Question

A clay layer of $4\mathrm{m}$ thickness is located below a sand layer of $10\mathrm{m}$ thickness. The volumetric weights are all $20\mathrm{kN}/{\mathrm{m}}^3$, and the groundwater table coincides with the soil surface. The compression constant of the clay is $C_{10}=20$. Predict the settlement of the soil by compression of the clay layer due to an additional load of $40\mathrm{kPa}$.

Short answer

Answer: The predicted settlement of the soil due to compression of the clay layer is approximately 2.857 meters.

Step-by-step solution

Identify the given data

We are given the following data: - Thickness of clay layer ($h_{clay}$) = $4\mathrm{m}$ - Thickness of sand layer ($h_{sand}$) = $10\mathrm{m}$ - Volumetric weight ($\gamma$) = $20\mathrm{kN}/{\mathrm{m}}^3$ - Groundwater table location coincides with the soil surface - Compression constant of the clay ($C_{10}$) = $20$ - Additional load applied ($\mathrm{\Delta }\sigma$) = $40\mathrm{kPa}$

Calculate the stress increase at the clay layer

First, we will calculate the increase in stress at the middle of the clay layer due to the additional load applied. The formula to calculate this is: $\mathrm{\Delta }{\sigma }_{clay}=\frac{\mathrm{\Delta }\sigma \ast h_{sand}}{h_{sand}+h_{clay}}$ Where: - $\mathrm{\Delta }{\sigma }_{clay}$ is the increase in stress at the middle of the clay layer - $\mathrm{\Delta }\sigma$ is the additional load applied - $h_{sand}$ is the thickness of the sand layer - $h_{clay}$ is the thickness of the clay layer Plugging in the given values: $\mathrm{\Delta }{\sigma }_{clay}=\frac{40\ast 10}{10+4}=\frac{400}{14}\approx 28.57\mathrm{kPa}$

Calculate the compression index ($C_c$)

Next, we will calculate the compression index ($C_c$) using the compression constant ($C_{10}$): $C_c=\frac{1}{C_{10}}=\frac{1}{20}=0.05$

Calculate the settlement of the clay layer

Now we can calculate the settlement of the clay layer ($\mathrm{\Delta }{H}_{clay}$) using the formula: $\mathrm{\Delta }{H}_{clay}=\frac{C_c\ast h_{clay}\ast \mathrm{\Delta }{\sigma }_{clay}}{1+e_0}$ Where: - $\mathrm{\Delta }{H}_{clay}$ is the settlement of the clay layer - $C_c$ is the compression index - $h_{clay}$ is the thickness of the clay layer - $\mathrm{\Delta }{\sigma }_{clay}$ is the stress increase at the clay layer - $e_0$ is the initial void ratio (assumed to be 1 for simplicity) Plugging in the values: $\mathrm{\Delta }{H}_{clay}=\frac{0.05\ast 4\ast 28.57}{1+1}=\frac{5.714}{2}\approx 2.857\mathrm{m}$ The predicted settlement of the soil due to compression of the clay layer is approximately $2.857\mathrm{m}$.