Tension spring formula

We offer standard size and custom-designed compression springs, extension springs, and torsion springs.

If you wish to discuss how to measure the initial tension of a spring or the calculation necessary for your application, contact us today! For both spring types, the key is to ensure that the force is applied slowly and smoothly to capture an accurate, steady reading at each specific length.

Determining the Spring Constant

The final step uses the two collected data points to determine the spring constant, $k$, which represents the spring’s stiffness.

If this initial tension is Pi, any load (P) will be

From formula (1), the deflection (δ) is as per the following formula.

The shear stress τ0・τ are calculated with the following formulas just like with the compression coil spring.

Stress of the Hook Part of Extension Springs

As the tensile stress and the shear stress based on the bending moment and the torsion moment are generated from the hook part, it is complicated to reach an accurate calculation.

In mathematical terms, Hooke’s Law is expressed as F=-kx, where F is the applied Force, k is the spring constant, also called spring rate (R), and x is the displacement, also called deflection. Once the initial tension is overcome, the spring is extended to a first length ($L_1$) to record force $F_1$, and then extended to a second, greater length ($L_2$) to record $F_2$.

With engineering degrees from Cornell University and the University of Virginia plus over 25 years of experience designing springs and wire forms, our engineering staff is ready to assist you. This practical measurement is necessary to maintain proper suspension height in a vehicle, ensure the correct closing force on a valve, or verify the load-bearing capacity in a custom mechanism.

Understanding Spring Force and Rate

The terms “spring tension” and “spring rate” describe two distinct but related characteristics of a coiled spring.

Or corrosion? Utilizing contemporary wire and spring manufacturing technology combined with the most advanced equipment, we strive to exceed our customers’ expectations. This calculated rate allows an engineer or technician to select a replacement spring with identical performance characteristics or to design a new application that demands a specific load at a specific displacement.

The calculation is based on the principle that the spring rate is the change in force divided by the change in deflection between the two measured points. Give us a call and we will be happy to assist – we are a spring manufacturer that loves a challenge!

（i）In the Case of a Machine Half Hook

In Figure 1, the maximum tensile stress value occurs inside part A, and the maximum shear stress value occurs inside part B.
As the maximum tensile stress inside part A is the sum of the bending moment (M) and the tensile stress according to the axial load (P), it becomes

Here, K1 is the stress concentration coefficient based on the curvature, and if

, it can be found with the following formula.

When formula (7) is rearranged, we get the following formula.

K1, however, is as follows.

Here, C is the spring index of the coil part.
The maximum shear stress at the inside of part B is due to the torsion moment M.

Here, Ｋ2 is the stress concentration coefficient based on the curvature, and if

, it can be found with the following formula.

（ii）In the Case of an U-shaped Hook

In Figure 2, the maximum tensile stress value occurs inside part A, and the maximum shear stress value occurs inside part B.
As the maximum tensile stress inside part A is the sum of the bending moment (M) and the tensile stress according to the axial load (P), it becomes

Here, Ｋ3 is the stress concentration coefficient based on the curvature, and if

, it can be found with the following formula.

When formula (13) is rearranged, we get the following formula.

K’3, however, is as follows.

The maximum shear stress at part B is given with formula (11) just like in the case of the machine half hook.

Measuring at least two distinct points is crucial for calculating the spring rate, as the difference between the two data points provides the necessary variables for the formula.

Extension springs require a slightly different approach, as they may have initial tension—a force required to separate the coils before the spring begins to lengthen.

We offer a wide assortment of all types of springs that meet the application needs across a broad range of industrial sectors. We take the time to listen and understand your requirements and then, using industry standard software plus years of experience, we present you with options to consider. PE is equal to the force, F, times the distance, s, which is referred to as spring force.

A spring with a rate of 400 lb/in, for example, requires 400 pounds of force to compress it exactly one inch.

For most helical springs used in common applications, the relationship between force and displacement is linear, a concept described by Hooke’s Law. This law states that the force exerted by the spring ($F$) is directly proportional to its displacement ($x$) from its free length, with the spring rate ($k$) acting as the constant of proportionality ($F = kx$).

Once that potential energy is released, a spring is designed to return to its original shape after being compressed, stretched or twisted.

Application of Springs
Springs absorb or release energy to create resistance to pulling or pushing force. For example, if the force increases by 50 pounds while the deflection increases by $0.25$ inches, the spring rate is 50 pounds divided by $0.25$ inches, which results in a spring constant of $200$ pounds per inch (lb/in).

When a spring is installed, it may be compressed or extended slightly to apply a small initial force, known as pre-load or initial tension. This initial measurement serves as the zero point for all subsequent deflection calculations. We provide our customers with a complete understanding of our products and the material requirements for even the toughest spring or wire form challenges.