In a direct shear test, a soil specimen is placed in a split shear box. A normal stress is applied first, then one half of the box is moved relative to the other until the specimen fails in shear.
$P$ is normal load, $V$ is shear force at failure, and $A$ is cross-sectional area of the shear plane.
The shear strength of soil is commonly represented by the Mohr-Coulomb failure equation.
Here $c$ is cohesion and $\phi$ is the angle of internal friction.
In a triaxial test, the specimen is enclosed in a membrane, subjected to confining pressure, then loaded axially until failure. The axial stress increment at failure is called deviator stress.
CD tests allow drainage during consolidation and shearing. CU tests consolidate first, then shear without drainage. UU tests do not allow drainage during either stage.
For saturated clay in UU loading, the total-stress friction angle is often taken as zero. The undrained shear strength is the radius of Mohr's circle.
A dry sand sample is tested in direct shear. The shear box is circular with diameter 50 mm. Normal load is 200 N and failure shear force is 130 N. Determine $\phi$.
Answer: $\phi = 33.02^\circ$.
A direct shear test gives several combinations of normal force $P$ and shear force $V$ at failure. Compute cohesion, friction angle, and $V$ for a new value of $P$.
The numerical table is embedded as an image in the PDF. Use this process once the table values are read:
Plot $\tau$ versus $\sigma$ or compute the slope from two points. The intercept is $c$ and the slope is $\tan\phi$.
A dry sand sample 60 mm by 60 mm by 25 mm high is tested in direct shear. Normal stress is 100 kPa and shear force at failure is 300 N. Determine the angle of friction.
Answer: $\phi=39.81^\circ$.
A series of undrained shear box tests gives normal loads and shear loads at failure. Determine cohesion of the soil sample.
The load table is embedded as an image in the PDF. Convert each load to stress using the given box area, then plot the failure envelope.
A dry sand sample has normal stress 200 kPa and fails at shear stress 135 kPa. Determine $\phi$, the failure shear stress for normal stress 145 kPa, and shear stress at 5 m depth if $\gamma=15.8$ kN/m3.
Answer: $\phi=34.02^\circ$, $\tau=97.88$ kPa, and $\tau=53.33$ kPa at 5 m depth.
A drained sand sample has normal stress 50 kPa and shear stress 30 kPa on the failure plane. Determine internal friction angle, failure plane angle with respect to the horizontal plane, and axial stress applied.
Answer: $\phi=30.96^\circ$, $\theta=60.48^\circ$, and axial/deviator stress is about 69.97 kPa.
A cohesive soil has $\phi=28^\circ$ and $c=31$ kPa. If maximum shearing stress is 65 kPa, compute lateral pressure in the cell at failure.
Answer: Lateral cell pressure is 15.15 kPa.
In a triaxial test for normally consolidated soil, normal stress at failure is 475 kPa and shear stress at failure is 350 kPa. Compute the maximum principal stress at failure.
Answer: $\sigma_1=1167.65$ kPa.
A cohesive soil has $\phi=28^\circ$ and $c=30$ kPa. If maximum shearing stress is 70 kPa, compute the normal stress at the point of maximum shear.
The normal stress at maximum shear is the center of Mohr's circle.
Answer: 92.68 kPa.
In a triaxial test on cohesionless soil, chamber pressure is 10 kPa and failure occurs when major compressive stress reaches 30 kPa. Compute the angle of internal friction.
Answer: $\phi=30^\circ$.
During an unconsolidated undrained test, the principal stress at failure is 114 kPa. Compute undrained cohesion for the unconfined compression test.
Answer: 57 kPa.
An unconfined compression test was conducted on a soil sample 50 mm in diameter. Failure load was 66 N. Find cohesion strength of clay.
Answer: 16.81 kPa.
For fully saturated clay, $c'=15$ kPa and $\phi'=29^\circ$. In a CU triaxial test, all-around pressure is 100 kPa and principal stress difference at failure is 170 kPa. Find expected pore water pressure at failure.
Answer: Expected pore water pressure is 36.73 kPa.
Loose sand samples were tested under consolidated undrained conditions. Failure stresses and excess pore water pressures are tabulated in the PDF. Compute drained friction angle.
