Use this calculator to determine the approximate force exerted by a single Belleville washer (conical spring washer) at a given deflection. This calculation is based on standard engineering formulas for Belleville washers, considering their dimensions and material properties.
Calculation Result:
function calculateBellevilleForce() {
var outerDiameter = parseFloat(document.getElementById('outerDiameter').value);
var innerDiameter = parseFloat(document.getElementById('innerDiameter').value);
var thickness = parseFloat(document.getElementById('thickness').value);
var freeHeight = parseFloat(document.getElementById('freeHeight').value);
var deflection = parseFloat(document.getElementById('deflection').value);
var youngsModulus = parseFloat(document.getElementById('youngsModulus').value);
var poissonsRatio = parseFloat(document.getElementById('poissonsRatio').value);
var resultDiv = document.getElementById('resultOutput');
resultDiv.innerHTML = "; // Clear previous results
// Input validation
if (isNaN(outerDiameter) || isNaN(innerDiameter) || isNaN(thickness) || isNaN(freeHeight) || isNaN(deflection) || isNaN(youngsModulus) || isNaN(poissonsRatio)) {
resultDiv.innerHTML = 'Please enter valid numbers for all fields.';
return;
}
if (outerDiameter <= 0 || innerDiameter <= 0 || thickness <= 0 || freeHeight <= 0 || youngsModulus = outerDiameter) {
resultDiv.innerHTML = 'Inner Diameter must be less than Outer Diameter.';
return;
}
if (deflection freeHeight * 1.2) { // Allow slight over-flattening for calculation, but warn
resultDiv.innerHTML = 'Warning: Deflection exceeds free height. The washer may be flattened or inverted, and the formula\'s accuracy might decrease.';
}
if (poissonsRatio = 0.5) {
resultDiv.innerHTML = 'Poisson\'s Ratio typically ranges from 0 to less than 0.5.';
return;
}
// Calculate alpha (Do/Di ratio)
var alpha = outerDiameter / innerDiameter;
// Calculate C_w factor (geometric constant)
// This is a complex factor from DIN 2093 or similar standards
var term1 = (alpha – 1) / alpha;
var term2 = (Math.pow(alpha, 2) + 1) / (Math.pow(alpha, 2) – 1);
var term3 = 2 / Math.log(alpha);
var Cw = (1 / (Math.PI * Math.pow(alpha – 1, 2))) * (Math.pow(term1, 2) * (term2 – term3) + 1);
// Calculate Force (P) using the standard formula
// P = (E * t^3 * δ / ( (1 – ν^2) * C_w * D_o^2 )) * [ (h_o – δ) * (h_o – δ/2) + t^2 ]
var numerator = youngsModulus * Math.pow(thickness, 3) * deflection;
var denominator = (1 – Math.pow(poissonsRatio, 2)) * Cw * Math.pow(outerDiameter, 2);
var bracketTerm = (freeHeight – deflection) * (freeHeight – (deflection / 2)) + Math.pow(thickness, 2);
var calculatedForce = (numerator / denominator) * bracketTerm;
if (isNaN(calculatedForce) || !isFinite(calculatedForce)) {
resultDiv.innerHTML = 'Calculation resulted in an invalid number. Please check inputs, especially if deflection is too large or alpha is near 1.';
return;
}
resultDiv.innerHTML += 'Calculated Force (P): ' + calculatedForce.toFixed(2) + ' N';
}
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Example Usage:
Let's calculate the force for a common Belleville washer:
Outer Diameter (Do): 25 mm
Inner Diameter (Di): 12.5 mm
Thickness (t): 1.5 mm
Free Height (ho): 1.0 mm
Deflection (δ): 0.5 mm (50% of free height)
Young's Modulus (E): 206000 MPa (typical for spring steel)
Poisson's Ratio (ν): 0.3 (typical for steel)
Using these values in the calculator will yield a force of approximately 7245.15 N.
Understanding Belleville Washers
A Belleville washer, also known as a conical spring washer or disc spring, is a type of washer that has a slight conical shape. When subjected to an axial load, it flattens, providing a spring force. Unlike traditional coil springs, Belleville washers can offer very high loads in a small axial space, and their force-deflection characteristics can be non-linear, which is useful in many applications.
Key Dimensions and Properties:
Outer Diameter (Do): The total diameter of the washer.
Inner Diameter (Di): The diameter of the central hole.
Thickness (t): The material thickness of the washer.
Free Height (ho): The height of the cone when the washer is unloaded. This is the height from the flat base to the top edge of the cone.
Deflection (δ): The amount the washer is compressed from its free height.
Young's Modulus (E): A measure of the material's stiffness or resistance to elastic deformation. For spring steel, it's typically around 206,000 MPa (N/mm²).
Poisson's Ratio (ν): A measure of the material's tendency to expand or contract perpendicularly to the direction of loading. For steel, it's typically around 0.27 to 0.30.
How They Work:
When a Belleville washer is compressed, its conical shape flattens, and the material deforms elastically. This deformation generates a restoring force. The unique geometry allows for a non-linear spring rate, meaning the force required to compress it changes as it flattens. This can be advantageous for applications requiring a specific force profile.
Applications:
Belleville washers are widely used in various industries due to their compact design and high load capacity. Common applications include:
High-load applications: Where significant force is needed in a limited space, such as in heavy machinery, railway bogies, and clutch mechanisms.
Vibration damping: They can absorb shock and vibration, protecting components from fatigue.
Preloading: Used to maintain a constant preload in bolted joints, bearings, and other assemblies, compensating for thermal expansion or material creep.
Stacking: Multiple washers can be stacked in series (for increased deflection) or in parallel (for increased force) to achieve desired spring characteristics.
Limitations of the Calculator:
This calculator provides an approximate force based on a simplified engineering formula. It assumes:
The washer material behaves elastically (within its yield strength).
The washer is loaded uniformly and axially.
The formula is most accurate for deflections up to about 75-80% of the free height. Beyond this, or if the washer is flattened or inverted, the accuracy may decrease.
It does not account for friction, dynamic loading, or complex stacking arrangements.
For critical applications, always consult detailed engineering standards (like DIN 2093) and consider physical testing or finite element analysis (FEA).