The Application of Reliability Methods in the Design of Tophat Stiffened

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1 The Application o Reliability Methods in the Design o Tophat Stiened Composite Panels under In-plane Loading Yang N. (1)() & Das P. K., () (1) Harbin Engineering University, China ()University o Strathclyde, UK Blake, J.I.R., Sobey, A.J. & Shenoi, R.A. University o Southampton, UK ABSTRACT: Composite materials have been widely used in modern engineering ields such as aircrat, space and marine structures due to their high strength-to-weight and stiness-to-weight ratios. However, structural eiciency gained through the adoption o composite materials can only be guaranteed by understanding the inluence o production upon as-designed perormance. In particular, topologies that are challenging to production including panels stiened with pi or tophat stieners dominate many engineering applications and oten observe complex loading. The design o stiened composite panels against buckling is a key point o composite structures. While a growing number o studies are related to the reliability analysis o composites ew o these relate to the local analysis o more complicated structures. Furthermore or the assessment o these structures in a design environment it is important to have models that allow the rapid assessment o the reliability o these local structures. This paper explores the use o a stochastic approach to the design o stiened composite panels or which typical applications can be ound in composite ship structures. A parametric study is conducted using Navier grillage theory and First-order Reliability Methods to investigate any detectable trend in the saety index with various design parameters. Finally, recommendations are made to provide guidance on applications. Keywords: Tophat-Stiened, Reliability Based Design, Sensitivity 1. INTRODUCTION Composite materials have been widely used in modern engineering ields such as aircrat, space and marine structures due to their high strength-to-weight and stiness-to-weight ratios. Stiened panels, comprised o a plate, longitudinal stieners and transverse rames, are very important components in ship and oshore structures, which can be ound in decks, bottoms, bulkheads, side shells and superstructures. The design o stiened composite panels against buckling is a key point o composite structures, particularly in bottom shell or deck units subjected to compressive load by longitudinal wave-induced or explosion-induced bending o the ship hull. The inherent uncertainties in geometry, materials, loads, and other aspects o any structure are unavoidable in structural responses. Because o the existence o such uncertainties, to ensure the structures can perorm their intended unction with desired conidence, these uncertainties or variabilities must be considered during structural design. Traditional methods o dealing with the uncertainties are to use conservative ixed values in equations to guard against the possibility o structural damage. Assumptions are made that all actors inluencing the load, strength and other 1

2 uncertainties are known, ignoring uncertainties that might occur such as variability o material properties or uncertainty in analysis models. The conventional deterministic design methods are simple but inlexible to adjust the prescribed saety margin and do not give a reliable indicator o satisactory perormance or the design o FRP structures. With the development o reliability technology, reliability methods have been used in reliability-based design or marine and oshore structures. Structural reliability methods allow the designers to limit the probability o undesirable events and lead to a balanced design. Reliability-based design is more lexible and consistent than corresponding deterministic analysis because it provides more rational saety levels over various types o structures and takes into account more inormation that is not considered properly by deterministic analysis. Reliability techniques have been in development or a number o years. These methods irst appeared in a mathematical orm in the 19 s by Mayer [1] and urther developed by Streletzki [] and Wierzbieki [3]. Practical usage o these methods was not developed until the late 196 s with the development o a second moment reliability index by Cornell [4]. Cassenti [5] urthered deterministic design methods by developing the probabilistic static ailure analysis procedure o unidirectional laminated composite structures. Yang [6] presented a reliability analysis o laminated plates based on the last-ply-ailure analysis concept. Cederbaum [7] presented work related to in-plane loads using irst ply-ailure on symmetric angle-ply laminates. Gurvich [8],[9] developed a probabilistic ailure model or the reliability o laminated composites subjected to combined lateral pressure and in-plane loads based on a ply group concept and this was urther developed to include both a ply group and a laminated plate subjected to uni-axial tensile loads. Speciic to a marine environment Jeong and Shenoi [1], [11] presented a simulation approach to assess the irst-ply ailure reliability o composite plates. Other marine speciic studies have concentrated on global assessment o ship hulls including Chen et al.[1] and Zhang [13]. Finally Blake et al. [14] looked at a method or assessing the reliability o composite grillages utilising Navier grillage theory with simple limit states under out o plane load. This research showed that grillage theory was good or assessing more complex composite structures however it or a ull analysis o composite structures there will be a requirement to analyse grillages with the addition o in-plane loads. While these studies have progressed the status o reliability analysis o composite structures the analysis has been perormed on simple structures, plates, cylindrical shells and others or in the case o the more complex analysis has concentrated on a global rather than local assessment o reliability. Furthermore this review shows that while there is a growing quantity o composite reliability literature this is generally not marine speciic and it is important to perorm reliability investigations using data similar to marine applications. This paper thereore ocuses on the reliability o composite grillage plates using Navier grillage theory or computationally inexpensive analysis under in-plane loading. The paper looks to incorporate reliability methods into the design o complex composite structures with rapid analysis techniques. Finally an assessment is made to detect any trend in the saety index with various design parameters and recommendations are made to provide guidance on applications.

