Aalto University School of Engineering

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Aalto University School of Engineering Kul-24.

1 Aalto University School of Engineering Kul Ship Structural Design (P) Lecture 11 Reliability and Optimization of Ship Structures

2 Kul Ship Structures Strength Design Framework Loads Response Strength Lecture 11: Reliability and optimization of ship structures

3 Weekly Exercise Exercise 11: Reliability and Optimization - Given , Return Define partial safety factors for your ship and check if your design is reliable. Optimize the hull girder in terms of weight, cost and/or VCG using hand calculations, Matlab or ConStruct Report the work Load Acting stress (load) Plate Safety factor Capacity Design stress Strengh stress (capacity) Buckling Fatigue Deck (OK) 60 (Faili).

4 Contents Motivation for reliability and optimization Reliability Analysis of Ship Structures σ fail Definition of optimization Ship structural optimization Literature 1. Ang, H-S. & Tang, W.H., Probability Concepts in Engineering Planning and Design, Wiley, 1975 σ all 2. Jensen, J., Load and Global Response of Ships. DTU Thoft-Christensen, P., ja Baker M., Structural Reliability Theory and its Applications. Springer-Verlag Hughes, O.F., Mistree, F., and Zanic, V., A Practical Method for the Rational Design of Ship Structures, Journal of Ship Research, Vol. 24, No. 2, 1980, Alan Klanac, Jasmin Jelovica, Vectorization and Constraint Grouping to Enhance Optimization of Marine Structures, Marine Structures, Volume 22, Issue 2, April 2009, Pages Ehlers, S., A Procedure to optimize ship side structures for crashworthiness, Journal of Engineering for the Maritime Environment, Remes, H., Romanoff, J., Varsta, P., Naar, H., Niemelä, A., Jelovica, J. and Klanac, A. Bralic, S., Hull/Superstructure-Interaction in Optimized Passenger Ships, Proceedings of the 3rd International Conference on Marine Structures, March 28th 30th 2011, Rostock, Germany, pp

5 Design Principles

6 Motivation for Optimization and Reliability Ships are performance based structures, thus the aim is to Maximize pay load Minimize price and operation costs Maximize speed Etc Reliability takes into account the fact that there are uncertainties both on load and capacity of the structure, e.g. Statistics on wave loads, Uncertainties in material and geometrical properties of the structure Etc

7 Reliability Motivation Safety Factors for Design Strength of materials analysis of failed and safely operated structures gives indication of the allowable stress and deformation limits for design These can be used to design new structures Brittle materials, comparison to fracture stress of the material Ductile materials, comparison to yield stress of the material The ratio of allowable and fracture/yield stress is called safety factor Gives indication of the safety level Does not give level of probability of failure The idea is to indicate how much the load could be increased to still safely operate the ship or structure With respect to yield typical value is 1.35 With respect to fracture typical value is 1.80 Two items affect this Objective uncertainty: loads, structural dimensions, material Subjective uncertainty: physical models, mathematical models Load Demand D σ fail σ all Strength Capacity C

Motivation for Optimization The structures are becoming more advanced Effectiveness often requires minimization or maximization of property(ies) of the structure under given load

search of the optimum We need strength constraints to make the structure feasible in practice The key issue is to balance both strength assessment and optimization algorithms cost vs.

8 Motivation for Optimization The structures are becoming more advanced Effectiveness often requires minimization or maximization of property(ies) of the structure under given load cases and constraints Structural optimization is mathematical method to find optimal structures Lightweight Sustainable Safe Cost effective Etc We need optimization algorithms for search of the optimum We need strength constraints to make the structure feasible in practice The key issue is to balance both strength assessment and optimization algorithms cost vs. accuracy In structural optimization you are expected to know about some of these Matrix algebra, calculus, mathematical tools (e.g. Matlab) Strength of materials, limit state analysis, FEM

9 Basic Concepts of the Reliability Level When Capacity is Constant, i.e. y=c We consider a case where Resistance or capacity C is constant c Load or demand D follows probability distribution f D (x). Then the reliability R is For example wave bending moment distribution R( c) = F D ( c) = c 0 f D ( x) dx and the probability of failure is Bending moment that the hull girder can tolerate P f (c) = 1 R(c)

