Screws 1

1 SCREWS

Threaded fasteners as screws, nuts and bolts are important components of

mechanical structures, and machines. Screws may be used as removable

fasteners or as devices for moving loads.

1.1 Screw thread

The basic arrangement of a helical thread wound around a cylinder is illus-

trated in Fig 1.1. The terminology of an external screw threads is, Fig. 1.1:

• pitch denoted by p is the distance, parallel to the screw axis, between

corresponding points on adjacent thread forms having uniform spacing;

• major diameter denoted by d is the largest (outside) diameter of a screw

thread.

• minor diameter denoted by dr or d1, is the smallest diameter of a screw

thread.

• pitch diameter denoted by dm or d2 is the imaginary diameter for which

the width of the threads and the grooves are equal.

Screws 2

The standard geometry of a basic profile of an external threads is shown in

Fig. 1.2, and it is basically the same for both Unified (inch series) and ISO

(International Standards Organization, metric) threads.

The lead denoted by l is the distance the nut moves parallel to the screw

axis when the nut is given one turn. A screw with two or more threads

cut beside each other is called multiple-threaded screw. The lead is equal to

twice the pitch for a double-threaded screw, and to 3 times the pitch for a

triple-threaded screw. The pitch p, lead l, and lead angle λ are represented in

Fig. 1.3. Figure 1.3(a) shows a single thread right hand screw and Fig. 1.3(b)

shows a double-threaded left hand screw. All threads are assumed to be

right-hand, unless otherwise specified.

A standard geometry of an ISO profile, M (metric) profile, with 60◦ sym-

metric threads is shown in Fig. 1.4. In Fig. 1.4 D(d) is the basic major

diameter of internal (external) thread, D1(d1) is the basic minor diame-

ter of internal (external) thread, D2(d2) is the basic pitch diameter, and

H = 0.5(3)1/2p.

Metric threads are specified by the letter M preceding the nominal major

diameter in millimeters and the pitch in millimeters per thread. For example:

M 14× 2

Screws 3

M is the SI thread designation, 10 mm is the outside (major) diameter, and

the pitch is 2 mm per thread.

Screw size in the Unified system is designated by the size number for

major diameter, the number of treads per inch, and the thread series, like

this:

5”8

− 18 UNF

5”8

is the the outside (major) diameter where the double tick marks mean

inches, and 18 threads per inch. Some Unified thread series are:

UNC Unified National Coarse

UNEF Unified National Extra Fine

UNF Unified National Fine

UNS Unified National Special

UNR Unified National Round (round root)

The UNR series threads have improved fatigue strengths.

1.2 Power screws

For application which require power transmission, the Acme, Fig. 1.5, and

square threads, Fig. 1.6, are used.

Power screws are used to convert rotary motion to linear motion of the

Screws 4

meting member along the screw axis. These screws are used to lift weights

(screw-type jacks) or exert large forces (presses, tensile testing machines).

The power screws can also be used to obtain precise positioning of the axial

movement.

A square-threaded power screw with a single thread having the pitch

diameter dm, the pitch p, and the helix angle λ is considered in Fig. 1.7.

Consider that a single thread of the screw is unrolled for exactly one turn.

The edge of the thread is the hypotenuse of a right triangle and the height

is the lead. The hypotenuse is the circumference of the pitch diameter circle

(Fig. 1.8). The angle λ is the helix angle of the thread.

The screw is loaded by an axial compressive force F , Figs. 1.7 and 1.8.

The force diagram for lifting the load is shown in Fig. 1.8(a), (the force

Pr acts to the right). The force diagram for lowering the load is shown in

Fig. 1.8(b), (the force Pl acts to the left). The friction force is

Ff = µN,

where µ is the coefficient of dry friction and N is the normal force. The

friction force is acting opposite to the motion.

Screws 5

The equilibrium of forces for raising the load gives

∑Fx = Pr − N sin λ − µN cos λ = 0, (1.1)

∑Fy = F + µN sinλ − N cosλ = 0. (1.2)

Similarly, for lowering the load one may write the equations

∑Fx = −Pl − N sin λ + µN cosλ = 0, (1.3)

∑Fy = F − µN sin λ − N cos λ = 0. (1.4)

Eliminating N and solving for Pr

Pr =F (sinλ + µ cosλ)

cos λ − µ sinλ, (1.5)

and for lowering the load

Pl =F (µ cos λ − sin λ)

cosλ + µ sin λ. (1.6)

Using the relation

tanλ = l/πdm,

and dividing the equations by cosλ one may obtain

Pr =F [(l/πdm) + µ]1 − (µl/πdm)

, (1.7)

Pl =F [µ − (l/πdm)]1 + (µl/πdm)

. (1.8)

Screws 6

The torque required to overcome the thread friction and to raise the load is

Tr = Prdm

2=

Fdm

2

(l + πµdm

πdm − µl

). (1.9)

The torque required to lower the load (and to overcome a part of the friction)

is

Tl =Fdm

2

(πµdm − l

πdm + µl

). (1.10)

When the lead, l, is large or the friction, µ, is low the load will lower itself. In

this case the screw will spin without any external effort, and the torque Tl in

Eq. (1.10) will be negative or zero. When the torque is positive, Tl > 0 Tl in

Eq. (1.10), the screw is said to be self-locking. The condition for self-locking

is

πµdm > l.

