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Numerical and experimental investigation on existing structures: two seminars

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2 - Structural optimisation and inverse analysis strategies for masonry structures
Corrado Chisari
Dept. of Civil and Environmental Engineering, Imperial College London

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Numerical and experimental investigation on existing structures: two seminars

  1. 1. STRUCTURAL OPTIMISATION AND INVERSE ANALYSIS STRATEGIES FOR MASONRY STRUCTURES Dr Corrado Chisari CSM Group – Department of Civil and Environmental Engineering Imperial College London
  2. 2. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Outline  Optimisation  Overview  Genetic Algorithms  Structural Optimisation  Examples  Calibration of model parameters  Calibration problems as optimization problems  Ill-posedness of inverse problems  Identification of mesoscale model parameters for masonry  Conclusions and ongoing research Structural optimisation and inverse analysis strategies for masonry structures 2
  3. 3. Optimisation
  4. 4. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Overview  Optimisation is the discipline that, starting from: - the input variable space - a model of the problem  tries to find the best solution considering - some objectives to achieve - some constraints to satisfy. Structural optimisation and inverse analysis strategies for masonry structures 4
  5. 5. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Overview In mathematical terms 𝒙 = arg min 𝒙 𝜔1(𝒙), 𝜔2(𝒙), … , 𝜔𝑠(𝒙) subjected to: 𝑔𝑗 𝒙 < 0, 𝑗 = 1, … , 𝑚 ℎ 𝑘 𝒙 = 0, 𝑘 = 1, … , 𝑛 where: - 𝒙 input vector, 𝒙 ∈ 𝛀 (input variable space) - 𝜔𝑖(𝒙) i-th objetive to minimise - 𝑔𝑗(𝒙) j-th inequality constraint - ℎ 𝑘 𝒙 k-th equality constraint Structural optimisation and inverse analysis strategies for masonry structures 5
  6. 6. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Overview When S=1 (mono-objective optimisation), some classical approaches are:  Linear programming: 𝒙 = argmin 𝒙 𝒄 𝑻 𝒙 with 𝑨𝒙 ≤ 𝒃  Quadratic programming: 𝒙 = argmin 𝒙 𝟏 𝟐 𝒙 𝑻 𝑸𝒙 + 𝒃𝒙 + 𝒄  Constrained quadratic programming: 𝒙 = argmin 𝒙 𝟏 𝟐 𝒙 𝑻 𝑸𝒙 + 𝒃𝒙 + 𝒄 with 𝑨𝒙 ≤ 𝒒  Convex programming: 𝒙 = argmin 𝒙 𝜔(𝒙) with 𝑔𝑖(𝒙) ≤ 0 and 𝜔(𝒙), 𝑔𝑖(𝒙) convex functions Under some strict hypotheses, these types of problems have closed-formn solutions that can be obtained by means of well- established methods (Simplex, Lagrange multipliers, ...). Structural optimisation and inverse analysis strategies for masonry structures 6
  7. 7. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Overview When the problem complexity increases iterative approaches: 𝒙 𝒕 = 𝒙 𝒕−𝟏 + Δ𝒙 𝒕 such that 𝜔 𝒙 𝒕 ≤ 𝜔 𝒙 𝒕−𝟏 They differ for the method to find the corrections Δ𝒙 𝒕:  Line search with steepest descend (Jacobian);  Line search with Newton direction (Hessian);  Trust region. They determine the path to follow by examining derivatives for ω (Jacobian and Hessian). For this reason they are called gradient-based methods. Structural optimisation and inverse analysis strategies for masonry structures 7
  8. 8. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Overview Real-world problems does not usually satisfy one or more requirements for the mentioned methods:  The objective function is not convex  The objective function and/or the constraints are not continuous  The function and/or the constraints are not differentiable  The variables x are discrete  Multi-objective problem  Black-box problem Structural optimisation and inverse analysis strategies for masonry structures 8 Need for more general optimisation methods
  9. 9. Genetic Algorithms
  10. 10. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Genetic Algorithms  They mimic the search for the optimum as observed in nature:  A species evolves during a number of generations improving its fitness towards environmental conditions, through recombination of genetic heritage of fittest individuals;  The least fit individuals become extinct during the evolution;  Casual mutations may bring novelties in an individual’s chromosome which, if positive, may propagate and open new evolutions paths;  It can happen that parents survive to their own offspring if these are not apt to the external environment (elitism). Structural optimisation and inverse analysis strategies for masonry structures 10 They belongs to the algorithm classes: zero-order: they do not use derivatives population-based: iterations regard populations, not just one individual stochastic: the process depends on some random components
