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machine elements Mechanical Engineering University of Gaziantep

### Transcript of Me307 machine elements formula sheet

• 1. Stress Analysis

Moment of Inertias

1. Atalet moment of inertia; 2. Polar moment of inertia;

2

xI y dA 2

yI x dA 2 2

( )zJ x y dA

Shape Ix Iy J

Rectangle bh3/12 hb3/12

2 212

bhb h

Triangle bh3/36 hb3/36 2 2

18

h bbh

Circle d4/64 d4/64 d4/32

Stresses

Normal Stresses Shear Stresses

Axial

Tensile F

A

Torsional

Tr

J

3

16 /T d for solid circular beam Compression F

A

Bending

b

Mc

I

3

32b

M

d

for solid circular beam

Transverse

(Flexural)

VQ

Ib , Q A y

max 4 / 3V A for solid circular beam

max 2 /V A for hollow circular section

max 3 / 2V A for rectangular beam

Principle stresses

2

2

1,22 2

x y x y

xy

2

tan 2xy

x y

Max. and min shear stresses

2

2

1,22

x y

xy

Von-Mises stresses 2 2

1 1 2 2' or 2 2

' 3x xy (for biaxial)

Stress States

Triaxial stress state

2 311

E E

1 322

E E

3 1 22

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 it

p a p b a b p p r

b a

2 2 2 2 2

2 2

( ) /i o o ir

p a p b a b p p r

b a

2

2 2

il

p a

b a

• If the external pressure is zero (po=0);

2 2

2 2 21it

a p b

b a r

2 2

2 2 21ir

a p b

b a r

r=a r ip 2 2

2 2t i

b ap

b a

r=b 0r

2

2 2

2t i

ap

b a

If the internal pressure is zero (pi=0);

2 2

2 2 21t o

b ap

b a r

2 2

2 2 21r o

b ap

b a r

r=a 0r 2

2 2

2t o

bp

b a

r=b r op

2 2

2 2t o

b ap

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

• 2. Deflection Analysis

Fk

y , k=spring constant

T GJk

AEk

l

Castiglianos Theorem:

Strain Energy

2

2

F LU

AE Direct Shear Force

2

2

F LU

AG

2

2

T LU

GJ Bending Moment

2

2

MU dx

EI

Flexural Shear

2

2

CFU dx

GA , C is constant

Buckling Consideration:

Slenderness ratio=l

k

, I

kA

1/2

1

l 2 EC

k Sy

2

2

1

Critical Unit Load = Euler Column/

crPl l C E

k k A l k

; 2

2Pcr

C EI

l

2 2

1

2

ycry=

SPl l 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

Uy

F

Tl

GJ

• 3.Design For Static Strength

Ductile Materials

1. Max. Normal Stress Theory (MNST):

If, 1 2 3

1

ySn

3. Distortion Energy Theory

If, 1 2 3

2 2 2

1 2 2 3 3 1( ) ( ) ( )'2

For baxial stress state;

2 2' 3x xy

1

ySn

2. Max. Shear Stress Theory (MSST):

Yield strength in shear (Ssy)=Sy/2

1 3

max2

, for biaxial stress state;

max

1 2 24

2x xy

max

sySn

Brittle Materials

1. Max. Normal Stress Theory (MNST): 3. The Modified Mohr Theory (MMT)

If, 1 2 3 1

utSn

or

3

ucSn

If, 1 2 3

31

3

1

uc

uc ut

ut

SS

S S

S

3

3

Sn

2. The Column Mohr Theory (CMT) or Internal

Friction Theory (IFT):

31

3

1

uc

uc

ut

SS

S

S

3

3

Sn

• 5. Design for Fatigue Strength

Endurance limit for test specimen (Se);

For ductile materials:

Se=0.5 Sut if Sut

• 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 mR

m

Fd L dT

2 d L

m mL

m

Fd d LT

2 d L

Or considering tan ;

mRFd

T tan2

mRFd

T tan2

If the friction between the stationary member and the collar of the screw is taken into consideration;

c cmRd FFd

T tan2 2

c cmR

d FFdT 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

dthen screw is self locking.

Bearing Stresses

b 2 2r4pF

h d d

b

m

Fp

d th

pt

2

Shear Stresses

s

r

2F

d h

n

2F

dh

Bending Stresses

The maximum bending stress, m

6F

d Np

N=h/p

• Tensile or Compressive stresses

x

t

F

A

2

tt

dA

4

r mt

d dd

2

Combined Stresses

Rxy 3

t

16T

d

Based on distortion energy theory;

Rxy 3

t

16T

d

2 2

x xy' 3 yS

n'

Based on maximum shear stres theory;

2 2

max x xy

14

2

sy

max

Sn

8. Design of Bolted Joints

Fe=Feb+Fep Feb=CFe Fep=(1-C)Fe b

b m

kC =

k k

Fb=Fi+CFe Fm=Fi-(1-C)Fe

b bb

A Ek

L

m 1 2 n

1 1 1 1..........

k k k k

ib

b

F

k im

m

F

k

Shigley and Mishke approach;

For cone angle of 030 ,

ii

i

i

1.813E dk

1.15L 0.5dln 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, m1.813Ed

k2.885L 2.5d

2ln0.577L 2.5d

• For cone angle of 045 ,

ii

i

i

E dk

5 2L 0.5dln

2L 2.5d

If L1=L2=L/2 and materials are same, mEd

kL 0.5d

2ln 5L 2.5d

Wileman approach;

i(B d/ L)

m ik 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

dB

5 L

m eq

2

1k E d e

2 1 B

1 2eq

1 2

E EE

E E

2

1

0.1dB

L

8

11

2

LB 1

L

b y tF S A or b p tF S A p yS 0.85S mF 0

e i p t e1 C nF F S A CnF n=load factor of safety

F1 C

eat

CnF

2A im a

t

F

A s

a m

e u

1n

S S

t u e uis e

A S CnF SF 1

n 2 S

Fi=the maximum value of preload for there is no fatigue failure.

Limitations:

p i p0.6F F 0.9F where p t pF A S

e utimax t ute

cF n SF A S 1

2N S

e ei t pF cF

(1 c) F A SN N

b3.5d c 10d b180

cN

• 9. Design of Riveted Joints

Shearing of Rivets:

F

A , F=Force on each rivet

2d

A4

Secondary Shear Force

ii N

2

i

1

MrF ''

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

FF'

A

10. Design of Welded Joints

Primary Shear Stress

F

'A

uJ 0.707hJ

Secondary Shear Stress

Mr

''J

uI 0.707hI

Bending Stress

Mc

I

• 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

• 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; a