ABSTRACT
A concrete pavement is subjected to warping stresses due to temperature gradient across the depth of the pavement. The temperature gradient is generally nonlinear for thick concrete pavements. Warping stresses computed from Bradbury’s equation might exceed the true stresses by as much as 100% in some cases. If warping stresses due moisture variation is taken into account, error in the computed stress by Bradbury’s equation can be much higher. The effect warping stresses on pavement design may be insignificant if the slab length is less than 4.5m.

* Professor of Civil Engineering, Indian Institute of Technology, Kharagpur, India.

1. INTRODUCTION
Warping stress caused by temperature gradient in a concrete pavement is an important factor affecting the performance of the pavement. The Guidelines of Indian Roads Congress, IRC: 581,2,3, uses Bradbury’s equation for warping stresses for thickness design of rigid pavements. Westergaard4 gave the first analytical solution for warping stresses in rigid pavements due to a linear temperature gradient across the depth of the pavement. He considered (i) a long and a broad slab and (ii) a slab of finite width but of infinite length. The slab was assumed to be glued to a Winkler foundation, which is equivalent to system of linearly elastic springs fixed to the slab so that there is no separation between the foundation and the slab when the slab warps up or down. Bradbury5, using the equation of Westergaard for a slab of finite width but of infinite length, derived equations for curling stresses in slabs of practical dimensions used in road construction.
During the day time when the slab is warped with convex upwards (Fig.l), the foundation pulls the slab down by spring action, the characteristic of the Winkler foundation. The stresses caused by the hypothetical tension in the spring are assumed to represent the curling or warping stress caused by selfweight of the slab. If the foundation is stiffer, the curling stress is large because the restraining force of the hypothetical spring foundation is also large though weight of the slab remains the same. The greater the temperature difference between the top and the bottom of the slab, greater is the warping and a large tension in the spring causes higher stresses. During the night hours, the slab curls as shown in Fig. 2. This condition is critical during the early life of concrete when it is very
weak. Curling stresses are sensitive to the stiffness of the idealised foundation. Author’s analysis shows that a concrete a slab laid on a Dry Lean Concrete (DLC) subbase warps up and leave the foundation even for a temperature difference of 10°C between the top and the bottom of the slab and the self weight of the slab is not large enough to deflect the slab to make it rest on the foundation. Any higher temperature difference causes additional warping but does not cause any additional curling stresses since the selfweight remains the same. It is, therefore, necessary to reexamine the validity of Bradbury’s approach for the computation of curling stresses due to linear temperature gradient since this approach has been recommended in all previous and the current guidelines of Indian roads Congress for design of rigid pavements1,2,3.
This Paper computes the values of stresses caused by the self weight of the slab when the slab is warped up during the day time since only this condition is considered in pavement design guidelines IRC: 582002^{3}. The charts and equations given by Timoshenko and Krieger6 have been used for stress computation. Results are compared with corresponding Bradbury’s approach adopted in IRC: 5820023. A few cases with nonlinear temperature distribution across the depth have also been presented.
2. LINEAR TEMPERAURE GRADIENT
2.1. Theoretical Analysis
The basic approach for the stress computation caused by the temperature gradient is as follows:
(i) Consider the slab of the rigid pavement to be weightless.
(ii) Allow the slab to curl when the temperature difference between the top and the bottom fibre is ‘t’ with top surface at a higher temperature. Only linear temperature gradient commonly considered in professional practice in India is assumed though it is known that for thicker slab, the temperature gradient is nonlinear7,8,9. The deflected shape will be similar to that shown in Fig. 3 and the slab is, by and large, free of flexural stresses. The slab rests on four comers during the curled up state.
(iii) Compute the upward lifting at the centre of the slab as well as at the centre of the edge due to the curling from geometrical consideration.
(iv) Apply uniformly distributed load equal to the weight of the slab on the curled surface.
(v) Compute the deflection at the centre due to the self weight to examine whether the curled up slab will deflect down to the foundation.
(vi) Compute the bending stresses along the edge and the centre caused by the uniformly distribute load if the slab does not rest on the foundation.
