Problem 5-15 & 16 Extra Part of Problem 5-15 pkwon/me471/Lect 5.2.pdfآ  5! Brittle Materials!...

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Transcript of Problem 5-15 & 16 Extra Part of Problem 5-15 pkwon/me471/Lect 5.2.pdfآ  5! Brittle Materials!...

  • 5

    Brittle Materials

    Modified I Mohr

    ( )

    BA uc

    B

    A

    B BA

    uc

    B

    utuc

    Autuc

    A

    B BA

    BA ut

    A

    n S

    nSSS SS

    n S

    σσσ

    σ σ

    σσ σσ

    σ σ

    σσ

    σσσ

    ≥≥−=

    >≥≥=− −

    ≤≥≥

    ≥≥=

    0

    1and01

    1and0

    0

    5-11 Selection of Failure Criteria

    Problem 5-15 & 16

    ( )

    ( ) ( )

    ( ) ksi175.9

    2375.0 375.0)4(190

    ksi52.27 4375.0

    375.0)6(190

    4

    4

    ===

    ===

    π τ

    π σ

    J Tc I Mc

    xz

    x

    967.0 076.33 32

    32076.33 54.16175.976.13

    76.13 2 52.27

    22

    ==

    =>=

    =+=

    ==

    n

    SD R

    C

    yA

    D

    A

    σx

    τxz

    ( ) ( )

    007.1 78.31 32

    78.31175.9352.273 222/122

    ==

    =+=+=

    n

    xyx τσσ

    Extra Part of Problem 5-15

    ( ) ksi94.4547.11328.41:Misesvon

    ksi23.4747.1164.2022;64.20 2 28.41:Tresca

    ksi47.11175.925.1175.9 ksi28.4152.275.152.27

    133.0/;33.1/

    22

    22

    '

    =+=

    =+×=×===

    =×==

    =×==

    ==

    req

    req y

    txz

    tsx

    y S

    RSC

    K K

    drdD

    τ

    σ

    ¾”

    1”

    0.1”

    Determine the required yield strength

    of a material after considering the stress

    concentration factors.

  • 6

    21

    Problem 5-5

    3/2

    )3/4)(2/( 3

    2 max

    r rryAQ

    =

    == ππ

    ( ) ksi21.1630.1609.0 4/)5.0( )5.0(1600

    5.0 75

    42

    =+−=

    +−=

    +−=

    ππ

    σ I cM

    A R

    x

    c

    y y

    Mz=125(8)-75(3) =775 lb-in

    Mx=200(8) =1600 lb-in

    A

    B

    Ty=200(3 ) =600 lb-in

    A

    ( ) ( ) ( ) ksi84.22/5.0

    )5.0(600 )1(4/5.0 3/)5.0(2125

    44

    3

    max

    −= −

    +=

    +=

    ππ

    τ J cM

    It QR yx

    xy

    B ( ) ksi78.7

    4/)5.0( )5.0(775

    5.0 75

    42

    =

    +−=

    +−=

    ππ

    σ I cM

    A R z c

    y y

    ( ) ( ) ( ) ksi4.3

    2/5.0 )5.0(600

    )1(4/5.0 3/)5.0(2200

    44

    3

    max

    =

    +=

    +=

    ππ

    τ J cM

    It QR yz

    yz

    A

    B

    y

    x

    z

    3’

    8’

    Fy=75lb Fx=125lb

    Fz=200lb

    d = 1”

    π3 4r

    Problem 5-5

    Mz=125(8)-75(3) =775 lb-in

    Mx=200(8) =1600 lb-in

    A

    B

    Ty=200(3) =600 lb-in

    75lb

    125lb

    200lb

    775 lb-in

    1600 lb-in

    125lb

    200lb

    inlb82.1777

    1600775 22

    −=

    +=RM

    ( ) lb9.2484.2532sin85.235 lb85.235200125 22

    =−⋅=

    =+=

    c

    R

    V V

    32o

    25.84o

    ( ) ( ) ksi195.18095.01.18

    5.0 75

    45.0 5.082.1777

    24

    −=−−=

    − ⋅

    −=−−= ππ

    σ A R

    I cM yR

    ( )( ) ksi095.3055.304.0

    2 600

    24 329.24

    44

    3

    =+=

    + ⋅

    =+= c c

    cc c

    J Tc

    Ib VQ

    ππ τ

    ksi34.214.52.922 22 =+== RD

    Problem 5-5

    1’

    1.1’

    0.05’

    D / d =1.1; r / d = 0.05 σ x ' = Kts27.52+Kt 0.095=1.85×18.1+1.87×0.095= 33.66ksi

    τ xz = Kt9.175=1.25×3.055= 3.82ksi

    Tresca :C = 33.66 2

    =16.85; Sy req = 2×R = 2× 16.852 +3.822 = 34.555ksi

    vonMises : S y

    req = 33.662 +3 3.82( )2 = 34.3ksi

    Determine the required yield strength of a material after considering the stress concentration factors.

    Rectangular plate with hole subjected to axial load. (a) Plate with cross-sectional plane; (b) one-half of plate with stress distribution; (c) plate with elliptical hole subjected to axial load.

    Rectangular Plate with Hole

    5-12 Introduction of Fracture Mechanics

  • 7

    (a) Mode I, opening; (b) mode II, sliding; (c) Mode III, tearing.

    Three Modes of Fracture

    Infinite Plate with Crack

    ( )⎩ ⎨ ⎧

    + =

    =

    ⎟ ⎠

    ⎞ ⎜ ⎝

    ⎛ +=

    ⎟ ⎠

    ⎞ ⎜ ⎝

    ⎛ −=

    yx z

    I xy

    I y

    I x

    r K r

    K r

    K

    σσν σ

    θθθ π

    τ

    θθθ π

    σ

    θθθ π

    σ

    0 2 3cos

    2 cos 2

    sin 2

    2 3sin

    2 sin1

    2 cos

    2

    2 3sin

    2 sin1

    2 cos

    2

    ToughnessFracture:ICI KaK ≥= πσ Crack Propagate when

    Yield stress and fracture toughness data for selected engineering materials at room temperature.

    Fracture Toughness

    propagateCrackICI KaK ≥= πβσ

    aKI πβσ=Other geometries

    I

    IC

    K Kn =Factor of Safety

  • 8

    Problem 5-43

    ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ −

    − =

    ⎟⎟ ⎠

    ⎞ ⎜⎜ ⎝

    ⎛ +

    − =

    2

    2

    22

    2

    2

    2

    22

    2

    1

    1

    r r

    rr pr

    r r

    rr pr

    o

    io

    ii r

    o

    io

    ii t

    σ

    σ