CAPÍTULO I: LA ONUDI EN ACCIÓN
C. Provocar un verdadero cambio
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 0m 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 x span = 0 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: 0 No. of tendons fully debonded at this section: 7 No. of tendons deflected at this section: 0
No. of tendons partially stressed: 14 (i.e. within the transmission length) The prestress force in these tendons is interpolated in accordance with EN 1992-1-1 clause 8.10.2.2.
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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 = 0.0 kN
Moment about the centroid of the precast beam: elastic modulus of concrete at transfer = 31.1307 GPa
height No of conc conc tendon tendon
After a further 0 iterations of the above process, the top and bottom stresses are as follows:
top stress = 0.0 MPa bottom stress = 0.0 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 10:15:31 Page: 4
© 2012 Bestech Systems Ltd
Max Prestress Force after transfer - EN 1992-1-1 Clause 5.10.3.(2) For tendon property Grade 1600 Ep 195.0
k7.fpk = 0.75*1860.0 = 1395.0 MPa k8.fp0,1k = 0.85*1600.0 = 1360.0 MPa
Maximum tendon stress after transfer = 1272.24 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 = 0.0 kN.m Corresponding stresses:
top stress = 0.0/1.2843E8 = 0.0 MPa bottom stress = 0.0/-1.614E8 = 0.0 MPa
Remove the dead load applied for transfer calculations Msw = 0.0 kN.m
Corresponding stresses:
top stress = 0.0/1.2843E8 = 0.0 MPa bottom stress = 0.0/-1.614E8 = 0.0 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 30.0 %
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
Age of concrete at time considered, t = ∞
EN 1992-1-1/3.3.2(8) for relaxation, t is taken as 500,000 hours Concrete age coefficient (Expression (3.2)), βcc:
βcc(t) = fcm(t)/fcm Expression (3.1) = exp{s[1-√(28/t)]} Expression (3.2) Coefficient for Class N cement, s = 0.25
βcc(t0) = exp{0.25[1.0-√(28/4.0)]} = 0.66269 βcc(t) = exp{0.25} = 1.28403 Characteristic strength of concrete, fck = 40.0 MPa Mean compressive strength of concrete, fcm = 40.0 + 8.0 (from Table 3.1) = 48.0 MPa fcm0 = 10.0 MPa fcm(t0) = βcc(t0) . fcm = 31.8094 MPa Ambient relative humidity = 80.0 %
Notional size of member, h0 = 2Ac/u = 2*9.051E5/7245.89 = 249.811 mm
Modulus of elasticity of concrete at 28 days, Ecm = 35.2205 GPa Modulus of elasticity of concrete at time considered,
Ecm(t) = βcc(t)0.3 . Ecm Expressions (3.5) & (3.1) = 1.284030.3 * 35.2205
= 37.9636 GPa
Area of concrete cross section, Ac = 9.05E5 mm² Perimeter of concrete cross section, u = 7245.9 mm Notional size, h0 = 2*Ac/u = 2*9.051E5/7245.89 = 249.81 mm
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) = φRH . β(fcm) . β(t0) . βc(t,t0) Expression (B.2)
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age is adjusted for expression (B.5) (for cement type and temperature) - for cement class N (α = 0)
At the level of the centroid of the tendons, the compressive stress in the concrete at time t0
= 0.0 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
and,
εcd(t) = 1.0*0.80018*238.54*10-6 = 190.877*10-6
Autogenous Shrinkage - Expression (3.11):
εca(t) = βas(t).εca(∞) βas(t) = 1.0 for t = ∞ εca(∞) = 2.5*(fck-10.0)*10-6 = 75.0*10-6
hence,
εca(t) = 1.0*75.0*10-6 = 75.0*10-6
Total Shrinkage:
εcs = εcd(t) + εca(t) = 190.87688 + 75.0 = 265.87688*10-6
Further Relaxation Clause 5.10.6(1)(b) Loss is calculated from clause 3.3.2(7) For tendon property Grade 1600 Ep 195.0
relaxation loss at 1000 hours, ρ1000 = 8.0 %
time after tensioning = 500000.0 hours
μ = 0.75921 (as calculated for initial relaxation loss above) for Class 1 relaxation, use Expression (3.28)
5.39 . ρ1000 . e6.7μ . [t/1000]0.75(1-μ) . 10-5 = 5.39 * 8.0 * 161.863 * 3.07185 * 10-5 = 0.21440
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With the initial relaxation deducted, the variation in tendon stress from relaxation becomes:
Δσpr / σpi = 0.21440 - 0.04571
In the table below the following vary with tendon height:
σc,QP = Stress in concrete adjacent to tendons
Traffic gr1a TS - for Bending design 1 Loading MApplied = 78.74031 kN.m
PApplied = -19.96643 kN Corresponding stresses:
top stress = -19.96643/905051.1 + 78.74031/5.8116E8 = -0.0221 + 0.13548
= 0.11342 MPa
bottom stress = -19.96643/905051.1 + 78.74031/-2.586E8 = -0.0221 + -0.304
= -0.3265 MPa
Traffic gr1a UDL - for Bending design 1 Loading MApplied = 32.7097 kN.m
PApplied = -10.9063 kN Corresponding stresses:
top stress = -10.9063/905051.1 + 32.7097/5.8116E8 = -0.0121 + 0.05628
= 0.04423 MPa
bottom stress = -10.9063/905051.1 + 32.7097/-2.586E8 = -0.0121 + -0.126
= -0.1385 MPa
Traffic gr1a Footway - for Bending design 1 Loading MApplied = 25.29156 kN.m
PApplied = -3.118579 kN Corresponding stresses:
top stress = -3.118579/905051.1 + 25.29156/5.8116E8 = -0.0034 + 0.04352
= 0.04007 MPa
bottom stress = -3.118579/905051.1 + 25.29156/-2.586E8 = -0.0034 + -0.098
= -0.1012 MPa
TOTAL LOSS OF PRESTRESS SUMMARY
Initial stressing force = 0.0 kN Prestress after all losses at t = ∞ = 0.0 kN Corresponding loss = 100 %
LIMITING STRESSES IN PRECAST BEAM
Compression
EN 1992-2 Clause 7.2(102) k1.fck = 0.6*40.0 = 24.0 MPa
In the presence of confinement or increase in cover this may be increased by up to 10%, i.e to: = 26.4 MPa
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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.