  11. 11. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Genetic Algorithms Structural optimisation and inverse analysis strategies for masonry structures 11 •Definition of chromosome and representation •Definition of the fitness function •Setting GA parameters Creation of the first population Stopping criterion satisfied? Ranking of the population Elitism New population Selection Recombination (crossover) Mutation Optimum Evaluation of the population yes no x1 x2 x3 x4 x5 x6
  12. 12. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Multi-objective optimisation 𝒙 = arg min 𝒙 𝝎 con 𝝎 = 𝜔1(𝒙) … 𝜔𝑠(𝒙)  The classical approach involves scalarising vector 𝝎 by means of weights wi: 𝒙 = arg min 𝒙 𝜔 𝑠𝑐𝑎𝑙 with 𝜔 𝑠𝑐𝑎𝑙 = 𝒘 𝒕 𝝎  The result depends on the choice of the weights. Structural optimisation and inverse analysis strategies for masonry structures 12
  13. 13. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Multi-objective optimisation  A solution x1 is said to dominate solution x2 (𝒙 𝟏 ≻ 𝒙 𝟐) if and only if:  𝜔𝑖 𝒙 𝟏 ≤ 𝜔𝑖 𝒙 𝟐 ∀𝑖 = 1, … , S  𝜔j 𝒙 𝟏 < 𝜔j 𝒙 𝟐 ∃𝑗 = 1, … , S  The set of non-dominated solutions is called Pareto Front (PF). Structural optimisation and inverse analysis strategies for masonry structures 13  «Dominates» and «Pareto Front solution» are the extensions of «is better than» and «optimal solution» to the case of multiple objectives.  Under some hypotheses, the solution of the scalarised problem belongs to the PF. This is however the general solution of the multi-objective problem.
  14. 14. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Non-dominated Sorting Genetic Algorithm (NSGA-II)  Population-based problems are naturally suited to looking for an ensemble of solutions (the Pareto Front) instead of a single solution Structural optimisation and inverse analysis strategies for masonry structures 14 0 1000 2000 3000 4000 5000 6000 7000 0 500 1000 f2 f1 Initial population 0 200 400 600 800 1000 1200 1400 20 40 60 80 f2 f1 Converged population Only the ranking method needs to be modified. x1 ω(x1) x2 ω(x2)≥ω(x1) ... xN ω(xN)≥ω(xN-1) x1 - x2 𝒙 𝟏 ≽ 𝒙 𝟐 ... xN 𝒙 𝑵−𝟏 ≽ 𝒙 𝑵 Mono-objective Multi-objective
  15. 15. Structural optimisation
  16. 16. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Optimal design Structural optimisation and inverse analysis strategies for masonry structures 16 Traditional design Optimal design Start Preliminary design External action analysis Internal stresses Section definition Verification Design modification End NoYes Start Parameterisation Trial structure External action analysis Internal stresses ObjectivesEnd No Yes Verification Optimisation process
  17. 17. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Structural optimisation Structural optimisation and inverse analysis strategies for masonry structures 17 Problem Variables Objectives Constraints Example Parametric optimisation - Dimensions and features of structural elements - Cost minimisation - Optimising performances - Structural constraints - Operative constraints Topological optimisation - Parameters identifying the shape of the structure - Weight minimisation - Stress maximisation - Structural constraints - Operative constraints Structural identification - Material parameters - Unknown boundary conditions - Minimising discrepancy with experimental data - Constraints due the physical or mathematical nature of the parameters Input variables Control code Output variables
  18. 18. Examples
  19. 19. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 1 – Rastrigin function Structural optimisation and inverse analysis strategies for masonry structures 19 Funzione di Rastrigin 𝑓 𝑥 = 𝐴𝑛 + 𝑖=1 𝑛 𝑥𝑖 2 − 𝐴𝑐𝑜𝑠(2𝜋𝑥𝑖 A=10, 𝑥𝑖 ∈ [−4.348,4.048] 𝑛 = 10 (numero di variabili) 𝒙 = 0.0115 0.0057 0.008 0.0101 −0.0012 −0.0024 0.0132 0.0087 −0.0055 0.0037 𝒙 𝑡𝑟𝑢𝑒 = 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
  20. 20. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 2 – Optimal design of bridges Structural optimisation and inverse analysis strategies for masonry structures 20 Impalcati ottimi individuati Spessori non vincolati Spessori vincolati H=cost. H=var. H=cost. H=var. Carpenteria [t] 760 769 840 733 Armatura L [t] 166 137 106 133 Calcestruzzo [t] 3168 3451 3168 3450 Costo [€] 2.562.089 2.551.675 2.802.303 2.492.916
