2 - Structural optimisation and inverse analysis strategies for masonry structures
Corrado Chisari
Dept. of Civil and Environmental Engineering, Imperial College London
Numerical and experimental investigation on existing structures: two seminars
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. 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
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. 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. 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. 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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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
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. 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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. 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. 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
16. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. 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
19. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. 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. 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. 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. 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. 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. 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. 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. 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. 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
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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. 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. 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. 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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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Multiple responses
Structural optimisation and inverse analysis strategies for masonry structures 39
Danger of overfittingModels are never perfect
40. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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
42. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. Outline Optimisation Calibration of model parameters
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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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. Outline Optimisation Calibration of model parameters
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Flat-jacks
Structural optimisation and inverse analysis strategies for masonry structures 45
http://www.expin.it/servizi/indagini-strutturali/?lang=en
46. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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Flat-jack test – preliminary study
Structural optimisation and inverse analysis strategies for masonry structures 47
kVkN
48. Outline Optimisation Calibration of model parameters
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Flat-jack test – preliminary study
Structural optimisation and inverse analysis strategies for masonry structures 48
• Instrumental layout
• Noise propagation
• Assessment of results
49. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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Flat-jack test – a new setup
Structural optimisation and inverse analysis strategies for masonry structures 50
51. Outline Optimisation Calibration of model parameters
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The flat-jack test - instruments
Structural optimisation and inverse analysis strategies for masonry structures 51
52. Outline Optimisation Calibration of model parameters
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The flat-jack test – test 1
Structural optimisation and inverse analysis strategies for masonry structures 52
53. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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The flat-jack test – test 2
Structural optimisation and inverse analysis strategies for masonry structures 54
56. Outline Optimisation Calibration of model parameters
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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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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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
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. 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
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. Outline Optimisation Calibration of model parameters
Conclusions and ongoing
research
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Engineering 100:257-260.
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Structural optimisation and inverse analysis strategies for masonry structures 64
65. THANK YOU FOR YOUR
ATTENTION
corrado.chisari@gmail.com
c.chisari12@imperial.ac.uk