1

Design of Nonpermanent Joints (Chapter 8)

2

Chapter 8 - Topics

3

Threaded Fasteners - Introduction

Budynas, p. 396

“… the curiosity of any person

interested in mechanical

engineering naturally results in the

acquisition of a good background

knowledge of fastening methods.”

“Contrary to first impressions, the

subject is one of the most

interesting in the entire field of

mechanical design.”

4

Screw Thread Terminology

5

Metric ‘M’ and ‘MJ’ Thread Profile

d – major diameter

dr – minor (or root) diameter

dp – pitch diameter

p – pitch

H = p2

3

6

Dimensions of Metric Threads

7

Dimensions of UNC and UNF Threads

8

Square and ACME Threads

9

Mechanics of Power Screws

worm gear

10

Mechanics of Power Screws

Objectives:

– Determine relation between applied torque

and axial load

– Consider both raising and lowering the load

– Compute stresses at the thread root

11

Recall from Elementary Mechanics

Static equilibrium:

0

0

M

F

12

Free-Body Diagrams

• Clear diagram of each body being analyzed

• Label coordinate system

• Isolate the body from it’s supports

• Replace supports with reaction forces

• Label all applied forces and moments

• Apply equilibrium equations

• Solve for unknown forces and moments

13

Power Screw

dm – mean thread diameter

p – pitch

- lead angle

- helix angle

14

Analysis of Thread Forces

pnl

l

where

n = 1 single thread

n = 2 double thread

n = 3 triple thread

15

Free Body Diagrams

Raising the Load Lowering the Load

PR, PL – net force due to applied torque

f - friction coefficient

N - normal force

16

Raising the Load

Equilibrium equations:

0cossin

0cossin

NNfFF

NfNPF

V

RH

17

Lowering the Load

Equilibrium equations:

0cossin

0cossin

NNfFF

NfNPF

V

LH

18

Torque Requirements

sincos

sincos

sincos

cossin

f

fFP

f

fFP

L

R

Eliminating N

and substituting

2

2

tan

mLL

mRR

m

dPT

dPT

d

l

19

Torque Requirements (cont.)

which gives

fld

ldfFdT

fld

dflFdT

m

mmL

m

mmR

2

2

20

Effect of Friction – Raising the Load

fld

dflFdT

m

mmR

2

20

FlT

RT

Te 0

if f = 0

and the efficiency, e, is given by

21

Effect of Friction (cont.)

Lowering the load:

Note, if

f dm > l

then

TL > 0

and an applied torque is needed to lower the load. For TL 0, the screw is “self-

locking.” Since l / dm = tan , the screw is self-locking if

f > tan

fld

ldfFdT

m

mmL

2

22

Power Screw Mechanics – ACME Threads

sec

sec

2 lfd

dflFdT

m

mmR

23

Additional Friction due to Thrust Collar

2

CCC

dfFT

24

Example 8-1 (a-c)

25

Stresses in Threads

26

Compressive axial stress

Shear stress due to torsion

2

4

rd

F

3

16

rd

T

Stresses in Body

27

Stresses in Threads

Bearing Stress

Bending Stress

Shear Stress

pnd

F

trb

6

pnd

F

tr

3

pnd

F

tmB

2

27

28

Distribution of Load in Engaged Threads

Experimental result:

First thread carries 38% of the load

Second thread carries 25% of the load

Third thread carries 18% of the load...

Seventh thread is stress free

29

Example 8-1 (d-f)

30

Mechanics of Bolted Joints(Sections 8-4 – 8-10)

Tensile load applied to joint

Bolt tension = preload due to

tightening of nut + stress due

to applied load

Member stress = compressive

stress due to tightening of nut

+ stress due to applied load

First, need to standard bolt dimensions => Section 8-3

31

Standard Dimensions of Threaded Fasteners (Section 8-3)

Hexagon-head Bolt

32

Hexagon-head Thread Length

inLind

inLindLT

6,2

12

6,4

12

200,252

200125,122

48,125,62

Lmmd

mmLmmd

mmdmmLmmd

LTLT

Note: Ideal bolt length – one or two threads project from the nut

33

Square and Hexagonal Bolt Dimensions

(cont.)

34

Square and Hexagonal Bolt Dimensions

(cont.)

