Civil, Structural & Geotechnical Engineering
Meyer-Peter & Muller (1948) Bedload Sediment Transport Formula: Shields Excess & q_b
Calculate gravel- and coarse-sand river bedload transport intensity Phi = 8 * MAX(0, theta_prime - 0.047)^1.5 and volumetric unit bedload rate q_b (m^2/s) using Eugen Meyer-Peter and Robert Muller’s (1948, ETH Zurich) flume law.
Predict river gravel bedload transport above critical Shields stress theta_c = 0.047!
Derived from 16 years of large-scale flume experiments at ETH Zurich by Eugen Meyer-Peter and Robert Muller (1948, IAHR Stockholm; revised cleanly into dimensionless Einstein-Shields form by Chien, 1954 & Wong-Parker, 2006), the Meyer-Peter & Muller (MPM) Bedload Equation states that rolling/saltating coarse bedload initiates ONLY once grain-effective Shields shear stress theta_prime = (R_h * S_f) / ((s - 1) * d_50) exceeds Critical Incipient Threshold theta_c = 0.047 (where s = rho_s / rho_w ~= 2.65 is submerged sediment specific gravity, R_h is hydraulic radius, S_f is energy slope, and d_50 is median grain diameter)!
Above threshold (theta_prime > 0.047), Dimensionless Einstein Bedload Intensity follows the 3/2 power law Phi = 8 * ((theta_prime - 0.047)^1.5), converting into Volumetric Unit Bedload Transport Rate (solid volume per meter of channel width, excluding pore voids) via q_b (m^2/s) = Phi * SQRT((s - 1) * 9.80665 * (d_50^3))!
| Mountain Gravel River State (d_50 = 0.025 m = 25 mm, s = 2.65) | Effective Shields Stress theta_prime | Dimensionless MPM Phi | Volumetric Bedload q_b (m^2/s) |
|---|---|---|---|
| Baseflow Below Threshold (R_h = 0.8 m, S = 0.0020) | 0.0388 <= 0.047 | 0.0000 | 0.00000 m^2/s (Armored Bed) |
| Bankfull Flood (R_h = 1.65 m, S = 0.0025) | 0.1000 | 0.0976 | 0.00155 m^2/s (4.11 kg/m/s) |
| Extreme Peak Flood (R_h = 2.475 m, S = 0.0025) | 0.1500 | 0.2644 | 0.00420 m^2/s (11.13 kg/m/s) |
Meyer-Peter & Muller Dimensionless Bedload Intensity Phi = 8 * MAX(0, theta - 0.047)^1.5
Computes Shields stress theta = (R_h * S) / ((s - 1) * d_50) from Hydraulic Radius B2 (m), Slope B3, Median Grain Size d_50 B4 (m), and Specific Gravity s B5 (> 1), returning 0 when theta <= 0.047.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 1), "Requires R_h > 0, Slope S > 0, Grain Size d_50 > 0, and s > 1", ROUND(8 * (MAX(0, (B2 * B3) / ((B5 - 1) * B4) - 0.047) ^ 1.5), 4))Dry Mass Unit Bedload Transport Rate g_b (kg / (m · s)) = rho_s * q_b
Converts Phi into dry mass transport rate per unit width (kg/m/s) using sediment density 1000 * s * Phi * SQRT((s - 1) * 9.80665 * d_50^3).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 1), "Requires R_h > 0, Slope S > 0, Grain Size d_50 > 0, and s > 1", ROUND(1000 * B5 * (8 * (MAX(0, (B2 * B3) / ((B5 - 1) * B4) - 0.047) ^ 1.5)) * SQRT((B5 - 1) * 9.80665 * (B4 ^ 3)), 3))Try the example
Use the original values or change them to test your understanding.