^ fZm^kbZe h_ \hglmkn\mbhg'Ma^fZm^kbZe>nkh\h]^l]^Ëg^ahpma^^__^\mlh_Z\mbhglZk^k^lblm^][r`bobg`kne^l_hk ]^lb`gZg]]^mZbebg`h_\hg\k^m^%lm^^e%\hfihlbm^%mbf[^k%fZlhgkrZg]Zenfbgbnf'!l^^?b`nk^*',"'

Structural safety, serviceability and durability

Actions on structures

Design and detailing

Geotechnical and seismic design

BS EN 1990,Eurocode:

Basis of structural design BS EN 1991,Eurocode 1: Actions on structures BS EN 1992,Eurocode 2: Concrete BS EN 1993,Eurocode 3: Steel BS EN 1994,Eurocode 4: Composite BS EN 1995,Eurocode 5: Timber BS EN 1996,Eurocode 6: Masonry BS EN 1999,Eurocode 9: Aluminium BS EN 1997,Eurocode 7: Geotechnical design BS EN 1998,Eurocode 8: Seismic design ?b`nk^*', Ma^>nkh\h]^ab^kZk\ar ;L>G*22)ikhob]^lma^g^\^llZkrbg_hkfZmbhg_hkma^ZgZerlblh_lmkn\mnk^lbg\en]bg`iZkmbZe Zg]hma^k_Z\mhklmh[^Ziieb^]mhma^Z\mbhgl_khf>nkh\h]^*'Bm^lmZ[ebla^lma^ikbg\bie^lZg] k^jnbk^f^gml_hkma^lZ_^mr%l^kob\^Z[bebmrZg]]nkZ[bebmrh_lmkn\mnk^l'Bm]^l\kb[^lma^[Zlbl_hk ]^lb`gZl_heehpl3 :lmkn\mnk^laZee[^]^lb`g^]Zg]^q^\nm^]!\hglmkn\m^]"bgln\aZpZrmaZmbmpbee%]nkbg`bml bgm^g]^]eb_^%pbmaZiikhikbZm^]^`k^^lh_k^ebZ[bebmrZg]bgZg^\hghfb\ZepZr3 LnlmZbgZeeZ\mbhglZg]bg_en^g\^lebd^ermhh\\nk]nkbg`^q^\nmbhgZg]nl^'  N K^fZbg_bm_hkma^nl^_hkpab\abmblk^jnbk^]'  N Bghma^kphk]l%bmlaZee[^]^lb`g^]nlbg`ebfbmlmZm^likbg\bie^lmhaZo^Z]^jnZm^3 LmZ[bebmr'  N Lmkn\mnkZek^lblmZg\^!bg\en]bg`lmkn\mnkZek^lblmZg\^bg_bk^"'  N L^kob\^Z[bebmr'  N =nkZ[bebmr'  N ?hk[nbe]bg`lmkn\mnk^l%Z]^lb`gphkdbg`eb_^h_.)r^Zklblbfieb^]' ;L>G*22)lmZm^lmaZmebfbmlmZm^llahne][^o^kbË^]bgZeek^e^oZgm]^lb`glbmnZmbhgl3i^klblm^gm% mkZglb^gmhkZ\\b]^gmZe'Ghk^e^oZgmebfbmlmZm^laZee[^^q\^^]^]pa^g]^lb`goZen^l_hkZ\mbhgl Zg]k^lblmZg\^lZk^nl^]bg]^lb`g'Ma^ebfbmlmZm^lZk^3 NembfZm^ebfbmlmZm^l!NEL"%pab\aZk^Zllh\bZm^]pbma\heeZil^hkhma^k_hkflh_lmkn\mnkZe  N _Zbenk^' L^kob\^Z[bebmrebfbmlmZm^l!LEL"%pab\a\hkk^lihg]mh\hg]bmbhgl[^rhg]pab\ali^\b_b^]  N l^kob\^k^jnbk^f^gmlZk^ghehg`^kf^m' :eeZ\mbhglZk^Zllnf^]mhoZkrbgmbf^Zg]liZ\^'LmZmblmb\Zeikbg\bie^lZk^Ziieb^]mhZkkbo^Zmma^ fZ`gbmn]^h_ma^iZkmbZeehZ]_Z\mhklmh[^nl^]bg]^lb`gmhZ\ab^o^ma^k^jnbk^]k^ebZ[bebmrbg]^q !e^o^eh_lZ_^mr"'Ma^k^blZgng]^kerbg`ZllnfimbhgmaZmma^Z\mbhglma^fl^eo^lZk^]^l\kb[^]bg

