POLYNOMIAL BASED RESPONSE SURFACE APPROACH FOR PROBABILISTIC MODELLING OF THE ULTIMATE STRENGTH OF STIFFENED PANELS

Transcription

1 Mng Ca Xu, A. P. Texera and C. Guedes Soares. Centre of Marne Technology and Engneerng (CENTEC), Insttuto Superor Técnco, Techncal Unversty of Lsbon, Portugal. POLYNOMIAL BASED RESPONSE SURFACE APPROACH FOR PROBABILISTIC MODELLING OF THE ULTIMATE STRENGTH OF STIFFENED PANELS Summary The purpose of present paper s to nvestgate the accuracy of dfferent polynomal based response surface models for probablstc modellng of the ultmate strength of stffened panels under unaxal compressve load. To reduce the number of varables consdered n the response surface approxmaton, a senstvty analyss s performed to estmate the mportance of the nput parameters and to dentfy whch ones contrbute more to the uncertanty on the ultmate strength of the stffened panels. The most mportant nput parameters are then used n the response surface approxmaton. The ultmate strength of stffened panels under unaxal compressve load s assessed by Fnte Element Analyss (FEA) for 100 and 1000 nput samples generated by Monte Carlo smulaton, whch are then used to ft the response surface models. The response surface models used nclude frst and second order polynomals wth and wthout nteracton terms. The accuracy of the dfferent response surface models s assessed by comparng the probablstc models of the ultmate strength of the stffened panels derved from the response surface approxmatons wth the one derved on the bass of a large sample of 5000 FEAs obtaned by Monte Carlo smulaton. Key words: Stffened panel. Ultmate strength. Monte Carlo smulaton method. Response Surface Method. POLINOMSKI ORIJENTIRAN PRISTUP ODZIVNE POVRŠINE ZA PROBABILISTIČKO MODELIRANJE GRANIČNE ČVRSTOĆE UKREPLJENIH PANELA Sažetak Svrha rada je stražt točnost vše modela odzvnh površna za probablstčko modelranje grančne čvrstoće jednoosno tlačno opterećenh ukrepljenh panela. Da b se smanjo broj varjabl u procjen odzvne površne provedena je analza senztvnost ulaznh parametara te se na taj načn odredlo koje varjable maju već utjecaj u procjen grančne čvrstoće ukrepljenog panela. Najvažnj ulazn parametr zatm su koršten u procjen odzvne površne. Grančna čvrstoća jednoosno tlačno opterećenh ukrepljenh panela odreñena je nelnearnom metodom konačnh elemenata za modela dobvenh Monte Carlo smulacjom, koj su zatm koršten za prlagodbu odzvne površne. Model odzvne površne uključuju polnome prvog drugog reda s bez nterakcjskog člana. Točnost razlčth modela odzvnh površna procjenjena je usporedbom probablstčh modela grančne čvrstoće ukrepljenh panela dobvenh aproksmacjom odzvne površne s velkm uzorkom od 5000 analza konačnh elemenata dobvenh Monte Carlo smulacjom. Ključne rječ: smulacja odzvna površna, ukrepljen panel, nelnearna MKE, Monte Carlo

