Let's break down the steps to solve the question based on the provided data: ### **Step 1: Interpret the Data** - **Sample dimensions:** 60 mm x 60 mm (area = 3600 mm² = 0.0036 m²) - **Initial thickness:** 11.23 mm (noted but not directly needed for stress calc) - **Vertical load:** 1.39 kN = 1390 N ### **Step 2: Calculate Displacements** - **Horizontal displacement (\(\delta_x\))** and **Vertical displacement (\(\delta_y\))** are already provided in mm. ### **Step 3: Calculate Shear and Normal Stress** - **Shear stress (\(\tau\))** at each step: \[ \tau = \frac{\text{Horizontal Load}}{\text{Area}} \] - **Normal stress (\(\sigma\))** (constant throughout, since vertical load is constant): \[ \sigma = \frac{\text{Vertical Load}}{\text{Area}} = \frac{1390}{3600} = 0.386 \text{ N/mm}^2 = 386 \text{ kPa} \] ### **Step 4: Identify Peak and Critical State** - **Peak state:** Maximum shear stress - **Critical state:** Final (constant) shear stress after peak ### **Step 5: Calculate Angles of Friction** - \(\phi = \arctan\left(\frac{\tau}{\sigma}\right)\) --- ### **Calculations** #### **Shear Stress (\(\tau\))** | Horizontal Load (N) | Shear Stress (kPa) | |---------------------|--------------------| | 0 | 0.0 | | 220 | 61.1 | | 446 | 123.9 | | 628 | 174.4 | | 886 | 246.1 | | 1026 | 285.0 | | 1098 | 305.0 | | 1081 | 300.3 | #### **Peak State** - **Peak Shear Stress (\(\tau_{peak}\)):** 305.0 kPa (at 1098 N) - **Corresponding Friction Angle:** \[ \tan \phi_{peak} = \frac{305}{386} = 0.790 \] \[ \phi_{peak} = \arctan(0.790) = 38.3^\circ \] #### **Critical State** - **Critical Shear Stress (\(\tau_{crit}\)):** 300.3 kPa (at 1081 N, last value) - **Corresponding Friction Angle:** \[ \tan \phi_{crit} = \frac{300.3}{386} = 0.778 \] \[ \phi_{crit} = \arctan(0.778) = 37.9^\circ \] --- ### **Plots** - **\(\delta_x\) vs. \(\delta_y\):** Plot horizontal displacement (x) vs. vertical displacement (y). - **\(\tau\) vs. \(\delta_y\):** Plot shear stress (y) vs. horizontal displacement (x). --- ### **Summary Table** | State | Shear Stress (kPa) | Friction Angle (deg) | |-------------------|--------------------|----------------------| | **Peak** | 305.0 | 38.3 | | **Critical State**| 300.3 | 37.9 | --- **If you need the plots or calculations for each step shown, let me know!**

Certainly! Here's a detailed step-by-step guide to generate the plots and perform the calculations based on the data provided. --- ### **Given Data Recap:** | Horizontal Load (N) | Horizontal Displacement (mm) | Vertical Displacement (mm) | |-----------------------|------------------------------|----------------------------| | 0 | 0 | 0 | | 220 | 0.054 | 0.123 | | 446 | 0.204 | 0.382 | | 628 | 0.354 | 0.483 | | 886 | 0.762 | 0.225 | | 1026 | 1.344 | -0.359 | | 1098 | 1.980 | -0.763 | | 1081 | 2.676 | -0.966 | --- ### **Step 1: Calculate Shear Stress (\(\tau\))** - **Area of the sample:** \[ A = 60 \text{ mm} \times 60 \text{ mm} = 3600 \text{ mm}^2 \] - **Vertical load:** \[ P_v = 1.39 \text{kN} = 1390 \text{ N} \] - **Normal stress (\(\sigma\)) (constant):** \[ \sigma = \frac{P_v}{A} = \frac{1390}{3600} \approx 0.386 \text{ N/mm}^2 = 386 \text{kPa} \] - **Shear stress at each step:** \[ \tau = \frac{\text{Horizontal Load}}{A} \] | Load (N) | \(\tau\) (kPa) | |----------|----------------| | 0 | 0.0 | | 220 | \(220/3600 \approx 61.1\) | | 446 | \(446/3600 \approx 123.9\) | | 628 | \(628/3600 \approx 174.4\) | | 886 | \(886/3600 \approx 246.1\) | | 1026 | \(1026/3600 \approx 285.0\) | | 1098 | \(1098/3600 \approx 305.0\) | | 1081 | \(1081/3600 \approx 300.3\) | --- ### **Step 2: Plot \(\tau\) vs. Horizontal Displacement (\(\delta_x\))** - **X-axis:** Horizontal Displacement (\(\delta_x\)) - **Y-axis:** Shear Stress (\(\tau\)) **Data points for plotting:** | \(\delta_x\) (mm) | \(\tau\) (kPa) | |-------------------|----------------| | 0.054 | 61.1 | | 0.204 | 123.9 | | 0.354 | 174.4 | | 0.762 | 246.1 | | 1.344 | 285.0 | | 1.980 | 305.0 | | 2.676 | 300.3 | Plot these points to visualize the shear