Flexible vs rigid; radius of relative stiffness, k

Differences between flexible and rigid pavement

Radius of relative stiffness (l)

The radius of relative stiffness $l$ is a measure of how stiff the slab is compared with the subgrade support. It defines the area of slab that takes part in resisting a load (Westergaard).

$$l=\left[\frac{E\,h^{3}}{12\,(1-\mu^{2})\,k}\right]^{1/4}$$

where $E$ is the modulus of elasticity of concrete (kg/cm2), $h$ the slab thickness (cm), $\mu$ Poisson's ratio and $k$ the modulus of subgrade reaction (kg/cm3). The unit of $l$ is cm.

• A thicker slab or higher $E$ gives a larger $l$ (a stiffer slab spreads the load over a wider area); a higher $k$ gives a smaller $l$.
• Typical values for 20-30 cm slabs are about 60-100 cm.
• It is used in all the stress equations (Westergaard, Kelly, Bradbury), in the ratios $L/l$ and $B/l$ for the warping coefficient, and to decide joint spacing and the spread of dowel load transfer (about $1.8\,l$).

Modulus of subgrade reaction (k)

The modulus of subgrade reaction $k$ is the pressure required to cause unit deflection of the subgrade (Winkler's foundation assumption):

$$k=\frac{p}{\Delta}\ \ (\text{kg/cm}^3)$$

It is found by a plate-load test: a rigid 75 cm diameter plate is loaded in increments, and $k$ is taken at 0.125 cm deflection (so $k = p_{0.125}/0.125$). The test value is corrected as follows:

• Correction for plate size: $k_{75}= k_{30}/2$ when a 30 cm plate is used.
• Correction for saturation (soaking) of the subgrade soil.
• A sub-base (e.g. lean concrete or granular) raises the effective $k$; IRC:58 gives tables of $k$ over granular or cement-treated sub-bases.
• Roughly, IRC:58 relates $k$ to the soaked CBR of the subgrade: about 2 kg/cm3 for CBR 2 and 5 to 6 kg/cm3 for CBR 10.

Rigid pavement stresses depend only weakly on $k$ (through $l$), so approximate values are acceptable.