Modelling of shrinkage induced curvature of cracked concrete beams

Transcription

1 Tailor Made Concrete Structures Walraven & Stoelhorst (eds) 2008 Taylor & Francis Group, London, ISBN Modelling of shrinkage induced curvature of cracked concrete beams R. Mu, J.P. Forth & A.W. Beeby The University of Leeds, Leeds, UK R. Scott University of Durham, Durham, UK ABSTRACT: Besides load and temperature, shrinkage and creep are the main factors that influence the curvature of reinforced concrete sections. For cracked sections, this effect is calculated semi-empirically in structural design codes such as BS8110 and EC2. In order to verify the accuracy of the calculation, the curvature of cracked section of concrete beams due to shrinkage was analyzed numerically and validated experimentally. The analysis in this paper is based on the mechanical equilibrium method and with basic assumptions of plane sections remaining plane and linear creep superposition. This method divides a section into a number of strips. The neutral 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 the time-dependent (creep and shrinkage) curvature of the experimental beams. Comparing the shrinkage curvatures determined using the codes (EC2 and 8110) with the curvatures of the measured beams and those predicted by the 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 curvature of spanning elements can be considered by using equation (1). This equation has been derived based on an uncracked section (section 3.6 of BS8110-2), although it does give the option to use cracked section properties. where: α is the deformation parameter considered which may be, for example, a strain, a curvature, or a rotation; α I, α II are the values of the parameter calculated for the uncracked and fully cracked conditions respectively; ζ is a distribution coefficient (allowing for tensioning stiffening at a section) given by where: 1/r cs is shrinkage curvature; ε cs is free shrinkage strain; α e is modular ratio; S s is first moment of area of the reinforcement about the centroid of the cracked or gross section; I is second moment of area of the cracked or gross section. In Eurocode 2 (British Standard Institute, 2004), the shrinkage curvature is again predicted based on an uncracked section (section of BSEN 1992). However, an equation is also suggested for the prediction of average curvature of the beam when the beam is cracked (section of BSEN 1992), ζ = 0 for uncracked sections; β is a coefficient taking account of the influence of the duration of the loading or of repeated loading on the average strain, and β = 1.0 for a single short-term loading and β = 0.5 for sustained loads or many cycles of repeated loading; σ s is the stress in the tension reinforcement calculated on the basis of a cracked section; σ sr is the stress in the tension reinforcement calculated on the basis of a cracked section under the loading conditions causing first cracking The approaches adopted in both of the codes for the calculation of curvature are clearly correct in relation to uncracked sections (Mosley et al. 1999, ACI Committee , Hobbs 1979, Ghali & Favre 573

2 1986) however the approaches have never been experimentally validated for cracked sections. The major experimental difficulty is separating the influence of shrinkage on curvature from the influence of creep. Consequently these approaches may be conservative. 2 NEW METHOD FOR SHRINKAGE CURVATURE DETERMINATION The calculation of shrinkage curvature is based on mechanical equilibrium and deformation compatibility. The cross section of a rectangular beam is divided horizontally into a number of strips. Assuming that plane sections remain plane and by considering the effects of the shrinkage and creep of the concrete, the neutral axis position and curvature of the section and thereafter the true strain and stress of every strip are determined by iteration. The method will be described in detail under three headings: assumptions and simplifications, analytical approach and the stress and strain in the concrete as affected by shrinkage and creep. 2.1 Initial simplifications adopted for this approach Two basic assumptions are employed. (1) Plane sections remain plane. (2) Linear creep is assumed (i.e. creep strain is proportional to stress). In the development of the model and in the calculations presented here, a number of simplifying assumptions were needed: (1) A simple-supported rectangular section beam was assumed with both top and bottom reinforcement. (2) The load was applied at time t o and kept constant for 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 normal assumption in design calculations but by no means necessarily true). (8) The analysis assumes that the concrete has zero tensile strength. Therefore this property is unnecessary. However, it is possible in future developments of this analysis, it may be required. The variation in elastic modulus, strength, creep coefficient and free shrinkage as functions of time are required to perform the calculation. These were obtained from the equations given in Eurocode 2 (and cross-checked with measured data recorded as part of this 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 sufficient horizontal strips to be accurate but not to slow the calculation process too much. A summary of the procedure for the analysis is as follows: (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 stress will be zero in any strip wholly within the tension zone. (3) Calculate the strains and hence the stresses in the top and bottom steel. (4) By equilibrium, where: σ s = stress in top steel; σ s = stress in bottom steel; σ i = stress in i th strip; y s = distance to top steel from whatever axis you choose; y s = distance to bottom steel from whatever axis you choose; y i = distance to centre of i th strip from whatever axis you choose; (Note: A convenient axis for the measurement of y is the centre of the section but this is not essential.) (5) If either or both equations (4) or (5) are untrue then adjust x and/or 1/r and check equilibrium again. When both equations are satisfied then the correct values of x and 1/r have been found and the analysis is complete. Attention has to be paid to find a way of adjusting x and 1/r in a systematic way so that the correct answer is found in relatively few cycles. (I.e. initially the increase in strain caused by the adjustment of x and/or 1/r was determined and then the corresponding change in the stress of the reinforcement and the strips was computed. The increase in force and moment of the section was derived as a function of increasing x and 1/r. Hence, by setting the increase in force and moment to the necessary values to meet the equilibrium condition of equations (4) and (5) gives the appropriate adjustment of x and 1/r.) 574