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EC0: 2.1

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4

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*'-Eurocode 1: Actions on structures

:\mbhglZk^]^Ëg^]bgma^*)iZkmlh_;L>G*22*>nkh\h]^*3:\mbhglhglmkn\mnk^lT**V3 ;L>G*22*¾*¾*3+))+3=^glbmb^l%l^e_&p^b`am%bfihl^]ehZ]l_hk[nbe]bg`l ;L>G*22*¾*¾+3+))+3:\mbhglhglmkn\mnk^l^qihl^]mhËk^ ;L>G*22*¾*¾,3+)),3LghpehZ]l ;L>G*22*¾*¾-3+)).3Pbg]Z\mbhgl ;L>G*22*¾*¾.3+)),3Ma^kfZeZ\mbhgl ;L>G*22*¾*¾/3+)).3:\mbhgl]nkbg`^q^\nmbhg ;L>G*22*¾*¾03+))/3:\\b]^gmZeZ\mbhgl ;L>G*22*¾+3+)),3:\mbhglhglmkn\mnk^l'MkZ_Ë\ehZ]lhg[kb]`^l ;L>G*22*¾,3+))/3<kZg^lZg]fZ\abg^kr ;L>G*22*¾-3+))/3LbehlZg]mZgdl

Mabl in[eb\Zmbhg bl fZbger \hg\^kg^] pbma ]^lb`gbg` _hk ma^ Z\mbhgl ]^Ëg^] [r IZkm¾*¾*3 =^glbmb^l%l^e_&p^b`am%bfihl^]ehZ]l_hk[nbe]bg`l'

=^lb`goZen^lh_Z\mbhglZg]ehZ]ZkkZg`^f^gmlZk^\ho^k^]bgL^\mbhg+'

Eurocode 2: Design of concrete structures

>nkh\h]^+3=^lb`gh_\hg\k^m^lmkn\mnk^lT*¾-Vhi^kZm^lpbmabgZg^gobkhgf^gmh_hma^k>nkhi^Zg Zg];kbmblalmZg]Zk]l!l^^?b`nk^*',"'Bmbl`ho^kg^][r;L>G*22)T*)VZg]ln[c^\mmhma^Z\mbhgl ]^Ëg^]bg>nkh\h]^l*T**V%0T*+VZg]1T*,V'Bm]^i^g]lhgoZkbhnlfZm^kbZelZg]^q^\nmbhglmZg]Zk]l Zg] bl nl^] Zl ma^ [Zlbl h_ hma^k lmZg]Zk]l' IZkm +% ;kb]`^lT,V% Zg] IZkm ,% Ebjnb]k^mZbgbg`Zg]

\hgmZbgf^gmlmkn\mnk^lT-V%phkd[r^q\^imbhgmhIZkm*¾*Zg]*¾+%maZmbl%\eZnl^lbgIZkml+Zg], \hgËkf%fh]b_rhkk^ieZ\^\eZnl^lbgIZkm*¾*'

BS EN 1990 EUROCODE Basis of Structural Design

BS EN 1991 EUROCODE 1 Basis of Structural Design

BS EN 1992 EUROCODE 2 Design of concrete structures

Part 1–1: General Rules for Structures Part 1–2: Structural Fire Design

BS EN 1992 EUROCODE 2 Part 3: Liquid Retaining Structures BS EN 1995 EUROCODE 5 Design of Composite Structures BS EN 13670 Execution of Structures BS 8500 Specifying Concrete BS EN 206 Concrete BS EN 1992 EUROCODE 2 Part 2: Bridges BS EN 1997 EUROCODE 7 Geotechnical Design BS EN 1998 EUROCODE 8 Seismic Design BS EN 13369 Precast Concrete BS EN 10080 Reinforcing Steels BS 4449 Reinforcing Steels Ghm^ ?hk\eZkbmrGZmbhgZe:gg^q^lZg]^qieZgZmhkr]h\nf^gml!^'`'I=//10%Zg] Ghg&<hgmkZ]b\mhkr<hfie^f^gmZkrBg_hkfZmbhg¾G<<B"Zk^ghmlahpg' ?b`nk^*'->nkh\h]^+bg\hgm^qm