  21. 21. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 3 – Optimal design of TMD for timber buildings Structural optimisation and inverse analysis strategies for masonry structures 21 Copyright Maurer Frequency optimisation Damping optimisation 𝜇 = 𝑚 𝑇𝑀𝐷 𝑚 𝑠 𝛼 = 𝜔 𝑇𝑀𝐷 𝜔𝑠 𝜉2 = 𝑐 𝑇𝑀𝐷 2 𝑚 𝑇𝑀𝐷 𝜔 𝑇𝑀𝐷 This approach does not account for the external input
  22. 22. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 3 – Optimal design of TMD for timber buildings Structural optimisation and inverse analysis strategies for masonry structures 22
  23. 23. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 3 – Optimal design of TMD for timber buildings Structural optimisation and inverse analysis strategies for masonry structures 23 Accounting for higher modes and more complex configurations
  24. 24. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 4 – Design of nonlinear viscous dampers Structural optimisation and inverse analysis strategies for masonry structures 24 𝑭 = 𝒄 ∙ 𝒖 𝜶 ∙ 𝒔𝒈𝒏( 𝒖) S. Silvestri, G. Gasparini, T. Trombetti. A five-step procedure for the dimensioning of viscous dampers to be inserted in building structures. J Earthq Eng, 14 (3) (2010), pp. 417-447
  25. 25. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 4 – Design of nonlinear viscous dampers Structural optimisation and inverse analysis strategies for masonry structures 25 Advantages: a. Flexibility in the objective definition; b. Possibility of embedding real- world constraints; c. Control on the forces transferred to the structure.
  26. 26. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 5 – Optimal sensor layout for structural parameter identification Structural optimisation and inverse analysis strategies for masonry structures 26
  27. 27. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 5 – Optimal sensor layout for structural parameter identification Structural optimisation and inverse analysis strategies for masonry structures 27
  28. 28. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 6 - Performance-based design of FRP retrofitting of existing RC frames Structural optimisation and inverse analysis strategies for masonry structures 28 Braga, F., R. Gigliotti and M. Laterza, 2006. Analytical stress- strain relationship for concrete confined by steel stirrups and/or FRP jackets. J. Struct. Eng., 132: 1402-1416. D’Amato, M., F. Braga, R. Gigliotti, S. Kunnath and M. Laterza, 2012. A numerical general-purpose confinement model for non-linear analysis of R/C members. Comput. Struct., 102-103: 64-75. Strength increase Ductility increase Effect of confinement: triaxial state of stress Toshimi Kabeyasawa, “Recent Development of Seismic Retrofit Methods in Japan”, Japan Building Disaster Prevention Association, January, 2005. FRP fabrics
  29. 29. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 6 - Performance-based design of FRP retrofitting of existing RC frames Structural optimisation and inverse analysis strategies for masonry structures 29
  30. 30. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Example 6 - Performance-based design of FRP retrofitting of existing RC frames Structural optimisation and inverse analysis strategies for masonry structures 30
  31. 31. Inverse analysis
  32. 32. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Inverse problems  Given: • The model space M; • The data space D; • The forward operator 𝒈(∙) ; • Some observations 𝒅 𝒐𝒃𝒔 ∈ 𝑫; Structural optimisation and inverse analysis strategies for masonry structures 32 Forward operator g (m) Observable data d Inverse operator g-1(d) Model parameters m Find 𝒎 ∈ 𝑴 such that: 𝒅 𝒐𝒃𝒔 = 𝒈( 𝒎)
  33. 33. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Solution of the inverse problem  Since the analytical expression of 𝒈(∙) (and 𝒈−𝟏 (∙)) is generally not known, the problem 𝒎 = 𝒈−𝟏 (𝒅 𝒐𝒃𝒔) is replaced by  𝒎 = arg min 𝒎∈𝑴 𝒅 𝒐𝒃𝒔 − 𝒈(𝒎) 𝑝 𝑝 Structural optimisation and inverse analysis strategies for masonry structures 33 Fernández-Martínez, J., Fernández-Muñiz, Z., Pallero, J. & Pedruelo-González, L., 2013. From Bayes to Tarantola: New insights to understand uncertainty in inverse problems. Journal of Applied Geophysics, Volume 98, pp. 62-72.
