Designing Slabs on Grade with Westergaard Theory

Industrial floors, airport pavements, and warehouse slabs are examples of slabs on grade, concrete slabs that rest directly on supporting soil or granular base. Their thickness must be sufficient to distribute wheel or rack loads so that the induced tensile stress at the bottom of the slab does not exceed the material’s flexural strength. Although many empirical methods exist, a classic analytical approach comes from Westergaard’s work on plate theory, which models a concrete slab as a thin elastic plate resting on a dense liquid foundation. The equations remain widely used for preliminary design and provide valuable insight into how soil stiffness and load magnitude interact.

This calculator implements Westergaard’s formula for an interior load, meaning the wheel load is applied far from free edges and corners so that the slab behaves as if it extends infinitely in all directions. The contact between wheel and slab is assumed to be circular with radius $a$. The slab’s response depends on its thickness $h$, the concrete’s modulus of elasticity $E$, Poisson’s ratio $\mathrm{\backslash nu}$, and the subgrade modulus $k$, which characterizes how the supporting soil reacts to deflection. Westergaard defined a radius of relative stiffness given by $$L={\frac{E{h}^3}{12k(1-{\mathrm{\backslash nu}}^2)}}^{}$$.

The maximum tensile stress at the bottom of the slab under an interior wheel load can be approximated by $$\mathrm{\backslash sigma}=\frac{0.316P(1+1.6\frac{a}{L})}{h^2}$$. Here $P$ is the wheel load and $a$ is the contact radius. To ensure safety, this stress must not exceed the concrete’s flexural strength $R$. The calculator iteratively solves for the slab thickness that satisfies $\mathrm{\backslash sigma}\le R$ by starting with an initial guess and refining it until the computed stress falls below the allowable limit.

Although Westergaard’s model makes several simplifying assumptions, such as a homogeneous, isotropic slab and a perfectly elastic support, it captures key trends. Increasing the subgrade modulus $k$ reduces deflections, allowing a thinner slab for the same wheel load. Conversely, heavy wheel loads or weak subgrades demand thicker slabs. The modulus of elasticity $E$ affects the radius of relative stiffness: stiffer concrete spreads the load over a broader area, reducing stresses.

Designers often estimate the subgrade modulus from plate load tests or correlations with soil type. Typical values are summarized below to aid preliminary studies:

Soil Type	k (MPa/m)
Soft clay	20
Medium clay	40
Dense sand	80
Gravel base	150

The calculator assumes the load is static and uniformly distributed over the contact area. Dynamic loads from forklifts or crane wheels may require additional impact factors. Edge or corner loading produces higher stresses than interior loading and would necessitate thicker slabs or reinforcement. For such conditions, other Westergaard equations should be used.

An example illustrates the process. Suppose a warehouse slab will support a 40 kN wheel load with a contact radius of 0.15 m on soil with k = 50 MPa/m. Using the default material properties and iterating, the calculator might return a required thickness of roughly 0.18 m. If the subgrade were improved to k = 100 MPa/m, the required thickness would drop because the stiffer support reduces bending. Alternatively, increasing the wheel load to 80 kN would raise the required thickness considerably.

The iterative algorithm seeks the smallest thickness satisfying the stress limit. Starting from a guess of 0.1 m, it repeatedly computes the radius of relative stiffness, evaluates the induced stress, and increases the thickness in small increments until $\mathrm{\backslash sigma}$ drops below $R$. This straightforward procedure mirrors manual design charts where engineers enter the wheel load and subgrade modulus to read off the required slab depth.

While the tool focuses on unreinforced slabs, many industrial floors incorporate reinforcing steel or fiber reinforcement. Steel does not significantly change the elastic analysis but provides post-cracking capacity and controls shrinkage. When reinforcement is present, designers may allow higher tensile stress than the plain concrete flexural strength because the steel carries part of the load after cracking.

Another practical consideration is curling due to temperature and moisture gradients through the slab thickness. Curling can lift corners off the subgrade, increasing stresses when wheel loads pass near edges. Adequate curing practices, moisture control, and proper joint spacing help mitigate curling and the associated cracking risk.

Finally, it is important to remember that slab design involves serviceability as well as strength. Excessive deflection can misalign machinery or cause ride quality issues for vehicles. Subgrade uniformity, joint load transfer, and surface finishing all influence long-term performance. The calculator provides a starting point, but engineers should supplement it with detailed analysis and field testing when necessary.