LECTURE 18 COE-3001-A

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Unformatted text preview: oments) ²་  We will learn how to predict the curvature of a beam In response to a load distribuaon. ²་  We will then consider the analysis and design of beams ²་  Ulamately this will learn to predict how beams deflect under load (module 8) M. Mello/Georgia Tech Aerospace 4 PURE BENDING AND NONUNIFORM BENDING PURE BENDING: FLEXURE OF A BEAM UNDER A PURE BENDING MOMENT NONUNIFORM BENDING: FLEXURE OF THE BEAM IN THE PRESENCE OF SHEAR FORCES M ( x) = M ( x ) = M1 Simply supported beam subjected to pure bending M2 Canalever beam subjected to pure bending NONUNIFORM BENDING M (x) = P a (interior span) 2/18/13 Simply supported beam subjected to four point bend M. Mello/Georgia Tech Aerospace We saw this last ame 5 BEAM CURVATURE CENTER OF CURVATURE O CURVATURE = 1 ⇢ 0 RADIUS OF CURVATURE (⇢) (normal to tangent lines at points m1, m2 on the curve) (5- 1) In actuality the deflecaons are typically quite small compared to the beam’s length. ds = ⇢d✓ = 1 d✓ = ⇢ ds (5- 2) For small deflecaons : ds ⇡ dx 1 d✓ Hence, = = (5- 3) ⇢ dx 2/18/13 M. Mello/Georgia Tech Aerospace 6 Curvature sign convenaon ²་  In general: = ( ) and ⇢ = ⇢(x) x ²་  Most importantly curvature depends on bending moment (we will see this very shortly)… ²་  As well as properaes of the beam itself (shape, cross secaon and type of material) ²་  If beam is prismaac and material is homogeneous, then curvature will vary only with bending moment. Just like the second derivaave and concavity of a curve in calculus 2/18/13 M. Mello/Georgia Tech Aerospace 7 LONGITUDINAL STRAIN IN BEAMS PRISAMTIC CROSS- SECTION ASSUMED (SYMMETRIC ABOUT THE Y- AXIS) ²་  FUNDAMENTAL ASSUMPTION OF BEAM THEORY: CROSS SECTONS REMAIN PLANE AND NORMALS TO THE LONGITUDINAL AXIS. ²་  CROSS SECTIONS MN AND PQ ROTATE WITH RESPECT TO EACH OTHER (ABOUT AXES PERPENDICULAR TO THE xy PLANE Neutral surface : longitudinal lines do not change length Neutral axis : intersecaon of neutral surface with any cross- secaonal plane 2/18/13 M. Mello/Georgia Tech Aerospace 8 LONGITUDINAL STRAIN IN BEAMS (cont.) ²་  Planes containing cross sections mn and pq in the deformed beam intersect in a line at O’ ²་  dΘ represents the angle between the two planes in the deformed beam ²་  ρ = distance from O’ to the neutral surface....
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This note was uploaded on 09/19/2013 for the course CEE 3001 taught by Professor Zhu during the Spring '09 term at Georgia Institute of Technology.

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