Elastic Strain, Deflection & Stability

Stress can not be measured but strain can Strain gage technology

Linearly elastic stress-strain relationship (Hooke’s Law)

strain: (uniaxial stress)

Single-Element (horizontal )

Two-Element (horiz. & vertic.)

Three-Element (all directions) equiangular rectangular

E1

1δ

=εE…Young’s Modulus

(Elasticity Modulus)

uniaxial: E

11

δ=ε

13,2 ε⋅ν−=ε biaxial:

EE21

1νδ

−δ

=ε

EE12

2νδ

−δ

=ε

EE21

3νδ

−νδ

−=ε triaxial:

EEE321

1νδ

−νδ

−δ

=ε

EEE312

2νδ

−νδ

−δ

=ε

EEE213

3νδ

−νδ

−δ

=ε

dy (neg.)

dz (neg.)

dx

Axial strain

also causes Lateral strain (Poisson’s Ratio)

strainaxialstrainlateral

=ν

Shear strain normally can’t be measured directly.

Shear strain: (Hook’s Law) G…shear modulus of elasticity

dx

γ Gτ

=γ

( )ν+=

12EG

uniaxial: E

11

δ=ε

dy (neg.)

dz (neg.)

dx

Axial strain

also causes Lateral strain (Poisson’s Ratio)

strainaxialstrainlateral

=ν

Uniaxial Linear Strain:

Strain Φ-direction: Φ⋅⎟⎠⎞

⎜⎝⎛ ε−ε

+ε+ε

=εΦ 2cos22

2121

Shear Strain Φ-direction: Φ⋅ε−ε

=γ Φ 2sin

2221

Mohr’s Circle:

•Half shear strain

+γ/2

+ε

•Angles twicethe real angles

substitute: σ? ε, τ? γ/2substitute: σ→ ε, τ→ γ/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

θ 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

...

...

...

δ

Slope dxdδ

=θ

2

2

dxd

EIM δ

=

Deflectio

Bending Moment EI

dxdM 2

2δ=

ShearForces EI

dxdV 3

3δ=

Load IntensityEI

dxdw 4

4δ=

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)