The numerical table is embedded as an image. Use the effective principal stresses at failure.
Using the SPT correlation table for cohesive soils, estimate approximate shear strength if blow count $N=6$.
For $N=6$, the soil is in the medium range, with $q_u$ approximately between 47.8 kPa and 95.7 kPa.
Answer: Approximate shear strength is about 35.9 kPa.
The Mohr-Coulomb criterion defines the shear stress at failure on any plane in a soil mass:
where $c'$ is effective cohesion and $\phi'$ is effective friction angle. For clean sands $c'=0$; for saturated clays under undrained loading $\phi_u=0$ and $\tau_f=S_u$. In a triaxial test the failure plane is inclined at $\alpha_f=45+\phi/2$ from the major principal plane. The Mohr circle at failure gives:
In a direct shear test the failure plane is forced horizontal, so normal and shear stresses are read directly. Plotting results from multiple tests gives the Mohr-Coulomb envelope: slope $=\tan\phi$, intercept $=c$. Sensitivity $S_t = S_{u,\text{undisturbed}}/S_{u,\text{remolded}}$ measures how much a clay loses strength when disturbed.
Three direct shear tests on a dry sand specimen yield the following failure data: Test 1 — normal stress 50 kPa, shear stress 29 kPa; Test 2 — normal stress 100 kPa, shear stress 58 kPa; Test 3 — normal stress 150 kPa, shear stress 87 kPa. Determine the cohesion $c$ and friction angle $\phi$.
Fit a straight line $\tau_f = c + \sigma_n\tan\phi$ through the three points. Use the slope from two end-points:
Intercept from Test 1: $29 = 0.580(50) + c \Rightarrow c = 29 - 29 = 0$ kPa.
Answer: $c = 0$ kPa and $\phi = 30.1°$ (clean sand, friction only).
A consolidated-undrained triaxial test on saturated sand ($c=0$) gives: cell pressure $\sigma_3=100$ kPa, deviator stress at failure $\sigma_1-\sigma_3=280$ kPa. Find (a) friction angle $\phi$, (b) angle of failure plane from horizontal, and (c) shear stress on the failure plane.
Shear stress on the failure plane using Mohr circle (radius $R=140$ kPa):
Answer: $\phi=35.7°$, failure plane at $62.85°$ from horizontal, $\tau_f=113.8$ kPa.
Three drained triaxial tests on a sandy clay give: Test 1 — $\sigma_3=40$ kPa, $\sigma_1=200$ kPa; Test 2 — $\sigma_3=80$ kPa, $\sigma_1=320$ kPa; Test 3 — $\sigma_3=160$ kPa, $\sigma_1=560$ kPa. Find the effective strength parameters $c'$ and $\phi'$.
Express as $\sigma_1 = N_\phi\,\sigma_3 + K$ where $N_\phi=\tan^2(45+\phi/2)$ and $K=2c\sqrt{N_\phi}$. Subtract equations for Tests 1 and 2:
$K = 200 - 40(3.0) = 80$, and $K = 2c\tan(60°) = 2c(1.732)$:
Verify with Test 3: $160(3)+80=560$ ✓
Answer: $c'=23.1$ kPa and $\phi'=30°$.
A vane shear test uses a vane with height $H=130$ mm and diameter $D=65$ mm. Torque at failure is $T=38$ N·m. Calculate undrained shear strength $S_u$, assuming equal shear mobilization on the cylindrical surface and both end caps.
Answer: $S_u \approx 37.7$ kPa.
An undisturbed clay sample has unconfined compressive strength $q_u=144$ kPa. After remolding at the same water content, $q_{u,\text{rem}}=18$ kPa. Compute undrained shear strength of each sample and the sensitivity $S_t$. Classify the clay.
$S_t=8$ falls in the "sensitive" category ($4 < S_t \leq 8$). Such clays can lose a large fraction of their strength when sampled or remolded, and must be handled carefully in the field.
Answer: $S_u=72$ kPa (undisturbed), $S_u=9$ kPa (remolded), $S_t=8$ — sensitive clay.