APPLICATION OF CHEBYSHEV- AND MARKOV-TYPE INEQUALITIES IN STRUCTURAL ENGINEERING ABSTRACT

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1 RT&A # (8) (Vol.8) 3, Mach APPLICATION OF CHEBYSHEV- AND MARKOV-TYPE INEQUALITIES IN STRUCTURAL ENGINEERING K. Balaji Rao. M.B. Anoop. Nagesh R. Iye 3 Chief Scientist, Head, Risk and Reliability of Stuctues Goup, Senio Pincipal Scientist, 3 Diecto (CSIR-Stuctual Engineeing Reseach Cente, Taamani, Chennai, Tamil Nadu, India) balajisec@gmail.com ABSTRACT This pape aims at binging out the usefulness of Chebyshev- and Makov- type inequalities in stuctual engineeing design decision making. By examining whethe the bounds aising fom Chebyshev - type inequality (associated with these ae weak uppe bound pobabilities) enclose the espective expeimental values fo deflections of six feocement I-beams and web shea fatigue life of a steel plate gide it is infeed that the bounds and the associated pobabilities estimated ae ealistic and hence can be used in stuctual engineeing design decision making. The pape also pesents some ecent developments in application of Makov type inequalities (which ae due to Steliga and Szynal ()) fo estimation of bounds on pobability of an event sought. The impotance of such bounds in stuctual engineeing applications is bought out. It is shown fom the esults of Monte Calo simulation that the bounds on pobability of an event, estimated using the method pesented by Steliga and Szynal, ae shap. One of the impotant advantages of the bounds pesented by Steliga and Szynal () is that the oiginal (hidden/intenal) andom vaiable need not have well defined moments. Possible engineeing applications ae also pointed out. Keywods: Chebyshev inequality, Makov inequality, deflection, fatigue life CHEBYSHEV INEQUALITY- SOME PRELIMINARIES Let X be a andom vaiable epesenting an action o esponse quantity. Example of action quantity can be load (o loading intensity), extenal bending moment o extenal taction foce. The esponse quantity can be deflection, otation, waping, stain, cack width. In most engineeing applications we may not be knowing the actual pobability density function (pdf) of X; yet, we will be asked to answe questions like P[g(x) ]=?. It may be noted that g(x) is a function of andom vaiable and is a specified value. Such decision making pobabilities ae equied in limit state design of stuctual components (viz. Bolotin, 969). In the face of non-availability of pdf of X can we make pobabilistic infeences about P[g(X) ]. It can be shown that (viz. Gnedenko, 976) P g X E gx () Let g(x) = [x - µ] ; = z σ whee µ and ae mean and standad deviation of X. Accoding to Chebyschev s inequality this pobability computed fom (Gnedenko, 976)

2 RT&A # (8) (Vol.8) 3, Mach E[( X ) P[( X ) z ] z (o) P[( X ) z ] z ] () Theefoe, Chebyshev s inequality gives weak uppe bound on the desied pobability. We note fom Eq.() that z should be geate than o equal to since if z < (though positive) the intepetation as the value of pobability will not be pope. Also, since we ae not having any infomation on type of pdf of X the bounds will be weak. Effots have been made in the liteatue to shapen the bounds and to detemine two-sided bounds and also, to detemine bounds fo multivaiate case (with- and without- coelation effect). While Eq. () gives the one-sided bound, let us conside two-sided case. When the distibution is symmetical about the mean, the symmetical bounds aound the mean ae given by (Steliga and Szynal, ), by, 4 P[ k k ] ( k k ) When the pdf of the andom vaiable is not symmetical about the mean, the bounds ae given P[ k Whee k k ; ( k )(k k ) k k 4[( k )( k ) ] ] ( k k ) (3) (4) APPLICATIONS In this section two example poblems demonstating the use of Chebyshev inequalities in detemining the weak uppe bound pobabilities, those equied fo engineeing decision making, ae pesented. One of the highlights of these examples is, to infuse confidence in engineeing applications, to compae the esults with the espective expeimental values. Example : In this example an attempt has been made to estimate the weak uppe bound pobabilities on andom cental deflection of feocement I-beams used fo oofing in low-cost housing. This example is consideed since the test data on cental deflection, at diffeent stages of loading, was available fo six specimens. These specimens wee tested at the stuctual engineeing laboatoy of Indian Institute of Science, Bangaloe, in 98s. The details of tests and the test esults ae available in (Pakash Desayi and Balaji Rao (988), Pakash Desayi and Balaji Rao (993), Balaji Rao (99)). Also, an effot was made to detemine statistical popeties of deflections using Monte Calo simulation technique. Moe details about basic andom vaiables consideed and details of simulation ae pesented in Balaji Rao (99). The final esults of simulation (viz. mean and standad deviations of deflection) fo six specimens consideed hee, at diffeent stages of loading, ae pesented in Table. Also pesented in this table ae expeimental values of cental deflections. The weak uppe bound pobabilities associated with bounds of lengths.5,.5,.75 and 3 ae computed using Chebyshev inequalities. These pobabilities ae computed fo two conditions : (a) assuming that the pdf of deflection, at diffeent stages of loading, ae symmetical about the mean, and, (b) assuming that the pdf of deflection is unknown o unsymmetical about the mean. The values of the bounds and thei coesponding pobabilities ae pesented in Table typically fo fist two inteval lengths. Since a bound of length 3 is vey often used in engineeing decision making, the same ae compaed fo the cases (a) and (b) in Figs.

