1.050 Content overview Engineering Mechanics I Content overview. Selection of boundary conditions: Euler buckling.

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1 .050 Content overview.050 Engineering Mechanics I Lecture 34 How things fail and how to avoid it Additional notes energy approach I. Dimensional analysis. On monsters, mice and mushrooms Lectures -3. Similarity relations: Important engineering tools Sept. II. Stresses and strength 3. Stresses and equilibrium 4. Strength models (how to design structures, foundations.. against mechanical failure Lectures 4-5 Sept./Oct. III. Deformation and strain 5. How strain gages work? 6. How to measure deformation in a 3D Lectures 6-9 structure/material? Oct. IV. Elasticity 7. Elasticity model link stresses and deformation 8. Variational methods in elasticity Lectures 0-3 Oct./Nov. V. How things fail and how to avoid it 9. Elastic instabilities 0. Fracture mechanics Lectures lasticity (permanent deformation Dec..050 Content overview I. Dimensional analysis II. Stresses and strength III. Deformation and strain Selection of boundary conditions: Euler buckling Clamped cantilever beam e IV. Elasticity V. How things fail and how to avoid it Lecture 33 (Mon: Buckling (loss of convexity Lecture 34 (Wed: Fracture mechanics I Lecture 35 (Fri teaching evaluation: Fracture mechanics II Lecture 36 (Mon: lastic yield Lecture 37 (Wed: Wrap-up plastic yield and closure < el crit π EI ( el effective length Single supported beam e Double clamped cantilever beam e 3 4

2 Example: Concrete column w/ circular cross-section Equation provides critical buckling load: Force must be below this load to avoid failure < crit Eπ d EI 64 π EI 4 ( l Circular cross-section I 5 m d 0. m Recall lecture 33 Analyzed buckling instability (displacement convergence at point where load is applied using two approaches: (i iterative solution using conventional small deformation beam theory (divergence of series (ii application of large- (nonexistence of solution since determinant of coefficient matrix is zero - bifurcation point crit 94 kn E 5 Ga 5 (iii instability is equivalent to loss of convexity (energy approach 6 Reminder: Governing equations: dm y Q dx z Slight imperfection Conventional, small Moment generated by rotation of beam Goal: Show that elastic instability corresponds to a loss of convexity New term large- Note: otential energy in large- depends also on rotation (because rotations create moments dm y Q + ω Q Large dx z y x Moment generated by rotation of beam 7 (EIϑ y + F x ω dx W y (ξr0 l 8 Calculation of potential energy

3 As studied previously: ' ' Assume K.A. displacement field: otential otential energy at any energy at other K.A. displacement arameter solution field Lower bound approach f (X / l must satisfy BCs : Solution corresponds to absolute min of potential energy for K.A. displacement field 9 0 l (EIϑ y + F x ω y r dx W (ξ 0 Now express potential energy as function of parameter δ : d f df δ EI l dx dx dx δ l Necessary condition for minimum of potential energy: δ EI d f df dx δ l dx dx l Also, the expression of Convexity: must be convex! ε pot > 0 Loss of convexity: 0 δ Elastic instability! δ! 0 δ δ 3

4 Loss of convexity: Instability occurs if 0 δ EI d f df dx δ l dx dx d f EI l dx dx crit df dx dx 3 l Notes: Critical load obtained from potential energy approach yields upper bound of actual critical buckling load Iterations with first order displacement field leads to identical series expansion as in the iterative approach with small Approximate solution: Actual solution: EI π EI EI crit.5 > crit.4674 l 4 l 4 l Summary Analyzed buckling instability (displacement convergence at point where load is applied using two approaches: (i iterative solution using conventional small deformation beam theory (divergence of series (ii application of large- (nonexistence of solution since determinant of coefficient matrix is zero - bifurcation point Fracture mechanics How brittle materials fail crack extension (iii instability is equivalent to loss of convexity (energy approach 5 6 4

5 Brittle fracture Brittle failure crack extension Brittle fracture typically uncontrolled response of a structure, often leading to sudden malfunction of entire system Images removed due to copyright restrictions. Images removed due to copyright restrictions: hotograph of fault line, World Trade Center towers, shattered wine glass, X-ray of broken bone. 7 Snapshots show microscopic processes as a crack extends. During crack propagation, elastic energy stored in the material is dissipated 8 by breaking atomic bonds M.J. Buehler, H. Tang, et al., hys. Rev. Letters, 007 5