2 0th Symposum SORTA01 Polynomal based response surface probablstc modellng of the ultmate strength of stffened panels 1. Introducton The fnte element method (FEM) s a well-accepted tool to model the response of both lnear and non-lnear of structures. Accountng for randomness and spatal varablty of the geometrcal and mechancal propertes of materals s one of the tasks of the stochastc or probablstc mechancs, whch has developed fast n the last decades (e.g. [1]). The avalable methods to deal wth uncertantes n the analyss of structures can be classfed n two man groups: those amed at estmatng the moments (usually the mean and varance) of response quanttes and those amed at quantfyng the probablty assocated wth a specfed falure crteron. The Response Surface Method (RSM) has been proved to be an effcent and wdely applcable method n structural relablty analyss and n propagaton and uncertanty analyses of response quanttes. The approach conssts of replacng the true lmt state functon or the structural response by an approxmaton, typcally frst- or second-order polynomals. Texera and Guedes Soares [] have adopted the response surface method to analyse the structural relablty of steel plates wth random felds of corroson and have compared ths approach wth the drect couplng of the FORM code wth the software for FEA. The response surface approach has been adopted by Kmeck and Guedes Soares [3] to determne the cumulatve probablty dstrbuton functon of the strength of compressed plates. Chen and Guedes Soares [4] assessed the relablty of the post-bucklng compressve strength of lamnated composte plates and stffened panels under axal compresson. The response surface method s effectve to evaluate the relablty of the ultmate strength of stffened panels when the lmt state functon s not explctly defned. In ths case t s useful to buld a smplfed response surface to avod the tme consumng evaluaton of complex FEM model. However, the accuracy of the response surface model s very mportant. The am of present paper s to nvestgate the accuracy of dfferent response surface models, whch nclude frst and second-order polynomals wth and wthout cross terms. A senstvty analyss s frst performed to dentfy whch parameters contrbute more to the uncertanty and then the most mportant ones are used n the response surface model. The Frst Order Second Moment (FOSM) method s used to analyse the senstvtes of the nput parameters adopted n the assessment of the ultmate strength of the stffened panels that nclude the thckness of the stffener and of the plate, the yeld stress and the elastc modulus of the materal and the ntal mperfectons of the plate. The ultmate strength of the stffened panels s calculated by a non-lnear FEM code. The Monte Carlo smulaton method s used to calculate the probablty densty functons of the ultmate strength based on the dfferent response surface models ftted to lmted number of FEAs. These probablstc models are then compared wth the ones derved from a sample of 5000 FEAs of the ultmate strength of the stffened panels.. Descrpton of the models.1. Geometry and materal propertes It s mportant to model the stffened panel edge condton n a relevant way. A stffened panel model wth two (1/+1+1/) bays n the longtudnal drecton s used n the present paper as shown n Fg. 1. The dmensons of the frames and the stffeners are L mm and 30 8 mm, respectvely. The thckness of the plate s 4 mm. Both, the geometrc and materal nonlneartes are taken nto account, ncludng elastc-plastc large deflecton. Where approprate, a b-lnear sotropc elastc-plastc materal model excludng stran rate effects s used. A plastc tangent modulus of 1000 MPa s acceptable for normal and hgh strength steel

3 Polynomal based response surface probablstc modellng of the ultmate strength of stffened panels 0th Symposum SORTA01 [5]. The others materal propertes are the Young s modulus E = 00 GPa, the Posson s rato ν = 0.3 and the yeld stress σ y = 35 MPa. Fg. 1 Geometry of the panel Fg. Coordnate system of the FE model The ANSYS nonlnear FE code s adopted to calculate the ultmate strength of the stffened panels. The shell 181 element s used to model the stffened panels, whch s a four nodes element wth sx degrees of freedom at each node and can account for lnear, large rotaton and large stran nonlnear. Both full and reduced ntegraton schemes are supported. Ths element s sutable for analysng thn-walled structures. The element sze s 1.5 mm n the longtudnal drecton for the plates and the stffeners, 16 elements for one span n the transverse drecton, 5 elements n the web and elements n the flange of the frame... Boundary condtons Fg. shows the coordnate system of the stffened panel, the dentfcaton of the plate edges and the appled load. Due to the contnuty of plate panel, the n-plane dsplacement at the edge of the model s consdered as unform n ther perpendcular drecton. The loadng s an mposed dsplacement n the longtudnal drecton at the B-B 1 edge. The plate deflectons n transverse drecton are symmetrc wth flat bar stffener havng symmetrc cross secton shape. When the stffened panel s subjected to load, the reacton force causes the rotaton wth respect to the shear centre of the cross-secton. The shear centre of the symmetrc cross secton of the stffener s on the web. The stffener rotates around the shear centre causng the symmetrc deformaton of the plate. Hence, the symmetrc boundary condtons are appled n the transverse drecton at the AB and A 1 B 1 edges, whch means that these edges have the same transverse dsplacement and rotaton θ x =0. The followng boundary condtons have been appled to the stffened panel: A-A 1 at the stffener and plate: u x =0, θ y =0, θ z =0; B-B 1 at the stffener and plate: u x = C d, θ y =0, θ z =0; A-B : u y =0, θ x =0, θ z =0; A 1 -B 1 : u y = constant, θ x =0, θ z =0; C-C 1 and D-D 1 : u y = constant, θ x =0, θ z =0..3. Intal geometrcal mperfectons It has generally been found that ntal geometrcal mperfectons tend to decrease the rgdty and the ultmate strength of plates. These ntal mperfectons affect sgnfcantly the ultmate strength of stffened panels and should be accounted for. The mperfectons are caused durng a complex fabrcaton process and are subject to sgnfcant uncertanty related 3

