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Unformatted text preview: AERSP 301 Bending of open and closed section beams Dr. Jose Palacios Today’s Items Today’s Items • HW 2 due today! • HW 3 due next Wednesday • Bending of Beams  Megson Chapter 16, Reference: Donaldson – Chapters Lambda and Mu – Direct stress calculation – Bending deflections • Direct stress at a point in the c/s depends on: – Its location in the c/s – The loading – The geometry of the c/s • Assumption – plane sections remain plane after deformation (No Warping), or crosssection does not deform in plane (i.e. σxx, σyy = 0) • Sign Conventions! Megson pp 461 Direct stress calculation due to bending M – bending moment S – shear force P – axial load T – torque W – distributed load Direct stress calculation due to bending (cont’d) • Beam subject to bending moments M x and M y and bends about its neutral axis (N.A.) • N.A. – stresses are zero at N.A. • C – centroid of c/s (origin of axes assumed to be at C). Neutral Surface Definition In the process of bending there is an axial line that do not extend or contract. The surface described by the set of lines that do not extend or contract is called the neutral surface. Lines on one side of the neutral surface extend and on the other contract since the arc length is smaller on one side and larger on the other side of the neutral surface. The figure shows the neutral surface in both the initial and the bent configuration. The axial strain in a line element a distance y above the neutral surface is given by: • Consider element δ A at a distance ξ from the N.A. • Direct Stress: • Because ρ (bending radius of curvature) relates the strain to the distance to the neutral surface: Direct stress calculation due to bending (cont’d) ( 29 ρ ξ θρ θρ ξ ρ θ ε = = = l l l z ξ ρ σ ε σ E E z z z = ∴ = First Moment of Inertia Definition • Given an area of any shape, and division of that area into very small, equalsized, elemental areas ( dA ) • and given an C xC y axis, from where each elemental area is located ( yi and xi ) • The first moment of area in the "X" and "Y" directions are respectively: dA x Ax I ydA Ay I y x ∫ ∫ = = = = • IF the beam is in pure bending, axial load resultant on the c/s is zero: • 1 st moment of inertia of the c/s about the N.A. is zero N.A. passes through the centroid, C • Assume the inclination of the N.A. to C x is α Direct stress calculation due to bending (cont’d) • Then The direct stress becomes: ∫ ∫ = ⇒ = A A z dA dA ξ σ α α ξ cos sin y x + = ( 29 α α ρ ξ ρ σ cos sin y x E E z + = = Direct stress calculation due to bending (cont’d) • Moment Resultants: • Substituting for σ z in the above expressions for...
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 GANDHI,FARHANLESIEUTRE,GEORGE
 Second moment of area, Direct stress calculation, xx yy xy, Mxy σz

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