Springs calculation

Table of Content:

• Control and syntax
• Information on the project
• Theory - Fundamentals
  - Compression springs
  - Conical springs
  - Belleville springs
  - Tension springs
  - Leaf springs
  - Torsion bar springs
  - Spiral springs
  - Torsion springs
• Process of calculation
• Selection of material, operational parameters
• Compression springs of round wires
• Compression springs of rectangular wires
• Conical springs of round wires
• Conical springs of rectangular wires
• Belleville springs
• Tension springs of round wires
• Tension springs of rectangular wires
• Spiral springs
• Torsion springs of round wires
• Torsion springs of rectangular wires
• Torsion bar springs with round section
• Torsion bar springs with rectangular section
• Leaf springs with constant profile
• Leaf springs with parabolic profile
• Laminated leaf springs
• Fatigue loading - check of loading capacity
• Setting, change the language
• Workbook modifications

Springs calculation

The calculation is to be used for geometrical and strength design of metal springs of various types and designs, subjected to static or cyclic loads. The program performs the following tasks:

1. Geometrical design and calculation of working cycle parameters for metal springs of the following types and designs:

   • Helical cylindrical compression springs of round wires and bars
   • Helical cylindrical compression springs of rectangular wires and bars
   • Helical conical compression springs of round wires and bars
   • Helical conical compression springs of rectangular wires and bars
   • Belleville springs
   • Helical cylindrical tension springs of round wires and bars
   • Helical cylindrical tension springs of rectangular wires and bars
   • Spiral springs
   • Helical cylindrical torsion springs made of round wires a bars
   • Helical cylindrical torsion springs made of rectangular wires and bars
   • Torsion bar springs with round section
   • Torsion bar springs with rectangular section
   • Leaf springs with constant profile
   • Leaf springs with parabolic profile
   • Laminated leaf springs

2. Automatic proposal (finding) of a spring with suitable dimensions.

3. Static and dynamic strength check.

4. The application includes a table of commonly used spring materials according to EN, ASTM/SAE, DIN, BS, JIS, UNI, SIS, CSN and others.

The calculation is based on data, procedures, algorithms and data from specialized literature and standards EN 13906, DIN 2088, DIN 2089, DIN 2090, DIN 2091, DIN 2092, DIN 2093, DIN 2095, DIN 2096, DIN 2097.

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Control, structure and syntax of calculations.

Information on the syntax and control of the calculation can be found in the document "Control, structure and syntax of calculations".

Information on the project.

Information on the purpose, use and control of the paragraph "Information on the project" can be found in the document  "Information on the project".

Theory - Fundamentals.

Springs are constructional elements designed to retain and accumulate mechanical energy, working on the principle of flexible deformation of material. Springs belong to the most loaded machine components and are usually used as:

• energy absorbers for drives and reciprocating devices
• interceptors of static and dynamic forces
• elements to create force joints
• shock absorbers in anti-vibration protection
• devices for controlling and measuring of forces

Spring function is evaluated according to the course and extent of its deformation depending on its load.

Based on the deformation pattern, springs can be divided into the following three types:

1. springs with linear characteristics
2. springs with degressive characteristics
3. springs with progressive characteristics

The W area under the spring characteristic curve represents the deformation work (energy) of a spring performed by the spring during its loading. Deformation energy of springs subjected to compression, tension or bending is specified by the formula:

for springs subjected to torsion:

The basic quantity specifying the spring functionality is its stiffness (spring constant). Spring constant k specifies the intensity of load (force or torque) which causes unit deformation (shift or turning) of the spring.

The spring with linear characteristics have invariable spring constant; other springs have variable spring constant.

Springs are mounted with initial stress, i.e. in the state when the spring is subjected to the minimum working load. In view of spring function, there are four basic states of springs:

The above-mentioned indexes are used in the calculation to specify individual parameters of the spring related to the given state of the spring.

The difference between the spring deformations in full load condition and initial stress condition is called the spring working stroke H, a_H.

As regards the strength check and the service life, there are the following two types of metal spring loads:

1. Static loading.
   Springs loaded statically or with lower variability, i.e. with cyclical changes of loading, with the requirement of a service life lower than 10⁵ working cycles.
2. Fatigue loading.
   Springs exposed to oscillating (dynamic) loading, i.e. with cyclical changes of loading, with the requirement of a service life from 10⁵ working cycles up.

Metal springs can be divided into groups according to many aspects. Division according to load type and structural design of a spring can be considered as basic. The most common spring types are described in detail as follows:

• Springs for axial forces load (compression/tension)
  - Helical (coil) springs
  - Belleville springs and washer springs
  - Ring (annular) springs
  - Constant force springs
• Springs for transversal forces load (flexion)
  - Leaf springs
  - Curved springs
• Springs for torque load
  - Torsion bar springs
  - Spiral springs
  - Helical (coil) springs

Helical cylindrical compression springs

Springs of cylindrical shape made of helically coiled wires, with constant clearance between the active coils, able to absorb external counter-acting forces applied against each other in their axis. Springs with wire diameter up to approx. 16 mm are usually cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with a diameter of the over 10 mm. Compression springs are usually made of wires and rods of round section. Springs of rectangular wire are most often used in applications where low constructional height of the spring (springs with b>h) is required together with relatively high load.

Specific properties

• suitable for low and medium load forces
• linear working characteristics
• relatively low spring constant
• easy mounting and dismantling
• low production costs

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]

Curvature correction factor
The coil bending causes additional bending stresses in coil springs. Therefore the calculation uses the correction coefficient to correct the tension.
For springs of round section wire, the correction coefficient is determined with the given spring coiling ratio by several empirically defined formulas (Wahl, Bergsträsserr, Göhner, ...). This calculation uses the following relation:

For springs of rectangular section wire, the correction coefficient is determined for the given spring index and b/h ratio from appropriate nomograms. In this calculation the correction coefficient is already included in the shape coefficient y.

Recommended spring dimensions

Design of spring ends

In case of compression springs, several various designs of spring ends are used. These differ in numbers of ends and machined coils and designs of supporting surfaces of the springs.

• End coils are edge coils of the spring, co-axial with the active coils, whose angle pitch does not change during functional deformation of the spring. End coils create a supporting surface for the spring and with compression springs, one end coil is usually used at both ends of the spring.
• Ground coils are edge coils of the spring, machined to a flat surface perpendicular to the spring axis. Usually machined from three-fourths of half of the end coil up to its free end. Machined coils are commonly used only with springs with diameters of wires d > 1 mm.

The most common types of spring end designs

1. Open ends not ground: the edge coil is not bent to the next one, the supporting surface is unmachined
2. Open ends ground: the edge coil is not bent to the next one, the supporting surface is machined to a flat end perpendicular to the spring axis
3. Closed ends not ground: the edge coil is bent to the next one (it usually adjoins its free end), the supporting surface is unmachined
4. Closed ends ground: the edge coil is bent to the next one, the supporting surface of the spring is machined

Check of buckling

In case of compression springs, it is always necessary to check its protection against side deflection. The check is performed by comparison of the maximum working deformation of the spring with the permitted deformation. The value of the permitted deformation is determined empirically for the given slenderness ratio of the spring L₀/D and the type of seating of the spring. Generally, the risk of possible side deflection increases with an increasing value of the slenderness ratio and increasing value of the working compression of the spring. The manner of seating of the spring has a significant effect on its possible side deflection.

1. Fixed - free ends
2. Pinned - pinned ends
3. Clamped - clamped ends with lateral restraint
4. Clamped - pinned ends
5. Clamped - clamped ends without lateral restraint

A spring which cannot be designed as secured against side deflection is usually installed on a pin or inside a sleeve. If there is a danger of damage of the spring due to friction, the spring can be divided into several shorter springs arranged in series.

Curves of permitted deformation according to the type of seating of the spring

Helical conical compression springs

Springs of conical shape made of helically coiled wires, with constant clearance between the active coils, able to absorb external counter-acting forces applied against each other in their axis. Springs with wire diameter up to approx. 16 mm are usually cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with a diameter of the over 10 mm. Conical springs are usually used if the spring constant is to rise together with its progressing compression.

Specific properties

• suitable for low and medium load forces
• nonlinear (progressive) working characteristics
• relatively low spring constant
• easy mounting and dismantling
• low production costs

Basic relations for spring calculation

With increasing compression of the conical spring, its active coils are brought into contact with adjacent coils gradually (first the coils with the largest diameter). These coils then do not participate in further compression of the spring which results in gradual increase in the spring constant. Working characteristics can therefore be divided into two areas:

1. Working area with linear characteristics (invariable spring constant) - F<F_C
2. Working area with progressive characteristics – after the contact of the first (largest) active coil - F>F_C

The limit force F_C depends on the pitch between the coils p, i.e. also on the selected size of free spring length L₀. The limit force F_C increases together with increasing spring length and the working area with linear spring constant rises.

Springs of round wire	Springs of rectangular wire
I. Working area with linear characteristics F≤FC
II. Working area with progressive characteristics F>FC

where:
c_min ... min. spring index (c_min=D_min/d; c_min=D_min/b) [-]
c_max ... max. spring index (c_max=D_max/d; c_max=D_max/b) [-]
b ... wire width [mm, in]
d ... wire diameter [mm, in]
d_x .. shift of coils [mm, in]
D_min .. min. mean spring diameter [mm, in]
D_max .. max. 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 rate [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]

Curvature correction factor
The coil bending causes additional bending stresses in coil springs. Therefore the calculation uses the correction coefficient to correct the tension.
For springs of round section wire, the correction coefficient is determined with the given spring coiling ratio by several empirically defined formulas (Wahl, Bergsträsserr, Göhner, ...). This calculation uses the following relation:

For springs of rectangular section wire, the correction coefficient is determined for the given spring index and b/h ratio from appropriate nomograms. In this calculation the correction coefficient is already included in the shape coefficient y.

