Rectangular cross section

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Unformatted text preview: : Bending- moment diagram 3/6/13 M. Mello/Georgia Tech Aerospace a) a) = P a Pa = M ( a) = P a = C 1 M ( x) = P a a!V = dM P= dx For x > L 26 a) And one more for the road… y P P a a x neutral axis (y=0) y Shear force |V|= P in regions to leG and right of center span Stress concentraYons where concentrated load is applied h/2 z neutral axis (y=0) h/2 Horizontal photoelasYc interference fringes reveal contours of constant flexural stress within region of constant bending moment M=Pa 3/6/13 M. Mello/Georgia Tech Aerospace b 27 One last look at four point bend BENDING MOMENT AND SHEAR FORCE DISTRIBUTIONS ARE SET BY STATICS ²་  ResulYng bending stresses, strains, and curvatures within the beam are dictated by the bending moment distribuYon [M(x)] in conjuncYon with the elasYc modulus (E) and moment of inerYa (I) ²་  ResulYng shear stresses within the beam are dictated by the shear force distribuYon [V(x)] in conjuncYon with the moment of inerYa (I) and 1st moment (Qa) of the cross- secYonal area of the region between the y value of the point of interest and one of the free surfaces of the beam (top or bo=om) 3/6/13 M. Mello/Georgia Tech Aerospace 28 SHEAR STRESSES IN BEAMS OF CIRCULAR CROSS SECTION ²་  FOR CIRCULAR CROSS SECTIONS CAN NO LONGER ASSSUME SHEAR STRESSES ACT PARALLEL TO THE y- axis (as they do in the case of a beam with a rectangular cross secYon) ²་  Advanced beam theory predicts the shear flow pa=ern shown below ²་  Although generally quite complicated to analyze, theory also predicts that that a parabolic shear stress profile results along the diameter (as depicted in the central figure) M V V y M x z neutral axis z z ²་  Shear stress must go to zero along the boundary ²་  Parabolic profile along the diameter (coincident with z axis in the figure on the leG) ²་  Max. shear stress at the center (r=0) is 4/3 the average verYcal shear stress V/A Shear flow diagram for a circular cross secYon 3/6/13 M. Mello/Georgia Tech Aerospace 29 Shear stress distribuYon along neutral axis of a beam with a circular cross secYon ²་  Parabolic shear stress profile along neutral axis of a beam with circular cross secYon strongly mimics the shear stress profile of a beam with a square cross...
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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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