Question

In a confined compression test a soil sample of $2\mathrm{cm}$ thickness has been preloaded by a stress of $100\mathrm{kPa}$. An additional load of $20\mathrm{kPa}$ leads to a vertical displacement of $0.030\mathrm{mm}$. Determine the value of the compression constant $C_{10}$.

Short answer

Question: Determine the compression constant, $C_{10}$, for a soil sample in a confined compression test with initial thickness of 2 cm, preload stress of 100 kPa, additional load of 20 kPa, and vertical displacement after the additional load of 0.030 mm. Answer: The compression constant $C_{10}$ for the soil sample is $3.846\times {10}^{-4}{\mathrm{m}}^2/\mathrm{kN}$.

Step-by-step solution

Convert the units to SI

To ensure consistency in calculations, it's important to convert all units to SI units. The given thickness of the soil sample is in centimeters and the vertical displacement is in millimeters. We need to convert them to meters: - Soil sample thickness: $2\mathrm{cm}\times \frac{1\mathrm{m}}{100\mathrm{cm}}=0.02\mathrm{m}$ - Vertical displacement: $0.030\mathrm{mm}\times \frac{1\mathrm{m}}{1000\mathrm{mm}}=0.00003\mathrm{m}$ Now, the soil sample thickness is $0.02\mathrm{m}$, and the vertical displacement is $0.00003\mathrm{m}$.

Calculate the vertical strain

The vertical strain is the change in the vertical dimension divided by the original dimension. In this case, we have the vertical displacement and the initial thickness of the soil sample: $$\mathrm{\Delta }\epsilon =\frac{\mathrm{\Delta }h}{h_0}=\frac{0.00003\mathrm{m}}{0.02\mathrm{m}}=0.0015$$ The vertical strain is 0.0015.

Calculate the change in stress

The change in stress is the difference between the initial preload stress and the total stress applied to the soil sample after the additional load: $$\mathrm{\Delta }\sigma ={\sigma }_{\text{total}}-{\sigma }_0=(100\mathrm{kPa}+20\mathrm{kPa})-100\mathrm{kPa}=20\mathrm{kPa}$$ The change in stress is 20 kPa.

Determine the compression constant $C_{10}$

Now, use the relationship: $$\mathrm{\Delta }\epsilon =\frac{C_{10}}{1+C_{10}{\sigma }_0}\mathrm{\Delta }\sigma$$ Plug in all the known values, and solve for $C_{10}$: $$0.0015=\frac{C_{10}}{1+C_{10}(100\mathrm{kPa})}(20\mathrm{kPa})$$ Rearrange the equation to isolate $C_{10}$: $$C_{10}=\frac{0.0015}{(1+100\times C_{10})\times 20\mathrm{kPa}}$$ Solving for $C_{10}$, we get: $$C_{10}=3.846\times {10}^{-4}{\mathrm{m}}^2/\mathrm{kN}$$ The compression constant $C_{10}$ is $3.846\times {10}^{-4}{\mathrm{m}}^2/\mathrm{kN}$.