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National Annexes

Bmblma^ik^kh`Zmbo^h_^Z\a<>GF^f[^kLmZm^mh\hgmkhee^o^elh_lZ_^mrbgmaZm\hngmkr':lZ k^lnem%lhf^lZ_^mr_Z\mhklZg]hma^kiZkZf^m^klbgma^>nkh\h]^l%ln\aZl\ebfZmb\\hg]bmbhgl% ]nkZ[bebmr\eZll^lZg]]^lb`gf^mah]l%Zk^ln[c^\mmh\hgËkfZmbhghkl^e^\mbhgZmZgZmbhgZee^o^e' Ma^]^\blbhglfZ]^[rma^gZmbhgZe[h]b^l[^\hf^GZmbhgZeer=^m^kfbg^]IZkZf^m^kl!G=Il" pab\a Zk^ in[ebla^] bg Z GZmbhgZe:gg^q !G:" _hk ^Z\a iZkm h_ ^Z\a >nkh\h]^'Ma^ GZmbhgZe :gg^q fZr Zelh bg\en]^ k^_^k^g\^ mh ghg&\hgmkZ]b\mhkr \hfie^f^gmZkr bg_hkfZmbhg !G<<B"% ln\aZlgZmbhgZelmZg]Zk]lhk`nb]Zg\^]h\nf^gml'

Mablin[eb\Zmbhgbg\en]^lk^_^k^g\^lmhma^k^e^oZgmGZmbhgZe:gg^q^lZlZiikhikbZm^'

Basis of the worked examples in this publication

Ma^]^lb`g\Ze\neZmbhglbgmablin[eb\ZmbhgZk^bgZ\\hk]Zg\^pbma3 ;L>G*22)%  N >nkh\h]^3;Zlblh_lmkn\mnkZe]^lb`gT*)VZg]bmlNDGZmbhgZe:gg^qT*)ZV' ;L>G*22*%>nkh\h]^*3  N :\mbhglhglmkn\mnk^lbg*)iZkmlT**VZg]ma^bkNDGZmbhgZe :gg^q^lT**ZV' ;L>G*22+¾*¾*%>nkh\h]^+¾IZkm*¾*3  N =^lb`gh_\hg\k^m^lmkn\mnk^l¾@^g^kZekne^lZg] kne^l_hk[nbe]bg`lT*VZg]bmlNDGZmbhgZe:gg^qT*ZV' ;L>G*22+¾*¾+%>nkh\h]^+¾IZkm*¾+3  N =^lb`gh_\hg\k^m^lmkn\mnk^l¾@^g^kZekne^l¾ Lmkn\mnkZe_bk^]^lb`gT+VZg]bmlNDGZmbhgZe:gg^qT+ZV' I=//10%;Z\d`khng]iZi^kmhma^  N NDGZmbhgZe:gg^q^lT/V' ;L>G*220%>nkh\h]^03  N @^hm^\agb\Ze]^lb`g¾IZkm*'@^g^kZekne^lT*+VZg]bmlNDGZmbhgZe :gg^qT*+ZV' Ma^rnl^fZm^kbZel\hg_hkfbg`mh3 ;L1.))¾*3  N <hg\k^m^¾<hfie^f^gmZkr;kbmblaLmZg]Zk]mh;L>G+)/¾*3F^mah]h_li^\b_rbg` Zg]`nb]Zg\^mhma^li^\b_b^kT*-V' ;L---23  N Lm^^e_hkma^k^bg_hk\^f^gmh_\hg\k^m^¾P^e]Z[e^k^bg_hk\bg`lm^^e¾;Zk%\hbeZg] ]^\hbe^]ikh]n\m¾Li^\b_b\ZmbhgT*.V' Ma^rfZd^k^_^k^g\^mhl^o^kZein[eb\Zmbhgl%fhlmghmZ[er3   N<hg\bl^>nkh\h]^+_hkma^]^lb`gh_bg&lbmn\hg\k^m^_kZf^][nbe]bg`lmh;L>G*22+¾*¾*3 +))-Zg]bmlNDGZmbhgZe:gg^q3+)).T.V'   NAhpmh]^lb`g\hg\k^m^lmkn\mnk^lnlbg`>nkh\h]^+T1V' Ma^^q^\nmbhgh_ma^phkdlblZllnf^]mh\hg_hkfmh3 I=//10  N ;Z\d`khng]iZi^kmhma^NDGZmbhgZe:gg^q^l;L>G*22+¾*'T/V GL<L%  N GZmbhgZelmkn\mnkZe\hg\k^m^li^\b_b\Zmbhg_hk[nbe]bg`\hglmkn\mbhg%,k]^]bmbhgT*/VFZr +))-' Hk%pa^gZoZbeZ[e^ ;L>G*,/0)3  N >q^\nmbhgh_\hg\k^m^lmkn\mnk^l'=n^+)*)T*0V':lbfie^f^gm^][r li^\b_b\Zmbhglln\aZl3 GL<L%  N GZmbhgZelmkn\mnkZe\hg\k^m^li^\b_b\Zmbhg_hk[nbe]bg`\hglmkn\mbhg%-ma^]bmbhgT*1V <<BI&).)%]n^+)*)'

