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Springs by the Numbers
Rate, stress, fatigue and fit of compression, extension, disc, torsion, bar and flat springs, on one machining centre
How to check and specify mechanical springs: rate and energy, compression, extension, disc, torsion, bar and flat springs, fatigue, and the springs that set forces in machines.
Start readingAbout this book
A spring is a part designed to deflect. Most machine parts are made stiff so that their deflection can be ignored; a spring is made flexible so that its deflection can be used, to store energy, to hold a force nearly constant while parts move, or to set a force or pressure that the rest of the machine depends on. That makes springs some of the most highly stressed parts in a machine, and some of the most often replaced with whatever fits.
This book shows how to check and specify mechanical springs with numbers. It starts with rate, energy and springs acting together, then works through helical compression springs, from checking one to designing one and checking it for fatigue, then square-wire and nested springs, extension springs and their hooks, disc spring stacks, torsion springs, torsion bars and flat springs. It finishes with the springs that set forces and pressures: detents, relief valves and spring-applied clamps.
It is for mechanical design and maintenance engineers, machine builders and fitters. It assumes engineering or trade training and comfort with force, stress and stiffness; the spring formulas are explained as they are used. Each chapter works the running case with real numbers, includes one plausible mistake and how it was found, links the KEVOS® tools that do the chapter’s calculations, and ends with an exercise and its worked answer.
The running case
Every chapter follows one machine. This is an illustration. A job shop in Ballarat, Victoria, has bought a used vertical machining centre, the VM-8, with a 40-taper spindle, a 20-pocket tool magazine with an arm-type tool changer and a fourth-axis rotary table. Before putting it to work, the shop is rebuilding it, and the rebuild has turned up a spring problem in nearly every assembly:
- Tool changer: the lock plunger that holds the changer arm parked, whose springs have broken twice.
- Tool magazine: the ball detents that hold tool holders in their pockets, and the shutter door between the magazine and the machining area.
- Spindle: the stack of disc springs that pulls the drawbar and clamps the tool holder.
- Enclosure: a hinged top hatch on torsion springs and a heavy rear service panel balanced by a torsion bar, with flat spring wipers on the way covers.
- Hydraulics: a small power unit with a direct-acting relief valve, and a spring-applied, hydraulically released brake on the rotary table.
- First job: a fixture whose pressure pads are pushed by heavy springs in tight pockets.
Loads, dimensions, materials, allowable stresses and test figures are the shop’s working figures or are calculated from stated assumptions. They do not describe KEVOS® work or any real business, machine maker or supplier’s data.
How to use the book
Read Chapter 1 first: it sets out rate, energy and how springs combine, which every later chapter uses. Chapters 2 to 4 cover helical compression springs and should be read in order. The later chapters can be read in any order once Chapter 2 is familiar. Chapter 10 also covers the inspection routine and records the shop adopted.
Every calculation in the book was checked by running it, and the KEVOS® tools linked from each chapter were checked against the calculations they perform. Wire strengths, allowable stresses and friction values are typical figures for illustration; a real design uses the spring maker’s data for the actual wire and process.
Source limits and professional responsibility
This book is educational. Springs store energy and can release it suddenly when a part is dismantled or breaks; a compressed drawbar stack or a wound counterbalance must be released with the right tools and procedure, and machines must be isolated and locked out before work. Springs in safety functions, such as brakes, guards and pressure relief, must be designed or checked by a competent person against the applicable standards and the makers’ data. Decisions about real machines remain with the competent people responsible for them.
Who this book is for
Mechanical design engineers, machine builders, maintenance engineers and fitters who specify, check, replace or troubleshoot springs in machines, fixtures, guards and hydraulic equipment.
What you will learn
- Work out a spring's rate, load, stored energy and the effect of springs and supports acting in series or in parallel.
- Check and design helical compression and extension springs for stress, solid length, buckling, hooks and fatigue.
- Size disc spring stacks, torsion springs, torsion bars and flat springs for the force, travel and space available.
- Check the springs that set a machine's forces and pressures: counterbalances, detents, relief valves and spring-applied clamps.
- Write a spring specification and an inspection routine that catch set, wear, wrong parts and wrong assembly.
Inside this book
Chapters
- 01Spring rate, stored energy and springs that work togetherWhat a spring's rate is, how much energy it stores, and how springs and their supports combine in series and in parallel, with the VM-8's spring register.
- 02Checking a helical compression springHow to check a round-wire compression spring: rate from wire and coils, Wahl-corrected stress, solid length and clash allowance, buckling and natural frequency.
- 03Designing a compression spring from two working pointsHow to design a compression spring from the forces needed at two lengths: rate, free length, wire size for a stress limit, coils, solid length, rounding to stock wire.