10 Basic Concepts of the Reliability Level Statistically Indipendent Demand D and Capacity C The reliability analysis aims to secure that the random event X < Y gets high enough probability level If the capacity C is independent of the demand D and has some probability density function f c (y), the reliability gets: R = y 0 0 f D ( x) dx fc ( y) dy = FD ( y) fc ( y) dy 0 Since the reliability is product of two probabilities P = f ( y) dy C c PD = FD ( y) accounting that all capacity events need to be considered Probability P C describes that capacity C is between y and y+dy and probability P D that load D is larger or smaller than y. So the probability of failure is P f 0 = 1 R = 1 F ( y) f ( y) dy D C Load (demand) D and capacity C density functions NOTE! Probability of failure P f IS NOT the area that the two distributions limit

11 Basic Concepts of the Reliability Level Probability of Failure in Terms of Density Distributions Since the probability has property f C ( y) dy = 1 0 It gives for failure probability Pf = 1 FD ( y) fc ( y) dy = [1 FD ( y)] fc ( y) dy = f D ( x) dx fc ( y) dy y If the integration is fist carried out with respect to capacity C, i.e. variable Y, we get x Pf = fc ( y) dy f D ( x) dx = FC ( x) f D ( x) dx Generally Closed form expressions cannot be found for this Some special cases exist where it can be found Numerical methods can be used always to get solution tedious

12 Basic Concepts of the Reliability Level Joint Distribution In general case the load X and response Y are not statistically independent Hull girder ultimate bending moment depend on flow around hull and hull can deform excessively to make non-linear problem Ice loads on frames and plates are affected by their flexibility Ship collision load are affected by structural response Then joint distribution f DC (x,y) is needed At limit state x = y At failure x > y. It is assumed that the limit-state function describes the failure G(X) = C(X) D(X) Which describes the difference between capacity C and load D, called safety margin Vector X describes the statistical factors affecting the reliability, e.g. External loads Material properties Geometrical properties Limit state G(x) = 0 defines the limit, when broken at G(x) < 0, the structure or its part looses load carrying capacity. Joint distribution of capacity C and load D Response Loads Strength hull girder collapse collision

13 Structural Reliability The reliability is defined with the help of limit state function as R = P{ g( x) > 0} = p( g) dg Respectively the probability of failure is P f = 1 R = P{ g( x) 0} = p( g) dg. The dimensiong must be done so that the probaility of damage P is smaller than the predefined probability of failure P f P(C D 0) P f. 0 0

14 Reliability Index β Let s assume that the load D and capacity C are normally distributed, then Then the safety marging mean and variance are and probability of failure is Cornell s reliability index β from 1969 is then If all variables are normally distributed and they can be described with mean and variance, the reliability method is called second moment method Normally distributed limit state function G(C, D) and safety index β D C g µ µ µ =, D C g σ σ σ + = ). ( c D C D C g g f P β σ σ µ µ σ µ = Φ + = Φ = Φ. 2 2 D C D C g g c σ σ µ µ σ µ β + = = ) ( 2 ) ( 2 1 ) (, 2 1 ) ( D D C C y D D x C C e y f e x f σ µ σ µ π σ π σ = =

15 Geometrical Explanation of Safety Index When normalized with The limit state is D D C C D D C C σ µ σ µ = ʹ = ʹ, 0 ) ( ) ( = + ʹ ʹ = = D C D C D C D C x g µ µ σ σ

16 Example 1 Cantilever beam with L = 6 m and point force P at the end The beam fails when the bending moment at the support (wall) M is equal or exceeds the critical value M F P In addition it is assumed that the force P and critical moment M F are statistically independent and the mean and variance values are: - Load µ P = 5 kn, σ P = 1 kn, - lujuus µ M = 50 knm, σ Μ = 5 knm, The limit state function is G( M F, P) = M F PL L=6m so µ σ G 2 G = 50 knm 5 kn 6 m = 20 knm = ( 5 knm) + ( 1kN 6 m) = 61(kNm) which gives the reliability index. µ g β c = = 2,57 σ g

17 The reliability Index for Several Variables 1 st Order Value Usually both the load and strength are dependent of several statistical variables Structural geometry Material properties The assumption of normally distributed variables is made. In general this dependency can be nonlinear. The basic variables are X = {X1, X2,...., Xn} The limit state n-dimensional surface is G(X)=0. In standardized normal distribution the variable is Z i = X i µ i σ i By linearization with respect to point Z 0 one gets n G G ( ) G ( ) Z Z + Z Z 0 0 i= 1 i ( Z Z i ) 0 i The linearization point is selected based on the average values Z 0 = µ. Then the first order reliability índex gets β FO = n i= 1 G( µ ) G Z i µ σ i 2 The problem is that the linearization point µ may be far away from the damage point and in addition the reliability index is not invariant (can change during the calculations).

