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Me307 machine elements formula sheet Erdi Karaçal Mechanical Engineer University of Gaziantep
1. 1. Stress Analysis
Moment of Inertias
1. Atalet moment of inertia; 2. Polar moment of inertia;
2
x I y dA
2
y I x dA
2 2
( ) z J x y dA
Shape Ix Iy J
Rectangle bh3/12 hb3/12 2 2
12
bh
b h
Triangle bh3/36 hb3/36 2 2
18
h b
bh
Circle πd4/64 πd4/64 πd4/32
Stresses
Normal Stresses Shear Stresses
Axial
Tensile
F
A
Torsional
Tr
J
3
Compression 16T / d for solid circular beam
F
A
Bending
b
Mc
I
3
32
b
M
d
for solid circular beam
Transverse
(Flexural)
VQ
Ib
, Q A y
max 4V / 3A for solid circular beam
max 2V / A for hollow circular section
max 3V / 2A for rectangular beam
Principle stresses
2
2
1,2
2 2
x y x y
xy
2
tan 2
xy
x y
Max. and min shear stresses
2
2
1,2
2
x y
xy
Von-Mises stresses
2 2
1 1 2 2 ' or 2 2
' 3 x xy (for biaxial)
Stress States
Triaxial stress state
1 2 3
1
E E
2 1 3
2
E E
3 1 2
2
E E
Stress in Cylinders
Thick-Walled (t/r>1/20) Wessels (internally and externally pressurized cyclinders):
2 2 2 2 2
2 2
( ) / i o o i
t
p a p b a b p p r
b a
2 2 2 2 2
2 2
( ) / i o o i
r
p a p b a b p p r
b a
2
2 2
i
l
p a
b a
2. If the external pressure is zero (po=0);
2 2
2 2 2
1 i
t
a p b
b a r
2 2
2 2 2
1 i
r
a p b
b a r
r=a r pi
2 2
t i 2 2
b a
p
b a
r=b 0 r
2
2 2
2
t i
a
p
b a
If the internal pressure is zero (pi=0);
2 2
2 2 2
1 t o
b a
p
b a r
2 2
2 2 2
1 r o
b a
p
b a r
r=a 0 r
2
2 2
2
t o
b
p
b a
r=b r o p
2 2
t o 2 2
b a
p
b a
a=inside radius of the cylinder b=outside radius of the cylinder pi=internal pressure po=external pressure
Thin-Walled Wessels(t/r<1/20):
2
i
t
pd
t
4
i
l
pd
t
Curved Members In Flexure:
A
r
dA
( )
My
Ae r y
o
o
o
Mc
Aer
, i
i
i
Mc
Aer
Press and Shrink Fit:
2 2
it 2 2
b a
p
b a
2 2
ot 2 2
c b
p
c b
2 2
o 2 2 o
o
bp c b
E c b
2 2
i 2 2 i
i
bp b a
E b a
2 2 2 2
o i 2 2 o 2 2 i
o i
bp c b bp b a
E c b E b a
2 2 2 2
2 2 2
; interface pressure
2
o i
E c b b a
if E E E = p =
b b c a
3. 2. Deflection Analysis
F
k
y
, k=spring constant
T GJ
k
l
,k=Torsional spring rate for tension or compression loading
AE
k
l
Castigliano’s Theorem:
Strain Energy
Axial Load
2
2
F L
U
AE
Direct Shear Force
2
2
F L
U
AG
Torsional Load
2
2
T L
U
GJ
Bending Moment
2
2
M
U dx
EI
Flexural Shear
2
2
CF
U dx
GA
, C is constant
Buckling Consideration:
Slenderness ratio=
l
k
,
I
k
A
1/2
1
l 2 EC
k Sy
2
2
1
Critical Unit Load = Euler Column
/
cr l l P C E
k k A l k
;
2
2
P
cr
C EI
l
2 2
1
1
Critital Unit Load Johnson's Column
2
cr y
y =
l l P S l
= S
k k A CE k
1. Both ends are rounded-simply supported C=1
2. Both ends are fixed C=4
