DIN 2093 Spring Stiffness: Tangent Slope of the Force‑Deflection Curve

1. Stiffness Definition

The spring stiffness $k$ of a disc spring is defined as the tangent slope of the force‑deflection curve at the operating point, i.e., the derivative of the axial force $F$ with respect to the deflection $s$:

Since the force‑deflection relationship of a disc spring is nonlinear (due to the geometric nonlinearity of disc bending), the stiffness $k$ is not constant but a function of the deflection $s$.

2. Load‑Deflection Basis (Almen‑Laszlo)

According to DIN 2093 / Almen‑Laszlo theory, the axial force at any deflection $s$ is:

Introducing dimensionless quantities:

And setting the constant:

The force formula simplifies to:

3. Derivation of the Tangent Stiffness Formula

Differentiating $F$ with respect to $\delta$, and using $\frac{d\delta}{ds} = \frac{1}{t}$, yields:

Expanding the bracket:

Multiplying by $\delta$:

Differentiating with respect to $\delta$:

Therefore, the final expression for the tangent stiffness is:

Or written in dimensional form:

Discussion:

• When $\delta = 0$ (uncompressed), the initial stiffness is $k_0 = \dfrac{C_F}{t}(\eta^2+1)$;
• When $\delta = \eta$ (fully flattened), the stiffness is $k_{flat} = \dfrac{C_F}{t}\left(1 - \frac{3}{2}\eta^2\right)$, which can be positive or negative depending on $\eta$.

4. Shape of the Stiffness Curve vs. $\eta = h_0/t$

The shape of the disc spring force‑deflection curve is entirely determined by the dimensionless cone height ratio $\eta$. The critical dividing point is $\eta = \sqrt{2} \approx 1.414$.

4.1 Positive Stiffness Region ($0 < \eta < \sqrt{2}$)

When $\eta < \sqrt{2}$, the force‑deflection curve is monotonically increasing, and the tangent stiffness remains positive throughout the stroke, though it gradually decreases with increasing deformation (softening characteristic), or is approximately linear for small $\eta$.

• Most applications in power machinery, preload compensation, etc., use this range.
• Typical $\eta = 0.6 \sim 0.9$ disc springs exhibit nearly constant or slowly decreasing stiffness.

4.2 Negative Stiffness Region ($\eta > \sqrt{2}$)

When $\eta > \sqrt{2}$, the force‑deflection curve is no longer monotonic: as the deflection increases, the force first rises, reaches a peak, then decreases, forming a "hump" and a "valley", before rising again. Between the peak and the valley, the tangent stiffness is negative.

Physical meaning of the negative stiffness segment: as displacement increases, the spring force decreases, making the system unstable, but this can be exploited for special purposes (e.g., constant force support, over‑center mechanisms, vibration control).

5. Application of Stiffness Characteristics in Engineering

• Constant or Soft Stiffness ($\eta < 1.4$): Used for precision loading, preload compensation, elastic energy storage.
• Negative Stiffness Constant Force Segment ($\eta > 1.4$ with partial stroke utilization): Can provide a nearly constant force over a certain displacement range, used in constant force clamps, overload protectors, vibration isolators, etc.
• Stiffness Matching: In bolted connections requiring elastic compensation, low $\eta$ disc springs (e.g., DIN 6796 washers) are typically chosen to obtain approximately linear and positive stiffness; misusing high $\eta$ disc springs may cause preload to decrease with settlement (negative stiffness), which should be avoided in design.

6. Stiffness of Stacked Disc Springs

When multiple disc springs are stacked in series or parallel, the combined stiffness follows:

• Parallel (same direction stacking): Total stiffness $k_{total} = n \cdot k_{single}$, total force doubles, stroke unchanged.
• Series (opposing direction stacking): Total stiffness $k_{total} = k_{single} / i$, total stroke doubles, force unchanged.

The combined system retains the nonlinear characteristics of a single disc, but the corresponding total force‑stroke curve is obtained through the above linear superposition.

Summary:
The stiffness of a DIN 2093 disc spring is determined by its geometry and material, with the core being the tangent stiffness formula $k = \frac{C_F}{t}[(\eta^2+1) - 3\eta\delta + \frac{3}{2}\delta^2]$. The dimensionless cone height $\eta$ allows prediction of the force‑deflection curve shape: $\eta < \sqrt{2}$ yields positive stiffness, suitable for most engineering springs; $\eta > \sqrt{2}$ introduces a negative stiffness region, enabling special constant force or snap‑through characteristics. Correctly understanding and calculating stiffness is key to disc spring selection and system design.