  34. 34. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Identification problems Structural optimisation and inverse analysis strategies for masonry structures 34 mtrial dc=g(mtrial) 𝜔 = 𝒅 𝒄 − 𝒅 𝒐𝒃𝒔 𝑡 𝑾 𝒅 𝒄 − 𝒅 𝒐𝒃𝒔 𝜔 𝑚𝑖𝑛? Updating m Optimisation algorithm No 𝒎 Yes dobs
  35. 35. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Identification of base-isolated bridges Structural optimisation and inverse analysis strategies for masonry structures 35 FE model: modal analysis Real structure: dynamic tests 𝜔2,𝑇 𝒑 = 𝑖=1 𝑁 𝑀 𝑓𝑟𝑒𝑓 ∆𝑓𝑖 𝑇𝑖,𝑐𝑜𝑚𝑝(𝒑) − 𝑇𝑖,𝑒𝑥𝑝 𝑇𝑒𝑥𝑝,𝑚𝑎𝑥 2 𝜔2,𝑀𝐴𝐶 𝒑 = 𝑖=1 𝑁 𝑀 𝑇𝑖,𝑒𝑥𝑝 𝑇𝑒𝑥𝑝,𝑚𝑎𝑥 1 − max 𝑗 (𝑀𝐴𝐶𝑖𝑗(𝒑)) 2 Chiara Bedon, Antonino Morassi, Dynamic testing and parameter identification of a base-isolated bridge, Engineering Structures, Volume 60, 2014, 85–99
  36. 36. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Identification of base-isolated bridges Structural optimisation and inverse analysis strategies for masonry structures 36
  37. 37. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Ill-posedness of the inverse problem Well-posed problem (Hadamard, 1902): 1. The solution exists; 2. It is unique; 3. It is stable. Structural optimisation and inverse analysis strategies for masonry structures 37 Input Output Input Output Forward problem Inverse problem While the forward problem is usually well- posed, it is not always the case of the corresponding inverse problem.
  38. 38. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Calibration of a model for steel members Structural optimisation and inverse analysis strategies for masonry structures 38 Monotonic test Cyclic test (AISC protocol) Pseudo-dynamic test (Spitak ground motion) S1 S2 S3 S4 Numerical model: smooth model, implemented in SeismoStruct (M. V. Sivaselvan and A. M. Reinhorn, “Hysteretic models for deteriorating inelastic structures,” J. Eng. Mech., vol. 126, no. 6, pp. 633-640, 2000)
  39. 39. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Multiple responses Structural optimisation and inverse analysis strategies for masonry structures 39 Danger of overfittingModels are never perfect
  40. 40. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Mono- vs multi-objective optimisation Structural optimisation and inverse analysis strategies for masonry structures 40 Calibration response Validation response Calibration by mono-objective optimisation Calibration by bi-objective optimisation
  41. 41. Experimental testing
  42. 42. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Motivation Structural optimisation and inverse analysis strategies for masonry structures 42 A. Borri, G. Castori, M. Corradi, E. Speranzini, Shear behavior of unreinforced and reinforced masonry panels subjected to in situ diagonal compression tests, Construction and Building Materials, 25(12), 2011, 4403 – 4414. Tests for macro-models are very invasive M. Corradi , A. Borri , A. Vignoli, Experimental study on the determination of strength of masonry walls, Construction and Building Materials, 17(5), 2003, 325 – 337.