4 0th Symposum SORTA01 Polynomal based response surface probablstc modellng of the ultmate strength of stffened panels to the magntude and spatal varaton. Three types of ntal deflectons are adopted as follows [6]: Intal deflecton of local plate panels, π x π x 3π x π y w0 pl = w0( 11) sn + w0( 1) sn + w0( 31) sn sn a a a b Column-type ntal deflecton of stffeners, a π x w0 c = sn 1000 a () Sde-ways ntal deflecton of stffeners due to angular rotaton about panel-stffener ntersecton lne, a π x w0 s = sn 1000hw a (3) where a and b are the length and wdth of the plate, h w s the heght of web of the stffeners, and the w 0 (11), w 0 (1) and w 0 (31) are the ampltudes of the Fourer components of the shape of the ntal dstortons of the plate. The shapes of ntal mperfectons are ncluded n the FE model at the APDL fle (ANSYS program desgn language) by shftng the nodes coordnates accordng to Eqns. (1)-(3). (1).4. Stochastc models of the random varables The probablstc models of the ultmate strength of the stffened panel derved n ths study consder the randomness on the geometrcal and materal propertes, as well as on the ntal dstortons of the panels. In partcular, the stochastc model of the ntal dstortons assumes that the ampltudes of the shape shown n Eq. (1) are random. The thcknesses of the stffeners (t s ) and of the plate (t p ) are also descrbed by random varables. The random varables consdered are as follows: X 1 - t s thckness of the stffeners (mm); X - t p thckness of the plate (mm); X 3 - σ y yeld stress of materal (MPa); X 4 - E elastc modulus; X 5 - X 6 - w 0(1) t p ; X 7 - w 0(31) t p Table1 Stochastc models of the random varables are the ampltudes of the shape of ntal deflecton of the plate. w 0(11) t p ; Random Varable X 1 X X 3 X 4 X 5 X 6 X 7 µ (Mean value) Cov (coeffcent of varaton) σ (standard devaton) Dstrbuton type Dstrbuton: 1 - Normal dstrbuton; - Lognormal dstrbuton; 3 - Webul dstrbuton. The mean value and the standard devaton of the random varables are shown n Table 1. The probablstc models of the ntal dstortons are derved from measurements of fabrcaton dstortons of 1998 plates of varous shp types, ncludng cargo shps, multpurpose tugs, bulk carrers, chemcal carrers, tanker, research vessels, passenger and cargo ferres [7]. Fg. 3 shows the stress-shortenng curve of the stffened panel obtaned for the 4

5 Polynomal based response surface probablstc modellng of the ultmate strength of stffened panels 0th Symposum SORTA01 mean value of the random varables presented n Table 1. The ultmate strength of the stffened panel s 49. MPa. 300 Stress(MPa) dl/l(10 3 ) Fg. 3 Stress-shortenng curve of the stffened panel for the mean value of the random varables.5. Senstvty analyss To reduce the number of varables consdered n the response surface approxmaton, a senstvty analyss has been performed to estmate the sgnfcance of the uncertanty of the nput parameters and to dentfy whch ones contrbute more to the uncertanty on the ultmate strength of the stffened panel. The frst order second moment (FOSM) method s adopted to perform the senstvty analyss, whch s based on the frst order Taylor seres expanson of the model output around the mean values of the random nput varables as follows: n g g( X1, X,..., X n ) g( µ x, µ,..., ) ( ) 1 x µ x + X n µ x = 1 (4) X µ The senstvty factors that represent the relatve mportance of each nput random varable n the model output, g(x), s defned as: by: α g x µ X n = 1 g X σ µ x σ X, = 1, n (5) The partal dervatves can be calculated numercally usng a fnte dfferences approach g x g g( µ x + x ) g( µ x ) = x x The ultmate strength of the stffened panels s calculated by nonlnear FEA and then, the senstvty factors are calculated accordng to Eqns. (5) and (6). Fg. 4 shows the senstvty factors of the nput parameters that shows that the ampltude W o11 of the ntal mperfectons and elastc modulus affect sgnfcantly the ultmate strength of the stffened panels. The senstvty factor of the yeld stress, Young modulus and W o11 are 0.18, 0.5 and -0.7, respectvely, whch llustrates that these varables are mportant. In partcular the ampltude W o11 of the shape of ntal deflecton of the plate s n fact the most mportant x (6) 5

