29.2. Wheel load stresses - Westergaard’s stress equation

The cement concrete slab is assumed to be homogeneous and to have uniform elastic properties with vertical sub-grade reaction being proportional to the deflection.

Westergaard's stress equation is built on specific assumptions. Firstly, the cement concrete slab is considered homogeneous, meaning it has uniform material properties throughout its volume. This simplifies the analysis. Secondly, it presumes that the elastic properties are consistent across the slab. Lastly, it states that the vertical reaction from the sub-grade supports the slab proportionally to the amount the slab deflects when loaded. This assumption is crucial because it allows engineers to calculate how much stress is distributed in the slab due to vehicle loads.

Imagine a rubber mat laid on a table. When you press down on it, it squishes slightly, distributing that pressure evenly across its surface. Just like how the mat responds uniformly to a weight, the concrete slab in pavements behaves similarly under the pressure from vehicles, distributing the load evenly due to its uniform properties.

Westergaard developed relationships for the stress at interior, edge and corner regions, denoted as σi, σe, σc in kg/cm2 respectively and given by the equation 29.3-29.5.

Westergaard derived specific equations to calculate stresses at three different positions on the pavement slab: the interior (σi), the edge (σe), and the corner (σc). These stresses are essential because each location experiences different conditions of load and support. For instance, the stress at the edge is affected by the slab's boundaries, while the interior stress reflects the slab's overall stiffness. The equations (29.3, 29.4, and 29.5) provide formulas that incorporate slab thickness, the load applied, and other variables like the radius of load distribution. Understanding these equations allows engineers to design pavements that can handle the expected stresses effectively.

Think about carrying a heavy backpack. If you hold it in the middle, the weight is supported evenly across your back (similar to the interior stress). However, if you shift it to one side, one shoulder bears more weight, increasing the stress on that side of your body (similar to edge stress). The formulas are like calculations for determining how much more weight one side endures when the backpack is unbalanced.

Where h is the slab thickness in cm, P is the wheel load in kg, a is the radius of the wheel load distribution in cm, l the radius of the relative stiffness in cm, and b is the radius of the resisting section in cm.

The stress equations rely on several key parameters: slab thickness (h), wheel load (P), the radius of the wheel load distribution (a), the radius of relative stiffness (l), and the radius of the resisting section (b). Each parameter plays a critical role in influencing the distribution and magnitude of stress within the slab. For example, a thicker slab (larger h) will typically withstand higher loads better than a thinner slab. Each variable contributes to how the slab will react under load, guiding engineers in their design decisions.

Consider a sponge on a table. If you press down hard with a little circle (the wheel load distribution), the sponge will compress differently based on its thickness. A thick sponge (greater slab thickness) will compress less than a thin sponge. This illustrates how adding more parameters into the mix affects the stress distribution.

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