3 . RELIABILITY-BASED ANALYSIS METHODS There are two types o design ormat that are normally used [15], namely direct reliability based design and Load and Resistance Factor Design (LRFD). Limit state equations are essential or conducting reliability analysis and the means by which a deinition o ailure is articulated mathematically. I the load applied to the structure is deined as L and the resistance o the structure to that load as R, the saety margin may be deined as M g( r, l) R L (1) Since R and L are random variants, M is also a random variant with corresponding probability density unction M (m). In this case, ailure is clearly the event ( M ) and thus the probability o ailure is Cornell proposed a reliability index deined as where P P[ M ] ( m) dm () M M / M (3) M and M are the mean value and the standard deviation o the saety margin, respectively. In engineering practice, the saety index,, instead o ailure probability, P, is oten used to represent the reliability level. 3. GENERAL MODEL OF STIFFENED COMPOSITE PANEL A stiened panel is a panel o plating bounded by, or example, transverse bulkheads, longitudinal bulkheads, side shell or large longitudinal girders. A typical stiened panel coniguration with the tophat-section stieners is shown in Figure 1. The stiened panel is reerred to x- and y- axis coinciding with its longitudinal and transverse edges, respectively, and a z-axis normal to its surace. The cross-section geometry is deined in terms o the six geometrical parameters b 1, b, b 3, b 4, b 5 and d. The length and breadth o the stiened panel are denoted by L and B, respectively. The spacing o the stieners is denoted by a between longitudinal stieners and b between transverse stieners. The numbers o longitudinal and transverse stieners are N g and N s, respectively. The web (sides), table (top) and lange (base) structures orming a tophat-stiener are made o FRP laminates. 3

4 b y L = (Ns+1)a A A a B = (Ng+1)b x y b4 b3 d t1 b t t3 z b b1 A-A Figure 1 Stiened panel coniguration with the tophat-section stieners During structural design and analysis, primary ailure modes should be considered careully as most orms o buckling will result sooner or later in complete collapse o the structure. According to [16, 17], the primary ailure modes or a stiened panel subject to compressive loads might arise in panels as ollows: Local buckling o the plating between stieners Where the lowest initial buckling stress corresponds to local buckling o the plate between stieners, a substantial postbuckling reserve o strength may exist. Generally, local buckling o the shell is associated with loss o eective width, which may cause a reduction in the lexural rigidity o the cross-section. Column-like buckling This buckling mode indicates a ailure pattern in which the collapse is reached by column or beam-column type collapse o the combination o the stiener with the eective plate. Collapse is possibly caused by material tensile or compressive ailure in the stieners. Tripping o stieners Tripping o a stiener can occur when the ultimate strength is reached by lateral torsional buckling o the stiener. This orm o instability is susceptible to open-section stieners. Tophat-section stieners which are usually used in composite ships have high torsional stiness and this buckling mode can be prevented by using stieners with good proportions. Overall instability o the stiened panel This ailure mode reers to the buckling o the gross panel involving longitudinal and transverse rames between the major support members. Overall instability ailure mode typically represents 4

