CAPÍTULO I: LA ONUDI EN ACCIÓN
A. La ONUDI como foro mundial
Design code: EN 1992-2:2005 with UK National Annex (modified) EN 1990 Equation 6.14 SLS Characteristic
Exposure Class: XD1, XD2, XS1, XS2, XS3
Load case: Traffic gr1a TS - for Bending design 1 Section Ref 1 at 10.5m from left end of beam
WARNING - A reduction of flange width to allow for shear lag effects may be appropriate for this beam. SAM makes no allowance for this.
Refer to EN 1992-1-1/5.3.2.1
Section details:
Ref 1 "Section 1"
at 0.5 x span = 10.5 m from left end of beam Analysis:
Traffic Actions: Bending for gr1a, loading I.D. 1 At time considered, t = ∞
Serviceability Limit State: Characteristic - EN 1990 Equation 6.14
ACTUAL STRESSES IN PRECAST BEAM
No. of tendons fully bonded at this section: 21 No. of tendons fully debonded at this section: 0 No. of tendons deflected at this section: 0
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In accordance with clause 5.10.9(1), for SLS, the Characteristic value must be used.
With rinf = 1.0, Pk,inf = 4448.25 kN
relaxation After relaxation
height No of area x loss force moment
mm tendons σpi % kN kN kN.m 60.0 11 2330.04 4.57 106.51239 2223.5233 133.4114 110.0 4 847.286 4.57 38.731779 808.55394 88.940933 210.0 2 423.643 4.57 19.365889 404.27697 84.898163 260.0 2 423.643 4.57 19.365889 404.27697 105.11201 1200.0 2 423.643 4.57 19.365889 404.27697 485.13236
TOTAL 21 4244.9082 897.49487 Moment about the centroid of the precast beam:
Mr = 897.49487-(4244.9082*0.5760392) = -1547.739 kN.m
Corresponding stresses:
top stress = 4244.9082/537225.68+-1547.739/1.2843E8 = 7.9015362+-12.05139
= -4.149853 MPa
bottom stress = 4244.9082/537225.68+-1547.739/-1.614E8 = 7.9015362+9.5890175
= 17.490554 MPa Self weight moment:
c.s.a. = 5.372E5 mm²
density = 24.0 kN/m³ + 1.0 kN/m³ = 25.0 kN/m³[1]
self weight = 5.372E5*25.0 = 13.4306 kN/m beam length = 21.0 m distance = 10.5 m
Msw = 0.5*13.4306*10.5*(21.0-10.5) = 740.364 kN.m
Corresponding stresses:
top stress = 740.364/1.2843E8 = 5.76481 MPa bottom stress = 740.364/-1.614E8 = -4.5869 MPa
Elastic Deformation - Clause 5.10.4(1)(iii)
stress at top of precast beam = 1.61496 MPa stress at bottom of precast beam = 12.9036 MPa depth of precast beam = 1300.0 mm elastic modulus of concrete at transfer = 31.1307 GPa
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top stress = 1.615-218.37943/537.22568--98.00285/1.2843E8 = 1.615-0.4064947--0.763094
= 1.9715546 MPa
bottom stress = 12.904-218.37943/537.22568--98.00285/-1.614E8 = 12.904-0.4064947-0.6071768
= 11.889955 MPa
After a further 2 iterations of the above process, the top and bottom stresses are as follows:
top stress = 1.94149211 MPa
Maximum tendon stress after transfer = 1330.61 MPa which is not greater than 1360.0 and therefore OK.
ACTIONS DURING EXECUTION
Erection of beam LoadingBending moment from erection loadcase at current span location:
MApplied = 738.00575 kN.m
Remove the dead load applied for transfer calculations Msw = -740.36 kN.m
Corresponding stresses:
top stress = -740.36/1.2843E8 = -5.7648 MPa bottom stress = -740.36/-1.614E8 = 4.58693 MPa
Construction stage 1A Loading MApplied = 512.3149 kN.m Corresponding stresses:
top stress = 512.3149/1.2843E8 = 3.98911 MPa bottom stress = 512.3149/-1.614E8 = -3.174 MPa
Construction stage 1B Loading MApplied = 21.87451 kN.m Corresponding stresses:
top stress = 21.87451/1.2843E8 = 0.17032 MPa bottom stress = 21.87451/-1.614E8 = -0.1355 MPa
Time Dependent Losses - EN 1992-1-1 Clause 5.10.6
Simplified method using Expression (5.46) ΔPc+s+r = Ap.Δσp,c+s+r
εcs.Ep + 0.8Δσpr + Ep/Ecm.φ(t,t0).σc,QP Δσp,c+s+r = ——————————————————————————————————————————————
1 + Ep/Ecm.Ap/Ac(1+Ac/Ic.zcp²)[1+0.8φ(t,t0)]
The calculated loss is apportioned partly to the precast beam alone and partly to the full composite section.
