General Calculation Formulas
Utilization factor of material
Outside spring diameter
D 1 = D + d [mm]
where:
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D |
mean spring diameter [mm] |
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d |
wire diameter [mm] |
Inside spring diameter
D 2 = D - d [mm]
where:
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D |
mean spring diameter [mm] |
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d |
wire diameter [mm] |
Working deflection
H = L 8 - L 1 = s 8 - s 1 [mm]
where:
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L 8 |
length of fully loaded spring [mm] |
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L 1 |
length of pre loaded spring [mm] |
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s 8 |
deflection of fully loaded spring [mm] |
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s 1 |
deflection of pre loaded spring [mm] |
Height of spring eye
where:
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L 0 |
length of free spring [mm] |
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L Z |
length of spring coiled part [mm] |
Spring index
c = D/d [-]
where:
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D |
mean spring diameter [mm] |
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d |
wire diameter [mm] |
Wahl correction factor
where:
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c |
spring index [-] |
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L Z |
length of spring coiled part [mm] |
Initial tension
where:
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d |
wire diameter [mm] |
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τ 0 |
free state stress [MPa] |
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D |
mean spring diameter [mm] |
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K w |
Wahl correction factor [-] |
General force exerted by the spring
where:
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d |
wire diameter [mm] |
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| τ |
torsional stress is force per unit area. of spring material in general [MPa] |
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D |
mean spring diameter [mm] |
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K w |
Wahl correction factor [-] |
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G |
modulus of elasticity of spring material [MPa] |
Spring constant
where:
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d |
wire diameter [mm] |
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G |
modulus of elasticity of spring material [MPa] |
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D |
mean spring diameter [mm] |
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n |
number of active coils [-] |
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F 8 |
working force in fully loaded spring [MPa] |
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F 1 |
working force in minimum loaded spring [MPa] |
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H |
working deflection [mm] |
Spring Design Calculation
Within the spring design, wire diameter, number of coils and spring free length L 0 are set for a specific load, material and assembly dimensions.
If the calculated spring does not match any wire diameter for the τ 0 stress according to the formula, the spring calculation is repeated with the corrected stress value in a free state within the recommended range.
The spring without initial tension is designed for a mean recommended pitch value t = 0.35 D [mm].
If the calculated spring does not match with any wire diameter of a selected pitch, the spring calculation is repeated with the corrected pitch value within the recommended 0.3 D ≤ t ≤ 0.4 D [mm] range.
The spring design is based on the τ 8 ≤ u s τ A strength condition and the recommended ranges of some spring geometric dimensions: L 0 ≤ D and L 0 ≤ 31.5 in and 4 ≤ D/d ≤ 16 and n ≥ 2.
Specified load, material, and spring assembly dimensions
First the input values for the calculation are checked and calculated.
Next the spring length at the free state is calculated.
After the calculation, the wire diameter, number of coils and spring diameters are designed so that the spring hook height conforms to the selected hook type. The previously mentioned strength and geometric conditions also must be fulfilled. The spring design must conform to any spring diameter value limited in the specification. If not so, the limits of spring diameter are determined by the geometric conditions for minimum and maximum allowable wire diameter.
All spring wire diameters that conform to the strength and geometric conditions are calculated, starting with the smallest, and working to the largest. Spring hook height and number of coils are tested. If all conditions are fulfilled, the design is finished with the selected values, irrespective of other conforming spring wire diameters, and a spring is designed with the least wire diameter and the least number of coils.
The calculated spring hook height must be within the d ≤ o ≤ 30 d range. A combination of the wire diameter, number of coils, and spring diameter must result in a calculated spring hook with a height that corresponds with the height of a basic hook type. The basic hook type is selected by first investigating the full loop, then the full loop inside, and then other types of hooks.
Specified load, material, and spring diameter
First the input values for the calculation are checked.
After the check, the wire diameter, number of coils, spring free length, and assembly dimensions are designed, so that the spring hook height conforms to the selected hook type. The strength and geometric conditions also must be fulfilled. If an assembly dimension L 1 or L 8 is stated in the specification, or the working spring deflection value is limited, then the spring design must conform to this condition. If not, the limits of assembly dimensions and free spring length are determined by the geometric conditions for the specified spring diameter and minimum or maximum allowable wire diameter.
Formula for designing a spring with a specified wire diameter.
where the τ 8 = 0.85 τ A value is used for the value of torsion stress for the spring material, in the spring fully loaded state.
If no suitable combination of spring dimensions can be designed for this wire diameter, geometric investigations proceed on all suitable spring wire diameters. They are tested, beginning with the smallest and working to the largest, for coil numbers that result in the spring hook height that conforms with the conditions. The design is finished with the selected values, irrespective of other suitable spring wire diameters, and the spring is designed with the least wire diameter and the least number of coils.
