Temperature stresses in rigid pavement; dowel bar design

Temperature stresses in rigid pavement

Temperature stresses in a concrete slab are of three kinds:

1. Warping stress - arises from the difference of temperature between the top and bottom of the slab (day: top hotter, bottom in tension; night: top in tension). The curling is resisted by the self-weight of the slab. Bradbury's formula for the edge is

$$\sigma_{te}=\frac{C\,E\,\alpha\,\Delta T}{2}$$

where $C$ is the coefficient from $L/l$ (chart), $E$ modulus of concrete, $\alpha$ coefficient of thermal expansion ($10\times10^{-6}$ per $^\circ$C) and $\Delta T$ the temperature differential across the slab. At the corner, $\sigma_{tc}=\dfrac{E\alpha\Delta T}{3(1-\mu)}\sqrt{\dfrac{a}{l}}$. 2. Frictional stress - due to overall expansion or contraction of the slab restrained by the sub-base:

$$\sigma_f=\frac{WLf}{2\times10^4}$$

3. Expansion stress - if free movement is blocked at joints (for example by sand or stones in joints), a compressive stress $E\alpha\,\Delta t$ develops and the slab may buckle or blow up, so expansion joints and sealants are provided.

Critical combinations: at the edge in summer (midday) $\sigma_e+\sigma_{te}-\sigma_f$; at the edge in winter (midday) $\sigma_e+\sigma_{te}+\sigma_f$; and at the corner at night $\sigma_c+\sigma_{tc}$ (friction is negligible at the corner).

Design procedure of dowel bars at expansion joints

Dowel bars transfer load across the expansion joint (smooth round mild steel bars, one end greased or sleeved with a cap). Design procedure (Bradbury, as in Khanna and Justo, with IRC:58 detailing):

1. Given data - wheel load $P$, joint width $\delta$, dowel diameter $d$, permissible stresses in shear ($F_s$), bending ($F_f$) and bearing ($F_b$), slab thickness, $E$, $k$, $\mu$.
2. Radius of relative stiffness $l=\left[\dfrac{Eh^3}{12(1-\mu^2)k}\right]^{1/4}$.
3. Load to be transferred by the dowel system is taken as 40 percent of the design wheel load.
4. Capacity of a single dowel (least of the three governs):
   • Shear: $P_s=0.785\,d^2F_s$
   • Bending: $P_f=\dfrac{2d^3F_f}{L_d+8.8\delta}$
   • Bearing: $P_b=\dfrac{F_b\,L_d^2\,d}{12.5\,(L_d+1.5\delta)}$
5. Length of dowel: the embedded length $L_d$ is found by equating the bending and bearing capacities (by trial); the total bar length is $L_d+\delta$, rounded up to a practical value.
6. Spacing - the dowel under the load transfers its full capacity; a dowel at distance $x$ transfers the proportion $(1-x/1.8l)$ and dowels beyond $1.8l$ transfer nothing. For a trial spacing $s$ add the proportions on both sides: $\Sigma = 1+2\sum(1-ks/1.8l)$.
7. Check $P_d\times\Sigma\ge 0.4P$; adjust spacing (usually 30-45 cm) until satisfied.
8. Detail the bar: one half coated with bitumen or greased, and a cap leaving an expansion gap at the free end.