For in-situ cast at 60 days, the proportion of the loss occurring before the in-situ is cast is calculated to be 28.63 %
Losses are calculated for time t = ∞
Age of concrete at end of curing, ts = 1.0 days Age of concrete at transfer, t0 = 4.0 days Age is adjusted for expression (B.5) (for cement type & temperature) - for cement class N (α = 0)
adjusted t0 = t0,T . [(9/(2+t0,T1.2)+1)α >=0.5 Expression (B.9) = 4.0 * [(9/(2+4.0 1.2)+1]0
= 4.0 days
Bestech Systems Limited Concrete age coefficient (Expression (3.2)), βcc:
βcc(t) = fcm(t)/fcm Expression (3.1) Modulus of elasticity of concrete at time considered,
Ecm(t) = βcc(t)0.3 . Ecm Expressions (3.5) & (3.1)
Creep coefficient for concrete - EN 1992-1-1 clause 3.1.4 and Annex B.1 φ(t,t0) = φ0 . βc(t,t0) Expression (B.1)
age is adjusted for expression (B.5) (for cement type and temperature) - for cement class N (α = 0)
9.0
t0 = t0,T . [ —————————————— + 1.0 ]α >=0.5 Expression (B.9) 2.0 + t0,T1.2
9.0
= 4.0 * [ —————————————— + 1.0 ]0 2.0 + 4.0 1.2
= 4.0 day
β(t0) = 1/(0.1+t00.2) Expression (B.5) = 1/(0.1+4.00.2)
= 0.70446
βc(t,t0) = 1.0 for time t = ∞ hence from (B.1) and (B.2):
φ(t,t0) = 1.17777*2.42487*0.70446 = 2.01193
Check for creep non-linearity EN 1992-1-1 clause 3.1.4(4)
At the level of the centroid of the tendons, the compressive stress in the concrete at time t0
= 8.31165 MPa.
This does not exceed 0.45*fck(t0), i.e. 18.0 MPa, so non-linear creep is not considered
Shrinkage Strain for concrete - EN 1992-1-1 clause 3.1.4(6) Total Shrinkage:
εcs = εcd + εca (3.8)
Drying Shrinkage - Expression (3.9):
εcd(t) = βds(t,ts).kh.εcd,0 βds(t,ts) = 1.0 for t = ∞ From Table 3.3:
kh = 0.80018
From Annex B, Expression (B.11):
εcd,0 = 0.85[(220+110.αds1).(exp(-αds2.fcm/fcm0)].10-6.βRH βRH = 1.55[1.0-(RH/100)³] (B.12) = 0.7564
For cement class N, αds1 = 4 αds2 = 0.12 hence,
εcd,0 = 0.85[(220+110*4.0)*exp(-0.12*48.0/10.0)]*10-6*0.7564 = 238.54*10-6
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With the initial relaxation deducted, the variation in tendon stress from relaxation becomes:
Δσpr / σpi = 0.21440 - 0.04571
Ep/Ecm = 195.0/37.9636 = 5.1365 Ep/Ecm.Ap/Ac = 5.1365*3150.0/9.051E5 = 0.01788 Ac/Ic = 9.051E5/2.33E11 = 3.8904
In the table below the following vary with tendon height:
σc,QP = Stress in concrete adjacent to tendons zcp = Section centre of gravity to tendons
φ(t,t0) = Creep Coefficient (if non-linear creep is considered)
shrink relax creep denom
height εcs.Ep φ(t,t0) Ep/Ecm.φ.σ
Ap 0.8Δσpr σc,QP zcp ΔPc+s+r
mm mm² MPa MPa MPa MPa mm kN 60.0 1650.0 51.846 190.57 2.012 8.605 88.927 839.705 1.175 465.43833 110.0 600.0 51.846 190.57 2.012 8.5109 87.954 789.705 1.16 170.90467 210.0 300.0 51.846 190.57 2.012 8.3226 86.008 689.705 1.133 86.962166 260.0 300.0 51.846 190.57 2.012 8.2284 85.035 639.705 1.121 87.637686 1200.0 300.0 51.846 190.57 2.012 6.4583 66.742 -300.3 1.063 87.248992
Total force loss: 898.19184
Total moment loss: 192.47246 Mcsr = 192.47246-(898.19184*0.8997047)
= -615.635 kN.m
Corresponding stresses - before composite:
