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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 Spring '09
 ZHU
 Force, Shear Stress, Stress, Shear strength

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