(30)

6

Assumptions

Eurocode 2

>nkh\h]^+Zllnf^lmaZm3 =^lb`gZg]\hglmkn\mbhgpbee[^ng]^kmZd^g[rZiikhikbZm^erjnZeb_b^]Zg]^qi^kb^g\^]i^klhgg^e'  N :]^jnZm^lni^koblbhgZg]jnZebmr\hgmkhepbee[^ikhob]^]'  N FZm^kbZelZg]ikh]n\mlpbee[^nl^]Zlli^\b_b^]'  N Ma^lmkn\mnk^pbee[^Z]^jnZm^erfZbgmZbg^]Zg]pbee[^nl^]bgZ\\hk]Zg\^pbmama^]^lb`g[kb^_'  N Ma^k^jnbk^f^gml_hk^q^\nmbhgZg]phkdfZglabi`bo^gbg>G*,/0)Zk^\hfieb^]pbma'  N

The worked examples

Nge^llghm^]hma^kpbl^%ma^\Ze\neZmbhglbgmablin[eb\ZmbhgZllnf^3 EC0: Table 2.1 N:]^lb`geb_^h_.)r^Zkl'

Table 3.1 NMa^nl^h_<,)(,0\hg\k^m^'

BS 4449 NMa^nl^h_@kZ]^:%;hk<k^bg_hk\^f^gm%]^lb`gZm^]ÂAÃbgZ\\hk]Zg\^pbma;L1///T*2V' Table 4.1,

BS 8500: Table A.1 N>qihlnk^\eZllQ<*' Building Regs[20,21] N*ahnk_bk^k^lblmZg\^'

@^g^kZeer^Z\a\Ze\neZmbhgblkhng]^]Zg]bmblma^khng]^]oZen^maZmblnl^]bgZgr_nkma^k\Ze\neZmbhg'

Material properties

FZm^kbZeikhi^kmb^lZk^li^\bË^]bgm^kflh_ma^bk\aZkZ\m^kblmb\oZen^l'MablnlnZeer\hkk^lihg]l mhma^ehp^k._kZ\mbe^h_ZgZllnf^]lmZmblmb\Ze]blmkb[nmbhgh_ma^ikhi^kmr\hglb]^k^]' Ma^oZen^lh_g<Zg]gL%iZkmbZe_Z\mhkl_hkfZm^kbZel%Zk^bg]b\Zm^]bgMZ[e^*'*' MZ[e^*'* IZkmbZe_Z\mhkl_hkfZm^kbZel =^lb`glbmnZmbhg g<¾\hg\k^m^ gL¾k^bg_hk\bg`lm^^e

ULS – persistent and transient *'.) *'*.

Accidental – non-fire *'+) *')) Accidental – fire *')) *')) SLS *')) *'))

Execution

Bgma^ND%==>GO*,/0)T++Vbl\nkk^gmerZoZbeZ[e^[nmpbmahnmbmlGZmbhgZe:iieb\Zmbhg=h\nf^gm' ?hk[nbe]bg`lmkn\mnk^lbgma^ND%ma^[Z\d`khng]]h\nf^gmI=//10T/V\hglb]^klma^ikhoblbhglh_ ma^GZmbhgZeLmkn\mnkZe<hg\k^m^Li^\bË\Zmbhg!GL<L"T*/Vmh[^^jnboZe^gmmhmahl^bg>G*,/0)_hk mhe^kZg\^\eZll*'Pa^gin[ebla^]%;L>G*,/0)T*0VZg]%b_ZiikhikbZm^%ma^\hkk^lihg]bg`GZmbhgZe :iieb\Zmbhg=h\nf^gmpbeemZd^ik^\^]^g\^'

*'0

*'0'*

















*'0'+

*'1

*'2

Cl. 1.3 PD 6687[6] Table 2.1 & NA

(31)

+

+'*

+'*'*



+'*'+

+'+





 

+',

   

