How Do You Calculate Rate/Loads for Disc Spring Stacks?
Calculating rate and loads for disc spring stacks requires a different approach than for helical springs. It's about combining individual disc spring[^1] properties.
To calculate rate and loads for disc spring[^1] stacks, you must first determine the load and deflection characteristics[^2] of a single disc spring[^1] using specific formulas that account for its outer diameter, inner diameter, thickness, and coned height[^3]. Then, for a stack, you sum individual deflections when springs are stacked in series to increase overall deflection, or you sum individual loads when springs are stacked in parallel to increase the total load capacity. Combinations of series and parallel stacking allow for highly customizable load-deflection curves[^4]](https://www.centuryspring.com/resources/what-is-spring-deflection?srsltid=AfmBOor31g5-LtWvyaPhlsfj00HKki5CMFhJhBL_GaNDNDZN-vZw6nw3)[^5]s.
I've seen the power of disc spring[^1]s to handle high loads in small spaces. But getting the calculations right for a stack is where the real engineering comes in.
What is a Disc Spring?
A disc spring[^1], also known as a Belleville washer[^6], is a conical shaped washer that acts like a spring.
A disc spring[^1], also known as a Belleville washer[^6], is a conical, ring-shaped spring designed to support high loads with small deflections in compact spaces. Unlike helical springs, they operate by flattening out under axial load. They can be stacked in various configurations (series, parallel, or combinations) to achieve specific load-deflection characteristics[^2], offering engineers a versatile solution for precise force and deflection requirements in constrained environments.
I consider disc spring[^1]s to be precision components. Their unique shape lets them handle loads that a helical spring of the same size simply couldn't touch.
Disc Spring Geometry and Material
The specific shape and material of a disc spring[^1] are crucial to its performance.
| Parameter | Description | Influence on Performance | Role in Calculation |
|---|---|---|---|
Outer Diameter (D_o) |
The largest diameter of the disc spring[^1]. | Larger D_o generally leads to higher load capacity[^7]. |
Primary dimension in load and deflection formulas. |
Inner Diameter (D_i) |
The smallest diameter of the hole in the center of the disc spring[^1]. | Smaller D_i generally leads to higher load capacity[^7]. |
Primary dimension in load and deflection formulas. |
Material Thickness (t) |
The thickness of the spring material. | Thicker t significantly increases load capacity[^7] and stiffness. |
Crucial exponential factor in load calculations[^8] (t^4). |
Coned Height (h) |
The height of the cone (free height minus thickness). | Dictates the maximum deflection and influences non-linearity. | Directly used in deflection and non-linearity calculations. |
| Material | Typically spring steel[^9] (e.g., 50CrV4, 301 Stainless, Inconel). | Affects Modulus of Elasticity[^10] (E) and allowable stress. | E is a key factor in all formulas. |
Modulus of Elasticity (E) |
A measure of the material's stiffness or resistance to elastic deformation. | Higher E means higher load capacity[^7] and stiffness. |
Directly used in load and deflection formulas. |
Poisson's Ratio[^11] (μ) |
A material property relating transverse strain to axial strain. | A minor factor in some refined calculations. | Typically assumed (e.g., 0.3) for common spring steels. |
Disc springs, unlike helical springs, get their unique properties from their specific conical geometry and the material they are made from. Understanding these parameters is the first step in any calculation.
- Outer Diameter (
D_o) and Inner Diameter (D_i): These define the overall size and ring shape of the disc spring[^1]. The ratio ofD_otoD_isignificantly influences the spring's load-deflection curve[^5] and stress distribution. - Material Thickness (
t): This is extremely critical. Even small changes in thickness have a large impact on load capacity[^7]. The load capacity[^7] of a disc spring[^1] is proportional to the thickness raised to the power of four (t^4), meaning a slight increase in thickness makes the spring much, much stiffer. - Coned Height (
h): This is the height of the cone, measured from the flat bottom surface to the top edge, before any load is applied. It is usually defined as the free height (L_o) minus the material thickness[^12] (t). The coned height[^3] determines the maximum available deflection of a single disc spring[^1] and contributes to the non-linear load-deflection curve[^13]e](https://www.centuryspring.com/resources/what-is-spring-deflection?srsltid=AfmBOor31g5-LtWvyaPhlsfj00HKki5CMFhJhBL_GaNDNDZN-vZw6nw3)[^5] characteristic of disc spring[^1]s. - Material: Disc springs are commonly made from high-strength spring steels[^14]](https://en.wikipedia.org/wiki/Spring_steel)[^9]s like 50CrV4 (SAE 6150), 301 Stainless Steel, or Inconel for high-temperature applications. The material's Modulus of Elasticity[^10] (
E) is a key mechanical property that defines its stiffness and is a direct input into the load and deflection formulas. Poisson's Ratio[^11] (μ) is another material constant, typically around 0.3 for steel, and is also used in the formulas.
