• r /n - I ii'. I 5 / ru I )( ::s u ~ - ...

.i
I

I, ( I •

I
' "

• _lcttAPTER - sl
• h • •

.. '

- '' '

'SHAFTS

Symbols ~ (With S.I. Units)

' •

. '

A Area of cross-sectio~, mm2

.. !'
b Section width, ·mm • .. • I_

. . .
.
'

C . ~uler's coefficient,· Key, Vfay ~eakening factor ; .

•
.
;, I 4

D Diameter of rod or shaft, mm . :

d Diameter of rod or shaft, •mm

J . I ,• .

~ •Inner diameter
. of hollow. shaft, ~' ; i
'. It ~
' .
'
' • ' •
\ .
' ,• I I

Outer _diameter of hollow shaft, mm

~ , . i. • • •

Modulus of ei'asticity· or' Youngs modulus, N/nun2

. ~ntricity, mm •·
:i'

- ._Force or·load~ N I
! .
' .
'

-
. l
Axial force,· N '
......
 (11
a,
DESIG N PARTICULARS EQUATIONS
7
T= 6x 10 xP .... (5.1)
Torque transmitted by the shaft .'
2xN
Pin kW
Nin rpm
Tin N-mm

(f) Diame ter of-eoli d shaft for steady loads:

When subject ed to =[
(I
d 16T ]113 .... (5.2)
,,
;9imple torque only 1t,:

I Tin N-mm
,: in N/mm
2

When subject ed to
d =[ 32 M ]l/3
1t ob .... (5.3)
simple bendin g momen t
only.
Min N-mm
2
ob in N/mm

113 113
When subjected to -
combined torque and
16 Te ]
d= [ _....;;;. (or) d= [ 32 Me ] .... (5.4)
.:. 7t 't --- 7t ab
bending moment including
. shock and fatigue fact.ors

DESIG N PARTI CULA RS

When subject ed to combin ed torque, bendin g momen t

and axi.al compreBBive load, includi ng
d= [ - T ]
16 -
7t "t
8
113
(or) d= [ 32 Me
7t ob
1 113
.... (5.5)

shock and fatigue factors. • I

' where

i1 I. -~ .
I

When IUbjeeted t.o ...~ (6.8)

simple t.orqae 'only
 Ut
DESIGN PARTIClJLARs EQUATIONS i»

When subjected to
aipiple bending do = [ 32 M r1s and tit = k do .... (5.7)
moment only x ob (1- k4}

ed torque and moment 113 113

Wh en subjected to combin factors. ] do =( 32Me ] .... (5.8 )
inc lud ing shock and fati gue do =[ 16Te (or)
. 1t ob (1 - k")
M 1t 't (1 - k")

C ~~r
and di =k do
2 • T)2
-0 wh ere Te = ✓ ('Kut • M) + (Kt

and Me = ~ [ Km •M + ✓c1<m
•M)2 + (Kt • T)~1

, and axi al com pre ssiv e loa

ed torq ue, ben din g mo me nt
Wh en sub ject ed to combin inc lud ing sho ck and
d, do= [ • 16 T e 4
]
118
(or) do= [ --32 -- M.,
=-
_1t o~ (1 - k")
1 113
, ... (5.9)
1t 't (1 - k )
fatigue factors.
and di =k do
Where

and .

''
M. = ![lK,. •l>! +. a F ~ ~i + k2\ 1+
2
: · . a F do Ci + k2) } ·+ (~
. •2
. T)
Km •.,M +. 8
... . .
I • . .

. .

.....
. •
.
I- .••• ;
, ....~-

.·2. 26 for find ends • •. · ·

llnined in bN rln p • ••..
"' L8 .;,. enda that .are pertly J'll ~
·.... . , . ·~
 DESI GN PART ICUL ARS EQUA TION S .·
aP
(iii) Duun eter of 80lid abaft for-· varial, Jt,.. hulin
(a) Soder berar equat ion for •impl e· load.f. ns:,
For variab le
torsio nal loads
,. Q Kr-tv
-
'tm • 1 •
-+---
""• 'ty - r.
.... (5.10).