3 RT&A # (8) (Vol.8) 3, Mach 6 fo the specimens consideed. Also shown in these figues ae the expeimentally obseved deflections. Fom these figues it is obseved that at almost all stages of loading, the estimated bounds contain the obseved deflection suggesting that the estimated weak uppe bound pobabilities ae acceptable and can be used in engineeing decision making. If it is felt that the length of inteval of 3 is high, fom Table it is noted that, at highe stages of loading, even though the lengths of inteval ae small, the bounds enclose the expeimentally obseved deflections. It may be noted that the weak uppe bound pobabilities fo the two intevals pesented in Table vay between to % (which is small though). These obsevations suggest that the Chebyshev s inequalities can be used fo engineeing decision making. Table. Bounds of diffeent inteval lengths and thei compaison with expeimental esults Specimen designation Applied load (kn) Exp. deflection Results of Monte Calo simulation, Balaji Rao (99) Mean, Standad deviation, Bounds symmetical (length of inteval =.5 ) Lowe bound (.5) Uppe bound ( +.5) Bounds unsymmetical (length of inteval =.5) Lowe bound ( ) Uppe bound ( +.5) MI MI MI3 MI4 MI5 MI Note:, associated weak uppe bound pobabilities ae. and., espectively

4 RT&A # (8) (Vol.8) 3, Mach Cental deflection,8,6,4,,8,6,4, Lowe bound Uppe bound expeimental lowe bound_unsymmetical uppe bound_unsymmetical,,4,6 Applied,8 load (t),,4,6 Figue. Load vesus deflection plot fo specimen MI with ( -.5, +.5 ) symmetical bounds (associated minimum pobability.56) and ( -, +. ) fo unsymmetical bounds (associated minimum pobability.44) with expeimental values Cental deflection 5 4,5 4 3,5 3,5,5,5 Lowe bound Uppe bound expeimental Lowe bound_unsymmetical Uppe bound_unsymmetical,,4,6,8,,4 Applied load (t) Figue. Load vesus deflection plot fo specimen MI with ( -.5, +.5 ) symmetical bounds (associated minimum pobability.56) and ( -, +. ) fo unsymmetical bounds (associated minimum pobability.44) with expeimental values 3

5 RT&A # (8) (Vol.8) 3, Mach,5 Cental deflection,5,5 Lowe bound Uppe bound expeimental Lowe bound_unsymmetical,5 Applied load (t),5 Figue 3. Load vesus deflection plot fo specimen MI3 with ( -.5, +.5 ) symmetical bounds (associated minimum pobability.56) and ( -, +. ) fo unsymmetical bounds (associated minimum pobability.44) with expeimental values 3 Cental deflection,5,5,5 Lowe bound Uppe bound expeimental Lowe bound_unsymmetical Uppe bound_unsymmetical,,4,6,8, Applied load (t) Figue 4. Load vesus deflection plot fo specimen MI4 with ( -.5, +.5 ) symmetical bounds (associated minimum pobability.56) and ( -, +. ) fo unsymmetical bounds (associated minimum pobability.44) with expeimental values 4

