Tailor Made Concrete Structures – Walraven & Stoelhorst (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-47535-8

Modelling of shrinkage induced curvature of cracked concrete beams

R. Mu, J.P. Forth & A.W. BeebyThe University of Leeds, Leeds, UK

R. ScottUniversity of Durham, Durham, UK

ABSTRACT: Besides load and temperature, shrinkage and creep are the main factors that influence the curva-ture of reinforced concrete sections. For cracked sections, this effect is calculated semi-empirically in structuraldesign codes such as BS8110 and EC2. In order to verify the accuracy of the calculation, the curvature ofcracked section of concrete beams due to shrinkage was analyzed numerically and validated experimentally. Theanalysis in this paper is based on the mechanical equilibrium method and with basic assumptions of plane sec-tions remaining plane and linear creep superposition. This method divides a section into a number of strips. Theneutral axis position and curvature of the section were determined by iteration until equilibrium was obtained.To verify the calculation results, two beams, cast with different shrinkage but similar creep concrete, were tested.In theory, any difference in curvature or deflection between the two beams was therefore caused by shrinkage.The results showed that the model proposed in this investigation for a cracked section adequately predicts thetime-dependent (creep and shrinkage) curvature of the experimental beams. Comparing the shrinkage curvaturesdetermined using the codes (EC2 and 8110) with the curvatures of the measured beams and those predicted bythe model proposed in this investigation, the code methods are suitably accurate for cracked beams.

1 INTRODUCTION

In the UK, BS8110 (British Standard Institute, 1985)recommends that the effect of shrinkage on the curva-ture of spanning elements can be considered by usingequation (1). This equation has been derived basedon an uncracked section (section 3.6 of BS8110-2),although it does give the option to use cracked sectionproperties.

where: 1/rcs is shrinkage curvature; εcs is free shrink-age strain; αe is modular ratio; Ss is first moment ofarea of the reinforcement about the centroid of thecracked or gross section; I is second moment of areaof the cracked or gross section.

In Eurocode 2 (British Standard Institute, 2004),the shrinkage curvature is again predicted based onan uncracked section (section 7.4.3 of BSEN 1992).However, an equation is also suggested for the predic-tion of average curvature of the beam when the beamis cracked (section 7.4.3 of BSEN 1992),

where: α is the deformation parameter consideredwhich may be, for example, a strain, a curvature, ora rotation; αI, αII are the values of the parameter cal-culated for the uncracked and fully cracked conditionsrespectively; ζ is a distribution coefficient (allowingfor tensioning stiffening at a section) given by

ζ = 0 for uncracked sections; β is a coefficient takingaccount of the influence of the duration of the load-ing or of repeated loading on the average strain, andβ = 1.0 for a single short-term loading and β = 0.5 forsustained loads or many cycles of repeated loading;σs is the stress in the tension reinforcement calculatedon the basis of a cracked section; σsr is the stress inthe tension reinforcement calculated on the basis of acracked section under the loading conditions causingfirst cracking

The approaches adopted in both of the codes forthe calculation of curvature are clearly correct in rela-tion to uncracked sections (Mosley et al. 1999, ACICommittee 435 1966, Hobbs 1979, Ghali & Favre

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1986) however the approaches have never been exper-imentally validated for cracked sections. The majorexperimental difficulty is separating the influence ofshrinkage on curvature from the influence of creep.Consequently these approaches may be conservative.

2 NEW METHOD FOR SHRINKAGECURVATURE DETERMINATION

The calculation of shrinkage curvature is based onmechanical equilibrium and deformation compatibil-ity. The cross section of a rectangular beam is dividedhorizontally into a number of strips. Assuming thatplane sections remain plane and by considering theeffects of the shrinkage and creep of the concrete, theneutral axis position and curvature of the section andthereafter the true strain and stress of every strip aredetermined by iteration. The method will be describedin detail under three headings: assumptions and simpli-fications, analytical approach and the stress and strainin the concrete as affected by shrinkage and creep.

2.1 Initial simplifications adopted for thisapproach

Two basic assumptions are employed.

(1) Plane sections remain plane.(2) Linear creep is assumed (i.e. creep strain is pro-

portional to stress).

In the development of the model and in the cal-culations presented here, a number of simplifyingassumptions were needed:

(1) A simple-supported rectangular section beam wasassumed with both top and bottom reinforcement.

(2) The load was applied at time to and kept constantfor the duration.

(3) There was no external axial load.(4) The analysis assumed a cracked section.(5) The tensile strength of concrete was assumed to be

zero and tension stiffening was ignored.(6) The relative humidity and temperature was

assumed to remain constant.(7) Creep and shrinkage were treated as independent

phenomena with no interaction (this is a normalassumption in design calculations but by no meansnecessarily true).