stress evolution with horizontal displacement. --- ### **Step 3: Plot \(\tau\) vs. Vertical Displacement (\(\delta_y\))** - **X-axis:** Vertical Displacement (\(\delta_y\)) - **Y-axis:** Shear Stress (\(\tau\)) **Data points:** | \(\delta_y\) (mm) | \(\tau\) (kPa) | |-------------------|----------------| | 0.123 | 61.1 | | 0.382 | 123.9 | | 0.483 | 174.4 | | 0.225 | 246.1 | | -0.359 | 285.0 | | -0.763 | 305.0 | | -0.966 | 300.3 | Plotting these points will help identify the peak shear stress (maximum \(\tau\)) and the corresponding states. --- ### **Step 4: Identify Peak and Critical States** - **Peak shear stress:** maximum \(\tau\) = **305.0 kPa** at \(\delta_x \approx 1.98\) mm and \(\delta_y \approx -0.763\) mm; corresponds to load 1098 N. - **Critical shear stress:** last value, \(\tau \approx 300.3\) kPa at \(\delta_x \approx 2.676\) mm and \(\delta_y \approx -0.966\) mm. --- ### **Step 5: Calculate Friction Angles** \[ \phi = \arctan\left(\frac{\tau}{\sigma}\right) \] - **Peak State:** \[ \phi_{peak} = \arctan\left(\frac{305}{386}\right) \approx \arctan(0.790) \approx 38.3^\circ \] - **Critical State:** \[ \phi_{crit} = \arctan\left(\frac{300.3}{386}\right) \approx \arctan(0.778) \approx 37.9^\circ \] --- ### **Summary:** | State | Shear Stress (kPa) | Friction Angle (°) | |-------------------|--------------------|--------------------| | **Peak** | 305.0 | **38.3°** | | **Critical State**| 300.3 | **37.9°** | --- ### **Final Notes:** - The plots of \(\tau\) vs. \(\delta_x\) and \(\delta_y\) help visualize the shear behavior. - The angles of friction indicate the soil's strength parameters at peak and critical states. If you'd like, I can also generate these plots for you or assist with further analysis!

Certainly! Here's how the plots would look based on the data provided. Since I can't create graphical plots directly here, I'll give you the instructions and sample Python code (using matplotlib) that you can run to generate the plots. --- ### Plot 1: Shear Stress (\(\tau\)) vs Horizontal Displacement (\(\delta_x\)) **Data points:** | \(\delta_x\) (mm) | \(\tau\) (kPa) | |-------------------|----------------| | 0.054 | 61.1 | | 0.204 | 123.9 | | 0.354 | 174.4 | | 0.762 | 246.1 | | 1.344 | 285.0 | | 1.980 | 305.0 | | 2.676 | 300.3 | --- ### Plot 2: Shear Stress (\(\tau\)) vs Vertical Displacement (\(\delta_y\)) **Data points:** | \(\delta_y\) (mm) | \(\tau\) (kPa) | |-------------------|----------------| | 0.123 | 61.1 | | 0.382 | 123.9 | | 0.483 | 174.4 | | 0.225 | 246.1 | | -0.359 | 285.0 | | -0.763 | 305.0 | | -0.966 | 300.3 | --- ### Python code to generate the plots: ```python import matplotlib.pyplot as plt # Data for Plot 1 delta_x = [0.054, 0.204, 0.354, 0.762, 1.344, 1.980, 2.676] shear_stress_x = [61.1, 123.9, 174.4, 246.1, 285.0, 305.0, 300.3] # Data for Plot 2 delta_y = [0.123, 0.382, 0.483, 0.225, -0.359, -0.763, -0.966] shear_stress_y = [61.1, 123.9, 174.4, 246.1, 285.0, 305.0, 300.3] plt.figure(figsize=(14, 6)) # Plot 1: tau vs delta_x plt.subplot(1, 2, 1) plt.plot(delta_x, shear_stress_x, marker='o', color='b') plt.xlabel('Horizontal Displacement δx (mm)') plt.ylabel('Shear Stress τ (kPa)') plt.title('Shear Stress vs Horizontal Displacement') plt.grid(True) # Plot 2: tau vs delta_y plt.subplot(1, 2, 2) plt.plot(delta_y, shear_stress_y, marker='o', color='r') plt.xlabel('Vertical Displacement δy (mm)') plt.ylabel('Shear Stress τ (kPa)') plt.title('Shear Stress vs Vertical Displacement') plt.grid(True) plt.tight_layout() plt.show() ``` --- ### **Summary:** - The first plot shows how shear stress evolves with horizontal displacement, reaching a peak at around 1.98 mm. - The second plot reveals how shear stress varies with vertical displacement, showing the peak and subsequent values indicating the critical state. **You can copy this code into a Python environment with matplotlib installed (e.g., Jupyter Notebook) to generate the plots.** If you'd like me to prepare images of the plots or help with any other part, just let me know!