Springs are essential to life. In fact everything on the planet acts like a spring, which follow Hooke’s law. When force is applied to an object, it will deflect. If more force is applied, it will deflect more.
When designing or selecting a compression spring the following items need to be considered:
- Spring Shape
- End Type
- Number of Coils
- Spring Set
- Allowable Spring Stress
- Spring Index
- Spring Calculations
- Spring Selection Approaches
One crucial thing to understand when designing compression springs is to be sure to operate them in their intended range. All springs work well in their linear range, but as soon as they are loaded past the yield strength, the spring stops behaving as intended and the spring life is severely shortened. It is little known that springs are one of the most energy efficient methods of storing energy as very little heat is created in the process.
One of the most fascinating things about springs is that the method they are used in is opposite of the load they see. A compression or extension spring is loaded in torsion and a torsion spring is loaded in bending. I liken this to driving on parkways and parking on driveways. It’s all backward!
1. Spring Shapes
Cylindrical compression springs are by far the most common, but there are other shapes to consider. The most popular are:
- Conical
- Hourglass
- Barrel
- Reduced Ends
Cylinderical Springs
Cylinderical springs are very common and come in a variety of sizes as an off the shelf part. The OD and ID of the spring is consistent and as a result, they are generally intended to be guided by being placed in a bore or over a shaft. The spring rate is constant and adding springs in series will enable more travel, while adding them in parallel allows for more force.
Conical Springs
Conical springs are widely used as well. They are prevalent in most battery powered electronics as they squeeze the negative terminal of the battery and hold it in place. A conical spring has a constantly decreasing radius. The main reason to use this type of spring is height constraint; the coils will nest inside each other until the collapsed height is roughly two thickness of the wire diameter. This allows more stored energy in a smaller space. Because the diameter of the spring is not constant, the spring rate is non-linear. When you push on a conical spring, the outer coils will compress first and the spring rate will continually increase as the diameter decreases. Conical springs also have some tolerance for lateral movement as one would see when a battery is inserted into the electrical device with a slight offset from the battery center-line. Along with hourglass and barrel springs, conical springs are often chosen when spring surge is an issue.
Hourglass Springs
Hourglass Springs are nose to nose conical springs. They allow for high loads in small heights. The main reason to choose an hourglass spring is for lateral stability. To demonstrate this, imagine holding the spring between the flat palms of your hands. If I push my hands together, I compress the spring. If I keep my hands parallel and the same distance apart, but shift them side to side, I add lateral loads. An hourglass spring will naturally resist this motion because of its wide base and shape. This is the reason that these springs are used in the railroad industry.
Barrel Spring
Barrel springs are used when buckling is an issue. Sometimes, to get the load capability needed the spring becomes very tall. If there is not a bore the spring fits into or a shaft it fits over, normal springs will buckle. A barrel spring has narrow ends and a much wider center. The wider middle section gives it more stability against buckling.
The second reason to use a barrel spring is it can be designed to reduce the total height just like a conical spring. Once again, the spring rate is non-linear.
Reduced End Spring
The final type of spring is a reduced end spring. You would select one of these for the same reason as a barrel spring. The main difference is that the reduced end spring is far more linear in nature.
2. Compression Spring End Types
There are four basic types of compression spring ends, they are:
- Open
- Open and Ground
- Closed
- Closed and Ground
An open end spring will have wire cut perpendicular and none of the coils touch each other in the relaxed state. The pitch is consistent throughout the length of the spring. An open and ground spring is different because the wire is ground at each end so that the spring can lay flush and not leave an indent in whatever it mates to. However, due to the nature of the open spring, there isn’t much material ground off the part.
A closed spring will have the last two coils touching. This offers more support at the end, but it will shorten the number of active coils by two thus changing the spring’s characteristics. They are also available in closed and ground versions so that they can lay flush.
Even if the ends of a spring are ground, they are not and cannot be perfectly perpendicular. As the spring is compressed, the ends will deform angularly thus causing the spring end to rock on a flat surface. This can lead to unevenness in the spring constant thus rendering it a variable defined by its height. In most cases, the application is not critical enough and the average spring rate can be used.
If this is an issue in your application there are two remedies to minimize the effects. Change the end of the spring from an open to a closed condition. As you can see from the diagrams, an open and ground spring will only have a small part of the spring ground. The closed condition grinds a large percentage of the last coil off and therefore offers more support to the end of the spring. If your wire diameter is smaller than 0.20” this will most likely solve your problem.
The other solution is to specify that the spring be ground at a specific height or load applied. This is a custom spring and you will pay for the vendor’s extra attention.
3. Number of Coils
Counting the number of coils on a spring can be tricky. There are two variables that specify the number of coils. The value Na is the number of active coils and Nt is the total number of coils. Only in the case of an open spring is Na equal to Nt.