Recommended spring dimensions

Design of spring ends

In case of compression springs, several various designs of spring ends are used. These differ in numbers of ends and machined coils and designs of supporting surfaces of the springs.

• End coils are edge coils of the spring, co-axial with the active coils, whose angle pitch does not change during functional deformation of the spring. End coils create a supporting surface for the spring and with compression springs, one end coil is usually used at both ends of the spring.
• Ground coils are edge coils of the spring, machined to a flat surface perpendicular to the spring axis. Usually machined from three-fourths of half of the end coil up to its free end. Machined coils are commonly used only with springs with diameters of wires d > 1 mm.

The most common types of spring end designs

1. Open ends not ground: the edge coil is not bent to the next one, the supporting surface is unmachined
2. Open ends ground: the edge coil is not bent to the next one, the supporting surface is machined to a flat end perpendicular to the spring axis
3. Closed ends not ground: the edge coil is bent to the next one (it usually adjoins its free end), the supporting surface is unmachined
4. Closed ends ground: the edge coil is bent to the next one, the supporting surface of the spring is machined

Belleville springs

Annular rings of hollow truncated cone, able to absorb external axial forces counter-acting against each other. The spring section is usually rectangular. Springs of larger sizes (t > 6 mm) are sometimes made with machined contact flats.

Belleville springs are designed for higher loads with low deformations. They are used individually or in sets. When using springs in a set it is necessary to take account of friction effects. Friction in the set accounts for 3 – 5% of loading per each layer. Working load must then be increased by this force.

Stress occurring in the Belleville spring is rather complex. Maximum stress (compressive) develops in the inner top edge. Tensile stress occurs on the bottom outer edge. Maximum compressive stress serves for strength check of springs subjected to static load. In the springs subjected to cyclic (fatigue) load the pattern of tensile stresses is checked.

Note: This calculation is designed for the Belleville springs without machined contact surfaces. In addition, the calculation does not take friction effects into consideration.

Specific properties

• suitable for large loading forces
• nonlinear (degressive) working characteristics
• high spring constant (stiffness)
• low space requirements
• easy mounting and dismantling
• low production costs

Working characteristics

The shape of the Belleville spring characteristic curve is strongly affected by the relative height h₀/t. For small values of the relative height the spring has nearly linear working characteristics; with rising ratio the characteristics are sharply degressive.

Design of a set

In the case of the Belleville springs there are three kinds of springs arrangement in the set.

1. Parallel arrangement: the springs are set parallel to each other, the resulting spring constant is higher than in a single spring
2. Serial arrangement: the springs are arranged against each other, the resulting constant of the set is lower than in a single spring
3. Combined arrangement

Basic relations for spring calculation

Single spring

Springs set

where:
D_e .. outside diameter [mm, in]
D_i .. inside diameter [mm, in]
E ... modulus of elasticity in tension [MPa, psi]
F ... spring force [N, lb]
F_S ... force of fully compressed spring [mm, in]
F_T ... total force of set [mm, in]
h ... disc height [mm, in]
h₀ ... inside height of disc (h₀=h-t) [mm, in]
i ... no. of sets (disc) in series in a stack [-]
k ... spring rate [N/mm, lb/in]
k_T ... total stiffness of set [N/mm, lb/in]
K₁, K₂, K₃ .. shape coefficient [-]
L₀ ... free spring length [mm, in]
L_S ... solid length [mm, in]
n ... no. of parallel discs in a set [-]
s ... spring deflection [mm, in]
s_T ... total deflection o set [mm, in]
t ... material thickness [mm, in]
d ... diameter ratio (d=D_e/D_i) [-]
m ... Poisson's ratio [-]
s_OM, s_I, s_II, s_III, s_IV ... material stress in the given point of the spring [MPa, psi]

Recommended spring dimensions

Friction

Friction has a significant effect on the function of the Belleville spring. Loading of the spring develops friction on the contact surface (edges) of the spring. In the case of parallel arrangement of the springs there is also surface friction between the discs. The effects of friction result in an increase of force during the loading and decrease of force during relieving of the spring.

Effects of friction on the loading of the spring

The amount of friction depends on many factors (spring design, material, surface treatment, number of springs in the set, lubricant type, etc.). Its impact on the spring loading cannot be theoretically exactly determined. The following formula is used to determine the approximate corrected force of the spring:

where:
m_M .. coefficient of surface friction [-]
m_R .. coefficient of edge friction[-]
- .... on loading
+ ... on unloading

Approximate values of friction coefficients

Springs with machined contact flats

Springs of larger sizes (t > 6 mm) are sometimes made with machined contact flats. Formulas for the calculation of these springs are slightly different and they can be found e.g. in DIN 2092.

Helical cylindrical tension springs

Springs of cylindrical shape made of helically coiled wires, with constant clearance between the active coils, able to absorb external axial forces counter-acting from each other. Springs with wire diameter up to approx. 16 mm are usually cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with a diameter of the over 10 mm. Tension springs are usually made of wires and rods of round section. Springs made of rectangular wire are used very rarely.

With regards to the considerable effects of the shape and design of fixing eyes on reduction of the spring's service life and impossibility of perfect shot peening of the spring, it is not advisable to use tension springs exposed to fatigue loading. If it is necessary to use a tension spring with fatigue loading, it is advisable to avoid use of fixing eyes and choose another type of fixing of the spring.

Specific properties

• suitable for low and medium load forces
• less suitable for cyclic (fatigue) load
• linear working characteristics
• relatively low spring constant
• easy mounting and dismantling
• low production costs

Spring design
Tension springs are used in two basic designs:

1. Spring with prestressing.
   Cold formed tension springs are preferably produced with prestressing, thus with close-coiled active coils. The spring prestressing has considerable effects on increase in the loading capacity of the spring. For deformation of the spring to the desired length, it is necessary to use a higher loading than with springs without prestressing. Prestressing appears in coils of the spring in the course of coiling of the spring wire, and its size depends on the used material, spring index and the manner of coiling
2. Spring without inner prestressing.
   If necessary due to technical reasons, it is possible to use loose-coiled tension springs without prestressing, with gaps between the active coils. The coil pitch of a free spring is usually in the range 0.2*D < p < 0.4*D.

Note: Hot formed springs and the springs of rectangular wire are always manufactured without initial tension.

Basic relations for spring calculation

Springs of round wire	Springs of rectangular wire
A. Spring with prestressing
B. Spring without inner prestressing

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]
F₀ ... initial tension [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_H ... height of spring hook [mm, in]
L_K ... length of active spring section [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]
t₀ ... initial stress [MPa, psi]
 

Curvature correction factor
The coil bending causes additional bending stresses in coil springs. Therefore the calculation uses the correction coefficient to correct the tension.
For springs of round section wire, the correction coefficient is determined with the given spring coiling ratio by several empirically defined formulas (Wahl, Bergsträsserr, Göhner, ...). This calculation uses the following relation:

For springs of rectangular section wire, the correction coefficient is determined for the given spring index and b/h ratio from appropriate nomograms. In this calculation the correction coefficient is already included in the shape coefficient y.

Spring initial stress
Initial stress arises in the spring coils during winding of the spring and its magnitude is dependent on the spring material, spring index and the way of winding. The usual values of the initial stress are within the range:

Higher values are technically difficult to achieve, the lower values are very difficult to measure with sufficient accuracy. To determine the spring initial stress for springs wound on a winding bench, the DIN 2089 standard specifies the following formula:

Initial tension is given by the formula:

Recommended spring dimensions

Design of spring ends

Tension springs are used in many different designs. The most common spring ends can be found in the following picture. The type of design of the spring ends depends on the desired method of fixing the spring, its dimensions and the amount of loading.

Design of spring ends

A ... Half loop
B ... Full loop
C ... Full loop on side

D ... Double twisted full loop
E ... Double twisted full loop on side
F ... Inside full loop

G ... Raised hook
H ... Raised hook on side
L ... Coned end with swivel eye

I ... Small eye
J ... Small eye on side
K ... Inclined full loop

M ... Coned end with swivel bolt
N ... Screwed
O ... Screwed in shackle

Tension springs are usually fixed using fixing eyes of several types (A .. J) with different heights of the eyes and differing properties. Fixing eyes are the best solution in the technological aspect, however, this brings certain problems in view of loading capacity of the spring. Loading of the spring creates a concentration of stress on the fixing eyes and this may be substantially higher than the calculated stress in the spring coils. In view of bending stress appearing in the fixing eye, small eyes (type I, J) or double eyes (type D, E) are the best solution. In view of concentration of stress in torsion at the point of transition of the coil into the loop, the full loops on the side (type C,E,I) are the best solution. For individual designs of fixing eyes, the following values of eye height are prescribed:

Hot formed springs, rectangular wire springs and cyclically loaded springs are usually used without spring hooks (M..O. design). With designs without fixing eyes the spring is fixed using end coils whose pitch does not change during functional deformation of the spring.

Check of spring hook stress

Loading of the spring creates a concentration of stress in the fixing eyes and this may be substantially higher than the calculated stress in the spring coils. It is therefore recommended to check such springs also in view of loading of the fixing eyes.