3 Figure 2. Effect of shrinkage on neutral axis position. Figure 4. Sketch of superposition of creep. Figure 3. Approximation of stress behaviour. Figure 4 is drawn for an element close to the compression face and hence all δσs are negative. Creep is assumed to occur only under the stress σ 0 from t o to t 1, under σ 1 from t 1 to t 2 and so-on. It is assumed that the creep from each increment in stress can be superposed thus the deformations can be assumed to develop as shown below. In the above sketch, A, the long-term increment in strain at time t 3,isgivenby: 2.3 Stress and strain relationship of concrete Effect of shrinkage The effect of shrinkage of the concrete can be removed as follows: The total strain can now be expressed more conveniently as: where: σ c is stress in concrete; ε tot is total strain of concrete; ε sh is shrinkage of concrete; E c is elastic modulus of concrete at the time of investigating. The shrinkage of concrete not only influences the stress-strain relationship, but also the neutral axis position. Figure 2 shows schematically the change in neutral axis position when shrinkage occurs Effect of creep If the compression or tension stress is constant, the effect of creep can be considered by incorporating a creep coefficient within the stress-strain relation. For a pure bending beam, however, even though the bending moment may be constant, the stress it produces within the cross section varies with time. An approximation is made to simplify the analysis as shown in Figure 3. As can be seen, the variation in stress is gradual. However, in order to develop the analysis, it is assumed that the change in stress occurs instantaneously and is equivalent to a new stress (δσ i ) being applied at the end of each time step. In this way, it can be considered that the stresses are constant, but new stresses are applied continuously. Incorporating the assumption of linear creep (Ghali and Favre 1986, Favre et al. 1983), the stress with consideration of creep can be analysed as below. where: ε sh(ti,t3) is the shrinkage from the time of the start of drying to t 3. This form of calculation can clearly be extended to any time, t n. Based on the approach described above, a Matlab programme was developed. Using the program, the profile of stress and strain of concrete, stress and strain of reinforcement, neutral axis position and curvature of cracked section can be determined. To analyze the effect of shrinkage, the comparison of curvatures can be made between beams with different shrinkage. The results of analysis will be presented below with experiment together. 3 EXPERIMENT To verify the predicted results from the analysis above, two concrete beams with different concrete mixes were tested. One (BEAM1) was a low shrinkage mix, which was prepared with cement content 550 kg/m 3, water/cement ratio 0.41 and with shrinkage reducing 575

