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BS EN  CLAUSE No.  SUBJECT 

199211 Design of Concrete Structures 
3.1.4(6)  Differential Shrinkage 
BS EN 19922 Design of Concrete Structures  

CLAUSE No.  SUBJECT 
3.1.4(6)  Differential Shrinkage 
Problem:
In determining the exposed perimeter 'u' for a pretensioned beam with insitu concrete deck slab:
1) Would the insitu deck soffit be considered as exposed with the presence of the permanent formwork?
2) Would the top of the insitu deck surface be considered as exposed with the presence of waterproofing?
Solution:
The response from the Eurocode Expert was:
“Most permanent formwork will prevent the majority of water loss from the insitu slab, but some drying shrinkage will occur before the waterproofing goes on from the upper surface.”
Having tried a few various solutions for when the waterproofing is applied and when the deck slab is cast, it is possible to present a Construction Programme which will, theoretically, produce negligible differential shrinkage between the insitu deck slab and precast beam.
This may sound desireable, but there could be disasterous consequencies if the beam has been manufactured and there is a delay in the Programme. It is reasonable to consider the Construction Programme as being similar to a design loading case and use the worse condition.
Below are examples of the results from different construction programmes.
Section Properties
Concrete Grades
Beam C40/50 f_{ck} = 40 N/mm^{2}
Beam at transfer C32/40 f_{ck} = 32 N/mm^{2}
Slab C32/40 f_{ck} = 32 N/mm^{2}
BS EN 199211
Table 3.1
Consider modular ratio effect for different concrete strengths between beam and slab then:
E_{cm,slab} = 33.3GPa
E_{cm,beam} = 35.2GPa
Converting slab to beam units (f_{ck} = 40 N/mm^{2}) then effective width of composite slab = 1000 × 33.3/35.2 = 946mm
Concrete Grade
C40/50 f_{ck} = 40 N/mm^{2}
Note: The section has been simplified. The slab should be divided into 3 layers to account for the beam nib within the slab and the thickness of the permanent formwork.
Property
Beam Section
Composite Section
Area(mm^{2})
449.22×10 ^{3}
591.12×10 ^{3}
Centroid(mm)
456
625
2nd Moment of Area(mm^{4})
52.905×10 ^{9}
101.476×10 ^{9}
Modulus @ Level 1(mm^{3})
116.020×10 ^{6}
162.362×10 ^{6}
Modulus @ Level 2(mm^{3})
89.066×10 ^{6}
238.768×10 ^{6}
Modulus @ Level 3(mm^{3})

176.480×10 ^{6}
Differential Shrinkage Effects
BS EN 199211
cl.3.1.4(6)
The theoretical differential shrinkage between the beam and the deck slab can be calculated at every stage of construction and throughout the design life of the bridge.
Example of Staged Construction Calculation
(Not recommended unless the construction programme has been well defined.)
Assume the time scale for the stages of construction relative to the initial curing of the beam to be:
i) Stress transfer after initial set of concrete beam : t = 1 day
ii) Cast deck slab onto beams = 1^{1}/_{2} months : say t = 45 days
iii) Apply waterproofing to deck slab = 4 months : say t = 120 days
iv) Complete bridge and open to traffic = 6 months : say t = 180 days
Total shrinkage strain = ε_{cs} = ε_{cd} + ε_{ca} .....................................................eqn.(3.8)
Where drying shrinkage strain ε_{cd}(t) = β_{ds}(t,t_{s}) × k_{h} × ε_{cd,0} ........eqn.(3.9)
And autogenous shrinkage strain ε_{ca}(t) = β_{as}(t) × ε_{ca}(∞) .................eqn.(3.11)
Shrinkage in Beam
Table 3.1
f_{cm} = f_{ck} + 8 = 40 + 8 = 48N/mm^{2}
Cross sectional area A_{c} = 449.22×10 ^{3}
Perimeter of beam in contact with atmosphere = u = 3100mm
