THE APPLICATION OF RELIABILITY BASED OPTIMIZATION OF TOPHAT - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS THE APPLICATION OF RELIABILITY BASED OPTIMIZATION OF TOPHAT STIFFENED COMPOSITE PANELS UNDER BIDIRECTIONAL BUCKLING LOAD X.G. Xue 1, 2 *, G.X. Li 1 , R.A. Shenoi 2 , A.J. Sobey 2 1 Department of Mechanical Engineering and Automation, National University of Defense Technology, Changsha, China, 2 School of Engineering Sciences, University of Southampton, Southampton, UK * Corresponding author (xuexiaoguang1985@gmail.com) Abstract V Elastic strain energy of the girders g A reliability based optimization method for V Elastic strain energy of the beams minimization the weight of tophat stiffened b composite panels with probabilistic deflection W The work done due to external load constraints is presented in this paper. An energy based grillage method is developed for the investigation of buckling problems under 1 Introduction bidirectional in-plane loads. The variables that have Composite materials have been increasingly used in large impact on structural safety have been identified, aircraft, space and marine due to their outstanding which will allow the optimization process be more strength, stiffness and light-weight properties [1]. In quantitative and efficient. Parametric study of plate the construction of marine vessels, stiffened panels dimension and loading ratio is conducted to comprised of a plate, longitudinal stiffeners and investigate the coupling effect on critical buckling transverse frame are the most commonly used load. structural elements, forming the deck, side shells and Keywords: Tophat-stiffened, Reliability based bulkheads. The composite structural behavior optimization, Grillage, Bidirectional load exhibits wide scatter as a result of the inherent uncertainties in manufacture and design variables. Structural reliability methods allow the engineers to Nomenclature reduce the probability of failure and lead to a L Length of stiffened panel balanced design. The reliability-based design optimization tries to find a highly reliable design by B Width of stiffened panel ensuring the satisfaction of probabilistic constraints,  Plate aspect ratio which is more flexible and consistent than corresponding deterministic analysis because they I Moment of inertia of girder provide more rational safety levels over various g types of structures and take into account more I Moment of inertia of beam information that is not considered properly by b deterministic analysis. Once the probabilistic model N Number of longitudinal girders is established, probabilistic analysis is run and then g the sensitivity factors are obtained in order to N Number of transverse beams determine the importance of a random variable, b which will allow the optimization process be more N Uniform load in x direction x quantitative and efficient. N Uniform load in y direction y 2 Analytical model  In-plane loading ratio 2.1 General configuration w Deflection of plate

The definition of a stiffened panel is a panel of where m and n represent half-range sine expansions plating bounded by, for example, transverse in x and y directions, respectively. The analysis of a bulkheads, longitudinal bulkheads, side shell or grillage based methods are capable of giving large longitudinal girders [2]. A typical stiffened comprehensive and adequate results. The potential panel configuration with the tophat-section stiffeners energy, V , in a deflected grillage can be written as is shown in Fig. 1. The stiffened panel is referred to    (2) V V V W x- and y- axis coinciding with its longitudinal and g b transverse edges, respectively, and a z-axis normal V and V are the strain energies in the girders and g b to its surface. The length and width of the stiffened beams respectively, which can be represented as [5], panel are denoted by L and B , respectively. The spacing of the stiffeners is denoted by a between 2    N 2 1  g w  L  longitudinal stiffeners and b between transverse V E I   dx (3)  g gi gi 2 2   0 x stiffeners. The numbers of longitudinal and  iB i 1  y N  1 transverse stiffeners are N g and N s , respectively. The g 2 web, table, and flange structures forming a tophat-    2 N 1  w b  B  V E I   dy (4) stiffener are made of FRP laminates and they are  b bj bj 2 2   0 y  jL assumed to be orthotropic plates. j 1  x N  1 b The work done W due to the bidirectional in-plane     L N 1 a b load, N and N , can be written as x y 2  N   1  g w L  N   dx  x   2 x 0 iB   i 1 y N  1  g W (5) N x   N x   1 B N b 2    g N 1 w  b L   N dy  y 2   0 y  jL 1  j x N  1 b By using the energy method in structural analysis    that maximum buckling load ( N N , N N ) N y N y x c y c often is approximated using energy conservation t f t f    0 V V W (6) g b b 2 b 1 b 2 b 1 A-A A-A Substitute Eq. 3, Eq. 4 and Eq. 5 into Eq. 6, the numerical critical buckling load can be obtained Fig.1 Tophat stiffened panel configuration            2 4 3 4 m D N 1 n D N 1   2.2 Analytical analysis  g g b b N (7)       c     2 2 2 The analysis of the stiffened composite panel is 1 1 L m N n N   g b carried out based on grillage model developed by     Vedeler [3]. The plate is compressed by in-plane where L B and N N are the plate aspect y x load resultant of N and N in x and y direction ratio and the in-plane loading ratio, respectively; y x ( , ) respectively. The deflection, w x y , at any point of N     N g b D E I and D E I are the flexural the grillage is expressed by the following double g gi gi b bj bj   summation of trigonometric series to Navier ’ s 1 i 1 j rigidity of the girder and beam, respectively. energy method [4]:       m x n y 3 Reliability based optimization ( , ) sin sin w x y f (1) mn L B   3.1 Optimization problem formulation m 1 n 1