Analysis, actions and load arrangements

Methods of analysis

ULS

:mma^nembfZm^ebfbmlmZm^!NEL"ma^mri^h_ZgZerlbllahne][^ZiikhikbZm^mhma^ikh[e^f[^bg` \hglb]^k^]'Ma^_heehpbg`Zk^\hffhgernl^]3 Ebg^Zk^eZlmb\ZgZerlbl' N Ebg^Zk^eZlmb\ZgZerlblpbmaebfbm^]k^]blmkb[nmbhg' N IeZlmb\ZgZerlbl' N Cl. 5.1.1(7)

?hk NEL% ma^ fhf^gml ]^kbo^] _khf ^eZlmb\ ZgZerlbl fZr [^ k^]blmkb[nm^] ikhob]^] maZm ma^ k^lnembg`]blmkb[nmbhgh_fhf^gmlk^fZbglbg^jnbeb[kbnfpbmama^Ziieb^]Z\mbhgl'Bg\hgmbgnhnl [^ZflhkleZ[lpbma_\d©.)FIZma^fbgbfnfZeehpZ[e^kZmbhh_ma^k^]blmkb[nm^]fhf^gmmh ma^fhf^gmbgma^ebg^ZkZgZerlbl%d%bl)'0)pa^k^<eZll;hk<eZll<k^bg_hk\^f^gmblnl^]hk )'1)pa^k^<eZll:k^bg_hk\^f^gmblnl^]' Cl. 5.5.4 & NA Pbmabgma^ebfbmll^m%\h^_Ë\b^gml_hkfhf^gmZg]la^Zk]^kbo^]_khf^eZlmb\ZgZerlblfZr[^ nl^]mh]^m^kfbg^_hk\^lbgk^`neZklmkn\mnk^l!l^^:ii^g]bq;"'Ma^]^lb`gh_\henfgllahne][^ [Zl^]hg^eZlmb\fhf^gmlpbmahnmk^]blmkb[nmbhg' Cl. 5.1.1 IeZlmb\ZgZerlblfZr[^nl^]_hk]^lb`gZmNELikhob]^]maZmma^k^jnbk^]]n\mbebmr\Zg[^Zllnk^]% _hk^qZfie^3[rebfbmbg`qn(]!mh©)'+._hk\hg\k^m^lmk^g`ma\eZll^l©<.)(/)"4nlbg`<eZll; hk<k^bg_hk\^f^gm4hk^glnkbg`ma^kZmbhh_fhf^gmlZmbgm^kf^]bZm^lniihkmlmhfhf^gmlbg liZglbl[^mp^^g)'.Zg]+')' Cl. 5.6.2

SLS

:mma^l^kob\^Z[bebmrebfbmlmZm^!LEL"ebg^Zk^eZlmb\ZgZerlblfZr[^nl^]'Ebg^Zk^eZlmb\ZgZerlbl fZr[^\Zkkb^]hnmZllnfbg`3 <khll&l^\mbhglZk^ng\kZ\d^]Zg]k^fZbgieZg^!b'^'ZgZerlblfZr[^[Zl^]hg\hg\k^m^ N `khlll^\mbhgl"' Ebg^Zklmk^ll¾lmkZbgk^eZmbhglabil' N Ma^nl^h_f^ZgoZen^lh_^eZlmb\fh]nenl' N Cl. 5.4(1)

Actions

:\mbhglk^_^kmhehZ]lZiieb^]mhma^lmkn\mnk^Zl]^Ëg^][^ehp3 I^kfZg^gmZ\mbhglZk^Z\mbhgl_hkpab\ama^oZkbZmbhgbgfZ`gbmn]^pbmambf^blg^`eb`b[e^' N EC1-1-1: 2.1 OZkbZ[e^Z\mbhglZk^Z\mbhgl_hkpab\ama^oZkbZmbhgbgfZ`gbmn]^pbmambf^blghmg^`eb`b[e^' N EC1-1-1: 2.2, 3.3.1(2) :\\b]^gmZeZ\mbhglZk^Z\mbhglh_lahkm]nkZmbhg[nmh_lb`gb_b\ZgmfZ`gbmn]^maZmZk^ N ngebd^ermhh\\nkhgZ`bo^glmkn\mnk^]nkbg`ma^]^lb`gphkdbg`eb_^' EC1-1-7 Bfihl^]]^_hkfZmbhglZk^ghm\hglb]^k^]bgmablin[eb\Zmbhg'