6 RT&A # (8) (Vol.8) 3, Mach Cental deflection,8,6,4,,8,6,4 Lowe bound Uppe bound expeimental,,5,5 Applied load (t) Figue 5. Load vesus deflection plot fo specimen MI5 with ( -.5, +.5 ) symmetical bounds (associated minimum pobability.56) and ( -, +. ) fo unsymmetical bounds (associated minimum pobability.44) with expeimental values 4 3,5 3,5 Cental deflection,5,5 Lowe bound Uppe bound expeimental Lowe bound_unsymmetical Uppe bound_unsymmetical,,4,6,8,,4,6 Applied load (t) Figue 6. Load vesus deflection plot fo specimen MI6 with ( -.5, +.5 ) symmetical bounds (associated minimum pobability.56) and ( -, +. ) fo unsymmetical bounds (associated minimum pobability.44) with expeimental values 5

7 RT&A # (8) (Vol.8) 3, Mach Example : This example shows how Chebyshev s inequality will help in fatigue esistant design of steel plate gides of a plate gide bidge. Moe details of this poblem can be found in Balaji Rao and Anoop (3). The basic equation used in pedicting the fatigue life using S N appoach is given by, N f C b (5) whee N f is the numbe of load cycles to the fatigue limit and is the applied stess ange, C and b ae the mateial paametes, known as the fatigue stength coefficient and the fatigue stength exponent, espectively. It is known that the numbe of cycles to failue (i.e. fatigue life, N f ), at a given applied stess ange is a andom vaiable. A typical plot showing the same is pesented in Fig. 7. It may be noted that the natue of pdf and the statistical popeties of N f may depend on stess ange. While it is desiable to establish the natue of these pobability distibutions using fatigue tests, it is expensive and time consuming. The median and the 5% and 95% factiles of fatigue life computed using the tansfomation of vaiable technique, at diffeent applied stess anges ae shown in the figue. Moe details of the pobabilistic analysis of the fatigue life of the plate gide ae pesented in Balaji Rao et.al. (3). Also shown in this figue ae expeimental fatigue lives epoted in liteatue. Except in few cases, expeimental scatte is enclosed by the estimated bounds. Let us apply the Chebyshev s inequality to detemine the bounds on fatigue life. At the applied stess ange of 7 MPa the mean (µ) and standad deviation () of N f ae espectively 7.47E+5 and 3.447E+5. Assuming the bounds to be symmetical and (k -k ) = 3, the pobability that the fatigue life will be between (.35E+5,.645E+5) is equal to o geate than.556. On the othe hand if the distibution is unsymmetical about the mean, even though we may keep (k -k ) = 3, assuming the bounds to be (4.4E+5, 4.365E+5), that satisfies the conditions associated with Eq. (4), then the pobability that the bound will contain actual life will be equal to geate than.444. This value of pobability is less than the case when the bounds ae symmetical fo the eason that in ode to assume that the bounds ae symmetical we should have had moe justification/confidence and this gets embedded in ou pedictions..e+8 Expeimental (Osman et al. [3]) pobabilistic (median (x) with bounds (I)) fatifue life (cycles).e+7.e+6.e pincipal suface stess ange (MPa) Figue 7. Compaison of esults of pobabilistic analysis with expeimentally obseved values of fatigue life fo plate gides epoted in liteatue 6