- 04Compression springs that cycle: stroke, stress range and fatigueHow to check a compression spring that cycles: the stress range from preload and stroke, wire endurance from Zimmerli's data, Goodman factors, and finding the real stroke.
- 05More force in less space: square-wire and nested springsWhat to do when a round-wire spring will not fit: square and rectangular wire, nested pairs, load sharing, coil clearance and hand of winding, compared on one fixture.
- 06Extension springs: initial tension and the hooks that breakHow to check an extension spring: rate and initial tension, body stress, bending and torsion at the hooks, natural frequency, and why the hook bend radius decides its life.
- 07Disc spring stacks: large forces in short spacesHow disc (Belleville) springs carry large forces in little space: the Almen–László characteristic, series and parallel stacking, stack design for force and travel, and stresses.
- 08Torsion springs and counterbalanced lidsHow helical torsion springs work and how to check one: bending stress, wind-up angle, legs and arbor clearance, and balancing a hinged hatch over its whole opening.
- 09Torsion bars and flat springsHow to size a torsion bar for stress and twist and balance a heavy panel with it, and how to check flat cantilever springs, with their limits at the ends and in large deflection.
- 10Springs that set a force: detents, relief valves and spring-applied clampsHow springs set forces and pressures in machines: ball detent release force, relief valve cracking pressure and overpressure, spring-applied hydraulic release clamps, and records.
Apply This in Practice
Use these KEVOS® tools to try the method from this reading with your own numbers.
Applies this method
Ball Detent Check (Spring Preload, Release Force, Spring Stress at Full Rise)
Checks a spring-loaded ball detent: preload and stress, release force from the notch angle and friction, and spring load and stress at full rise, against a release force range.
Used in Springs that set a force: detents, relief valves and spring-applied clamps
Applies this method
Compression Spring Designer (Two Working Points)
Designs a compression spring from loads at two lengths: rate and free length, the wire size for an allowable stress, coils, solid length, clash allowance, stress at solid and buckling ratio.
Used in Designing a compression spring from two working points
Applies this method
Compression Spring Fatigue Check (Working Stroke, Goodman Factor)
Checks a compression spring cycling between two lengths: alternating and mean stresses with the Wahl factor, and Goodman, Soderberg and Gerber factors with first-cycle yield, for infinite life.
Used in Compression springs that cycle: stroke, stress range and fatigue
Applies this method
Disc (Belleville) Spring Load and Stress
Gives a disc spring's force, force at flat and top-edge stresses by the Almen–László method, with stack force and travel, used to check a drawbar stack at its released position.
Applies this method
Disc Spring Stack Design
Designs a disc spring stack: groups in series for the stroke within a share of cone height, discs in parallel for the force, stack force, free and solid heights, and top inner-edge stress.
Applies this method
Extension Spring Designer (Body, Hooks, Initial Tension, Frequency)
Checks an extension spring with loop hooks: body stress, hook bending and torsional stress at the bend radius, initial-tension stress against a range, and natural frequency.
Used in Extension springs: initial tension and the hooks that break
Applies this method
Flat and Leaf Spring
Gives the deflection, rate and bending stress of a flat cantilever, multi-leaf cantilever or semi-elliptic spring, used here for the flat strips that press way cover wipers.
Used in Torsion bars and flat springs
Applies this method
Helical Compression Spring Design Check
Checks a round-wire compression spring: rate, working load, Wahl-corrected stress at the working length and at solid, solid length, clash allowance, natural frequency and buckling ratio.
Applies this method
Helical Extension Spring Check
Checks an extension spring's body: rate, load including initial tension, Wahl-corrected body stress and initial-tension stress, for the springs that pull a magazine shutter closed.
Used in Extension springs: initial tension and the hooks that break
Applies this method
Helical Torsion Spring Check
Checks a helical torsion spring's body: angular deflection, rate, bending stress with the curvature factor and wound inside diameter against the arbor, for a hatch counterbalance spring.
Applies this method
Lid Counterbalance with Torsion Springs (Balance, Handle Force, Spring Stress)
Follows a hinged hatch balanced by torsion springs through its opening: out-of-balance torque shut, open and at its worst, largest handle force, and spring stress at full wind-up.
Applies this method
Nested Compression Spring Check (Load Share, Stresses, Solid Length, Clearance)
Checks two nested compression springs: each spring's rate, load and stress, the combined rate and load share, solid lengths, and the radial clearance between them against the load needed.
