Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1200
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
25
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
222
εfree(ea) = T1αc+ εca
0.50
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
72 1.05
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
29
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.40
Restraint to drying shrinkage
R3
0.40
Long term restrained strain
εfree(lt)
µε
82
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
45
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
537
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
154
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
74
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
20
Bar spacing
s
mm
110
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
2856
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
390
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
1637
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
mm
150
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.01904
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
Sr,max
mm
527
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.02
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.04
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
2737
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
51
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
430
εfree(ea) = T1αc+ εca
0.50
Use restraint calculator for walls or adjacent slabs; or historical data
140
εr(ea) = R1 K1 (T1 αc+ εca)
2.04
Low risk of early age cracking if εr(ea)/εctu < 1.
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε µε
97
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.40
Restraint to drying shrinkage
R3
0.40
Long term restrained strain
εfree(lt)
µε
82
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
45
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
745
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
222
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
142
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.05
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.08
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
20
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
182
εfree(ea) = T1αc+ εca
0.50
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
59 0.86
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
16
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.40
Restraint to drying shrinkage
R3
0.40
Long term restrained strain
εfree(lt)
µε
82
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
45
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
497
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
141
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
61
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.01
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.03
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
20
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
182
εfree(ea) = T1αc+ εca
0.55
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
65 0.95
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
22
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.44
Restraint to drying shrinkage
R3
0.44
Long term restrained strain
εfree(lt)
µε
90
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
53
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
497
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
155
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
75
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.01
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.04
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
20
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
182
εfree(ea) = T1αc+ εca
0.60
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
71 1.04
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
28
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.48
Restraint to drying shrinkage
R3
0.48
Long term restrained strain
εfree(lt)
µε
98
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
61
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
497
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
169
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
89
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.02
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.05
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
20
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
182
εfree(ea) = T1αc+ εca
0.65
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
77 1.12
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
34
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.52
Restraint to drying shrinkage
R3
0.52
Long term restrained strain
εfree(lt)
µε
106
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
69
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
497
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
183
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
104
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.02
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.06
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
20
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
182
εfree(ea) = T1αc+ εca
0.70
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
83 1.21
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
40
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.56
Restraint to drying shrinkage
R3
0.56
Long term restrained strain
εfree(lt)
µε
115
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
78
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
497
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
197
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
118
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.02
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.06
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
20
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
182
εfree(ea) = T1αc+ εca
0.75
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
89 1.30
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
46
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.60
Restraint to drying shrinkage
R3
0.60
Long term restrained strain
εfree(lt)
µε
123
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
86
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
497
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
212
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
132
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.03
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.07
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
20
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
182
εfree(ea) = T1αc+ εca
0.80
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
95 1.38
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
52
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.64
Restraint to drying shrinkage
R3
0.64
Long term restrained strain
εfree(lt)
µε
131
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
94
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
497
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
226
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
146
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.03
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.08
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
25
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
222
εfree(ea) = T1αc+ εca
0.50
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
72 1.05
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
29
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.40
Restraint to drying shrinkage
R3
0.40
Long term restrained strain
εfree(lt)
µε
82
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
45
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
537
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
154
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
74
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.02
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.04
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
25
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
222
εfree(ea) = T1αc+ εca
0.55
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
79 1.16
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
37
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.44
Restraint to drying shrinkage
R3
0.44
Long term restrained strain
εfree(lt)
µε
90
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
53
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
537
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
169
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
90
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.02
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.05
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
25
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
222
εfree(ea) = T1αc+ εca
0.60
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
87 1.26
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
44
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.48
Restraint to drying shrinkage
R3
0.48
Long term restrained strain
εfree(lt)
µε
98
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
61
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
537
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
185
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
105
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.02
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.06
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
25
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
222
εfree(ea) = T1αc+ εca
0.65
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
94 1.37
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
51
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.52
Restraint to drying shrinkage
R3
0.52
Long term restrained strain
εfree(lt)
µε
106
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
69
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
537
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
200
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
120
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.03
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.07
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
25
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
222
εfree(ea) = T1αc+ εca
0.70
Use restraint calculator for walls or adjacent slabs; or historical data
101
εr(ea) = R1 K1 (T1 αc+ εca)
1.48
Low risk of early age cracking if εr(ea)/εctu < 1.
