Rectangular Section Coil Spring

Is this a rectangular shaped spring?

...or rectangular metal round shaped spring..?

It's like an ordinary round coil spring, with rectangular wire. In proportion, much like the blue one in your picture.

This type of spring is commonly called a "die spring" and are sold by most major spring companies and die maker's supply companies (for example Danley, Producto, and many others) Their catalogs provide detailed spring specifications for dimensions, spring rates, and maximum deflection. They all use a standard color code for force range, blue for medium, red for medium heavy, gold for heavy, and green for extra heavy.

When you compress (or stretch) a coil spring, you are actually applying a torsion (twist). The rate of a torsion spring is proportional to the polar moment of inertia. For a solid cross section it varies as the fourth power of dimension. Here are some formulas for various cross sections:

http://www.roymech.co.uk/Useful_Tables/Torsion/Torsion.html

You should be able to take values for a circular cross section spring and relate it to a non circular cross section.

I hope this helps.

The original equipment manufacturer deserves the first telephone call. In this sort of enquiry, an internet search engine will be found invaluable for researching telephone numbers, a practice that is highly recommended. The manufacturer would even be able to quote for (a) replacement(s). Make the call!

The problem with that is it's the manufacturer who has asked me to help them! They're a manufacturer of conveyors, not springs, and want to allow adequate clearance for the screw to extend under load. At 18.3m long and 500mm OD it's bigger than what they're used to. They can do standard conveyors OK but short on mechanical engineering input for something outside their usual range.

I found a site called OilZone which has a spring rate calculator (formulas not shown), but have to enter a figure for k₂. By trying k₂ = 1, I can check it gives same result as rate = G*w*b^3/(n*D^3)*k₂ as on other sites, but with k₂ = 0.36 I'm sceptical about the result.

Look in Roark Formula for Stres and Strain chapter Torsion (you can have the book on the net for free) there are equations for EXACTLY what you need

Perhaps you've already visited this site, the info below is an extract

Basic relations for spring calculation

Springs of round wire	Springs of rectangular wire

where:
c ... spring index (c=D/d; c=D/b) [-]
b ... wire width [mm, in]
d ... wire diameter [mm, in]
D ... mean spring diameter [mm, in]
F ... loading of spring [N, lb]
G ... modulus of elasticity in shear [MPa, psi]
h ... wire height [mm, in]
k ... spring constant [N/mm, lb/in]
K_s ... curvature correction factor [-]
L₀ ... free spring length [mm, in]
L_S ... solid length [mm, in]
n ... number of active coils [-]
p ... pitch between coils [mm, in]
s ... spring deflection [mm, in]
e,y ... shape coefficient [-] (e.g. DIN 2090)
t ... torsional stress of the spring material [MPa, psi]

Thanks, but yes I have. That's the MIT site data.

Problem is values for ε and ψ are not given. Last line but one says "e,y ... shape coefficient [-] (e.g. DIN 2090". Somebody failed to check the font!

If you let me have the material, the total coils, the free length & the end type, I could run it through our IST spring programme for you. That would give you the rate (plus other info) but wouldn't give you the formulas used.

Drew

That would be great!

Material is C45 non-alloy high grade steel. R_m 580 - 820MPa. Data sheet doesn't give elastic modulus but this will be 200GPa like most steels.

Screw is 500mm OD, 243mm ID giving radial width 128.5mm. Thickness 35mm. Length 18.3m, pitch 333mm giving 55 turns.

Expected load 50kN.

Mainly interested in spring rate and maximum stress.

Following nick name's post #7, I've looked at it using Roark. I have Issue 7 on the computer but hadn't thought to look there. I was hoping for a result something like my 1st-principles figure for spring rate, but it gave about 1/4 of that, similar to website calculators. I'm even more reluctant to query Roark, but I'm still sceptical. For one thing in the formula he uses a and b which are 1/2 the width and thickness. The reason escapes me, but it could cause confusion. Also he gives a formula for square section, and if I put in b = 39mm, which = 1/2 the side to give same torsional I-value as my rectangle, the spring rate comes to ~ 80% of my estimate.

Look forward to your figures, thanks again.

Sorry, seem to have got ahead of myself here, IST doesn't recognise C45 as a spring material, so it cannot calculate the spring parameters.

Apologies for false raising of hope.

Drew

No problem. I got on the IST website, so thanks for pointing me in that direction.

Downloaded a trial version of the software. I just used one of the carbon steels on the list as E and G don't vary significantly with steel grade and type, and I'm mainly interested in spring rate.

The results are somewhat higher, 20 - 25%, than given by Roymech/Leespring, but still much lower than my first-principles figures so I was clearly out there!

Thanks again

You are a lot smarter than me, i can't do what you have done. That said you could compare the formula for a round wire spring against a die spring where the manufacturer supplies the data. Obviously if there is an agreeance you could scale it up. If there isn't agreeance you could add a multiplier to get the required result. If you do the maths over as large a range as possible the 'test' results would be more reliable. The only fly in the ointment here is that die springs can come in different strengths but i am sure you can work through that.

Jim

Following gringogreg's #4, I looked up die springs, and found the colour-coded springs. Biggest was 2.5" OD, 1.5" ID, which is much smaller than my spring, but it would have given a useful check. Trouble is it doesn't give thickness or pitch so not a lot of use. Thanks anyway!