First of all, you cannot count any coils where the coils are closed or touching. These are inactive coils which don’t cause the spring to be a spring. Start counting each coil from one end to the other starting where the section first opens. For an open end spring, this is the first coil. Keep counting until you get to the other end of the spring. Generally speaking, coils are counted as full turns, but half and quarter turns are also widely used. Many spring manufactures will allow two digits of precision on the number of coils. For torsion springs, like clothes pins, you will specify the end condition in degrees.
In the examples shown, each spring has 6 coils even though the closed end spring has 8 total coils. The table below shows the relationship between active coils, total coils, the solid height and the pitch of each spring type. Allow a 3% tolerance on the solid height to account for the end conditions, wire diameter tolerance and helical wind waviness.
4. Spring Set
Some springs need to be “set” so that there is a uniform initial or relaxed height. For compression springs, the spring is “set” or “take a set” to the correct height when it is compressed for the first time. To do this, compress it to its collapsed height and release. As you can imagine, the setting process will leave the spring height shorter. At this point the spring will have been set and it will return to this height every time as long as the loads are within the springs limits.
For standard springs, this is done at the manufacturer so they can provide consistent product and not have to deal with the question, “Why are my springs too tall?” from every customer.
However, if you are ordering a custom spring, the vendor may not be able to (or just not want to) set your spring. If this is the case, you have two options. Option 1: you can assemble the spring and let the first operation set the spring. This may work in some cases, but if the application doesn’t compress the spring enough or if the spring won’t physically fit option two is needed.
Option 2: set the springs yourself. For small springs, you can simply squeeze them in your hands. For larger springs, you may need a pneumatic or hydraulic press. This can be a very time consuming process. When designing a custom spring, be sure to consult your vendor for how much set is expected so that you can design your spring to be taller when wound and the right height when set. Your vendor should have plenty of data to get you an accurate spring design.
5. Allowable Spring Stress
All springs function according to their dimensions. The dimensions will specify the stresses seen in the spring. Based on general stress calculations, we know that the largest stresses in a compressive spring will be on the outside surface of the wire as load is applied.
When designing a spring, the dimensions are critical to spring characteristics. Often we are confined by the form factor where the spring is applied, i.e., the spring needs to fit in a ¾” hole or over a ½” rod. As the designer, if you find that there is only one or two choices for springs available, you may want to ask yourself if you are being too strict with the design. I have often found that in my design I needed to go up in capacity, but the form factor was already set in stone. Now it is really hard to get the performance I need. Had I adjusted my design to have four or five choices of springs, I could simply change out a spring. In practice, I desire to have five choices; my ideal spring plus two stronger and two weaker spring alternatives.
The life of the spring is dictated by fatigue. As the spring is cycled, grains in the steel rub against each other. As they repeatedly rub, small cracks will get larger and larger until failure occurs. There are two main components that dictate the life of a spring; the number of cycles and the stress intensity.
The number of cycles is pretty easy to determine, it is something that is easily counted or estimated. We find that most materials have an endurance limit, where if the spring makes it to that number of cycles, it will last forever. For steel, this occurs around 1-2 million cycles. Unless your spring is custom, vendors only make springs with infinite life. Even then, vendors will fight you on a finite life spring.
Stress intensity is much more complicated. A lot of it has to do with the mean (average) and alternating stresses. The average stress, alternating stress and stress range are defined by the following equations. We will use a maximum stress of 50 ksi and minimum stress of 10 ksi.
Since we are only dealing with compression springs, the minimum stress is usually zero unless the spring is always compressed. In the case above, we will always have a preload on our spring. This is usually the case in applications where a handle needs to be centered. We want preload on it so that the handle doesn’t vibrate when at rest and it also requires some force to move when leaving the center position. (They are also a pain to install as well.) In any case, our stress range is 40 ksi. To extend our spring life, we will want to decrease this as much as possible. If a spring was to be used as both an extension and compression spring, the minimum stress would be negative thus increasing the stress range greatly and decreasing life span.
One step that spring manufacturers perform to limit the magnitude of stress is to stress relieve. The stress relieving process heats the spring to high temperatures so that permanent or residual stresses from the forming process are eliminated. The temperature here is not so high that the spring would lose its initial temper.
There are generally three different categories that spring design fits in to:
- The first category is for springs that can be compressed solid without set or yielding. These are usually pretty strong springs and are designed with lower allowable stresses. These stresses are generally less than 40% of the materials tensile strength. Forty percent is chosen because the endurance limit of most materials is around 45% of tensile strength. The extra 5% is added so that material defects or defects from the forming process do not limit the spring’s life.
- The second category is for springs that are ‘set’ when fully compressed. Once the initial load is removed, there is no further yielding of the material and the relaxed spring length will remain the same. These springs are designed for infinite life, but the target stress is higher but generally lower than 60% of the tensile strength.
- The final category is for springs that will yield the material if fully compressed. This means that some level of set is to be expected every time it is fully compressed until it results in failure or a collapsed spring. These springs are not designed for infinite life and/or have a maximum load or compressed height recommended by the vendor. These springs are obviously designed to more than 60% of the tensile strength.