Table 3.3
k_{h} = 0.76 (by interpolation)
Eqn.3.10
β_{ds}(t,t_{s}) = (t  t_{s}) / [(t  t_{s}) + 0.04 √(h_{o}^{3})]
At begining of drying shrinkage: Age of concrete = t_{s} = 1 day
β_{ds}(1,1) = 0 ......................(at transfer)
β_{ds}(1,45) = (451) / [(451) + 0.04 √(290^{3})] = 0.18 ......................(at casting deck)
β_{ds}(1,180) = (1801) / [(1801) + 0.04 √(290^{3})] = 0.48 .......................(at open for traffic)
β_{ds}(Long term) = 1.0
Annex B(B.11)
Basic drying shrinkage strain = 0.85[(220 + 110 ⋅ α_{ds1}) ⋅ exp( α_{ds2} ⋅ f_{cm} / f_{cmo})] ⋅ 10^{6} ⋅ β_{RH}
Annex B(B.12)
β_{RH} = 1.55[1  (RH / RH_{o})^{3}]
RH = 70%
RH_{o} = 100% ∴ β_{RH} = 1.02
It is probable that the beam will be constructed using rapid hardening cement ∴ assume cement Class R which gives:
α_{ds1} = 6 α_{ds2} = 0.11
f_{cm} = 48N/mm^{2} f_{cmo} = 10N/mm^{2}
Hence Basic drying shrinkage strain = ε_{cd,o} = 450 με
Eqn 3.9
Drying shrinkage strain ε_{cd}(t) = β_{ds}(t,t_{s}) × k_{h} × ε_{cd,0}
At Stress Transfer
At Deck Cast
At Open to Traffic
Long Term
Units
t
1
45
180
∞
(days)
β_{ds}(t,t_{s})
0
0.18
0.48
1.0
ε_{cd}(t)
0
62
164
342
(με)
Eqn 3.12
Long term autogenous shrinkage strain = ε_{ca}(∞) = 2.5(f_{ck}  10)10^{6} = (2.5 × 30)10^{6} = 75 με
Eqn. 3.13
β_{as}(t) = 1  exp(  0.2t^{0.5})
Eqn. 3.11
Autogenous shrinkage strain ε_{ca}(t) = β_{as}(t) × ε_{ca}(∞)
At Stress Transfer
At Deck Cast
At Open to Traffic
Long Term
Units
t
1
45
180
∞
(days)
β_{as}(t)
0.18
0.74
0.93
1.0
ε_{ca}(t)
14
56
70
75
(με)
Eqn. 3.8
ε_{cs} = ε_{cd} + ε_{ca}
Σ Shrinkage Strain
_________
14
_________
118
_________
234
_________
417
(με)
Shrinkage in Deck Slab
Following the same procedure for the beam the shrinkage in the deck slab can be estimated.
Table 3.1
f_{cm} = f_{ck} + 8 = 32 + 8 = 40N/mm^{2}
Cross sectional area A_{c} = 1000 × 150 = 150×10 ^{3}
Note:
Permanent formwork and waterproofing will reduce the perimeter of the deck which is exposed to drying.
An additional stage can be introduced for waterproofing between ‘casting the deck’ and ‘bridge open to traffic’. If the period between casting the deck and bridge opening is short then this stage could be omitted.
There is no moisture loss through the waterproofing so the top surface of the deck is assumed to be at the mid notional thickness of the member. Similarly with the permanent formwork, so for the notional size of the cross section:
Before waterproofing h_{o1} = 150 + 150 = 300mm
After waterproofing h_{o2} = 150 + 150 + 150 = 450mm
Table 3.3
Before waterproofing k_{h1} = 0.75
After waterproofing k_{h2} = 0.71
Assume an initial curing period t_{s} of 3 days and Class N concrete.
Annex B(B.11)
Basic drying shrinkage strain = 0.85[(220 + 110 ⋅ α_{ds1}) ⋅ exp( α_{ds2} ⋅ f_{cm} / f_{cmo})] ⋅ 10^{6} ⋅ β_{RH}
Annex B(B.12)
β_{RH} = 1.02 (as before)
α_{ds1} = 4 α_{ds2} = 0.12
f_{cm} = 40N/mm^{2} f_{cmo} = 10N/mm^{2}
Hence Basic drying shrinkage strain = ε_{cd,o} = 354 με
Initial period for drying shrinkage in deck slab is between deck casting and waterproofing (between day 45 and day 120 on the programme) so (t  t_{s}) = (75  3) = 72 days.
β_{ds}(t,t_{s}) = 72 / [72 + 0.04 √(300^{3})] = 0.257
ε_{cd}(t) = β_{ds}(t,t_{s}) ⋅ k_{h1} ⋅ ε_{cd,0} = 0.257 × 0.75 × 354 = 68.2 με
The next stage of the drying shrinkage, between waterproofing and open to traffic, has a reduced exposure surface. An estimated artificial start time for this period can be obtained which would give the same drying shrinkage (68.2 με) for the reduced exposure as was produced during the initial drying period.
