ABSTRACT: The B3 model for concrete creep and shrinkage was first published in 1995 (Baweja & Bazant1995). This model encompasses all basic features of hardening concrete. It performs well within medium loadduration in years. On the other hand, for longer time frames, the model tends to provide underestimations. Itimplausibly predicts excessive deformation or deflections on several kinds of engineering structures, such asbridges. A recalibration of the B3 model parameters brings only partial remedy, and a new phenomenon needsto be introduced: An additional increase of the creep rate after approximately 1000 days of load duration. Thisbehavior may be explained by over-stressing cement paste and interfacial transition zone around aggregatescaused by a redistribution of the load from the creeping cement paste to the stiffer non-creeping aggregates.Another factor is the swelling of the paste at high relative humidity levels. This redistributing effect is significantat w/c > 0.5. Consequently, all creep-sensitive structures need to remain below this limit. We propose anextension of the B3 model based on basic creep data (Brooks 2005) from specimens which had been loaded for30 years.

1 INTRODUCTION

1.1 Historical background

The B3 model for concrete creep and shrinkage wasfirst published in 1995 (Baweja & Bazant 1995,Bazant & Baweja 1995a, Bazant & Baweja 1995b).The model considers concrete as an aging viscoelas-tic material according to solidification theory for con-crete creep (Bazant & Prasannan 1989). This ap-proach requires extensive calibration of twelve pa-rameters q1, q2 . . . q7, λ0,m,n,α1, α2, several of themdepending on concrete composition.

From the micromechanical point of view, concretecreep and shrinkage originate at the level of ce-ment paste on the scale of micrometers. Descend-ing even further to heterogeneous composition of ce-ment paste, creep is dominantly located at the levelof C-S-H, the main hydration product of alite and be-lite. Such micromechanical analysis of concrete creepwas given by Pichler and Lackner (Pichler & Lack-ner 2008) by means of correspondence principle orvia numerical FFT-based method (Smilauer & Bazant2010). To elucidate the creep transition across fourscales, Figure 1 considers 30-year basic creep of con-crete. For simplification, concrete is treated here asa non-aging material and the corresponding princi-ple in Laplace-Carson domain is utilized (Smilauer2012).

Since creep depends strongly on the behavior of

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sic

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ep

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mp

lian

ce

[(G

Pa

)-1]

Time (t-t’) [days]

C-S

-H

Pas

te

Mortar

Concrete

30 years

Experiment

Figure 1: Creep at four scales as predicted by continuum homog-enization methods, w/c = 0.5.

cement paste, the water-to-cement ratio (w/c) playsa significant role in concrete creep evolution and isa natural choice as a parameter for creep models. Ad-ditional effects need to be taken into account: aggre-gate stiffening role with creep mitigation and interfa-cial transition zone (ITZ) around aggregates. For thecase of the B3 model, the tedious calibration proce-dure was described (Bazant & Baweja 1995a), utiliz-ing about 600 time curves data of various concretecompositions from the Northwestern University databank.

Today, it is clear that long-term creep data are

Updating B3 model for long-term basic creep

V. Šmilauer, M. Lepš & M. GregorováFaculty of Civil Engineering, Department of Mechanics, Czech TechnicalUniversity Prague, Prague, Czech Republic

Life-Cycle and Sustainability of Civil Infrastructure Systems – Strauss, Frangopol & Bergmeister (Eds)© 2013 Taylor & Francis Group, London, ISBN 978-0-415-62126-7

1395

sparse, raising questions about long-term creep pre-diction of the B3 model. From the current, sightlyappended Northwestern creep database (Bazant & Li2008), containing 675 datasets (and 12258 points),only 154 datasets have at least one measurement after3 years and only 20 datasets after 10 years. Brooks(Brooks 2005) measured 30-year creep using 18 con-crete types from normal and light-weight aggregates.He concluded that the creep compliances were under-estimated by all prediction models (CEB, GL, B3,ACI, BS). These data were appended to the creepdatabase later.

It is evident that measured excessive long-termcreep within lab specimens will have consequencesfor existing concrete structures. By today, 66 large-span, segmentally erected bridges were gathered, ex-hibiting excessive deflections (Bazant, Hubler, & Yu2011). What does the term “excessive deflection”mean? Generally, it describes the situation wherea model prediction and real measurements stronglydeviate, say by a factor of two, during the service lifeof a bridge. Excessive deflections, exceeding approxi-mately 1/800 of the bridge span, may cause geometri-cally non-linear effects on the structure, decrease ser-viceability or traveling comfort.

