Elastic Strain, Deflection & · PDF fileLinearly elastic stress-strain relationship...
Transcript of Elastic Strain, Deflection & · PDF fileLinearly elastic stress-strain relationship...
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)