1.4 Optimization of an orthogonally stiffened plate considering fatigue constraints

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1 1.4 Optimization of an orthogonall stiffened plate considering fatigue constraints Luis M.C. Simões 1, József Farkas and Karol Jarmai 1 Universit of Coimbra, Portugal,lcsimoes@dec.uc.pt Universit of Miskolc, Hungar,altjar@uni-miskolc.hu Abstract The aim of this work is the optimization of a uniaiall compressed stiffened plate subjected to static and fatigue loading. The design variables are the thickness of the base plate, the number and stiffeners of the orthogonall stiffened plate. The constraints deal with the static overall plate buckling, the stiffener failure and the fatigue strength of the welded connections between the stiffeners and the interaction of the two tpes of failure. The cost function includes the cost of material, assembl, welding and painting. Randomness is considered both in loading and material properties. A level II reliabilit method (FORM) is emploed. The overall structural reliabilit is obtained b using Ditlevsen method of conditional bounding. The costs of the plate designed to ensure a stipulated probabilit of failure will be compared with the solutions obtained for a code based method, which emplos partial safet factors. Kewords: reliabilit-based optimization, stiffened plates, fatigue 1. Introduction Stiffened plates are often the main structural components of load-carring structures such as bridges, columns, towers, platforms, vehicles etc. The aim of this work is the optimization of a uniaiall compressed stiffened plate. The thickness of the base plate as well as the numbers and dimensions of the longitudinal and transverse stiffeners are sought, which fulfil the design and fabrication constraints and minimize the cost function. The constraints relate to the static overall plate buckling, to the stiffener induced failure and to the fatigue strength of welded connections between the stiffeners. Interaction of the two tpes of failure, buckling and fatigue can be more dangerous than each individuall: the fatigue crack propagation might affect the development of buckling. The buckling constraints are formulated according to the Det Norske Veritas design rules, the fatigue strength constraint is epressed using the data of Eurocode 3. The fabrication constraints limit the maimal number of stiffeners in one direction to ensure the welding of welds connecting the stiffeners to the base plate. The cost function includes the cost of material, assembl, welding and painting and is formulated according to Farkas,J., Jármai,K.(003). Stresses and displacements can be computed given the deterministic parameters of loads, geometr and material behaviour. Some structural codes specif a maimum probabilit of failure within a given reference period (lifetime of the structure). This probabilit of failure is ideall translated into partial safet factors and combination factors b which variables like strength and load have to be divided or multiplied to find the so called design values. The structure is supposed to have met the reliabilit requirements when the limit states are not eceeded. The advantage of code tpe level I method (using partial safet factors out of codes) is that the limit states are to be checked for onl a small number of combinations of variables. The safet factors

2 are often derived for components of the structure disregarding the sstem behaviour. The disadvantage is lack of accurac. This problem can be overcome b using more sophisticated reliabilit methods such as level II (first order second order reliabilit method, FOSM [4] and level III (Monte Carlo) reliabilit methods. In this work FOSM was used and the sensitivit information was obtained analticall. Besides stipulating maimum probabilities of failure for the individual modes, the overall probabilit of failure which account for the interaction b correlating the modes of failure is considered. A branch and bound strateg coupled with a entrop-based algorithm is used to solve the reliabilit-based optimization. The entrop-based procedure is emploed to find optimum continuous design variables giving lower bounds on the decision tree and the discrete solutions are found b implicit enumeration. Results are given comparing deterministic and reliabilit-based solutions and show how the optimum solution changes with the aial force and loading amplitude used to describe fatigue. b 0 =n s N a 0=n s s t a w s e s N s e h 1/ h / G t w z G t h 1 / h / a w G t w zg t f b t f b Figure 1. Orthogonall stiffened plate loaded b uniaial compression. Design variables

