Bulk Solids Mechanics, Eurocode EN 1991-4 Grain/Cement Silo Design & Janssen Theory
Janssen Silo Wall Friction & Pressure Saturation Formula
Calculate Janssen Characteristic Screening Depth z0 (m), Saturated Vertical Floor Pressure Pv(z) (kPa), Lateral Silo Wall Hoop Pressure Ph(z) (kPa), and Naive Hydrostatic Fluid Overprediction (kPa).
Why an hourglass ticks at constant rate—and why grain silo floor pressure saturates instead of growing likerho*g*z!
Designing a tall grain, polymer-pellet, or cement silo as a hydrostatic liquid tank (`P = rho * g * z`) wildly overdesigns the bottom hopper while missing wall-buckling vertical friction drag! H. A. Janssen (1895) proved that horizontal active/passive earth-pressure ratio `K = P_h / P_v` pushes granular solids outward against the cylinder wall, mobilizing upward Wall Shear Friction `tau_w = mu_w * P_h = mu_w * K * P_v` across Hydraulic Radius `R_h = Area / Perimeter = D / 4` (for a circular bin of diameter `D`)! Integrating the differential slice balance `dP_v/dz + (P_v / z_0) = rho * g` defines Characteristic Janssen Screening Depth `z_0 = R_h / (mu_w * K)` (m) and Exponential Saturation `P_v(z) = (rho * g * z_0 / 1,000) * (1 - EXP(-z / z_0))` kPa, with Lateral Hoop Pressure `P_h(z) = K * P_v(z)` kPa—transferring most of the column weight into vertical compression down the steel silo shell!
Verify on a 6.0 m Diameter Corrugated-Steel Wheat Silo Filled to 20.0 m—Silo_Diameter `D` = 6.00 m (`R_h = 1.50` m, B2), Grain_Fill_Depth `z` = 20.00 m (B3), Bulk_Density `rho_b` = 780.0 kg/m³ (B4), Grain-on-Wall_Friction `mu_w` = 0.40 (B5), and Janssen_Lateral_Ratio `K` = 0.48 (B6): Characteristic Depth `z_0` equals `7.81` m (E2), Vertical Bottom Pressure `P_v` saturates at `55.14` kPa (E3, nearly `3x` below naive `152.98` kPa hydrostatic fluid pressure in E5!), and Lateral Wall Hoop Pressure `P_h` equals `26.47` kPa (E4)!
| Eurocode EN 1991-4 Silo KPI (D=6.0 m, z=20 m, rho=780 kg/m³, mu_w=0.40, K=0.48) | Formula | Verified Output |
|---|---|---|
| Characteristic Screening Depth z0 | (D_m / 4) / (Wall_Friction_mu * K) | 7.81 m (12.50 m @ smooth steel 0.25) |
| Vertical Silo Pressure Pv(z) (kPa) | (rho*g*z0/1000) * (1 - EXP(-z / z0)) | 55.14 kPa (76.31 kPa @ smooth 0.25!) |
| Lateral Wall Hoop Pressure Ph (kPa) | K_Ratio * Pv_z_kPa | 26.47 kPa (36.63 kPa @ smooth 0.25!) |
| Naive Hydrostatic Fluid rho*g*z | (rho_kg_m3 * 9.80665 * z_m) / 1000 | 152.98 kPa (2.77x overprediction!) |
Janssen Differential-Slice Exponential Silo Pressure Formula
Validate `Silo_Diameter D > 0` m, `Depth z > 0` m, `Bulk_Density > 0` kg/m³, `Wall_Friction mu_w` in `(0.05, 1.0]`, and `Lateral_Ratio K` in `(0.15, 1.0]`. Note EN 1991-4 tests BOTH upper-bound `mu_w` (maximizing vertical shell buckling drag) AND lower-bound smooth `mu_w` (maximizing hopper & hoop burst pressure)!
=LET(dm_s, B2, dp_z, B3, rb_k, B4, mw_f, B5, kr_j, B6, IF(OR(dm_s <= 0, dp_z <= 0, rb_k <= 0, mw_f <= 0.05, mw_f > 1, kr_j <= 0.15, kr_j > 1), "Invalid Silo Input", ROUND((rb_k * 9.80665 * dp_z) / 1000, 2)))Polished Smooth Stainless Wall Lower-Friction Burst Check (Set Wall Friction mu_w = 0.25)
Polish the bin liner so Wall Friction `mu_w` B5 drops from `0.40` down to `0.25`: less weight hangs on the walls, lengthening Characteristic Depth E2 to `12.50` m and driving Floor Pressure E3 up `+38.4%` to `76.31` kPa and Lateral Hoop Burst Pressure E4 up to `36.63` kPa!
=IF(ISTEXT(E5), E5, LET(zc_m, (B2 / 4) / (B5 * B6), pv_k, ((B4 * 9.80665 * zc_m) / 1000) * (1 - EXP(-B3 / zc_m)), ROUND(B6 * pv_k, 2)))Try the example
Use the original values or change them to test your understanding.