The amounts of possible concentrations depend on the type, design and dimensions of the eye and it is very difficult to calculate them theoretically. Despite this, at least approximate calculations are used to provide some orientation information on any possible exceeding of strength limits of the chosen material of the spring. Two basic strength checks are performed with regards to the design of the fixing eye:

Check of bending stress in spring hook

The amount of the bending stress which appears in the bend of the eye depends on the radius of the spring hook r_b. The amount of stress increases with an increasing radius and vice versa. The following formula can be used to determine bending stress:

Check of stress in transition bend

In the case of tension springs, the highest stress concentrations appear in points of transitions of coils to spring hook. The size of these stresses depends on the transition bend radius r_s. Generally speaking, the size of stress in the transition bend decreases with an increasing radius of the bend and vice versa. The following formula can be used to determine peak stress:

Leaf springs

Springs based on the principle of long slander beams of rectangular section subjected to bending. They are used as cantilever springs (fixed at one end), or as simple beams (fixed at both ends). The leaf springs can be used either independently or in sets (laminated leaf springs).

Specific properties

Single springs

• suitable for low and medium load forces
• linear working characteristics
• relatively low spring constant
• considerable length requirements, otherwise minimum space needed
• low production costs

Laminated leaf springs

• suitable for higher loading forces
• theoretically linear working characteristics (friction between the leaves causes hysteretic pattern of the working curve)
• relatively higher spring constant (stiffness)
• high space requirements
• demanding maintenance (lubrication and cleanness)

Spring design
Leaf springs are used in many different designs and shapes. They can be divided into three groups for calculation purposes:

1. Single springs with constant profile: usually springs in rectangular, triangular or trapezoidal shapes
2. Single springs with parabolic profile: usually of rectangular shape, sometimes springs thicker in the middle and at the end of the leaf are used
3. Laminated leaf springs: manufactured in many designs. They use leaves with constant and parabolic profile, in rectangular, triangular or trapezoidal shapes. Precise calculation of the laminated springs is very complex. Taking simplified conditions into account, this calculation deals with two basic types of laminated springs:
   - springs with constant section leaves of triangular shape
   - springs with constant section leaves of rectangular shape

Basic relations for spring calculation

A. Single springs with constant profile

B. Single springs with parabolic profile

C. Laminated leaf springs

where:
b ... width of spring leaf [mm, in]
b' ... leaf width at end of spring [mm, in]
E ... modulus of elasticity in tension [MPa, psi]
F ... loading of spring [N, lb]
k ... spring constant [N/mm, lb/in]
L ... functional spring length [mm, in]
L' ... length of leaf with constant thickness [mm, in]
n ... total number of spring leaf [-]
n' ... number of extra full-length leaves [-]
s ... spring deflection [mm, in]
t ... thickness of spring leaf [mm, in]
t' ... leaf thickness at end of spring [mm, in]
y ... shape coefficient [-]
s ... bending stress of the spring material [MPa, psi]

Extra leaves
Spring leaves of full length, rectangular shape with constant profile. These leaves are added to the spring for two reasons:

• to increase the spring stiffness and load capacity
• they are often ended with hooks to fix the spring

Torsion bar springs

Springs based on the principle of long slender bars of circular or rectangular section subjected to torsion. The ends of bars with circular section are mostly fixed by means of grooving. Sometimes one end is square-shaped in order to facilitate attachment. Torsion bar springs must be secured against bending stress.

Specific properties

• suitable for higher loading torques
• linear working characteristics
• high spring constant
• considerable length requirements, otherwise minimum space needed
• low production costs

Basic relations for spring calculation

Bar with round section	Bar with rectangular section

where:
b ... bar width [mm, in]
d ... bar diameter [mm, in]
M ... loading of spring [Nmm, lb in]
G ... modulus of elasticity in shear [MPa, psi]
k ... torque spring rate [Nmm/°, lb in/°]
L ... functional spring length [mm, in]
t ... bar thickness [mm, in]
a ... angular deflection [°]
b,g ... shape coefficient [-]
t ... torsional stress of the spring material [MPa, psi]

Shape coefficients
These coefficients take stress distribution in the bar section b/t into consideration. Their value can be found in the table:

Spiral springs

The spring made of a strip with rectangular section wound into the shape of Archimedes spiral, with constant spacing between its active coils, loaded with torque in the direction of the winding.

Note: This calculation is designed for spiral springs with fixed ends of the spring.

Specific properties

• suitable for low loading torques
• linear working characteristics
• low spring constant
• low production costs

Basic relations for spring calculation

where:
a ... space between coils [mm, in]
b ... width of spring strip [mm, in]
M ... loading of spring [Nmm, lb in]
E ... modulus of elasticity in tension [MPa, psi]
k ... torque spring rate [Nmm/°, lb in/°]
K_b ... curvature correction factor [-]
L ... functional spring length [mm, in]
n ... number of active coils [-]
t ... thickness of spring strip [mm, in]
R_e ... outer radius [mm, in]
R_i ... inner radius [mm, in]
a ... angular deflection [°]
d₀ ... leg angle of free spring [°]
s ... bending stress of the spring material [MPa, psi]

Curvature correction factor
Correction coefficient represents the spring additional stress resulting from its curvature. Its value can be found in the graph:

Recommended spring dimensions

Helical cylindrical torsion springs

Springs of cylindrical shape made of helically coiled wires, with constant spacing between the active coils, able to absorb external forces applied in the planes perpendicular to the winding axis through a torque in the direction of winding or unwinding. Springs with wire diameter up to approx. 16 mm are usually cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with a diameter of the over 10 mm.

Note: This calculation is designed for the torsion springs loaded in the direction of coil winding, with fixed arms. The calculation does not take into account the effects of supporting of the spring against the inner or outer guiding part, nor the effects of friction that appears with it. The effects of possible friction between the coils of the spring are also not considered.

Specific properties

• suitable for low and medium loading torques
• linear working characteristics
• relatively low spring constant
• low production costs

Spring design

Torsion springs are produced in two basic designs: tight-coiled and loose-coiled (with clearance between the coils). If the springs are exposed to a static loading, the tight-coiled springs are recommended. However, if friction appears between the coils of these springs while they are working, this may cause the service life of the springs to decrease. In addition to this, the close distance of the coils prevents perfect shot peening of the spring. Therefore loose-coiled springs are suitable for use with fatigue loading. The pitch of the spring is usually in the range of 0.3*D < p < 0.5*D.

Note: The length of close wound spring loaded in the direction of coil winding grows during its loading. Hot formed springs shall be usually produced with a clearance between the coils.

Basic relations for spring calculation

Springs of round wire	Springs of rectangular wire

where:
c ... spring index (c=D/d; c=D/t) [-]
b ... wire width [mm, in]
d ... wire diameter [mm, in]
D ... mean spring diameter [mm, in]
M ... loading of spring [Nmm, lb in]
E ... modulus of elasticity in tension [MPa, psi]
k ... torque spring rate [Nmm/°, lb in/°]
K_b ... curvature correction factor [-]
L_K ... Length of coiled section [mm, in]
n ... number of active coils [-]
p ... pitch between coils [mm, in]
t ... wire thickness [mm, in]
a ... angular deflection [°]
d₀ ... leg angle of free spring [°]
s ... bending stress of the spring material [MPa, psi]

Curvature correction factor
Correction coefficient represents the spring additional stress resulting from its curvature.

Functional dimensions of the spring
Functional deformation (shift of the arm) of the torsional spring leads to the change of its dimensions. The diameter of springs loaded in the direction of coil winding decreases during its loading:

In addition, the length of close wound spring grows:

Recommended spring dimensions

Design of spring ends

With regards to the possible occurrence of stress concentrations, the shape of the legs of the torsion spring should be as simple as possible. The basic types of legs used with torsion springs are given in the illustration. The option of the leg design depends on the desired method of setting the spring, its dimensions and desired distance of the loading application point from the spring axis, while the supporting and working legs of the spring may be different.

Basic types of legs

1. Straight tangential leg
2. Straight axial leg
3. Radial external leg
4. Radial internal leg

Method of fixing the leg

If both legs of the torsion spring are fixed, the working angle is given only by twisting the spring coils. If the leg is supported freely (loaded) at one point, the leg only bends when the spring is loaded. This causes an increase in the actual functional angular deflection of the leg. The amount of bending in the leg increases with increased distance of the application point of the force from the coils of the spring (length of the leg). Fixed mounting of the legs increases the accuracy of the calculation and improves the functions of the spring.

Actual (adjusted) angular deflection of the spring with a free leaning arm will then be for:
- radial arms

- tangential arms

Check of the spring arm stress

The springs with bended arms are subjected to concentrations of tension at the bends which can be much higher than the calculated stress in the spring coils. The amount of these concentrations depends on the leg bending radius. The smaller the bending radius, the higher the values of stress peaks in the spring legs. The following formula can be used to determine approximate peak tension:

Process of calculation.

The way of design procedure used in this book allows defining dimensions of a spring with a certain degree of looseness. Therefore, in the "Spring design" paragraph for each of the input parameters, their exact values corresponding to the other parameters of the spring are calculated in real time. These values are displayed in green fields situated to the right of the input cells.

Typical calculation/design of the spring consists of the following steps:

1. Set up the desired calculation units (SI/Imperial). [1.3]
2. Select the corresponding standard [1.1] and the type of material [1.2].
3. Select the suitable material of the spring [1.6] according to the recommended areas of use [1.7-1.10].
4. Define the operation and production parameters of the spring in paragraph [1.21].
5. Set the required safety level [1.27].
6. Select the relevant chapter with required spring type. The actual design of the spring dimensions is provided in the "Spring design" paragraph.
7. Enter the required parameters of the working cycle (working load and spring stroke) into the first three input fields.
8. Set preliminary dimensions of the spring in the other input fields, alternatively use one of the design (optimization) functions of the calculation.
9. Based on the calculated recommended values (green cells) adjust the spring dimensions so that the calculated working load and spring stroke best correspond to the required values.
10.  In the "Check data" paragraph of the designed spring check the calculated level of safety (check for strength of spring subjected to static loading).
11. Check the other values of the spring in the paragraphs "Design values" and "Parameters of working cycle".
12. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.
13. Save the book with the suitable solution under a new name.