8 RT&A # (8) (Vol.8) 3, Mach SOME RECENT DEVELOPMENTS Extension of Makov-type inequalities to a class of andom vaiables without moment condition equiement was poposed by Steliga and Szynal (). This is a flexible fomulation that enables the computation of bounds on pobabilities fo the andom vaiable which does not have well defined moments. Some of the examples of andom vaiables that belong to this categoy ae alpha stable andom vaiables. These andom vaiables have factional moments and the existence of the ode of moments depends on the value of the exponent. Recent R&D at CSIR- SERC has evealed that fo descibing the vaiations in some of the engineeing quantities, alphastable distibutions ae moe appopiate (Balaji Rao and Anoop, ). Hence, the esults of Steliga and Szynal () ae impotant and the same ae consideed futhe. Steliga and Szynal () assumed the following conditions: () X is a positive andom vaiables (X a.s.), () G is a class of all positive, stictly inceasing functions g with g() =. Let N be set of integes. Making use of Makov inequalities, the following inequality has been poved by them. Fo a given > and k in N g( kx ) E g( kx ) g(( k )X ) I [ X ] P[ X g( kx ) ] E g( kx ) g(( k )X ) g( kx ) E g( kx ) g(( k )X ) I[ X ] (6) Whee I[.] denotes the indicato function. The above bounds ae valid only when X is a positive andom vaiable. The computation of bounds does not equie moments of X. Howeve, we should know the functional fom (stictly inceasing) of g(.) and an idea about the ealizations of X. We should also be in a position to detemine the expected values of g(.). The stictly inceasing function g(.) can be fomulated based on phenomenological modeling involving X. Possible stuctual engineeing application: Fo example, the aveage (smoothed) oof loaddisplacement cuve of a moment esisting fame subjected to lateal loads, obtained using pushove analysis, ove the ange of engineeing design inteest, can be consideed as stictly inceasing function of load. In a gavity contolled expeiment, the vaiations obseved in oof displacements of nominally simila fames can be attibuted to the vaiations in dimensions and stengths of mateials. These vaiations can be aggegated into oveall igidity/compliance of the fame (X, which is always positive, and thus satisfying the condition equied to estimate the bounds using the above equation). We can always assign an acceptable pobability distibution to the deflection, geneate andom values of deflection following the assigned pdf and indiectly estimate the igidity and examine the indicato function. This execise would cicumvent us fom diectly geneating andom ealizations of X (as aleady indicated this andom vaiable may not have well defined moments). Still we will be able to compute the bounds on the equied pobability with egad to X. This also suggests that the fomulations pesented by Steliga and Szynal () can be used in an invese poblem to chaacteize, pobabilistically, the intenal vaiable. Howeve, the condition kn may pehaps need to elaxed though pope fomulations. Computation of bounds on pobability of equied event using Eq. (6) equies X to be positive. Let X be the andom vaiable which is eal valued. Then, the bounds pesented above fo positive andom vaiables can be used by substituting X in the place of X. Let us conside some special cases wheein does X take on negative values whose moments may o may not exist and a stictly inceasing function of X, g(x), is obsevable and whose distibution is known (in such a case X can be consideed as a hidden/intenal vaiable whose value is infeed based on a physical elationship g(.) and X). 7

9 RT&A # (8) (Vol.8) 3, Mach The following functional fom is assumed fo >. We will now conside special cases and povide necessay bounds fo P(X ). ( a ) g( x ) x m, mn ( b ) g( x ) x, In ode to compute denominato in Eq. (6), we need the following bounds (Gut, 5), wheneve x > and y >, x y fo ( x y ) (7) ( x y ) fo The following ae the bounds deived by Steliga and Szynal (). These would be useful in engineeing applications some of which ae pointed out in the next section. k X I LBk ; E I X, k N, k X k X LB I k MB m; E I k ; E m k X I X, m m k X k X k,m N k X I X k X k X, k N, k X I X k X k X, k N, I MBk ; E The pedictive powe of the above equations, chaacteised by shapness of bounds, fo g(x) = x, when the undelying andom vaiable is following nomal and lognomal distibutions ae pesented in Tables and, espectively, fo diffeent values of k and. To estimate the lowe and uppe bounds, Monte Calo simulation technique involving 5 simulation cycles ae used. The actual pobabilities whose bounds ae being estimated ae pesented in foot note of these tables. Fom the esults pesented in these tables it is infeed that the bounds developed by Steliga and Szynal ae tight and can be used in the engineeing applications fo making pobability statements about the intenal vaiable which is esponsible fo geneating the andom obsevable g(x). Howeve, it should be noted that the assumptions made in deiving the bounds should be satisfied. As stated ealie, one of the limitations, pehaps, in engineeing application would be the need imposed by kn. Possible stuctual engineeing application: In many engineeing poblems, X may take negative values and also moments of X may not exist. Fo example, ecent studies by Balaji Rao and Anoop (), at CSIR-SERC, have shown that the desciption of evolution of suface stain field of a einfoced concete flexual membe follows a Levy pocess. Accodingly, at any stage of loading, the fluctuations in suface stains may be descibed using an alpha-stable distibution. It is known that handling such andom vaiables can be difficult and may be desiable to make pobabilistic infeences of these vaiables based on the pobabilistic vaiations in obsevables such as deflections and/o cackwidths which ae functions of intenal vaiables such as stains. This study is being futheed at CSIR-SERC. 8

