1.050 Content overview Engineering Mechanics I Content overview. Outline and goals. Lecture 28
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1 .5 Content overvew.5 Engneerng Mechancs I Lecture 8 Introucton: Energy bouns n lnear elastcty (cont I. Dmensonal analyss. On monsters, mce an mushrooms Lectures -. Smlarty relatons: Important engneerng tools Sept. II. Stresses an strength. Stresses an equlbrum 4. Strength moels (how to esgn structures, founatons.. aganst mechancal falure Lectures 4-5 Sept./Oct. III. Deformaton an stran 5. How stran gages work? 6. How to measure eformaton n a D Lectures 6-9 structure/materal? Oct. IV. Elastcty 7. Elastcty moel lnk stresses an eformaton 8. Varatonal methos n elastcty Lectures - Oct./Nov. V. How thngs fal an how to avo t 9. Elastc nstabltes. Plastcty (permanent eformaton Lectures -7. Fracture mechancs Dec..5 Content overvew I. Dmensonal analyss II. Stresses an strength III. Deformaton an stran IV. Elastcty Lecture : Applcatons an examples Lecture 4: Beam elastcty Lecture 5: Applcatons an examples (beam elastcty Lecture 6: cont an closure Lecture 7: Introucton: Energy bouns n lnear elastcty (D system Lecture 8: Introucton: Energy bouns n lnear elastcty (D system, cont Lecture 9: Generalzaton to D, examples Outlne an goals Use concept of concept of convexty to erve contons that specfy the solutons to elastcty problems Obtan two approaches: Approach : Base on mnmzng the ental energy Approach : Base on mnmzng the plementary energy Last part: Combne the two approaches: Upper/lower boun V. How thngs fal an how to avo t Lectures to 7 4
2 Remner: convexty of a functon f (x f (b a f (b f (a x x=a Total external work v r v r W = ξ F + ξ R f (x secant Work one by prescrbe forces Dsplacement s unknown Work one by prescrbe splacements, force unknown a b tangent Free energy an plementary free energy functons x ψ, ψ are convex! 5 6 N Total nternal work State equatons Combnng t v r v r! W = ξ F + ξ R =ψ +ψ Complementary free energy ψ N = δ v (ψ ξ r! v r R = ψ ξ F ψ Free energy δ = N Complementary energy Potental energy =: ε =: ε δ δ N =ψ (N +ψ (δ 7 Soluton to elastcty problem ε = ε 8
3 Example system: D truss structure N N N Rg bounary Mnmum ental energy approach Conser two knematcally amssble (K.A. splacement fels ( Approxmaton δ = δ = δ = ξ to soluton (K.A. N N N ( Rg bar δ δ δ P ξ 9 δ Actual soluton δ P δ δ = δ + (ξ δ ξ Prescrbe force Unknown δ = δ + 4 (ξ δ splacement δ Mnmum ental energy approach ( ξ P = N δ Mnmum ental energy approach ε (δ,ξ =ψ (δ Pξ ψ (δ Pξ = ε (δ,ξ ( ξ P = N δ (-( P(ξ ξ = N (δ δ = (δ δ δ N = δ Convexty: (δ δ ψ (δ ψ (δ δ P(ξ ξ ψ (δ ψ (δ ε ( δ, ξ = ψ ( δ P ξ ψ ( δ Pξ = ε ( δ, ξ Potental energy of actual soluton s always smaller than the soluton to any other splacement fel Therefore, the actual soluton realzes a mnmum of the ental energy: ε (δ,ξ = mn ε (δ,ξ δ K.A. To fn a soluton, mnmze the ental energy for a selecte choce of knematcally amssble splacement fels We have not nvoke the EQ contons!
4 Mnmum plementary energy approach Contons for statcally amssble (S.A. N + N + N = R N + N N = Conser two statcally amssble force fels Approxmaton to soluton Stll S.A. N N R N ( N, N, N N N N ( δ δ R δ N N N N = /R ξ Prescrbe Actual soluton (obtane n N = / R splacement lecture N = 7 /R Unknown force R Mnmum plementary energy approach ( ξ R = N δ ( ξ R = N δ (-( ξ (R R = δ (N N = (N (N N N δ = N ψ Convexty: (N N ψ ( N ψ (N N ξ (R R ψ (N ψ (N ε ( N, R = ψ ( N ξ R ψ ( N ξ R = ε ( N, R 4 Mnmum plementary energy approach Combne: Upper/lower boun ε ( N, R = ψ ( N ξ R ψ ( N ξ R = ε Complementary energy of actual soluton s always smaller than the soluton to any other splacement fel Therefore, the actual soluton realzes a mnmum of the plementary energy: ε ( N, R = mn ε( N, R N S.A. ( N, R To fn a soluton, mnmze the plementary energy for a selecte choce of statcally amssble force fels We have not nvoke the knematcs of the problem! 5 Recall that the soluton to elastcty problem ε = ε Therefore ε (N, R = max( ε (N, R (change sgn N S.A. ε (δ,ξ = mn ε (δ,ξ δ K.A. max( ε (N, R ε (N, R N S.A. s equal to ε (δ,ξ mn ε (δ,ξ δ K.A. Upper boun Lower boun At the soluton to the elastcty problem, the upper an lower boun conce 6 4
5 Approach to approxmate/numercal soluton of elastcty problems Mnmum ental energy approach: Select a guess for a splacement fel; the only conton that must be satsfe s that t s knematcally amssble. In a numercal soluton, ths splacement fel s typcally a functon of some unknown parameters (a, Express the ental energy as a functon of the unknown parameters a, Mnmze the ental energy by fnng the approprate set of parameters (a, for the mnmum generally yels approxmate soluton The actual soluton s gven by the splacement fel that yels a total mnmum of the ental energy. Otherwse, an approxmate soluton s obtane Mnmum plementary energy approach: Select a guess for a force fel; the only conton that must be satsfe s that t s statcally amssble. In a numercal soluton, ths force fel s typcally a functon of some unknown parameters (b,b, Express the plementary energy as a functon of the unknown parameters b,b, Mnmze the plementary energy by fnng the approprate set of parameters (b,b, for the mnmum generally yels approxmate soluton The actual soluton s gven by the force fel that yels a total mnmum of the plementary energy. Otherwse, an approxmate soluton s obtane At the elastc soluton, the mnmum ental energy approach soluton an the negatve of the soluton of the mnmum plementary energy approach conce 7 5
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