(8) The analysis assumes that the concrete has zerotensile strength. Therefore this property is unnec-essary. However, it is possible in future develop-ments of this analysis, it may be required.

The variation in elastic modulus, strength, creepcoefficient and free shrinkage as functions of timeare required to perform the calculation. These wereobtained from the equations given in Eurocode 2 (andcross-checked with measured data recorded as part ofthis investigation).

Figure 1. Sketch of cross-section and strain.

2.2 Analytical approach

Figure 1 illustrates the notations of the cross section.The first step is to divide this cross-section into suf-ficient horizontal strips to be accurate but not to slowthe calculation process too much.

A summary of the procedure for the analysis is asfollows:

(1) Guess a value for x and 1/r. (defined in Figure 1)(2) Calculate the average strain and hence the average

stress in each strip. Note that the concrete stresswill be zero in any strip wholly within the tensionzone.

(3) Calculate the strains and hence the stresses in thetop and bottom steel.

(4) By equilibrium,

where: σ’s = stress in top steel; σs = stress in bot-tom steel; σi = stress in ith strip; y’s = distanceto top steel from whatever axis you choose;ys = distance to bottom steel from whatever axisyou choose; yi = distance to centre of ith strip fromwhatever axis you choose;(Note: A convenient axis for the measurement of yis the centre of the section but this is not essential.)

(5) If either or both equations (4) or (5) are untrue thenadjust x and/or 1/r and check equilibrium again.When both equations are satisfied then the cor-rect values of x and 1/r have been found and theanalysis is complete. Attention has to be paid tofind a way of adjusting x and 1/r in a system-atic way so that the correct answer is found inrelatively few cycles. (I.e. initially the increase instrain caused by the adjustment of x and/or 1/r wasdetermined and then the corresponding change inthe stress of the reinforcement and the strips wascomputed. The increase in force and moment ofthe section was derived as a function of increas-ing x and 1/r. Hence, by setting the increase inforce and moment to the necessary values to meetthe equilibrium condition of equations (4) and (5)gives the appropriate adjustment of x and 1/r.)

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Figure 2. Effect of shrinkage on neutral axis position.

Figure 3. Approximation of stress behaviour.

2.3 Stress and strain relationship of concrete

2.3.1 Effect of shrinkageThe effect of shrinkage of the concrete can be removedas follows:

where: σc is stress in concrete; εtot is total strain ofconcrete; εsh is shrinkage of concrete; Ec is elasticmodulus of concrete at the time of investigating.

The shrinkage of concrete not only influences thestress-strain relationship, but also the neutral axisposition. Figure 2 shows schematically the change inneutral axis position when shrinkage occurs.

2.3.2 Effect of creepIf the compression or tension stress is constant, theeffect of creep can be considered by incorporating acreep coefficient within the stress-strain relation. For apure bending beam, however, even though the bendingmoment may be constant, the stress it produces withinthe cross section varies with time.An approximation ismade to simplify the analysis as shown in Figure 3. Ascan be seen, the variation in stress is gradual. However,in order to develop the analysis, it is assumed that thechange in stress occurs instantaneously and is equiv-alent to a new stress (δσi) being applied at the end ofeach time step. In this way, it can be considered that thestresses are constant, but new stresses are applied con-tinuously. Incorporating the assumption of linear creep(Ghali and Favre 1986, Favre et al. 1983), the stresswith consideration of creep can be analysed as below.

Figure 4. Sketch of superposition of creep.

Figure 4 is drawn for an element close to the com-pression face and hence all δσs are negative. Creep isassumed to occur only under the stress σ0 from to to t1,under σ1 from t1 to t2 and so-on. It is assumed that thecreep from each increment in stress can be superposedthus the deformations can be assumed to develop asshown below.

In the above sketch, A, the long-term increment instrain at time t3, is given by:

The total strain can now be expressed more conve-niently as:

where: εsh(ti,t3) is the shrinkage from the time of thestart of drying to t3.

This form of calculation can clearly be extended toany time, tn.

Based on the approach described above, a Matlabprogramme was developed. Using the program, theprofile of stress and strain of concrete, stress and strainof reinforcement, neutral axis position and curvatureof cracked section can be determined. To analyzethe effect of shrinkage, the comparison of curvaturescan be made between beams with different shrinkage.The results of analysis will be presented below withexperiment together.

3 EXPERIMENT

To verify the predicted results from the analysis above,two concrete beams with different concrete mixeswere tested. One (BEAM1) was a low shrinkage mix,which was prepared with cement content 550 kg/m3,water/cement ratio 0.41 and with shrinkage reducing

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Figure 5. details of reinforcement of concrete beams.

admixture, the other (BEAM2) a high shrinkage mix,which was cast with cement content 275 kg/m3 andwater/cement ratio 0.58. However, both mixes hadvery similar creep properties. The two beams weresubjected to the same cracking load. Therefore, anydifference in the curvature of the beams is the resultof shrinkage.