3 The design variables are the base plate thickness t, sizes and number of stiffeners in both directions: h, h, n, n. Ranges of unknowns: 4 < t < 0 mm, 15 < h < 1016 mm, 4<n<n ma. The maimum values of n i is given b the fabrication constraints Eq. (1). b0 a b 300 mm, 0 b 300 n mm. (1) n 3 Constraints 3.1 Overall buckling constraint according to DNV N f 1 f cr, f 1 () n A 4 e f 1 NEs b0 a0, B E,NE B (3) A b E e 0 a0 b0 It can be seen from the load-carring capacit formula N E that, when a 0 >b 0, to have a larger N E, B (h ) should be larger than B (h ). From the theoretical buckling strength σ E the critical strength σ cr is calculated b using a slenderness λ to take into account the effect of initial imperfections. The factored compressive force is calculated as N = γ stat N stat + γ F N/, (4) where γ stat = 1.1 and γ F = 1.35 are safet factors, N stat is the static component and a variable component has an amplitude of N/. 3. Constraint on stiffener torsional buckling according to DNV The constraint is formulated as N k 1 acr (5) n Ae1 S 3.3 Constraint on fatigue strength of welded connections of stiffeners The constraint on fatigue strength is defined b N N 0 (6) n A e Mf where α 0 is the interaction factor to avoid the danger of interaction of the buckling and fatigue phenomena, N is the variable load range, σ N is the fatigue stress range corresponding to the number of ccles N C, γ Mf is the safet factor for fatigue. 4. Cost function The cost function includes the cost of material, assembl, welding as well as painting and is formulated according to the fabrication sequence. The cost of material KM km V ;km 1. 0 $/kg. (7) Welding of the base plate from butt welds (3 in direction of a 0 and 3 in direction of b 0 ) (SAW - submerged arc welding) Farkas,J., Jármai,K.(007): The fabrication cost factor is taken as k F = 1.0 $/min, the factor of compleit of the assembl : W

4 K n 0 kf W 16V0 1. 3CW t 3a0 3b0, (8) Welding (n -1) stiffeners to the base plate in direction with double fillet welds (GMAW-C - gas metal arc welding with CO ): K 3 W1 kf W 1 1 w 0 n V a b n 1 (9) Welding of (n 1) stiffeners to the base plate in direction with double fillet welds. These stiffeners should be interrupted and welded with fillet welds to the stiffeners in the direction. K 3 n n n V a a n W kf W w T (10) which is rounded to 0.4t w. Painting KP kp PSP (11) k P = $/mm, Θ P =, Surface to be painted S P = a 0 b 0 + a 0 (n 1)(h 1 + b ) + b 0 (n 1)(h 1 + b ) (1) The total cost K = K M + K 0 + K W1 + K W +K P (13) 1 5. Reliabilit-based optimization A failure event ma be described b a functional relation, the limit state function, in the following wa F={g() 0} (14) In the case the limit state function g () is a linear function of the normall distributed basic random variables the probabilit of failure can be written in terms of the linear safet margin M as: P F Pg 0 PM 0 (15) which reduces to the evaluation of the standard normal distribution function P (16) F where is the reliabilit inde given as M / M (17) The reliabilit inde has the geometrical interpretation as the smallest distance from the line (or the hperplane) forming the boundar between the safe domain and the failure domain. The evaluation of the probabilit of failure reduces to simple evaluations in terms of mean values and standard deviations of the basic random variables. When the limit state function is not linear in the random variables, the linearization of the limit state function in the design point of the failure surface represented in normalised space u. was proposed in Hasofer, A.M. & Lind, N.C. (1974), u / (18) i i i i As one does not know the design point in advance, this has to be found iterativel in a number of different was. Provided that the limit state function is differentiable, the following simple iteration scheme ma be followed: n i g / ui g / ui (19) j1

