It’s Spring Time Part 3: Load-Deflection Relationships

5 min read

We’ve been introduced to materials and characteristics of springs. Let’s dive into helical springs more deeply specifically compression and extension springs. Their load deflection relationships will be evaluated for both.

Load-Deflection Relationship for Helical Compression Springs

When the load on the spring: P and the deformation: δ are proportional to each other (in a linear relation), they are said to behave under “Hooke’s law”. The constant of proportion: k, is called the “spring constant”. [Fig.1] shows a relationship between the load and deformation. The slope of this figure represents spring constant: k.

P = k x δ

k: Spring constant

Examples of products designed and manufactured by taking advantage of this property include spring balances (scales for load weight measurement) and springs for safety valves operating at a required force.

Springs with Various Loading Characteristics

Contrary to the linear characteristics shown in (1), some of the springs’ load and deformation are in a non-linear relationship. Helical compression springs having non-linear characteristics of load and deflection are available in the following three types:

The functions of helical compression springs with non-linear characteristics are achieved when the line or lines and the seating surface contact each other as the load increases. This occurs because the coil spring position causes a change in at least one of the following design parameters: [1] coil diameter, [2] pitch, or [3] wire diameter.

The following table summarizes advantages and disadvantages of typical springs having the non-linear characteristics introduced earlier.

Type of spring with non-linear characteristics Advantages Disadvantages
Conical springs
Contact with surrounding objects can be avoided when the spring deforms.
The seating surface contact type has the lower solid height.
For the equal wire diameter type, the energy absorption rate becomes smaller as the coil diameter decreases.
The line contact type has the higher solid height.
Variable pitch springs
Higher solid height.
Large in mass.
Coil springs with a tapered material
Lower solid height.
Smaller in mass compared to variable pitch springs.

Load-Deflection Relationship for Helical Extension Springs

Unlike helical compression springs, non-linear characteristics cannot be added to helical extension springs. However, the initial tension can be put into the springs.

Helical Extension Springs with Initial Tension

Initial tension, which is a force holding the coils against each other, can be put into helical extension springs even under a no-load condition. The initial tension is created by twisting the wire in the direction of the coils in contact with one another during the spring winding process. Cold-formed and solid-coiled springs are wound with some tension. In general, springs that are purposely designed to have initial tension are referred to as springs with initial tension.

  • The load and deflection characteristics of springs with and without the initial tension are shown in [Fig.1].
  • [Formula A] represents the relationship between loads and deflection of an extension spring shown in [Fig.1]. [Formula B] is a relational expression between loads and deflection of an extension spring with the initial tension.

[Formula A]
Load P (N) = Spring constant k (N/mm) × Deflection δ (mm)

[Formula B]
Load P (N) = Initial tension Pi (N) + Spring constant k (N/mm) × Deflection δ (mm)

Initial tension Pi will be calculated by the following formula:


Advantages Disadvantages
·        Instability of springs under a no-load condition can be minimized.

·        Designing a smaller spring (larger load with smaller spring constant) is possible.

·        Load variation tends to be large at a specified length.

·        Even if it is necessary to perform low-temperature annealing to remove distortion occurred by coiling, the annealing result will not be sufficient.

Various Shapes of Helical Extension Springs

Most of the helical extension springs are without non-linear characteristics on the spring plane and classified into the cylinder type and the spindle ends type. In addition to this classification by the outer shape, the springs can be also categorized by the hook shape on both ends.

Integrated double hooks are the most common and come in the following styles:

  • Semicircular hook
  • Circular hook
  • Reverse circular hook
  • Side circular hook
  • Square hook
  • U-hook & V-hook

Separate hooks

  • Tapered end with a circular hook
  • Plug type
  • Plate type

How to Secure Springs

 Fixing both ends of a spring is an important measure for stabilizing behavior. There are well-suited ways of fixing both ends depending on the type of spring, such as helical compression springs and helical extension springs.

How to Secure Helical Compression Springs

In considerations of the degree of freedom of the secured spring, avoiding buckling, and prevention of load-point displacement, the methods shown in [Fig.1] are adopted for securing both ends of helical compression springs. As for the tips, use a Coil Spring Washer (SPGCC, for example) by MISUMI for its ease of production if you use the securing method shown in (b). Otherwise, prepare a guide hole on both ends and place the spring into it.

How to Secure Helical Extension Springs

Hooks on both ends of helical extension springs are available in various shapes. a) is the most standard shape. See the characteristics of b) and c) in the following table.

Hook shape Characteristics
V-hook This shape is frequently adopted for measuring equipment because it can minimize the allowance (backlash) between the mating side and spring hook area.
Square hook This shape is adopted only when the mating side is shaped like a flat plate.

The linear and non-linear characteristics of springs based on the load-deflection formulas show the advantages/disadvantages of the different types of helical springs. Whether in compression or extension, the load capabilities will differ. Securing and properly installing springs are also different depending on the type and application. Up next, in our last post in the Spring series, we will cover energy absorption, frequency and the surging phenomenon of springs. Come back next week!

About the Author

Carlicia Layosa

Carlicia is the Marketing Automation Manager at MISUMI. She holds a bachelor's degree in Mechanical Engineering and a master's degree in Energy Engineering from the University of Illinois at Chicago. She is a Certified SOLIDWORKS Associate, Marketo Certified Expert, and is passionate about education and training.

4 thoughts on “It’s Spring Time Part 3: Load-Deflection Relationships

  1. Thank you for your useful tutorial about springs. Every engineering problem is more interesting, and complicated, than I supposed.

    I am designing a photographic jig that moves ca 1.2 m on a rail, which needs power supply 12V and control wires.
    One method of supporting these cables is to hang them from the top of a vertical wire ca 0.7 m high with a vertical wound spring attached to the baseboard to allow easy bending to accommodate the jig’s movement. I have seen these, but do not know the name so cannot find a supplier.

    If you know what I am trying to describe, I should be grateful if you could let me know its proper description/name.

    1. Hi CV,

      Would you be able to provide a simple sketch of what you’re looking for?
      You can email it to our engineering team at, perhaps we can name what you’re looking for!

      Thank you!

  2. I need an IRM that would be a Hookean Spring Load at 1 in deflection. Which would be the most suitable? I usually perform puncture tests according to ASTM D4833.

    1. Hi Carolina,

      Be sure to check out our tension springs. We have a variety to choose from at different lengths and loads.

      Thank you!

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