  43. 43. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Motivation Structural optimisation and inverse analysis strategies for masonry structures 43 Tests on single components are less invasive… …but it is difficult to extract representative specimens from existing structures http://www.matest.com/ ≠
  44. 44. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Features of the test • To be performed in-situ (existing structures) • Low-invasive • Involving a sufficient volume of masonry • Able to capture elastic and strength material parameters for interfaces Structural optimisation and inverse analysis strategies for masonry structures 44
  45. 45. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Flat-jacks Structural optimisation and inverse analysis strategies for masonry structures 45 http://www.expin.it/servizi/indagini-strutturali/?lang=en
  46. 46. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Flat-jack test – preliminary study Structural optimisation and inverse analysis strategies for masonry structures 46 The experimental setup consists of different phases: 1. Horizontal cut; 2. Horizontal flat-jack; 3. Two vertical cuts and restraining; 4. Vertical flat-jack.
  47. 47. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Flat-jack test – preliminary study Structural optimisation and inverse analysis strategies for masonry structures 47 kVkN
  48. 48. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Flat-jack test – preliminary study Structural optimisation and inverse analysis strategies for masonry structures 48 • Instrumental layout • Noise propagation • Assessment of results
  49. 49. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Flat-jack test – preliminary study Structural optimisation and inverse analysis strategies for masonry structures 49 • The effect of horizontal cut/flat jack • Horizontal tension • Yielding at the border • Decreasing with distance 𝜎𝑥 = 𝑝 sinh 2𝜉 − 2 2𝑒 𝜉 cosh3 𝜉 𝜎 𝑦 = 𝑝 tanh3 𝜉 𝜎 𝑥 = 𝑝 sinh2 𝜉 𝑒 𝜉 cosh3 𝜉 𝜎 𝑦 = 𝑝 coth 𝜉 𝑢 𝑥 = − 3𝑁 𝑚 𝐸𝑡ℎ𝛽(1 + 2𝑘) sinh 𝛽𝑥 + 3𝑁 𝑚 𝐸𝑡ℎ𝛽(1 + 2𝑘) tanh 𝛽𝑏1 cosh 𝛽𝑥 𝑢 𝑥 = − 3𝑁 𝑚 𝐸𝑡ℎ𝛽(1 + 2𝑘) sinh 𝛽𝑥 + 3𝑁 𝑚 𝐸𝑡ℎ𝛽(1 + 2𝑘) coth 𝛽𝑏1 cosh 𝛽𝑥 • The effect of vertical flat jack • Horizontal deformability • Boundary conditions
  50. 50. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Flat-jack test – a new setup Structural optimisation and inverse analysis strategies for masonry structures 50
  51. 51. Outline Optimisation Calibration of model parameters Conclusions and ongoing research The flat-jack test - instruments Structural optimisation and inverse analysis strategies for masonry structures 51
  52. 52. Outline Optimisation Calibration of model parameters Conclusions and ongoing research The flat-jack test – test 1 Structural optimisation and inverse analysis strategies for masonry structures 52
  53. 53. Outline Optimisation Calibration of model parameters Conclusions and ongoing research The flat-jack test – test 1  Bricks in tension: 1. Micro-cracking leads to premature decreasing of stiffness that is difficult to identify; 2. A vertical crack propagates without involving mortar joint nonlinear behaviour. Structural optimisation and inverse analysis strategies for masonry structures 53 Local reinforcement by means of CFRP strips