6 0th Symposum SORTA01 Polynomal based response surface probablstc modellng of the ultmate strength of stffened panels varable and has a negatve contrbuton for the ultmate strength of the stffened panel under unaxal compresson. Hence, the followng random varables are chosen for response surface approxmatons: t s, σ y, w 0 (11) / t p and E. Fg. 4 Senstvty factors (α ) of the random varables 3. Response surface method The response surface method s an effectve approach to evaluate the relablty of complex structures when the lmt state functon does not exst n an explct form. The basc dea of ths approach s to use a smple mathematcal functon that can be easly calculated, nstead of the true lmt state of the structure. The approxmate functon η represents a surface n the k-dmensonal space and s usually called a response surface defned as follows: Y = η( X, X, KK, X k ) + ε, ( = 1, k) (7) 1 where X s are the nput random varables, η s the approxmate functon and ε s a zero-mean random error term. The response surface models η can represent the lmt state functon used n the relablty analyss or only the response of the structure that s typcally responsble for the hgh computatonal tme nvolved n the evaluaton of the lmt state functon. The functon η s determned by usng a least squares fttng procedure that adjusts the regresson coeffcents ensurng that the approxmate functon s, as much as possble, close to the sample ponts used n the regresson process. Once the zero-mean random error term s very small, Eq. (7) can be used to predct Y,.e., the response of the structure or the lmt state functon. Several response surface models are avalable wth dfferent complexty and accuracy. In general two dfferent types, regresson models and nterpolaton models are wdely used. Regresson analyss s adopted n ths study, whch s a statstcal technque for modelng and nvestgatng the relatonshp between two or more varables and s often used to defne response surface approxmatons. Accordng to the senstvty analyss, four random varables are used to buld the response surface models, whch are the thckness of the stffeners, the ampltude W o11 of the ntal mperfectons of the plate, the elastc modulus and the yeld stress of materal. The probablstc models of these random varables are shown n Table 1. To compare the accuracy of dfferent regresson functons, four fttng models Eqns. (8)-(11), are used to approxmate the ultmate strength assessed by the nonlnear FEA, as followng: 4 Frst-order polynomal: η = β + β x (8) 1 0 = 1 6

7 Polynomal based response surface probablstc modellng of the ultmate strength of stffened panels 0th Symposum SORTA Frst-order polynomal wth nteracton terms: η = β + β x + β x x = 1 = 1 Second-order polynomal: η = β + β x + β x 3 0 Complete second-order polynomal wth nteracton terms: x j x x j x = 1 = 1 j= + 1 = 1 η = β + β + β + β (9) j j = 1 = 1 j= + 1 (10) (11) Snce there are four ndependent random varables n the four functons, the number of coeffcents βs to be determned s fve, eleven, nne and ffteen for Eqns. (8)-(11), respectvely. The assumpton of ndependence of nput varables s not a restrcton snce a set of dependent random varables can always be transformed nto a set of ndependent ones [8, 9]. For a gven polynomal, ts coeffcents are estmated from samples of the nput random varables X generated accordng to ther probablty dstrbutons. Samples szes of N f =100 and 1000 realzatons of the nput random varables X have been adopted n the fttng process to evaluate the ultmate strength of the stffened panel by means of nonlnear FEA. Y s the vector of ultmate strength of the stffened panels: ( 1 ) T Y y y y nf = L (1) The least squares method s used to estmate the regresson coeffcents, whch s based on the mnmzaton of the sum of squares of the dfferences between the predcted and the observed values. The unknown coeffcents β are obtaned by: [ β ] ( ) 1 T T T r r r = X X X Y (13) r where r =1,, 3, 4 correspond to the dfferent response surface models η gven n Eqns. (8-11). The response surface models are then derved usng the two samples of N f = 100 and 1000 values of the ultmate strength (denoted by η 1,r and η,r, respectvely) and the four dfferent regresson models of Eqns. (8)-(11). Fg. 5 and Fg. 6 show the ultmate strength (σ u,fem ) calculated by FEA and the one predcted by the response surface models (σ u,rsm ) for N f =100 and 1000, respectvely, whch are all normalzed by the mean value of the ultmate strength of the plates ( µ ). The angles of the normalzed (σ u,fem - σ u,rsm ) curves are almost equal to 45 degree, showng that the values of σ u,fem and σ u,rsm are very close. As n smple lnear regresson, the adequacy of the regresson model can be assessed by the coeffcent of determnaton R, whch s a statstcal measure that descrbes the correlaton between the data and the predctons of the regresson models. The coeffcent of determnaton R are all greater than σ u R = R =0.981 R = R = (a) Eq. (8) (b) Eq. (9) (c) Eq. (10) (d) Eq. (11) Fg. 5 Normalzed ultmate strength calculated by FEM and predcted by the RSM, σ, / µ (N f =100) u σ u 7