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε µε
58
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.56
Restraint to drying shrinkage
R3
0.56
Long term restrained strain
εfree(lt)
µε
115
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
78
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
537
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
216
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
136
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.03
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.07
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
25
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
222
εfree(ea) = T1αc+ εca
0.75
Use restraint calculator for walls or adjacent slabs; or historical data
108
εr(ea) = R1 K1 (T1 αc+ εca)
1.58
Low risk of early age cracking if εr(ea)/εctu < 1.
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε µε
65
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.60
Restraint to drying shrinkage
R3
0.60
Long term restrained strain
εfree(lt)
µε
123
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
86
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
537
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
231
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
151
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.04
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.08
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
25
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
222
εfree(ea) = T1αc+ εca
0.80
Use restraint calculator for walls or adjacent slabs; or historical data
115
εr(ea) = R1 K1 (T1 αc+ εca)
1.69
Low risk of early age cracking if εr(ea)/εctu < 1.
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε µε
73
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.64
Restraint to drying shrinkage
R3
0.64
Long term restrained strain
εfree(lt)
µε
131
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
94
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
537
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
246
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
167
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.04
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.09
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
30
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
262
εfree(ea) = T1αc+ εca
0.50
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
85 1.24
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
42
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.40
Restraint to drying shrinkage
R3
0.40
Long term restrained strain
εfree(lt)
µε
82
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
45
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
577
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
167
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
87
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.02
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.05
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
30
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
262
εfree(ea) = T1αc+ εca
0.55
Use restraint calculator for walls or adjacent slabs; or historical data
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε
94 1.37
εr(ea) = R1 K1 (T1 αc+ εca) Low risk of early age cracking if εr(ea)/εctu < 1.
µε
51
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.44
Restraint to drying shrinkage
R3
0.44
Long term restrained strain
εfree(lt)
µε
90
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
53
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
577
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
184
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
104
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.03
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.06
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
30
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
262
εfree(ea) = T1αc+ εca
0.60
Use restraint calculator for walls or adjacent slabs; or historical data
102
εr(ea) = R1 K1 (T1 αc+ εca)
1.49
Low risk of early age cracking if εr(ea)/εctu < 1.
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε µε
59
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.48
Restraint to drying shrinkage
R3
0.48
Long term restrained strain
εfree(lt)
µε
98
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
61
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
577
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
200
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
121
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.03
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.07
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
30
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
262
εfree(ea) = T1αc+ εca
0.65
Use restraint calculator for walls or adjacent slabs; or historical data
111
εr(ea) = R1 K1 (T1 αc+ εca)
1.62
Low risk of early age cracking if εr(ea)/εctu < 1.
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε µε
68
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.52
Restraint to drying shrinkage
R3
0.52
Long term restrained strain
εfree(lt)
µε
106
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
69
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
577
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
217
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
137
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.04
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.08
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
30
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
262
εfree(ea) = T1αc+ εca
0.70
Use restraint calculator for walls or adjacent slabs; or historical data
119
εr(ea) = R1 K1 (T1 αc+ εca)
1.74
Low risk of early age cracking if εr(ea)/εctu < 1.
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε µε
76
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.56
Restraint to drying shrinkage
R3
0.56
Long term restrained strain
εfree(lt)
µε
115
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
78
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
577
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
234
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
154
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.04
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.08
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
30
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
262
εfree(ea) = T1αc+ εca
0.75
Use restraint calculator for walls or adjacent slabs; or historical data
128
εr(ea) = R1 K1 (T1 αc+ εca)
1.87
Low risk of early age cracking if εr(ea)/εctu < 1.
