Load stress in a concrete pavement slab

3.4 Load stress in a concrete pavement slab

If a concrete pavement slab can be idealized as a thin plate resting on a Winkler foundation, the solution of Equation 3.33 will provide an estimate of the stresses due to load. The dimension of the plate is assumed as L × B and all edges are considered as free. It is assumed that a uniform loading of magnitude q o per unit area is acting on a rectangular area of dimension l × b. The center of this area is located at (¯ x, ¯ y). The arrangement is schematically presented in Figure 3.12.

Let a loading expressed in the form of Equation 3.41 be applied 2 to the plate [99, 134, 276]

where, m and n are any numbers. Then, Equation 3.41 is a possible solution to Equation 3.33. The

boundary conditions, as all edges are free (Refer to Section 3.3.4 for

2 That is, the loading is assumed to be different than uniformly distributed loading on the rectangular area shown in Figure 3.12.

68 Chapter 3. Load stress in concrete pavement

Figure 3.12: A slab resting on a Winkler spring acted upon by a rectangular patch loading.

a discussion on boundary conditions) are satisfied if ω is expressed as [99, 134, 276, 295],

where, c is a constant. The value of c can be obtained by putting Equations 3.41 and 3.42 in Equation 3.33, as [134, 276]

c= (3.43)

L 2 + B 2 +k

Thus, the expression for deflection ω becomes,

nπy ω=

mπx

2 2 2 sin

L 2 + B 2 +k

3.4. Load stress in a concrete pavement slab

Equation 3.44 provides a solution for single sinusoidal loading (Equation 3.41). It is possible to express any loading function q = f (x, y) (acting over a given area), as the sum of a series of sinusoidal loadings [99, 134, 276], as per Navier’s transformation. That is,

q = f (x, y) (3.45)

dxdy (3.46) LB

f (x, y) sin

sin

B Area

The expression for ω in the present case, therefore, becomes [99, 134, 276],

In the present case, the load is uniformly distributed over a rect- angular area of l × b, the center of which is located at (¯ x, ¯ y). That

is q = f (x, y) = Q lb within the region bound between x = ¯ x− b 2

and ¯ l x−

2 and y = ¯ y− 2 and ¯ y− 2 (refer to Figure 3.12). The value of c mn is calculated as [99, 134, 276],

LBlb x− ¯ b 2 l y− ¯ 2 L

sin (3.48) LBlbmn

sin

sin

sin

2B

B 2L

70 Chapter 3. Load stress in concrete pavement

The expression for ω is therefore [99, 134, 276],

1 mπ¯ x ω= π 2 LBlb m=1 n=1 mn π 4 m 2 n 2 D 2

B Once the expression for ω is obtained (from Equation 3.49), it can

B 2L

2B L

be used to find the bending moment (refer to Equations 3.24, 3.25 and 3.26) or the bending stress (refer to Equation 3.23). This is one of the possible approaches for obtaining the solution of Equa- tion 3.33, and one may refer to, for example, [247, 276, 295] for alternative approaches and numerical methods of solution.

Figure 3.13 presents a schematic diagram showing the variation of maximum bending stress with slab thickness (h) and a modulus of subgrade reaction (k) (these terms appear in the expressions of

D and q ∗ respectively in Equation 3.33). It may be noted that the maximum bending stress will always occur below the wheels. Fig- ure 3.13 shows that the maximum bending stress (at the interior) decreases with (i) the increase in the slab thickness and/or (ii) the increase in the modulus of the subgrade reaction.

In the above formulation the springs can take tension as well; how- ever, in reality underlaying layer does not pull back the concrete slab when it tries to bend upward. Thus tension, if it arises in the analysis, should be made equal to zero; yet, the portion (in which tension would arise) is not known a priori. Thus, the solu- tion needs to be obtained iteratively, and achieving a closed-form solution may become difficult.

Further, it is assumed that the rectangular loaded area approx- imately represents the tire imprint. However, actual tire imprint may not necessarily be rectangular in shape, nor be uniform in loading (refer to Figure 5.7).

lus of subgra

ction = k'

ction = k''

Thickness of the concrete slab

Figure 3.13: Schematic diagram showing variation of bending stress (due to load at interior) with slab thickness and modulus of subgrade reaction.