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Unformatted text preview: Mechanics of Solids Lecture 8 by Dr Emre Erkmen Lecturer, School of Civil and Environmental Engineering, The University of Technology Sydney Office: 2.520 Phone:9514 9769 Email: [email protected] Lecture hours: Tuesdays 11:0014:00, Wednesdays 15:0018:00 Office hours: Tuesdays 15:0017:00 1 Mechanics of Solids  Lecture 8 Outline Outline Deflections of beams Buckling of columns 2 Torsion of circular members Mechanics of Solids  Lecture 8 Deflections Deflections are very important when designing structures and machines Load 3 STRENGTH b is related to STRESSES . SERVICEABILITY b is related to DEFLECTIONS . Mechanics of Solids  Lecture 8 Elastic Curve 4 (i) all deflections are elastic (ii) deflections due to shear are negligible and are ignored (iii) plane sections remain plane Mechanics of Solids  Lecture 8 Rotationdeflection M θ ( x 1 ) ( x 2 ) θ ( x ) θ ( x 2 ) θ 5 d tan lim d x v v x x θ θ Δ → Δ ≈ = = Δ x 1 x 2 1 θ ( x 1 ) v ( x 2 ) v d d v x θ = Mechanics of Solids  Lecture 8 Strain – curvature relation lim lim x B A B A A B AB d y y AB x dx θ θ ε → → ′ − Δ = = − = − Δ 6 x ε = 2 2 d v y dx − Mechanics of Solids  Lecture 8 Momentcurvature relations x x Crosssection d y x z For a linear elastic material 7 x x M E y I σ ε = = − Mechanics of Solids  Lecture 8 Momentcurvature relations M EI 2 2 d v dx = M y I σ = − x E σ ε = 8 d d v x θ = x d y dx θ ε = − Mechanics of Solids  Lecture 8 • Possible boundary conditions are shown here. Boundary Conditions v v v θ 9 Mechanics of Solids  Lecture 8 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. Exampleelastic curve 10 Mechanics of Solids  Lecture 8 From freebody diagram, we have the moment expression Exampleelastic curve Px M − = 11 By integrating twice 2 2 2 1 3 1 2 2 6 d EI Px dx d Px EI C dx Px EI C x C υ υ υ = − = − + = − + + Mechanics of Solids  Lecture 8 Using boundary conditions d ν / dx = 0 at x = L, and ν = 0 at x = L , 3 1 2 2 PL C PL + − = Exampleelastic curve 12 2 1 6 C L C + + − = Thus, C 1 = PL 2 /2 and C 2 = PL 3 /3. Mechanics of Solids  Lecture 8 Substituting these results into Eqns with θ = d ν / dx , we get ( ) 3 2 3 3 2 6 P x L x L EI υ = − + − Exampleelastic curve 2 2 3 dv P L = − + 13 ( ) 3 3 6 x L dx EI θ = = − + Mechanics of Solids  Lecture 8 Exampleelastic curve Determine the elastic curve. 14 Mechanics of Solids  Lecture 8 Exampleelastic curve Beam is indeterminate to first degree as indicated from the freebody diagram. We can express the internal moment M in terms of the redundant force at A using segment shown below....
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 Three '11
 BROWN
 mechanics, Environmental Engineering, Trigraph, Column

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