Embed Size (px)
DESCRIPTION
Calculates crack widths in reinforced concrete.
Citation preview
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1200
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 25
Autogenous shrinkage µε 22
µε 222
Restraint R 0.50 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 72
Risk of early age cracking 1.05
Early-age crack-inducing strain µε 29
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.40
Restraint to drying shrinkage 0.40
Long term restrained strain µε 82
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 45
Total strain (early-age + long term)
Free contraction µε 537
Restrained contraction µε 154
Crack-inducing strain µε 74
Reinforcement details
Bar diameter φ mm 20
Bar spacing s mm 110
Cover c mm 50
Area of steel per face per m 2856
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 390
Minimum area of steel per face per m 1637
Crack spacing and width
mm 150
Steel ratio for estimating crack spacing 0.01904
Coefficient for bond characteristics 0.8
Crack spacing mm 527
Early age crack width mm 0.02
Long term crack width mm 0.04
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 2737
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 51
Autogenous shrinkage µε 22
µε 430
Restraint R 0.50 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 140
Risk of early age cracking 2.04
Early-age crack-inducing strain µε 97
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.40
Restraint to drying shrinkage 0.40
Long term restrained strain µε 82
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 45
Total strain (early-age + long term)
Free contraction µε 745
Restrained contraction µε 222
Crack-inducing strain µε 142
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.05
Long term crack width mm 0.08
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 20
Autogenous shrinkage µε 22
µε 182
Restraint R 0.50 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 59
Risk of early age cracking 0.86
Early-age crack-inducing strain µε 16
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.40
Restraint to drying shrinkage 0.40
Long term restrained strain µε 82
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 45
Total strain (early-age + long term)
Free contraction µε 497
Restrained contraction µε 141
Crack-inducing strain µε 61
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.01
Long term crack width mm 0.03
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 20
Autogenous shrinkage µε 22
µε 182
Restraint R 0.55 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 65
Risk of early age cracking 0.95
Early-age crack-inducing strain µε 22
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.44
Restraint to drying shrinkage 0.44
Long term restrained strain µε 90
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 53
Total strain (early-age + long term)
Free contraction µε 497
Restrained contraction µε 155
Crack-inducing strain µε 75
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.01
Long term crack width mm 0.04
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 20
Autogenous shrinkage µε 22
µε 182
Restraint R 0.60 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 71
Risk of early age cracking 1.04
Early-age crack-inducing strain µε 28
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.48
Restraint to drying shrinkage 0.48
Long term restrained strain µε 98
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 61
Total strain (early-age + long term)
Free contraction µε 497
Restrained contraction µε 169
Crack-inducing strain µε 89
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.02
Long term crack width mm 0.05
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 20
Autogenous shrinkage µε 22
µε 182
Restraint R 0.65 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 77
Risk of early age cracking 1.12
Early-age crack-inducing strain µε 34
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.52
Restraint to drying shrinkage 0.52
Long term restrained strain µε 106
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 69
Total strain (early-age + long term)
Free contraction µε 497
Restrained contraction µε 183
Crack-inducing strain µε 104
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.02
Long term crack width mm 0.06
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 20
Autogenous shrinkage µε 22
µε 182
Restraint R 0.70 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 83
Risk of early age cracking 1.21
Early-age crack-inducing strain µε 40
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.56
Restraint to drying shrinkage 0.56
Long term restrained strain µε 115
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 78
Total strain (early-age + long term)
Free contraction µε 497
Restrained contraction µε 197
Crack-inducing strain µε 118
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.02
Long term crack width mm 0.06
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 20
Autogenous shrinkage µε 22
µε 182
Restraint R 0.75 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 89
Risk of early age cracking 1.30
Early-age crack-inducing strain µε 46
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.60
Restraint to drying shrinkage 0.60
Long term restrained strain µε 123
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 86
Total strain (early-age + long term)
Free contraction µε 497
Restrained contraction µε 212
Crack-inducing strain µε 132
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.03
Long term crack width mm 0.07
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 20
Autogenous shrinkage µε 22
µε 182
Restraint R 0.80 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 95
Risk of early age cracking 1.38
Early-age crack-inducing strain µε 52
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.64
Restraint to drying shrinkage 0.64
Long term restrained strain µε 131
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 94
Total strain (early-age + long term)
Free contraction µε 497
Restrained contraction µε 226
Crack-inducing strain µε 146
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.03
Long term crack width mm 0.08
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 25
Autogenous shrinkage µε 22