5 the collapse pattern when the stieners are relatively weak. This ailure mode should be proportioned so that this orm o ailure is preceded by that interrame collapse mode because this ailure involves a large portion o structure and is likely to be more catastrophic. 4. ANALYTICAL MODELS Generally speaking, olded plate methods and numerical methods are capable o giving comprehensive and adequate results [17]. However, they are not computationally eicient rom a design point o view or the considerable preparation and computational time involved, particularly i the repeated analyses are required at the preliminary design stage due to the involvement o a large number o variables. Simpliied analytical methods provide a more time-eective means o calculating the strength o stiened panels. For engineering practice, most o the necessary evaluation o stresses and deormations can be carried out by means o simple ormulae based on beam and plate theory on idealized geometries and boundary conditions. 4.1 Column-like Buckling As discussed above, initial buckling may be considered by the orm o column-like instability o longitudinal stieners together with the eective plating so that they would behave as a beam-column when the stieners are dominant. Ignoring the torsional rigidity o the gross panel, the Poisson s ratio and the eect o the intersecting beams, assuming the panel is simply supported, column buckling strength (without consideration o initial imperection and lateral pressure) is given using the Euler ormula [17] D D E /( 1 ) (4) Aa a GA s in which D N i1 ( EI ) is the lexural rigidity o a stiener with associated eective width o plate; i i is the web (sides), table (top) and lange (base) structures; A is the total cross-sectional area o a stiener with an attached strip o plate; a is the spacing between transverse rames; GA s is the shear rigidity in which A s may be taken as the area o the stiener webs, G is the shear modulus o the web (sides) In real applications, composite structures exhibit some unavoidable initial imperections due to the manuacturing process or heavy load connected to the hull, these initial imperections may trigger buckling or premature strength ailure at a load ar below those corresponding to elastic buckling. The initial deormation w are assumed to be an idealization o initial delection shapes or a one-dimensional member, which can be approximately expressed in mathematical orm as x w sin (5) a which takes the hal sinusoidal wave pattern. The total (initial plus added) delection,w, may also take a similar shape to the initial delection as shown in equation (6) 5

6 x w sin (6) a where is the maximum initial imperection and is amplitude o the total delection. The bending moment equilibrium is given by, d D ( w w dx ) Pw The strain-energy-based approach is employed to determine the initially delected column. The total potential energy can be given by, П = U + W (8) The elastic strain energy U and the external potential energy W are calculated as, (7) U a D w w ( ) dx (9) x x P W Pu a w ( ) x w ( ) dx x Applying the principle o minimum potential energy, the amplitude o the total delection can be ound as ollows, 1 / where Ф=1/(1-/ E ) is called the magniication actor. E The maximum stress max at the outer ibre o the cross-section can thereore be obtained by the sum o axial stress and bending stress as ollows, P M max max (1) A W where M P max ; D W ; E i is the membrane equivalent Young s modulus o the element E y i considered (Appendix A); y is the vertical distance rom the neutral axis to the point in question. For plate-beam under combined axial compression P and lateral line load q, the internal bending moment along the span can be obtained by the sum o the bending moment due to lateral load and geometric eccentricity, which may include lateral delection caused by external load as well as an initial imperection. M (11) (1) M P( w ) (13) max q max q max where M qmax and w qmax are maximum bending moment and maximum delection due to lateral load alone. The maximum stress max at the outer ibre o the cross-section can thereore be obtained by 6

7 substituting M max rom Eq.(13) into Eq.(1). 4. Overall Buckling I support members are relatively weak, they will delect together with the plate so that the stiened panel can buckle together. This mode is termed overall buckling. The overall buckling o a stiened composite plate is perormed based on a modiied grillage model, in which the stiened panel is treated as a grillage through substituting equivalent elastic properties o its laminate components into the analysis (Appendix A) The double series expression or the delection o the stiened panel can be assumed mx ny w mn sin sin (14) L B m1 n1 which ulil the end conditions when the plate is simply supported along all edges. The coeicient mn in the series or the delection may be determined by the condition that the change in potential energy due to the assumed delection is minimum. The total potential energy o the longitudinal girders and transverse stieners are given by V g 1 N g i1 D L gi w x y y i dx (15) V s 1 N s j 1 D B sj w y x x j dy (16) ib where y i, N 1 g jl x j when all girders and stieners are arranged at equal distance; D gi N 1 s and D sj are the lexural rigidity o the i th girder and the j th stiener, respectively. The eect o the plate is to act as eective lange or girder and beams. The total potential energy will be V=V g +V b (17) When the stiened panel is subjected to a uniorm longitudinal compressive stress σ in the x-direction, the work o the external orce will be W gi l 1 w x Ag yi dx (18) x Ng W g W gi i1 (19) When the stiened panel is subjected to a uniorm pressure load q alone, the work o the external orce will be W q LB q m1 n1 mn mx ny sin sin L B () 7

