FUZZY FINITE ELEMENT METHOD

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1 FUZZY FINITE ELEMENT METHOD RELIABILITY TRUCTURE ANALYI UING PROBABILITY 3.. Maxmum Normal tress Internal force s the shear force, V has a magntude equal to the load P and bendng moment, M. Bendng moments are then creatng the normal stress on the cross secton, whle the shear force, V produces shear stress on the secton. Maxmum normal stress, s a parameter that s commonly used n structural desgn. Maxmum normal stress depends on the bendng moment and secton modulus, s and then: = (3.) and = (3.) where c s the maxmum dstance from the neutral surface and I s the moment of nerta of the cross-secton. The bendng moment equaton for a unform load beam structure havng a support and a smple support s shown below: = 8 (4 5 + ) (3.3) where w, x, dan l are the dstrbuted loadng, dstance and length of the beam. 3.. Relablty Assessment Approach Usng Normal Probablty Relablty equaton can be wrtten as below: P r = P f (3.4) where P f s the probablty of structural falure and P r can be nterpreted as a measure for the relablty.

2 afety margn equaton, Z s: Z = y max (3.5) The probablty of a structural falure s when the Z value s less than or equal to zero and can be represented by the equaton: P f =P(Z ) (3.6) where y and are normal ndependent varables. Thus, the mean value of yeld strength and maxmum stress are respectvely m max. Whle the standard devaton of the yeld strength and maxmum stress are m y and respectvely y dan max. The combnaton of the two ndependent varate wll generate a new varate wth mean and standard devaton dfferent from the orgnal. The equaton of mean and standard devaton of the margn of safety s: m Z =m y m (3.7) Z = y / (3.8) The probablty of structural falure can be dentfed wth the functon as below: P = Φ m Z Z (3.9) where Φ s the standard normal dstrbuton functon and m Z and Z are the mean and standard devaton of the safety margn.

3 3 3.3 TRUCTURAL ANALYI UING FUZZY THEORY Procedures for structural relablty analyss usng fuzzy sets theory begns wth the fuzzfcaton of uncertanty nput and followed up wth cuttng-α, defuzzfcaton and relablty assessment fuzzfcaton of Uncertanty nput Fuzzfcaton can be nterpreted as a specfcaton of the membershp functon x of an sets of uncertanty. The uncertanty of each parameter that s nterpreted by a membershp functon that wll brng value to the trend. Two parameters of the modulus of secton, s and loadng, w are used as fuzzy parameters. Fuzzy normal stress s the result after the α-cut. It was found that, the normal stress depends on and bendng moment and modulus of secton, whle the bendng moment depends on the load. nce both loadng and bendng moment s the dependent varable, then the bendng moment s a fuzzy parameter. Trangular fuzzy numbers are used for understandng the functon of all parameters of fuzzy membershp. Upper lmt value (s u, w u, M u ) and the lower lmt (s l, w l, M l ) of these functons wll be determned by expert opnon. Whle the mddle value (s t, w t, M t ) s between the upper and lower lmt. Then the two parameters are mapped to a decson (output) wth the α-cut. s w Fuzzy secton modulus, Fuzzy loadng, s t s l s u s w l w t w u w Fgure 3. Membershp functons wth trangular fuzzy numbers for the modulus of secton and loadng

4 4 M Fuzzy bendng moment, M ~ M l M t M u M Fgure 3. Membershp functons wth trangular fuzzy numbers of bendng moment Fgure 3. and 3. shows that the tendency for the upper lmt of the bendng moment, M l and the lower lmt, M u s zero, e no tendency for the moment, whereas the trend wth a value of mples that the trend of the moment s gettng a hundred per cent, and symbolzed by the symbol M t α-cut The α-cut s one way of mappng that maps fuzzy nput to fuzzy output wth specfc functons (Moller et al., ). The term mappng s specfed here to mean logcal relatonshp between two or more enttes. Mappng of the nput (modulus of secton, s and the bendng moment, M) to output (the maxmum normal stress, max ) performed after the fuzzfcaton. In ths process, all the fuzzy secton modulus, ~ s and fuzzy bendng moment, M ~ at each stage of the trend, α mapped by usng equatons 3.. Ths mappng resulted n four sgnfcant values for each level of the trend, α at the normal stress space,. The maxmum and mnmum values are selected from a combnaton of these results and used as the upper lmt and lower lmt for the output fuzzy.