It is also clear that the excessive long-term creepof concrete is not the only cause of excessively de-flecting bridges. There are other reasons, which aredifficult to identify especially due to missing data onconcrete composition, difficult communication withauthorities and contacting involved people witnessedthe construction stage. Nevertheless, the detailed in-vestigation revealed so far that the following issuesplay effects for the long-time deflections:

1. Using inappropriate creep models. There is noindication from existing structures and labora-tory tests that concrete creep ever stabilizes, i.e.has a finite upper bound on creep. Unfortunately,this is the feature of current ACI model, CEB-fibmodel, GL model, and Japanese JSCE and JRAmodels (Bazant, Hubler, & Yu 2011), widelyused in a bridge design. The excessive deflec-tions occur as a consequence of assumed finitecreep values. This is probably the cause of sev-eral Japanese bridges, such as Tsukiyono bridgewith w/c = 0.39 (Maekawa, Chijiwa, & Ishida2011).

2. Simultaneous creep and shrinkage. Several creepmodels, which have been formulated in theframework of the beam theory, introduce an aver-age strain over the entire cross section. Nonuni-form distribution of moisture leads to varyingshrinkage. Therefore, a coupled finite elementanalysis should be performed if the cross sectiondiffers in the thicknesses of its parts (web anddeck dimensions of box-girders).

3. Insufficient prestress/excessive steel relaxation.

Slender box-girder bridges have higher live/deadload ratio, resulting in decreased resistance todynamic effects. A pair of La Lutrive bridges inSwitzerland, finished in 1973 is a typical repre-sentative falling into this group. Figure 2 showscracks appearing on the bridge up to width of 0.1mm. It clearly demonstrates that prestressed con-crete was loaded in tension. The initiated crackshad never close up completely during the loadcycling and this explains their orientation on theweb. Post-tensioning in 1989 (both bridges) andin 2000 (one bridge) stabilized deflections onboth bridges (Burdet 2010).

4. Nonlinear creep. This paper further shows thatconcretes with w/c > 0.5 exhibit a significantnon-linear creep at times after approximately1000 days of loading (Brooks 2005). Theseconcretes were uniaxially loaded at the 0.3stress/strength ratio from 14 days of hardening.Increased creep rate becomes evident on con-cretes with w/c > 0.5, not affecting the ma-jority of contemporary bridges, but affectingolder bridges, e.g. the Koror-Babeldaob bridgein Palau had supposedly w/c = 0.62 (Bazant, Yu,Li, Klein, & Krıstek 2010).

Figure 2: Cracks appearing on the girder wall in La Lutrivebridge before additional prestressing.

2 B3 CREEP MODEL

B3 model expresses the strain evolution under a con-stant stress as (Bazant & Baweja 2000)

ε(t) = J(t, t′)σ + εsh(t) + α∆T (t), (1)

in which J(t, t′) is the compliance function = strain(creep plus elastic) at time t caused by a unit uni-axial constant stress applied at age t′ , σ = uniaxialstress , ε(t) = strain (both σ and ε are positive if ten-sile), εsh(t) = shrinkage strain (negative if volume de-creases), ∆T (t) = temperature change from a refer-ence temperature at time t, and α = thermal expansioncoefficient. The compliance function J(t, t′) may fur-ther be decomposed as

J(t, t′) = q1 +C0(t, t′) +Cd(t, t

′, t0), (2)

in which q1 = instantaneous strain due to a unit stress,C0(t, t

′) = compliance function for basic creep (creep

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at constant moisture content and no moisture move-ment through the material), and Cd(t, t

′, t0) = addi-tional compliance function due to simultaneous dry-ing (so-called drying creep). Note that autogeneousshrinkage is neglected in the current version of the B3model.

The total basic creep compliance C0(t, t′) is ex-

pressed as

C0(t, t′) = Cv(t, t

′) + q4 ln(t

t′

)(3)

Cv(t, t′) =

∫ t

t′v−1(t)Cg(τ − t′) dτ =

=∫ t

t′v−1(t)

n(τ − t′)n−1

λn0 + (τ − t′)ndτ (4)

v−1(t) =

[q2

(λ0t

)m

+ q3

](5)

The numerical integration of Equation (4) bringsproblems due to Cg(0) = ∞, so the numerical inte-gration has to be carried out with extremely fine timesteps after the introduction of load. For this reason,it is more advantageous to assume that v−1(tn+1/2)remains a constant within a sufficiently short time in-terval, where subscript n+ 1/2 refers to time tn+1/2

at the mid-step on log-time scale, i.e. the geometricalmean of tn and tn+1. The integral of Cg yields∫ n(τ − t′)n−1