8 where A g is the cross-sectional area o a girder. The critical load may be determined by the minimum potential energy ( V W ) mn (1) The coeicient mn can be expressed when the stiened panel is subjected to a uniorm pressure load alone mn 6 4 mnm ( N g 16qLB Dg 4 1) n ( N 3 L b D 1) B I the stiened panel is subjected to a lateral load q as well as to axial compression σ, the delection parameter mn are multiplied by the magniication actor 1 1 E b 3 (). (3) where σ E is the critical compression or the same m and n as the index o the parameter. I the stiened panel has the initial delection w, it may approximately take a similar shape by equation (14) with the amplitude δ. The maximum stress σ max at the outer ibre o the cross-section o the girder can be obtained by Eq.(1) and Eq.(13). For each element o the stiened panel, the average direct stress value acting through the thickness o the element o a particular laminate can be predicted at any point in the section using Eq.(1). The corresponding direct orce intensity per unit width N and moment intensity per unit width M o the laminate section about its own mid-plane can be obtained. Then the ply-to-ply stress analysis can be perormed by Appendix B. Laminate composites may ail by ibre breakage, matrix cracking or shear ailure depending on the geometry, stacking sequence and the load. In the present study, the maximum stress criterion is being used in the principal material direction o each layer, in which the individual stress components are compared with the corresponding material allowable strength values. Failure is deined as First-Ply-Failure (FPF): cracking or crazing o the surace resin, which is usually detected in marine structures, and should be avoided because such cracks allow ingress o water to the laminate, leading to degradation o strength and stiness [18]. The ultimate strength is assumed to be reached when the maximum stress in any layer is reached: rom this point onwards, rapid progressive collapse under compression is expected to ensue. This method requires an iterative procedure but usually only a ew iterations are required. 4.3 Eective Width The lange is usually not ully eectively induced by the plate buckling or shear lag, which results in a non-uniorm stress distribution. Many dierent eective width equations or steel have been extensively studied since von Karman et al. [19] irst introduced the concept. As a matter o act, GRP reinorced plates have peculiar dierences with respect to steel plates and not many ormulae are 8

9 available in literature except those presented by Classiication Societies in which only simple relationships are provided. Boote [] summarised the ormulae or eective width calculation rom dierent Classiications Societies. The ormula rom Bureau Veritas is chosen or this calculation where, b e b or.a b (4) and b e is eective width between stieners, b is the physical width between longitudinal stieners, b is the stiener base width (no overlap) and a is the distance between the transverse stieners (see Figure 1). The choice o eective width is dependent on the consideration o either the transverse beam stieners or the longitudinal girders. 5. APPLICATION OF THE RELIABILITY METHOD 5.1 Introduction In this section, a stochastic approach to the design o a stiened composite panel under compressive load and the combination o compressive and lateral loads or ship structures is applied and the importance o dierent stochastic parameters on the reliability index and ailure probability is investigated. The panel with rectangular tophat-sections consisting o webs, crown and base plate is shown in Table 1. This is typical o the topology in the bottom panel o composite ship structures. The shell and stiener laminates are assumed to be reinorced by woven rovings, which are balanced laminates o the type commonly used in ship construction. Mechanical properties or a unidirectional layer are dependent on lay-up and ibre-volume raction and calculated or practical application in Appendix C the material properties used within these equations or E-Glass and Epoxy are listed in Table. Table 1. Geometric Properties (mm) Panel thickness Crown width Crown height Web width Web height Longitudinal Transverse Thickness, t, o single layer =.56 Table. Material properties o resin and ibre [17] Shear Compressive Tensile Young s modulus Poisson s Tensile strength modulus strength ailure strain E (GPa) ratio ν (MPa) G (GPa) (MPa) (%) Epoxy E-Glass Formulation o Limit States and Random Variables Deinition The ollowing two limit states are generally considered in the design o the ship structures []. 9

