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

• Published on
12-Mar-2018

• View
219

6

Embed Size (px)

Transcript

• 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

• 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

=

dw 44

=

• Test:

Monday, February 21

Chapter 1