Chapter 5 Elastic Strain, Deflection, and notes...Chapter 5 Elastic Strain, Deflection, ... Chapter...

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Transcript of Chapter 5 Elastic Strain, Deflection, and notes...Chapter 5 Elastic Strain, Deflection, ... Chapter...

  • Chapter 5 Elastic Strain, Deflection, and Stability 1

    Chapter 5 Elastic Strain, Deflection, and Stability

    Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction

    by x also causes a strain in the y-direction & z-direction by

  • Chapter 5 Elastic Strain, Deflection, and Stability 2

    E =

    Resulting Strain Each Direction

    Stress x y z

    x

    y

    z

  • Chapter 5 Elastic Strain, Deflection, and Stability 3

    Adding the columns to obtain the total strain in each direction Shear strain Note: shear strain on a given plane is _________________by the shear stresses on other planes.

    Generalized Hookes Law Only ______________________elastic constants are needed for an__________ material.

    x

    y

    z

    xy = , yz = , zx =

    G =

  • Chapter 5 Elastic Strain, Defl

    Only two are independent elastic constant

    Mohrs circle /2

    from G =

    Pure shear stress

    ection, and Stability 4

    1 =

    2 =

    3 =

    =

    /

  • Chapter 5 Elastic Strain, Deflection, and Stability 5

    Example:

    1. The stress that develops in the y-direction.

    2. The strain in the z-direction.

    3. The strain in the x-direction.

    4. The stiffness E = z / z in the z-direciton.

    Is this equal to E ?

    y = & x =

  • Chapter 5 Elastic Strain, Deflection, and Stability 6

    1. y = = 0

    _______________________________________ y = 2. z =

    _______________________________________ =

    3. x = _______________________________________ =

    4. E = z / z =

  • Chapter 5 Elastic Strain, Deflection, and Stability 7

    Volumetric Strain & Hydrostatic Stress

    Volume changes associated with __________________________. Shear strains cause only __________________________ Since V=LWH

    =dV

    =VdV

    =

    ==VdV

    v

    == V 5.0 volumedecreasestresstensile> 5.0

  • Chapter 5 Elastic Strain, Deflection, and Stability 8

    Hydrostatic stresses Invariant Volumetric strain hydrostatic stress

    Constant ___________ modulus

    B=

    h = v =

  • Chapter 5 Elastic Strain, Deflection, and Stability 9

    Castiglianos Method

    Useful in computing elastic deflection and redundant reactions

    Deflection Figure 5.15 General load deflection curve for elastic range

    stored elastic energy is equal to _________________ times ___________________. Deflection, In general case,

    == UU

    == UddU

    =

    =

  • Chapter 5 Elastic Strain, Deflection, and Stability 10

    Axial Loading Case

    Sample problem 5.4

    =

    =

    =

    =

    =

    U

    U

    U

  • Chapter 5 Elastic Strain, Deflection, and Stability 11

    Sample problem 5.4 cont. 1. 2. 3.

    202LxxonlyvalidPxM

    VM

    ===

    ==

    Q

    =

    =

    =

    =U

    ==PU

  • Chapter 5 Elastic Strain, Deflection, and Stability 12

    Problem 5.15 (page233) What are the angular and linear displacements of point A of Figure 5.15? Known: Figure P.15 is given. Find: Calculate the angular and linear displacements of point A.

  • Chapter 5 Elastic Strain, Deflection, and Stability 13

    Problem 5.19 (page 234) Figure 5.19 shows a steel shaft supported by self-aligning bearings and subjected to a uniformly distributed load. Using Castiglianos method, determine the required diameter d to limit the deflection to 0.2mm.

    Known: A steel shaft supported by self-aligning bearings is subjected to a uniformly distributed load. Find: Using Castiglianos Method, determine the required diameter, d, to limit the deflection to 0.2mm. Assumption: 1. The steel shaft remains in the elastic region. 2. The transverse shear deflection is negligible. Analysis:

  • Chapter 5 Elastic Strain, Deflection, and Stability 14

    Problem 5.23(page 235) In order to reduce the deflection of the I-beam cantilever shown, a support is to be added at S. (a). What vertical force at S is needed to reduce the deflection at this point to zero? (b). What force is needed to cause an upward deflection at S of 5mm? (c). What can you say about the effect of these forces at S on the bending stresses at the point of beam attachment? Assumptions: 1. The beam remains elastic. 2. Transverse shear deflection is negligible. Analysis:

  • Chapter 5 Elastic Strain, Deflection, and Stability 15

    Redundant Reactions by Castiglianos Method

    Reduntant reaction: _____________ force or moment that is ___________________

    for equilibrium. As magnitude of a redundant reaction is varied, _____________________ changes,

    But _________________ remains. Castiglianos theorem states that the _________________associated with any

    reaction that can be varied without upsetting equilibrium. The deflection = ______________________.

  • Chapter 5 Elastic Strain, Deflection, and Stability 16

    Sample Problem 5.9

    Find: Determine the tension in the guy wire Assumption: 1. 2. 3. Analysis:

    Figure 5.22

    At point a

  • Chapter 5 Elastic Strain, Deflection, and Stability 17

    M=

    = Bending energy below point a The horizontal deflection at point a

    =

    == dyEIMu

    3

    0

    2

    2

    ===Fu0

    = F

  • Chapter 5 Elastic Strain, Deflection, and Stability 18

    Euler Column Buckling

    Figure 5.24 B=0 x=L, y=0

    Q

    0sin = LA

    =cr

    ==EIM

    dxyd2

    2

    =

    2AI =

    =crS

    =ESor cr

  • Chapter 5 Elastic Strain, Deflection, and Stability 19

    Fig5.25 Log-log plot of Euler Eq. 5.11 (dimensionless, hence applies to all materials within their elastic range).

    Fig5.26 Euler column buckling curves illustrated for two values of E and Sy.

    ESpLe

    cr 1.010/

    ==

  • Chapter 5 Elastic Strain, Deflection, and Stability 20

    Figure 5.27 Equivalent column lengths for various end conditions

    Figure 5.28 Euler and Johnson column curves illustrated for two valuses of E and Sy

  • Chapter 5 Elastic Strain, Deflection, and Stability 21

    Secant formula for the ______________loading, taking the _______________________ into account. Where c denotes the distance from the neutral bending plane to the extreme fiber.

    +==

    AEPLec

    SAPS

    cre

    ycrcr

    4)(sec)(1 2

  • Chapter 5 Elastic Strain, Deflection, and Stability 22

    Draft paper 1/2

  • Chapter 5 Elastic Strain, Deflection, and Stability 23

    Draft paper 2/2

    Chapter 5Elastic Strain, Deflection, and StabilityElastic Stress-Strain RelationshipA stress in the x-direction causes a strain in the x-directionResulting Strain Each Direction

    Adding the columns to obtain the total strain in each directionGeneralized Hookes Law

    Example:Volumetric Strain & Hydrostatic StressCastiglianos MethodDeflection

    Axial Loading Case

    Sample problem 5.4Redundant Reactions by Castiglianos Method