35

Cap-screws

36

Hexagon Head Cap Screw Dimensions (Appendix A-30)

37

Machine Screws

38

Hexagonal Nuts

a) end view

b) washer faced regular nut

c) regular nut chamfered on both faces

d) jam nut with washer face

e) jam nut chamfered on both faces

39

Hexagonal Nut Dimensions(Appendix A-31)

40

Washer Dimensions(Appendix A-32 and A-33)

.

.

....

41

Mechanics of Bolted Joints(Sections 8-4 – 8-10)

Tensile load applied to joint

Bolt tension = preload due to

tightening of nut + stress due

to applied load

Member stress = compressive

stress due to tightening of nut

+ stress due to applied load

Grip length, l

42

Alternate Bolted Connection

Effective grip

length, l’

O-ring seal

43

Fastener Stiffness(Section 8-4)

44

Bolt Stiffness

d

dd

l

EAk

t

tt

l

EAk

dttd

td

td

tdb

lAlA

EAA

kk

kkk

Unthreaded portion

Threaded portion

Effective bolt stiffness (springs in series)

45

Procedure for Finding Fastener Stiffness(Table 8-7)

46

At for threaded members - Table 8-1 (see Table 8-2 for UNC and UNF)

47

Procedure for Finding Fastener Stiffness(cont.)

48

Member Stiffness(Section 8-5)

For multiple members, treat as springs in series

If one of the members is a gasket with low stiffness

im kkkkk

1...

1111

321

gasketm kk

49

Member Stiffness (no gasket)

Pressure cone method

Frustum - portion of a cone which lies

between two parallel planes

- half-apex angle

50

Member Stiffness (cont.)

Frustum stiffness – two members with same material and thickness and taking = 30º, l = 2t (see text for derivation)

dl

dl

dEkm

5.25774.0

5.05774.05ln2

5774.0

51

Comparison to Other Methods

Use of =30º gives

good agreement

with FEA

52

Member Stiffness:

Exponential Curve Fit to FEA Results (Eq. 8-23)

l

BdA

dE

km exp

53

Example 8-2

54

Example 8-2 part (a)

55

b) Both members are steel

c) Repeat using Eq. 8-23

Example 8-2 parts (b & c)

dl

dl

dEkm

5.25774.0

5.05774.05ln2

5774.0

l

BdA

dE

km exp

56

Example 8-2 part (d)

d) Bolt stiffness (use Table 8-7)

Table 8-7

57

Bolt Strength(Section 8-6 )

Sy – yield strength

Sut – ultimate tensile strength

Sp - The proof strength is the stress

above which permanent set is

induced

Permanent set – first measurable

deviation from linear elastic behavior

58

Variability of Strength

Statistical measures:

Mean strength = 145.1 ksi

Standard deviation = 10.3 ksi

Minimum strength = 120 ksi

59

SAE Specifications for Steel Bolts

(Table 8-9)

Note: Head Markings

Also note: grade of

nut must match the

bolt grade

60

ASTM Specifications for Steel Bolts(Table 8-10)

61

Metric Classes for Steel Bolts, Screws and Studs(Table 8-11)

62

Tension Joints – The External Load

Preload: Fi

Load applied to joint: P

Portion of P taken by bolt: Pb

Portion of P taken by members: Pm

P = Pb + Pm

Resultant bolt load: Fb = Pb + Fi

Resultant member load: Fm = Pm - Fi

Fraction of external load P carried by bolt: C

Fraction of external load P carried by members: 1 - C

63

Deformation of Bolt and Members

Pb

kbPm

km

Pm Pbkm

kb

P k kb km

P

kb km

Pb kb

Pb kbP

kb km

64

Bolt and Member Loads

Pm 1C P

C kb

kb km

Fb Pb Fi CP Fi Fm 0

Fm Pm Fi 1C P Fi Fm 0

65

Example Bolt and Member Stiffnesses(1/2 in – 13 NC steel bolts)

C – fraction of load carried by bolt (< 20%)

(1-C) - fraction of load carried by members

66

Relation between Bolt Torque and Bolt Tension

The torque required to raise a fastener and for overcoming collar

friction were derived in section 8-2.

sec

sec

2 lfd

dflFdT

m

mmR

2

CCC

dfFT

67

Relation between Bolt Torque and Bolt Tension

(cont.)