(t  t_{s}) ⋅ k_{h2} ⋅ ε_{cd,0} = [(t  t_{s}) + 0.04 √(h_{o2}^{3})] ⋅ ε_{cd}
(t  t_{s}) × 0.71 × 354 = [(t  t_{s}) + 0.04 × √(450^{3})] × 68.2
(t  t_{s}) = 381.8 / 2.685 = 142 days
Artificial time when open to traffic = 142 + (180  120) = 202 days
So period for drying shrinkage in deck slab up to ‘open to traffic’ in terms of reduced exposure condition = (t  t_{s}) = (202  3) = 199 days.
β_{ds}(t,t_{s}) = 199 / [199 + 0.04 √(450^{3})] = 0.34
ε_{cd}(t) = β_{ds}(t,t_{s}) ⋅ k_{h1} ⋅ ε_{cd,0} = 0.34 × 0.71 × 354 = 85 με
Eqn 3.12
Long term autogenous shrinkage strain = ε_{ca}(∞) = 2.5(f_{ck}  10)10^{6} = (2.5 × 22)10^{6} = 55 με
At Waterproofing
At Open to Traffic
Long Term
Units
t
75
135
∞
(days)
t  t_{s}
72
132
∞
(days)
Eqn. 3.13
β_{as}(t)
0.82
0.90
1.0
Eqn. 3.10
β_{ds}(t,t_{s})
0.26
0.34
1.0
Eqn. 3.11
ε_{ca}(t)
45
50
55
(με)
Eqn. 3.9
ε_{cd}(t)
68
85
251
(με)
Eqn. 3.8
ε_{cs} = ε_{cd} + ε_{ca}
Σ Shrinkage Strain
_________
113
_________
135
_________
306
(με)
Differential shrinkage when bridge open to traffic = (135  0)  (234  118) = 19με
Long Term differential shrinkage = (306  0)  (417  118) = 7με
Note: If the deck is cast sooner than 45 days then the differential shrinkage is reduced. For example, if the deck is cast at 30 days then the long term differential shrinkage works out to be 10με which suggests the beam shrinks more than the insitu deck slab (a new phenomenon !).
However, at design stage, it could be a false economy to tie the contractor to a rigid construction programme in order to limit the differential shrinkage between the beam and the deck slab.
Any delays in pouring the deck slab, or applying the waterproofing would increase the differential shrinkage and this should be considered at design stage.
Assumptions
Total shrinkage strain = ε_{cs} = ε_{cd} + ε_{ca} .....................................................eqn.(3.8)
Where drying shrinkage strain ε_{cd}(t) = β_{ds}(t,t_{s}) × k_{h} × ε_{cd,0} ........eqn.(3.9)
And autogenous shrinkage strain ε_{ca}(t) = β_{as}(t) × ε_{ca}(∞) .................eqn.(3.11)
Shrinkage in Beam
Table 3.1
f_{cm} = f_{ck} + 8 = 40 + 8 = 48N/mm^{2}
Cross sectional area A_{c} = 449.22×10 ^{3}
Perimeter of beam in contact with atmosphere = u = 3100mm
Table 3.3
k_{h} = 0.76 (by interpolation)
Eqn.3.10
β_{ds}(t,t_{s}) = (t  t_{s}) / [(t  t_{s}) + 0.04 √(h_{o}^{3})]
At begining of drying shrinkage: Age of concrete = t_{s} = 1 day
β_{ds}(1,1) = 0 ......................(at transfer)
β_{ds}(1,180) = (1801) / [(1801) + 0.04 √(290^{3})] = 0.48 ......................(at casting deck)
β_{ds}(1,360) = (3601) / [(3601) + 0.04 √(290^{3})] = 0.65 .......................(at open for traffic)
β_{ds}(Long term) = 1.0
Annex B(B.11)
Basic drying shrinkage strain = 0.85[(220 + 110 ⋅ α_{ds1}) ⋅ exp( α_{ds2} ⋅ f_{cm} / f_{cmo})] ⋅ 10^{6} ⋅ β_{RH}
Annex B(B.12)
β_{RH} = 1.55[1  (RH / RH_{o})^{3}]
RH = 70%
RH_{o} = 100% ∴ β_{RH} = 1.02
It is probable that the beam will be constructed using rapid hardening cement ∴ assume cement Class R which gives:
α_{ds1} = 6 α_{ds2} = 0.11
f_{cm} = 48N/mm^{2} f_{cmo} = 10N/mm^{2}
Hence Basic drying shrinkage strain = ε_{cd,o} = 450 με
Eqn 3.9
Drying shrinkage strain ε_{cd}(t) = β_{ds}(t,t_{s}) × k_{h} × ε_{cd,0}
At Stress Transfer
At Deck Cast
At Open to Traffic
Long Term
Units
t
1
180
360
∞
(days)
β_{ds}(t,t_{s})
0
0.48
0.65
1.0
ε_{cd}(t)
0
164
222
342
(με)
Eqn 3.12
Long term autogenous shrinkage strain = ε_{ca}(∞) = 2.5(f_{ck}  10)10^{6} = (2.5 × 30)10^{6} = 75 με
Eqn. 3.13
β_{as}(t) = 1  exp(  0.2t^{0.5})
Eqn. 3.11
Autogenous shrinkage strain ε_{ca}(t) = β_{as}(t) × ε_{ca}(∞)
At Stress Transfer
At Deck Cast
At Open to Traffic
Long Term
Units
t
1
180
360
∞
(days)
β_{as}(t)
0.18
0.93
0.98
1.0
ε_{ca}(t)
14
70
74
75
(με)
Eqn. 3.8
ε_{cs} = ε_{cd} + ε_{ca}
Σ Shrinkage Strain
_________
14
_________
234
_________
296
_________
417
(με)
Shrinkage in Deck Slab
Following the same procedure for the beam the shrinkage in the deck slab can be estimated.
