Non-Linear Analysis of Beams with Large Deflections An Interval Finite Element Approach

Transcription

1 Non-inear nalsis of Beams with arge Deflections n Interval Finite Element pproach R..Mullen, R..Muhanna an M.V. Rama Rao School of Civil an Environmental Engineering, Universit of South Carolina, Columbia, SC 908, US rlm@sc.eu School of Civil an Environmental Engineering, Georgia Institute of echnolog, tlanta, G-0, US rafi.muhanna@gatech.eu Vasavi College of Engineering, Heraba-, INDI r.mvrr@gmail.com bstract: his work illustrates the evelopment of Nonlinear Interval Finite Element Metho for Euler-Bernouli Beams with large eflections uner interval loa. B incluing the von Karman strains Re 00, the secant stiffness is a function of the loa. n interval loa results in interval secant stiffness. Iterations using successivel upate secant stiffness are use to obtain the large eflection solution. In this paper, the formulation of non-linear interval secant solution strateg is presente. Several eample problems will be solve using the evelope metho. he behaviour of the solution is stuie in terms of convergence, computational efficienc, an sharpness of interval bouns. Kewors: Interval; Interval Finite Elements; Uncertaint; Nonlinear Strain 04 b authors. Printe in US.. Introuction Structures must be esigne to safel respon to etreme loaings. Etreme loaings often results in larger eformations in structures which conflict with small strain analsis. Both analtical an numerical methos have been evelope to perform structural analsis using large strains Bathe 99, Re 00. o the authors knowlege, numerical methos have not been presente that will preict response of a structure, incluing nonlinear strain, when subject to uncertain interval loaing. Of course, one coul alwas conuct a Monte Carlo simulation to investigate the large strain response of a structure. However, MC simulations cannot provie guarantee bouns on a solution while interval base methos can. In other applications, interval methos have been foun to be computationall more efficient compare to conventional sensitivit methos or MC simulations. his paper is complementar to the authors previous work on treating non-linear material behavior using interval escription of uncertaint. In the moeling of non-linear materials, the constitutive relationships ha behavior of a ecreasing tangent, or secant stiffness as the material nonlinearities progresse. In the present moel of a large eflection beam, the structure becomes stiffer as the loaing increase. hus, this paper epans nonlinear interval solutions methos to both softening an harening sstems. In this paper, we will first review general interval methos for linear finite element methos. We will then formulate a large eflection beam in an interval sense. Sample calculations illustrating the application of REC 04 - R.. Mullen, R.. Muhanna an M.V. Rama Rao

2 R.. Mullen, R.. Muhanna an M.V. Rama Rao the interval metho are then presente along with observations on the behavior of the evelope formulation. Finall, conclusions are given base on this work... Review of Interval finite element metho Since the earl evelopment of Interval Finite Element Methos IFEM uring the mi nineties of last centur, several researchers worke on various aspects of Interval Finite Element Methos. lefel an Herzberger, 98; Neumaier, 987; Kölüoglu, Cakmak, hmet, Soren, 995; Koluoglu, Cakmak an Nielsen,995; Rao an Sawer, 995; Muhanna an Mullen,995; Nakagiri an Yoshikawa,99; Mullen an Muhanna,99, Rao an Berke,997;Koluoglu, an Elishakoff, 998; Rao an Chen, 998,Nakagiri an Suzuki,999; Muhanna an Mullen,999; Mullen an Muhanna,999. In particular, researchers have focuse among other issues on two major problems; the first is how to obtain solutions for the resulting linear interval sstem of equations with reasonable bouns on the sstem response that make sense from practical point of view, or in other wors with the least possible overestimation of their bouning intervals, the secon is how to obtain reasonable bouns on the erive quantities that are functions of the sstem response. For eample, when the sstem response is the isplacement, the erive quantities might be forces or stresses which are functions of the isplacements. Obtaining tight bouns on the erive quantities has been a tougher challenge ue to the eisting epenenc of these quantities on the primar epenent variables which are alrea overestimate. So far, the erive quantities are obtaine with significantl increase overestimation. During the last ecae, several significant avances have been mae in the application of Interval Finite Element Methos IFEM to problems of uncertaint structural mechanics. Important contributions have been mae b several researchers uring the past one ecae Muhanna an Mullen, 00;Corliss, Fole an Kearfott,004; Zhang,005; Muhanna, Zhang an Mullen,005; Popova, Iankov, Bonev, 00;Neumaier an Pownuk,007; Rama Rao, Mullen an Muhanna,00, Rama Rao, Mullen an Muhanna,0. hese researchers focuse on the issue of computing structural response in the presence of uncertaint loa, stiffness, an element cross sectional area. significant effort has been mae in the work of Zhang 005 to control the aitional overestimation in the values of the erive quantities; the erive quantities have been calculate b an implicit substitution of the primar quantities. In aition to calculating rigorous bouns on the solution of the resulting linear interval sstem, a special treatment has been evelope to hanle the overestimation in the erive quantities. Instea of first evaluating the primar quantities an then substituting the obtaine values in the epression for the erive quantities, the epression for the primar quantities has been substitute before its evaluation in the erive quantities epression an both were evaluate simultaneousl preventing a large amount of overestimation in the values of erive quantities. In spite of the avancement provie b this approach, still it is conitione b the original IFEM formulation an the special treatment of require transformations. significant improvement in the formulation of IFEM with application to truss problems has been introuce in the work of Neumaier an Pownuk 007. his work has presente an iterative metho for REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao

