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.
Additional board-style practice items for this topic.
In a triaxial test on a normally consolidated soil (cohesion $c=0$), failure occurs on a plane where the normal stress is 500 kPa and the shear stress is 280 kPa.
Compute the angle of internal friction, in degrees.
Compute the angle of the failure plane with the major principal plane, in degrees.
Compute the major principal stress at failure, in kPa.
Part 1 — Friction angle. The failure envelope passes through the origin, so
$$\tan\phi=\frac{\tau_f}{\sigma_f}.$$Part 2 — Failure-plane angle.
$$\theta=45^{\circ}+\frac{\phi}{2}.$$Part 3 — Major principal stress. From the Mohr circle, the radius and center are $R=\dfrac{\tau}{\cos\phi}$ and $\dfrac{R}{\sin\phi}$, so
$$\sigma_1=\frac{R}{\sin\phi}+R.$$A cohesionless specimen has confining pressure 65 kPa and fails when the additional axial (deviator) stress reaches 150 kPa. Find its internal friction angle.
A drained triaxial test has confining pressure 130 kPa and total axial stress at failure 260 kPa. Find the internal friction angle.
A cohesionless soil has confining pressure 65 kPa and maximum principal stress 87.75 kPa in a triaxial test. Evaluate its internal friction angle.
On the failure plane of a cohesionless soil, normal stress is 130 kPa and shear stress is 60 kPa. Determine the internal friction angle.
Evaluate the soil friction angle if the Rankine passive earth-pressure coefficient is 3.75.
Evaluate the soil friction angle if the Rankine active earth-pressure coefficient is 0.45.
An unconfined compression specimen has diameter 60 mm and fails at load 85 N.
Find undrained cohesion.
Maximum shear stress acts on a plane at what angle to a principal plane?
In triaxial loading, deviator stress creates shear on which planes?
Which statements describe Mohr-Coulomb shear strength?
Which statement about the failure plane is correct?
Which test is normally performed in the field?
Use the reference SPT-clay correlations qu≈12.5N kPa and cu=qu/2.
Find approximate qu for N1.
Find approximate cu for N2.
Report the SPT value used for consistency classification.
A Mohr circle has center stress 230 kPa and radius 160 kPa.
Find friction angle at tangency.
Find shear stress at tangency.
Find normal stress at tangency.
A cohesive soil with unconfined strength 100-200 kPa is classified as what?
A cohesive soil with unconfined strength 25-50 kPa is classified as what?
A cohesive soil with unconfined strength 200-400 kPa is classified as what?
At fixed c and φ, what happens to shear strength when normal effective stress decreases?
An unconfined compression specimen has cross-sectional area 45 cm2 and fails under an axial load of 5.5 kN.
Find cohesion.
Find unconfined compressive strength.
Find the maximum shear stress (the Mohr-circle radius).
A saturated soil slice on a plane inclined 15° has thickness 7.5 m, unit weights γd=16.5 and γsat=19 kN/m3, and friction angle 26°.
Find dry factor of safety.
Find saturated factor of safety.
Estimate critical saturated thickness.