top stress = ( 898.192/5.372E5+-615.64/1.284E8 )* 0.286 = ( 1.6719079+-4.793611 )* 0.286
= -0.893716 MPa
bottom stress = ( 898.192/5.372E5+-615.64/-1.61E8 )* 0.2862 = ( 1.6719079+3.8141679 )* 0.286
= 1.5706162 MPa
- after composite:
top stress = ( 898.192/9.051E5+-615.64/5.812E8 )*(1.0- 0.286) = ( 0.9924210+-1.059313 )*(1.0-0.286)
= -0.047742 MPa
bottom stress = ( 898.192/9.051E5+-615.64/-2.59E8 )*(1.0-0.286 ) = ( 0.9924210+2.3809161 )*(1.0-0.286)
= 2.4075798 MPa
Surfacing 1 Loading
MApplied = 99.65918 kN.m Corresponding stresses:
top stress = 99.65918/5.8116E8 = 0.17148 MPa bottom stress = 99.65918/-2.586E8 = -0.3854 MPa
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bottom stress = -43.8224/905051.1 + 934.3025/-2.586E8 = -0.0484 + -3.613
bottom stress = -4.365749/905051.1 + 324.4073/-2.586E8 = -0.0048 + -1.255
bottom stress = 1.418796/905051.1 + 19.32731/-2.586E8 = 0.00157 + -0.075
In the presence of confinement or increase in cover this may be increased by up to 10%, i.e to: = 26.4 MPa
Tension
Tension is governed by crack width considerations, and reinforcement provided for crack width control.
Exposure Class is XD1, XD2, XD3, XS1, XS2, XS3 ...
... for which decompression is checked for the Frequent combination of loads.
Decompression requires all of the tendon to be at least 65.0 mm above the level of the neutral axis.
LIMITING STRESSES FOR IN SITU CONCRETE
Compression
EN 1992-2-2 Clause 7.2(102)
To avoid longitudinal cracking, compressive stress is limited to:
σc = k1.fck = 0.6*31.875 = 19.125 MPa Tension
Tension is governed by crack width considerations, and reinforcement provided for crack width control.
EN 1992-1_1 Clause 7.3
However, no tensile stress is present at this section.
TRANSMISSION LENGTH
Bond stress at release, EN 1992-1-1 Clause 8.10.2.2(1)
fbpt = ηp1.η1.fctd(t) Expression (8.15) where
fctd(t) = αct.0.7fctm(t)/γc fctm(t) = -2.3253 MPa[2]
αct = 1.0 - from EN 1992-1-1/3.1.6(2) tendon type coefficient, ηp1 = 3.2 bond condition coefficient, η1 = 1.0 hence
fctd(t) = 1.0*0.7*-2.3253/1.5 = -1.0851 MPa
and
fbpt = 3.2*1.0*-1.0851 = -3.4724 MPa
Bestech Systems Limited 2 Slaters Court
Princess Street Knutsford WA16 6BW
Job: Sample Reports Job No.: 6.5d
Calc. By: dlg
Beam: Prestress Beam - Inner span 1 Checked:
Eurocode + UK NA
Data File: J:\...\6.50d Data Files\inner beam span 1.sam 02/02/2012 09:39:44
SAM v6.50d 06/02/2012 10:09:59 Page: 12
© 2012 Bestech Systems Ltd
Basic transmission length, EN 1992-1-1 Clause 8.10.2.2(2)
lpt = α1.α2.φ.σpm0/fbpt Expression (8.16) where
speed of release coefficient, α1 = 1.0 tendon surface coefficient, α2 = 0.19 nominal diameter of tendon, φ = 16.0 mm tendon stress after release, σpm0 = 1440.0 MPa hence
lpt = 1.0*0.19*16.0*1440.0 / 3.47242 = 1.26068 m
Design value of transmission length, EN 1992-1-1 Clause 8.10.2.2(3) lpt1 = 0.8*lpt
= 0.8*1.26068 = 1.00854 m
STRUCTURAL EFFECTS OF TIME DEPENDENT BEHAVIOUR EN 1992-2 Annex KK.7
Age of concrete at first loading, t0 = 4.0 days Age of concrete when first composite, tc = 60.0 days Age of concrete at time considered, t = ∞