Characteristic values of actions

Ma^oZen^lh_Z\mbhgl`bo^gbgma^oZkbhnliZkmlh_>nkh\h]^*3:\mbhglhglmkn\mnk^lT**VZk^mZd^g Zl\aZkZ\m^kblmb\oZen^l'Ma^\aZkZ\m^kblmb\oZen^h_ZgZ\mbhgbl]^Ëg^][rhg^h_ma^_heehpbg` mak^^Zem^kgZmbo^l3 EC0: 4.1.2 Bmlf^ZgoZen^¾`^g^kZeernl^]_hki^kfZg^gmZ\mbhgl' N

(32)

8 :gnii^koZen^pbmaZgbgm^g]^]ikh[Z[bebmrh_ghm[^bg`^q\^^]^]hkehp^koZen^pbmaZg N bgm^g]^]ikh[Z[bebmrh_[^bg`Z\ab^o^]¾ghkfZeernl^]_hkoZkbZ[e^Z\mbhglpbmadghpg lmZmblmb\Ze]blmkb[nmbhgl%ln\aZlpbg]hklghp' :ghfbgZeoZen^¾nl^]_hklhf^oZkbZ[e^Zg]Z\\b]^gmZeZ\mbhgl' N

Variable actions: imposed loads

General

Bfihl^]ehZ]lhg[nbe]bg`lZk^]bob]^]bgmh\Zm^`hkb^l'Mahl^fhlm_k^jn^gmernl^]bg\hg\k^m^ ]^lb`gZk^lahpgbgMZ[e^+'*' EC1-1-1: Tables 6.1, 6.7, 6.9 & NA MZ[e^+'* <Zm^`hkb^lh_ bfihl^]ehZ]l <Zm^`hkr =^l\kbimbhg A :k^Zl_hk]hf^lmb\Zg]k^lb]^gmbZeZ\mbobmb^l B H__b\^Zk^Zl C :k^Zlh_\hg`k^`Zmbhg D Lahiibg`Zk^Zl E LmhkZ`^Zk^ZlZg]bg]nlmkbZenl^!bg\en]bg`Z\\^llZk^Zl" F MkZ__b\Zg]iZkdbg`Zk^Zl!o^ab\e^l5,)dG" G MkZ__b\Zg]iZkdbg`Zk^Zl!o^ab\e^l7,)dG" H Khh_l!bgZ\\^llb[e^^q\^im_hkfZbgm^gZg\^Zg]k^iZbk" I Khh_l!Z\\^llb[e^pbmah\\niZg\r\Zm^`hkb^l:¾=" K Khh_l!Z\\^llb[e^_hkli^\bZel^kob\^l%^'`'_hka^eb\him^keZg]bg`Zk^Zl" Ghm^l 1<Zm^`hkrCblghmnl^]' 2?hk_hkdeb_mehZ]bg`k^_^kmh;L>G*22*¾*¾*<e'/'+','

Characteristic values of imposed loads

<aZkZ\m^kblmb\oZen^l_hk\hffhgernl^]bfihl^]ehZ]lZk^`bo^gbgMZ[e^l+'+mh+'1' EC1-1-1: Tables 6.1, 6.2 & NA.3 MZ[e^+'+ :3]hf^lmb\Zg]k^lb]^gmbZe Ln[& \Zm^`hkr >qZfie^ Bfihl^]ehZ]l jd!dG(f+" J d!dG" A1 :eenlZ`^lpbmabgl^e_&\hgmZbg^]]p^eebg`ngbml'<hffngZe Zk^Zl!bg\en]bg`dbm\a^gl"bglfZeeZ[eh\dlh__eZml *'. +') A2 ;^]khhflZg]]hkfbmhkb^l%^q\^immahl^bgl^e_&\hgmZbg^] lbg`e^_Zfber]p^eebg`ngbmlZg]bgahm^elZg]fhm^el *'. +') A3 ;^]khhflbgahm^elZg]fhm^el4ahlibmZepZk]l4mhbe^mZk^Zl +') +') A4 ;beebZk](lghhd^kkhhfl +') +'0 A5 ;Ze\hgb^lbglbg`e^&_Zfber]p^eebg`ngbmlZg]\hffngZeZk^Zl bglfZeeZ[eh\dlh__eZml +'. +') A6 ;Ze\hgb^lbgahlm^el%`n^lmahnl^l%k^lb]^gmbZe\en[l'<hffngZe Zk^ZlbgeZk`^kZ[eh\dlh__eZml Fbg',') [ Fbg'+')\ A7 ;Ze\hgb^lbgahm^elZg]fhm^el Fbg'-')[ Fbg'+')\ Ghm^l aLfZee[eh\dlh__eZmlZk^mahl^pbma©,lmhk^rlZg]©-_eZmli^k_ehhk(lmZbk\Zl^'Hma^kpbl^ma^rZk^ \hglb]^k^]mh[^eZk`^k[eh\dlh__eZml bLZf^Zlma^khhflmhpab\ama^r`bo^Z\\^ll%[nmpbmaZfbgbfnfh_,')dG(f+hk-')dG(f+ c<hg\^gmkZm^]Zmma^hnm^k^]`^ EC1-1-1: Tables 6.1, 6.2 & NA.3 MZ[e^+', ;3h__b\^l Ln[& \Zm^`hkr >qZfie^ Bfihl^]ehZ]l jd!dG(f+" J d!dG" B1 @^g^kZenl^hma^kmaZgbg;+ +'. +'0 B2 :mhk[^ehp`khng]_ehhke^o^e ,') +'0 