Near the bottom of this link:

http://www.roymech.co.uk/Useful_Tables/Springs/Springs_helical.html

is a section on rectangular springs, I would be surprised if this did not answer your question....good luck!

Your MAJOR error was to consider, for torsion similitude, the polar inertia moment. In torsion there is a MAJOR difference between the torsional stiffness modulus and the polar inertia moment.

If you look at the litterature you will see the membrane similitude and understand your error.

The question is:

What do you want a correct answer or an answer to confirm your results ?

I'm sure there is a MAJOR difference between the torsional stiffness modulus and the polar inertia moment. One is length³, the other is length⁴, for a start.

As the twist angle varies as torque/moment of inertia I'd expect the spring rate to do likewise. It works for round section! Clearly there might be adjustments for section shape, but I'm surprised if the factor is about 4. I can see the maximum stress is higher for rectangular section than circular (for a given MoI) as the modulus is smaller, but that's not what I'm asking about just now.

I want the correct answer of course, I just want to be sure it is the correct answer.

BTW thanks for your suggestion to try Roark, I hadn't thought of him.

The ANGULAR deflexion being it in flexion or in torsion is a quantity proportional to a product torque [ F*L]x length[L] and divided by a material intrinsic stiffness (E or G) with the dimensions [ F/L²] and with a section geometric characteristic let us call it J. An angle has a dimension 1 so that the equation in its dimensional form is

1 = F*L^2/(F/L^2*[J]) it results that the dimensions of [J] has to be L^4.

And this which ever is the loading type.

Dimensional analysis is a very powerful tool to verify if a final algebraic result is correct.

I wouldn't disagree with that.

Also stress = torque/Z where Z = torsional modulus, which is length³.

I use dimension checks all the time. One of the advantages of Mathcad is it gives an error if you try to add or subtract things with different dimensions, and if you multiply or divide and the result doesn't have the units you expect you know immediately you've done something wrong.

You will find in the LEE SPRING Engineering Guide ALL formula and coefficients for the computation of round, square AND rectangular sections. Coming from a spring manufacturer I presume that the equations and graphs are correct.

Sorry to go on about this nick, but is this the guide you refer to? (I can't see how it would not be, and I've looked at it before). I can't see anything in it about rectangular section springs, or much in the way of formulas.

http://leespring.com/downloads/uk/leespring_engguide.pdf

It gives the basic formula spring rate = G*d^4/(8*n*D^3) but everybody's happy with that!

If I'm missing something, can you point me to the page number?

In my edition page 1 & 2.

Send me on the private channel your mail address and I send you the doc.

As for the differences you should think that in bending the deformed section leaves the original plane. According to the hypothesis that it remains plane one gets the equations where the I figures since the stress is proportional to strain which proportional to the distance to the neutral line. Integrating on the surface for the total bending you come to the J value proportional to b*h^3.

In case of torsion the section remains in the original plane and you should imagine a series of concentric flows in which flows the shear stress. The farther the flow tube section is the lower is the intensity this the reason why the maximal shear stress occurs at the middle of the longest side because it is the narrowest point to the center.

This you can see very well in the computation of stress in closed thin walled profiles. It is the same in principle for other profiles if you assume a series of flow tubes running in parallel.

OK thanks. Message sent

I am curious as to why you feel the the rectangular spring design calculation in the below reference offered by Rixter is not appropriate for your rectangular spring design.
http://www.roymech.co.uk/Useful_Tables/Torsion/Torsion.html

When I started looking at this I did it from first principles, but then thought probably a good idea to check on the internet for other opinions. When I did I found the discrepancies mentioned in original post, I couldn't see why the difference and asked if anybody could comment. I got some useful hints about calculation sources.

But having looked again at your (and Rixter's) link, I see where I was going wrong. I used the formula θ = T*L/(J*G) and assumed J is the 2^nd moment of area in torsion. In the case of a solid circular bar it's the same, pi*r⁴/2, but evidently not otherwise. So for a rectangle 2a x 2b I used J = 4/3*(a*b³ + a³*b). Doh!

The on-line calculators e.g. Oilzone gives the formula spring rate k = (K₂*G*b*t³)/(N_a*D³) but it doesn't say how to find K₂. That of course is the difficult bit, it's a trivial matter to use the formula for k when K₂ is known. (I mistakenly said earlier the Oilzone calculator doesn't give the formula). Maybe there are on-line calculators that do it all for you but I haven't found one.

There are formulas in Roymech, Lee Spring and Roark, which reassuringly give results within a shout of each other. Roymech and Lee Spring give same result for spring rate but differ somewhat for maximum stress. Roark gives a rate ~ 10% higher for the particular spring I looked at for comparison.

If anybody's interested, I find Roark easiest to use. Can put the formulas in XL and it's all there, no need to find K₁ and K₂ from tables as with Roymech and Lee Spring. Need to put in a calculation to tell it to use the smaller of wire width and thickness, but that's not too hard.

Thanks for everybody's help, my original calculation was way out so I've learnt something.

Thanks for the follow up. It doesn't always happen.

I would like to say i learnt something; perhaps i can say that i learnt to employ an engineer to do this stuff for me!

Jim

You may have a point there!

Incidentally while on the internet I came across this, so the subject has been discussed before.

http://cr4.globalspec.com/thread/75459/Spring-Calculation