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LECTURE+18+COE-3001-A

Prisamtic cross section assumed symmetric

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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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