10 RT&A # (8) (Vol.8) 3, Mach Table. Results of simulation (N = 5 cycles) fo the bounds on P[X ] * fo X being nomally distibuted with mean =. and standad deviation =.; g(x) = x k ¾ ½ ¼ /9 /6 LBI(k;;) MBI(k;; ) LBI(k;; ) MBI(k;; ) LBI(k;;) MBI(k;; ) LBI(k;; ) MBI(k;; ) LBI(k;; ) UBI(k;; ) LBI(k;; ) MBI(k;; ) LBI(k;; ) MBI(k;; ) Note: * Actual pobability value =.455 Table 3. Results of simulation (N = 5 cycles) fo the bounds on P[X 3] * fo X being lognomally distibuted with mean =. and standad deviation =.5 ** ; g(x) = x k ¾ ½ ¼ /9 /6 LBI(k;; ) MBI(k;; ) LBI(k;; ) MBI(k;; ) LBI(k;; ) MBI(k;; ) LBI(k;; ) MBI(k;; ) LBI(k;; ) MBI(k;; ) LBI(k;; ) MBI(k;; ) LBI(k;; ) MBI(k;; ) Note : * Actual pobability value =.38; ** paametes of lognomal: - lamda = ; exi = ; 3 SUMMARY This pape aims at binging out the usefulness of Chebyshev- and Makov- type inequalities in stuctual engineeing design decision making. By examining whethe the bounds aising fom Chebyshev - type inequality (associated with these ae weak uppe bound pobabilities) encloses the espective expeimental values fo: (a) pediction of cental deflection of six feocement I-beams, and, (b) fatigue life of a steel plate gide of a plate-gide bidge, against the limit state of web shea buckling, it is infeed that the bounds and the associated pobabilities estimated ae ealistic and hence can be used in stuctual engineeing design decision making. The pape also pesents ecent developments in detemination of inequalities of the type of Makov, which ae due to 9

11 RT&A # (8) (Vol.8) 3, Mach Steliga and Szynal (). The impotance of such bounds in stuctual engineeing applications is bought out. It is shown fom the esults of Monte Calo simulation that the bounds on pobability of an event sought, estimated using the method pesented by Steliga and Szynal, ae shap. One of the impotant advantages of the bounds pesented by Steliga and Szynal () is that the oiginal (hidden/intenal) andom vaiable need not have well defined moments. Possible engineeing applications ae also pointed out. ACKNOWLEDGEMENT One of the authos, KBR, is gateful to Pof. Pakash Desayi, fomely pofesso, Depatment of Civil Engineeing, Indian Institute of Science, Bangaloe, unde whose guidance he did his Ph.D. The authos thank the Diecto, CSIR-SERC, fo giving pemission fo publishing this pape. REFERENCES. Balaji Rao, K and Anoop, M.B. Why do we need pobability distibutions with fat tails to descibe the suface stain evolution in einfoced concete flexual membes?. Meccanica, Spinge, DOI.7/S Balaji Rao, K., Anoop, M. B., Raghava, G., Pakash, M. and Rajaduai, A 3. Pobabilistic fatigue life analysis of welded steel plate ailway bidge gides using S-N cuve appoach. Jounal of Risk and Reliability, Pat O, IMecH, UK Accepted fo Publication 3. Balaji Rao. K 99. Studies on Reliability of Reinfoced Concete Beams in Cacking and Feocement Elements in Tension and Flexue. Ph. D thesis submitted to Depatment of Civil Engineeing, Indian Institute of Science, Bangaloe, 87 pp. 4. Bolotin, V. V 969. Statistical Methods in Stuctual Mechanics. Holden-Day, Inc., USA. 5. Gnedenko, B.V 976. The theoy of pobability. English Tanslation, Mi Publishes, Moscow. 6. Gut, A 5. Pobability: A gaduate couse. Spinge texts in Statistics, Spinge, New Yok. 7. Pakash Desayi and Balaji Rao, K 993. Estimation of Spacing and Width of Cacks in Feocement Monolithic I-Joists. Jounal of Feocement, Vol.3, No., pp: Pakash Desayi and Balaji Rao, K 988. Pediction of Cacking and Ultimate Moments, and, Load-Deflection Behaviou of Feocement Elements. Thid Intenational Symposium on Feocement, New Delhi, Decembe 8-, Poceedings, pp: Steliga, K and Szynal, D. On Makov-type inequalities. Intenational Jounal of Pue and Applied Mathematics, Vol. 58, No., pp:37 5.