Elastic Strain, Deflection elastic stress-strain relationship (Hooke’s Law) strain: ... Deflection in direction of Load p186 Deflection ... Shear, Moment Deflection for Beams

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    12-Mar-2018

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  • Elastic Strain, Deflection & Stability

    Stress can not be measured but strain can Strain gage technology

    Linearly elastic stress-strain relationship (Hookes Law)

    strain: (uniaxial stress)

    Single-Element (horizontal )

    Two-Element (horiz. & vertic.)

    Three-Element (all directions) equiangular rectangular

    E1

    1

    =EYoungs Modulus

    (Elasticity Modulus)

  • uniaxial: E

    11

    =

    13,2 = biaxial:

    EE21

    1

    =

    EE12

    2

    =

    EE21

    3

    = triaxial:

    EEE321

    1

    =

    EEE312

    2

    =

    EEE213

    3

    =

    dy (neg.)

    dz (neg.)

    dx

    Axial strain

    also causes Lateral strain (Poissons Ratio)

    strainaxialstrainlateral

    =

  • Shear strain normally cant be measured directly.

    Shear strain: (Hooks Law) Gshear modulus of elasticity

    dx

    G

    =

    ( )+= 12EG

    uniaxial: E

    11

    =

    dy (neg.)

    dz (neg.)

    dx

    Axial strain

    also causes Lateral strain (Poissons Ratio)

    strainaxialstrainlateral

    =

    Uniaxial Linear Strain:

  • Strain -direction:

    +

    += 2cos22

    2121

    Shear Strain -direction: = 2sin22

    21

    Mohrs Circle:

    Half shear strain

    +/2

    +

    Angles twicethe real angles

    substitute: ? , ? /2substitute: , /2

    Mohr Strain Circle

  • Deflection or stiffness, rather than stress, is controlling factor in design

    satisfying rigidity preventing interference or disengagement of gears

    Elastic stable systems: small disturbance corrected be elastic forces

    Tension Bending Torsion Compression

    short column

    Elastic unstable systems: small disturbance can cause buckling (collapse)

    Compression slender column

    Elastic Strain, Deflection & Stability

  • Deflection in direction of Load p186

    Deflection not in direction of Load

    K Section Property (Table 5.2)

    Deflection Spring Rate

    Rigidity Property Section Material

    in radiant

  • Table 5.2: Section Properties for Torsional Deflection

  • Appendix D: Shear, Moment & Deflection for Beams

    Use Method of Superposition At any point you can sum the deflection due to individual loads

    Simply Supported Beams D-2

    Cantilever Beams D-1

    Beams with Fixed Ends D-3

    ...

    ...

    ...

  • Slope dxd

    =

    2

    2

    dxd

    EIM

    =

    Deflectio

    Bending Moment EIdx

    dM 22

    =

    ShearForces EIdx

    dV 33

    =

    Load IntensityEIdx

    dw 44

    =

  • Test:

    Monday, February 21

    Chapter 1

    Chapter 2.1-2.5 (Chapter 2.6 additional reading)

    Chapter 4.1-4.5 (Chapter 4.7 additional reading)

    Chapter 4.8-4.12 (Chapter 4.13-4.17 additional reading)