Lectures_LEC_36-Chapt 8-1

Urban and Civil

Nov 29, 2013 (4 years and 7 months ago)

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8
-
1

8
-
2

The Mechanics of Power Screws

8
-
3

Strength Constraints

8
-
4

Joints
-
Fasteners Stiffness

8
-
5

Joints
-
Member Stiffness

8
-
6

Bolt Strength

8
-
7

Tension Joints
-

8
-
8

Relating Bolt Torque to Bolt Tension

8
-
9

8
-
10

Joints

8
-
11

8
-
12

Shear Joints

8
-
13

Setscrews

8
-
14

Keys and Pins

8
-
15

Stochastic Considerations

8
-
3

Strength Constraints

8
-
4

Joints
-
Fasteners Stiffness

8
-
5

Joints
-
Member Stiffness

8
-
6

Bolt Strength

8
-
4

Joints: Fastener Stiffness

When

a

connection

is

desired

that

can

be

disassembled

without

destructive

methods

and

that

is

strong

enough

to

resist

external

tensile

moment

and

shear

or

a

combination

of

these,

then

the

simple

bolted

joint

using

hardened

steel

washers

is

a

good

solution
.

Twisting

the

nut

stretches

the

bolt

to

produce

the

clamping

force
.

This

clamping

force

is

called

the

pretention

or

bolt

.

This

force

exists

in

the

connection

after

the

nut

has

been

properly

tightened
.

Figure

8
-
13

A

bolted

connection

in

tension

by

the

forces

P
.

Note

the

use

of

two

washers
.

Note

how

the

extend

into

the

body

of

the

connection
.

This

is

usual

and

is

desired
.

L
G

is

the

grip

of

the

connection
.

Figure

8
-
14

Section

of

cylindrical

pressure

vessel
.

Hexagon
-

caps

crews

are

used

to

fasten

the

cylinder

to

the

body
.

Note

the

use

of

an

O
-
ring

seal
.

LG’

is

the

effective

length

of

the

connection

(See

Table

8
-
7
)

.

Joints: Fastener Stiffness

Figure

8
-
14

shows

another

tension
-

connection
.

This

joint

uses

cap

screws

into

one

of

the

members
.

Joints: Fastener Stiffness

An

alternative

approach

to

this

problem

(of

not

using

a

nut)

would

be

to

use

studs
.

A

stud

is

a

rod

on

both

ends
.

The

stud

is

screwed

into

the

lower

member

first
;

then

the

top

member

is

positioned

and

fastened

down

with

hardened

washers

and

nuts

Studs

are

regarded

as

permanent
,

and

so

the

joint

can

be

disassembled

merely

by

removing

the

nut

and

washer
.

Joints: Fastener Stiffness

Spring

Rate

:

The

ratio

between

the

force

applied

to

the

member

and

the

deflection

produced

by

that

force
.

The

grip

L
G

of

a

connection

is

the

total

thickness

of

the

clamped

material
.

Total

distance

between

the

underside

of

the

nut

to

the

bearing

face

of

the

bolt

;

includes

washer,

thickness

etc
.

The

grip

L
G

here

is

the

sum

of

the

thicknesses

of

both

members

and

both

washers
.

Table

8
-
7

Suggested

Procedure

for

Finding

fastener

Stiffness

To find different parameters use table 8
-
7

1 1 1
,
b d t
d t
b
d t
d t
d t
d t
d t
b
d t t d
k k k
k k
k
k k
A E A E
k k
l l
A A E
k
A l Al
 

 

A
t
: tensile stress area (Tables 8
-
1, 8
-
2),
l
t
: length of threaded portion of the grip;
A
d
:
major diameter area of fastener;
l
d
: length of unthreaded portion in grip.
k
b
: is the
estimated effective stiffness of the bolt or cap screw in the clamped zone.

For short bolts

k
b
=
k
t

From Chapter 5

(Springs in series)

In

joint

under

tension

the

members

are

under

compression

and

the

bolt

under

tension
:

k
b

=

equivalent

spring

constant

of

bolt

composed

of

k
t

and

k
d

parts

acting

as

springs

in

series
.

d

t

There may be more than two members included in the grip of the
fastener.

All together these act like compressive springs in series.

Equivalent spring constant
k
m

If one of the members is a soft gasket, its stiffness relative
to the other members is usually so small that for all
practical purposes the others can be neglected and only

With no gasket, the stiffness of the members is difficult to
obtain, except by experimentation.

nut and area is not uniform.

