Unformatted text preview:

1.050 Engineering Mechanics I Summary of variables/concepts Lecture 27 - 37 1Variable Definition f (x) secant ∂ff (x) | (b − a) ≤ f (b) − f (a)tangent ∂xx=a Notes & comments Convexity of a function External work Free energy and complementary free energy 1 32 N1 N2 N3 δ1 δ2 P δ3 ξ0 ab Wd ψ * i ψi x Wd =ξ v ⋅ F r d +ξ vd ⋅ R r Ni = ∂ψi ∂ψi * Ni ∂δi δi =∂Ni iψ* iψ Complementary free energy Free energy δi ∑δiNi =ψi *(Ni ) +ψi (δi ) i Lectures 27 and 28: Basic concepts: Convexity, external work, free energy, complementary free energy, introduced initially for truss structures (see schematic show in the lower right part). 2Variable Definition Truss problems * d d−(ψ −ξ v ⋅ R r)= ! ψ −ξ v ⋅ F r −εcom =εpot Complementary Potential energy energy com: ε= pot: ε= ' '⎧max(−ε (N , R ))⎫ ' com i i ' ' ⎪⎪N S.A. ⎪⎪' '−εcom(Ni , R ) ≤⎨ is equal to ⎬≤εpot(δi ,ξi ) ⎪ min ε (δ ',ξ ') ⎪Lower bound ⎪⎩ δi ' K.A. pot i i ⎪⎭ Upper bound Notes & comments At elastic solution: Potential energy is equal to negative of complementary energy Upper/lower bound At the solution to the elasticity problem, the upper and lower bound coincide Consequence of convexity of elastic potentials ψ ,ψ * Lectures 27 and 28: Introduction to potential energy and complementary energy, definition at the elastic solution, upper/lower bound, example of energy bounds for truss structures. The upper/lower bounds of the expressions are a consequence of the convexity of the elastic potentials (see previous slide). 3Variable Definition Notes & comments Complementary free energy ψ * (1-D) ψ Free energy (1-D) Contributions from external W =∑Fv id ⋅ξ r iW =∑Rr id ⋅ξ r idW ,W * i=1..Ni=1..N work ψ= 1 (W * +W ) Clapeyron’s formulas 2 Significance: Enables one ψ * = 1 (W * +W ) calculate free energy, 2 complementary free energy, potential energy and ε pot = 1 (W * − W )2 complementary energy directly from the boundary εcom = 1 (W − W * ) conditions (external work), 2 at the solution (“target”)! Lectures 27-29: The equations for free energy and complementary free energy for truss structures are summarized. Lower part: Clapeyron’s formulas, used to calculate the “target” solution, that is, the results at the solution. These equations are generally valid, not only for truss structures (but the expressions of how to calculate the individual terms that appear in these equations are different). 4Variable Definition Notes & comments ( ) .A.K' pot pot.A.K' com .A.S' .A.S' com )'( )'(min toequalis )'(max )'( ξ ξ σ σ ξε ξε σε σε r r r r ≤ ⎪ ⎪ ⎭ ⎪⎪⎬ ⎫ ⎪ ⎪ ⎩ ⎪⎪⎨ ⎧ − ≤− Lower bound Upper boundSolution Complementary energy approach Potential energy approach )()'()'( ** com dTW r −= σψσε Upper/lower bound for 3D elasticity problems Complementary free energy (3-D, isotropic material) Free energy (3-D, isotropic material) *ψ ψ Complementary energy and potential energy External work contributionsVolume force contribution Stress vector contribution )()'()'(pot dW ξεψξε rr −= Displacement contribution Ω⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ += ∫ Ω dG s K m 22 * 2 1 σψ )trace(3 1 σσ = m ( )22 3: 2 1 ms σσσ −= ( ) Ω+= ∫ Ω dGK dv 22 2 1 εεψ )trace(εε = v ⎟ ⎠ ⎞⎜ ⎝ ⎛ == 22 3 1:2 vd εεεε Lecture 30: Energy bounds for 3D isotropic elasticity. Note that the external work contribution under force (stress) boundary conditions involves a volume integral due to the volume forces (gravity). The lower part summarizes the equations used to calculate the free energy and complementary free energy, as well as the external work contributions (external work contribution part). 5Variable Definition Notes & comments *ψ dxEI M ES N lx y∫ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ += 0.. 22 * 2 1 2 1ψ Complementary free energy (for beams) ψ () ()dxEIES lx yxx∫ = ⎥⎦ ⎤ ⎢⎣ ⎡ += 0.. 0 220 2 1 2 1 ϑεψ Free energy (for beams) Note 1: For 2D, the only contributions are axial forces & moments and axial strains and curvatures Note 2: Target solution using Clapeyron’s formulas P l/2 l/2δ δ= unknown displacement at point of load application Target solution δε Pcom 2 1 = [ ] [ ]∑∑ ++=+⋅= i y Ri d yzi d zxi d x i y Riyi d MxRxRxMxRxW ,, * )()()()()( ωξξωξ rr External work by prescribed displacements [ ] [ ] [ ]∑∫ ∑∫ ++++= +⋅+⋅= = = i i d yyi d zzi d xx lx i d zzi d xx i i d yyi d lx d xMxFxFdxxfxf xMxFx dxfW )()()()()( )()(( ) 00 0.. 00 0 0.. 0 ωξξξξ ωξξ rrrr External work by prescribed force densities/forces/moments Lecture 31: How to calculate free energy, complementary energy and external work for beam structures. 6Variable Definition Notes & comments ⎧ max (−ε (F ', M '))⎫ ⎪Fx',My 'S.A. com x y ⎪ −εcom(Fx', M y ') ≤ ⎨⎪is equal to ⎬⎪≤ εpot(ξx ',ωy ') F ',M 'S.A. ξ ',,ω 'K.A. ⎩ x y pot x y ⎭ x y ⎪ min ε (ξ ',ω ') ⎪ x y ⎪ξ ',ω ' K.A. ⎪ Lower bound Solution Upper bound Fx', M y ' Complementary that provide Potential energy approach energy absolute approach max of −ε “Displacement “Stress approach” com approach) ξx ',ωy ' Work with unknown that provide Work with unknown but S.A. moments and absolute but K.A. forces min of εpot displacements Step 1: Express target solution (Clapeyron’s formulas) – calculate complementary energy AT solution Step 2: Determine reaction forces and reaction moments Step 3: Determine force and moment distribution, as a function of reaction forces and reaction moments (need My and N) Step 4: Express complementary energy as function of reaction forces and reaction moments (integrate) Step 5: Minimize complementary energy (take partial derivatives w.r.t. all unknown reaction forces and reaction moments and set to zero); result: set of unknown reaction forces and moments that minimize the complementary energy Step 6: Calculate complementary energy at the minimum (based on resulting forces and moments obtained in step 5) Step 7: Make comparison with target solution = find solution displacement Step-by-step procedure – how to solve beam problems with complementary energy approach Lectures 31-32: How to solve beam problems using the complementary approach. This slide shows the overview over the upper/lower bounds. The lower part summarizes a step by step procedure of how to solve statically


View Full Document
Download Lecture Notes
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture Notes and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture Notes 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?