18 Application in Practice The reability of side structure of ice strengthened ship was presented in RINA 1990 for Baltic Sea conditions The load was estimated from frame measurements of MS Kemira from several years Baltic Sea regions were classified based on time and place. From 12 hour periods (during 1 year) the mean µ and standard deviation σ of ice-load was calculated and Gumbel I fit was carried out { exp[ C( y )]} F( y) = exp U Based on this long term distribution was formed for the lifetime of 20 years G(y) = F(y) N Where N takes into account the lifetime and number of ice operation days in specific sea area The limit stage is plastic capacity of a frame and the statistical parameters are yield strength (large impact on results) and geometrical variables (low impact on results). The strength was assumed to follow normal distribution.

19 Why Optimization? Example 5 variable problem 15 different design possibilities for every variable 1 s for evaluating structural response 5 15 = s ª 8.7days t f1 h c t w t f2 2p 19/02/2009

20 Definition - mathematical Optimization is a mathematical process of finding an extreme of a function applying mathematical programming Extreme f( x) x 10

21 Definition - engineering Optimization is an automated definition of best design practice

22 Definition business/managerial Optimization is creation of more efficient business solutions/products Maximizing profits Minimizing costs Maximizing efficiency Minimizing 19/02/2009

23 History of optimization Karush and Kuhn & Tucker defined the basic principles of finding the minimum of a function 1930 s to 50 s Danzig Linear programming Numerical procedure for efficient solution to linear problems SIMPLEX method considered among 10 best mathematical concepts of the 20 th century Structural optimization in shipbuilding Until 60 s optimization exists only in manual forms with the help of diagrams that explain structural behaviour In 60 s NTNU Trondheim Application of computers Weight optimization of beams, girders, frames, grillages and main frames of various ships, mostly tankers 19/02/2009

24 History(2) In 80 s MAESTRO FE analysis and SLP Weight optimization of ship hull Multicriterion optimization In 90 s till present day Optimization using genetic algorithms Optimization in the fuzzy, probabilistic, and rough set environment Topology and shape optimization = /02/2009

25 Topics of optimization Sizing optimization Thickness Height Geometry (position) Shape optimization Curvature Angle Topology optimization Position Existence/non-existence of an element Kul Laivan rakenteiden optimointi 19/02/2009

26 Example of shape optimization n n Minimization of stress concentration in the corners of window openings Despite of questionable window shape obtained after optimization, max von Mises stress was reduced from 230 to 200 MPa

27 Types of optimization Single-criteria optimization Optimize for one particular criteria Optimal solution No design selection needed just search for the optimum Multicriterion optimization Optimize for more than one criteria Pareto optimum 0,53 A [m 2 ] 0,51 0,49 0,47 0,45 0,43 0,41 w(w)=0,01; w(vcg)=0,5 g=2000 w(w)=0,45;w(vcg)=0,45 g=2000 w(w)=0,5;w(vcg)=0,01 g=2000 w(w)=0,01; w(vcg)=0,5 g=500 w(w)=0,45;w(vcg)=0,45 g=500 w(w)=0,5;w(vcg)=0,01 g=500 Reference design Concurrent optimum Design selection for the mostefficient solution 0,39 4,7 4,8 4,9 5,0 5,1 5,2 VCG 5,3[m] 5,4

objective hyperspace (dots indicate Pareto-optimal

28 Pareto front In case of multiple objectives, Pareto front is the set of best non-dominated solutions in objective hyperspace (dots indicate Pareto-optimal solutions) 19/02/2009 Kul Laivan rakenteiden optimointi

29 Optimization problem Cantilever Minimum weight/volume/area design problem Cantilever has to sustain the loading F F L b h 19/02/2009

30 Objective function By optimizing the cantilever we minimize the objective function A = b h 19/02/2009 Kul Laivan rakenteiden optimointi

31 Design variables Variables that are changed during the process of optimization b, h 19/02/2009 Kul Laivan rakenteiden optimointi

32 Constraints Normal stress σ := 3 F b h 2 L h + 1 Shear stress τ := F b h Displacement w : = 4 F L Ebh 3 3

33 Optimization problem and parameters min Abh (, ) st.. σ τ w max max max σ 0 τ 0 w 0 90 bh, 120 Parameters: F = 10 kn L = 2m E = 200 GPa σ τ w max max max = 200 MPa = 50 MPa = 10 mm 19/02/2009