1 2 3
1 1 1 1 1
....
m i
k k k k k
   
8
-
5

Joints
-

Member Stiffness

8
-
5

Joints Member Stiffness

Joint pressure distribution theoretical models

Ito

used

ultrasonic

techniques

to

determine

pressure

distribution

at

the

member

interface
.

Results

show

that

pressure

stays

high

out

to

1
.
5

bolt

.

Ito

suggested

the

use

of

Rotscher’s

pressure

cone

method

for

stiffness

calculations

with

a

variable

cone

angle
.

This

method

is

quite

complicated
.

8
-
5

Joints Member Stiffness

Figure 8
-
15

We choose a simpler approach using a fixed cone
angle.

The contraction of an element of the cone of
thickness
dx

is subjected to a compressive force
P

is, from Eq. (5
-
3),

The area of the element is

Pdx
d
EA

8
-
5

Joints Member Stiffness

Figure 8
-
15b general cone
geometry using a half
-
apex
angle

.

2 2
2 2
0
tan
2 2
tan tan
2 2
i
D d
A r r x
D d D d
x x
  
  
 
   
    
 
   
   
 
 
 
  
  
  
  

Substituting this into the previous equation and integrating the resulting equation from
0

to
t

gives.

For Members made of Aluminum, hardened steel and cast iron
25
o
<

<33
o

With

=30
o
, this becomes

8
-
5

Joints Member Stiffness

(2 tan )( )
ln
tan (2 tan )( )
tan
(2 tan )( )
ln
(2 tan )( )
P t D d D d
Ed t D d D d
P Ed
k
t D d D d
t D d D d

  
 

  

  
 
  
  
(8
-
19)

0.574
(1.55 )( )
ln
(1.55 )( )
E d
k
t D d D d
t D d D d


  
  
(8
-
20)

Substituting this into the previous equation and integrating the resulting equation from
0

to
t

gives.

With

=30
o
, this becomes

8
-
5

Joints Member Stiffness

(2 tan )( )
ln
tan (2 tan )( )
tan
(2 tan )( )
ln
(2 tan )( )
P t D d D d
Ed t D d D d
P Ed
k
t D d D d
t D d D d

  
 

  

  
 
  
  
(8
-
19)

0.574
(1.55 )( )
ln
(1.55 )( )
E d
k
t D d D d
t D d D d


  
  
(8
-
20)

8
-
5

Joints Member Stiffness

CH
-
8 LEC 36 Slide
16

If members of the joint have the
same E with symmetrical frusta (l=2t)
,
then they act as two identical springs in series
k
m

=
k/2
.
For

=
30
°

and

D =
d
w

= 1.5 d,
this can be written as
,

Finite element analysis agree with

= 30
o

recommendation coinciding
exactly at the
aspect ratio
d/l

= 0.4
.

Additionally, FEM offered an exponential curve
-
fit of the form

where A and B are given in Table 8
-
8.

0.5774
0.5774 0.5
2ln 5
0.5774 2.5
m
E d
k
l d
l d

 

 

 
(8
-
22)

(/)
Bd l
m
k
Ae
Ed

(8
-
23)

8
-
5

Joints Member Stiffness

Figure 8
-
16

The dimensionless plot
of stiffness versus aspect
ratio of the members of
a bolted joint, showing
the relative accuracy of
method of
Rotsher
,
Michke

and
Motosh
,
compared to Finite
Element Analysis (FEA)
conducted by
Wileman
,
Choudury
, and Green.

8
-
5

Joints Member Stiffness

Bolt strength is specified by:

1.
minimum proof strength S
p

2.
F
p
,

3.
and minimum tensile strength,
S
ut

The

is the maximum load (force) that a
bolt can withstand without acquiring a
permanent set.

The
proof strength
is the
quotient

of the
proof
and the
tensile
-
stress area
.

The
proof strength
90%

of the
0.2%

offset yield strength.

8
-
6

Bolt Strength

The SAE specifications are given in

Table 8
-
9

according to
minimum tensile strength.

The ASTM Specs for steel bolts (structural)
are in
Table 8
-
10
.

Metric Specs are in
Table 8
-
11.

If
S
p

not available use:

S
p

=0.85
S
y

F
p

=
A
t

S
p

8
-
6

Bolt Strength