Unformatted text preview: ²་ Initial distance dx between two the two planes
remains unchanged at the neutral surface.
²་ Lines above or below the neutral surface will
either lengthen or shorten.
²་ This gives rise to normal strains εx within the beam
2/18/13 M. Mello/Georgia Tech Aerospace 9 LONGITUDINAL STRAIN IN BEAMS (cont.) ²་ Consider longitudinal line segment ef between planes mn and pq ²་ The longitudinal line ef in the deformed beam remains at a distance y from the neutral surface ²་ Length of line ef aHer bending: L1 = (⇢ ²་ but, d✓ = y ) d✓
dx
⇢ ²་ and so, dx
⇢
elongation = L1 L1 = dx y y dx
⇢ dx = ) normal strain: ✏x = y
⇢ ²་ Note: y>0 above the neutral surface and y<0 for points below the neutral surface ²་ ALSO, THE SIGN CONVENTION FOR ρ preserves tradiaonal sign convenaon for strain 2/18/13
M. Mello/Georgia Tech Aerospace 10 NORMAL STRESSES IN LINEAR ELASTIC BEAMS ²་ LONGITUDINAL ELEMENTS IN A BEAM ARE SUBJECTED ONLY TO TENSION OR COMPRESSION ²་ THIS IS CALLED A UNIAXIAL STATE OF STRESS ( WE ONLY HAVE ONE STRESS COMPONENT WHICH ACTS ON THE BEAM ELEMENTS (σx) ²་ INVOKE HOOKE’S LAW (1D) VERSION WHICH WE HAVE ALREADY USED FOR AXIAL MEMBERS (AND THE SHEAR STRESS VERSON FOR AXIAL TORSION) Hooke’s law: x = E ✏x But we have shown that: ✏x = y
⇢ y
Hence, x = E (5 7) ⇢ Or, equivalently: x = E y NEXT STEPS: (1) how do locate the neutral axis of a cross secaon? (2) Need to establish a relaaon between stress and bending moment (3) BENDING MOMENT IS RELATED TO CURVATURE (AS WE WILL SEE) AND THIS FORGES THE LINK BETWEEN STRESS AND BENDING MOMENT 2/18/13 M. Mello/Georgia Tech Aerospace 11 LOCATION OF THE NEUTRAL AXIS • Force acang on element: x dA = E ydA • Resultant force acang on cross secaon: Z
Z
F=
E ydA = 0
x dA =
A A • Curvature (κ) and elasac modulus (E) are nonzero constants • Hence, z axis must pass through the centroid of the cross secaon Z ydA = 0 (5 8) A • But the z axis is the neutral axis • Neutral axis must pass through the centroid when the beam material is linear elasac (obeys Hooke’s law) and when no axial forces act on the cross secaon. 2/18/13 M. Mello/Georgia Tech Aerospace 12...
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 Spring '09
 ZHU
 Force, neutral axis, Radius of curvature

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