4 Figure 5. details of reinforcement of concrete beams. admixture, the other (BEAM2) a high shrinkage mix, which was cast with cement content 275 kg/m 3 and water/cement ratio However, both mixes had very similar creep properties. The two beams were subjected to the same cracking load. Therefore, any difference in the curvature of the beams is the result of shrinkage. The dimensions and reinforcement details of the beams are shown in Figure 5.The concrete cover (from concrete surface to bar surface) is 30 mm. No link bar is placed in the constant moment zone. After casting, the beams were cured in moulds covered with wet gunnysack and plastic sheet until they were loaded at 3 days. Companion cubes for compressive strength tests and prisms for creep and free shrinkage tests were cast from the same mixes used for the beams. They were also cured in the same way. Three days after casting, the beams were demoulded and placed in the test rigs. A four-point bending load was applied to the beams. The span of the beams is 4000 mm and the two loading points are 1500 mm apart. The total load applied to the two beams by hydraulic jack is 36.0 kn. The load is checked and adjusted frequently to maintain the load constant. The ambient temperature is in the range of C and the relative humidity is between 40 and 60 percent. In order to monitor the depth of neutral axis / curvature with time, Demec gauges (200 mm) were used to measure the horizontal strain on the side of the beam in the region of the constant moment zone. The strain was measured at five different depths, two of which corresponded to the position of the main tension and top steel and one the mid-depth of the beam. Readings were taken twice per day for the first two days. Then the frequency of readings was reduced to once per day, once every two days and finally once every four days at 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 the concrete mixes were measured on companion prisms. For the analysis, the code (CEB Model Code 1990) equations for shrinkage and creep are used and the coefficients are determined by experimental data fitting. Figure 6. Shrinkage of concrete mixes. Figure 6 compares the experimental and predicted shrinkage where it can be seen that initially the model under-predicts the experimental shrinkage. However, after approximately two months, the predicted shrinkage results agree well with the experimental data. From the measured trends of shrinkage in Figure 6 it is possible to expect the long-term shrinkage to drop below that predicted by the model. The ultimate measured shrinkage was therefore obtained by extrapolation and compared with ultimate shrinkage predicted from the model. It was found that the ultimate shrinkage may be expected to reach 750 and 850 microstrain for BEAM1 and BEAM2, respectively, whereas the model predicts values of 820 and 950 microstrain. This short-term under prediction and over prediction of ultimate values by the model can possibly be explained by the nonstandard concrete used in this investigation and the curing and early age at which the tests commenced. The model is capable of accommodating a range of concrete mix designs and ages at loading however these concretes and test procedures may have been slightly beyond its capacity. More importantly, the difference in shrinkage between the two mixes is correctly predicted by the model for the duration of the tests, which is about 82, 124 and 128 microstrain at the age 28, 90 and 180 days, respectively. This difference is smaller than expected. Creep under constant load was also measured from 3 days. The creep test method is the same as that described in reference (Neville 1996). The creep coefficient corresponding to the initial load is compared with the experimental results in Figure 7. Figure 7 shows that the creep coefficient of the two mixes is very similar and agrees well with the code model. In the analysis by the authors, identical creep is used for the two mixes. 576

5 Figure 7. Creep coefficients. Figure 8. Curvature of concrete beams. 4.2 Curvature Figure 8 shows the time dependant curvature (resulting from shrinkage and creep) of the experimental beams and as predicted by the code models (shrinkage curvature only). It can be seen that the curvature of the two test beams is almost the same. This was unfortunate, although not surprising, as it was explained earlier that the difference in the free shrinkage between the two mixes as measured on the control prisms was only 100 microstrain (Figure 6). Also although the prisms were sealed to represent the volume to surface ratio of the experimental beams the authors are not entirely confident of this approach (Aitcin et al. 1997, Almudaiheem & Hansen 1987) and hence this may also help to explain this lack of difference in the curvature of the two types of beam. From Figure 8 it can be seen that the curvature of high shrinkage beam (BEAM2) is very slightly higher than that of the low shrinkage beam (BEAM1) however realistically they are the same. The prediction results from the shrinkage model presented in this investigation do show that the high shrinkage beam exhibits higher curvature than low shrinkage beam and again perhaps confirms the inadequacy of estimating full-scale shrinkage from sealed prism samples. The experimental curvature actually represents the mean curvature of the beams (a combination of curvatures at the cracked and uncracked sections of the beams). However, the predicted curvatures shown in the figure are the curvatures at the position of a crack only. In theory, the curvature of a section at a crack should be greater than the mean curvature and than that of an uncracked section. However, Figure 8 indicates that the mean curvatures from the beam tests are higher than that from the modelling of a section at crack, especially for the first two months. A possible reason for this may be the difference in shrinkage between the beam and the prism as mentioned above. Figure 9. Comparison of shrinkage curvatures. Also it may be because the elastic modulus of concrete, which is converted from compressive strength using the equation in the code, is over estimated in the model. Figure 8 also illustrates the difference between the measured time-dependant curvatures of the beams (which includes shrinkage curvature and creep curvature) and shrinkage curvature predicted by the codes. This difference is due to creep and from this investigation it can be inferred that creep accounts for approximately 50% of the total time dependent curvature. This is not unreasonable as if we compare the specific creep of this concrete obtained from the prism tests and the maximum compressive stress in the beam, with the shrinkage of the free prisms, it is found that the ratio of creep curvature to shrinkage curvature is roughly the same as that of creep to shrinkage. Figure 9 compares the shrinkage curvatures derived from codes (BS 8110 and EC2) and from the calculation method proposed in this paper. The curvature determined according to the codes is exactly the same as the calculation of this paper for a cracked section. In the calculation of mean curvature (curvatures at cracked and uncracked sections) using Equation 577

