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

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

View Full Document Right Arrow Icon
± ± ± 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
Background image of page 1

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

View Full DocumentRight Arrow Icon
±± ± 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
Background image of page 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.
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
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 Right Arrow Icon
Ask a homework question - tutors are online