Silo Vertical Pressure (Janssen)
Calculate the vertical pressure of stored product at a silo cross-section by the Janssen equation, p_v = (γ·D)/(4·μ·K)·(1 − e^(−4·μ·K·z/D)), from the product unit weight γ (N/m³), silo diameter D (m), product-wall friction coefficient μ, lateral pressure ratio K and depth z (m). The Janssen equation (1895) is the basis of silo structural design and reveals a counterintuitive, fundamental fact: pressure at the bottom of a silo does NOT grow indefinitely with product height like a liquid (p = γ·h). Instead it tends to a LIMIT (asymptotic) value. This is because granular product (grain, cement, ore) transmits part of its weight LATERALLY to the walls, and product-wall friction 'holds' that load, relieving the bottom. The deeper it goes, the larger the fraction of weight carried by wall friction, until all added weight is absorbed by the walls and bottom pressure stabilizes. So silos can be very tall without bottom pressures proportional to height. This arching and wall-friction effect is the heart of silo design. Enter the unit weight, diameter, friction coefficient, lateral pressure ratio and depth.
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Silo vertical pressure (Janssen)
The Janssen equation (1895), p_v = (γ·D)/(4·μ·K)·(1 − e^(−4·μ·K·z/D)), gives the vertical pressure of the stored material in a silo at a depth z, and reveals one of the most counterintuitive and fundamental facts in the engineering of storage structures: the pressure at the bottom of a silo does not grow without bound with the height of the stored material, the way it would in a liquid (p = γ·h). Instead, it tends toward a limiting value (an asymptote). The reason is that granular material — grain, cement, ore — transfers part of its own weight laterally to the walls, and the friction between material and wall holds that load up, relieving the bottom. The deeper you go, the larger the share of the weight carried by wall friction, until practically all the additional weight gets absorbed by the walls and the pressure at the bottom levels off. This is why silos can be built very tall and slender without the bottom and the foundation seeing pressures proportional to the height — something impossible with liquids. That combination of arching and wall friction, captured in the formula by γ (unit weight), D (diameter), μ (material-wall friction), K (lateral pressure ratio) and z (depth), is the heart of structural silo design. Enter the unit weight, the diameter, the friction coefficient, the lateral pressure ratio and the depth.
Related Tools
Silo Asymptotic Pressure
Calculate the asymptotic (saturation) vertical pressure of a deep silo, p_∞ = (γ·D) ÷ (4·μ·K), from the product unit weight γ, diameter D, product-wall friction coefficient μ and lateral pressure ratio K. This is the LIMIT value the Janssen vertical pressure tends to at great depth — the maximum bottom pressure a silo can reach, however tall the stored product. It is Janssen's most striking result: while in a liquid pressure would grow without limit with height (p = γ·h), in granular product WALL FRICTION absorbs all added weight beyond a certain depth, making bottom pressure SATURATE. So a 30 m silo of grain may have a bottom pressure equal to only a few metres of product column. This asymptotic pressure is fundamental in design: it sets the maximum bottom and wall load the structure must bear, regardless of height, and explains why silos can be built slender and tall with relatively modest foundations. Note it is proportional to diameter and inversely proportional to friction — wide, smooth-walled silos generate higher pressures. Enter the unit weight, diameter, friction coefficient and lateral pressure ratio.
Silo Horizontal Pressure (Janssen)
Calculate the horizontal pressure the stored product exerts on a silo wall by the Janssen equation, p_h = (γ·D)/(4·μ)·(1 − e^(−4·μ·K·z/D)), from the unit weight γ, diameter D, product-wall friction coefficient μ, lateral pressure ratio K and depth z. Horizontal pressure is the outward thrust grains apply against the silo walls — the load that sizes the wall for hoop tension (in cylindrical silos, the wall acts as a ring under internal pressure). It relates to vertical pressure by the lateral pressure ratio K (p_h = K·p_v), typically 0.3-0.6 for granular products and depending on the product's internal friction angle. Like vertical pressure, horizontal pressure tends to an asymptotic value with depth, by the same wall-friction effect of Janssen theory. Horizontal pressure is decisive for the thickness and reinforcement of concrete silo walls and the plating of steel silos, and rises significantly during DISCHARGE (dynamic overpressure), which codes handle with amplification factors. Enter the unit weight, diameter, friction coefficient, lateral pressure ratio and depth.
