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Machine Elements · Chapter 7 of 10 · Intermediate

Screws, Fasteners, and Joints

A thread is an inclined plane wrapped around a cylinder. That one idea sets the torque a power screw needs to lift a load and the preload that keeps a bolted joint tight.

01

Readiness check

This chapter is about threads under load. Tick only what you can do closed-notes.

  • Relate torque, force, and a radius.
  • Recall the coefficient of friction in a contact.
  • Read pitch and lead from a thread.
  • Use a stress area to get force from a stress.
  • Picture an inclined plane and its mechanical advantage.
0 or 1 weak itemsContinue with this chapter.
2 weak itemsReview friction in Statics: Friction.
3 or more weak itemsRevisit torque and force before continuing.
02

The core idea

A thread turns rotation into linear force through an inclined plane. The torque to raise a load fights both the load's climb and friction, and a tightened bolt stores that effort as a clamping preload.

TR = (Fdm/2)(l + πμdm)/(πdm − μl) + collarFi = 0.75 At SpT = K Fi d

Unwrapping one turn of a thread gives a ramp of rise l (the lead) over a circumference πdm. Pushing a load up that ramp against friction needs a torque; that is a power screw lifting a load, or a bolt being tightened. In a bolt the tightening torque creates a preload Fi, the clamp force that holds the joint together and keeps an external load from ever fully reaching the bolt. Preload is what makes a bolted joint reliable, especially under fatigue.

The skill works when: you treat the thread as an inclined plane and account for friction at the thread and the collar.
The skill breaks down when: preload is ignored, or collar friction is left out of a power-screw torque.
The concept. A bolt clamps two members with a preload set during tightening. The thread is an inclined plane: the same geometry that raises a load on a power screw stores clamp force in a bolt.
03

The skills, taught in order

Five skills cover thread geometry, power-screw torque and efficiency, bolt preload, and tightening torque.

7.1 Thread geometry

A thread is defined by its pitch (axial distance between threads), lead (axial advance per turn), and mean diameter dm. Square and Acme threads carry power; the V-shaped ISO metric thread fastens. The tensile stress area At, slightly larger than the minor-diameter area, is what carries the load in a bolt.

7.2 Power-screw torque

Raising a load needs TR = (Fdm/2)(l + πμdm)/(πdm − μl) plus a collar term Fμcdc/2. Lowering reverses the sign of the lead term. The thread torque overcomes both the climb and thread friction; the collar torque overcomes friction at the thrust face.

7.3 Efficiency and self-locking

The efficiency of a power screw is e = Fl/(2πT), the useful lift over the work put in per turn, often only 20 to 40 percent. A screw is self-locking when friction exceeds the lead angle's tendency to back-drive, which is why a jack holds its load without a brake.

7.4 Bolt preload

Tightening a bolt stretches it to a preload Fi. A common target is Fi = 0.75 At Sp for reused connections (0.90 for permanent ones), where Sp is the proof strength of the bolt's property class. Preload keeps the members in compression so the joint does not separate or loosen.

Property classProof strength Sp (MPa)Typical use
4.8310low-carbon, general
5.8380medium-carbon, general
8.8600quenched and tempered
10.9830high-strength structural

Metric property classes (Shigley, Table 8-11). Tensile stress area for M10 is At = 58.0 mm² (Table 8-1).

7.5 Tightening torque

The wrench torque that produces a target preload is estimated as T = K Fi d, with a nut factor K of about 0.2 for as-received steel threads. Lubrication lowers K and changes the torque, which is why torque specs assume a thread condition.

Engineering connection: preload and the bolt-versus-member stiffness ratio decide how much of an external load the bolt feels, the key to fatigue-resistant joints.

04

Worked example 1: torque to raise a load on a power screw

A square-thread power screw has mean diameter dm = 22 mm and lead l = 5 mm (single thread). It raises a 6 kN load with thread friction μ = 0.15 and collar friction μc = 0.12 at a collar mean diameter dc = 40 mm. Find the raising torque.

Figure 1. Raising the load takes torque at the thread (climb plus friction) and torque at the collar (thrust-face friction); the two add to the wrench torque.
  1. ProblemFind the torque to raise the load on the screw in Figure 1.
  2. Given / findF = 6 kN, dm = 22 mm, l = 5 mm, μ = 0.15, μc = 0.12, dc = 40 mm. Find TR.
  3. AssumptionsSquare thread (no thread-angle correction); uniform friction; collar friction acts at the mean collar diameter.
  4. ModelAdd the thread-raising torque and the collar torque.
  5. EquationsTthread = (Fdm/2)(l + πμdm)/(πdm − μl)Tcollar = Fμcdc/2
  6. SolveTthread = (6000·22/2)(5 + π·0.15·22)/(π·22 − 0.15·5) = 66 000(15.37)/(68.37) = 14.84 N·m. Tcollar = 6000·0.12·40/2 = 14.40 N·m. TR = 14.84 + 14.40 = 29.2 N·m.
  7. CheckThe collar contributes nearly half the torque, a reminder that thrust-face friction is not a small correction. The efficiency e = Fl/(2πTR) = 30 000/183 700 = 0.16, typical for a power screw.
  8. ConclusionMost of the wrench effort fights friction, not the lifting itself. That same friction is what makes the screw self-locking.
Result. Tthread = 14.84 N·m, Tcollar = 14.40 N·m, so TR = 29.2 N·m.
05

Worked example 2: bolt preload and tightening torque

An M10 bolt (tensile stress area At = 58.0 mm²) of property class 5.8 (proof strength Sp = 380 MPa) is used in a reusable joint. Find the recommended preload and the tightening torque, using Fi = 0.75 At Sp and a nut factor K = 0.2.

