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: $E,\sigma_{0.2},\nu$ 

Featured image spring shapes
Spring typeEquations
Buckled plate / wire spring
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
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
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)
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
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
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*
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
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 $

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