Let's solve the steps systematically as described in the problem. The task is to derive and explain **Terzaghi's 1D consolidation equation** and related concepts. --- ## **Step 1: Volumetric Strain and Effective Stress** #### **Fundamental Relationship** For a saturated soil element: - **Total Stress**: \(\sigma\) (constant) - **Pore Water Pressure**: \(u\) - **Effective Stress**: \(\sigma' = \sigma - u\) Assume incompressible grains and water, so volume change is due to soil skeleton only. **Volumetric strain** (\(\epsilon_v\)) is related to the change in effective stress by the coefficient of volume compressibility (\(m_v\)): \[ \epsilon_v = m_v \Delta \sigma' \] **Rate of change**: \[ \frac{\partial \epsilon_v}{\partial t} = m_v \frac{\partial \sigma'}{\partial t} \] But since \(\sigma\) is constant: \[ \frac{\partial \sigma'}{\partial t} = -\frac{\partial u}{\partial t} \] So: \[ \frac{\partial \epsilon_v}{\partial t} = -m_v \frac{\partial u}{\partial t} \] --- ## **Step 2: Darcy's Law and Conservation of Mass** For a soil slice of thickness \(dz\): - **Inflow**: \(q_{in}\) - **Outflow**: \(q_{out}\) - **Rate of change of volume of water in element**: \(q_{in} - q_{out}\) Darcy's law for vertical flow: \[ q = -k \frac{\partial h}{\partial z} \] where \(k\) is the permeability, \(h\) is the hydraulic head. In consolidation, change in head is due to change in excess pore water pressure \(u\): \[ q = -k \frac{1}{\gamma_w} \frac{\partial u}{\partial z} \] **Net outflow** from element: \[ q_{in} - q_{out} = - \frac{\partial q}{\partial z} dz \] --- ## **Step 3: Equate Soil Skeleton Volume Change to Net Outflow** Volume change per unit volume per unit time is: \[ \frac{\partial \epsilon_v}{\partial t} dz \] But this must equal the net outflow per unit volume: \[ \frac{\partial \epsilon_v}{\partial t} dz = - \frac{\partial q}{\partial z} dz \] \[ \frac{\partial \epsilon_v}{\partial t} = - \frac{\partial q}{\partial z} \] Substitute for \(q\): \[ \frac{\partial \epsilon_v}{\partial t} = - \frac{\partial}{\partial z} \left( -k \frac{1}{\gamma_w} \frac{\partial u}{\partial z} \right) \] \[ \frac{\partial \epsilon_v}{\partial t} = \frac{\partial}{\partial z} \left( k \frac{1}{\gamma_w} \frac{\partial u}{\partial z} \right) \] If \(k\) is constant: \[ \frac{\partial \epsilon_v}{\partial t} = k \frac{1}{\gamma_w} \frac{\partial^2 u}{\partial z^2} \] Recall from Step 1: \[ \frac{\partial \epsilon_v}{\partial t} = -m_v \frac{\partial u}{\partial t} \] So: \[ -m_v \frac{\partial u}{\partial t} = k \frac{1}{\gamma_w} \frac{\partial^2 u}{\partial z^2} \] \[ \frac{\partial u}{\partial t} = \frac{k}{m_v \gamma_w} \frac{\partial^2 u}{\partial z^2} \] --- ## **Step 4: Coefficient of Consolidation** Define **coefficient of consolidation** \(c_v\): \[ c_v = \frac{k}{m_v \gamma_w} \] So the partial differential equation becomes: \[ \frac{\partial u}{\partial t} = c_v \frac{\partial^2 u}{\partial z^2} \] --- ## **Step 5: Non-Dimensional Variables** Let: - \(Z = \frac{z}{H}\) (Depth ratio) - \(T_v = \frac{c_v t}{H^2}\) (Time factor) Transform the equation: \[ \frac{\partial u}{\partial t} = c_v \frac{\partial^2 u}{\partial z^2} \] to: \[ \frac{\partial u}{\partial T_v} = \frac{\partial^2 u}{\partial Z^2} \] --- ## **Step 6: Analytical Solution and Degree of Consolidation** The general analytical solution (for double drainage, zero initial excess pressure at boundaries): \[ u(z, t) = u_ \sum_{n=1}^{\infty} \frac{2}{n\pi} \sin\left(\frac{n\pi z}{H}\right) \exp\left(-c_v \left(\frac{n\pi}{H}\right)^2 t\right) \] **Local degree of consolidation** at depth \(z\): \[ U_z = \frac{u_ - u(z, t)}{u_} \] --- ## **Step 7: Average Degree of Consolidation** \[ U_{avg} = \frac{1}{H} \int_^H U_z \, dz \] This results in: \[ U_{avg} = 1 - \sum_{n=1,3,5,...}^{\infty} \frac{2}{n\pi} \exp\left(-n^2 \pi^2 T_v\right) \] where the sum is over odd values of \(n\). --- ## **Summary Table** | Step | Equation/Concept | Final Formula/Explanation | |------|------------------|--------------------------| | 1 | Volumetric strain & \(\frac{\partial \epsilon_v}{\partial t} = -m_v \frac{\partial u}{\partial t}\) | | 2 | Darcy's Law | \(q = -k \frac{1}{\gamma_w} \frac{\partial u}{\partial z}\) | | 3 | Mass conservation | \(\frac{\partial \epsilon_v}{\partial t} = - \frac{\partial q}{\partial z}\) | | 4 | Consolidation eqn | \(\frac{\partial u}{\partial t} = c_v \frac{\partial^2 u}{\partial z^2}\) | | 5 | Non-dimensional | \(Z = \frac{z}{H}, \; T_v = \frac{c_v t}{H^2}\) | | 6 | Solution | \(u(z, t) = u_ \sum ...\), \(U_z = \frac{u_ - u(z, t)}{u_}\) | | 7 | \(U_{avg}\) | \(U_{avg} = 1 - \sum_{n=1,3,5,...}^{\infty} \frac{2}{n\pi} \exp(-n^2 \pi^2 T_v)\) | --- ## **Final Answers** **Governing PDE (Terzaghi’s 1D consolidation equation):** \[ \frac{\partial u}{\partial t} = c_v \frac{\partial^2 u}{\partial z^2} \] **Coefficient of consolidation:** \[ c_v = \frac{k}{m_v \gamma_w} \] **Dimensionless variables:** \[ Z = \frac{z}{H}, \quad T_v = \frac{c_v t}{H^2} \] **Average degree of consolidation:** \[ U_{avg} = 1 - \sum_{n=1,3,5,...}^{\infty} \frac{2}{n\pi} \exp(-n^2 \pi^2 T_v) \] --- ### **Diagram Explanation** - The diagram shows a soil layer of thickness \(H\) under load. - \(u(z,t)\): excess pore pressure at depth \(z\) and time \(t\). - \(q_{in}\) and \(q_{out}\): flow in and out of an element \(dz\). - \(k\): permeability, \(m_v\): compressibility. --- **Let me know if you need the step-by-step derivation for a particular part or further clarification!**