Me307 machine elements formula sheet

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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

    l ,k=Torsional spring rate for tension or compression loading

    AEk

    l

    Castiglianos Theorem:

    Strain Energy

    Axial Load

    2

    2

    F LU

    AE Direct Shear Force

    2

    2

    F LU

    AG

    Torsional Load

    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

    1Critital Unit Load Johnson's Column

    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

    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

  • 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

    Static loading;

    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

    Critical load= iceF

    F1 C

    Dynamic Loading:

    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