The dimensions and reinforcement details of thebeams are shown in Figure 5.The concrete cover (fromconcrete surface to bar surface) is 30 mm. No link baris placed in the constant moment zone.

After casting, the beams were cured in moulds cov-ered with wet ‘gunnysack’ and plastic sheet until theywere loaded at 3 days. Companion cubes for com-pressive strength tests and prisms for creep and freeshrinkage tests were cast from the same mixes usedfor the beams. They were also cured in the same way.

Three days after casting, the beams were demouldedand placed in the test rigs. A four-point bending loadwas applied to the beams. The span of the beams is4000 mm and the two loading points are 1500 mmapart. The total load applied to the two beams byhydraulic jack is 36.0 kN. The load is checked andadjusted frequently to maintain the load constant. Theambient temperature is in the range of 20–25◦C andthe relative humidity is between 40 and 60 percent.

In order to monitor the depth of neutral axis / curva-ture with time, Demec gauges (200 mm) were used tomeasure the horizontal strain on the side of the beamin the region of the constant moment zone. The strainwas measured at five different depths, two of whichcorresponded to the position of the main tension andtop steel and one the mid-depth of the beam. Readingswere taken twice per day for the first two days. Thenthe frequency of readings was reduced to once per day,once every two days and finally once every four daysat the age of one week, three weeks and six weeks,respectively.

4 RESULTS AND DISCUSSION

4.1 shrinkage and creep of concrete

In this investigation the shrinkage and creep of theconcrete mixes were measured on companion prisms.For the analysis, the code (CEB Model Code 1990)equations for shrinkage and creep are used and thecoefficients are determined by experimental datafitting.

Figure 6. Shrinkage of concrete mixes.

Figure 6 compares the experimental and predictedshrinkage where it can be seen that initially the modelunder-predicts the experimental shrinkage. However,after approximately two months, the predicted shrink-age results agree well with the experimental data. Fromthe measured trends of shrinkage in Figure 6 it is pos-sible to expect the long-term shrinkage to drop belowthat predicted by the model. The ultimate measuredshrinkage was therefore obtained by extrapolation andcompared with ultimate shrinkage predicted from themodel. It was found that the ultimate shrinkage may beexpected to reach 750 and 850 microstrain for BEAM1and BEAM2, respectively, whereas the model predictsvalues of 820 and 950 microstrain. This short-termunder prediction and over prediction of ultimate valuesby the model can possibly be explained by the non-standard concrete used in this investigation and thecuring and early age at which the tests commenced.The model is capable of accommodating a range ofconcrete mix designs and ages at loading howeverthese concretes and test procedures may have beenslightly beyond its capacity.

More importantly, the difference in shrinkagebetween the two mixes is correctly predicted by themodel for the duration of the tests, which is about82, 124 and 128 microstrain at the age 28, 90 and180 days, respectively. This difference is smaller thanexpected.

Creep under constant load was also measured from3 days. The creep test method is the same as thatdescribed in reference (Neville 1996). The creep coef-ficient corresponding to the initial load is comparedwith the experimental results in Figure 7.

Figure 7 shows that the creep coefficient of the twomixes is very similar and agrees well with the codemodel. In the analysis by the authors, identical creepis used for the two mixes.

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Figure 7. Creep coefficients.

4.2 Curvature

Figure 8 shows the time dependant curvature (resultingfrom shrinkage and creep) of the experimental beamsand as predicted by the code models (shrinkage curva-ture only). It can be seen that the curvature of the twotest beams is almost the same. This was unfortunate,although not surprising, as it was explained earlier thatthe difference in the free shrinkage between the twomixes as measured on the control prisms was only100 microstrain (Figure 6). Also although the prismswere sealed to represent the volume to surface ratioof the experimental beams the authors are not entirelyconfident of this approach (Aitcin et al. 1997, Almu-daiheem & Hansen 1987) and hence this may also helpto explain this lack of difference in the curvature ofthe two types of beam. From Figure 8 it can be seenthat the curvature of high shrinkage beam (BEAM2)is very slightly higher than that of the low shrink-age beam (BEAM1) however realistically they are thesame. The prediction results from the shrinkage modelpresented in this investigation do show that the highshrinkage beam exhibits higher curvature than lowshrinkage beam and again perhaps confirms the inad-equacy of estimating full-scale shrinkage from sealedprism samples.

The experimental curvature actually represents themean curvature of the beams (a combination of cur-vatures at the cracked and uncracked sections of thebeams). However, the predicted curvatures shown inthe figure are the curvatures at the position of a crackonly. In theory, the curvature of a section at a crackshould be greater than the mean curvature and thanthat of an uncracked section. However, Figure 8 indi-cates that the mean curvatures from the beam testsare higher than that from the modelling of a sectionat crack, especially for the first two months. A possi-ble reason for this may be the difference in shrinkagebetween the beam and the prism as mentioned above.