Creep coefficient when first composite, φ(tc,t0) = 0.89250 Final creep coefficient, φ(∞,t0) = 2.00881 Creep coefficient increment, φ(∞,tc) = 1.20422 Specified value of Ageing coefficient, χ = 0.8 From Expression (KK.119):
φ(∞,t0) - φ(tc,t0) 2.00881-0.89250 ————————————————— = —————————————————————
1 + χ.φ(∞,tc) 1.0 + 0.8*1.20422 = 0.56856
SLS STRESS SUMMARY TABLE
Concrete Stresses (MPa)
force moment In situ Precast
kN kN.m top bottom top bottom CHARACTERISTIC PERMANENT ACTIONS AND PRESTRESS
Prestress[3] 4244.91 -1547.7 -4.1499 17.4906 Self Weight 740.364 5.76481 -4.5869 ——————————————————————————————————————————————————————
Prestress + Self Weight 1.61496 12.9036
Elastic Def -203.96 90.6953 0.32653 -0.9415
TRANSFER 4040.95 -716.68 1.94149 11.9621 Cr+Sh+Rlx B -257.14 176.251 0.89371 -1.5706 Erection -2.3584 -0.0184 0.01461 In situ 1A 512.315 3.98911 -3.174 In situ 1B 21.8745 0.05072 0.03293 0.03764 -0.0846 0.0 0.0 0.0 0.0 0.0
TOTAL PERMANENT EFFECTS, S0 6.8436 7.14742 Cr+Sh+Rlx A -641.05 439.384 0.34886 -0.0084 0.04774 -2.4076
TOTAL PERMANENT EFFECTS, S0,∞ 0.39958 0.0245 6.89134 4.73984 CREEP REDISTRIBUTION according to EN 1992-2 Annex KK.7
Construction On Centering, Sc = G + P1 + P2
Permanent G 606.626 1.40663 0.91333 1.04381 -2.3461 Prestress P1 -2584.1 -5.992 -3.8906 -4.4465 9.99388[4]
3780.83 3.95141 3.95141 4.17748 4.17748 Prestress P2 2329.9 5.40251 3.50787 4.00902 -9.0107[5]
(Sc - S0)*0.56856 2.68239 2.52957 -1.1711 -2.4635
Hence from KK.119,
TOTAL CONSTRUCTION EFFECTS, S∞ 3.08198 2.55407 5.72024 2.27635 SDL 99.6592 0.23108 0.15004 0.17148 -0.3854 Diff. Shr. 1 605.383 286.182 -0.2604 -0.4932 1.16132 -0.4378 Diff. Shr. 2 -101.97 -0.2364 -0.1535 -0.1754 0.39436 [Differential shrinkage is included when adverse ]
TOTAL PERMANENT EFFECTS 3.31306 2.70412 5.89172 1.89092
[including diff. shrinkage 2.81612 2.05738 6.87758 1.8474]
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The following combinations of variable actions are evaluated in accordance with EN 1990 Equation 6.14b:
a) Traffic as leading action + ψ0(Thermal + Other) b) Thermal as leading action + ψ0(Traffic + Other) c) Other as leading action + ψ0(Traffic + Thermal) For thermal actions the following cases are considered:
i) temperature restraint alone ii) differential temperature: Heating iii) differential temperature: Cooling
iv) temperature restraint + differential temperature: Heating v) temperature restraint + differential temperature: Cooling
The most adverse case is with Traffic as leading action
Traffic 2.9146 1.87532 2.14742 -4.9944 ψ0 x Thermal 1.88238 -0.2735 0.04654 -0.8270 ψ0 x Other 0.0 0.0 0.0 0.0
TOTAL VARIABLE ACTIONS 4.79698 1.60173 2.19396 -5.8215
TOTAL PERMANENT (from above) 3.31306 2.70412 6.87758 1.8474
——————————————————————————————————
TOTAL COMBINATION 8.11005 4.30585 9.07154 -3.9741
WARNING - The flexural tensile stress exceeds the value of fct,eff so the section cannot be assumed to be uncracked. (EN 1992-1-1/7.1(2)) A cracked section analysis must be performed to derive the true compression stress in the concrete.