The precise combination of these geometric dimensions and material properties allows disc spring[^1]s to achieve very high load capacities within minimal axial space. I always start by gathering these exact specifications from the spring's drawing or datasheet.
Load-Deflection Curve of a Single Disc Spring
A single disc spring[^1] has a unique, often non-linear, load-deflection curve[^5].
| Deflection Point | Description | Characteristics of Curve | Application Implications |
|---|---|---|---|
| Initial Deflection | From fully open to approximately 75% of its coned height[^3]. | Load increases relatively linearly, but less steeply than the middle. | Good for initial preloading or low-force applications. |
| Mid-Deflection | Around 75% to 100% of its coned height[^3]. | Load increase flattens out or even decreases slightly near 100% flat (for h/t > 1.4). | Can provide constant force over a range, or even snap action. |
| Flattened Deflection | When the spring is almost completely flat. | Load increases very steeply as the spring approaches flat. | Ideal for high-load applications where small deflection changes lead to large force changes. |
| Non-linearity | The curve is not a straight line, especially for h/t ratios greater than 0.4. |
Allows for constant force over a range or very stiff behavior at ends. | Versatile for custom force requirements. |
| Load (P) vs. Deflection (δ) | Load P is a function of deflection δ, h, t, D_o, D_i, E, and μ. |
Defined by a complex formula involving these geometric and material parameters. | Requires precise calculation for specific deflection points. |
The load-deflection curve[^5] for a single disc spring[^1] is quite distinctive, especially compared to the linear behavior of many helical springs. It's often non-linear, meaning the force required to compress it by a certain amount isn't constant throughout its deflection range.
The formula for the load P for a given deflection δ of a single disc spring[^1] is complex and involves several constants and parameters:
P = (4 * E / (1 - μ^2)) * (t^4 / (K * D_o^2)) * [δ * (h - δ) * (h/t) + t^2]
Where:
E= Modulus of Elasticity[^10]μ= Poisson's Ratio[^11]t= Material thicknessD_o= Outer diameterh= Coned heightδ= DeflectionK= A constant that depends onD_o/D_iratio.
This formula shows that:
- Initial Deflection (up to about 75% of
h): The load increases as deflection increases, but often at a somewhat moderate rate. - Near Flat Deflection (around
δ = h): As the disc spring[^1] approaches a completely flat position, the load can increase very steeply. For certainh/tratios (specifically,h/t > 1.4), the load can even decrease slightly before rapidly increasing as it flattens. This "flattening" or "snap-through" behavior can be useful for applications requiring a relatively constant force over a small range or even a "snap" action.
Understanding this curve is crucial. It allows engineers to predict the exact force a single disc spring[^1] will provide at any point of its deflection. This knowledge is then applied to design stacks that achieve specific overall load-deflection characteristics[^2]. I use specialized software to plot these curves accurately, as manual calculation for every point can be tedious.
How to Calculate for Series Stacks?
Stacking disc spring[^1]s in series increases the total deflection of the stack while maintaining the load of a single spring.
To calculate for disc spring[^1] stacks in series, where each spring is stacked in the same direction, you sum the deflections of the individual disc spring[^1]s to find the total deflection of the stack. The load capacity[^7] of the series stack[^15] remains approximately the same as that of a single spring. For example, if 'n' springs with individual deflection 'δ_single' are stacked in series, the total stack deflection 'δ_stack' will be 'n × δ_single' for a given load 'P_single'.
I often use series stack[^15]s when a compact design needs more travel than a single disc spring[^1] can provide. It's an efficient way to get more deflection without increasing the load requirement.
What is a Series Stack?
