Stress Conc Factors

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    Stress concentration factorsStress concentration factors

    L. VerganiL. Vergani

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    POLITECNICO DI MILANO

    Load and sign convention

    Any applied load can be classified with respect to time in the following

    ways:

    Static load- load is gradually applied and equilibrium is reachedin a relatively short time. The structure experiences no dynamic

    effect.

    Sustained load- Load (such as the weight of a structure) is

    constant over a long time

    Impact load- Load is rapidly applied An impact load is usuallyattributed to an energy imparted to a system

    Cyclic load- Load can vary and even reverse itself in sign and

    has a characteristic period with respect to time

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    Load and sign convention

    And with respect to the area over which it is applied in the following ways:

    Concentrated load- Load is applied to a small area

    Distributed load- Load is spread along the entire area

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    Load and sign convention

    Classification with respect the direction

    Normal load: the load passes through the centroid of the

    resisting areaPositive: tensile

    Negative:compression

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    Load and sign convention

    Shear force: the force P is in the same direction of V.

    The shear force is positive if the force direction and the

    normal direction are both positive or negative

    positive

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    POLITECNICO DI MILANO

    Load and sign convention

    Bending moment: load is applied

    transversely to the longitudinal axis

    of the member.

    Topper half of the

    member positive

    If the tensile stresses are in the

    positive direction the bendingmoment is positive

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    Load and sign convention

    Torsion moment: load is applied in order to apply a

    twisting motion

    Right hand rule

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    Load and sign convention

    Support reaction:

    If the support preventstranslation in a given

    direction, a force is

    developed on the memberin that direction.

    If the support prevents

    rotation a coupled moment

    is averted on the member.

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    Load and sign convention

    Static equilibrium:

    Equilibrium of a body requires:A balance of forces to prevent the body from translating along to a

    straight or curved path;A balance of of moments to prevent to prevent the body from rotating

    Equation:

    Py= 0Px= 0 P

    z= 0M x= 0 M y= 0 Mz= 0

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    POLITECNICO DI MILANO

    Stress concentration factors

    If the cross section of a load-carrying member

    varies gradually the stresses are accuratelycalculated by the previous equations.

    If there are abrupt changes ( as: holes,

    notches, fillets) in the cross section themechanics of materials can not predict the high

    values of the stresses.

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    POLITECNICO DI MILANO

    Stress concentration factors

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    Stress concentration factors

    Kt=maximum stressnominal stress

    = max

    nom

    nom= F

    (h d)b

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    Stress concentration factors

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    POLITECNICO DI MILANO

    Stress concentration factors

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    POLITECNICO DI MILANO

    Stress concentration factors

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    POLITECNICO DI MILANO

    Stress concentration factors

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    POLITECNICO DI MILANO

    Stress concentration factors

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    POLITECNICO DI MILANO

    Stress concentration factors

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    Stress concentration factors

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    Stress concentration factors

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    POLITECNICO DI MILANO

    Stress concentration factors

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    Stress concentration factors

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    Stress concentration factors

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    Stress concentration factors

    Kt is the elastic stress concentration factor

    The values are not valid if there is yielding

    Kt is used to obtain the peak stress near a

    notches Kt is not related to the failure of a crcaked

    member

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    Stress concentration factors

    d=20 mm

    D=26r=2 mm

    T=100 Nm

    n=?

    max=?

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    POLITECNICO DI MILANO

    Stress concentration factors

    max=Ktn

    max=Ktn

    -n-n

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    Stress concentration factors

    schema

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    Stress concentration factors

    If we neglect the curvature

    If we consider a small

    thickness vessel The most stressed zones

    are near the nozzles

    a

    c

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    POLITECNICO DI MILANO

    Stress concentration factors

    max sn

    max=Ktn=Kt

    n

    E

    max=Ktn

    max=Ktn

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    Fully plastic yielding

    Ductile material

    model

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    If we apply P>Pi ?

    h=110 mm

    d=10 mm

    b=10 mm

    Fully plastic yielding

    h

    d

    P

    P

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    Fully plastic yielding

    The fully plastic yielding can be estimated

    considering a perfect plastic material

    Ppt=sn(h d)b

    sn

    Ppt

    Ppt

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    Fully plastic yielding

    If the material is ductile, beyond the point of

    yielding the local strains are larger thanwould be estimated from elastic analysis.

    When the applied load increases a larger

    volume of material yields.

    When yielding spreads to the entire cross

    sectional area is the fully plastic yielding.

    Notch stress and strain

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    Notch stress and strain

    Neubers rule 2t

    KKK =

    nom

    eff

    nom

    eff

    K

    K

    max

    max

    =

    =

    E

    KK nomtnomnomteffeff

    222

    maxmax

    ==

    In order to evaluate the strain value:

    Notch stress and strain

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    Notch stress and strain

    Perfect plastic material

    r

    r

    maxeffmaxeff

    max

    =snloading

    r=

    sn- K

    t

    nom

    r=

    max- K

    t

    nom/E

    unloading

    E

    K

    sn

    nomt

    2

    max

    )(=

    EK

    sn

    nomt

    2

    max)(=

    Notch stress and strain

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    Notch stress and strain

    loading

    unloading

    E

    K nt2

    om )( =

    E

    K

    E

    K

    sn

    ntnt

    2

    )()( 2om2

    om =

    =

    r

    r

    maxeff

    maxeff

    E

    K

    E

    K

    E

    K

    sn

    nomt

    sn

    nomt

    sn

    nomt

    r

    2

    )(

    2

    )()( 222max

    =

    ==

    snr =

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    Example