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
Bestech Systems Limited CREEP REDISTRIBUTION according to EN 1992-2 Annex KK.7
Construction On Centering, Sc = G + P1 + P2
VARIABLE ACTIONS - CHARACTERISTIC COMBINATION Traffic
Selected case: ψ0 Traffic gr1a TS 1 78.7403 0.18258 0.11855 0.13548 -0.3045 0.75 -19.966 -0.0209 -0.0209 -0.0221 -0.0221 Traffic gr1a UDL 1 32.7097 0.07585 0.04925 0.05628 -0.1265 0.75 -10.906 -0.0114 -0.0114 -0.0121 -0.0121 Traffic gr1a FT 1 25.2916 0.05865 0.03808 0.04352 -0.0978 0.4 -3.1186 -0.0033 -0.0033 -0.0034 -0.0034 Total (Leading) : 0.28154 0.17035 0.19773 -0.5663 Total (in Combination) : 0.19177 0.11557 0.13427 -0.3893 Other traffic cases for comparison:
Traffic gr1a TS 2 -330.15 -0.7655 -0.4970 -0.5680 1.27682 0.75 51.2513 0.05356 0.05356 0.05663 0.05663 Traffic gr1a UDL 2 -136.32 -0.3161 -0.2052 -0.2345 0.52720 0.75 23.2038 0.02425 0.02425 0.02564 0.02564 Traffic gr1a FT 2 25.2916 0.05865 0.03808 0.04352 -0.0978 0.4 -3.1186 -0.0033 -0.0033 -0.0034 -0.0034 Total (Leading) : -0.9484 -0.5896 -0.6803 1.78504 Total (in Combination) : -0.7307 -0.4544 -0.5242 1.37422 Temperature Restraint
None defined
Differential Temperature - Heating
Diff. Tmp H1 -1016.0 -413.84 2.47603 -0.8853 -0.8625 1.35841 Diff. Tmp H2 0.0 0.0 0.0 0.0 0.0 0.6 Differential Temperature - Cooling
Diff. Tmp C1 976.559 143.384 -1.4373 0.52916 0.24758 -1.7606 Diff. Tmp C2 2.0E-4 4.63E-7 3.01E-7 3.44E-7 -7.7E-7 0.6 Other Variable Action
None defined
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
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The most adverse case is with Traffic as leading action
Traffic 0.28154 0.17035 0.19773 -0.5663
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
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
Differential Temperature - Heating ψ1 = 0.6
Diff. Tmp H1 -1016.0 -413.84 2.47603 -0.8853 -0.8625 1.35841 Diff. Tmp H2 0.0 0.0 0.0 0.0 0.0 0.5 Differential Temperature - Cooling
ψ1 = 0.6
Diff. Tmp C1 976.559 143.384 -1.4373 0.52916 0.24758 -1.7606 Diff. Tmp C2 2.0E-4 4.63E-7 3.01E-7 3.44E-7 -7.7E-7 0.5 Other Variable Action
None defined
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 0.16962 0.10164 0.11824 -0.3488 ψ2 x Thermal 1.23801 -0.4426 0.12379 -0.8803 ψ2 x Other 0.0 0.0 0.0 0.0
TOTAL VARIABLE ACTIONS 1.40764 -0.3410 0.24203 -1.2292
TOTAL PERMANENT (from above) 0.0 -0.4932 1.16132 -0.4378
——————————————————————————————————
TOTAL COMBINATION 1.40764 -0.8342 1.40336 -1.6671
SLS FLEXURE
Precast Curvature Deflection
Stress E Strain (x10-6) (mm)
(MPa) (x10-6) (rad/m) Here Max.
After Transfer T 0.04918 ET 1.57982 2.16062 0.0 15.0561 B -0.0383 -1.229
After Erection T -0.2676 EI -14.415 -32.784 0.0 19.6107 B 0.52367 28.2042
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 10:15:31 Page: 16
© 2012 Bestech Systems Ltd
After in situ 1A T -0.2562 EI -13.799 -31.933 0.0 6.14087 B 0.51456 27.7139
After in situ 1B T -0.2562 EI -13.799 -31.933 0.0 5.84195 B 0.51456 27.7139
Long-term Dead T 1.16132 EL 99.2094 105.09 0.0 -4.8888 B -0.4378 -37.408
Diff. Temp H1 T -0.5175 ES -14.695 -29.104 0.0 1.59073 B 0.81504 23.1413
Diff. Temp H2 T 1.1E-6 ES 3.12E-5 7.79E-5 0.0 -1.1726 B -2.5E-6 -7.0E-5
Diff. Temp C1 T 0.14855 ES 4.2178 26.3161 0.0 -1.4413 B -1.0564 -29.993
Diff. Temp C2 T 2.06E-7 ES 5.86E-6 1.46E-5 0.0 0.40625 B -4.6E-7 -1.3E-5
Traffic gr1a 1 T 0.19773 ES 5.61411 16.6888 0.0 -6.7548 B -0.5663 -16.081
Extreme in-service 92.6748 0.0 -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.