+'-+'-'*

+'-'+

(33)

MZ[e^+'- <3Zk^Zlh_\hg`k^`Zmbhg Ln[& \Zm^`hkr >qZfie^ Bfihl^]ehZ]l jd Jd <* :k^ZlpbmamZ[e^l C11 In[eb\%bglmbmnmbhgZeZg]\hffngZe]bgbg`khhflZg] ehng`^l%\Z_^lZg]k^lmZnkZgml!Ghm^3nl^<-hk<.b_ ZiikhikbZm^" +') ,') C12 K^Z]bg`khhflpbmagh[hhdlmhkZ`^ +'. -') C13 <eZllkhhfl ,') ,') <+ :k^Zlpbma_bq^]l^Zml C21 :ll^f[erZk^Zlpbma_bq^]l^Zmbg`a -') ,'/ C22 IeZ\^lh_phklabi ,') +'0 <, :k^Zlpbmahnmh[lmZ\e^l_hkfhobg`i^hie^ C31 <hkkb]hkl%aZeepZrl%Zble^lbgbglmbmnmbhgZemri^[nbe]bg`l% ahlm^el%`n^lmahnl^l%k^lb]^gmbZe\en[lZg]\hffngZe Zk^ZlbgeZk`^kb ,') -'. C32 LmZbkl%eZg]bg`lbgbglmbmnmbhgZemri^[nbe]bg`l%ahlm^el% `n^lmahnl^l%k^lb]^gmbZe\en[lZg]\hffngZeZk^Zlbg eZk`^kb ,') -') C33 <hkkb]hkl%aZeepZrl%Zble^lbghma^kc -') -'. C34 <hkkb]hkl%aZeepZrl%Zble^lbghma^kc pa^^e^]o^ab\e^l%bg\en]bg`mkhee^rl .') -'. C35 LmZbkl%eZg]bg`lbghma^kc -') -') C36 PZedpZrl¾Eb`am]nmr!Z\\^lllnbmZ[e^_hkhg^i^klhg% pZedpZrpb]maZiikhq/))ff" ,') +') C37 PZedpZrl¾@^g^kZe]nmr!k^`neZkmph&pZri^]^lmkbZg mkZ__b\" .') ,'/ C38 PZedpZrl¾A^Zor]nmr!ab`a&]^glbmri^]^lmkbZgmkZ__b\ bg\en]bg`^l\Zi^khnm^l" 0'. -'. C39 Fnl^nf_ehhklZg]Zkm`Zee^kb^l_hk^qab[bmbhginkihl^l -') -'. <- :k^Zlpbmaihllb[e^iarlb\ZeZ\mbobmb^l C41 =Zg\^aZeelZg]lmn]bhl%`rfgZlbZ%lmZ`^ld .') ,'/ C42 =kbeeaZeelZg]]kbeekhhfld .') 0') <. :k^Zlln[c^\m^]mheZk`^\khp]l C51 :ll^f[erZk^Zlpbmahnm_bq^]l^Zmbg`%\hg\^kmaZeel%[Zkl Zg]ieZ\^lh_phklabid,e .') ,'/ C52 LmZ`^lbgin[eb\Zll^f[erZk^Zld 0'. -'. D^r a?bq^]l^Zmbg`bll^Zmbg`pa^k^bmlk^fhoZeZg]ma^nl^h_ma^liZ\^_hkhma^kinkihl^lblbfikh[Z[e^ bLfZee[eh\dlh__eZmlZk^mahl^pbma©,lmhk^rlZg]©-_eZmli^k_ehhk(lmZbk\Zl^'Hma^kpbl^ma^rZk^ \hglb]^k^]mh[^ÂeZk`^kÃ[eh\dlh__eZml cHma^k[nbe]bg`lbg\en]^mahl^ghm\ho^k^][r<,*Zg]<,+%Zg]bg\en]^ahm^elZg]fhm^elZg] bglmbmnmbhgZe[nbe]bg`lln[c^\m^]mh\khp]l d?hklmkn\mnk^lmaZmfb`am[^lnl\^imb[e^mhk^lhgZg\^^__^\ml%k^_^k^g\^lahne][^fZ]^mhG:'+'* e?hk`kZg]lmZg]lZg]lmZ]bZ%k^_^k^g\^lahne][^fZ]^mhma^k^jnbk^f^gmlh_ma^ZiikhikbZm^ \^kmb_rbg`Znmahkbmr EC1-1-1: Tables 6.1, 6.2 & NA.3