A series stack[^15] is formed by placing disc spring[^1]s in the same direction, one on top of the other.
| Characteristic | Description | Primary Effect on Stack Performance | Analogy |
|---|---|---|---|
| Springs in Same Direction | Each disc spring[^1] is oriented identically, conical side facing the same way. | Allows each spring to deflect independently under load. | Stacking multiple soft helical springs end-to-end. |
| Increased Deflection | The total deflection of the stack is the sum of individual spring deflections. | Achieves greater travel for the overall stack. | Like adding extra segments to a flexible ruler. |
| Same Load Capacity | The load capacity[^7] of the stack remains essentially the same as a single spring. | The force required to compress the stack is no more than for one spring. | A chain is only as strong as its weakest link (load-wise). |
| Non-Linear Summation | The individual non-linear curves sum up to a larger non-linear curve for the stack. | Preserves the desired non-linear behavior over a greater deflection range. | The combined flexibility extends over a longer range. |
| Stack Height | The overall free height of the stack increases with the number of springs. | Requires more axial space for the spring assembly. | A taller stack for greater movement. |
A series stack is created by placing multiple disc spring[^1]s on top of each other, all oriented in the same direction (e.g., all cones pointing up). When an axial load is applied to this stack, each individual disc spring[^1] deflects independently.
The primary effect of a series stack[^15] is to increase the total deflection of the spring system. If you have 'n' disc spring[^1]s, and each spring deflects by δ_single under a certain load P_single, then the total deflection of the stack (δ_stack) will be n times δ_single.
δ_stack = n × δ_single (for a given load P_single)
Crucially, the load capacity[^7] of the series stack[^15] remains approximately the same as the load capacity[^7] of a single disc spring[^1]. This is because the load is effectively transferred through each spring in sequence; each spring bears the full load. So, if a single disc spring[^1] can handle a maximum load of 1000 N, a stack of five identical springs in series will still only handle 1000 N, but it will deflect five times as much.
The overall load-deflection curve[^5] of a series stack[^15] will reflect the non-linear curve of a single spring, but stretched out over a greater deflection range. This allows designers to achieve specific load-deflection characteristics[^2] (like a relatively constant force over a range) across a larger travel distance. I use series stack[^15]ing when my primary goal is to increase the range of motion of the spring system while keeping the applied force within limits.
Calculating Deflection and Load in Series Stacks
The calculations for series stack[^15]s are straightforward: sum deflections, keep load constant.
| Calculation Aspect | Single Disc Spring | Series Stack (n springs) | Implications |
|---|---|---|---|
Deflection (δ) |
δ_single (from single spring formula for load P). |
δ_stack = n × δ_single (for the same load P). |
Enables greater travel in compact designs. |
Load (P) |
P_single (from single spring formula for deflection δ). |
P_stack = P_single (for a given deflection δ_single). |
The total force exerted by the stack is equal to one spring. |
Spring Rate (k) |
k_single = P_single / δ_single (often non-linear). |
k_stack = k_single / n (the stack is 'softer' overall). |
Lower overall stiffness, easier to compress. |
| Stack Height | L_o_single (Free height of one spring). |
L_o_stack = n × L_o_single (Total free height). |
Requires careful consideration of available axial space. |
| Stress | Str |
[^1]: Understanding disc springs is essential for applications requiring high load capacity in compact spaces.
[^2]: Explore how deflection characteristics influence spring selection for applications.
[^3]: Understand the role of coned height in determining spring performance.
[^4]: Learn how to design springs with tailored load-deflection characteristics.
[^5]: Learn how to analyze load-deflection curves to optimize spring performance.
[^6]: Explore the versatility of Belleville washers in various engineering applications.
[^7]: Understand the calculations behind determining load capacity for effective design.
[^8]: Learn the essential formulas for accurate load calculations in spring design.
[^9]: Learn about different spring steels and their applications in engineering.
[^10]: Discover the significance of Modulus of Elasticity in material selection and spring design.
[^11]: Gain insights into how Poisson's Ratio influences material behavior under stress.
[^12]: Explore the critical impact of material thickness on load capacity and stiffness.
[^13]: Understand the implications of non-linear behavior in spring design.
[^14]: Discover the properties of high-strength spring steels used in disc springs.
[^15]: Discover how series stacking can enhance deflection while maintaining load capacity.