Dividing both sides of this inequality by πdm, and using l/πdm = tanλ,

yields

µ > tan λ. (1.11)

The self-locking is obtained whenever the coefficient of friction is equal to or

greater than the tangent of the thread lead angle.

Screws 7

The torque, T0, required only to raise the load when the friction is zero,

µ = 0, is obtained from Eq. (1.9)

T0 =F l

2π. (1.12)

The screw efficiency e can be defined as

e =T0

Tr=

F l

2πTr. (1.13)

For square threads the normal thread load, F , is parallel to the axis of the

screw, Figs 1.6 and 1.7. The preceding equations can be applied for square

threads.

For Acme threads, Figs 1.5, or other threads, the normal thread load is

inclined to the axis due to the thread angle 2α and the lead angle λ. The lead

angle can be neglected (is small) and only the effect of the thread angle is

considered, Fig. 1.9. The angle α increases the frictional force by the wedging

action of the threads. The torque required for raising the load is obtained

from Eq. (1.9) where the frictional terms must be divided by cos α

Tr =Fdm

2

(l + πµdm sec α

πdm − µl sec α

). (1.14)

Equation (1.14) is an approximation because the effect of the lead angle has

been neglected. For power screws the square thread is more efficient than

Screws 8

the Acme thread. The Acme thread adds an additional friction due to the

wedging action. It is easier to machine an Acme thread than a square thread.

In general, when the screw is loaded axially, a thrust bearing or thrust

collar may be used between the rotating and stationary links to carry the

axial component, Fig. 1.10. The load is concentrated at the mean collar

diameter dc. The torque required is

Tc =Fµcdc

2, (1.15)

where µc is the coefficient of collar friction.

Example

A double square-thread power screw has the major diameter d = 64 mm

and the pitch p = 8 mm. The coefficients of friction is µ = 0.08 and the

coefficient of collar friction µc = 0.08. The mean collar diameter is dc = 80

mm. The external load on the screw is F = 10 kN.

Find:

1. the lead, the pitch (mean) diameter and the minor diameter;

2. the torque required to raise the load;

3. the torque required to lower the load;

4. the efficiency.

Screws 9

Solution

1. From Fig. 1.6(a):

the minor diameter is

dr = d − p = 64 − 8 = 56 mm,

the pitch (mean) diameter is

dm = d − p/2 = 64 − 4 = 60 mm.

The lead is

l = 2 p = 2 (8) = 16 mm.

2. The torque required to raise the load is

Tr =Fdm

2

(l + πµdm

πdm − µl

)+

Fµcdc

2

=104(60)(10−3)

2

[16 + π0.08(60)π60 − 0.08(16)

]+

104(0.08)(80)(10−3)2

= 49.8 + 32 = 81.8 N m.

3. The torque required to lower the load is

Tl =Fdm

2

(πµdm − l

πdm + µl

)+

Fµcdc

2

=104(60)(10−3)

2

[π0.08(60) − 16π60 + 0.08(16)

]+

104(0.08)(80)(10−3)2

= −1.54 + 32 = 30.45 N m.

Screws 10

The screw is not self-locking (the first term in the above expression is nega-

tive).

4. The overall efficiency is

e =F l

2πTr=

104(16)(10−3)2π(81.8)

= 0.31.

Major diameter

Pitch diameter

Minor diameter

Pitch p

45◦ chamfer

RootCrest

Thread angle 2α

Figure 1.1

dm

d

dr

, d2

, d1

p

4

60◦

30◦

Roo

t(o

rm

inor

)di

amet

erd

r

Pit

chdi

amet

erd

m

Maj

ordi

amet

erd

p

8p

Root

Axis of thread

Figure 1.2

Crest

l

p

λ

l

p

λ

(a)

Single thread-right hand

(b)

Double thread-left hand

Figure 1.3

H

8

3H8

p

8

30◦60◦

External threads

Internal threads

Figure 1.4

60◦H

5H8

H

4

p

2p

2

p

4

p

D1

d1

,

D2

d2

,

Dd ,

p

2

drdm

d

0.3p

(a) Acme

2α = 29◦

Figure 1.5

dr

p

2

2α = 29◦

p

p

2

p

ddm

(b) Acme stub

drd

Figure 1.6

p

2

p

2

p

(a)

5◦α = 5◦

dm dr d

p

p

2

p

2

(b) Modified square

Square

dm

dm

Nut

λ

F/2

p

Figure 1.7

F/2

F

µN

N

F

λ

πdm

l

Figure 1.8

πdm

N

F

µN

l

λ

(a) (b)

Pr Pl

yy

x x

αF

cos α

2α =Threadangle

Figure 1.9

F

dc

F

2

Nut

Collar

F

2

F

2F

2

Figure 1.10