5 5 s M Fuzzy secton modulus, Fuzzy bendng moment, M ~ α k α k s k,u s k,l s u s M k,l M k,u M u M mappng Normal stress, α k k,l k,u u Fgure 3.3 mappng of s ~ and M ~ at at all stages of the trend, α Defuzzfcaton of Normal stresses Defuzzfcaton of Normal stress s usng the center of gravty or centrod. Ths technque determnes the pont at whch t wll dstrbute one area (area graph) nto two parts whch have the same value. Mathematcally, the pont s called center of gravty, COG (Negnevtsky, 5). COG s expressed as equaton 3..

6 6 u l u l COG d d By applyng the numercal soluton methods, namely trapezodal rule, the equaton of the new COG would be formed as shown n equaton 3.. n o ) ( h COG n u l where h o Fuzzy tructural relablty defnton The defnton of fuzzy relablty of the structure begns wth the determnaton of safety margns, Z wth COG along wth the real maxmum normal stress, max. Gven that the yeld strength of the varable dstrbuton s normal. o the probablty s used n the calculaton P r wth a mean value, m y and standard devaton, y s fxed. In ths context, the probablty of structural falure s the maxmum normal stress exceeds the yeld strength and can be represented by the followng equaton: y max max - m P P y y f Then the relablty, P r can be evaluated usng equaton 3.4. Maxmum normal stress: s M I Mc m (3.) (3.3) (3.9) (3.) (3.)

7 7 where s s the modulus of secton. Table 3. hear, moment and deflecton of the beam structure statcally determnate and ndetermnate R V wl wl M wl x M l x w x y 4lx x 4E I 4 w l ymax 8E I 6l w R 5wl 8 wl M 8 w M 8 4x R 3wl 8 5wl V wx 8 5lx l w x y 3 48E I l xx l R R wl w V l x w M 6lx 6x w x y 4E I l x l M y max M wl 4 w l 384E I

8 Centrod The equaton can be wrtten as: ~ xda A x da A (3.4) 3.5 ANALYI OF MAXIMUM NORMAL TRE The loadng, w s 5.55 MN/m and moment of nersa, I s 6.58 x -5 m 4. Heght of the beam structure s.6m and the structure s made of alumnum alloy 4-T4 where the elastc modulus s 73. GPa. The statstcal dstrbuton of alumnum 4-T4 where the mean value and standard devaton of the yeld strength are 34 MPa and 3.4 MPa respectvely. w M.58 m R R Fgure 3.5 Free body dagram of the beam structure 3.5. Determnstc method The determnstc method s the most common method to help engneers to solve problems relatng to the loadng on the beam structure. The stress values obtaned usng the method s 9 MPa. The stress value s then compared to the statstcal dstrbuton of alumnum 4-T4 n whch the mean and standard devaton of the yeld strength s 34 MPa dan 3.4 MPa respectvely.

9 tochastc Methods tochastc methods are also derved from the determnstc method n whch all common terms are also used n stochastc methods. The dfference between the two methods are stochastc methods nvolvng the dstrbuton of data for nput parameters, such as n ths study were unformly dstrbuted load parameter, w and moment of nerta, I. The nput parameters for unformly dstrbuted load and moment of nerta are n normal form wth constant varance of.. By usng stochastc methods and takng nto account the effects of error propagaton n the beam of maxmum stress s 366 MPa and the structural relablty s Fuzzy methods Normal stresses can be determned usng equaton 3.9 and bendng beam structure can be determned usng equaton 3.3. Fgure 3.6 shows the upper, mddle and lower lmt for each nput parameter (unformly dstrbuted load and moment of nerta) and the fuzzy normal stress output. Fgure 3.7 shows the locaton of COG n the fuzzy normal stress profle. Fgure 3.6 Fgure membershp functons for loadng, w, secton modulus, s and the bendng moment, M

10 Darjah Kepercayaan.8.6 COG.4. 5 MPa tress (MPa) Fgure 3.7 Fuzzy maxmum normal stress dagram Results showed that the maxmum normal stress for the structural support beam s 5 MPa. The relablty of the structure s.9879 and the value obtaned by comparng the value of maxmum stress wth the dstrbuton of data n 4-T4 alumnum materal as descrbed n secton FUZZY FINITE ELEMENT METHOD (FFEM) Fgure 3.8 shows the flow chart for structural analyss. Mappng functon s based on the fnte element method.