Used in More force in less space: square-wire and nested springs
Applies this method
Relief Valve Spring Check (Cracking Pressure, Full-Lift Pressure, Overpressure, Spring Stress)
Checks a direct-acting relief valve's spring: cracking and full-lift pressures from the seat area, overpressure, spring stress at full lift, room before solid, and the length for a set pressure.
Used in Springs that set a force: detents, relief valves and spring-applied clamps
Applies this method
Spring Clamp Hydraulic Release Check (Clamp Force, Release Pressure, Spring Stress)
Checks a spring-applied, hydraulically released clamp or brake: clamping force of the springs, load and pressure to release with seal friction, and spring stresses clamped and released.
Used in Springs that set a force: detents, relief valves and spring-applied clamps
Applies this method
Springs in Series or Parallel
Gives the equivalent rate of up to four springs in series or parallel and each spring's load and deflection, used here for a plunger spring on a flexible bracket and for paired springs.
Used in Spring rate, stored energy and springs that work together
Applies this method
Square versus Round Wire Spring Comparison (Rate, Load, Stress, Solid Length)
Compares square-wire and round-wire springs on the same coil and lengths: rates, loads, stresses, shared solid length and which one gives the load needed within the allowable.
Used in More force in less space: square-wire and nested springs
Applies this method
Square-Wire Compression Spring
Gives the rate, load at the working length and corrected torsional stress of a square-wire compression spring, for a pressure pad spring that must fit a shallow pocket.
Used in More force in less space: square-wire and nested springs
Applies this method
Torsion Bar Spring
Checks a solid round torsion bar under an arm load: torque, twist, torsional rate, arm-tip deflection and shear stress, used to compare a correct and a wrong replacement bar.
Used in Torsion bars and flat springs
Applies this method
Torsion Bar Spring Design (Diameter for Stress and Twist)
Sizes a solid torsion bar for stress and twist from a load and arm: governing diameter, and for the chosen bar its stress, twist, rate, arm-tip travel and energy stored.
Used in Torsion bars and flat springs
Applies this method
Torsion Spring Designer (Angle, Stress, Legs and Arbor)
Checks a torsion spring including its legs: equivalent coils, deflection, rate, bending stress, the largest safe angle, wound inside diameter and body length over the arbor.
Related Tools and Examples
Useful alongside
Beam of Uniform Strength: Safe Load, Size and Outline
Sizes the tapered cantilevers of uniform strength that this chapter says a leaf spring approaches.
Related to Torsion bars and flat springs
Useful alongside
Disc Spring Forces, Stresses and Fatigue Life (Corrected Outside Diameter, Critical Points, Life Equations)
Takes the disc spring stacks in this chapter on to fatigue: the corrected outside diameter, the stresses at the bottom edges and the cycles the stack will last.
Useful alongside
Semi-Elliptic Leaf Spring Design (Leaves for Stress and Deflection)
Sizes the number of leaves of a semi-elliptic leaf spring for stress and deflection; mentioned as the related tool for vehicle-type leaf springs, which the case machine does not use.
Related to Torsion bars and flat springs
References and source notes
The manuscript and worked examples are original KEVOS® writing. References inform particular concepts; their authors do not endorse this book. Web references were consulted on 1 November 2026.
- A. Altland and B. Simons. Concepts of Theoretical Solid State Physics (2001). Lecture notes released for verbatim copying only, so treated here as a protected work. Used only as a map of ideas: the harmonic approximation, in which every small deformation about equilibrium behaves as a spring storing energy in proportion to the square of the displacement, and the way springs combine into chains and networks. No text, equations, figures or problems are reproduced, and all cases and calculations are KEVOS® originals.
- R. G. Budynas and J. K. Nisbett, McGraw-Hill Education. Shigley's Mechanical Engineering Design, tenth edition (2015). Background for helical compression, extension and torsion spring analysis, the Wahl and curvature correction factors, spring wire strengths fitted as A ÷ d^m, Zimmerli's endurance data and the Goodman fatigue check, used in Chapters 1 to 8.
- W. C. Young, R. G. Budynas and A. M. Sadegh, McGraw-Hill Education. Roark's Formulas for Stress and Strain, eighth edition (2012). Background for flat cantilever springs, torsion of round bars and the Almen–László treatment of disc springs, used in Chapters 7 and 9.
- R. C. Juvinall and K. M. Marshek, John Wiley & Sons. Fundamentals of Machine Component Design, fifth edition (2012). Background for spring materials, end types, buckling, surge and the design of springs from two working points, used in Chapters 2 to 6.
- Safe Work Australia. Model Code of Practice: Managing the risks of plant in the workplace (2022). Named in Chapters 1, 7 and 10 for the duties to manage the risks of plant, including stored energy in springs and the isolation of machines before maintenance.