  54. 54. Outline Optimisation Calibration of model parameters Conclusions and ongoing research The flat-jack test – test 2 Structural optimisation and inverse analysis strategies for masonry structures 54
  55. 55. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Elastic parameters Structural optimisation and inverse analysis strategies for masonry structures 55 Parameter Lower bound Upper bound Step Solution 𝑘 𝑉 10 N/mm3 4000 N/mm3 1 N/mm3 443 N/mm3 𝑟ℎ𝑗,𝑏𝑗 0.001 0.7 10-4 0.1 𝑘 𝑁 10 N/mm3 4000 N/mm3 1 N/mm3 218 N/mm3 kVkN Solutions 𝑡𝑜𝑙 = 𝑘2 𝜔 𝑟𝑒𝑓 k=10% 𝜔 𝑟𝑒𝑓 = 𝒖 𝒐𝒃𝒔 𝑡 𝑾𝒖 𝒐𝒃𝒔 Eb=11200 MPa υb=0.19 (Experimental)
  56. 56. Outline Optimisation Calibration of model parameters Conclusions and ongoing research The flat-jack test - strength parameters Structural optimisation and inverse analysis strategies for masonry structures 56 Interface Property Symbol Simplified assumption Bedjoints initial cohesion 𝑐0,𝑏𝑗 - initial friction coefficient tan 𝜙0,𝑏𝑗 - initial tensile strength 𝜎𝑡0,𝑏𝑗 𝑐0,𝑏𝑗 20 + 0.25 𝑐0,𝑏𝑗 𝑡𝑎𝑛 𝜙0,𝑏𝑗 residual cohesion 𝑐 𝑟,𝑏𝑗 0 residual friction coefficient tan 𝜙 𝑟,𝑏𝑗 tan 𝜙0,𝑏𝑗 residual tensile strength 𝜎𝑡𝑟,𝑏𝑗 0 energy fracture, mode I 𝐺𝑓𝐼,𝑏𝑗 Very high energy fracture, mode II 𝐺𝑓𝐼𝐼,𝑏𝑗 Very high energy fracture, compression 𝐺𝑓𝐶,𝑏𝑗 Very high dilatancy angle tan 𝜙 𝑑,𝑏𝑗 0 Headjoints initial cohesion 𝑐0,ℎ𝑗 0 initial friction coefficient tan 𝜙0,ℎ𝑗 tan 𝜙0,𝑏𝑗 initial tensile strength 𝜎𝑡0,ℎ𝑗 0 residual cohesion 𝑐 𝑟,ℎ𝑗 0 residual friction coefficient tan 𝜙 𝑟,ℎ𝑗 tan 𝜙 𝑟,𝑏𝑗 residual tensile strength 𝜎𝑡𝑟,ℎ𝑗 0 energy fracture, mode I 𝐺𝑓𝐼,ℎ𝑗 Very high energy fracture, mode II 𝐺𝑓𝐼𝐼,ℎ𝑗 Very high energy fracture, compression 𝐺𝑓𝐶,ℎ𝑗 Very high dilatancy angle tan 𝜙 𝑑,ℎ𝑗 0 Unknowns: • 𝑐0,𝑏𝑗 • tan 𝜙0,𝑏𝑗=tan 𝜙 𝑟,𝑏𝑗
  57. 57. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Strength parameters – simplified approach Structural optimisation and inverse analysis strategies for masonry structures 57 𝑝 𝑢 ⋅ ℎ = 𝜏 𝑢 ⋅ 𝑏 tan 𝜙 𝑟 = 𝜏 𝑢 𝜎𝑣 = 𝑝 𝑢 𝜎𝑣 ℎ 𝐵 ≅ 1.0
  58. 58. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Strength parameters – simplified approach Structural optimisation and inverse analysis strategies for masonry structures 58 rhj,bj kV (N/mm3) kN (N/mm3) c0,bj (MPa) ω 0.0 1061 333 0.454 0.001014 0.1 669 221 0.559 0.001201 0.2 578 185 0.606 0.0014 0.3 465 171 0.641 0.001628 0.4 348 174 0.670 0.001889 0.5 275 179 0.696 0.002233 0.6 256 170 0.720 0.002506 Optimum 𝑘 𝑉 = 1061𝑁/𝑚𝑚3 𝑘 𝑁 = 333𝑁/𝑚𝑚3 𝑟ℎ𝑗,𝑏𝑗 = 0.0 𝑐0,𝑏𝑗 = 0.454 tan 𝜙0,𝑏𝑗 = 1.0
  59. 59. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Validation – phase 1 Structural optimisation and inverse analysis strategies for masonry structures 59
  60. 60. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Validation – test 1 Structural optimisation and inverse analysis strategies for masonry structures 60 (a) (b) first crack 𝒑 𝒗𝒇𝒋 = 𝟏. 𝟏𝑴𝑷𝒂 𝒑 𝒗𝒇𝒋 = 𝟏. 𝟏𝟔𝑴𝑷𝒂𝒑 𝒗𝒇𝒋 = 𝟎. 𝟗𝟗𝑴𝑷𝒂
  61. 61. Outline Optimisation Calibration of model parameters Conclusions and ongoing research URM perforated wall Structural optimisation and inverse analysis strategies for masonry structures 61 𝑟ℎ𝑗,𝑏𝑗 = 0.6 (𝛿ℎ = 1.3 𝑚𝑚) 𝑟ℎ𝑗,𝑏𝑗 = 0.0 (𝛿ℎ = 1.3 𝑚𝑚) 𝑟ℎ𝑗,𝑏𝑗 = 0.0 (𝛿ℎ = 5 𝑚𝑚) Elastic Stiffnesses (kN/mm) rhj,bj=0.0 231.08 10% rhj,bj=0.1 209.20 0% rhj,bj=0.2 197.70 -6% rhj,bj=0.3 191.50 -9% rhj,bj=0.4 191.88 -9% rhj,bj=0.5 192.26 -8% rhj,bj=0.6 188.54 -10% Central value 209.81 rhj,bj kV (N/mm3) kN (N/mm3) c0,bj (MPa) 0.0 1061 333 0.454 0.1 669 221 0.559 0.2 578 185 0.606 0.3 465 171 0.641 0.4 348 174 0.670 0.5 275 179 0.696 0.6 256 170 0.720