11 start Fuzzy nput Input mappng n fuzzy graph Mappng and FEM calculatons, Ku = F α-cut < α = α Fuzzy output Defuzzcaton Relablty end Fgure 3.8 The flow chart of structural analyss

12 REULT AND DICUION Table 4. descrbe the physcal propertes nvolved n ths relablty analyss. Table 4. Mechancal Propertes of Alumnum 4-T4 Physcal Propertes I unt Densty 78 kg/m 3 Ultmate strength (tenson) Yeld trength (Tenson) Elastc Modulus 469 MPa 34 MPa 73. GPa w 3 Fgure 4. poston of the nodes on the beam structure Table 4. Types of entry Method Unform Dstrbuted load Input Type Moment of Inerta Determnent tochastc Average Value Normal dstrbuton a Average Value Normal dstrbuton b Fuzzy FFEM a COV =. b COV =. Trangular Membershp Functon c Trangular Membershp Functon c Trangular Membershp Functon c Trangular Membershp Functon c c wdth = 6σ normal dstrbuton

13 3 4.3 FUZZY FINITE ELEMENT METHOD (FFEM) 4.3. Deflecton From the graph of deflecton aganst the nodes, the hghest possble deflecton s located at node 3. Black lnes represent the mnmum and maxmum, the green lne s the hghest confdence level. Whle the red lne graph represents the deflecton for the COG. COG value s not equal to the value at the hghest peak. The value of the maxmum deflecton of the beam structure s equvalent to the value of the COG deflecton components at node 3 s.5mm.. α = deflecton (mm) α = α = COG -. Nod Fgure 4. fuzzy deflecton aganst beam length confdence level COG COG..5mm deflecton (mm) Fgure 4.3 maxmum deflecton of the fuzzy

14 rotaton. rotatons (x -3 rad) Putaran Postf 5 5 Putaran Negatf length Fgure 4.4 fuzzy rotaton aganst beam length confdence level COG.38 rad rotatons (x -3 rad) Fgure 4.5 maxmum fuzzy rotaton

15 bendng moment..5. bendng moment (MN.m) length Fgure 4.6 fuzzy bendng moment aganst beam length confdence level COG 78.9 kn.m bendng moment (MN.m) Fgure 4.7 maxmum fuzzy bendng moment

16 shear force shear force (MN) length Fgure 4.8 fuzzy shear force aganst beam length confdence level COG..69 MN shear force (MN) Fgure 4.9 maxmum fuzzy shear force

17 Bendng tress 6 5 Bendng stress (MPa) length Fgure 4. fuzzy bendng stress aganst beam length confdence level COG. 5 MPa bendng stress (MPa) Fgure 4. maxmum fuzzy bendng stress

18 node number falure probablty (x - ) node number Fgure 4. falure probablty aganst node number α-cut relablty α-cut Fgure 4.3 relablty aganst α-cut

19 Pemalar Varan relablty COV of w COV of bendng moment Fgure 4.4 Relatonshp between the relablty of COV of moment of nerta and COV of dstrbuted load 4.5 ANALYI OF FFEM relablty Ketentuan Determnstc tokastk tochastcs Kabur Fuzzy FFEM dstrbuted load (MN/m) Fgure 4.5 Comparson of analytcal methods n structural relablty analyss

20 4.6 OTHER TYPE OF BEAM deflectons (mm) length Fgure 4.6 fuzzy deflectons confdence level COG = 7MPa bendng stress (MPa) Fgure 4.7 membershp functon of bendng stress

21 deflectons (mm) length Fgure 4.8 fuzzy deflectons confdence level COG = 8MPa 3 4 bendng stress (MPa) Fgure 4.9 membershp functon of bendng stress

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