34 Graphical solution general case What is a design??? Every design point represents a point in a n- dimensional hyperspace, where n is a number of design variables Objective functions x 2 Constraints Feasible region x 1 19/02/2009

35 Graphical solution to cantilever h [mm] b [mm] 19/02/2009

36 Mathematical programming Global search methods Exploration of whole design space Robust methods Time consuming Local search methods Gradient based methods Search for local optimum Quick Hybrid methods Combination of good properties of global and local search methods 19/02/2009

37 Local search methods Classic gradient methods are not applicable for complex engineering problems Approximation of design space linearization Sequential linear programming Optimization is performed by moving the single point thru design space according to the negative gradient of objective function 19/02/2009

38 Global methods Direct search methods, dealing with the objective function and not requiring calculation of gradients Genetic algorithms, genetic programming, evolutionary programming, evolution strategy, particle swarm optimization etc.

39 Alternative approach to optimization Application of genetic algorithms Simulating the evolutionary principles Do not work with functions, but their values Nonlinear and non-derivable functions can be treated with ease Do not work with one design alternative (DA), but with a population of DAs Rank the DAs according to the particular measure Consideration of discrete variables is simple if variables are binary coded Variable values are represented as strings of bits (0/1) Possibility for mixed consideration of variables 19/02/2009

40 Working principle of genetic algorithm GAs are computerized search and optimization algorithms designed based on mechanics of natural genetics and natural selection START Parametrs of genetic algorithm Coding of cromosoms Parent choosing NO Fitness calculation Tournament of survival Mutations Crosingover NGEN = MAXGEN Next generation YES Output of results END

PSO Relatively new method proposed in 1995 Concept based on bird or fish swarm behavior and how knowedlge is tranferred Belongs to the group of

Population-based optimisation technique developed by Kennedy and Eberhart (1995) Best particle in current calculation round redirects particles of

41 PSO Relatively new method proposed in 1995 Concept based on bird or fish swarm behavior and how knowedlge is tranferred Belongs to the group of evolutionary algorithms similar principles as genetic algorithm Particle swarm optimisation (PSO) Identify the conceptual design alternatives Population-based optimisation technique developed by Kennedy and Eberhart (1995) Best particle in current calculation round redirects particles of next round to previous best particle One-way sharing mechanism, which looks only for the best solution only All particles tend to converge to the best solution

42 PSO Relatively new method proposed in 1995 Concept based on bird or fish swarm behavior and how knowedlge is tranferred Belongs to the group of evolutionary algorithms similar principles as genetic algorithm Identify the conceptual design alternatives Population-based optimisation technique developed by Kennedy and Eberhart (1995) Best particle in current calculation round redirects particles of next round to previous best particle One-way sharing mechanism, which looks only for the best solution only All particles tend to converge to the best solution

43 Exploration of the design space Speed of particle i at iteration k in design space - Particle s best location until iteration k - Swarm s best location until iteration k - Weight factors for the three direction components - i p k g p k i v k wc, 1, c2 x = x + v i i i k+ 1 k k+ 1 v i = 1 v i 1 1( p i x i ) 2 2( p g x i k w + k cr + + k k c r k k)

Structural optimisation methods and framework Local response and strength Euler-Bernoulli beam and Kirchhoff plate theory for local response evaluation

44 Structural optimisation methods and framework Local response and strength Euler-Bernoulli beam and Kirchhoff plate theory for local response evaluation Analytical strength criteria for yielding and buckling Optimization Large number of design variables Novel ship concepts with GA Steel GA Design criteria Strength Weight and cost Other... Analysis guided by optimisation Updating of scantling ConStruct Technoeconomical analysis Database of feasible design alternatives Non-convex design space Genetic Algorithm with vectorization and constraint grouping to create Pareto surface Optimisation framework ConStruct-software environment 44

Structural optimisation - Case description Prismatic Post-Panamax cruise ship L=275.73 m, B=42.68 m, and H=43.