λn0 + (τ − t′)ndτ = ln

[1 + λ−n0 (τ − t′)n

], (6)

which can be translated to Equation (4) when integrat-ing within a time interval tn and tn+1

Cv(tn+1/2, t′)tn+1tn =

[q2

(λ0

tn+1/2

)m

+ q3

]· (7)

{ln[1 + λ−n0 (tn+1 − t′)n]− ln[1 + λ−n0 (tn − t′)n]}.The advantage of Equation (7) is a very coarse sam-

pling (up to a few hundreds intervals for years) witha high accuracy of results.

The parameters q1, q2, q3, q4 in Equations (1)-(5)have such physical meaning; q1 asymptotic elas-tic compliance, q2 aging viscoelastic compliance,q3 non-aging viscoelastic compliance, q4 aging flowcompliance. Estimations of q1 to q4 parameters re-lated to concrete composition are given elsewhere,e.g. (Jirasek & Bazant 2002, pp. 668).

2.1 Long-term linear creep

Inverse analysis of the long-term basic creep needsonly these parameters: q1, q2, q3, λ0, n, m, q4. For thisspecific case

J(t, t′) = q1 +Cv(t, t′) + q4 ln

(t

t′

), (8)

where Cv(t, t′) is integrated from Equation (4). For

sufficiently long times (≈ years), the final creep func-tion yields

J(t− t′) = q1 + q3 ln[1 + λ−n0 (t− t′)n] +

+q4 ln(t− t′), (9)

J(t− t′) = nq3 + q4. (10)

Since n = 0.1 and q3 < q4 for ordinary concretes,the slope at a sufficiently long time is determinedmainly by q4. Neglecting q3 gives a relative error lessthan 10%, which is acceptable.

2.2 Creep recovery

Creep recovery provides further constraints to theB3 model parameters. The most relevant data seemto come from Brooks (Brooks 2005), who measuredcreep recovery after 30 years of loading. He con-cluded that the levels of creep recovery were smallcompared to the final values of creep. For the NorthNotts specimens stored in water, the creep recoverywas between 5-14 % at six months after the unload-ing. Let us define times as t′ = 14 days, end of load-ing t1 = 30 years = 10950 days, end of experimentsix months after the end of loading t2 = 30.5 years= 11133 days. The creep recovery leads to the follow-ing equations when considering the long-term loadingstage followed with the unloading

J(t2, t1)

J(t1, t′)≈

q3 ln[1 + (180)0.1] + q4 ln 1113310950

q3 ln[1 + (10936)0.1] + q4 ln 1095014

=0.986q3 + 0.0165q41.263q3 + 6.662q4

≈

≈ 0.986q31.263q3 + 6.662q4

= 0.05÷ 0.14 (11)

For ordinary concretes with w/c < 0.6, q3 is nor-mally smaller than q4. Equation (11) can be furthersimplified to

0.986q31.263q3 + 6.662q4

≈ 0.986q36.662q4

= 0.05÷ 0.14 (12)

q4 = (1.06÷ 2.97)q3 (13)

Equation (13) provides the ratio estimate betweenq3 and q4. This ratio may be even higher since the ag-ing viscoelastic compliance, introduced with q2, wasneglected.

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2.3 Micromechanically-nonlinear creep

Experiments lasting 30 years demonstrated that afterapproximately 1000 days of loading, the final slopeof basic creep increases (Brooks 2005). These exper-iments covered ordinary concretes with w/c ratii be-tween 0.5 and 0.8. The higher the w/c, the higherthe slope change. All specimens were loaded at the0.3 stress/strength ratio. Figure 3 shows remedy onthe Brooks experimental data with w/c = 0.54. In-creasing q3 or q4 from the original parameters threetimes leads to significant over-prediction of short-time creep while matching the final slope of long-termcreep data. The transition in creep slope occurs closeto (t− t′) = 1000 days, which is related to λ0 parame-ter, calibrated to much shorter 1 day. It is obvious thatthe B3 model required additional parameter to enableslope transition after certain time.

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B3 q3, q4 increased

B3 default

21.8*log(t-t’) - 76.4

Figure 3: Performance of the original B3 model and the predic-tion when increasing q3 and q4 three times.