For variab le
bendin g Joads 1· j) Kr 0 vb +
a'eb
Omb
---
1
Oy - f8
.... (5.11)

~
Kr 0 va +Oma 1 .... (5.12)
For variab le ---
axial loads
F<
I a'ea Oy - fa

Ob max - Ob min . _ Oa max - Oa min

'tmax - 'tmin , 2
'tv = 2 ; Ovb = 2 Ova -

+ 'tmin <Jb max + Ob min Oa max + Oa min

'tm =
'tmax
2 ;0 mb = 2 ; Oma = 2

16 T max 32 Mmax 4 F max

ds ; Obmax .= 3 ; Oama x= 2
'tmax= 1td 1td.
1t
16 Tmin 32 Maim 4 F mm .
't . - ---= = . a min -- _ 3 . , va min =___;;=
~
• ,.

min - . d3 , b
_ 1t d _ ~- ~, . _ ,t d
1t -

<e_= Ge· X. · X.. · X.s; a'lb~ Ge· Ka,·· Kar, ·.Ka; •

a.· .. 0 .-,.

Diam eter of solid shaft for simpl e variab le d =[ r.{ 8 Kr(Tm ax -T~n > + (Tmax + Tmin
O'e ·Ke· K.r.-: Kaz - · 'ty ·
l y/3
JJ
.... (5.13)

torsio nal load 1t

.
·'•

·- [ 16 r. { Kr <Mmu - Muwi> ' <Mmu + ~ ) } ]l/3'

Diam eter of solid shaft for simpl e variab le d = ~ . Ge ·. JG, . x_. . ~ ~ - . Gy . •••• (5.14)
bendi ng load

· Diameter of eolid abaft for simpl e variable uial .

tant:
e· !f:.
\i·!. ,-. -~---
T

Ti •
,- ·;

./
(ii)The result ant I "" ..
DOl'lll81iA~ 1111 . . . .
._~·: 1·
~ •

F~·~ ;.i,in ed ~ t . •:dal and tonf o~ loada

~=~ ~+ .;: ~
=7. •.~ .. (5.18)

.... (5.19)
.. ,'
 U1
DESIGN PARTICULARS :...
N

For combined variable be.;_dina ~d axial

loads:
i
I
I (J
The resultant. <Jrb -ab+<J __y_
•- r r•-r. .... (5.20)
normal stress . . 8

For combined variable bendinar, axial and

tonional Joada:

The resultant shear stress

'trbae = ½✓~ha + 4 ,t? = ~ .... (5.21)

The resultant ·normal stress

arbae = ½[ Orba + ✓~ba + 4 ,t? = ~ 1 .... (5.22)

where

- ~ • Ovb • Oy .
orb - Oe . Kt, . Kar .
Kaz + Omb

DESIGN PARTICULARS

16 ) 16 ( )
Omb = 1t ds ( Mmax + Mmin ; ~vb= 1t ds \ Mmu-:- Mmm

2 (Fmu- Faun
2 (Fmu + Fmin); Ova =--2
Oma =--2
)
•d ' •d

8 - '• · · 8 •.
3 (Tmu + Tnun); 'fv =~ ( Tmu -Tmm)
~=-
Kd : Kd
.' ' ., . .
, .
Substitute •d3 (1- k4) in the place ot d in. the
MocJiflcationa for the hollow· abaft . . above equations.

do
... •· ·.- do• Outer diameter of hollow _abaft , •.. .... ,. ·.•.
·, .
,. • I cit=- Inner diameter of hollow~.
• ' • • r ~ ••
-... , : d = Diameter of 10lid abaft . •
 Table 5.1: Shock and Fatigue Factors
- - - - - - - - - - - - - - - - - - ------- --· - ----- ----. .---- ---
Nature of load
Kt

1. Stationary shafta:
1.0 1.0
(i) Gradually applied load
1.5 to 2.0 1.5 to 2.0
(ii) Suddenly applied load

2. Rotatin g shafts:
1.5 1.0
(i) Gradually applied load
1.5 to 2.0 1.0 to 1.5
(ii) Suddenly applied load with minor shock
2.0 to 3.0 1.5 to 3.0
(iii) Suddenly applied load with major shock

DESIG N PARTICULARS EQUATIONS

Weakening factor due to key way on shafts C = 1.0- 0.2 ( ; )- 1.1 ( ~) .... (5.23)

Equation for torsional rigidity T G8 .... (5.24)

J=-, -

(iv) Critica l or 'whirli ng speeds of shafts: (roe)

w
For a shaft of negligible .... (5.25)
self-weight with single ~c=~ rad/s
attached disc
2
.. g=9.81 m/s
6 = Deflection at the disc. location in metres
•. .>
•

• limp)y 8Up

.... (6.26)

m. • Oritical apeed ot lbaft when .all the loada are actme

qether . '.
. . ·.
COct, CDc2 •·· Olen • Critical apeeda of ·abaft when the loads
W1 , W2 ••• Wn are acting individually .

