Engineering
Terzaghi Effective Soil Stress & Rankine Earth Pressure (Ka & Kp) formulas
Explore a simplified hydrostatic effective-stress calculation and Rankine coefficients for level cohesionless backfill.
State the soil profile and wall assumptions first
For this teaching example total vertical stress is gamma*z, with one prescribed constant bulk unit weight over the whole depth. Real layered profiles require summing the weights of each layer, using suitable moist and saturated unit weights; a rising water table may change total stress as well as pore pressure.
Pore pressure is hydrostatic below a nonnegative water-table depth. Above it, positive pore pressure is set to zero; suction, capillary effects, seepage, surcharge and excess pore pressure are not modeled. Unit weights are in kN/m³ and depths in metres, so stress is in kPa.
The Ka and Kp formulas apply to the simple Rankine idealization: level, cohesionless backfill and a vertical smooth wall, with enough movement to mobilize the relevant limit state. They are not at-rest coefficients. Water pressure is separate from effective lateral soil stress; these coefficients alone do not calculate a complete wall load or establish design adequacy.
| Geotechnical Metric | Formula | z = 6 m, z_w = 2 m, gamma = 20, phi = 30 deg | Dry Soil (Water Table > 6 m) |
|---|---|---|---|
| Total Overburden Stress sigma_v | B3 * B2 | 20 * 6 = 120.00 kPa | 120.00 kPa |
| Pore Water Pressure u | MAX(0, (B2 - B4) * B5) | 4 * 9.81 = 39.24 kPa | 0.00 kPa |
| Effective Stress sigma'_v | (B3 * B2) - MAX(0, (B2-B4)*B5) | 120 - 39.24 = 80.76 kPa | 120.00 kPa |
| Rankine Active / Passive Ka, Kp | (1 -/+ SIN(phi)) / (1 +/- SIN(phi)) | Ka = 0.3333 | Kp = 3.0000 | Ka = 0.3333 | Kp = 3.0000 |
Terzaghi Effective Vertical Stress sigma'_v (kPa)
This simplified soil configuration requires bulk unit weight above water unit weight, both positive, and nonnegative depths. Change layer modeling rather than forcing unusual profiles into this single-weight example.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<0,B4<0,B3<=0,B5<=0,B3<=B5),"Invalid inputs",ROUND((B3 * B2) - MAX(0, (B2 - B4) * B5), 2)))Rankine active coefficient Ka
Converts friction angle phi (B6) from degrees to radians inside SIN.
=IF(NOT(ISNUMBER(B6)),"Invalid inputs",IF(OR(B6<0,B6>=90),"Invalid inputs",ROUND((1 - SIN(RADIANS(B6))) / (1 + SIN(RADIANS(B6))), 4)))Rankine passive coefficient Kp
Compute from the unrounded trigonometric expression. Do not take the reciprocal of the four-decimal displayed Ka.
=IF(NOT(ISNUMBER(B6)),"Invalid inputs",IF(OR(B6<0,B6>=90),"Invalid inputs",ROUND((1 + SIN(RADIANS(B6))) / (1 - SIN(RADIANS(B6))), 4)))Hydrostatic pore pressure
Only depth below the specified water table contributes in this model.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<0,B4<0,B3<=0,B5<=0,B3<=B5),"Invalid inputs",MAX(0,(B2-B4)*B5)))Total vertical stress
The prescribed uniform unit weight times depth.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<0,B4<0,B3<=0,B5<=0,B3<=B5),"Invalid inputs",B3*B2))Try the example
Use the original values or change them to test your understanding.