Janssen Lateral Pressure

\[P_h=\frac{\rho gD}{4\mu}\left(1-e^{-\frac{4\mu Kz}{D}}\right)\]

Variables

Phlateral pressure on silo wall (Pa)

rho[ρ]bulk density of stored material (kg/m³)

ggravitational acceleration (m/s²)

Dsilo diameter (m)

mu[μ]wall friction coefficient (-)

Klateral pressure coefficient (-)

zdepth below material surface (m)

Description

What is this formula?

The Janssen equation estimates the lateral pressure exerted by granular materials stored inside vertical silos.

Unlike liquids, granular materials transfer part of their weight to the silo walls through friction, causing lateral pressure to increase with depth and eventually approach a limiting value.

When to use it

Use this equation when designing grain silos, feed storage structures, agricultural storage bins, and bulk material handling facilities.

Example

Given:

rho = 750 kg/m³

g = 9.81 m/s²

D = 6 m

mu = 0.40

K = 0.45

z = 10 m

Formula:

Ph = (rho·g·D·(1 − exp(-(4·mu·K·z)/D))) / (4·mu)

Substitution:

Ph = (750 × 9.81 × 6 × (1 − exp(-(4 × 0.40 × 0.45 × 10)/6))) / (4 × 0.40)

Ph = 19.28 kPa

Result:

The estimated lateral wall pressure is approximately 18.5 kPa.

Applications

- Grain silo design

- Agricultural storage engineering

- Feed storage structures

- Bulk solids handling

- Structural load calculations

Note

The Janssen equation is an engineering model derived from force equilibrium and experimental observations of granular materials.

The coefficients K and mu depend strongly on the stored material and wall properties.

Different design standards may use modified Janssen formulations or additional safety factors.

The equation is most suitable for static storage conditions and may underestimate loads produced during filling, discharge, eccentric flow, or arching phenomena.