Let's break down the requirements and guide you through each step: ### 1. **Plotting the Data** You are asked to plot: - **Peak shear stress (\(\tau_p\)) vs. Normal stress (\(\sigma'_n\))** - **Critical state shear stress (\(\tau_{cs}\)) vs. Normal stress (\(\sigma'_n\))** #### Data (from Table 2): | \(\sigma'_n\) (kPa) | \(\tau_p\) (kPa) | \(\tau_{cs}\) (kPa) | |---------------------|------------------|---------------------| | 20 | 23.8 | 12.6 | | 41.7 | 43.1 | 26.3 | | 100 | 83.9 | 63.2 | | 200 | 132.0 | 126.4 | ### 2. **Sketching Failure Envelopes** The failure envelope is typically a straight line for dense sand: \[ \tau = \sigma' \tan \phi \] Where: - \(\tau\) = shear stress - \(\sigma'\) = normal effective stress - \(\phi\) = angle of friction #### For the **critical state**, the envelope passes through the origin (c = 0). ### 3. **Finding the Critical State Angle of Friction (\(\phi_{cs}\))** Use the equation of the straight line passing through the origin: \[ \tau_{cs} = \sigma' \tan \phi_{cs} \] So: \[ \tan \phi_{cs} = \frac{\tau_{cs}}{\sigma'} \] Calculate \(\phi_{cs}\) for each data point: | \(\sigma'_n\) (kPa) | \(\tau_{cs}\) (kPa) | \(\tau_{cs}/\sigma'_n\) | \(\phi_{cs}\) (deg) | |---------------------|---------------------|-------------------------|---------------------| | 20 | 12.6 | 0.63 | 32.3° | | 41.7 | 26.3 | 0.63 | 32.3° | | 100 | 63.2 | 0.632 | 32.3° | | 200 | 126.4 | 0.632 | 32.3° | - \(\phi_{cs} = \tan^{-1}(0.632) = 32.3^\circ\) (matches the answer given). --- ## **How to Sketch** 1. **Draw axes**: Horizontal axis = \(\sigma'_n\), Vertical axis = shear stress (\(\tau\)). 2. **Plot both sets of points**: - One for peak (\(\tau_p\)) vs. \(\sigma'_n\) - One for critical state (\(\tau_{cs}\)) vs. \(\sigma'_n\) 3. **Draw lines through the origin** for each set (peak and critical state lines). - The slope of the critical state line = \(\tan 32.3^\circ\). - The peak envelope will have a higher slope, indicating a higher friction angle. --- ### **Summary Table for Calculation** | Stress (\(\sigma'_n\)) | Peak Shear (\(\tau_p\)) | Critical State (\(\tau_{cs}\)) | |------------------------|-------------------------|----------------------------------| | 20 | 23.8 | 12.6 | | 41.7 | 43.1 | 26.3 | | 100 | 83.9 | 63.2 | | 200 | 132.0 | 126.4 | ### **Key Equations** - **Critical state envelope slope**: \(\tan \phi_{cs}\) - \(\phi_{cs} = \tan^{-1}( \frac{\tau_{cs}}{\sigma'_n} )\) --- ## **Final Answers** - **Critical state angle of friction:** \(\phi_{cs} = 32.3^\circ\) - **Failure envelopes:** Straight lines through the origin with slopes corresponding to \(\tan \phi_{peak}\) and \(\tan \phi_{cs}\). If you need a plotted graph, let me know!