5 G 1,,... v (0) which will provide the design point u* as well as the reliabilit inde β. The reliabilit assessment requires an enumeration of the reliabilit indices associated with limit state functions to evaluate the structural sstem probabilit of failure. Collapse modes are usuall correlated through loading and resistances. For this reason, several investigators considered this problem b finding bounds for p F. B taking into account the probabilities of joint failure events such as PF i F j which means the probabilit that both events Fi and F j will simultaneousl occur. The resulting closed-form solutions for the lower and upper bounds are as follows: m i1 p F F1 MaPFi PFi Fj ;0 (1) i j1 p F m m F F P MaP F i i i1 i ji j The above bounds can be further approimated using Ditlevsen (1979) method of conditional bounding [10] to find the probabilities of the joint events. This is accomplished b using a Gaussian distribution space in which it is alwas possible to determine three numbers 1, and the correlation coefficient ij for each pair of collapse modes F i and F j such that if ij >0 ( F i and P( F F ) Ma { ( ) ( ) ; ( ) ( )} i j ij j i ij i j j i 1 ij 1 ij P( F F ) ( ) ( ) ( ) ( ) i j ij j i ij i j j i 1ij 1ij F j positivel correlated): In which i and j are the safet indices of the ith and the jth failure mode and [ ] is the standardized normal probabilit distribution function. The probabilities of the joint events P(F i F j ) in (8) and 89) are then approimated b the appropriate sides of (3) and (4). For eample, if F i and F j are positivel dependent for the lower (1) and upper () bounds it is necessar to use the approimations given b the upper (4) and lower (3) bounds, respectivel. 6. Optimization Strateg 6.1 Branch and Bound The problem is non-linear and the design variables are discrete. Given the small number of discrete design variables an implicit branch and bound strateg was adopted to find the least cost solution. The two main ingredients are a combinatorial tree with appropriatel defined nodes and some upper and lower bounds to the optimum solution associated the nodes of the tree. It is then possible to eliminate a large number of potential solutions without evaluating them. Three levels were considered in the combinatorial tree. The plate thickness is fied at the top of the tree, the remaining levels corresponding to n (and the appropriate UB profile h ) and n associated with h. A strong branching rule was emploed. Each node can be branched into n s new nodes, each of these being associated with the number of stiffners needed in the direction. This requires using continuous values close to the geometric characteristics of an UB section, A, b, t, t, which are s f w () (3) (4)

6 approimated b curve-fitting functions written as a function of h. The stiffener height is also obtained from a curve fitting of the heights h. Care has to be taken to find geometrical properties leading to conve underestimates of the actual UB section, so that the solution obtained b using the real UB geometric characteristics is more costl than the solution given b using continuous approimations. In the second level of the tree the branches correspond to different stiffener UB profiles. At the third level the resulting minimum discrete solution becomes the incumbent solution (upper bound). An leaf of the tree whose bound is strictl less than the incumbent is active. Otherwise it is designated as terminated and need not to be considered further. The B&B tree is developed until ever leaf is terminated. The branching strateg adopted was breadth first, consisting of choosing the node with the lower bound. 6. Optimum design with continuous design variables For solving each relaed problem with continuous design variables the simultaneous minimization of the cost and constraints is sought. All these goals are cast in a normalized form. For the sake of simplicit, the goals and variables described in the following deal with stiffened shells. If a reference cost K 0 is specified, this goal can be written in the form, g1 t, n, h K t, n, h / K0 1 0 (5) Another two goals arise from the constraint on overall buckling and single panel buckling: g3 t, n, h c / cr 1 0 (6) g t, n, h l / acr 1 0 (6) The remaining goal deals with the fatigue strength of the stiffeners connections: g4 t, h / n 1 0 (8) The objective of this Pareto optimization is to obtain an unbiased improvement of the current design, which can be found b the unconstrained minimization of the conve scalar function Simões, L.M.C. & Templeman, A.B.(1989): F 1 3 t, h.1n ep gt, h jl This form leads to a conve conservative approimation of the objective and constraint boundaries. Accurac increases with ρ. The strateg adopted was an iterative sequence of eplicit approimation models, formulated b taking Talor series approimations of all the goals truncated after the linear term. This gives: 1 3 Min F t, h.1n ep g t, h jl 0 g 0 j t t, h g t, h dt 0 j h dh This problem has an analtic solution giving the design variables changes dt and dh. Solving for a particular numerical value of g oj forms an iteration of the solution to problem (30). Move limits must be imposed on the design variable changes to guarantee the accurac of the approimations. Given the small number of design variables an analtic solution is available. During the iterations the control parameter ρ, which should not be decreased to produce an improved solution, is increased. (9) (30)