VARIABLE ACTIONS - FREQUENT COMBINATION Traffic
Selected case: ψ2 ψ1 = 0.75
Traffic gr1a TS 1 700.727 1.62483 1.05501 1.20573 -2.71 0.0 -32.867 -0.0343 -0.0343 -0.0363 -0.0363 ψ1 = 0.75
Traffic gr1a UDL 1 243.305 0.56417 0.36631 0.41865 -0.9409 0.0 -3.2743 -0.0034 -0.0034 -0.0036 -0.0036 ψ1 = 0.4
however, EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 Traffic gr1a FT 1 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 Total (Leading) : 2.15123 1.38355 1.58445 -3.6909 Total (in Combination) : 0.0 0.0 0.0 0.0 Other traffic cases for comparison:
ψ1 = 0.75
Traffic gr1a TS 2 -125.77 -0.2916 -0.1893 -0.2164 0.48640 0.0 10.5473 0.01102 0.01102 0.01165 0.01165 ψ1 = 0.75
Traffic gr1a UDL 2 -67.26 -0.1559 -0.1012 -0.1157 0.26012 0.0 3.91875 0.0041 0.0041 0.00433 0.00433 ψ1 = 0.4
however, EN 1991-2 Clause 4.5.2 excludes the footway loading from the frequent action of LM1 Traffic gr1a FT 2 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 Total (Leading) : -0.4324 -0.2755 -0.3161 0.76251 Total (in Combination) : 0.0 0.0 0.0 0.0 Temperature Restraint
None defined
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The following combinations of variable actions are evaluated in accordance with EN 1990 Equation 6.15b:
a) ψ1(Traffic) as leading action + ψ2(Thermal + Other) b) ψ1(Thermal) as leading action + ψ2(Traffic + Other) c) ψ1(Other) as leading action + ψ2(Traffic + Thermal) For thermal actions the following cases are considered:
i) temperature restraint alone ii) differential temperature: Heating iii) differential temperature: Cooling
iv) temperature restraint + differential temperature: Heating v) temperature restraint + differential temperature: Cooling The most adverse case is with Traffic as leading action
ψ1 x Traffic 2.15123 1.38355 1.58445 -3.6909
After in situ 1A T 6.80596 EI 366.561 -17.651 6.14087 6.14087 B 7.23201 389.507
After in situ 1B T 6.8436 EI 368.588 -12.587 5.84195 5.84195 B 7.14742 384.951
Long-term Dead T 8.04868 EL 687.582 245.625 -4.8888 -4.8888 B 4.31089 368.27
Diff. Temp H1 T -0.5175 ES -14.695 -29.104 1.59073 1.59073 B 0.81504 23.1413
Diff. Temp H2 T 0.29442 ES 8.35951 20.8834 -1.1442 -1.1726 B -0.6617 -18.789
Diff. Temp C1 T 0.14855 ES 4.2178 26.3161 -1.4413 -1.4413 B -1.0564 -29.993
Diff. Temp C2 T -0.1020 ES -2.8964 -7.2357 0.39644 0.40625 B 0.22928 6.51001
Traffic gr1a 1 T 2.14742 ES 60.9709 155.98 -6.7548 -6.7548 B -4.9944 -141.8
Extreme in-service 420.685 -12.688 -12.688 Curvatures here are derived from precast section height: 1300.0mm ET = Elastic Modulus at Transfer = 31130.7MPa
[EN1992-1-1 Clauses 3.1.3-(3) and 3.1.2-(6) with age 4 days]
EI = Intermediate Term Elastic Modulus = 18567.1MPa
[EN1992-1-1 Clause 7.4.3-(5) at 60.0 days (φ=0.89693)]
EL = Long Term Elastic Modulus = 11705.8MPa
[EN1992-1-1 Clause 7.4.3-(5) at infinite time (φ=2.00881)]
ES = Short Term Elastic Modulus [Ecm] = 35220.5MPa
________
[1] Refer to EN 1991-1-1 Table A.1 Note 1)
[2] For the derivation of this value refer to the limiting stress calculations for transfer [3] includes draw-in and initial relaxation
[4] With immediate losses and shrinkage / creep / relaxation losses until time at which insitu is cast.
[5] Secondary effects arising from prestress in continuous section.