6 (2), ζ could be affected by creep as that influences the centroid of the section and consequently leads to the change in critical moment of the section. In EC2, it is not clearly explained whether ζ should be fixed. Figure 9 compares the mean curvature with fixed ζ (EC2-Mean-Fixed) with variable ζ (EC2- Mean-Variable). The mean curvature with variable ζ tends to approach the curvature of state1 with time. This is unreasonable as, with the development of cracking with time the mean curvature should approximate to the curvature of state2 (fully cracked). The mean curvature with fixed ζ is slightly lower than that for cracked section but is much closer to the measured beam values. 5 CONCLUSIONS The analytical approach proposed in this investigation horizontally divides the cross section of beams under pure bending into a number of strips. With the assumption of plane sections remaining plane and with consideration of the effect of shrinkage and creep of concrete, the neutral axis position and curvature of the section and thereafter the true strain and stress of every strip are determined by iteration. The model proposed adequately predicts shrinkage curvature (uncracked and cracked sections) and matches the code predictions. The time-dependent (creep and shrinkage) curvatures of the measured beams are adequately predicted by the model proposed in this investigation for a cracked section. When using Equation (2) to determine the average curvature, the constant ζ should be fixed at its initial value. For the mixes used in this investigation, 50% of the curvature is due to shrinkage and 50% due to creep. In practice, shrinkage is likely to be lower so creep may be expected to dominate curvature. The results illustrate the ability of MC90 to approximately predict the movement of a range of concretes but not always to accurately predict these movements (the accuracy is influenced by unusual mix designs as was the case in this investigation and early ages at loading). When using equation (1) to calculate the shrinkage curvature, the effect of applying the shrinkage later is to reduce the modular ratio, α e, reduce I and slightly increase S. These changes almost exactly balance each other with the apparent result that (1/r) is unaffected (and independent of creep). REFERENCES British Standard Institute BS : 1985 Structural use of concrete Part 2: Code of practice for special circumstances. London: British Standard Institute. British Standard Institute BS EN : 2004 Eurocode 2: Design of concrete structures Part 1-1: General rules and rules for buildings. London: British Standard Institute. Mosley, W. H. Bungey. J. H. & Hulse, R Reinforced concrete design (5th edition), London: Macmillan Press Ltd. ACI Committee Deflections of reinforced concrete flexural members. Journal of theamerican Concrete Institute Hobbs, D. W Shrinkage-induced curvature of reinforced concrete members. London: Cement and Concrete Association. Ghali, A. & Favre R Concrete structures: stresses and deformations. London and NewYork: Chapman and Hall. Favre, R. Beeby, A. W. Falkner, H. Koprna, M. & Schiessl, P Cracking and deformations. CEM Mannual. Lausanne: Printed and distributed by the Swiss Federal Institute of Technology. Neville, A. M Properties of concrete (4th edition). London: Longman Science and Technology. Aitcin, P.C. Neville, A.M. & Acker, P Integrated view of shrinkage deformation, Concrete International 19(9): Almudaiheem, J. A. & Hansen, W Effect of specimen size and shape on drying shrinkage of concrete. ACI Materials Journal 84(2):