Basal Stress Equations for Granular Debris Masses on Smooth or Discretized Slopes

Knowledge of basal stresses is essential for analyzing slope stability and modeling the dynamics and erosive potential of debris flows and avalanches. Here we derive and test new algebraic formulas for calculating the shear stress τ and normal stress σ at the base of variable‐thickness granular debris masses in states of static or dynamic equilibrium on slopes. The formulas include a lateral pressure coefficient κ , but use of a fixed value κ  = 0.7 yields predictions of σ that on average err by less than 3% and of τ that on average err by less than 13% in matching basal stresses measured in six large‐scale experiments involving wet debris masses with varying geometries and compositions. Much larger prediction errors result from use of infinite‐slope or shallow‐debris approximations. Specialized versions of the new formulas apply if basal topography is discretized and represented by a “staircase” function in a digital elevation model. Use of these formulas to assess static limiting equilibrium conditions shows that the apparent basal Coulomb friction angle ϕ tread of debris that engages friction acting on the horizontal surfaces (or “treads”) of a staircase sloping at an angle θ is generally described by tan ϕ tread  =  tan ( ϕ  −  θ )+ κ  tan  θ , where ϕ is the true basal friction angle of the same debris in contact with a uniformly sloping bed. Differences between the values of ϕ and ϕ tread can greatly influence the results of numerical simulations that use unsmoothed digital elevation model topography to calculate the stability or dynamics of debris masses on slopes.