Non-linear creep can be one possible explanationof this phenomenon, which could be supported bya micromechanical analysis. At loading, the stress re-distributes to different components of concrete, thestiffer components bear generally higher stress. Creepcauses stress redistribution away from the creepingto the non-creeping components. In the case of con-crete, the stress is transferred from creeping cementpaste to non-creeping aggregates. Such behavior wasdemonstrated at the level of cement paste, wherenon-creeping solid phases take away the load fromcreeping C-S-H, see Figure 4. Further details of FFT-based algorithm, representative volume element andparameters for creep function are described elsewhere(Smilauer & Bazant 2010).

Figure 5 shows the idea of stress redistribution ona swelling-shrinking specimen simultaneously loadedin compression. The cement paste, created to a largeextent by C-S-H, responds to high humidity withswelling. This redistributes stress away from aggre-gates to the paste so that the integrated stress givesback a uniform macroscopic stress. Opposite situa-tion happens under shrinking, causing drying creep

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ctio

n [

MP

a]

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Creeping C-S-H All solids except C-S-H

Figure 4: Stress redistribution at the level of cement paste.

and microcracking.

-

--

-

-

ITZ

Paste

Aggregate

Swelling Shrinking

Figure 5: Idea of stress redistribution to different components ofconcrete during swelling and shrinking of cement paste whilesimultaneously loaded in compression.

This micromechanically-nonlinear creep can be in-troduced to the B3 model. It must be mentioned thatonly 20 datasets out of 681 in the ITI creep databasecontain creep data older that 10 years. The flow term,quantified with q4 parameter, is modified with a func-tion taking w/c into account. For simplicity, the non-linear creep is introduced around 1000 days after theloading. The flow term is increased in the followingmanner

q4 ln(t

t′

)· (14)

[1 + ll · 13(w/c)3 ·

(1− 1

1 + (0.0005(t− t′))3

)]

ll =|σ(t′)|

0.3|fc(t′)|(15)

where ll is the load level for long-term creep. For aspecimen loaded at the load/strength ratio of 0.3 at thetime of loading, the ll = 1. Such ratio corresponds toanalyzed Brooks’ data in this paper. Figure 6 showsthe flow term multiplier from Equation (14), whichwas calibrated to Brooks’ data on basic creep (Brooks

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2005). No further calibrations to long-term creep datawith various load/strength ratii were taken into ac-count, especially due to missing long-term creep data.Therefore, Equation (14) gives only limited, yet up-dated, predictions.

Figure 7 demonstrates the differences betweenoriginal version of the B3 model and the updated ver-sion with the modification according to Equation (14).The figure plots the basic creep data from two types ofaggregates, where North Notts gravel is a good aggre-gate and rounded Stourton is a poor quality aggregate.It is clear that Stourton aggregates contribute addi-tionally to concrete compliance and shows the sametrend of increased creep rate. The ITZ seems to beunchanged in both aggregates since the slope changeoccurs roughly at the same time and the same cementfineness was used.

1

2

3

4

5

6

7

8

102

103

104

105

Th

e f

low

te

rm m

ultip

lier

(-)

Time t-t’ (days)

0.80.50.3

Figure 6: The flow term multiplier according to Equation (14),capturing the micromechanically-nonlinear creep.

Introducing micromechanically-nonlinear creepchanges the phenomenological law in B3 model.Macroscopically, the creep formulation and linearitymay be assumed with standard solution techniques.The load level introduced in Equation (15) needs aconstant value over long term. Introducing severalloading steps and actual load-level in each load in-crement leads to wrong results.

Brooks’ specimens subjected to simultaneous dry-ing at ambient relative humidity of 0.646 exhibit asmaller slope change after 1000 days (Brooks 2005).Drying causes C-S-H to shrink which distributesmacroscopic load to aggregates sooner (and thereforenonlinear creep becomes part of the drying creep) andthe effect is similar as in the case of long-term creep.On the other hand, C-S-H close to RH 1.0 (basiccreep) expands and hence takes away the load fromaggregates and over-stresses C-S-H.

2.4 Relating strength to concrete composition

The parameters q1, q2, q3 depend on concrete com-pressive strength at 28 days. It is well known thatcompressive strength is a function of mainly w/c and

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ep J

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Stourton

B3 updated

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B3 updated

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ep J

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ep J

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Stourton

B3 updated

Figure 7: Prediction of the original B3 creep model and updatedB3 model. The North Notts aggregate is a good aggregate whilethe Stourton aggregate is of poor quality.

