lec20

# lec20 - 2.001 MECHANICS AND MATERIALS I Lecture 22 Prof...

This preview shows pages 1–4. Sign up to view the full content.

± ± ± 2.001 - MECHANICS AND MATERIALS I Lecture # 11/27/2006 Prof. Carol Livermore Beam in pure bending ρ = radius of curvature ± xx = y σ xx = Ey ρρ Locating the neutral axis dA =0 A ρ Moment-Curvature M = ± 2 dA A ρ Special Case: E = constant Neutral Axis: ydA A Moment-Curvature M = EI I = y 2 dA ρ A Neutral Axis Shortcut 1. Symmetric cross section neutral axis in the center for E = constant For E = constant: Area 1: ² ( w w 1 dA 1 A 1 Area 2: ² ( w w 2 ) dA 2 A 2 Total: ²² ( w w 1 dA 1 +( w w 2 ) dA 2 A 1 A 2 1 22

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
±± ± wdA w 1 dA 1 w 2 dA 2 =0 A 1 + A 2 A 1 A 2 wA = w 1 A 2 + w 2 A 2 i w i A i w = A i i Foragenera lbeam : i w i A i E 0 w = A i E i i 3. Parallel Axis Theorem: I y =0 = I 0 + y 2 A EXAMPLE: bh 3 I top =+ y 2 bh 12 I totatl = I top + I mid + ... Efective Bending Stifness ( EI ) eff 2
± ( EI ) eff = E i I i y =0 i Beam Deﬂection (Displacement and slope) Recall for a one material beam 1 M ( x )= ρ ( x ) Note for E ± =constant : 1 M ( x )=( ) ρ ( x ) Note for gradual change in cross section, this works.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 05/05/2010 for the course MSE 2.001 taught by Professor Carollivermore during the Fall '06 term at MIT.

### Page1 / 10

lec20 - 2.001 MECHANICS AND MATERIALS I Lecture 22 Prof...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online