(vii) Compare the results with Bradbury’s equation.
The above approach appears to be the most rational since DLC3 is provided as a subbase below the concrete slab in India to give a nonerodible foundation. There is no bonding between the subbase and the concrete slab because of a 125micron plastic sheet at the interlayer. The assumption of spring like foundation bonded to the slab is no more valid because of absence of any bond and the slab is free to warp up and down causing separation between the DLC and the concrete slab.
2.2. Computation of Deflection and Stress
Square slabs of size 3.5mX3.5m, 4.0mX4.0m, 4.5mX4.5m and 5.0mX5.0m for thickness of 25cm and 30cm have been considered for comparison of curling stresses computed by author’s approach with those by IRC:582002^{3}.Square slabs have been taken for illustrations since solutions for such slabs are readily available^{6}.
If the difference of temperature between the top and the bottom fibre is ‘t’ with linear temperature gradient along the depth of a slab, the coefficient of thermal expansion , thickness of the slab==h the radius of curvature of a weightless curled slab.R, is approximately given as :
R = . . . Equn. 1
The deflected shape during the daytime will be as shown in Fig.3.

To calculate the curling stresses, the selfweight of the slab is applied in the form of a uniform distributed load over the warped up surface of the slab supported at the comers. Timoshenko and Krieger^{6} have given a solution for bending moment and the deflection for square slabs supported on four comers.
Figs. 6 and 7 show the stresses computed by Bradbury’s equations as well as by the author (Tim) using Timoshenko’s approach for temperature differences of 13ºC, 17ºC and 21ºC, respectively.
The stresses computed by the author’s approach remains the same irrespective of the upward warping since the bending moment due to the selfweight does not depend upon the upward deflection. When Bradbury’s approach is adopted, greater the difference of temperature between the top and the bottom fibres, greater is the warping stresses because of higher restraining force provided by the hypothetical foundation. Only for the temperature difference of 13 C are the stresses close to each other (Figs. 6 and 7).
It is further seen that for temperature differences of 17°C and 21°C, the Bradbury’s equation over estimates the stresses by a large amount, the maximum amount may be higher than 60 per cent for the above mentioned slab sizes. Though only square slabs areconsidered, the results cannot be much different for rectangular slabs of comparable dimensions used in India. It may .however, be noted that the analysis has been done for a rigid foundation which does not allow any deflection at the comers due to self weight but in actual practice, there would be deformation along the comers due to self weight and the warping stresses would be further lower than those computed by the author. This clearly indicates that the Bradbury’s values are in error by a large amount and the actual stresses would be much lower than the Bradbury’ values.
It is further seen from Eqs. 4&5 that the edge stresses are about 40 per cent greater than the interior stresses, just the opposite to those given by Bradbury’s equation given by Eqs. 7 and 8. This appears reasonable since when the slab warps up, the self weight of the slab will cause less bending moment in the interior because of two way action of the slab as compared to the edge.
It is thus clear that when the slab is laid over the 125micron plastic sheet placed over DLC, it is inappropriate to apply the recommendation of Indian Roads Congress3 for using Bradbury’s equation for computation of temperature warping stresses. Author’s approach presented in the paper is rational and can be used to correctly estimate warping stresses for the construction practice in India.
3. NONLINEAR temperature GRADIENT
Experimental observations7,8,9, indicate that the temperature gradient in slabs having thickness of 200m and higher is nonlinear. A simplified analysis for estimation of internal stresses due nonlinear variation of temperature is presented below. On examination of temperature data7,8, it is found that the temperature variation between the top and the bottom fibre of a concrete pavement is close to a parabola and can be approximated as a bilinear distribution across the depth.
Temperature measurement in 20 cm concrete slab laid on water bound macadam during the peak summer at Khargpur^{8} indicated the maximum temperature difference between the top and bottom fibre of the slab during the day time was about 13°C. This occurred in the afternoon around 1 PM and the temperature variation with depth was parabolic. The difference of temperature between the top and mid depth was almost twice the difference between the middle and the bottom fibre.