Janssen Characteristic Depth
Calculate a silo's Janssen characteristic depth, z₀ = D ÷ (4·μ·K), from the diameter D, product-wall friction coefficient μ and lateral pressure ratio K. The characteristic depth governs how fast silo pressures approach their asymptotic (limit) value: in the Janssen equation, it is the depth at which pressure reaches about 63% (1 − 1/e) of the maximum. Depths of a few times z₀ practically reach the limit pressure. Conceptually, z₀ shows how 'deep' the silo must be for wall friction to dominate: silos with small z₀ (small diameter, high friction) quickly reach the constant-pressure regime and behave as slender (tall) silos; silos with large z₀ (large diameter) saturate slowly and behave more like squat silos, where much of the weight still reaches the bottom. The characteristic depth is thus a natural measure of the vertical 'scale' of the silo's pressure behavior, useful to classify it and understand its load profile. Enter the diameter, friction coefficient and lateral pressure ratio.
Cylindrical Silo Capacity
Calculate the storage capacity (mass) of a silo's cylindrical part, M = ρ · (π·D²/4) · H, from the product bulk density ρ (kg/m³), the silo inner diameter D (m) and the cylindrical body height H (m). The calculation combines the cylinder volume (section area times height) with the product's bulk density — the product mass per unit apparent volume, which includes the voids between particles and differs from the solid particle density. Bulk density varies with product and state: soybeans ~720 kg/m³, corn ~720, wheat ~770, cement ~1500, and it also changes with moisture and compaction. Capacity is a silo's most basic commercial and operational parameter: it sets how much product it stores, and thus the logistics of receiving, dispatch and stock turnover. This computes the cylindrical part; total capacity also includes the lower hopper volume and, in grain silos, the upper product cone above the transition line (formed by the angle of repose during filling). Correctly estimating capacity is essential in designing storage units, cooperatives and grain terminals. Enter the bulk density, diameter and cylindrical body height.
Silo Discharge Overpressure
Calculate the DISCHARGE horizontal pressure in a silo, p_d = C_d · p_h, from the discharge overpressure coefficient C_d and the static horizontal pressure p_h (from Janssen for the full silo at rest). One of the most important phenomena — historically responsible for many silo failures — is that wall pressures during DISCHARGE are SIGNIFICANTLY HIGHER than static pressures with the silo merely full. When the product starts flowing toward the outlet, flow zones and dynamic arches form, and stress redistribution generates pressure peaks (overpressures) on the wall, especially at the transition from the cylindrical body to the hopper. The overpressure coefficient C_d (typically 1.3-2.0 or more, per the code, flow type — mass or funnel — and geometry) amplifies the static pressure to cover these dynamic peaks. Silo design codes (such as EN 1991-4 / Eurocode and ANSI) prescribe these factors precisely because designing a silo only for static loads, ignoring discharge overpressure, is a classic cause of structural collapse. Enter the overpressure coefficient and the static horizontal pressure.
Silo Slenderness Ratio
Calculate a silo's slenderness ratio, λ = H ÷ D, from the stored product height H (m) and the silo diameter D (m). This simple ratio is the fundamental criterion that CLASSIFIES silos and determines how their pressures behave and how codes treat them. Silos with HIGH slenderness (typically H/D ≥ 1.5-2, called slender or 'tall') are dominated by the Janssen wall-friction effect: pressure saturates quickly, most weight transfers to the walls, and the bottom receives a much lower pressure than the product column would suggest. Silos with LOW ratio (H/D < 1.0-1.5, called squat) behave intermediately between Janssen and a tank: wall friction has less extent to act, and a larger fraction of weight reaches the bottom. This distinction changes the applicable pressure formulas, the discharge overpressure factors and even the expected flow type. The slenderness ratio is thus the first decision in silo analysis — it sets which load model to use and influences the whole structural concept, from foundation to walls. Enter the product height and the silo diameter.
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