8 0th Symposum SORTA01 Polynomal based response surface probablstc modellng of the ultmate strength of stffened panels R =0.991 R = R = R = (a) Eq. (8) (b) Eq. (9) (c) Eq. (10) (d) Eq. (11) Fg. 6 Normalzed ultmate strength calculated by FEM and predcted by the RSM, σ, / µ (N f =1000) u σ u 4. Probablstc modellng of the ultmate strength of stffened plates The assessment of the uncertanty on the ultmate strength of stffened panels s very mportant for relablty analyss and specfcaton of desgn values for the strength of these structural elements. Probablstc models of the ultmate strength are now derved based on Monte Carlo smulaton usng the dfferent response surface models ftted to samples of sze 100 and 1000 of the ultmate strength of the stffened plate sample ponts are used to derve the probablty densty functon by the regresson models. The accuracy of the dfferent response surface models s assessed by comparng the probablstc models of the ultmate strength of the stffened panels derved from the response surface approxmatons wth the one derved on the bass of a large sample of 5000 FEAs obtaned by Monte Carlo smulaton. Table shows the summary statstcs of the ultmate strength of the stffened panel usng the two approaches. Fg. 7 shows the rato between the statstcs derved from the dfferent regresson models and the FE analyses, obtaned by Monte Carlo Smulaton. The mean values of the ultmate strength are very close ndependently of response models and of the sample sze (N f ) used n the fttng process. The same behavour s observed for the coeffcent of varaton (cov) of the ultmate strength, whch s also smlar when usng the varous response surface models (Eqns. (8)-(11)). However, the cov obtaned usng the response surface models η,r derved from N f = 1000 sample ponts s larger than the one obtaned wth η 1,r (wth N f = 100) and closer to the one obtaned by FEA. For nstance the dfference on the cov between η 14 and η 4 s around 33%. Ths llustrates that the N f s very mportant for estmatng the cov when the response surface method s adopted. The cov obtaned by FE analyses s larger than the ones calculated from MCS of the regresson models. The 95% and 5% fractles of the ultmate strength of the stffened panels were also calculated. The X 95% derved from η 1,r and η,r are very close, however the X 5% values exhbt larger varablty, e.g. around 6% between η 14 and η 4. It can be seen that n general the regresson models η,r provde better predctons of the fractles of the response. In fact for N f =1000 the range of the nput samples s wder and thus the regresson models η,r are capable of predctng wth more accuracy the values at the lower and upper tals of the probablty densty functon of the ultmate strength of the stffened panels. Ths better accuracy of the η,r response surfaces s especally mportant to predct the skewness γ of the response. In ths case t s also mportant to nclude cross terms (η 4 ). When second-order polynomals, Eq. (10), and complete second-order polynomals wth nteracton terms, Eq. (11), are used, the skewness of the response models η 13, η 14, η 3 and η 4 are smlar and around 0.7 of the one obtaned by FEAs. For the frst-order polynomals, the skewness of the response models η 11, η 1, η 1 and η are less than 0.. Ths llustrates that the regresson formulas wth the second-order terms are more accurate than the frst-order approxmaton. From the comparson of the summary statstcs of response obtaned by η 1,r, 8

