Engineering Theory Appendix

Equations, transformations, assumptions, limitations, and references used by the app.

Mohr-Coulomb

σ₁ = σ₃(1 + sinφ)/(1 - sinφ) + 2c cosφ/(1 - sinφ)

JLab uses the direct shear form and the triaxial principal-stress form. The tensile cutoff is plotted at negative tensile strength when tensile strength is positive.

Generalized Hoek-Brown Constants

m_b = m_i exp((GSI - 100)/(28 - 14D))

s = exp((GSI - 100)/(9 - 3D))

a = 0.5 + (exp(-GSI/15) - exp(-20/3))/6

The Hoek-Brown tensile limit used by JLab is based on s sigmaCi / mb when mb is positive. The app also computes rock-mass compressive strength and closed-form equivalent Mohr-Coulomb parameters over the active range.

Rock-Mass Modulus

Generalized Hoek-Diederichs uses intact modulus, GSI, and D. The simplified Hoek-Diederichs method estimates deformation modulus from GSI and D. The HCTC 2002 method uses sigmaCi, GSI, and D. The app reports modulus in the active stress unit.

Normal/Shear Conversion

For Hoek-Brown and Power Curve/Barton-Bandis principal plots, JLab derives tangent Mohr circles from the envelope slope using the exact Balmer derivative transform. Steep valid Hoek-Brown derivatives are not clipped.

Power Curve

The app implements tau = a(sigma_n + d)^b + c without a waviness angle term. Values are ignored where sigma_n + d is not positive. The instantaneous slope is a b (sigma_n + d)^(b - 1).

Barton-Bandis

Barton-Bandis is evaluated only while 0 < sigma_n <= JCS and the apparent friction angle is between 0 and 70 degrees. Out-of-domain samples are unavailable rather than extrapolated. The model is most appropriate for discontinuity shear strength rather than intact rock-mass failure.

Calibration

Calibration minimizes the mean of the selected L2, L1, or Huber loss applied to absolute, floored-relative, or log-ratio residuals. Principal mode compares predicted sigma1 with entered sigma1; shear mode compares predicted tau with entered tau. Raw RMSE remains sqrt(SSE / n) in stress units regardless of the selected objective. Hoek-Brown shear and Power Curve principal calibration use sampled/interpolated curves and Auto therefore selects pattern search.

Calculation Kernel Contract

The WebAssembly calculation kernel returns NaN for non-finite or invalid-domain inputs. Kernel initialization fails with a contextual error if a required calculation export is missing or not callable.

Limitations

Envelope calculations are deterministic and do not model uncertainty.
Water pressure, effective stress corrections, anisotropy, scale corrections, and dilation are not modelled directly.
Equivalent Mohr-Coulomb values depend on the computed fit range and sample count. Power Curve and Barton-Bandis minimize shear RMSE over positive normal stress.
Calibration is bounded and deterministic, but a numerical optimum does not establish that one envelope represents pooled tests with different failure mechanisms.
Relative and robust objectives encode engineering choices; they do not infer measurement uncertainty or provide confidence intervals.

References

The following publications and numerical-method references support the equations and methods described in this appendix.