µε 222
Restraint R 0.50 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 72
Risk of early age cracking 1.05
Early-age crack-inducing strain µε 29
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.40
Restraint to drying shrinkage 0.40
Long term restrained strain µε 82
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 45
Total strain (early-age + long term)
Free contraction µε 537
Restrained contraction µε 154
Crack-inducing strain µε 74
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.02
Long term crack width mm 0.04
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 25
Autogenous shrinkage µε 22
µε 222
Restraint R 0.55 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 79
Risk of early age cracking 1.16
Early-age crack-inducing strain µε 37
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.44
Restraint to drying shrinkage 0.44
Long term restrained strain µε 90
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 53
Total strain (early-age + long term)
Free contraction µε 537
Restrained contraction µε 169
Crack-inducing strain µε 90
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.02
Long term crack width mm 0.05
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 25
Autogenous shrinkage µε 22
µε 222
Restraint R 0.60 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 87
Risk of early age cracking 1.26
Early-age crack-inducing strain µε 44
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.48
Restraint to drying shrinkage 0.48
Long term restrained strain µε 98
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 61
Total strain (early-age + long term)
Free contraction µε 537
Restrained contraction µε 185
Crack-inducing strain µε 105
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.02
Long term crack width mm 0.06
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 25
Autogenous shrinkage µε 22
µε 222
Restraint R 0.65 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 94
Risk of early age cracking 1.37
Early-age crack-inducing strain µε 51
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.52
Restraint to drying shrinkage 0.52
Long term restrained strain µε 106
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 69
Total strain (early-age + long term)
Free contraction µε 537
Restrained contraction µε 200
Crack-inducing strain µε 120
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.03
Long term crack width mm 0.07
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 25
Autogenous shrinkage µε 22
µε 222
Restraint R 0.70 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 101
Risk of early age cracking 1.48
Early-age crack-inducing strain µε 58
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.56
Restraint to drying shrinkage 0.56
Long term restrained strain µε 115
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 78
Total strain (early-age + long term)
Free contraction µε 537
Restrained contraction µε 216
Crack-inducing strain µε 136
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.03
Long term crack width mm 0.07
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 25
Autogenous shrinkage µε 22
µε 222
Restraint R 0.75 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 108
Risk of early age cracking 1.58
Early-age crack-inducing strain µε 65
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.60
Restraint to drying shrinkage 0.60
Long term restrained strain µε 123
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 86
Total strain (early-age + long term)
Free contraction µε 537
Restrained contraction µε 231
Crack-inducing strain µε 151
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.04
Long term crack width mm 0.08
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 25
Autogenous shrinkage µε 22
µε 222
Restraint R 0.80 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 115
Risk of early age cracking 1.69
Early-age crack-inducing strain µε 73
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.64
Restraint to drying shrinkage 0.64
Long term restrained strain µε 131
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 94
Total strain (early-age + long term)
Free contraction µε 537
Restrained contraction µε 246
Crack-inducing strain µε 167
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.04
Long term crack width mm 0.09
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 30
Autogenous shrinkage µε 22
µε 262
Restraint R 0.50 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 85
Risk of early age cracking 1.24
Early-age crack-inducing strain µε 42
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.40
Restraint to drying shrinkage 0.40
Long term restrained strain µε 82
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 45
Total strain (early-age + long term)
Free contraction µε 577
Restrained contraction µε 167
Crack-inducing strain µε 87
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.02
Long term crack width mm 0.05
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 30
Autogenous shrinkage µε 22
µε 262
Restraint R 0.55 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 94
Risk of early age cracking 1.37
Early-age crack-inducing strain µε 51
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.44
Restraint to drying shrinkage 0.44
Long term restrained strain µε 90
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 53
Total strain (early-age + long term)
Free contraction µε 577
Restrained contraction µε 184
Crack-inducing strain µε 104
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.03
Long term crack width mm 0.06
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 30
Autogenous shrinkage µε 22
µε 262
Restraint R 0.60 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 102
Risk of early age cracking 1.49
Early-age crack-inducing strain µε 59
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.48
Restraint to drying shrinkage 0.48
Long term restrained strain µε 98
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 61
Total strain (early-age + long term)
Free contraction µε 577
Restrained contraction µε 200
Crack-inducing strain µε 121
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.03
Long term crack width mm 0.07
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 30