Figure 8. Curvature of concrete beams.

Figure 9. Comparison of shrinkage curvatures.

Also it may be because the elastic modulus of con-crete, which is converted from compressive strengthusing the equation in the code, is over estimated in themodel.

Figure 8 also illustrates the difference between themeasured time-dependant curvatures of the beams(which includes shrinkage curvature and creep curva-ture) and shrinkage curvature predicted by the codes.This difference is due to creep and from this inves-tigation it can be inferred that creep accounts forapproximately 50% of the total time dependent cur-vature. This is not unreasonable as if we compare thespecific creep of this concrete obtained from the prismtests and the maximum compressive stress in the beam,with the shrinkage of the free prisms, it is found thatthe ratio of creep curvature to shrinkage curvature isroughly the same as that of creep to shrinkage.

Figure 9 compares the shrinkage curvatures derivedfrom codes (BS 8110 and EC2) and from the calcu-lation method proposed in this paper. The curvaturedetermined according to the codes is exactly the sameas the calculation of this paper for a cracked section.

In the calculation of mean curvature (curvaturesat cracked and uncracked sections) using Equation

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(2), ζ could be affected by creep as that influencesthe centroid of the section and consequently leadsto the change in critical moment of the section. InEC2, it is not clearly explained whether ζ should befixed. Figure 9 compares the mean curvature withfixed ζ (EC2-Mean-Fixed) with variable ζ (EC2-Mean-Variable). The mean curvature with variable ζtends to approach the curvature of state1 with time.This is unreasonable as, with the development of crack-ing with time the mean curvature should approximateto the curvature of state2 (fully cracked). The meancurvature with fixed ζ is slightly lower than that forcracked section but is much closer to the measuredbeam values.

5 CONCLUSIONS

The analytical approach proposed in this investiga-tion horizontally divides the cross section of beamsunder pure bending into a number of strips. Withthe assumption of plane sections remaining planeand with consideration of the effect of shrinkage andcreep of concrete, the neutral axis position and curva-ture of the section and thereafter the true strain andstress of every strip are determined by iteration. Themodel proposed adequately predicts shrinkage curva-ture (uncracked and cracked sections) and matches thecode predictions.

The time-dependent (creep and shrinkage) curva-tures of the measured beams are adequately predictedby the model proposed in this investigation for acracked section.

When using Equation (2) to determine the averagecurvature, the constant ζ should be fixed at its initialvalue.

For the mixes used in this investigation, 50% of thecurvature is due to shrinkage and 50% due to creep. Inpractice, shrinkage is likely to be lower so creep maybe expected to dominate curvature.

The results illustrate the ability of MC90 to approx-imately predict the movement of a range of concretes

but not always to accurately predict these movements(the accuracy is influenced by unusual mix designsas was the case in this investigation and early ages atloading).

When using equation (1) to calculate the shrinkagecurvature, the effect of applying the shrinkage later isto reduce the modular ratio, αe, reduce I and slightlyincrease S. These changes almost exactly balance eachother with the apparent result that (1/r) is unaffected(and independent of creep).

REFERENCES

British Standard Institute. 1985. BS 8110-2: 1985 Structuraluse of concrete – Part 2: Code of practice for specialcircumstances. London: British Standard Institute.

British Standard Institute. 2004. BS EN 1992-1-1: 2004Eurocode 2: Design of concrete structures – Part 1-1:General rules and rules for buildings. London: BritishStandard Institute.

Mosley, W. H. Bungey. J. H. & Hulse, R. 1999. Reinforcedconcrete design (5th edition), London: Macmillan PressLtd.

ACI Committee 435. 1966. Deflections of reinforced con-crete flexural members. Journal of theAmerican ConcreteInstitute 637–674.

Hobbs, D. W. 1979. Shrinkage-induced curvature of rein-forced concrete members. London: Cement and ConcreteAssociation.

Ghali, A. & Favre R. 1986. Concrete structures: stresses anddeformations. London and NewYork: Chapman and Hall.

Favre, R. Beeby, A. W. Falkner, H. Koprna, M. & Schiessl,P. 1983. Cracking and deformations. CEM Mannual.Lausanne: Printed and distributed by the Swiss FederalInstitute of Technology.

Neville, A. M. 1996. Properties of concrete (4th edition).London: Longman Science and Technology.

Aitcin, P.C. Neville, A.M. & Acker, P. 1997. Integrated viewof shrinkage deformation, Concrete International 19(9):35–41.

Almudaiheem, J. A. & Hansen, W. 1987. Effect of speci-men size and shape on drying shrinkage of concrete. ACIMaterials Journal 84(2): 130–135.

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