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε µε
85
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.60
Restraint to drying shrinkage
R3
0.60
Long term restrained strain
εfree(lt)
µε
123
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
86
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
577
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
251
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
171
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.05
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.09
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
Lee Tunnel Client: Location:
Thames Water Overflow Shaft - Results
255664
BLC ARM
Early Age Thermal Crack Control Reinforcement Design
May - 2010
RD
Risk and control of cracking due to continuous edge restraint Input parameters
Symbol
Unit
Value
h
mm
1500
Strength class
fck / fck,cube
MPa
C40/50
Age at cracking
tc
days
3
Creep factor
K1
0.65
Sustained load factor
K2
0.80
Coefficient of thermal expansion of concrete
αc
µε/oC
8.0
If aggregate is unknown use 12 µε / oC
Characteristic yield strength of reinforcement
fyk
MPa
500
500 Mpa
Tensile strength at cracking
fctm(tc)
MPa
2.10
Mean value of tensile strength fctm(tc)
Elastic modulus
Ecm(tc)
GPa
30.2
Mean value of elastic modulus Ecm(tc)
Tensile strain capacity
εctu(ea)
µε
86
Tensile strength
fctm
MPa
3.51
Mean 28-day value
Elastic modulus
Ecm
GPa
35.2
Mean 28-day value
Tensile strain capacity (sustained loading)
εctu(lt)
µε
123
εctu(lt) = [ fctm / Ecm ] x [K2 / K1]
o
C
30
T1 = Peak temperature - mean ambient temperature
Section details and material properties Section thickness
Assume 3 days unless more reliable information is available K1 = 0.65 if R is calculated; K1 = 1 if R is assumed to be 0.5 (including creep to EN1992-1-1)
Early age concrete properties
εctu(ea) = [ fctm(tc) / Ecm(tc) ] x [K2 / K1]
Long term concrete properties
Early-age strain Temperature drop
T1
Autogenous shrinkage
εca(ea)
µε
22
EN1992-1-1 εca(ea) = 2.5 (fck - 10) x (1-exp(- 0.2 tc0.5)
Free contraction
εfree(ea)
µε
262
εfree(ea) = T1αc+ εca
0.80
Use restraint calculator for walls or adjacent slabs; or historical data
136
εr(ea) = R1 K1 (T1 αc+ εca)
1.99
Low risk of early age cracking if εr(ea)/εctu < 1.
Restrained early-age strain and risk of cracking Restraint
Early-age restrained contraction Risk of early age cracking
R εr(ea) εr(ea)/εctu
Early-age crack-inducing strain
εcr(ea)
Autogenous shrinkage (residual up to 28 days)
δεca(lt)
Long term strain (excluding early-age strain)
µε µε
93
εc(ea) = R1 K1 (T1αc + εca) - 0.5 εctu
µε
27
δεca(lt) = εca(28) - εca(ea)
Long term temperature change
T2
o
C
15
Drying shrinkage
εcd
µε
168
T2 and εcd only apply when causing differential contraction or when the sections acting integrally are subject to external restraint.
εfree(lt)
µε
315
εfree(lt) = δεca + T2 αc + εcd
Long term free contraction
Restrained long term strain Restraint to long term thermal strains
R2
0.64
Restraint to drying shrinkage
R3
0.64
Long term restrained strain
εfree(lt)
µε
131
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)}
Increase in tensile strain capacity
δεctu
µε
37
δεctu = εctu(28) - εctu(ea)
Long term crack-inducing strain
εcr(lt)
94
εc(lt) = K1 {R2T2αc + R3(δεca + εcd)} - δεctu
Restraint will reduce as En / Eo approaches 1 in the long term
Total strain (early-age + long term) Free contraction
εr(total)
µε
577
εfree(total) = εfree(ea) + εfree(lt)
Restrained contraction
εr(total)
µε
267
εr(total) = εr(ea) + εr(lt)
Crack-inducing strain
εcr(total)
µε
187
εcr(total) = εcr(ea) + εcr(lt)
Bar diameter
φ
mm
25
Bar spacing
s
mm
140
Reinforcement details
Cover
c
mm
50
Area of steel per face per m
As
mm2
3506
Cracking initiated at early age strain Minimum area of reinforcement As,min Steel ratio for early age cracking Coefficient Coefficient
fctm/fyk
0.00420
k
0.65
kc
1
fctm / fyk = ρcrit k = 1.0 for h ≤ 300mm; k = 0.65 for h ≥ 800mm; intermediate values are interpolated For pure tension kc = 1
Surface zone used in calculating As,min
hs,min
mm
488
hs,min = k kc h/2
Minimum area of steel per face per m
As,min
mm2
2047
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
mm
156.25
he,ef = 2.5 (c + φ/2) [NOTE: hs,min and he,ef are not the same]
Steel ratio for estimating crack spacing
ρp,eff
0.02244
ρp,eff = As / Ac,eff = As/ (he,ef x 1000)
k1
0.8
EN1992-1-1 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
Crack spacing and width
Coefficient for bond characteristics
Sr,max
mm
549
Sr,max = 3.4c + 0.425 k1 φ/ρp,eff
Early age crack width
wk
mm
0.05
wk = εc(ea) Sr,max
Long term crack width
wk
mm
0.10
wk = εc(total)Sr,max
Crack spacing
Minimum reinforcement requirement for late-life cracking only Steel ratio for late-life cracking
fctm/fyk
Minimum area of steel per face
As,min
0.0070 mm2
3421
fctm / fyk = ρcrit Highlighted if As < As,min