Selection of material, spring operational and production parameters. [1]

The purpose of this paragraph is to select suitable material of the spring. It also defines the basic operational and production values of the spring.

1.1 Material standard.

Select the required national standard from the list to determine the spring material.

Recommendation: Most European countries are currently substituting or have already substituted the local standards (DIN, BS, UNI, UNE, ...) in the area of spring materials with corresponding equivalents of standards EN. Therefore we recommend using only the appropriate European norms EN.

1.2 Material type.

According to the spring design select from the list the corresponding material type (intermediate product) from which the spring will be produced.

1.3 Calculation units.

Select the desired calculation units in the selection list. When switching over the units, all values will be recalculated immediately.

1.4 Graph type.

In the list select the required graph type which you want to be displayed in the spring calculation.

1.5 Spring material.

This paragraph can be used for selection of the spring material.

Choose the spring material from the list [1.6]. The first five rows of the list is reserved for materials defined by the user. Information and settings of proper materials can be found in the document "Workbook (calculation) modifications". Other rows of the list include a selection of materials for the actually specified standard [1.1] and material type [1.2].

Rows [1.7 - 1.10] includes information on the recommended use of the chosen material. The spring material should be designed with regards to the method of loading the spring and the operational conditions. If you must use a material less suitable, this fact should be reflected in the increased level of safety in the design of the spring (see paragraph [1.21]) . Properties of the chosen material, described in rows [1.7, 1.9] are evaluated in five degrees (excellent, very good, good, poor, insufficient), and the relative strength is described in row [1.8] in three degrees (high, medium, low).

Note: In case the checkbox to the right of the selection list is enabled, the necessary parameters for the chosen material are determined automatically. Otherwise, fill in the material characteristics manually.

1.12, 1.14 Modulus of elasticity.

The value specified at the basic temperature of 20°C (68°F).

1.18 Ultimate tensile strength.

Enter the ultimate tensile strength of the selected material. When selecting the check box in line [1.6] the minimum value of the ultimate strength defined for the selected material will be automatically set here.

Warning: The ultimate tensile strength of the cold drawn spring wires of some materials is considerably dependent on the wire diameter. Material strength increases with decreasing diameter of the wire. In automatic setting of material values the calculation uses minimum ultimate stress values of the selected material for wires of the largest diameters (approx. 15 mm, 5/8 in). The spring that you have designed will therefore most probably be oversized. That is why in the case of final calculations we recommend you to set this value manually depending on the wire diameter used for the spring being designed. Approximate values of the ultimate strength depending on the wire diameter can be found in the graphs:

Ultimate tensile strength - ASTM

Ultimate tensile strength - EN

1.21 Operational parameters, safety.

This paragraph is designed to set the operational and production parameters of the spring and to them related safety coefficients.

Set the corresponding operational conditions in the selection lists. The input field for entering the appropriate safety coefficient is situated to the right of each list. This coefficient expresses the influence of the given parameter on possible decrease in the load capacity of the spring.

Note: If the check box to the right of the input field is selected, the safety coefficient value will be designed by the program. In this automatic design the calculation also incorporates in the coefficient value the suitability of the selected material and the effects (interactions), if any, of the other values of the spring.

1.22 Working temperature.

Temperature of the working environment affects the spring relaxation, i.e. decrease in the force from the spring with its deformation to a constant length, depending on time. It is advisable to take this fact into account when designing the spring, and increase the level of safety during strength checks of the spring in case of temperatures over 80°C (180°F). It is necessary to respect the working temperature also with selection of the spring material.

1.23 Method of loading.

As regards the strength check and the service life, there are the following two types of metal spring loads:

1. Static loading.
   Springs loaded statically or with lower variability, i.e. with cyclical changes of loading, with the requirement of a service life lower than 10⁵ working cycles.
2. Fatigue loading.
   Springs exposed to oscillating (dynamic) loading, i.e. with cyclical changes of loading, with the requirement of a service life from 10⁵ working cycles up.

Note: Springs subjected to cyclic load must always be checked for potential fatigue damage – see chapter [17]. If the spring is to satisfy the fatigue check, it is usually necessary to oversize it considerably as compared to its static strength.

1.24 Operational mode of loading.

Choose the loading mode which best meets your entered data.

1. Light service.
   Continuous loading without shocks, with a course according to a sinusoid, loading with small deformations or low frequency, little or rarely loaded springs with a service life up to 1000 cycles. For example, springs used in measuring instruments, safety and relief devices, etc.
2. Medium duty service.
   Continuous loading with lower or medium variations, loading with normal frequency of deformations. Commonly used springs in machine tools, machine products or electrical components.
3. Heavy duty service.
   Springs with discontinuous course of loading, loading with strong shocks, loading with high frequency of deformations or sudden deformations over longer or irregular time periods. For example, springs used in pneumatic hammers, hydraulic machines, valves, etc.

1.25 Working environment.

The service life of springs decreases significantly due to corrosion effects. Corrosion has very powerful effects particularly on springs exposed to fatigue loading. It is advisable to take this fact into account when designing the spring, and increase the level of safety during strength checks of the spring in case of a corrosion-aggressive environment. It is also necessary to consider corrosion effects with selection of the spring material.

1.26 Surface treatment of the spring.

Shot peening of the spring increases the fatigue limit by approx. 15 to 25%. In case of springs with shot peening exposed to fatigue loading, this allows users to reduce the consumption of material for production of the spring, reduce its dimensions and installation space, increase the working stroke or increase protection of the spring against fatigue breaks. Therefore, it is advisable to apply the technical requirement of shot peening to all springs exposed to oscillating loading.

Springs with galvanic coating have significantly higher corrosion resistance. On the other hand galvanic coating reduces the load capacity of the spring by about 10%.

Note 1: For technology reasons only springs with the wire diameter over 1 mm are shot peened in the case of coil springs.

Note 2: In the case of springs subjected to static loads (apart from the Belleville springs) spring shot peening does not have any significant effects on the strength calculation.

Note 3: In view of the compression stresses occurring in the Belleville springs shot peening of these springs has a negative impact, i.e. decrease in their strength. Therefore it should not be used for statically loaded springs.

1.27 Total level of safety.

It specifies the minimum permissible ratio between the limit permissible stress of the selected spring material and the actual stress of the spring at the maximum working load.

The required level of safety is used in the check calculation of strength of the statically loaded spring. The value specified here therefore actually eliminates any potential negative impact of the operational conditions on the spring load capacity decrease. Apart from the above mentioned facts the required value of safety should also incorporate some other factors (such as accuracy and reliability of input information, significance of the equipment, production quality, ...). Common springs are usually designed with the level of safety within the range of <1...2>.

Warning: Springs subjected to cyclic load must always be checked for potential fatigue damage – see chapter [17]. If the spring is to satisfy the fatigue check, it is usually necessary to oversize it considerably as compared to its static strength. The level of safety specified here should then by about twice as high as for the spring with static load.

Note: When selecting the check box the demanded safety will be determined automatically based on the entered partial coefficients of safety [1.22 - 1.26].

Helical cylindrical compression springs of round wires and bars. [2]

Springs of cylindrical shape made of helically coiled wires, with constant clearance between the active coils, able to absorb external counter-acting forces applied against each other in their axis. Springs with wire diameter up to approx. 16 mm are usually cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with a diameter of the over 10 mm.

Spring design procedure.

1. Enter the required parameters of the working cycle (working load and spring stroke). [2.2 - 2.4]
2. Set preliminary dimensions of the spring in the input fields [2.6 - 2.8], alternatively use one of the design (optimization) functions of the calculation [2.9].
3. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
4. Check the calculated value of safety [2.16] for the designed spring.
5. Select the required design of the spring ends in the list [2.18].
6. Set the corresponding length of the unloaded spring. [2.21]
7. Check the parameters of the spring working cycle. [2.27]
8. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Warning: In case of compression springs, it is always necessary to check its protection against side deflection.

Tip: Detailed information on the calculation of compression springs can be found in the theoretical section of help.

2.1 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

2.9 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation.

• The spring design for the given spring index D/d is started by moving the scroll bar.
• When pressing the appropriate button you will design the spring while keeping the values of the selected spring dimensions in paragraph [2.1].

When designing the spring the calculation is trying to optimize the dimensions so that the wire diameter is as small as possible while keeping the required safety [1.27].

Note: When designing a spring for the given spring index D/d the calculation will select the nearest suitable wire diameter out of the preferred series.

2.10 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [2.16] should not drop under the required value [1.27].

Lines [2.14, 2.15] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

2.18 Design of spring ends.

Select the required design of the spring ends from the list.

Tip: Detailed information can be found in the theoretical section of the help.

2.21 Free spring length.

Set the corresponding length of the unloaded spring.

Note: If the check box to the right of the input field is selected, the spring free length will be set automatically within the range of the recommended values.

2.27 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Helical cylindrical compression springs of rectangular wires and bars. [3]

Springs of cylindrical shape made of helically coiled wires, with constant clearance between the active coils, able to absorb external counter-acting forces applied against each other in their axis. Springs made of rectangular wire are cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with wire thickness of the over 10 mm.

Spring design procedure.

