Operations, Logistics & Engineering
Schmertmann Strain Influence Factor (I_z) Formula: Sand Footing Immediate & Creep Settlement
Calculate shallow foundation settlement on granular sand soils using John H. Schmertmann’s (1970/1978) strain-influence method S = C_1 * C_2 * delta_q * SUM(I_z * dz / E_s) with embedment relief C_1 and creep factor C_2.
Why vertical strain peaks at z = B/2 beneath a sand footing—not directly at the base z = 0!
Classical Boussinesq elastic theory predicts maximum vertical stress directly beneath a footing base (z = 0), yet John H. Schmertmann (1970; Schmertmann, Hartman & Brown, 1978, ASCE Journal of the Geotechnical Engineering Division) proved via finite-element analysis and sand model tests that lateral confinement at the footing interface suppresses strain at z = 0—causing vertical strain influence I_z to peak at depth z = B / 2 (for square/circular footings, reaching I_zp = 0.5 + 0.1 * SQRT(delta_q / sigma'_vp) and tapering linearly to 0 at depth 2B)!
Total footing settlement (mm) across sublayers of thickness dz_i (m), Cone Penetration Young’s modulus E_s,i (kPa), and mid-layer strain influence factor I_z,i is S_mm = 1000 * C_1 * C_2 * delta_q * SUM(I_z,i * dz_i / E_s,i), where Embedment Relief Factor C_1 = MAX(0.5, 1 - 0.5 * (q_0 / delta_q)) accounts for overburden stress q_0 = gamma * D_f at foundation level (delta_q = q_gross - q_0 is net bearing pressure!) and Time-Creep Factor C_2 = 1 + 0.2 * LOG10(t_years / 0.1) captures secondary sand creep over t >= 0.1 years!
| Schmertmann Factor (q_gross=220 kPa, q_0=20 kPa -> delta_q=200 kPa) | Schmertmann (1970/1978) Equation | Computed Value |
|---|---|---|
| Embedment Relief Factor C_1 | MAX(0.5, 1 - 0.5 * (20 / 200)) | 0.9500 (5% Overburden Relief) |
| 5-Year Time Creep Factor C_2 (t = 5 yr) | 1 + 0.2 * LOG10(5 / 0.1) | 1.3398 (+34.0% 5-Year Creep over 0.1 yr) |
| Peak Strain Influence I_zp (sigma'_vp = 38 kPa) | 0.5 + 0.1 * SQRT(200 / 38) | 0.7294 (At Depth z = B/2) |
Schmertmann Total Footing Settlement S (mm) = 1000 * C_1 * C_2 * delta_q * SUM(I_z * dz / E_s)
Evaluates C_1, C_2, and multi-layer strain summation across 3 sand sublayers (dz in D2:D4, I_z in E2:E4, E_s in F2:F4) for net pressure delta_q = B2 - B3.
=IF(OR(B2 <= B3, B3 < 0, B4 < 0.1, MIN(D2:D4) <= 0, MIN(F2:F4) <= 0), "Invalid Schmertmann Settlement Inputs (Requires q_gross > q0 & t >= 0.1 yr)", ROUND(1000 * MAX(0.5, 1 - 0.5 * (B3 / (B2 - B3))) * (1 + 0.2 * LOG10(B4 / 0.1)) * (B2 - B3) * SUMPRODUCT(E2:E4, D2:D4 / F2:F4), 2))Schmertmann (1978) Peak Strain Influence Factor I_zp = 0.5 + 0.1 * SQRT((q_gross - q_0) / sigma'_vp)
Computes the peak triangular strain influence value I_zp using effective overburden stress sigma'_vp (B5, kPa) at peak-strain depth.
=IF(OR(B2 <= B3, B5 <= 0), "Invalid Pressures", ROUND(0.5 + 0.1 * SQRT((B2 - B3) / B5), 4))Try the example
Use the original values or change them to test your understanding.