# Spring shapes

## Construction Fundamentals

#### Introduction

Various shapes can be used as a spring to provide certain stiffness. Below several shapes are described. Also the stroke according to the maximum occurring stress in the particular shape is added; maximum stroke at yield stress.
$$\tau_{0.2}\approx0.4\cdot\sigma_{0.2} \textsf{ (JPE estimate for metals)}$$

Below materials are characterized with: $E,\sigma_{0.2},\nu$ Spring typeEquations
Buckled plate / wire spring $F_{buckling}=\frac{\pi^2EI}{L^2}=\frac{\pi^2Ebt^3}{12L^2}$

$F_{buckling}=\frac{\pi^2EI}{L^2}=\frac{\pi^3Ed^4}{64L^2}$

$C\approx0$ (Constant force F)

$\delta\ s_{max}=\ (not\ known)$
Torsion wire spring $M=Fr$

$K=\frac{M}{\delta\varphi}=\frac{Ed^4}{64nD}$

$\delta\varphi=\frac{2\pi nD}{Ed}\sigma$
Spiral plate / wire spring $M=Fr$

$K=\frac{M}{\delta\varphi}=\frac{Ebt^3}{12L}$

$K=\frac{M}{\delta\varphi}=\frac{\pi Ed^4}{64L}$

$\delta\varphi=\frac{2L}{Et}\sigma$

$\delta\varphi=\frac{2L}{Ed}\sigma$
Disc spring (DIN 2092) $r_1=\frac{r_2}{2}$

$C=\frac{F}{\delta s}=\frac{Et^3}{0.69{r_2}^2\left(1-\nu^2\right)}$

$\delta s_{max}=h$
Compression / tension spring $C=\frac{F}{\delta s}=\frac{d^4E}{16nD^3(1+\nu)}$

$\delta s=\frac{2\pi n D^2\left(1+\nu\right)}{dE}\tau$
Inclined compression spring $C=\frac{F}{\delta s}=\frac{d^4E}{32n(r_1+r_2)(r_1^2+r_2^2)(1+\nu)}$

$\delta s=\frac{2\pi n(r_1+r_2)(r_1^2+r_2^2)\left(1+\nu\right)}{dr_2E}\tau$
Ring plate / wire spring* $C=\frac{F}{\delta s}=4.48\frac{Ebt^3}{D^3(1-\nu^2)}$

$C=\frac{F}{\delta s}=2.64\frac{Ed^4}{D^3(1-\nu^2)}$

$\delta s=6.72\frac{D^2}{2Et}\sigma$

$\delta s=6.72\frac{D^2}{2Ed}\sigma$
Plate / wire spring $C=\frac{3EI}{L^3}=\frac{Ebt^3}{4L^3}$

$C=\frac{3EI}{L^3}=\frac{3\pi E\ d^4}{64L^3}$

$\delta s=\frac{4L^2}{3Et}\sigma$

$\delta s=\frac{4L^2}{3Ed}\sigma$