. .. :,
O>m - Critical speed of shaft by ita self-weight only.
•- I • • .,!J•

Critical speed in rpm 30 •

Nc=-Cl >c .... (5.28)
. X
 DESIGN PARTICULARS EQUATIONS
...
.'
For a shaft of negligible • • •N •945.8 h ~ d L • (5 29)
c= ✓ rplll: V: en u1_~ "2 are m mm .... .
self-weight and two loads only 61 +.&i , . ·. •
. (v) Deftectiona on 1hafta due to varioua loadlnt
conditiona: W
wz3 .... (5.30)
For a shaft of negligible 6c= 48 EI
weight with central load A~ .~ DfB
in simple supports
•~- - - I ----'--ti
w

~
For the above conditions wz3
6c= 192 EI .... (5.31)
with fixed supports A~ ~8
,. I
w
Wa2 b2
For a shaft of negligible A7
le 6c= 3EU
.... (5.32)

'B
weight with a non-central
load in simple supports I• a ➔• b ,.,. .
i-
I "' (6c - Deflection at C)
"'
•·•• • I .w
•· For the .above
"~ conclitiona with
le

For a shaft of negligible .

r self-~eight with a . _ w per unit
· uniformly distributed / length . 5wz4
6c= 384 EI .... (5.34)
. load or a shaft _with 1°0000.~ 00M ZS' 8 .-
no load except it.a · A ...,_ C___...,_T'-
....____
aelf-weight only
t- ' "
. -.)

..
m.o 118.0 ..
t•
Bot rolled or forpc1 • . 0.25-0.8& , 482.0 . 482.0 2'1.0 . 2715.0 ' 271,0 . w.o 2'.
. ~ , !f•. •. ~
,
... o.86-0."5 ~.o •. 1520.0 280.o a1,.o a1,.o 130.0 22

l I~•, ~"'\ l"i ··~ ; . \lj~ •: J~!••~ ., :~• · :0.46-0.65. °!'..g&3.0• t ·.t 813.0 278.15 8".0 • •• 34&.0 .138.0 20

.294.0 • 382.0 .__. ·, ,382.~ ... 147.0 26

8 ~ 'Ji Nickel • . . -·
-------~----+----+-----r""-,-
Chrome Vanadium . •
0:16-0.25 ••588.0
0.26-0.35 620.0
588.0
620.0 sio:•o: 414.0 414.0
.
155.0 25
l
..,
J
j
J
J
 2
Table 5.3: l\JasjmllD:l Permissible Working Stresses for Shafts, N/mm • : ;
a,
• . ·,. . . . . ; .. ·;
:"_,.·':!· . . _ - . f '!
--P'
·: Grade of shafting. k' S~~,ll~e_: ~~~clin•:/ ~!m.ple ~~~on_ / ,Co~~~ed stress
-·
"Commercial Steel" 55: 55
110
shafting without allowance for-keyways
"Commercial Steel" 83 . 41 41
• shafting with ~owance for keyways

~ ........-: ,r-
- """'" _Table 5.4: (Column factor a)
- ~
, - ...- ~
•T ;- i,_' •.,
- - -
;o. ~ ··•r
.: - 50 ,._
115-
• • - 4 ... ,. ...:.
.. • 100 -·
•· .L'K . -. . 25 . - . - • - - 75 -~- .. -- - .
- --
: -
' - -
- ,- · 1.49 ' 1.78 2.02
- - a - -
1.00 • -- -
1.12 - 1.28 -- -_-
I l

:Table: 5.5:. Standard -Diameter of Shafts (mm)

-
6, 8, 10,. 12, 14, 16, 18, 20, 22,,. . 25, 28, 32, _36, _40, 45, 50, 56, 63, 71, 80, 90, 190, 110, 125, 140, 160, 180, 200,
220, 240, 260, 280, 300, 320, 34'0, r360, 380, 400, · 420, 440, 450, 480, 500, 530, 560, 600.

':l i . ' .'"-~:;·

:,