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a/c ratii. Other factors, such as air entrainment, con-tribute to strength as well. The aim was fitting allcompressive strength to concrete composition, yield-ingfc = 29.18(w/c)−1.09(a/c)−0.328. (16)

Figure 8 plots the results for over 100 pointsfrom the creep database. The points were equallyweighted and yielded the correlation coefficient 0.821and the standard deviation of the strengths difference±11.72 MPa.

0

20

40

60

80

100

120

0 20 40 60 80 100 120

Pre

dic

ted

str

en

gth

(M

Pa

)

Measured strength (MPa)

29.18(w/c)-1.09

(a/c)-0.328

Figure 8: Measured and fitted 28-day compressive strength fromconcrete composition.

3 CONCLUSIONS

Basic creep experiments lasting 30 years had demon-strated a compliance slope increase when plotted onthe logarithmic time scale. The B3 model for basiccreep was enhanced to capture this feature. From astructural point of view, it is suggested to use con-cretes with w/c < 0.5 on creep-sensitive structuressuch as bridges, exhibiting smaller slope changes atlonger times.

The parameter q4 needs to be re-calibrated and re-lated more to concrete composition. Tentatively, ac-cording to Figure 6, the q4 parameter needs to be in-creased 2.5 times for the long-term creep at w/c = 0.5and 7.5 times for w/c = 0.8. This is a similar findingwith Bazant et al. (Bazant, Hubler, & Yu 2011) whosuggested to multiply q3 and q4 by a factor of 1.6 tomatch the final slopes.

4 ACKNOWLEDGEMENT

We acknowledge the financial support from the Min-istry of Industry and Trade of the Czech Republicunder the project FR-TI1/612, from the Czech Sci-ence Foundation project 105/10/2400 and from theTechnology Agency of the Czech Republic throughthe project TA01030733. Olivier Burdet from EPFL,Switzerland is acknowledged for private conversationand valuable comments on La Lutrieve bridges.

REFERENCES

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Bazant, Z. & G.-H. Li (2008). Comprehensive database on con-crete creep and shrinkage. ACI Materials Journal 105(6), 635– 637.

Bazant, Z., Q. Yu, G.-H. Li, G. Klein, & V. Krıstek (2010). Ex-cessive Deflections of Record-Span Prestressed Box Girder.Lessons learned from the collapse of the Koror-BabeldaobBridge in Palau. Concrete International 32(6), 44 – 52.

Bazant, Z. P. & S. Baweja (1995a, August). Justification andrefinement of Model B3 for concrete creep and shrinkage.1. Statistics and sensitivity. Materials and Structures 28(7),415–430.

Bazant, Z. P. & S. Baweja (1995b, Oct). Justification and refine-ment of Model B3 for concrete creep and shrinkage. 2. Up-dating and theoretical basis. Materials and Structures 28(8),488–495.

Bazant, Z. P. & S. Baweja (2000). Creep and shrinkage predic-tion model for analysis and design of concrete structures:Model B3. Technical report, Am. Concrete Institute.

Bazant, Z. P. & S. Prasannan (1989). Solidification theory forconcrete creep: I. Formulation. J. Engng. Mech. 115(8),1691–1703.

Baweja, S. & Z. P. Bazant (1995, July). Creep and shrinkageprediction model for analysis and design of concrete struc-tures — model B3. Materials and Structures 28(6), 357–365.RILEM, Paris.

Brooks, J. J. (2005). 30-year creep and shrinkage of concrete.Magazine of Concrete Research 57(9), 545–556.

Burdet, O. (2010). Experience in the Long-Term Monitoring ofBridges. In 3rd fib International Congress, Washington D.C.,USA, pp. 108–113.

Jirasek, M. & Z. P. Bazant (2002). Inelastic analysis of struc-tures. John Wiley & Sons, Ltd.

Maekawa, K., N. Chijiwa, & T. Ishida (2011). Long-term defor-mational simulation of pc bridges based on the thermo-hygromodel of micro-pores in cementitious composites. Cementand Concrete Research 41(12), 1310 – 1319.

Pichler, C. & R. Lackner (2008). A multiscale creep model asbasis for simulation of early-age concrete behavior. Comput-ers and Concrete 5(4), 295 – 328.

Smilauer, V. & Z. P. Bazant (2010). Identification of viscoelasticC-S-H behavior in mature cement paste by FFT-based ho-mogenization method. Cem. Concr. Res. 40(2), 197 – 207.

Smilauer, V. (2012). Multiscale hierarchical modeling of hydrat-ing concrete. Saxe-Coburg Publications. In preparation.

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