1. Enter the required parameters of the working cycle (working load and spring stroke). [3.2 - 3.4]
2. Set preliminary dimensions of the spring in the input fields [3.6 - 3.9], alternatively use one of the design (optimization) functions of the calculation [3.10].
3. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
4. Check the calculated value of safety [3.17] for the designed spring.
5. Select the required design of the spring ends in the list [3.19].
6. Set the corresponding length of the unloaded spring. [3.22]
7. Check the parameters of the spring working cycle. [3.28]
8. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Warning: In case of compression springs, it is always necessary to check its protection against side deflection.

Tip: Detailed information on the calculation of compression springs can be found in the theoretical section of help.

3.1 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

3.10 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation.

• Spring design for the given ratios D/b, b/h is started by moving one of the scroll bars.
• When pressing the appropriate button you will design the spring while keeping the values of the selected spring dimensions in paragraph [3.1].

When designing the spring the calculation is trying to optimize the dimensions so that the wire section is as small as possible while keeping the required safety [1.27].

3.11 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [3.17] should not drop under the required value [1.27].

Lines [3.15, 3.16] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

3.19 Design of spring ends.

Select the required design of the spring ends from the list.

Tip: Detailed information can be found in the theoretical section of the help.

3.22 Free spring length.

Set the corresponding length of the unloaded spring.

Note: If the check box to the right of the input field is selected, the spring free length will be set automatically within the range of the recommended values.

3.28 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Helical conical compression springs of round wires and bars. [4]

Springs of conical shape made of helically coiled wires, with constant clearance between the active coils, able to absorb external counter-acting forces applied against each other in their axis. Springs with wire diameter up to approx. 16 mm are usually cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with a diameter of the over 10 mm.

 With increasing compression of the conical spring, its active coils are brought into contact with adjacent coils gradually (first the coils with the largest diameter). These coils then do not participate in further compression of the spring which results in gradual increase in the spring constant. Working characteristics can therefore be divided into two areas:

1. Working area with linear characteristics (invariable spring constant) - F<F_C
2. Working area with progressive characteristics – after the contact of the first (largest) active coil - F>F_C

The limit force F_C depends on the selected size of free spring length L₀. The limit force F_C increases together with increasing spring length and the working area with linear spring constant rises.

Spring design procedure.

1. Enter the required parameters of the working cycle (working load and spring stroke). [4.2 - 4.4]
2. Set preliminary dimensions of the spring in the input fields [4.5 - 4.9], alternatively use one of the design (optimization) functions of the calculation [4.11].
3. Based on the preliminary design values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke correspond to the required values best.
4. Select the required design of the spring ends in the list [4.24].
5. Set the corresponding length of the unloaded spring. [4.27]
6. By pressing the [4.10] button carry out the spring calculation.
7. Check the results in paragraph [4.10]. In the case of unsatisfactory design adjust the spring dimensions [4.1, 4.27] and repeat the calculation.
8. Check the calculated value of safety [4.19] for the designed spring.
9. Check the parameters of the spring working cycle. [4.33]
10. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Warning: This calculation cannot be used for non-telescopic springs with floating (supported) ends.

Tip: Detailed information on the calculation of conical springs can be found in the theoretical section of help.

4.1 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the approximate value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

Warning: The parameters of the "pre-design" are specified for the spring working in the area with constant stiffness (spring constant) (F8<F_C). In the case of spring exposed to higher force the actual value of the working stroke will be lower than the value specified here.

Tip: The value of the limit force F_C is among others influenced by the length of unloaded spring set in line [4.27].

4.10 Calculation.

Complexity of the conical spring design does not allow calculating all the spring parameters in real time. Therefore, after each change of the input data, it is necessary to start the calculation manually by pressing the "Calculate" button.

Note: If the spring has not been calculated, some of the input parameters will have approximate values ("~" symbol) or no values have been specified.

4.11 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation. Spring design for the given ratios D_max/D_min, D_min/d is started by moving one of the scroll bars. When designing the spring the calculation is trying to optimize the dimensions so that the wire diameter is as small as possible while keeping the required safety [1.27].

Note: The calculation will select the nearest suitable wire diameter out of the preferred series.

4.12 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [4.19] should not drop under the required value [1.27].

Lines [4.18] is used to calculate theoretical value of the maximum working load at which the level of safety for the designed spring will still be met.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

4.21 Maximum loading.

This value specifies the maximum load (limit force) at which the spring will still work with constant stiffness (spring constant). The spring constant increases with growing load.

Tip: The value of the limit force F_C is among others influenced by the length of unloaded spring, set in line [4.27].

4.24 Design of spring ends.

Select the required design of the spring ends from the list.

Tip: Detailed information can be found in the theoretical section of the help.

4.27 Free spring length.

Set the corresponding length of the unloaded spring.

Free spring length considerably influences the maximum force value [4.21], and therefore also the working characteristic of the spring. The limit force increases together with increasing spring length, and the working area with the linear spring constant rises.

Note: If the check box to the right of the input field is selected, the spring free length will be set automatically within the range of the recommended values.

4.33 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Note: Working cycle parameters will be displayed only after performing the spring calculation (see [4.10]).

Helical conical compression springs of rectangular wires and bars. [5]

Springs of conical shape made of helically coiled wires, with constant clearance between the active coils, able to absorb external counter-acting forces applied against each other in their axis. Springs made of rectangular wire are cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with wire thickness of the over 10 mm.

With increasing compression of the conical spring, its active coils are brought into contact with adjacent coils gradually (first the coils with the largest diameter). These coils then do not participate in further compression of the spring which results in gradual increase in the spring constant. Working characteristics can therefore be divided into two areas:

1. Working area with linear characteristics (invariable spring constant) - F<F_C
2. Working area with progressive characteristics – after the contact of the first (largest) active coil - F>F_C

The limit force F_C depends on the selected size of free spring length L₀. The limit force F_C increases together with increasing spring length and the working area with linear spring constant rises.

Spring design procedure.

1. Enter the required parameters of the working cycle (working load and spring stroke). [5.2 - 5.4]
2. Set preliminary dimensions of the spring in the input fields [5.5 - 5.10], alternatively use one of the design (optimization) functions of the calculation [5.12].
3. Based on the preliminary design values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke correspond to the required values best.
4. Select the required design of the spring ends in the list [5.25].
5. Set the corresponding length of the unloaded spring. [5.28]
6. By pressing the [5.11] button carry out the spring calculation.
7. Check the results in paragraph [5.11]. In the case of unsatisfactory design adjust the spring dimensions [5.1, 5.28] and repeat the calculation.
8. Check the calculated value of safety [5.20] for the designed spring.
9. Check the parameters of the spring working cycle. [5.34]
10. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Warning: This calculation cannot be used for non-telescopic springs with floating (supported) ends.

Tip: Detailed information on the calculation of conical springs can be found in the theoretical section of help.

5.1 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the approximate value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

Warning: The parameters of the "pre-design" are specified for the spring working in the area with constant stiffness (spring constant) (F8<F_C). In the case of spring exposed to higher force the actual value of the working stroke will be lower than the value specified here.

Tip: The value of the limit force F_C is among others influenced by the length of unloaded spring set in line [5.28].

5.11 Calculation.

Complexity of the conical spring design does not allow calculating all the spring parameters in real time. Therefore, after each change of the input data, it is necessary to start the calculation manually by pressing the "Calculate" button.

Note: If the spring has not been calculated, some of the input parameters will have approximate values ("~" symbol) or no values have been specified.

5.12 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation. Spring design for the given ratios D_max/D_min, D_min/b, b/h is started by moving one of the scroll bars. When designing the spring the calculation is trying to optimize the dimensions so that the wire section is as small as possible while keeping the required safety [1.27].

5.13 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [5.20] should not drop under the required value [1.27].

Lines [5.19] is used to calculate theoretical value of the maximum working load at which the level of safety for the designed spring will still be met.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

5.22 Maximum loading.

This value specifies the maximum load (limit force) at which the spring will still work with constant stiffness (spring constant). The spring constant increases with growing load.

Tip: The value of the limit force F_C is among others influenced by the length of unloaded spring, set in line [5.28].

5.25 Design of spring ends.

Select the required design of the spring ends from the list.

Tip: Detailed information can be found in the theoretical section of the help.

5.28 Free spring length.

Set the corresponding length of the unloaded spring.

Free spring length considerably influences the maximum force value [5.22], and therefore also the working characteristic of the spring. The limit force increases together with increasing spring length, and the working area with the linear spring constant rises.

Note: If the check box to the right of the input field is selected, the spring free length will be set automatically within the range of the recommended values.

5.34 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Note: Working cycle parameters will be displayed only after performing the spring calculation (see [5.11]).

Belleville springs. [6]

Annular rings of hollow truncated cone, able to absorb external axial forces counter-acting against each other. The spring section is usually rectangular. Springs of larger sizes (t > 6 mm) are sometimes made with machined contact flats.

Belleville springs are designed for higher loads with low deformations. They are used individually or in sets. When using springs in a set it is necessary to take account of friction effects. Friction in the set accounts for 3 – 5% of loading per each layer. Working load must then be increased by this force.

The shape of the Belleville spring characteristic curve is strongly affected by the relative height h₀/t. For small values of the relative height the spring has nearly linear working characteristics; with rising ratio the characteristics are sharply degressive.

Spring design procedure.