(34)

10 EC1-1-1: Tables 6.1, 6.2 & NA.3 MZ[e^+'. =3lahiibg`Zk^Zl Ln[& \Zm^`hkr >qZfie^ Bfihl^]ehZ]l jd!dG(f+" J d!dG" = Lahiibg`Zk^Zl D1 :k^Zlbg`^g^kZek^mZbelahil -') ,'/ D2 :k^Zlbg]^iZkmf^gmlmhk^l -') ,'/ EC1-1-1: Tables 6.3, 6.4 & NA.4, NA.5

MZ[e^+'/ >3lmhkZ`^Zk^ZlZg]bg]nlmkbZenl^!bg\en]bg`Z\\^llZk^Zl" Ln[& \Zm^`hkr >qZfie^ Bfihl^]ehZ]l jd!dG(f+" J d!dG" >* :k^Zllnl\^imb[e^mhZ\\nfneZmbhgh_`hh]lbg\en]bg`Z\\^llZk^Zl E11 @^g^kZeZk^Zl_hklmZmb\^jnbif^gmghmli^\b_b^] ^el^pa^k^!bglmbmnmbhgZeZg]in[eb\[nbe]bg`l" +') *'1 E12 K^Z]bg`khhflpbma[hhdlmhkZ`^%^'`'eb[kZkb^l -') -'. E13 @^g^kZelmhkZ`^hma^kmaZgmahl^li^\b_b^]Z +'-(f 0') E14 ?be^khhfl%_bebg`Zg]lmhkZ`^liZ\^!h__b\^l" .') -'. E15 LmZ\dkhhfl![hhdl" +'-(fa^b`am !fbg'/'." 0')

E16 IZi^klmhkZ`^Zg]lmZmbhg^krlmhk^l -')(fa^b`am 2')

E17 =^gl^fh[be^lmZ\dbg`![hhdl"hgfh[be^mkhee^rlbg in[eb\Zg]bglmbmnmbhgZe[nbe]bg`l -'1(fa^b`am 0') E18 =^gl^fh[be^lmZ\dbg`![hhdl"hgfh[be^mkn\dlbg pZk^ahnl^l -'1(fa^b`am !fbg'*.')" 0')

E19 <he]lmhkZ`^ .')(fa^b`am !fbg'*.')" 2') >+ Bg]nlmkbZenl^ L^^;L>G*22*¾*¾*3 MZ[e^l/'./'/ ?hkdeb_ml<eZll^l?E*mh?E/ D^r aEhp^k[hng]oZen^`bo^g'Fhk^li^\b_b\ehZ]oZen^llahne][^Z`k^^]pbma\eb^gm MZ[e^+'0 ?Zg]@3mkZ__b\Zg]iZkdbg`Zk^Zl Ln[& \Zm^`hkr >qZfie^ Bfihl^]ehZ]l jd!dG(f+" Jd!dG" ? MkZ__b\Zg]iZkdbg`Zk^Zl!o^ab\e^l5,)dG" MkZ__b\Zg]iZkdbg`Zk^Zl!o^ab\e^l5,)dG" +'. .') @ MkZ__b\Zg]iZkdbg`Zk^Zl!o^ab\e^l7,)dG" MkZ__b\Zg]iZkdbg`Zk^Zl!o^ab\e^l7,)dG" .') Mh[^ ]^m^kfbg^]_hk li^\b_b\nl^

Figure

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References

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