The calculated spring hook height must be within the d ≤ o ≤ 30 d range. The corresponding hook type is selected for the hook height that is calculated in this way. A combination of the wire diameter, number of coils, free spring length, and assembly dimensions must result in a calculated spring hook with a height that corresponds with the height of a basic hook type. The basic hook type is selected by first investigating the full loop, then the full loop inside, and then other types of hooks.
Specified maximum working force, determined material, assembly dimensions, and spring diameter
First the input values for the calculation are checked and calculated.
Then the wire diameter, number of coils, spring free length, and the F 1 minimum working force are designed, so that the spring hook height conforms the selected hook type. The strength and geometric conditions also must be fulfilled.
Formula for designing a spring with a specified wire diameter.
where the τ 8 = 0.9 τ A value is used for the value of torsion stress for the spring material, in the spring fully loaded state.
If no suitable combination of spring dimensions can be designed for this wire diameter, geometric investigations proceed on all suitable spring wire diameters. They are tested, beginning with the smallest and working to the largest, for coil numbers that result in the spring hook height that conforms with the conditions. The design is finished with the selected values, irrespective of other suitable spring wire diameters, and the spring is designed with the least wire diameter and the least number of coils.
Spring Check Calculation
Calculates corresponding values of assembly dimensions and working deflection for the specified load, material, and spring dimensions.
First, the input values for the calculation are checked. Then the assembly dimensions are calculated using the following formulas.
Length of preloaded spring
Length of fully loaded spring
where:
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L 0 |
length of free spring [mm] |
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F 1 |
working force in minimum loaded spring [mm] |
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D |
mean spring diameter [mm] |
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n |
number of active coils [-] |
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G |
modulus of elasticity of spring material [MPa] |
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d |
wire diameter [mm] |
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F 8 |
working force in fully loaded spring [MPa] |
Working deflection
H = L 1 - L 8 [mm]
Calculation of Working Forces
Calculates corresponding forces produced by springs in their working states for the specified material, assembly dimensions, and spring dimensions. First the input data is checked and calculated, and then the working forces are calculated using to the following formulas.
Minimum working force
Maximum working force
Calculation of spring output parameters
Common for all types of spring calculation, and calculated in the following order.
Hook height factor
Spring constant
Length of coiled part
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Spring without initial tension |
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L z = t n + d [mm] |
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Spring with initial tension |
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L z = 1.03 (n + 1) d [mm] |
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Pre loaded spring deflection
s 1 = L 1 - L 0 [mm]
Total spring deflection
s 8 = L 8 - L 0 [mm]
Torsional stress of spring material in the preloaded state
Torsional stress of spring material in the fully loaded stress
Spring limit force
Deflection in limit state
where:
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k |
spring constant [N/mm] |
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F 9 |
working force of spring loaded at limit [N] |
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F 0 |
spring initial tension [N] |
Limit spring length
L 9 = L 0 + s 9 [mm]
Spring deformation energy
Developed wire length
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l = 3.2 D n + l 0 [mm] |
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Where the developed hook length l 0 : |
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for half hook |
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l 0 = π D + 4 o - 2 D - 2 d [mm] |
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for full loop |
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l 0 = 2 (π D - 2 d) [mm] |
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for full loop on side |
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l 0 = 2 (π D - 2 d) [mm] |
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for full loop inside |
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l 0 = 2 (π D - d) [mm] |
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for raised hook |
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l 0 = π D + 2 o - D + 3 d [mm] |
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for double twisted full loop |
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l 0 = 4 π D [mm] |
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for double twisted full loop on side |
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l 0 = 4 π D [mm] |
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for non-specified hook type |
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l 0 = 0 [mm] |
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Spring mass
Natural frequency of spring surge
Check of spring load
τ 8 ≤ u s τ A
Overview of used variables:
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d |
wire diameter [mm] |
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k |
spring constant [N/mm] |
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D |
mean spring diameter [mm] |
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D 1 |
spring outside diameter [mm] |
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D 2 |
spring inside diameter [mm] |
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F |
general force exerted by the spring [N] |
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G |
shear modulus of elasticity of spring material [MPa] |
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H |
working deflection [mm] |
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c |
spring index [-] |
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K w |
Wahl correction factor [-] |
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l |
developed wire length [mm] |
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L |
spring length in general [mm] |
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L Z |
length of coiled spring part [mm] |
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m |
spring mass [N] |
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n |
number of active coils [-] |
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o |
spring hook height [mm] |
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t |
pitch of active coils in free state [mm] |
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s |
spring deflection (elongation) in general [mm] |
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u s |
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| ρ |
density of spring material [N/mm 3 ] |
| τ |
torsional stress is force per unit area. of spring material in general [MPa] |
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τ A |
allowable torsion stress of spring material [MPa] |