Figure 2. The proof strength and stress area set the bolt's capacity; preloading to 75 percent of proof load gives the clamp force, and the nut factor converts it to a wrench torque.
  1. ProblemFind the preload and tightening torque for the bolt in Figure 2.
  2. Given / findAt = 58.0 mm², Sp = 380 MPa, K = 0.2, d = 10 mm, reusable (factor 0.75). Find Fi and T.
  3. AssumptionsReused connection (0.75 of proof load); nut factor 0.2 for as-received steel threads.
  4. ModelProof load Fp = AtSp; preload Fi = 0.75 Fp; torque T = K Fi d.
  5. EquationsFp = AtSpFi = 0.75 FpT = K Fi d
  6. SolveFp = 58.0 × 380 = 22 040 N = 22.0 kN. Fi = 0.75 × 22 040 = 16.5 kN. T = 0.2 × 16 530 × 10 = 33 060 N·mm = 33.1 N·m.
  7. CheckThe preload is a large fraction of the bolt's capacity, which is intended: a high clamp force is what keeps the joint from separating and resists fatigue. The torque is a sensible value for an M10 bolt.
  8. ConclusionPreload, not the external load, is the heart of bolted-joint design. The nut factor ties a clamp force target to a wrench setting.
Result. Fi = 16.5 kN (75 percent of the 22.0 kN proof load), tightened with T = 33.1 N·m.
06

Misconceptions and diagnostics

MistakeSymptomDiagnostic questionCorrection
Ignoring collar frictionPower-screw torque too low"Is there a thrust collar?"Add the collar term Fμcdc/2.
Skipping preloadJoint loosens or fatigues"What clamp force holds this joint?"Design a preload Fi, usually 0.75 of proof load.
Using the shank area for a boltBolt strength over-estimated"Is the load on the threaded section?"Use the tensile stress area At, not the full shank.
One torque value for all conditionsWrong preload after lubrication"What thread condition does K assume?"Match the nut factor K to the actual lubrication and finish.
07

Practice ladder

Level 1 · Direct skill

An M12 bolt (At = 84.3 mm²) of class 8.8 (Sp = 600 MPa) is used permanently. Find the preload at 0.90 of proof load.

Show answer

Fp = 84.3 × 600 = 50 580 N. Fi = 0.90 × 50 580 = 45.5 kN. Permanent joints use the higher 0.90 fraction.

Level 2 · Mixed concept

Find the tightening torque for the Level 1 bolt with K = 0.2.

Show answer

T = K Fi d = 0.2 × 45 520 × 12 = 109 200 N·mm = 109 N·m. Larger, stronger bolts need substantially more torque.

Level 3 · Independent problem

For the power screw of Worked Example 1, find the torque to lower the load (replace l with −l in the thread term).

Show answer

Tthread = (66 000)(πμdm − l)/(πdm + μl) = 66 000(10.37 − 5)/(69.12 + 0.75) = 66 000(5.37/69.87) = 5.07 N·m. Adding the collar, TL = 5.07 + 14.40 = 19.5 N·m. Lowering needs less thread torque, and the positive value means the screw is self-locking.

Level 4 · Transfer to real engineering

Find a bolted joint that must not loosen (a wheel, a flange, an engine head). Explain how preload and the bolt-to-member stiffness ratio protect it from fatigue.

What good work looks like

The idea that a stiff member and a preloaded bolt make the bolt feel only a small share of the external load, so its alternating stress stays low and fatigue life is long.

08

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I included both thread and collar friction in the screw torque."
"Give me three property classes; I will find the preload and torque for each."
"What torque raises this load?" Summing the thread and collar terms yourself is the skill.
"How tight should this bolt be?" Setting the preload and converting it is the point.

Portfolio task

Analyse a real threaded element: for a power screw, find the raising torque and efficiency; for a bolt, find the preload and tightening torque from its property class.

Must include: thread geometry, friction values, and either a torque-and-efficiency pair or a preload-and-torque pair.
09

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. What is a thread, geometrically?

An inclined plane wrapped around a cylinder, with rise equal to the lead per turn.

2. What two torques raise a power-screw load?

The thread torque (climb plus friction) and the collar torque (thrust-face friction).

3. Give a target bolt preload.

Fi = 0.75 At Sp for reused joints, 0.90 for permanent ones.

4. Estimate tightening torque.

T = K Fi d, with K about 0.2 for as-received steel threads.

5. Why does preload matter for fatigue?

A preloaded joint keeps the members in compression, so the bolt feels only a small share of an external load.

TodayFinish this quiz and Levels 1 and 2 of the ladder.
+1 dayRe-derive the screw torque and bolt preload from a blank page.
+3 daysWork a lowering torque and a second preload.
+7 daysCarry preload thinking into springs, Chapter 8.
+30 daysReuse the stress area and proof strength in joint fatigue.
10

Textbook mapping

ItemMapping
Primary sourceBudynas and Nisbett, Shigley's Mechanical Engineering Design, Chapter 8 (Screws, Fasteners, and the Design of Nonpermanent Joints)
Cross-referenceNorton, Ch. 15 · Statics: Friction
Core topics7.1 Thread geometry · 7.2 Power-screw torque · 7.3 Efficiency and self-locking · 7.4 Bolt preload · 7.5 Tightening torque
Engineering connectionPreload and stiffness ratio govern bolted-joint fatigue.
Read nextChapter 8: Mechanical Springs.