Calculate spring force using the Almen-Laszlo equations
Dimensions
Material & Load
Calculation Results
Estimated Spring Force (P)
0.00 N
Dish Height (h0)0.00 mm
Diameter Ratio (δ)0.00
Spring Rate (k)0.00 N/mm
Force (kgf)0.00 kgf
Warning: Deflection exceeds dish height (washer is compressed past flat).
Understanding Belleville Washer Load Calculations
A Belleville washer, also known as a conical spring washer or disc spring, is a type of spring shaped like a frustum of a cone. Unlike standard coil springs, Belleville washers provide very high loads in small spaces. This calculator utilizes the industry-standard Almen-Laszlo equations to predict the load-deflection characteristics.
Key Geometry and Terms
Outside Diameter (Do): The maximum span of the washer.
Inside Diameter (Di): The diameter of the center hole.
Thickness (t): The material thickness of the disc.
Dish Height (h0): Calculated as the Total Height (H) minus the Thickness (t). This represents the available travel until the washer is flattened.
Deflection (f): The distance the washer is compressed from its free state.
The Almen-Laszlo Formula
The load (P) exerted by a disc spring at a specific deflection (f) is calculated using the following relationship:
Where M is a constant derived from the ratio of the diameters (δ = Do / Di).
Stacking Belleville Washers
One of the primary advantages of these washers is their ability to be stacked to modify spring constants:
Parallel Stacking: Washers are stacked in the same direction. This increases the total load capacity (Load x Number of Washers) while keeping the deflection the same as a single washer.
Series Stacking: Washers are stacked back-to-back (opposing directions). This increases the total deflection (Deflection x Number of Washers) while the load remains the same as a single washer.
Parallel-Series: A combination used to achieve specific high-load, high-travel requirements.
Example Calculation
Consider a steel washer with an OD of 50mm, ID of 25mm, Thickness of 2mm, and a Total Height of 3.5mm. If we compress it by 0.75mm:
Dish Height: 3.5mm – 2mm = 1.5mm
Diameter Ratio: 50 / 25 = 2.0
Calculated Force: Approximately 4,500 Newtons (varies based on precise material modulus).
This illustrates how a small component can manage significant industrial loads, making them ideal for thermal expansion compensation or high-vibration bolting applications.
function calculateBelleville() {
var od = parseFloat(document.getElementById("wash_od").value);
var id = parseFloat(document.getElementById("wash_id").value);
var t = parseFloat(document.getElementById("wash_t").value);
var H = parseFloat(document.getElementById("wash_H").value);
var E = parseFloat(document.getElementById("wash_E").value);
var v = parseFloat(document.getElementById("wash_v").value);
var f = parseFloat(document.getElementById("wash_f").value);
// Validation
if (isNaN(od) || isNaN(id) || isNaN(t) || isNaN(H) || isNaN(E) || isNaN(v) || isNaN(f)) {
alert("Please enter valid numeric values for all fields.");
return;
}
if (id >= od) {
alert("Outside diameter must be larger than inside diameter.");
return;
}
if (t >= H) {
alert("Total height must be greater than thickness.");
return;
}
var h0 = H – t;
var delta = od / id;
// Almen-Laszlo Constant M
// M = (6 / (pi * ln(delta))) * ((delta – 1) / delta)^2
var lnDelta = Math.log(delta);
var M = (6 / (Math.PI * lnDelta)) * Math.pow((delta – 1) / delta, 2);
// Load Calculation (P)
// Formula: P = (4E/(1-v^2)) * (t^4/(M*D_o^2)) * (f/t) * [ (h0/t – f/t)(h0/t – f/2t) + 1 ]
var term1 = (4 * E) / (1 – Math.pow(v, 2));
var term2 = Math.pow(t, 4) / (M * Math.pow(od, 2));
var term3 = f / t;
var term4 = ( (h0 / t) – (f / t) ) * ( (h0 / t) – (f / (2 * t)) ) + 1;
var load = term1 * term2 * term3 * term4;
// Results Processing
var springRate = f > 0 ? load / f : 0;
var kgf = load / 9.80665;
// Display
document.getElementById("wash_force_out").innerHTML = load.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 2}) + ' N';
document.getElementById("wash_ho_out").innerText = h0.toFixed(2) + " mm";
document.getElementById("wash_delta_out").innerText = delta.toFixed(2);
document.getElementById("wash_rate_out").innerText = springRate.toFixed(2) + " N/mm";
document.getElementById("wash_kgf_out").innerText = kgf.toFixed(2) + " kgf";
// Warning
var warning = document.getElementById("wash_warning");
if (f > h0) {
warning.style.display = "block";
} else {
warning.style.display = "none";
}
}
// Initial Run
window.onload = function() {
calculateBelleville();
};