1. Enter the required parameters of the working cycle (working load, spring stroke and relative spring deflection). [6.2 - 6.5]
2. Enter spring dimensions in the input fields [6.8 - 6.12], alternatively select the spring from the database [6.14] or use one of the find functions [6.15].
3. Based on the preliminary design values (green cells) select the numbers of discs in the set [6.6, 6.7], so that the calculated working load and spring stroke best correspond to the required values.
4. Set the permissible stress of the spring. [6.30]
5. By pressing the [6.13] button carry out the spring calculation.
6. Check the results in paragraph [6.13]. In the case of unsatisfactory design adjust the spring dimensions and repeat the calculation.
7. Check the calculated value of safety [6.34] for the designed spring.
8. Check the parameters of the spring working cycle. [6.35]
9. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Note: This calculation is designed for the Belleville springs without machined contact surfaces. In addition, the calculation does not take friction effects into consideration.

Tip: Detailed information on the calculation of Belleville springs can be found in the theoretical section of help.

6.1 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore, to facilitate the design, the approximate values corresponding to the other parameters of the spring are calculated in real time for some of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

6.5 Max. permissible spring deflection.

Select the permissible proportionate spring deflection in the selection list.

In the case of statically loaded springs the operating deflection of the spring should not exceed 75 – 80% of the maximum (full) deflection [6.19]. In the springs with cyclic (fatigue) loading 50% value is usually accepted.

6.6 Number of parallel discs in the set.

Discs arranged in the same direction.

Tip: Parallel arrangement of discs increases the total stiffness (spring rate) of the spring. At the same time the load capacity of the spring grows.

6.7 Number of sets (disc) in series in a stack.

Discs or disc sets arranged against each other.

Tip: The serial arrangement of discs reduces the total stiffness (spring rate) of the spring.

6.13 Calculation.

Complexity of the Belleville spring design does not allow calculating all the spring parameters in real time. Therefore, after each change of the input data, it is necessary to start the calculation manually by pressing the "Calculate" button.

Note: If the spring has not been calculated, some of the input parameters will have approximate values ("~" symbol) or no values have been specified.

6.14 Spring selection.

In the selection list you will find the database of the Belleville springs with commonly manufactured dimensions. Spring dimensions in the list are specified as "D_e x D_i x t x h".

Note: For the calculation in SI units the list specifies dimensions of the springs supplied by Schnorr GmbH (springs designated with "*" symbol correspond to DIN 2093). The calculation in the "Imperial" units uses springs made by the producers Key Bellevilles, Inc and WCL company.

Warning: After selecting the spring from the list, the program will automatically design the satisfactory values of disc numbers in a set [6.6, 6.7].

6.15 Searching for a spring.

This paragraph serves for automatic design (finding) of a Belleville spring of satisfactory dimensions.

In the selection lists set the permissible deviation from the required working stroke of the spring [6.4] and maximum permissible numbers of discs in a set. After pressing the "Find first" button the program will find the first spring from the list [6.14], which meets all the specified requirements while keeping the minimum safety. If the selected spring does not meet your expectations, use the "Find next" button to find a spring of different dimensions.

6.22 Maximum permissible loading.

Theoretically determined value of the maximum working loading at which the maximum compressive stress [6.31] of the designed spring does not exceed the permissible limit [6.30] while satisfying the permissible spring deflection at the same time [6.5].

6.29 Strength check.

Stress occurring in the Belleville spring is rather complex. Maximum stress (compressive) develops in the inner top edge. Tensile stress occurs on the bottom outer edge. Maximum compressive stress serves for strength check of springs subjected to static loading. In the springs subjected to cyclic (fatigue) loading the pattern of tensile stresses is checked.

This paragraph specifies the results of the strength check of a statically loaded spring. The check is carried out by comparing the permissible stress of the material used [6.30] with the maximum compressive stress of a fully loaded spring [6.31]. The resulting level of safety [6.34] should not drop under the recommended value [6.33].

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

6.30 Permissible compressive stress.

Enter the permissible compressive stress of the spring material.

Note: If the check box to the right of the input field is selected, the permissible stress for the selected material will be set automatically [1.6].

6.33 Recommended level of safety.

Recommended level of safety for the selected material [1.6] is estimated based on the operational conditions defined in paragraph [1.21].

6.35 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Note: Working cycle parameters will be displayed only after performing the spring calculation (see [6.13]).

Helical cylindrical tension springs of round wires and bars. [7]

Springs of cylindrical shape made of helically coiled wires, with constant clearance between the active coils, able to absorb external axial forces counter-acting from each other. Springs with wire diameter up to approx. 16 mm are usually cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with a diameter of the over10 mm.

Spring design procedure.

1. Select the suitable spring design from the list. [7.1]
2. Enter the required parameters of the working cycle (working load and spring stroke). [7.4 - 7.6]
3. Set preliminary dimensions of the spring in the input fields [7.8 - 7.10], alternatively use one of the design (optimization) functions of the calculation [7.11].
4. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
5. Check the calculated value of safety [7.18] for the designed spring.
6. In list [7.20] select the required design of the spring hook, set its height in line [7.22].
7. In lines [7.24, 7.25] set the corresponding initial stress of the spring (for springs with initial stress) or the length of unloaded spring (for the springs without initial stress).
8. Check and adjust, if needed, the spring design parameters in paragraph [7.2].
9. Check the parameters of the spring working cycle. [7.31]
10. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Recommendation: With regards to the considerable effects of the shape and design of fixing eyes on reduction of the spring's service life and impossibility of perfect shot peening of the spring, it is not advisable to use tension springs exposed to fatigue loading. If it is necessary to use a tension spring with fatigue loading, it is advisable to avoid use of fixing eyes and choose another type of fixing of the spring.

Warning: Loading of the spring creates a concentration of stress in the fixing eyes and this may be substantially higher than the calculated stress in the spring coils. It is therefore recommended to check such springs also in view of loading of the fixing eyes.

Tip: Detailed information on the calculation of tension springs can be found in the theoretical section of help.

7.1 Spring type.

Tension springs are used in two basic designs:

1. Spring with prestressing.
   Cold formed tension springs are preferably produced with prestressing, thus with close-coiled active coils. The spring prestressing has considerable effects on increase in the loading capacity of the spring. For deformation of the spring to the desired length, it is necessary to use a higher loading than with springs without prestressing. Prestressing appears in coils of the spring in the course of coiling of the spring wire, and its size depends on the used material, spring index and the manner of coiling.
2. Spring without inner prestressing.
   If necessary due to technical reasons, it is possible to use loose-coiled tension springs without prestressing, with gaps between the active coils. The coil pitch of a free spring is usually in the range 0.2*D < p < 0.4*D.

Note: Hot formed springs always manufactured without initial tension.

7.2 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

7.11 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation.

• The spring design for the given spring index D/d is started by moving the scroll bar.
• When pressing the appropriate button you will design the spring while keeping the values of the selected spring dimensions in paragraph [7.2].

When designing the spring the calculation is trying to optimize the dimensions so that the wire diameter is as small as possible while keeping the required safety [1.27].

Note: When designing a spring for the given spring index D/d the calculation will select the nearest suitable wire diameter out of the preferred series.

7.12 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [7.18] should not drop under the required value [1.27].

Lines [7.16, 7.17] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Note: The calculation used here performs a strength check of the spring only for stress in active coils, and does not take into consideration any possible stress concentrations in the fixing eye.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

7.20 Design of spring ends.

Select the required design of the spring ends from the list.

Loading of the spring creates a concentration of stress in the fixing eyes and this may be substantially higher than the calculated stress in the spring coils. This is why sometimes a different way of spring attachment is used.

Note: Hot formed springs and cyclically loaded springs are usually used without spring hooks.

Tip: Detailed information can be found in the theoretical section of the help.

7.22 Height of spring hook.

The height of the spring hook depends on its type and for individual types, their recommended limits are prescribed. In case of springs without fixing eyes, this term means the distance between the end of active coils and the point of fixing of the spring (see the illustration).

Tip: Detailed information can be found in the theoretical section of the help.

7.24 Initial stress.

Initial stress appears in coils of the spring in the course of coiling of the spring wire, and its size depends on the used material, spring index and the manner of coiling. The initial stress will be zero for springs with spaces between its coils.

Note: If the check box to the right of the input field is selected, the spring initial stress will be set automatically within the range of the recommended values.

Tip: Detailed information can be found in the theoretical section of the help.

7.25 Free spring length.

Enter the free spring length for a spring without initial stress.

Note: If the check box to the right of the input field is selected, the spring free length will be set automatically within the range of the recommended values.

7.31 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Helical cylindrical tension springs of rectangular wires and bars. [8]

Springs of cylindrical shape made of helically coiled wires, with constant clearance between the active coils, able to absorb external axial forces counter-acting from each other. Springs made of rectangular wire are cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with wire thickness of the over 10 mm.

Spring design procedure.

1. Enter the required parameters of the working cycle (working load and spring stroke). [8.2 - 8.4]
2. Set preliminary dimensions of the spring in the input fields [8.6 - 8.9], alternatively use one of the design (optimization) functions of the calculation [8.10].
3. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
4. Check the calculated value of safety [8.17] for the designed spring.
5. In list [8.19] select the required design of the spring hook, set its height in line [8.21].
6. Set the corresponding length of the unloaded spring. [8.23]
7. Check the parameters of the spring working cycle. [8.29]
8. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Recommendation: With regards to the considerable effects of the shape and design of fixing eyes on reduction of the spring's service life and impossibility of perfect shot peening of the spring, it is not advisable to use tension springs exposed to fatigue loading. If it is necessary to use a tension spring with fatigue loading, it is advisable to avoid use of fixing eyes and choose another type of fixing of the spring.

Warning: Loading of the spring creates a concentration of stress in the fixing eyes and this may be substantially higher than the calculated stress in the spring coils. It is therefore recommended to check such springs also in view of loading of the fixing eyes.

