An engineer is designing a Jointed Plain Concrete Pavement (JPCP) for an urban arterial. The design ESALs ($W_{18}$) are $15 \times 10^6$. The concrete has a Modulus of Rupture ($S_c'$) of $650 \text{ psi}$ and an Elastic Modulus ($E_c$) of $4,000,000 \text{ psi}$. The effective modulus of subgrade reaction ($k$) is $200 \text{ pci}$. The load transfer coefficient ($J$) is $3.2$, and the drainage coefficient ($C_d$) is $1.0$.

While the full AASHTO equation requires iterative solving or a nomograph, assume the engineer has narrowed the choice down to an $8\text{-inch}$ slab or a $10\text{-inch}$ slab. Using a simplified fatigue check where the allowable ESALs roughly scale with $D^4$ (thickness to the fourth power), if an $8\text{-inch}$ slab can theoretically carry $5 \times 10^6$ ESALs, calculate the theoretical capacity of the $10\text{-inch}$ slab and determine which thickness to use.

A concrete slab with a thickness ($h$) of $250 \text{ mm}$ ($0.25 \text{ m}$) is subjected to an interior wheel load ($P$) of $50,000 \text{ N}$. The radius of the contact area ($a$) is $0.15 \text{ m}$. The concrete has an elastic modulus ($E$) of $30,000 \text{ MPa}$ ($30,000 \times 10^6 \text{ N/m}^2$) and a Poisson's ratio ($\mu$) of $0.15$. The modulus of subgrade reaction ($k$) is $50 \text{ MN/m}^3$ ($50 \times 10^6 \text{ N/m}^3$). Calculate the radius of relative stiffness ($l$) and the maximum interior tensile stress ($\sigma_i$).

A concrete pavement slab is $9.0 \text{ meters}$ long and $0.20 \text{ meters}$ thick. During a sudden drop in ambient temperature, the slab attempts to contract. The coefficient of friction ($f$) between the slab and the subgrade is $1.5$. The unit weight of concrete ($\gamma_c$) is $24,000 \text{ N/m}^3$. Calculate the maximum tensile stress developed in the concrete due to subgrade friction.

In flexible pavement design, the subgrade is typically characterized by its California Bearing Ratio (CBR) or Resilient Modulus ($M_r$). However, rigid pavement design relies on the Modulus of Subgrade Reaction ($k$). Explain the physical difference between these metrics and why $k$ is used for concrete pavements.

A highway agency is inspecting an older Jointed Plain Concrete Pavement (JPCP) that was constructed without dowel bars at the transverse joints. The inspectors observe severe "faulting"—a condition where the "leave" slab is depressed lower than the "approach" slab, creating a bump at every joint. Explain the mechanical process that causes this failure and how dowel bars prevent it.