7 5. Numerical Eamples Numerical data (Figure 1): a 0 = 4000, b 0 = 8000 mm, steel ield stress f = 355 MPa, elastic modulus E = MPa, shear modulus G = , densit ρ = kg/mm 3, selected rolled I-sections UB profiles. Consistent with the traditional limit state design (level 1 approach), the following solutions consider a deterministic behaviour of all the variables for several load combinations of the original compressive force N stat and the load range for fatigue N. N stat N h h t n n Cost B adopting coefficients of variation of 0.15 for the compressive force, 0.5 for the load amplitude and 0.10 for the design stress, solution 1 was used to tune mean values for the static force, load amplitude and design stresses. The Gaussian distribution was adopted for all the random variables. Although the randomness of Young modulus also plas an important role in the structural reliabilit, this was not considered here for the sake of simplicit. In this eample the probabilit of failure will be describe with the overall buckling stresses, the stiffener torsional buckling and the fatigue strength of the welded connections of stiffeners induced b the loadings. A maimum individual probabilit of failure p f 1.0E-4 (beta larger than 3.7) was established. The following reliabilit based optimum solutions were obtained: Nstat N h h t n n Cost The reliabilit-based solutions are generall least costl than the deterministic and ensure the safet level adopted. If the all the modes are considered and the overall probabilit of failure p f 1.0E-4 is imposed b using Ditlevsen improved second order bounds, the reliabilit based solutions for problems 4 and 5 are changed as in the first both the local buckling and fatigue have small values and in the later the overall buckling and fatigue are critical. Nstat N h h t n n Cost The solutions are 1% more costl, being thicker but reducing the number and sizes of the stiffeners in the direction. The influence of the coefficient of variation of the static compressive force on solution 1 was also studied. If the coefficient of variation is increased b 10% to the reliabilit-based solution becomes:

8 Nstat N h h t n n Cost The fatigue constraint is now more important, replacing the local bucking constraint obtained in the previous reliabilit-based solution. There is almost no change in the cost, the solution now being thicker and with larger stiffeners in the direction. Fatigue is usuall associated with a Weibull tpe II probabilit distribution and it is usuall more demanding in terms of design than the normal distribution. If the probabilit distribution functions of the random variables are not Gaussian, the Rosenblatt transformation ma be used. It consists of finding for each random value an equivalent Gaussian distribution function. An increased coefficient of variation for load variation of of 0.75 was specified in problem 1. The following result was obtained: Nstat N h h t n n Cost This reliabilit-based solution is now 1% more costl. However this is obtained b increasing the number of stiffeners in the direction. References Farkas,J., Jármai,K.(007): Economic orthogonall welded stiffening of a uniaiall compressed steel plate. Welding in the World 51 No.3-4, Det Norske Veritas (1995): Buckling strength analsis. Classification Notes No Høvik, Norwa. Eurocode 3 (00), Part 1-9. Fatigue strength of steel structures. Brussels. Farkas,J., Jármai,K.(003): Economic design of metal structures. Rotterdam, Millpress 003. Farkas,J., Jármai,K. (1997): Analsis and optimum design of metal structures. Rotterdam, Balkema. Section Farkas,J., Jármai,K. (1996) Fatigue constraints in the optimum design of welded structures. Int.Conf. Fatigue of Welded Components and Structures. Senlis (France). Eds. Lieurade,H.P. and Rabbe,P. Les Editions de Phsique, Les Ulis, France, Hasofer, A.M. & Lind, N.C. (1974): Eact and invariant second moment code format, J.Eng. Mech Div. 100(1), Profil Arbed Sales program (001), Structural shapes. Arcelor Long Commercial. Farkas,J.(0) Thickness design of aiall compressed unstiffened clindrical shells with circumferential welds. Welding in the World, 46 No.11/ European Convention of Constructional Steelwork (ECCS) (1988), Recommendations for Construction. Buckling of steel shells. No.56. Brussels. Ditlevsen, O.(1979): Narrow reliabilit bounds for structural sstems, J.Struct. Mech, 7 (4), Simões, L.M.C. & Templeman, A.B.(1989): Entrop-based snthesis of pretensioned cable net structures, Engineering Optimization, 15, Rosenblatt, M (195) Remarks on a Multivariate Transformation, Annals of Math. Stat., Simões,L.M.C., Farkas, J & Jarmai, K. (006), Reliabilit-based optimum design of a welded stringer-stiffened steel clindrical shell subject to aial compression and bending, Struct. Multidisciplinar Optim, 30 Farkas,J., Jármai,K.and Kozesh, Z. (007) Cost minimization of an orthogonall stiffened welded steel plate subject to static and fatigue load, Welding in the world, 51, special issue, pp