3. One end fixed, one end rounded and guided C=2
4. One end fixed, one end free C=1/4
U Total energy
F Force on the deflection point
Angular deflection
U
y
F
Tl
GJ
4. 3.Design For Static Strength
Ductile Materials
1. Max. Normal Stress Theory (MNST):
If, 1 2 3
1
y S
n
3. Distortion Energy Theory
If, 1 2 3
2 2 2
1 2 2 3 3 1 ( ) ( ) ( )
'
2
For baxial stress state;
2 2
' 3 x xy
1
y S
n
2. Max. Shear Stress Theory (MSST):
Yield strength in shear (Ssy)=Sy/2
1 3
max
2
, for biaxial stress state;
max
1 2 2
4
2
x xy
max
sy S
n
Brittle Materials
1. Max. Normal Stress Theory (MNST): 3. The Modified Mohr Theory (MMT)
If, 1 2 3
1
ut S
n
or
3
uc S
n
If, 1 2 3
3
1
3
1
uc
uc ut
ut
S
S
S S
S
3
3
S
n
2. The Column Mohr Theory (CMT) or Internal
Friction Theory (IFT):
3
1
3
1
uc
uc
ut
S
S
S
S
3
3
S
n
5. 5. Design for Fatigue Strength
Endurance limit for test specimen (Se’);
For ductile materials:
Se’=0.5 Sut if Sut<1400 MPa
Se’=700 MPa if Sut 1400 MPa
For irons:
Se’=0.4 Sut if Sut<400 MPa
Se’=160 MPa if Sut 400 MPa
For Aliminiums:
Se’=0.4 Sut if Sut<330 MPa
Se’=130 MPa if Sut 330 MPa
For copper alloys:
Se’ 0.4 Sut if Sut<280 MPa
Se’ 100 MPa if Sut 280 MPa
Se = ka kb kc kd ke Se’
Sf=10c Nb
u
e
1 0.8S
b log
3 S
2
u
e
0.8S
c log
S
ka= surface factor, ka=aSut
b
Surface Finish Factor a Factor b
Ground 1.58 -0.065
Machined or Cold Drawn 4.51 -0.265
Hot Rolled 57.7 -0.718
As Forged 272 -0.995
kb= size factor;
kb=1 if d 8 mm and kb= 1.189d-0.097 if 8 mm<d 250 mm for bending & torsional loading.
For non-rotating element, 0.097
b eq k 1.189d deq=0.37d
For pure axial loading, kb=1 and Se’=0.45Sut
For combined loading, =1.11 if Sut 1520 MPa and =1 if Sut 1520 MPa for ductile materials.
kc=reliability factor
kd=temperature effects, kd=1 if T 3500 and kd=0.5 if 3500<T 5000
ke=stress concentration factor, ke=1/Kf Kf=1+q(Kt-1)
Kt=geometric stress concentration factor, q=notch sensitivity.
Modified Goodman Soderberg
Infinite Life Finite Life Infinite Life Finite Life
a m
e u
1
n =
σ σ
+
S S
a m
f u
1
n =
σ σ
+
S S
a m
e y
1
n =
σ σ
+
S S
a m
f y
1
n =
σ σ
+
S S
Fa=(Fmax-Fmin)/2 Fm=(Fmax+Fmin)/2
6. 6. Tolerances and Fits
TF=Cmax-Cmin dL=DU-c Cmax=DU-dL Cmin=DL-dU
TF=Imax+Cmax dU=dL+TS Imax=dU-DL Imin=dL-Du
TF=Imax-Imin dU=DL+Imax
TS=dU-dL TH=DU-DL TF=TH+TS
7. Design of Power Screws
m m
R
m
Fd L d
T
2 d L
m m
L
m
Fd d L
T
2 d L
Or considering tan ;
m
R
Fd
T tan
2
m
R
Fd
T tan
2
If the friction between the stationary member and the collar of the screw is taken into consideration;
m c c
R
Fd d F
T tan
2 2
m c c
R
Fd d F
T tan
2 2
o
R R
T FL
T 2 T
when collar friction is negligible, we obtain as,
tan
tan
If tan or
m
L
d
then screw is self locking.