10 Ultimate Limit States (ULS) The ULS generally considered the maximum load at which the structure collapses and can no longer serve its intended unction. Serviceability Limit State (SLS) The SLS is usually related to ailure modes leading to service interruptions or restrictions. Repair is usually required to return the structure to an acceptable state. The ailure due to instability or buckling o longitudinal stieners (lexural or tripping) or overall buckling is related to the ultimate limit state. The ailure margin o structures can be evaluated when the applied compressive load reaches or exceeds its ultimate compressive strength as deined in Eq.(5). g X P P (5) u where X u is the model uncertainty o the strength prediction; P ult is the ultimate compressive strength o a stiened composite panel; P is the applied compressive load. ult Table 3 Typical Distributions or Variables rom DNV [] Variable Distribution Type Current Long Term Speed (Pressure) Weibull Properties Yield Strength (Steel) Normal Properties Young s Modulus Normal Properties Initial Deormation o Panels Normal The reliability is dependent upon the statistical distributions o the inputs. Dierent inputs are generally grouped together with statistical distributions as determined by structural codes e.g. CIRIA [1], DNV [] or EUROCOMP [3]. Typical distributions or pressure and material deinitions are Weibull distributions and Normal distributions respectively, as can be seen rom Table 3 given by the DNV design rules and used or the analyses presented later. In general, the basic variables concerned with external load and geometric values have the largest and smallest coeicients o variation respectively. Thereore, the geometric properties such as dimension o panel a, b, b 3, b 4 and the thickness o laminae t, which may luctuate in the vicinity o the given values depending on the manuacturing processes, are considered as random variables. All geometric properties are assumed to have a COV (Coeicient o Variation) o 3%. Initial imperection is also taken into account as this problem can never be totally eliminated. The material properties o ibre and matrix, ibre volume raction, which may aect the mechanical properties o the laminate, are treated as random variables with a 5% COV. The modelling uncertainty is generally associated with idealizations in ormulating mathematical models and on the like. The modelling uncertainty is usually incorporated into a reliability analysis by the ratio between the actual response and predicted modelling response. Faulkner et al.[4] suggested that a normal distribution is usually assumed, the mean value and coeicient o variance or strength parameters are assumed to be 1. and 1% or simplicity, respectively. It is assumed that a saety actor o is applied to the analysis and thereore the mean value has been chosen as hal o the ailure load with the loads used in the analysis ound rom the resulting distribution. All these 1

11 variables are assumed as independent variables and they are randomly generated according to their assumed probability distribution as shown in Table 4 where the values have been estimated based on experience in the marine industry and correlated against the values ound in Sriramula and Chryssanthopoulos [5]. Table 4. Statistical properties o basic design variables 5.3 Sensitivity Analysis Symbol Distribution Mean value C.O.V a Normal 55mm.3 b Normal 5mm.3 t Normal.56mm.3 b 3 Normal 5mm.3 b 4 Normal 39mm.3 Normal.55mm.3 E Normal 7.GPa.5 E m Normal 3.GPa.5 G Normal 3.GPa.5 G m Normal 1.9Gpa.5 V Normal.55.5 X u Normal 1..1 P Weibull.5P ult.15 In a practical structural design, knowing the most important design parameters and their impact on saety index enables the designers to know where to look to improve reliability. In a deterministic analysis, the sensitivities o design variables can only be computed by quantiying the change in the perormance measure due to a change in the variable value. On the other hand, i a design is based on reliability theory, each random variable is deined by the mean value, coeicient o variance and distribution type. Once the probabilistic model is established, probabilistic analysis is run and then the sensitivity actors are obtained in order to determine the importance o a random variable. In addition, the complexity o the mathematical model is greatly inluenced by the dimensionality o the space o variables in the analysis, thereore it is important to reduce the number o variables and thereby increase the eiciency o the reliability analysis. The variable having a small sensitivity actor might be assumed to be o ixed value rather than being a random variable in subsequent analyses. The ollowing three important actors were considered in this paper [6]. Sensitivity actor is generally considered as a measure o the sensitivity o the reliability index with respect to the standard normal variable u i *. It provides some insight into the relative weight that each one has in determining the inal reliability o the structures. A larger i implies more sensitivity o reliability index to the standard variate u i * i (6) * u i This actor is usually providing an importance ranking o input variables. However, it is not useul or 11

12 design purpose as they are dependent on mean value, standard deviation and distribution type o random variables. Another two sensitivity parameters and scaled sensitivity o with respect to the mean and the standard deviation o each basic random variable in question are more useul or design as deined in Eqs.(7)-(8). i (7) i where and respectively. i (8) i represent the mean value and standard deviation o basic random variables, 5.4 Results and Discussions Table 5 shows the three sensitivity actors, and η or the dominant variables. The important actors or dominant variables are also shown in Figure. The saety index = 3.67 and ailure probability P = are obtained via the proposed method together with the irst order reliability method (FORM), calculated directly rom the limit state equation. From Figure, the importance o the dominant variables, by order, is modelling uncertainty o the strength prediction X u, applied load P, ibre volume V, the height o section b 4, the thickness o laminae t, the length a, Young s modulus o ibre E, and shear modulus o resin G m. However, unrepresented in this igure are the sensitivities o other variables, which play such small roles in contributing to the probability o ailure and can be replaced by deterministic values in the urther analysis. The sensitivity actor represents the sensitivity o with respect to the mean values. The positive sensitivity actors such as geometric parameters t, b 4 and material properties o ibre and resin E, G m, V are obtained and treated as strength parameters. That means the saety index increases with increasing mean value o the variables. The negative sensitivity actors are treated as load parameters such as the length o stiener a and compressive load P. This indicates that the saety index decreases with increasing mean value. The combination o in-plane and lateral loading is also considered because lateral loading rom sea water pressure or cargo is always present on plates and stiened plates elements. Pressure load o kPa with the uncertainty 1% is considered and a Weibull distribution is assumed in the reliability analysis. The direction o lateral pressure is assumed to be the same with the initial imperection towards the stieners. By comparison o the panel with and without lateral pressure, the reliability index decreases rom 3.67 to.5 and the probability o ailure increases rom to The eect o lateral pressure on the stiened plates is to lower the ultimate collapse load and thereore reduce the reliability index compared with the stiened plate under in-plane loading alone. Table 5 Sensitivity actors o basic variables 1