45 Structural optimisation - Case description Prismatic Post-Panamax cruise ship L= m, B=42.68 m, and H=43.75 m Web frame spacing 2730mm Load according to DNV rules M=-4400/8400MNm External pressure loads Deck loads Objectives for optimisation Weight Vertical centre of gravity (VCG) 45

46 Structural optimisation Selected Designs WEIGHT OPTIMUM Vertical centre of gravity [m] W opt. Initial design Weight [ton/m] VCG opt. STIFF OUTER SHELL OR BULKHEAD FLEXIBLE OUTER SHELL VCG OPTIMUM STIFF OUTER SHELL OR BULKHEAD FLEXIBLE OUTER SHELL 46

Results FEM comparison 6120mm t=7mm, s=680mm HP100x6 t=13mm, s=680mm HP100x7 T480x8/200x10 S=2730mm T480x8/200x10 S=2730mm t=7mm, s=680mm HP100x6, p side =10kPa, p deck =5kPa t=6mm, s=680mm HP100x6,

47 Results FEM comparison 6120mm t=7mm, s=680mm HP100x6 t=13mm, s=680mm HP100x7 T480x8/200x10 S=2730mm T480x8/200x10 S=2730mm t=7mm, s=680mm HP100x6, p side =10kPa, p deck =5kPa t=6mm, s=680mm HP100x6, p side =10kPa, p deck =5kPa t=5 mm, s=680mm HP100x6, p side =10kPa, p deck =5kPa t=5 mm, s=680mm HP100x6, p side =10kPa, p deck =5kPa t=5 mm, s=680mm HP100x6, p side =10kPa, p deck =5kPa t=5 mm, s=680mm HP100x6, p side =10kPa, p deck =5kPa t=5 mm, s=680mm HP100x6, p side =10kPa, p deck =5kPa t=5 mm, s=680mm HP100x6, p side =10kPa, p deck =20kPa t=5 mm, s=680mm HP100x6, p side =11kPa, p deck =5kPa t=5 mm, s=680mm HP100x6, p side =18kPa, p deck =5kPa t=5.5mm, s=680mm HP100x6, p side =24kPa, p deck =5kPa t=6.5mm, s=680mm HP100x6, p side =29kPa, p deck =5kPa t=7.5mm, s=680mm HP100x6, p side =41kPa, p deck =5kPa t=13mm, s=680mm HP240x12, p side =69kPa, p deck =5kPa t=16mm, s=680mm HP280x11 p side =94kPa p deck =102kPa z-coordinate [m] L/8 L/4 L/2 SUPERSTRUCTURE RECESS HULL x-normal stress [MPa] Initial design Coupled Beam method FEM 47

48 Results Normal stress and force distribution D11 D10 D9 D8 D7 D6 D5 D4 D3 D2 D1 D0 TD TT Normal Stress [MPa] z-coordinate [m] x=l/8 8 4 x=l/4 x=l/2 SUPERSTRUCTURE RECESS HULL z-coordinate [m] Force [MN] Initial design Weight optimum VGC optimum 8 4 x=l/8 x=l/4 x=l/2 48

49 Results Moment distributions M/M max [-] 1 0,75 0,5 Total moment M = M h + M s Weight optimum, M h 50% Initial Design, M h Weight Optimum Initial Design VCG optimum Superstructure Moment, M s 0,25 30% VCG optimum Hull Moment, M h x-coordinate [m] 49

50 Results - Variation of neutral axis Change in load carrying mechanism affects on the location of the neutral axis Shape of stress distribution in z-direction is varied Share of bending moment in x-direction is varied due to different vertical and shear stiffness between decks (scantlings) x=0 SUPERSTRUCTURE x=l/4 x=l/2 RECESS Weight optimum, VCG = N.A. VCG optimum, VCG Initial Design, N.A. HULL VCG optimum, N.A

design? What happens to our design if we are uncertain of the loads, dimension or material strength? That is how reliable is our design?

51 Summary The classical division of loads, response and strength work on assessment of single design x=x(t,s,b,l.,,,) In practice we are interested at least on two variations around this design point x+dx Where is the optimum x*, i.e. the lightest, fastest, safest, etc. design? What happens to our design if we are uncertain of the loads, dimension or material strength? That is how reliable is our design? Depending on design stage reliability and optimization can be carried out using only limited Accuracy of the model (2D vs 3D) Degrees of freedom in design (how much dimension are allowed to change due to other design disciplines, design spiral)

Sensitivity and Reliability Analysis of Nonlinear Frame Structures

Sensitivity and Reliability Analysis of Nonlinear Frame Structures Michael H. Scott Associate Professor School of Civil and Construction Engineering Applied Mathematics and Computation Seminar April 8,