Autogenous shrinkage µε 22
µε 262
Restraint R 0.65 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 111
Risk of early age cracking 1.62
Early-age crack-inducing strain µε 68
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.52
Restraint to drying shrinkage 0.52
Long term restrained strain µε 106
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 69
Total strain (early-age + long term)
Free contraction µε 577
Restrained contraction µε 217
Crack-inducing strain µε 137
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.04
Long term crack width mm 0.08
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 30
Autogenous shrinkage µε 22
µε 262
Restraint R 0.70 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 119
Risk of early age cracking 1.74
Early-age crack-inducing strain µε 76
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.56
Restraint to drying shrinkage 0.56
Long term restrained strain µε 115
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 78
Total strain (early-age + long term)
Free contraction µε 577
Restrained contraction µε 234
Crack-inducing strain µε 154
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.04
Long term crack width mm 0.08
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 30
Autogenous shrinkage µε 22
µε 262
Restraint R 0.75 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 128
Risk of early age cracking 1.87
Early-age crack-inducing strain µε 85
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.60
Restraint to drying shrinkage 0.60
Long term restrained strain µε 123
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 86
Total strain (early-age + long term)
Free contraction µε 577
Restrained contraction µε 251
Crack-inducing strain µε 171
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.05
Long term crack width mm 0.09
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
Risk and control of cracking due to continuous edge restraint
Input parameters Symbol Unit Value
Section details and material properties
Section thickness h mm 1500
Strength class MPa C40/50
Age at cracking days 3 Assume 3 days unless more reliable information is available
Creep factor 0.65
Sustained load factor 0.80
Coefficient of thermal expansion of concrete 8.0
Characteristic yield strength of reinforcement MPa 500 500 Mpa
Early age concrete properties
Tensile strength at cracking MPa 2.10
Elastic modulus GPa 30.2
Tensile strain capacity µε 86
Long term concrete properties
Tensile strength MPa 3.51 Mean 28-day value
Elastic modulus GPa 35.2 Mean 28-day value
Tensile strain capacity (sustained loading) µε 123
Early-age strain
Temperature drop 30
Autogenous shrinkage µε 22
µε 262
Restraint R 0.80 Use restraint calculator for walls or adjacent slabs; or historical data
Early-age restrained contraction µε 136
Risk of early age cracking 1.99
Early-age crack-inducing strain µε 93
Autogenous shrinkage (residual up to 28 days) µε 27
Long term temperature change 15
Drying shrinkage µε 168
Long term free contraction µε 315
Restrained long term strain
Restraint to long term thermal strains 0.64
Restraint to drying shrinkage 0.64
Long term restrained strain µε 131
Increase in tensile strain capacity µε 37
Long term crack-inducing strain 94
Total strain (early-age + long term)
Free contraction µε 577
Restrained contraction µε 267
Crack-inducing strain µε 187
Reinforcement details
Bar diameter φ mm 25
Bar spacing s mm 140
Cover c mm 50
Area of steel per face per m 3506
Cracking initiated at early age strain
Steel ratio for early age cracking 0.00420
Coefficient k 0.65
Coefficient 1
mm 488
Minimum area of steel per face per m 2047
Crack spacing and width
mm 156.25
Steel ratio for estimating crack spacing 0.02244
Coefficient for bond characteristics 0.8
Crack spacing mm 549
Early age crack width mm 0.05
Long term crack width mm 0.10
Minimum reinforcement requirement for late-life cracking only
Steel ratio for late-life cracking 0.0070
Minimum area of steel per face 3421
fck / fck,cube
tc
K1 K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
K2
αc µε/oC If aggregate is unknown use 12 µε / oC
fyk
fctm(tc) Mean value of tensile strength fctm(tc)
Ecm(tc) Mean value of elastic modulus Ecm(tc)
εctu(ea) εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
fctm
Ecm
εctu(lt) εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
T1oC T1 = Peak temperature - mean ambient temperature
εca(ea) EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction εfree(ea) εfree(ea) = T1αc+ εca
Restrained early-age strain and risk of cracking
εr(ea) εr(ea) = R1 K1 (T1 αc+ εca)
εr(ea)/εctu Low risk of early age cracking if εr(ea)/εctu < 1.
εcr(ea) εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
Long term strain (excluding early-age strain)
δεca(lt) δεca(lt) = εca(28) - εca(ea)
T2oC T2 and εcd only apply when causing differential contraction or when the
sections acting integrally are subject to external restraint. εcd
εfree(lt) εfree(lt) = δεca + T2 αc + εcd
R2Restraint will reduce as En / Eo approaches 1 in the long term
R3
εfree(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
δεctu δεctu = εctu(28) - εctu(ea)
εcr(lt) εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
εr(total) εfree(total) = εfree(ea) + εfree(lt)
εr(total) εr(total) = εr(ea) + εr(lt)
εcr(total) εcr(total) = εcr(ea) + εcr(lt)
As mm2
Minimum area of reinforcement As,min
fctm/fyk fctm / fyk = ρcrit
k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated
kc For pure tension kc = 1
Surface zone used in calculating As,min hs,min hs,min = k kc h/2
As,min mm2 As,min = (hs,min x 1000) (fctm / fyk) Highlighted if As < As,min
Surface zone defining the effective area of concrete in tension, Ac,eff
he,ef he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
ρp,eff ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1EN1992-1-1 recommends k1= 0.8 but provides a factor of 0.7 where good bond cannot be guaranteed. Hence k1 = 0.8/0.7 = 1.14
Sr,max Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
wk wk = εc(ea) Sr,max
wk wk = εc(total)Sr,max
fctm/fyk fctm / fyk = ρcrit
As,min mm2 Highlighted if As < As,min
Early Age Thermal Crack Control Reinforcement Design
Lee Tunnel
Client: Thames WaterLocation: Overflow Shaft - Results
BLC
ARM
RD
255664
May - 2010
