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Unformatted text preview: Lecture 8 1 Lecture 8 1 Outline: Deflection of beams Buckling of slender columns Torsion of circular bars 1. Deflection of beams Similar to axially loaded bars, beams deflect under vertical loading. Determining the beam deflections is very important for design considerations. It is often helpful to sketch the deflected shape of the beam under loading as shown in Fig.1. Fig. 1 Deflected shape (elastic curve) When determining the deflected shape following assumptions are made (i) all deflections are elastic ( therefore deflected shape is also called elastic curve) (ii) plane sections remain plane (iii) deflections are due to internal bending moment only and deflections due to shear are negligible Under these assumptions, the relation between the deflection and bending moment can be established. As shown in the Fig. 2, take a beam under uniform bending moment. On its deflected shape, at two different points on the axis of the beam 1 x and 2 x , there will be two different vertical deflections 1 ( ) v x and 2 ( ) v x , and two different corresponding rotations of the sections 1 ( ) x θ and 2 ( ) x θ . 1 Mechanics of Solids Dr Emre Erkmen Lecture 8 2 M θ x 1 x 2 ( x 1 ) ( x 2 ) θ ( x 1 ) θ ( x 2 ) θ ( x 1 ) v ( x 2 ) v Fig. 2 Deflection and rotation of the crosssections under bending moment If 1 x and 2 x are selected to be very close 2 1 x x x − = Δ , we can relate the rotation of the cross section θ at any x to the change in deflections v Δ as shown in Fig. 3. Since the rotation angle θ is the slope when x Δ is infinitesimally small, it is equal to deflection derivative. x v x v ( x ) Fig. 3 Relation between rotation and deflection From Lecture 4, remember the relation between the strain and the second derivative of the deflection (curvature). Fig. 4 Relation between strain and the curvature (from Lecture 4) 2 2 x dv d d d v dx y y y dx dx dx θ ε = − = − = − Also from Lecture 4, remember the relation between the stress and the internal moment. x x M E y I σ ε = = − 2 2 d v M v dx EI ′′ ⇒ = = lim lim x B A B A A B AB y AB x θ ε → → ′ − Δ = = − Δ d tan lim d x v v x x θ θ Δ → Δ ≈ = = Δ Lecture 8 3 Boundary conditions Supports that resist to applied forces prevent deflections. Similar to axially loaded bar problems, compatibility conditions related to boundary conditions can be used in combination with equilibrium equations to solve statically indeterminate problems. Note that horizontal deflection is not considered in the analysis at all. Example Cantilevered beam shown is subjected to a vertical load P at its end. Determine the elastic curve. EI is constant. Find the slope and the deflection at point A....
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 Three '11
 BROWN
 Shear Stress, Moment Of Inertia, Torsion, Column, Buckling

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