Assuming bilinear temperature variation as shown in Fig. 9, the slab was considered as two unbonded slabs kept one above the other as shown in Fig. 8. During the day time, the upper slab will tend to bend more due to the higher temperature difference between the top and the middle surfaces than the lower half and consequently the upper and the lower slabs will tend to have different radii of curvature. The real slab is a monolithic mass and will warp up as one unit and internal stresses will be set up due to internal bending moments to annul the different curvature of the upper and the lower parts. This causes compressive stresses at the top and the bottom and tensile stresses at mid depth the values of stresses can be computed from geometrical compatibility. The magnitude of internal stresses at the edge and the interior together with warping stresses due to self weight and Bradbury are given in Table 1 for a 30 cm slab for temperature differences of 13°C, 17ºC and 21°C between the top and the bottom fibre. The maximum temperature difference for a 30 cm slab recommended in IRC58: 2002^{3} for computation of warping stresses is 21°C. The slab dimensions considered are 4.5mX4.5m.
4. TOTAL STRESSES
Table 1 clearly shows that Bradbury’s solutions for temperature differences of 13°C, 17°C and 21°C are higher by about 14 per cent, 93 per cent and 168 per cent, respectively from those of the sums of temperature warping stresses and the internal stresses due to nonlinear variation of temperature. It is also well known that the concrete expands with increase in moisture content and shrinks with lowering of moisture content. In the day time when the temperature at the top surface is higher as compared to the bottom surface, the moisture content variation with depth will just be opposite to that of temperature variation and if both moisture and temperature variations with depth are considered, the warping stresses may annual each other. In night time, much less fatigue life is consumed due to negative temperature gradient as compared to the life consumed in the afternoon when temperature gradient is positive. High temperature gradients occur for a short time only in the afternoons and hence, if warping stresses due to temperature gradient are neglected in design of rigid pavement; there is little likely hood of any serious error for slabs of dimensions less than 4.5m.
5. CONCLUSIONS
Warping stresses computed by Bradbury’s formula are too high to be applicable for design of rigid pavements when plastic sheets are placed between the DLC and the rigid pavements. Author’s approach using Timoshenko’s formulation gives an upper bound estimate of the warping stresses. Bradbury’s equations may give warping stresses which may be higher by as much as 168 per cent of the actual warping stresses. The actual stress values may even be further lower if moisture induced warping stresses can be accounted for. Temperature variation across the depth of a slab is nonlinear and the internal stresses lowers the warping stresses considerably. There may not be much effect on thickness design if temperature warping stresses are neglected in the analysis for lengths less than 4.5m.
REFERENCES
1. IRC:581974, Guidelines for the Design of Rigid Pavements for Highways, Indian Roads Congress, New Delhi.
2. IRC:581989, 1st Revision, Guidelines for the Design of Rigid Pavements for Highways, Indian Roads Congress, New Delhi.
3. IRC:582002, 2nd Revision, Guidelines for the Design of Plain Jointed Rigid Pavements for Highways, Indian Roads Congress, New Delhi.
4. Westergaard, H.M. ‘Analysis of Stresses in Concrete Pavements due to Variations of Temperatures’ Highway Research Board Proc.Vol.6,1926.
5. Bradbury, R.D. ‘Reinforced Concrete Pavements’ Wire Reinforcement Institute, Washington.D.C., 1947.
6. Timoshenko, S and WoinowskyKrieger, ‘Theory of Plates and Shells’, McGrawHill Book Company 2nd edition, 1959.
7. Croney. David and Croney, Paul ‘The Design and Performance of Road Pavements’ McGrawHill Book Company, 1992.
8. Subramanian,V.V. ‘Investigation on Temperature and Friction Stresses in Bonded Cement Concrete Pavements’, Ph.D. thesis, Indian Institute of Technology, 1964.
9. Choubane. B. and Tia.M ‘Nonlinear Temperature Gradient Effect on Maximum Warping Stresses in Rigid Pavements’. Transportation Research Record 1370,Washington, 1992. 