6. Spring Index
The spring index is a simple ratio of the mean diameter to the wire diameter. It is a variable used in many spring calculations and give us an idea of what type of spring it is. The index should be kept between 4 and 10.
If it is lower than 4 and your spring is coiled too tight, this will require special tooling to form. When forming a tight radius, you might also have problems with the material cracking internally and that will lead to premature failure.
Spring indexes greater than 10 means the spring will be real flimsy and lead to problems with packaging and tangling. Think of a slinky. You also won’t be able to grind or perform other operations like plating.
This should be a giant red flag that something is not right with your design. Don’t design yourself into a hole!
7. Spring Calculations
So this is a little bit of a backwards approach. There are five main ways of selecting a spring, but they only make sense once you understand the equations behind spring design. For this we will go through the calculations and then come back to the methods of selection.
Most spring design is done based on dimensions so it is likely that you will have the outside diameter (OD) and the wire diameter. From these two pieces of information, we can find the inside diameter (ID) and the mean diameter, Dm using the following equations:
From here we can calculate the stress on the spring (assuming the wire is round).
Where C is the spring index, P is the applied load and K is the stress concentration factor. Note K is shown for extension and compression springs. Other types of springs have different values.
The final step is to calculate the spring rate, R
Where G is the shear modulus of elasticity and δ is the deflection. Many times you will be given the spring rate, so we can rearrange this equation to calculate the number of active coils in the spring.
The last thing that you may want to know about your compression spring is how much it expands as it is loaded. This is important to know if your spring is compressing in a bore or extending over a rod. At the solid height (SH), the diameter is defined as:
8. Spring Selection Approaches
There are five basic methods for selecting a compression spring. You will need to select one of the five that best fits your application.
- Design based on physical dimensions
- Design based on spring rate
- Design based on two loads
- Design based on one load and spring rate
- Design based on one load and free length
Case 1: Design based on physical dimensions
This is probably the most common method of designing a spring. When approaching the design you probably already have a certain envelope that the spring must fit into or around. When selecting a spring by physical dimensions, you need to specify two of these three things:
- Outside Diameter – OD
- Inside Diameter – ID
- Wire Diameter – d
Case 2: Design based on spring rate
When you know the spring rate you desire, you can start your selection there. You will need other information such as free length, load capacity or some dimension of the spring to complete the selection. Once you have selected this information you will probably be able to calculate the number of active coils and the free length using the following equations.
Case 3: Design based on two loads
For this design criteria, you will specify the spring rate based on two loads that are applied to the spring. At each load, the spring will deflect differently. Many spring manufacturers will have this option of choosing a spring on their website. The variable, L, is the total height of the spring and not the deflection.
(Note: Spring rates are always positive. Swap L1 and L2 if needed). Once you have the rate, you are basically back in the previous design case (Case 2) and need to make other decisions to select your spring.
Case 4: Design based on one load and spring rate
This is really another subset of load Case 2. The only difference is we will be able to calculate the free length without making any other selections.
Case 5: Design based on one load and free length
This is really just a subset of Case 3 where we assume that P2 is 0 lb and L2 is the free length. Then use the equations in Case 2.
So let’s run an example
I need a spring that will fit in a 1.25” hole and have a free length of 5.00 in. I need a spring that will support at least 100 lb without bottoming out. I then go to Lee Spring’s website and search for what I know: fits in 1.25” hole and free length of 5.00 in. I get the following results.
Looking through the list, I have multiple options which means that my design is reasonable and I’m not designing myself into a corner. I look through the spring rate column and notice that the first four springs are just not going to cut it. A spring rate of 5.3 lb/in is going to take 20” of compression to get me to 100 lb! The 22.25 lb/in spring rate is probably just outside of where I need to be. I have selected the 53 lb/in spring as a starting place because my deflection at 100 lb will be just less than 2”. At this time, I will perform my hand calculations (left) and determine the results. I have also displayed the additional information from the manufacturer (right). Notice how my calculations of solid height, active and total coils match with the manufacturer. We also have a maximum load capability of 140 lb. I have also added a calculation for W which is the working range of the spring. I’m not sure why manufacturers don’t provide this, but it would be nice.
At this point, we can move on to the stress calculations for this spring. Since we are using music wire, we will assume that for this diameter, the tensile strength is 225 ksi.
As you can see from the stress calculations, the intended load is only 10% of the tensile strength. Also, our deflection is 1.89 in as we estimated. If you are curious as to where the 0.577 comes from, please watch the video on Mohr’s Circle.
Conclusion
Spring design seems complicated because there are so many different ways to approach it. Just remember that there is no perfect solution for your application. With all the off-the-shelf sizes, shapes and materials to choose from, you will select a very good spring for your application. Just remember to watch for the red flags of improper spring index and always have plenty of alternative springs ready.
You got this!
Glossary of Terms