9 Polynomal based response surface probablstc modellng of the ultmate strength of stffened panels 0th Symposum SORTA01 η,r and FEM that ncludes µ, cov, γ, X 95% and X 5%, t can be concluded that the predctons of η,r are closer to FEM than the ones of η 1,r. Therefore, ncreasng the N f enhance the accuracy of the response surface models. Ths accuracy can be further mproved by usng regresson models wth second-order and cross terms. Table Statstc of the ultmate strength of the stffened panel obtaned by RSM and FEM RSM FEM N s η η 11 η 1 η 13 η 14 η 1 η η 3 η 4 - N f µ cov (%) γ x 95% x 5% Note: N s - number smulatons used by the RSM; N f sample sze used to ft the response surface model; µ- mean value; cov - coeffcent of varaton; γ - skewness. X 5% and X 95% - response fractles. (a) N f =100 (b) N f =1000 Fg. 7 Rato of statstcs calculated by RSM (10 5 ) and FEM (N s = 5000 FEAs) 5. Concluson Ths paper has nvestgated the accuracy of varous polynomal response surface models for probablstc modellng of the ultmate strength of stffened panels under unaxal compressve load. A prelmnary FOSM senstvty analyss has been performed to dentfy the nput parameters that more contrbute to the uncertanty on the ultmate strength of the stffened panels. The yeld stress, the Young modulus and the ampltude of the ntal mperfectons of the plate affect sgnfcantly the ultmate strength of the stffened panels and therefore were ncluded n the response surface approxmatons. The probablstc models of the ultmate strength of the stffened panel were derved based on Monte Carlo smulaton usng dfferent response surface models, ncludng frst and second order polynomals wth and wthout cross terms. The coeffcents of determnaton R for the eght response surface models consdered were all greater than Ths would 9

10 0th Symposum SORTA01 Polynomal based response surface probablstc modellng of the ultmate strength of stffened panels suggest that all response surface models can predct wth hgh level of accuracy the ultmate strength of the panel. However ther ablty for predctng statstcs of the response such as the skewness and strength fractles ncreases when second order polynomals are used as response surface models. In fact although the mean values of the ultmate strength obtaned wth the dfferent response models are very close, the order of the approxmaton and the sample sze (N f ) used to derve the regresson coeffcents of the response surface models nfluence the predctons of the fractles of the response. When the sample sze ncreased from N f =100 to N f =1000 the second order regresson models were capable of predctng wth more accuracy the values at the lower and upper tals of the probablty densty functon of the ultmate strength of the stffened panels. Ths better accuracy of the second order response surfaces s especally mportant to predct the skewness γ of the response. It was also concluded that t s mportant to nclude cross terms to further mprove the accuracy of the regresson models, partcularly when hgher moments of the response are sought. 6. Acknowledgements The frst author has been fnanced by the Portuguese Foundaton for Scence and Technology (Fundação para a Cênca e Tecnologa), under contract SFRH / BD / 6510/ 009. Ths work has been made n the scope of the project Adaptve methods for relablty analyss of complex structures that s funded by the Portuguese Foundaton for Scence and Technology (Fundação para a Cênca e a Tecnologa - FCT) under the contract PTDC/ECM/ 11593/ References [1]. Matthes, H.G., Brenner, C.E., Bucher, C.G. and Guedes Soares C.: Uncertantes n probablstc numercal analyss of structures and solds - stochastc fnte elements, Structural Safety (1997) 19, 3, p []. Texera, A. P., Guedes Soares, C. : "Response surface relablty analyss of steel plates wth random felds of corroson", Safety Relablty and Rsk of Structures, Infrastructures and Engneerng Systems, Furuta, Frangopol & Shnozuka (Eds.), Taylor & Francs Group, London, U.K., 010, p [3]. Kmeck, M., Guedes Soares, C.: "Response surface approach to the probablty dstrbuton of the strength of compressed plates", Marne Structures (00)15,, p [4]. Chen, N.Z., Guedes Soares, C.: "Relablty assessment of post-bucklng compressve strength of lamnated composte plates and stffened panels under axal compresson", Internatonal Journal of Solds and Structures (007) 44, p [5]. IACS: "Common Structural Rules for Double Hull Ol Tankers". Internatonal Assocaton of Classfcaton Socetes, London, 006. [6]. Fujkubo, M., Yao, T., Khedmat, M.R., Harada M.: "Estmaton of ultmate strength of contnuous stffened panel under combned transverse thrust and lateral pressure Part : Contnuous stffened panel", Marne Structures (005)18, pp [7]. Kmeck, M., Jastrzebsk T., Kuznar, J.: "Statstcs of shp platng dstortons", Marne Structures (1995) 8, p [8]. Dtlevsen, O., Madsen, H.: "Structural relablty methods". John Wley & Sons Ltd, Chchester, [9]. Melchers, R.:"Structural relablty analyss and predcton (nd Ed.)", Ells Horwood, Chchester, UK,