13 Random Random η variable variable η a G m t V b X u E P Parametric Study Figure Important actor Although the probabilistic method provides more inormation than the corresponding deterministic counterparts in the analysis, this method also requires more comprehensive inormation. Reliability analysis shows that not only the mean value but also COV o random variables play a signiicant role in determining the reliability or saety. However, such inormation is generally indeterminate. Furthermore this data can be used to inorm designers or production engineers where eort can be best utilised or the greatest cost to beneit ratio. Thereore, it is worth studying the eects o the statistical distribution o the various variables having the largest sensitivity actors calculated in the previous section. From Table 5, the modelling uncertainty X u, the ibre volume V, the thickness o laminae t, Young s modulus o ibre E and shear modulus o resin G m are chosen to study the eect o coeicient o variation. The results are computed by varying each o the parameters in turn with other variables held the same as the previous analysis, shown in table 4. The results are presented in Figures 3-5. The model has also been analysed or a wider range o load uncertainties by ignoring the actual source o loading. Axial load COVs o 1%, % and 3% are considered. The results, in terms o P/P ult versus the saety index, are presented in Figure 6. The reliability index o the model uncertainty and applied load reduces rapidly as the uncertainty increases. In other words, these two variables are very sensitive to the statistics o variables. Thus more precisely knowing the statistics o these two design variables will induce more meaningul results in the reliability analysis. For example, increasing the COV rom 1% to 15% in the model 13

14 uncertainty o the strength prediction leads to a reduction in reliability index o.8% and around a -old increase in the ailure probability. The variations in component thickness and the ibre content within a component rom design is caused by production deects, which may arise in the manuacture o composite structures and are diicult to eliminate. The inluences o variation in the volume raction V and thickness t on reliability are shown in Figure 4. It is evident that the reliability indices are strongly dependent on the variation o these two quantities. That is to say that the minor variation in these two variables has a major eect on the reliability. Thereore, i the uncertainties o such random variables can be reduced through appropriate care or stricter quality control, reliability can be increased. Uncertainties in material properties E and G m are also investigated as shown in Figure 5. It is evident that the inluence o the variation in the ibre modulus E is stronger than the shear modulus G m on reliability. This enables the designers to know where to look to improve reliability. For example, i the coeicient o variation o the G m is reduced rom 15% to 11%, the ailure probability is reduced around 4%. However, the ailure probability can then be reduced to almost hal i the coeicient o variation o the E is reduced by the same percent. This indicates that it is better to reduce the uncertainty o E rather than G m in terms o their relative importance with respect to variations in their standard deviations. 5 4 β COV o X u (%) Figure 3 Variation o with change o COV o X u 4 COV (%) 3.8 β t V 3 COV (%) COV (%) Figure 4 Variation o with change o COV o V and t 14

15 4 β G m E COV (%) Figure 5 Variation o with change o COV 1 o E and G m COV (%) P /P ult % COV o P % 3%. β Figure 6 Variation o design axial load with saety index 6. CONCLUSIONS In the present paper, a simpliied analytical method was presented or the reliability assessment o the ultimate compressive strength o tophat-stiened laminate panels under axial compression combined with uniorm lateral pressure. A reliability-based design method is used or the practical design o composite stiened panels. From the study the ollowing speciic conclusions can be drawn: A unique eature o ibre-reinorced polymers is the lexibility in their composition which enables a designer to design a structure to speciically meet constraints. The simpliied analytical method provides a rapid means or the initial design purpose and explores quantitatively the inluence o the various constituent properties on reliability. This has enabled the variables having the greatest inluence on reliability to be identiied and may allow the engineer to concentrate on these more important variables. The uncertainties o variables are generally caused by the lack o data, modelling simpliications, human errors or inadequate knowledge o physical phenomena and it is important to have an 15

16 understanding or how these will aect a given structure. The parametric study provided an insight into COV o various parameters, provided by dierent sensitivity actors, to the eect on the reliability index allowing an understanding o how the variations, whatever their cause, aect this value. The results show that the model uncertainty and applied loads are sensitive to the distribution and variation o the variables. Thus more precisely knowing the statistics o these design variables will induce more meaningul results in the reliability analysis. Improving the accuracy o the analysis method, such as by applying more advanced methods can minimise the modelling error and its variation so as to increase the saety index. Furthermore in a practical structural design, knowing the most important design parameters and their impact on saety index enables the designers to know where to look to improve reliability. The variation in component thicknesses and the ibre content within a component has a large impact on the reliability. Thereore, i the uncertainties o such random variables can be reduced through appropriate care or stricter quality control, the reliability can be increased substantially. The analysis indicates that this area will give a good cost to beneit ratio in considering increasing the reliability either at the design or production stages. The inluence o the variation in the ibre modulus E is stronger than in the shear modulus G m on reliability. That indicates that it is better to reduce the uncertainty o E rather than G m in terms o their relative importance with respect to variations in their standard deviations. More testing would be required to control the scatter o this signiicant variable. ACKNOWLEDGEMENTS This work has been undertaken with the support o the UK MoD Sea Systems Group (Bath), BMT and the European network activity, MarSTRUCT. REFERENCES 1. M. MAYER. Die sicherheit der bauwerke und ihre berechnung nach grenzkraten statt nach zulassigenspannung. Berlin:Springer-Verlag, N.S. STRELETZKI. Statistical basis or the evaluation o the structural saety actor. Stroizdat,Moscow: State Publishing House or Buildings, W. WIERZBICKI. Saety o structures as a probabilistic problem. Technical Report, Przeglad Techniczny,Warsaw, Poland, C.A. CORNELL. A probability-based structural code. ACI Journal, pp , B.N. CASSENTI. Probabilistic static ailure o composite materials. AIAA Journal, vol. : pp , L. YANG. Reliability o composite laminates. Mechanics o Structures and Materials, vol. 16:pp ,

17 7. G. CEDERBAUM, I. ELISHAKOFF, and L. LIBRESCU. Reliability o laminated plates via the irst order second-moment method. Composite Structures, vol. 15:pp , M.R. GURVICH and R.B. PIPES. Probabilistic analysis o multi-step ailure process o a laminated composite in bending. Composite Science and Technology, vol. 55:pp , M.R. GURVICH and R.B. PIPES. Probabilistic strength analysis o our-directional laminated composites. Composite Science and Technology, vol. 56:pp , H.K. Jeong and R.A. Shenoi. Reliability analysis o mid-plane symmetric laminated plates using direct simulation method. Composite Structures, vol. 43:pp. 1 13, H.K. Jeong and R.A. Shenoi. Probabilistic strength analysis o rectangular rp plates using monte carlo simulation. Computers and Structures, vol. 76:pp ,. 1. Chen, N-Z and Guedes Soares, C., Reliability assessment or ultimate longitudinal strength o ship hulls in composite materials. Probabilistic Engineering Mechanics, Vol, Issue 4, pp.33-34, Zhang. W., Tang, W., Chen, N. and Zhang, S. The eects o stochastic characteristics o materials on the reliability o a composite ship hull, Journal o Marine Science and Application, Vol.1, Issue 1, pp J.I.R. BLAKE, R.A. SHENOI, O. DAS, and N. YANG. The application o reliability methods in the design o stiened rp composite panels or marine vessels. Ship and Oshore Structures, vol.4():pp.87 97, ISO 394. General Principles on reliability or structures. Second Edition. International Organization or Standardization, 1998, Geneva. 16. Paik J.K. Ultimate Limit State Design o Steel-Plated Structures. John Wiley& Sons, LTD Smith, C.S. Design o Marine Structures in Composite Materials. Elsevier Science Publishers Ltd, 199.London. 18. Shenoi R.A. & Wellicome J.F. Composite Materials in Maritime Structures. 1993, Cambridge. 19. von Karman T, Sechler EE, Donnel LH. The Strength o thin plates in compression. ASME Appl Mech Trans 193; Vol.54:pp Dario Boote. Parametric evaluation o the eective breadth or GRP beams with FEM calculation. Maritime Industry. Ocean Engineering and Coastal Recources. IMAM 7, Varna, Bulgaria. 1. J.M.C. CADAI, T.J. STRATFORD, L.C. HOLLAWAY, and W.G. DUCKETT. Strengthening metallic structures using externally bonded ibre-reinorced polymers. Classic House, 4.. DNV. Design o oshore steel structures load and resistance actor design method. Technical Report OSC11, Oslo,. 3. Report o a Concrete Society Committee. Design guidance or strengthening concrete structures using ibre composite materials. Technical Report No. 55, The Concrete Society, Faulkner, D., Soares, C. G. & Warwick, D.M. Modelling requirements or structural design and sssessment. In: D. Faulkner, A. Incecik and M.J. Cowling, Editors, Integrity o Oshore Structures, 1988, Vol. 3, pp Sriramula, S. and Chryssanthopoulos, M.K. (9) Quantiication o uncertainty modelling in stochastic analysis o FRP composites, Composites Part A: Applied Science and Manuacturing, Vol. 4, PP Mansour A.E. & Wirsching P.H. Sensitivity actors and their application to marine structures. Marine Structures, 1995, pp

18 7. Datoo M. H. Mechanics o Fibrous Composites. Elsevier science publishers Ltd, Reddy, J.N. Mechanics o Laminated Composite Plates and Shells: Theory and Analysis. U.S.CRC Press. 4. Appendix A For stiened plates, the section is made rom an assembly o lat layered laminated composite element such as crown, web and lange, which will be reerred to as elements o section. The material properties vary rom element to element, depending on the laminate coniguration in each element. In order to perorm the analysis o structures made rom composite laminated plate using the methods mentioned above, the equivalent Young s modulus value is required or each element. Symmetric laminates are considered here only as they are the majority o laminate conigurations used in practice. The coupling stiness terms B ij are zero or symmetric laminates: implying that there are no membrane-bending coupling eects. From Datoo [7], the membrane equivalent Young s modulus value o the laminate plate in the x-direction and the y-direction are E E m x m y (A (A A A - A - A 1 1 )/A )/A 11 t t (A.1) where A ij are called extensional stiness; t is the total thickness o the laminate element under consideration. Appendix B For each element, the average direct stress value acting through the thickness o the element o a particular laminate can be predicted at any point in the section using Eq.(1), then the corresponding direct orce intensity per unit width N and moment intensity per unit width M o the laminate section about its own mid-plane can be obtained. Then the ply-to-ply stress analysis can be perormed by using plate theory [8]. The orce and moment resultants to the strains o a laminate are given in a compact orm as where and 1 N M ( A B B D ( ) 1 ( ) xx yy ) xy 1 are vectors o the membrane and bending strains. (B.1) ( 1 ) xx ( 1 ) yy (B.) ( 1 ) xy The extensional stinesses A ij, the bending-extensional coupling stinesses B ij and the bending stinesses D ij, are deined as 18

19 ( A,B,D ) Strains at any point in the plate can be computed ij ij ij ( xx xx ( yy yy ( xy xy t / t / ) ) ) Q ( 1,z,z )dz (B.3) ij ( ( z ( With these values, the strains,, ) through the laminate thickness can be determined and the ( xx yy xy ply-to-ply stresses can be calculated based on the constitutive relations and the ailure criteria is used to determine whether the plate is ailure. Appendix C Elastic properties or a unidirectional layer should be established ideally by tests, however, or initial design purpose, it may be obtained by several simple approximations to the elastic constants with reasonable accuracy. Semi-empirical equations o moduli derived by Halpin and Tsai are chosen here [17], i.e. Young s modulus E 1 in the ibre direction, Young s modulus E in the transverse direction, major Poisson s ratio v 1 and shear modulus G 1 as ollows E 1 ) xx 1 ) yy 1) xy (B.4) 1 E V EmVm (C.1) 1 V mvm (C.) M M m 1 V 1V (C.3) M M / M / M m m 1 (C.4) where M = E or G 1, M = E or G, M m = E m or G m respectively. Reinorcing actor ξ G = 1 or the prediction o shear G 1 and ξ E = or the Young s modulus E approximately. The strengths o a unidirectional composite layer can be obtained using simple equations as ollows Longitudinal tensile strength X t * E V When * * > m (C.5) X t * ( E V EmVm ) When * < m * (C.6) Transverse tensile strength and in-plane shear strength Y t Em * 1 ( V V ) 1 m E (C.7) Em * St 1 ( V V ) 1 m E (C.8) 19

20 where m * and * is strains to ailure o the resin and ibres respectively. m * is the matrix tensile strength, m * is the matrix shear strength. The longitudinal and transverse compressive strength can be approximately estimated by the same equation.

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