Tip: Detailed information on the calculation of tension springs can be found in the theoretical section of help.

8.1 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

8.10 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation.

• Spring design for the given ratios D/b, b/h is started by moving one of the scroll bars.
• When pressing the appropriate button you will design the spring while keeping the values of the selected spring dimensions in paragraph [8.1].

When designing the spring the calculation is trying to optimize the dimensions so that the wire section is as small as possible while keeping the required safety [1.27].

8.11 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [8.17] should not drop under the required value [1.27].

Lines [8.15, 8.16] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Note: The calculation used here performs a strength check of the spring only for stress in active coils, and does not take into consideration any possible stress concentrations in the fixing eye.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

8.19 Design of spring ends.

Due to high concentrations of stress occurring in the hook, the springs made of rectangular wire usually use different way of spring fixing.

Tip: Detailed information can be found in the theoretical section of the help.

8.21 Height of spring hook.

The height of the spring hook depends on its type and for individual types, their recommended limits are prescribed. In case of springs without fixing eyes, this term means the distance between the end of active coils and the point of fixing of the spring (see the illustration).

Tip: Detailed information can be found in the theoretical section of the help.

8.23 Free spring length.

Set the corresponding length of the unloaded spring.

Note: If the check box to the right of the input field is selected, the spring free length will be set automatically within the range of the recommended values.

8.29 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Spiral springs. [9]

The spring made of a strip with rectangular section wound into the shape of Archimedes spiral, with constant spacing between its active coils, loaded with torque in the direction of the winding.

Spring design procedure.

1. Enter the required parameters of the working cycle (working load and spring stroke). [9.2 - 9.4]
2. Set preliminary dimensions of the spring in the input fields [9.6 - 9.10], alternatively use one of the design (optimization) functions of the calculation [9.11].
3. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
4. Check the calculated value of safety [9.18] for the designed spring.
5. Check the parameters of the spring working cycle. [9.26]
6. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Note: This calculation is designed for spiral springs with fixed ends of the spring.

Tip: Detailed information on the calculation of spiral springs can be found in the theoretical section of help.

9.1 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

9.11 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation. Spring design for the given ratios R_i/t, b/t, a₀/t is started by moving one of the scroll bars. When designing the spring the calculation is trying to optimize the dimensions so that the strip thickness is as small as possible while keeping the required safety [1.27].

Note: The calculation will select the nearest suitable strip thickness out of the preferred series.

9.12 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [9.18] should not drop under the required value [1.27].

Lines [9.16, 9.17] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

9.26 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Helical cylindrical torsion springs made of round wires a bars. [10]

Springs of cylindrical shape made of helically coiled wires, with constant spacing between the active coils, able to absorb external forces applied in the planes perpendicular to the winding axis through a torque in the direction of winding or unwinding. Springs with wire diameter up to approx. 16 mm are usually cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with a diameter of the over 10 mm.

Torsion springs are produced in two basic designs: tight-coiled and loose-coiled (with clearance between the coils). If the springs are exposed to a static loading, the tight-coiled springs are recommended. Loose-coiled springs are suitable for use with fatigue loading.

Spring design procedure.

1. Enter the required parameters of the working cycle (working load and spring stroke). [10.2 - 10.4]
2. Set preliminary dimensions of the spring in the input fields [10.6 - 10.8], alternatively use one of the design (optimization) functions of the calculation [10.9].
3. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
4. Check the calculated value of safety [10.16] for the designed spring.
5. Select the required design of the spring arms in the list [10.18]
6. Set the corresponding length of the coil part of the spring. [10.21]
7. Check the parameters of the spring working cycle. [10.27]
8. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Note: This calculation is designed for the torsion springs loaded in the direction of coil winding, with fixed arms. The calculation does not take into account the effects of supporting of the spring against the inner or outer guiding part, nor the effects of friction that appears with it. The effects of possible friction between the coils of the spring are also not considered.

Warning: The springs with bended arms are subjected to concentrations of tension at the bends which can be much higher than the calculated stress in the spring coils. It is therefore recommended to check such springs also in view of loading of the arms.

Tip: Detailed information on the calculation of torsion springs can be found in the theoretical section of help.

10.1 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

10.9 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation.

• The spring design for the given spring index D/d is started by moving the scroll bar.
• When pressing the button you will design a spring for the specified spring diameter [10.6].

When designing the spring the calculation is trying to optimize the dimensions so that the wire diameter is as small as possible while keeping the required safety [1.27].

Note: When designing a spring for the given spring index D/d the calculation will select the nearest suitable wire diameter out of the preferred series.

10.10 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [10.16] should not drop under the required value [1.27].

Lines [10.14, 10.15] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Note: The calculation used here performs a strength check of the spring only for stress in active coils, and does not take into consideration any possible stress concentrations in the spring arms.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

10.18 Design of spring ends.

Select the required design of the spring ends from the list.

Tip: Detailed information can be found in the theoretical section of the help.

10.21 Length of coiled section.

Set the corresponding length of coiled section.

Note: If the check box to the right of the input field is selected, the spring length will be set automatically within the range of the recommended values.

10.25 Dimensions of fully loaded spring.

Functional deformation (shift of the arm) of the torsional spring leads to the change of its dimensions. The diameter of springs loaded in the direction of coil winding decreases during its loading. In addition, the length of close wound spring grows.

10.27 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Helical cylindrical torsion springs made of rectangular wires and bars. [11]

Springs of cylindrical shape made of helically coiled wires, with constant spacing between the active coils, able to absorb external forces applied in the planes perpendicular to the winding axis through a torque in the direction of winding or unwinding. Springs made of rectangular wire are cold wound. Hot forming shall be used for the production of heavily loaded springs of greater sizes with wire thickness of the over 10 mm.

Torsion springs are produced in two basic designs: tight-coiled and loose-coiled (with clearance between the coils). If the springs are exposed to a static loading, the tight-coiled springs are recommended. Loose-coiled springs are suitable for use with fatigue loading.

Spring design procedure.

1. Enter the required parameters of the working cycle (working load and spring stroke). [11.2 - 11.4]
2. Set preliminary dimensions of the spring in the input fields [11.6 - 11.8], alternatively use one of the design (optimization) functions of the calculation [11.10].
3. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
4. Check the calculated value of safety [11.17] for the designed spring.
5. Select the required design of the spring arms in the list [11.19]
6. Set the corresponding length of the coil part of the spring. [11.22]
7. Check the parameters of the spring working cycle. [11.28]
8. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Note: This calculation is designed for the torsion springs loaded in the direction of coil winding, with fixed arms. The calculation does not take into account the effects of supporting of the spring against the inner or outer guiding part, nor the effects of friction that appears with it. The effects of possible friction between the coils of the spring are also not considered.

Warning: The springs with bended arms are subjected to concentrations of tension at the bends which can be much higher than the calculated stress in the spring coils. It is therefore recommended to check such springs also in view of loading of the arms.

Tip: Detailed information on the calculation of torsion springs can be found in the theoretical section of help.

11.1 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

11.10 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation.

• Spring design for the given ratios D/t, b/t is started by moving one of the scroll bars.
• When pressing the button you will design a spring for the specified spring diameter [11.6].

When designing the spring the calculation is trying to optimize the dimensions so that the wire thickness is as small as possible while keeping the required safety [1.27].

Note: When designing a spring for the given spring index D/t the calculation will select the nearest suitable wire thickness out of the preferred series.

11.11 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [11.17] should not drop under the required value [1.27].

Lines [11.15, 11.16] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Note: The calculation used here performs a strength check of the spring only for stress in active coils, and does not take into consideration any possible stress concentrations in the spring arms.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

11.19 Design of spring ends.

Select the required design of the spring ends from the list.

Tip: Detailed information can be found in the theoretical section of the help.

11.22 Length of coiled section.

Set the corresponding length of coiled section.

Note: If the check box to the right of the input field is selected, the spring length will be set automatically within the range of the recommended values.

11.26 Dimensions of fully loaded spring.

Functional deformation (shift of the arm) of the torsional spring leads to the change of its dimensions. The diameter of springs loaded in the direction of coil winding decreases during its loading. In addition, the length of close wound spring grows.

11.28 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Torsion bar springs with round section. [12]

Springs based on the principle of long slender bars of circular section subjected to torsion. The ends of bars are mostly fixed by means of grooving. Sometimes one end is square-shaped in order to facilitate attachment.

Spring design procedure.

1. Enter the required parameters of the working cycle (working load and spring stroke). [12.2 - 12.4]
2. Set preliminary dimensions of the spring in the input fields [12.6, 12.7], alternatively use one of the design (optimization) functions of the calculation [12.8].
3. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
4. Check the calculated value of safety [12.15] for the designed spring.
5. Check the parameters of the spring working cycle. [12.16]
6. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Note: Torsion bar springs must be secured against bending stress.

Tip: Detailed information on the calculation of torsion bar springs can be found in the theoretical section of help.

12.1 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

12.8 Spring optimization.

When pressing the button you will design a spring of satisfactory dimensions. When designing the spring the calculation is trying to optimize the dimensions so that the bar diameter is as small as possible while keeping the required safety [1.27].

Note: The calculation will select the nearest suitable bar diameter out of the preferred series.

12.9 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [12.15] should not drop under the required value [1.27].

Lines [12.13, 12.14] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

12.16 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Torsion bar springs with rectangular section. [13]

Springs based on the principle of long slender bars of rectangular section subjected to torsion.

Spring design procedure.