Table 3.1
f_{cm} = f_{ck} + 8 = 32 + 8 = 40N/mm^{2}
Cross sectional area A_{c} = 1000 × 150 = 150×10 ^{3}
Note:
Permanent formwork and waterproofing will reduce the perimeter of the deck which is exposed to drying.
However, as the nature of formwork is not known and the timescale for waterproofing has not been defined, then, in this case, it is safer to assume these effects can be ignored.
So for the notional size of the cross section h_{o} = 2 × 150×10 ^{3} / 2000 = 150mm
Table 3.3
k_{h} = 0.93
Assume an initial curing period t_{s} of 3 days and Class N concrete.
Eqn.3.10
β_{ds}(t,t_{s}) = (t  t_{s}) / [(t  t_{s}) + 0.04 √(h_{o}^{3})]
At begining of drying shrinkage: Age of concrete = t_{s} = 3 days
β_{ds}(3,3) = 0 ......................(at end of curing)
β_{ds}(3,180) = (1803) / [(1803) + 0.04 √(150^{3})] = 0.71 .......................(at open for traffic)
β_{ds}(Long term) = 1.0
Annex B(B.11)
Basic drying shrinkage strain = 0.85[(220 + 110 ⋅ α_{ds1}) ⋅ exp( α_{ds2} ⋅ f_{cm} / f_{cmo})] ⋅ 10^{6} ⋅ β_{RH}
Annex B(B.12)
β_{RH} = 1.02 (as before)
α_{ds1} = 4 α_{ds2} = 0.12
f_{cm} = 40N/mm^{2} f_{cmo} = 10N/mm^{2}
Hence Basic drying shrinkage strain = ε_{cd,o} = 354 με
Eqn 3.9
Drying shrinkage strain ε_{cd}(t) = β_{ds}(t,t_{s}) × k_{h} × ε_{cd,0}
At Deck Cast
At Open to Traffic
Long Term
Units
t
3
180
∞
(days)
Eqn. 3.10
β_{ds}(t,t_{s})
0
0.71
1.0
Eqn. 3.9
ε_{cd}(t)
0
234
329
(με)
Eqn 3.12
Long term autogenous shrinkage strain = ε_{ca}(∞) = 2.5(f_{ck}  10)10^{6} = (2.5 × 22)10^{6} = 55 με
Eqn. 3.13
β_{as}(t) = 1  exp(  0.2t^{0.5})
Eqn. 3.11
Autogenous shrinkage strain ε_{ca}(t) = β_{as}(t) × ε_{ca}(∞)
At Deck Cast
At Open to Traffic
Long Term
Units
t
3
180
∞
(days)
Eqn. 3.13
β_{as}(t)
0.29
0.93
1.0
Eqn. 3.11
ε_{ca}(t)
16
51
55
(με)
Eqn. 3.8
ε_{cs} = ε_{cd} + ε_{ca}
Σ Shrinkage Strain
_________
16
_________
285
_________
384
(με)
Differential shrinkage when bridge open to traffic = (285  16)  (296  234) = 207με
Long Term differential shrinkage = (384  16)  (417  234) = 185με
(It is worth noting that a value of 200με, for the Differential Shrinkage Strain between the slab and beam, was assumed in the BS 5400 design).