Bearing Stresses
b 2 2
r
4pF
h d d
b
m
Fp
d th
p
t
2
Shear Stresses
For Screw Thread For Nut Thread
s
r
2F
d h
n
2F
dh
Bending Stresses
The maximum bending stress,
m
6F
d Np
N=h/p
7. Tensile or Compressive stresses
x
t
F
A
2
t
t
d
A
4
r m
t
d d
d
2
Combined Stresses
R
xy 3
t
16T
d
Based on distortion energy theory;
R
xy 3
t
16T
d
2 2
' x 3xy y S
n
'
Based on maximum shear stres theory;
2 2
max x xy
1
4
2
sy
max
S
n
8. Design of Bolted Joints
Fe=Feb+Fep Feb=CFe Fep=(1-C)Fe b
b m
k
C =
k k
Fb=Fi+CFe Fm=Fi-(1-C)Fe
b b
b
A E
k
L
m 1 2 n
1 1 1 1
..........
k k k k
i
b
b
F
k
i
m
m
F
k
Shigley and Mishke approach;
For cone angle of 0 30 ,
i
i
i
i
1.813E d
k
1.15L 0.5d
ln 5
1.15L 2.5d
m 1 2 n
1 1 1 1
..........
k k k k
If L1=L2=L/2 and materials are same, m
1.813Ed
k
2.885L 2.5d
2ln
0.577L 2.5d
8. For cone angle of 0 45 ,
i
i
i
i
E d
k
5 2L 0.5d
ln
2L 2.5d
If L1=L2=L/2 and materials are same, m
Ed
k
L 0.5d
2ln 5
L 2.5d
Wileman approach;
(Bid/L)
m i k EdA e
Where Ai and Bi are constants related to the material. For Steel Ai=0.78715 and Bi=0.62873, for
Aliminium Ai=0.79670 and Bi=0.63816, for Gray cast iron Ai=0.77871 and Bi=0.61616.
Filiz approach;
1
d
B
5 L
m eq
2
1
k E d e
2 1 B
1 2
eq
1 2
E E
E
E E
2
1
0.1d
B
L
8
1
1
2
L
B 1
L
Static loading;
b y t F S A or b p t F S A p y S 0.85S mF 0
e i p t e 1C nF F S A CnF n=load factor of safety
Critical load= i
ce
F
F
1 C
Dynamic Loading:
e
a
t
CnF
2A
i
m a
t
F
A
s
a m
e u
1
n
S S
t u e u
i
s e
A S CnF S
F 1
n 2 S
Fi=the maximum value of preload for there is no fatigue failure.
Limitations:
p i p 0.6F F 0.9F where p t p F A S
e ut
imax t ut
e
cF n S
F A S 1
2N S
e e
i t p
F cF
(1 c) F A S
N N
b 3.5d c 10d b
180
c
N
9. 9. Design of Riveted Joints
Shearing of Rivets:
F
A
, F=Force on each rivet
2 d
A
4
Secondary Shear Force
i
i N
2
i
1
Mr
F ''
r
Bearing (compression) Failure:
F
A
, A=td, t=thickness of the plate
Plate Tension Failure:
F
A
, A w Nd t
w=width of plate
N=number of rivets on the
selected cross section
Primary Shear Force
N
i
1
F
F'
A
10. Design of Welded Joints
Primary Shear Stress
F
'
A
u J 0.707hJ
Secondary Shear Stress
Mr
''
J
u I 0.707hI
Bending Stress
Mc
I
10. Table 9-3 Minimum weld-metal properties
AWS electrode
Number
n
Tensile Strength
MPa
Yield Strength
MPa
Percent
Elongation E60xx 420 340 17-25 E70xx 480 390 22 E80xx 530 460 19 E90xx 620 530 14-17 E100xx 690 600 13-16 E120xx 830 740 14
Table 9-5 Fatigue-strength reduction factors
Type of Weld
Kf Reinforced butt weld 1.2 Toe of transverse fillet weld 1.5 End of parallel fillet weld 2.7 T-butt joint with sharp corners 2.0
11. Table 9-1 Torsional Properties of Fillet Welds*
Weld
Throat Area
Location of G
Unit Polar Moment of Inertia
*G is centroid of weld group; h is weld size; plane of torque couple is in the plane of the paper; all welds are of the same size.
12. Table 9-2 Bending Properties of Fillet Welds*
Weld
Throat Area
Location of G
Unit Moment of Inertia
*Iu, unit moment of inertia, is taken about a horizontal axis through G, the centroid of the weld group; h is weld size; the plane of the bending couple is normal to the paper; all welds are of the same size