1. Enter the required parameters of the working cycle (working load and spring stroke). [13.2 - 13.4]
2. Set preliminary dimensions of the spring in the input fields [13.6 - 13.8], alternatively use one of the design (optimization) functions of the calculation [13.9].
3. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
4. Check the calculated value of safety [13.16] for the designed spring.
5. Check the parameters of the spring working cycle. [13.17]
6. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Note: Torsion bar springs must be secured against bending stress.

Tip: Detailed information on the calculation of torsion bar springs can be found in the theoretical section of help.

13.1 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

13.9 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation.

• The spring design for the given ratio b/t is started by moving the scroll bar.
• When pressing the appropriate button you will design the spring while keeping the values of the selected spring dimensions in paragraph [13.1].

When designing the spring the calculation is trying to optimize the dimensions so that the bar thickness is as small as possible while keeping the required safety [1.27].

Note: When designing a spring for the given ratio b/t the calculation will select the nearest suitable bar thickness out of the preferred series.

13.10 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [13.16] should not drop under the required value [1.27].

Lines [13.14, 13.15] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

13.17 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Leaf springs with constant profile. [14]

Springs based on the principle of long slander beams of rectangular section subjected to bending. They are used as cantilever springs (fixed at one end), or as simple beams (fixed at both ends). Springs of rectangular, triangular or trapezoidal shape are used.

Spring design procedure.

1. Select the corresponding design and shape of the spring in lists [14.1, 14.2].
2. Enter the required parameters of the working cycle (working load and spring stroke). [14.4 - 14.6]
3. Set preliminary dimensions of the spring in the input fields [14.8 - 14.11], alternatively use one of the design (optimization) functions of the calculation [14.12].
4. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
5. Check the calculated value of safety [14.19] for the designed spring.
6. Check the parameters of the spring working cycle. [14.20]
7. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Recommendation: The spring deflection should not exceed about 30% of the spring functional length. In the case of springs with high deflection, the actual deflection values may considerably differ from the theoretically determined values.

Tip: Detailed information on the calculation of leaf springs can be found in the theoretical section of help.

14.3 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

14.12 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation.

• The spring design for the given ratio L/b is started by moving one of the scroll bars.
• When pressing the appropriate button you will design the spring while keeping the values of the selected spring dimensions in paragraph [14.3].

When designing the spring the calculation is trying to optimize the dimensions so that the leaf thickness is as small as possible while keeping the required safety [1.27].

Note: When designing a spring for the given ratio L/b the calculation will select the nearest suitable leaf thickness out of the preferred series.

14.13 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [14.19] should not drop under the required value [1.27].

Lines [14.17, 14.18] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

14.20 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Leaf springs with parabolic profile. [15]

Springs based on the principle of long slander beams of rectangular section subjected to bending. They are used as cantilever springs (fixed at one end), or as simple beams (fixed at both ends). Usually of rectangular shape, sometimes springs thicker in the middle and at the end of the leaf are used

Spring design procedure.

1. Set the corresponding design and shape of the spring. [15.1]
2. Enter the required parameters of the working cycle (working load and spring stroke). [15.3 - 15.5]
3. Set preliminary dimensions of the spring in the input fields [15.7 - 15.11], alternatively use one of the design (optimization) functions of the calculation [15.12].
4. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
5. Check the calculated value of safety [15.19] for the designed spring.
6. Check the parameters of the spring working cycle. [15.20]
7. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Recommendation: The spring deflection should not exceed about 30% of the spring functional length. In the case of springs with high deflection, the actual deflection values may considerably differ from the theoretically determined values.

Tip: Detailed information on the calculation of leaf springs can be found in the theoretical section of help.

15.2 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

15.12 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation.

• The spring design for the given ratio L/b is started by moving the scroll bar.
• When pressing the appropriate button you will design the spring while keeping the values of the selected spring dimensions in paragraph [15.2].

When designing the spring the calculation is trying to optimize the dimensions so that the leaf thickness is as small as possible while keeping the required safety [1.27].

Note: When designing a spring for the given ratio L/b the calculation will select the nearest suitable leaf thickness out of the preferred series.

15.13 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [15.19] should not drop under the required value [1.27].

Lines [15.17, 15.18] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

15.20 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Laminated leaf springs. [16]

Springs based on the principle of a bunch of long slander beams of rectangular section subjected to bending.

Spring design procedure.

1. Select the corresponding shape of the spring leaves in list [16.1].
2. Enter the required parameters of the working cycle (working load and spring stroke). [16.3 - 16.5]
3. Set preliminary dimensions of the spring in the input fields [16.7 - 16.11], alternatively use one of the design (optimization) functions of the calculation [16.12].
4. Based on the calculated recommended values (green cells) adjust the spring dimensions, so that the calculated working load and spring stroke best correspond to the required values.
5. Check the calculated value of safety [16.19] for the designed spring.
6. Check the parameters of the spring working cycle. [16.20]
7. Chapter [17] serves to check the spring subjected to cyclic (fatigue) loading.

Recommendation: The spring deflection should not exceed about 15% of the spring functional length. In the case of springs with high deflection, the actual deflection values may considerably differ from the theoretically determined values.

Note: This calculation does not take into consideration the influence of friction between the spring leaves.

Tip: Detailed information on the calculation of leaf springs can be found in the theoretical section of help.

16.2 Spring design.

The way of design procedure used in this book allows to define dimensions of a spring with a certain degree of looseness. Therefore the exact value corresponding to the other parameters of the spring is calculated in real time for each of the input parameters. These values are displayed in green fields situated to the right of the input cells. Enter the calculated value in the input box using the appropriate "<" button.

16.7 Number of extra full-length leaves.

Spring leaves of full length, rectangular shape with constant profile. These leaves are added to the spring for two reasons:

• to increase the spring stiffness and load capacity
• they are often ended with hooks to fix the spring

16.12 Spring optimization.

The controls located in this paragraph serve for starting the design (optimization) functions of the calculation.

• The spring design for the given number of spring leaves and ratio L/b is started by moving one of the scroll bars.
• When pressing the appropriate button you will design the spring while keeping the values of the selected spring dimensions in paragraph [16.2].

When designing the spring the calculation is trying to optimize the dimensions so that the leaf thickness is as small as possible while keeping the required safety [1.27].

Note: When designing a spring for the given ratio L/b the calculation will select the nearest suitable leaf thickness out of the preferred series.

16.13 Check data.

This paragraph specifies the results of the strength check of the designed spring. The check is carried out by comparing the permissible stress of the material used [1.20] with the actual stress of the fully loaded spring. The resulting level of safety [16.19] should not drop under the required value [1.27].

Lines [16.17, 16.18] are used to calculate theoretical values of the maximum working load and spring stroke where the required level of safety for the designed spring will still be met.

Warning: Spring subjected to cyclic load must also be checked for potential fatigue damage – see chapter [17].

16.20 Parameters of the working cycle.

This paragraph specifies the basic working parameters of the designed spring in the prestressed (index 1), fully loaded (index 8) and limit condition (index 9).

Meaning of the parameters:

Check of loading capacity of a spring exposed to fatigue loading. [17]

This paragraph is designed to carry out the strength check of springs subjected to cyclic (fatigue) loading, i.e. the springs with the service life requirement higher than 10⁵ of working cycles. The check is carried out by comparing the permissible stress of the material used [17.8] with the actual stress of a fully loaded spring [17.4].

Check procedure

1. Select the appropriate type of the spring in list [17.1].
2. Choose the desired spring service life [17.2]
3. Set the ultimate fatigue strength of the spring material [17.6]
4. Check the level of safety of the designed spring against the fatigue damage [17.10]

If the spring does not satisfy the endurance check, repeat its design while adhering to the following recommendations:

• use a material with the highest possible strength, suitable for dynamic loading (see [1.7])
• design the spring with higher (overdesigned) level of static safety [1.27]
• design the spring with as small a difference between the maximum and minimum loading as possible

17.2 Desired service life of the spring.

Two fields of fatigue loading of springs can be distinguished with springs exposed to fatigue loading. In the first field, with limited service life of springs (lower than approx. 10⁷ working cycles, the fatigue strength of the spring decreases with an increasing number of working cycles. In the field of unlimited service life (the desired service life of the spring is higher than 10⁷ working cycles), the fatigue limit of the material and thus the strength of the spring remains approximately constant.

17.6 Ultimate fatigue strength.

Set the maximum permissible stress of the spring material for infinite life and zero-to-maximum stress fluctuation. If the check box to the right of the input field is selected, the minimum value of fatigue strength specified for the selected material will be set automatically here [1.6] as well as the selected surface treatment of the spring [1.26].

Warning: Fatigue strength is usually dependent not only on the material properties but also on the size of the part subjected to stress. Fatigue strength grows with decreasing dimensions of the part. This is why in the case of final calculations we recommend specifying the material strength taking account of the spring dimensions according to the bill of material or the manufacturer’s specifications.

17.8 Fatigue strength of the spring for the given loading.

Determination of the maximum fatigue strength of the spring is based on the ultimate fatigue strength of the chosen material and the given course of loading of the spring using a Goodman's fatigue diagram.

17.9 Recommended minimum level of safety.

Recommended level of safety for the selected material [1.6] is estimated based on the operational conditions defined in paragraph [1.21].

Note: The value specified here does not incorporate some other factors such as accuracy and reliability of the input information, significance of the equipment, production quality, etc.

Setting calculations, change the language.

Information on setting of calculation parameters and setting of the language can be found in the document "Setting calculations, change the language".

Workbook modifications (calculation).

General information on how to modify and extend calculation workbooks is mentioned in the document "Workbook (calculation) modifications".