Mohr-Coulomb Failure Calculator: Unconfined Compressive Strength, Shear Failure Envelope, and Confining Stress

Solve the Mohr-Coulomb shear failure criterion for rock and soil: find unconfined compressive strength and the maximum principal stress at failure from cohesion and friction angle, compare strength across rock types, or solve the minimum confining stress needed to prevent failure at a given applied stress. Everything updates live.

The Mohr-Coulomb Failure Criterion

Rock and soil fail in shear once the shear stress acting on some internal plane exceeds the strength of that plane. The Mohr-Coulomb criterion models that strength as a straight line, tau = c + sigma_n x tan(phi), where tau is the shear stress at failure, sigma_n is the normal stress clamping the plane shut, c is cohesion, the inherent shear strength at zero normal stress, and phi is the internal friction angle, which controls how much extra strength comes from confinement.

This tool solves the Mohr-Coulomb criterion in principal-stress form, compares strength parameters across rock types, and solves the confining stress needed to keep a given applied stress below the failure envelope.

Principal-Stress Form

Rearranged in terms of the maximum and minimum principal stresses, the same criterion becomes sigma1 = UCS + q x sigma3, where UCS is the unconfined compressive strength, UCS = 2c x tan(45 + phi/2), and q is a slope factor, q = tan^2(45 + phi/2). Confining stress sigma3 raises the stress sigma1 needed to cause failure, which is why triaxial confinement strengthens rock far beyond its unconfined strength, and why breakout and wellbore failure are sensitive to the effective confining stress at the borehole wall.

At the moment of failure, the Mohr circle for the stress state is exactly tangent to the failure envelope, and the failure plane forms at an angle of 45 minus phi over 2 measured from the sigma1 direction, a relationship that follows directly from the geometry of the Mohr circle construction.

Comparing Rock Strength

Weaker, more compressible rocks like shale carry low cohesion and a modest friction angle, while dense igneous rocks like granite carry both high cohesion and a high friction angle, compounding to a much larger unconfined compressive strength. The comparison tab builds a table across any set of rock types from cohesion and friction angle alone, with UCS and the slope factor q solved automatically for the chart.

Solving for Required Confining Stress

Given a known applied stress sigma1 that a rock mass must withstand, the principal-stress form of the Mohr-Coulomb criterion rearranges directly to solve for the minimum confining stress sigma3 that keeps the rock below its failure envelope. This is the same logic that underlies wellbore stability analysis, where mud weight supplies part of the confining stress at the borehole wall, and dropping mud weight below a critical value lets the effective stress state cross the failure envelope and produce breakout.

Common Mistakes

Using total stresses instead of effective stresses, since the Mohr-Coulomb criterion for rock and soil strength is properly applied to effective stress, total stress minus pore pressure, not total stress alone.

Assuming cohesion alone controls strength at high confinement, when in practice the friction term, sigma_n x tan(phi), dominates strength once confining stress becomes large.

Confusing the internal friction angle with the failure plane angle, the failure plane forms at 45 minus phi over 2 from the sigma1 direction, not at the friction angle itself.

Applying a single cohesion and friction angle across a heterogeneous rock mass, when natural fractures, bedding planes, and weathering can lower the effective strength well below intact rock values measured on small lab specimens.

Who Uses This

Geomechanics and geotechnical engineers evaluating rock and soil shear strength for slope, foundation, and excavation design.

Drilling and wellbore stability engineers relating confining stress and mud weight to breakout risk near the borehole wall.

Rock mechanics students and instructors working through Mohr circle and failure envelope exercises.

Mining engineers assessing pillar and slope stability from intact rock strength parameters.

Try It Above

Use the quick solve tab to convert cohesion and friction angle into unconfined compressive strength and the failure stress at a given confinement, move to the rock type comparison tab to see how strength varies across lithologies, or use the solve tab when you need the confining stress required to prevent failure at a known applied stress.

Related Calculators and References

Continue building your workflow with the True Stratigraphic Thickness Calculator. For background on the formulas used above, see Wikipedia: Mohr-Coulomb Theory.