Setting the mean collar diameter to:

leads to

where

and

dc d 1.5d 2

T KFid

K dm

2d

tan f sec

1 f tan sec

0.625 fc

tan l

dm,

68

Torque Factors for Various Materials

When bolt condition is not given, use K = 0.2

69

Example 8-3

70

Example 8-3

(cont.)

71

Table 8-2

At = 0.373 in2

Ar = 0.351 in2

72

Table 8-9

Sp = 85 ksi

73

Tension Joint with Preload

b CP

AtFi

At

n SpAt Fi

CP

PnP

The total tensile stress in bolt

Introducing a load factor, n, where

and solving for n gives

where n is the load factor associated with bolt strength

74

Tension Joint with Preload (cont.)

We also need to check for joint separation where Fm = 0

Introducing a load factor, n0, where

gives

where n0 is the load factor associated with joint separation

PnP 0

n0 Fi

P 1C

75

Recommended Preload

sconnectionpermanentforF

fastenersreusedsconnectionntnonpermaneforFF

p

p

i ,90.0

,,75.0

ptp SAF

Where the proof load is given by

76

Example 8-4

77

Example 8-4 (cont.)

78

Example 8-4 (cont.)

79

Example 8-4 (cont.)

80

Example 8-4 Summary

0.0

10.0

20.0

30.0

40.0

50.0

0.0 10.0 20.0 30.0 40.0 50.0

Bo

lt L

oa

d (

ksi)

External Load (kip)

For P < 22.8 kip

Bolt load = 14.4 + 0.368 * P

Plates separate when

P=3.80 (6 kip) = 22.8 kip

For P > 22.8 kip

Bolt load = P

81

Finite Element Validation of Example 8-4

Bolt

Bottom Plate

Top plate

Assembly

Contact

Surfaces

Element type - Axisymmetric

Three parts: bolt, top plate and bottom plate

(all axisymmetric)

82

Finite Element Validation of Example 8-4 (cont.)

Three Steps

Step 1 - Apply tensile stress to bolt to stretch bolt beyond plate thickness

(undeformed bolt length smaller than plates)

Step 2 - Define and activate contact surfaces

Release tensile stress to squeeze members

(simulates tightening of bolt)

Step 3 - Apply external tensile load on joint

Measure bolt load as function of external load

83

Finite Element Results - Animation

http://personal.egr.uri.edu/taggart/courses/mce301/ex84/ex84.html

84

Finite Element Results

0.0

10.0

20.0

30.0

40.0

50.0

0.0 10.0 20.0 30.0 40.0 50.0

Bo

lt L

oa

d (

ksi)

External Load (kip)

FEA

Theory

85

Gasketed Joints

Gaskets – used to prevent leaks at joints

86

Gasketed Joints

The bolt tension determines the pressure acting on the gasket. For N bolts, the

pressure is

Substituting

Gives

For uniform pressure on the gaskets, the recommended distance between the

bolts should be between 3 and 6 times the bolt diameter

where Db is the bolt circle diameter

g

iA

NCnPFp 1

NA

Fp

g

m

im FnPCF 1

63 dN

Db

87

Bolted and Riveted Joints in Shear

F

F

88

Failure Due to Bending of

Rivet or Member

2

tFM

cI

M

F

F

89

Failure Due to Shear of Rivet

A

F

F

F

90

Tensile Failure of Members

A

F

where A is the net section area.

May need to include stress

concentration effects – to be

discussed later in course

91

Bearing Failure

A

F

Is the average stress acting on the

contact area, A = t d.

92

Tear-out Failures

Shear Tear-out Failure Tensile Tear-out Failure

Tear-out failure can usually be avoided by placing the bolt or

rivet far enough from the edge

at

F

2

93

Design of Bolted and Riveted Shear Joints

Must consider all possible failure modes: Example 8-6

94

Shear Joints with Eccentric Loading

Joint must transmit both shear load and bending moment

95

Joint Loads

Direct load (primary shear)

Moment load (secondary shear)

n

VF 1'

DDCCBBAA rFrFrFrFM """"1

96

Example 8-7

97

Example 8-7

Resultant forces acting on each joint: