Structural Safety Evaluation Using Modified Latin Hypercube Sampling Technique

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1 International Journal o Perormability Engineering Vol. 9, No. 5, eptember 203, pp AM Consultants Printed in India tructural aety Evaluation Using Modiied Latin Hypercube ampling Technique P. BHATTACHAJEE, K. AMEH KUMA 2,and T. A. JANADHAN EDDY 3 eliability Engineering Division, DDL, Hyderabad , INDIA 2 Production Planning Division, DDL, Hyderabad , INDIA 3 Proessor, Mechanical Engineering, smania University, INDIA (eceived on December 2, 202 and revised on June 02, 203) Abstract tructural saety and high reliability are most important requirements o aerospace product. Until recently, a design was considered to be robust i all the variables that aected its lie had been taken into account and brought under control. o ar designers have traditionally handled variability with saety actors. ome saety actors are derived rom observation and analysis and in many cases it use to be pure guesswork. In those cases, the bigger the guess, the bigger the risk, the bigger the saety actor resulting in over design which is undesirable in case o aerospace product. In the present paper Latin Hypercube ampling (LH) based quasi random polar sampling technique is suggested or structural saety evaluation. The data generated rom LH are selected quasi-randomly to establish structural load & resistance dispersion and it has been used or saety evaluation. This technique is illustrated with a thin cylindrical pressure vessel, which is used etensively in aerospace applications. The proposed technique ound to be simple and this technique will be useul to pressure vessel designer during initial design phase. Keywords tructural saety, reliability inde, Latin hypercube sampling, polar sampling and pressure vessel.. Introduction ince modern structures require more critical and comple designs, the need or accurate approaches to assess uncertainties in loads, geometry, material properties, analysis processes and designing has increased signiicantly. However, computational structural mechanics has evolved to a level o sophistication that merits consideration o newer methods or incorporating uncertainty in the design process. In all engineering structural system design, uncertainties are unavoidable due to stochastic nature o material and imperect nature o mathematical model. These uncertainties can be accounted through reliability analysis i.e., through ailure probability distribution o load and resistance. In general, moment methods such as Mean Value (MV), First rder econd Moment (FM), Advance First rder econd Moment (AFM), and econd rder eliability Method (M) are etensively used or structural saety (reliability) assessment. These moment methods require the derivatives o ailure unctions (Limit tate Function). imulation techniques which oer the most versatile means o computing the probability o ailure by using either the evaluation o multiold integration, or directly by simulating. tructure with very low probability o ailure, a direct or straightorward Monte Carlo imulation (MC) procedure may be made computationally more eicient by incorporating Variance eduction Technique (VT). The Latin Hypercube ampling is one among the VT and is discussed in this paper. To make the LH more eective, quasi-random polar *Corresponding author s pradeep9_ rqa@yahoo.com 55

2 56 P. Bhattacharjee, K. amesh Kumar, and T. A. Janardhan eddy sampling is suggested. An attempt is made to evaluate structural saety by using LH based quasi-random polar sampling technique. This technique is illustrated with practical eample o a thin cylindrical pressure vessel which is used in aerospace vehicles. 2. Nomenclature Di g(), G() p p t β μ σ Φ Internal diameter Perormance unction Pressure Probability o ailure esistance Ultimate strength o material/ Load Thickness aety inde Mean tandard deviation Normal Pd Normal Cd Limit tate unction a set deined by locus o points g() = 0 p is deined as p ( ) = F G( ) 0 X d where G( ) = ( {( p D 2t) 0.6 p}) () is the joint probability density o design variables i aety Inde β, the aety Inde, is deined as the scalar distance in standard normal space 3. tructural eliability g β = σ g Uncertainties and randomness associated with both loads and structural characteristics introduce variability in structural response. The predicted response o structure based on mathematical and statistical models with imperect knowledge tends to depart rom reality. ecent developments in the area o probabilistic methods and statistical inerence oer a mathematical basis that will enable designer to incorporate the inluence o variability and uncertainties arising rom a variety o sources more eectively into design process. ome o these developments are discussed in this paper rom the stand point o reliability o thin maraging steel cylindrical pressure vessel structure. The structural analysis is accomplished by eamining the limit state which describes the conditions that render a structure unsatisactory or one o the intended roles under one load eect or a combination o load eects. From the ultimate strength stand point, the limit state equations is generally epressed in terms o structural resistance and load eects, where as, or serviceability, the limit state represent the evaluation o a perormance criterion. The two basic variables representing action and resistance are unction o a number o other variables. An increase in number o variables in the limit state equation and the

3 tructural aety Evaluation using Modiied Latin Hypercube ampling Technique 57 departure o their distributions rom the normal together with a nonlinear ailure unction introduces compleity to the problem. In such case, this limit state, (G = - = 0) is a hyper- surace o n-dimensional space, and separates the ailure and sae regions. The probability o ailure is equal to the volume integral over the ailure region [6-0]. p = ( ) G( X ) <0 X X d () In which (X) is the joint probability density unction o the n-dimensional vector X. The preceding integral may be evaluated by means o one o the ollowing techniques or combination o either viz; a) numerical integration b) approimation methods c) simulation. In the present paper, simulation methods have been used. In reliability analysis, random variables occur in modeling Loads () and esistance (). Their randomness is characterized by means and ; standard deviations, σ and r. σ ; and corresponding probability density unction ( s) and ( ) The probability o ailure, p is given by p F ( s) ( s)ds = 0 The perormance unction is described as Z = - = g(, 2 n ), where i are the basic variables. The reliability inde or saety inde β is evaluated as Z β = (2) σ Z The probability o ailure is given by p = Φ( β) (3) where Ф is the cumulative standard normal distribution unction 3. imulation Methods imulation based algorithms [-3] were the irst developed methods or reliability (probability o ailure) analysis and included techniques such as Monte Carlo imulation and Latin Hypercube ampling. imulation techniques perorm discrete series o numerical eperiments to estimate the multi dimensional integral. These techniques are preerable or comple problems where accuracy is a primary concern since simulation tends to converge to theoretical solution. However, the major disadvantage is the large number o simulation run required to obtain an accurate result. In this paper, Latin Hypercube ampling, and LH based quasi-random Polar amplings are discussed. 3.. Latin Hyper Cube ampling The Latin Hyper Cube ampling (LH) method [3-5] is a technique or reducing a number o MC simulation run. It was irst proposed by Mc Kay in 979 and then urther reined by onald L. Iman in 98. In Latin Hypercube, the samples are orced to be in interested region. This can be done by dividing standard normal probability density unction into desired interval. The area under PDF curve or each interval is equal, thereore, the divided Cumulative Density Function (CDF) can be developed. The random number is generated by choosing standard normal CDF at each interval. The chosen CDF is then appropriately transormed onto basic domain according to desired distribution. nce all samples are generated, the set o random variables is organized independently by uniormly selecting each o variables generated values such that the selected value must be used only once, as depicted in Fig -. The simulation steps are as ollows For each variable generate one point rom each o the interval u ij, j=,2,,m or variable i.

4 58 P. Bhattacharjee, K. amesh Kumar, and T. A. Janardhan eddy The irst point u j in the LH sample is generated by sampling one value uij rom each ais u i. The second point is generated in the same way, ecept that the values u are deleted rom the sample. In this way m points are generated. ij The probability o ailure rom this sample is estimated rom m Jˆ p = I[ g( u )] m j= This procedure is repeated N times and the inal estimate o p is N m kj p = I[ g( u )] (5) Nm k= j= kj u is the realization number, Ĵ is the k th LHC sample. Where There is no simple orm or the standard error o this simulation but in general the standard error is o the magnitude times the standard error o crude MC. mn (4) F Ui ( u i ) u j u i F Ui ( u i ) Figure Latin Hypercube imulation Method 3..2 Modiied Latin Hypercube ampling The modiied LH termed, as Quasi andom Polar ampling [0-2] is an improvised method o LH technique. In polar sampling division o range o possible value o each random input variable depends on the number o sample required. In this method irst random samples are generated using LH or each variable. These variables are then arranged in ascending order as given in matri Fig-2. The variables o each column are then selected randomly using discrete uniorm distribution. A quasi-random matri is generated which is column wise reshule o the original ascending order matri as shown in Fig-3. The modiied LH matri is used or saety analysis.

5 tructural aety Evaluation using Modiied Latin Hypercube ampling Technique 59 n n2 m 2m nm Figure 2 Ascending rder LH Matri n 2 n nm m 2m Figure 3 Quasi andom Modiied LH Matri 4. Design Data The pressure vessel design is carried out as per AME pressure vessel code. The design data o the present thin maraging steel cylindrical pressure vessel obtained rom the design documents are given in Table. Table Design Parameters Table 2 Parameter Variability Internal Diameter Working P r Design P r Thickness Design Code Construction mm.00 Kg/ mm sq.35 Kg/ mm sq.4 mm AME Pressure Vessel Maraging steel Flow ormed l. No Parameter Pressure (p) Kg/ mm sq Internal Diameter (Di) mm Thickness (t) mm Material trength () Kg/ mm sq Mean (μ) tandard Deviation (σ) Manuacturing Data Pressure vessel data, viz; internal diameter, thickness, ovality etc., are collected time to time rom manuacturing centre. ther material properties like ultimate tensile strength, yield strength and racture toughness have been generated in material testing lab. The details o various parameters and their variability are placed in Table Analysis In the present case the saety evaluation is carried out as per section ** or proposed thin maraging steel cylindrical pressure vessel. The ailure mode considered is plastic ailure, that is when load induced hoop stress eceed the material strength which is epressed as below Hoop tress (H) = pd i 0. 6 p (6) 2 t The limit state unction pd i (7) G = 0. 6 p 2t

6 520 P. Bhattacharjee, K. amesh Kumar, and T. A. Janardhan eddy The aety (eliability) Inde is evaluated as β =, H, σ and H H (8) 2 2 ( σ + σ ) σ are calculated rom Table 4 data o Modiied LH Matri, where = , H = 07. 9, σ = 2. 3, σ H = ubstituting above values in (8) we get β = p = Φ β we get p = The probability o ailure is evaluated as ( ) 6. esults and Discussion The simulation method is the best method or solving comple structural saety problem. Also it has been noticed that Monte Carlo imulation (MC) is the best and most accurate technique. However MC is epensive, incase o low probability o ailure, which requires a large number o simulation run. The common methods used or structural saety assessment are FM, AFM, M etc., and these require the evaluation o derivatives o perormance unctions, which ound to be diicult in comple cases. The LH is one o the variation reduction technique used in the present case is simple. The modiied LH (quasi random) polar sampling technique makes it very eective, as the random sample are generated rom ull range o the distribution. In the present case only 20 samples o each variable is considered or illustration as shown in Table-3 & Table-4. Like every simulation method, in this case also a repeat and large number o simulation run will yield better result. Table 3 Ascending LH Matri l. No Material trength Pressure Diameter Thickness H

7 tructural aety Evaluation using Modiied Latin Hypercube ampling Technique 52 Table 4 Modiied LH (Polar ampling) Matri l. No Material trength Pressure Diameter Thickness Conclusion A simpliied variation reduction technique i.e., Modiied Latin Hypercube ampling is presented with a practical illustration. LH reduces dramatically number o simulation runs. The modiied quasi-random polar sampling ound to be eective as it cover ull range o distribution. Discrete uniorm random number has been used to reshule the ascending LH matri also ound to be equally eective or evaluation o comple structure. The proposed paper will be useul to structural saety designer during initial phase o design incorporating the manuacturing variability. eerences [] orensen, John Dalsgaard. Notes in tructural eliability Theory and isk Analysis. Aalborg University, Denmark, [2] ubinstein, euven Y. imulation and Monte Carlo Method. John Wiley & ons, New York, 98. [3] Young-Kyun Lee., Dae-ik Hwang. A study on the techniques o estimating the probability o ailure. Jr. o Chungcheong Mathematical ociety. 2008; 20(4) [4] Bhattacharjee, P., K. amesh Kumar., and T.A. Janardhan eddy. A Comparative tudy o Probabilistic tructural aety Analysis. 2 nd International Conerence on eliability, aety and Hazard isk-based Technologies and Physics-o Failure Methods. Mumbai, India, December 4-6, 200; [5] Bhattacharjee, P., K. amesh Kumar., and T.A. Janardhan eddy. tructural eliability Analysis o a Pressure Vessel using Multiple egression. Proceedings o International Conerence on Computational Methods in Engineering and ciences. Hyderabad, India, January 8-0, 2009 ;

8 522 P. Bhattacharjee, K. amesh Kumar, and T. A. Janardhan eddy [6] Bhattacharjee, P., K. amesh Kumar., and T.A. Janardhan eddy. tructural aety Assessment o a Pressure Vessel Using Perormance Function Approimation. International Journal o Perormability Engineering. 20; 7(4) [7] Bhattacharjee, P., K. amesh Kumar, and T.A. Janardhan eddy. tructural aety Evaluation Using Inverse Measure Approach.. International Journal o Perormability Engineering. 202; 8(5) [8] David G. obinson. A urvey o Probabilistic Methods used in eliability, isk and Uncertainty Analysis Analytical Techniques-I. andia National Laboratories, Albuquerque, Caliornia, AND [9] Feli. Wong. First-rder econd-moment Methods. Computers & tructures, 985; 20(4) [0] Melchers,.E. tructural eliability Analysis and prediction. Ellis Harwood Limited, England, 987. [] adha, P., and K. ajagopalan. andom Polar ampling Technique or the eliability Analysis o ubmarine Hull. Jr. o Naval Architecture and Marine Engineering. 2004; () [2] losson, A., G. andberg., and. Dahlblom n Latin Hypercube ampling or tructural eliability Analysis. tructural aety, 2003; 25() P. Bhattacharjee currently heads Directorate o eliability and Quality Assurance, DDL Hyderabad. He has contributed immensely in the ield o eliability and Quality or various aerospace product development programmes. Currently he is involved in the development o Probabilistic Design methods and their application to aerospace structures. He has done his Ph.D. in tructural eliability rom smania University Hyderabad. He has published several technical papers in National and International conerences and Journals. K. amesh Kumar currently heads Production Planning & Control o Directorate o Engineering DDL Hyderabad. He has done his Ph.D. rom Melbourne University Australia. He has developed many techniques in the ield o low orming. He has contributed immensely in the ield o aerospace product development. He has several papers in the ields o metal orming to his credit. T. A. Janardhan eddy is a ormer Proessor o Mechanical Engineering Dept., smania University, Hyderabad. He has completed his Ph.D. rom IIT Madras. His area o interest and research are Manuacturing Engineering & NC Machines. He has established a cell in smania University or promoting the spirit o entrepreneurship among enthusiastic students. everal students are being guided or their Ph.D programs under his supervision. He has participated in several International conerences at UA, Canada and India.