  62. 62. Conclusions
  63. 63. Outline Optimisation Calibration of model parameters Conclusions and ongoing research Conclusions and ongoing research  Full characterisation by means of flat-jacks  Meta-model approximation for expensive simulations  Use of full-field measurements  Multi-level model calibration (MultiCAMS project) Structural optimisation and inverse analysis strategies for masonry structures 63
  64. 64. Outline Optimisation Calibration of model parameters Conclusions and ongoing research References  Chisari C, Bedon C, 2017. Performance-based design of FRP retrofitting of existing RC frames by means of multi-objective optimisation. Bollettino di Geofisica Teorica e Applicata, 58(4), pp. 377-394.  Parcianello E, Chisari C, Amadio C, 2017. Optimal design of nonlinear viscous dampers for frame structures. Soil Dynamics and Earthquake Engineering 100:257-260.  Chisari C, Macorini L, Amadio C, Izzuddin BA, 2016. Optimal sensor placement for structural parameter identification. Structural and Multidisciplinary Optimization, 55(2): 647-662, DOI: 10.1007/s00158-016-1531-1.  Chisari C, Francavilla AB, Latour M, Piluso V, Rizzano G, Amadio C, 2017. Critical issues in parameter calibration of cyclic models for steel members. Engineering Structures, 132:123-138, DOI: 10.1016/j.engstruct.2016.11.030.  Chisari C, Bedon C, 2016. Multi-objective optimization of FRP jackets for improving seismic response of reinforced concrete frames. American Journal of Engineering and Applied Sciences 9(3): 669-679, DOI:10.3844/ajeassp.2016.669.679.  Poh’sié GH, Chisari C, Rinaldin G, Amadio C, Fragiacomo M, 2016. Optimal design of tuned mass dampers for a multi-storey cross laminated timber building against seismic loads. Earthquake Engineering and Structural Dynamics, DOI: 10.1002/eqe.2736.  Chisari C, Macorini L, Amadio C, Izzuddin BA, 2015. An Experimental-Numerical Procedure for the Identification of Mesoscale Material Properties for Brick-Masonry. Proceedings of the 15th International Conference on Civil, Structural and Environmental Engineering Computing, Paper 72  Chisari C, Bedon C, Amadio C, 2015. Dynamic and static identification of base-isolated bridges using Genetic Algorithms. Engineering Structures 102:80-92.  Poh'sié GH, Chisari C, Rinaldin G, Fragiacomo M, Amadio C, Ceccotti A, 2015. Application of a translational tuned mass damper designed by means of genetic algorithms on a multistory cross-laminated timber building. Journal of Structural Engineering ASCE, DOI: 10.1061/(ASCE)ST.1943- 541X.0001342, E4015008.  Chisari C, Macorini L, Amadio C, Izzuddin BA, 2015. An Inverse Analysis Procedure for Material Parameter Identification of Mortar Joints in Unreinforced Masonry, Computer and Structures 155:97-105.  Amadio C, Chisari C, Panighel F, 2015. Analysis and optimisation of tapered steel-concrete composite bridges. Proceedings of the 25th Conference on Steel Structures C.T.A., Salerno (in Italian). Structural optimisation and inverse analysis strategies for masonry structures 64
  65. 65. THANK YOU FOR YOUR ATTENTION corrado.chisari@gmail.com c.chisari12@imperial.ac.uk

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