Question 1 of 5: Joint Shear Strength — Patton Bilinear Criterion and Pore-Pressure-Induced Failure

Question 1: Joint Shear Strength — Patton Bilinear Criterion and Pore-Pressure-Induced Failure (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Square shear box $160\times160\ \text{mm}$, area $A=0.0256\ \text{m}^2$; six force pairs $(F_N,F_{S,peak},F_{S,ult})$ in kN as tabulated below.

Find. (a) Peak Patton bilinear fit ($c_1',\phi_1'$, $c_2',\phi_2'$, transition $\sigma_n$) and the ultimate strength line; explain the ultimate criterion's linearity. (b) Pore pressure $u$ to bring the most critically-oriented joint to peak failure at $\sigma_1=3,\ \sigma_3=1.2$ MPa, plus the effective normal and shear stress on that plane. (c) Discuss the scale effect on joint roughness/strength.

Approach. Convert each force to stress by dividing by the box area (0.0256 m²), least-squares fit two straight lines to the peak data (specimens 1–3 through the origin, specimens 4–6 with an intercept) per the Patton model, fit one line to the ultimate data, then use the fitted high-stress peak line in a standard effective-stress Mohr-circle tangency construction for part (b).

1. Part (a) — convert forces to stresses. $\sigma_n=F_N/A$, $\tau=F_S/A$ with $A=0.0256\ \text{m}^2$ (values tabulated above, e.g. specimen 1: $\sigma_n=1.3/0.0256=\boxed{50.8\ \text{kPa}}$, $\tau_{peak}=2.5/0.0256=97.7\ \text{kPa}$).
2. Fit the low-stress peak segment (specimens 1–3). These three points lie close to a line through the origin (no measurable cohesion at low $\sigma_n$, consistent with pure dilation over asperities). Least-squares through the origin: $$\tau=\sigma_n\tan(\phi_1'),\qquad \tan(\phi_1')=\frac{\sum\sigma_n\tau}{\sum\sigma_n^2}=1.423\ \Rightarrow\ \boxed{\phi_1'\approx54.9^\circ,\ c_1'=0}$$
3. Fit the high-stress peak segment (specimens 4–6). These three points sit on a distinctly flatter line offset above the origin (asperities now shearing through rather than riding over). Ordinary least-squares: $$\tau=c_2'+\sigma_n\tan(\phi_2'),\qquad \boxed{\phi_2'\approx27.0^\circ,\ c_2'\approx396\ \text{kPa}}$$
4. Transition normal stress. Intersecting the two fitted lines, $\sigma_n\tan\phi_1' = c_2'+\sigma_n\tan\phi_2'$: $$\sigma_{n,t}=\frac{c_2'}{\tan\phi_1'-\tan\phi_2'}=\frac{395.8}{1.423-0.510}\approx\boxed{434\ \text{kPa}}\ (\tau_t\approx617\ \text{kPa})$$ This falls cleanly between specimen 3 (391 kPa, on line 1) and specimen 4 (781 kPa, on line 2), confirming the two three-point groups were split correctly.
5. Ultimate strength — single line, all six points. $$\tau_{ult}=40.6+0.412\,\sigma_n\ \Rightarrow\ \boxed{\phi_u\approx22.4^\circ}$$ A single straight line fits all six ultimate points well — no break in slope is apparent. Why no bilinearity at ultimate: the peak envelope is bilinear because two different failure mechanisms compete with normal stress — at low $\sigma_n$ the joint fails by dilating over intact asperities ($\phi_b+i$, a large apparent friction angle with no cohesion), while at high $\sigma_n$ the asperities are sheared off before the surfaces can ride up, giving a lower friction angle but a real cohesive intercept (asperity shear strength). By the time the ultimate (fully post-peak, residual) state is reached, the asperities on BOTH sides of the transition have already been ridden over or sheared down to a common, smoothed sliding surface with no further dilation left to mobilize — so a single, roughly constant friction angle (close to the rock's basic friction angle) governs at every normal stress, and the envelope stays linear.
6. Part (b) — critical joint under total stresses $\sigma_1=3$, $\sigma_3=1.2$ MPa. With joints in all orientations, the critical joint is the one tangent to the peak envelope as the effective-stress Mohr circle is pushed left by rising pore pressure $u$. The circle's radius $R=(\sigma_1-\sigma_3)/2=0.900$ MPa is unaffected by $u$; only its centre $C_c=(\sigma_1+\sigma_3)/2-u$ moves. Because $R=0.9$ MPa already places the circle's likely tangency point above the 0.434 MPa transition, use the high-stress segment ($c_2'=0.396$ MPa, $\phi_2'=27.0^\circ$). Tangency condition: $$C_c\sin\phi_2'+c_2'\cos\phi_2'=R\ \Rightarrow\ C_c=\frac{R-c_2'\cos\phi_2'}{\sin\phi_2'}=\frac{0.900-0.396(0.891)}{0.454}=1.204\ \text{MPa}$$ $$u = \frac{\sigma_1+\sigma_3}{2}-C_c = 2.100-1.204=\boxed{0.90\ \text{MPa}}$$
7. Stresses on the critical plane. $$\sigma_n'=C_c+R\sin\phi_2'=1.204+0.900(0.454)=\boxed{1.61\ \text{MPa}},\qquad \tau_f=R\cos\phi_2'=0.900(0.891)=\boxed{0.80\ \text{MPa}}$$ Check: $\sigma_n'=1.61\ \text{MPa} = 1610$ kPa exceeds the 434 kPa transition, confirming the high-stress segment was the correct choice.
   Check: the critical plane is taken at $\theta=45^\circ+\phi_2'/2\approx58.5^\circ$ from the $\sigma_1$ direction, standard Mohr-Coulomb construction; effective stresses use Terzaghi's principle $\sigma'=\sigma-u$ with the SAME pore pressure acting equally on $\sigma_1$ and $\sigma_3$.

Part (c) — scale effect on joint shear strength. A 160 mm shear box samples only the first-order (small-wavelength) asperities of a rough joint; the very high $\phi_1'\approx55^\circ$ measured at low normal stress reflects riding up over these small, steep asperities almost undamaged. A field-scale joint of the same rock exposes many metres of surface, over which larger-wavelength, lower-amplitude undulations ("waviness") dominate the effective roughness, while the small asperities that controlled the lab test are progressively sheared off before the whole surface can mobilize its peak resistance simultaneously (progressive, non-simultaneous peak mobilization along the joint). The net result, well documented by Barton & Choubey's JRC (joint roughness coefficient) framework and by Bandis, Lumsden & Barton's scale-effect studies, is that the effective peak friction angle and JRC both DECREASE as sample length increases, converging toward something closer to the ultimate/residual value measured here (≈22–27°) rather than the lab peak value of 55°. Practically, this means lab peak-strength envelopes such as the low-stress segment fitted above should never be applied directly to metre-scale field joints without a roughness/scale correction (e.g. Barton's $\text{JRC}_n = \text{JRC}_0(L_n/L_0)^{-0.02\text{JRC}_0}$); the high-stress segment and the ultimate line, being controlled by asperity shearing rather than fine-scale dilation, are comparatively less scale-sensitive.