3 Non-inear nalsis of Beams with arge Deflections n Interval Finite Element pproach computing rigorous bouns on the solution of linear interval sstems, with a computable overestimation factor that is frequentl quite small. his approach has been emonstrate b solving truss problems with over 5000 variables an over 0000 interval parameters, with ecellent bouns for up to about 0% input uncertaint. lthough, no calculate erive quantities have been reporte in this work, a formulation has been introuce for the calculation of erive quantities b intersecting the simple enclosure z = Zu, where z epens linearl or nonlinearl on the solution u of the uncertain sstem with another enclosure obtaine from the centere form Neumaier an Pownuk, 007, Eq. 4., pp 57. In spite of the provie improvement in this formulation, the two-step approach will result in aitional overestimation when evaluating the erive quantities. It is quite clear that among other factors, the issue of obtaining tight enclosures for the primar variables as well as for the erive quantities is conitione b IFEM formulation an the methos use for the evaluation of the erive quantities. new mie formulation for Interval Finite Element Methos was evelope b the authors Rama Rao, Muhanna an Mullen, 00, 0 where the erive quantities of the conventional formulation are treate as epenent variables along with the primar variables. he formulation uses the mie variational approach base on the agrange multiplier metho. he sstem solution provies the primar variables along with the agrange multipliers which represent the erive quantities themselves.. Formulation he kinematic equation of Euler-Bernoulli beam theor uner the assumption that the beam is loae in - plane of smmetr, the aial strain, for large eformation Bauchau an Craig, 009 is epresse as follows: U V U V V, where U an V are the aial an transverse isplacements, respectivel. ssuming U V to be small, the epression for, is simplifie as,,, U, V V, REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao

4 R.. Mullen, R.. Muhanna an M.V. Rama Rao REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao Within the contet of finite element, the shape functions of the aial isplacement can be approimate b a linear function an those of the transversal isplacement b a cubic one. hus the shape functions for cubiclinear beam are efine as N V U U U where 0 0 N 4 an V U V U 5 he first erivative of U an first an secon erivatives of V are given as U V N V N where N 7 an N 8 Using Eqs., 7, an 8, an, can be epresse as B B,, 9 where 4 N N B B 0 he aial strain,0 at the neutral ais is obtaine b substituting = 0 in Eq. 0 as B mi,0

5 Non-inear nalsis of Beams with arge Deflections n Interval Finite Element pproach B mi where he total potential energ Π of a beam with geometric strain effects an subjecte to surface traction q on portion of the surface is given b Π, E, q 0 Eq. can be use in Eq. to obtain,, E,, q 4 0 his can be epane as,,,,,, E E 0 0 or, E,, q 0 q he iscrete structural moel can be given as Π K P In orer to obtain the stiffness matri K for the nonlinear case, from equation 9 we obtain: B E, B 0 B E, B 0 B E, B 8 0 Invoking Π 0 an ropping the loa term for simplicit, this gives 5 REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao

6 i B E, B j R.. Mullen, R.. Muhanna an M.V. Rama Rao i B E, B j i B E, B j 0 0 i BE B j 0 Since is arbitrar, we obtain K P Where K = K +K +K an K B E, B K E B B B B 0 K E B B, 0 In the current interval formulation, the loa is assume to be given in an interval form which results in an interval solution for isplacements {} an consequentl interval strains ε. Here stiffness matri is obtaine using numerical integration, in orer to avoi membrane locking, K an K are evaluate using reuce integration in irection onl. Now we rewrite ε, an, ε in Eq. 9 as interval ε, B ε Bmi ε B N N N η where η is a scalar interval multiplier In light of the reuce integration of secon an thir terms of Eq., the iniviual stiffness matrices can be rewritten as K B E B 0 K mi mi 4 B E N N E B 0 0 K N E N 0 7 REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao

7 Non-inear nalsis of Beams with arge Deflections n Interval Finite Element pproach hus K matri in the present work is compute using integration points, K an K matrices are evaluate using reuce integration at one point. Matrices K, K an K can be shown in a simplifie form as follows: 4 K D K ηd.5 ηd mi mi K η D mi 8 Using Eq. 8, the overall stiffness matri K can be efine as follows D D K mi 9 D mi D mi where [D ] is a iagonal matri whose iagonal terms are the prouct of weights w an E at each of m n integration points use for the ouble integral. lso [D mi ] is a iagonal matri whose iagonal terms are the prouct of weights w an E at each of the p integration points use for the single integral. Each entr η occurring in the interval iagonal matri contains the interval multipliers at each of the p integration points use for the single integral. his can be shown as K G η D 0 G Consiering the mie finite element formulation evelope earlier b the authors Rama Rao, Muhanna an Mullen, 0, the following sstem of equations is consiere: K C C U P 0 λ 0 Here C is the constraint matri, λ is the vector of interval multipliers, P is the vector of eternal loas an U is the vector of interval noal isplacements. Substituting Eq. 0 in Eq., we obtain G η D G C C U P 0 λ 0 REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao

8 It is observe that Eq. has the structure R.. Mullen, R.. Muhanna an M.V. Rama Rao B Dgiven b Neumaier Neumaier an Pownuk, Eample problems wo eample problems are chosen to illustrate the applicabilit of the present interval approach to hanle geometric nonlinearit in case of beam element. hese eamples are chosen to emonstrate the abilit of the current approach to obtain sharp bouns to the isplacements an forces even in the presence of large uncertainties an large number of interval variables. It is assume that for each element in the structure obes a linear constitutive relationship. Interval uncertaint of loa is consiere for the two eample problems. he eample problems are solve for various levels of interval withs of the loas centre at their nominal values. ll interval variables are assume to var inepenentl. wo approaches are use, namel; interval an combinatorial. he first eample chosen is a fie-fie beam of =00 in..54m span as shown in Figure. Four elements consecutivel numbere from left to right are use. he beam is subjecte to a nominal istribute loa of w=0 lb/in..75 kn/m in the verticall ownwar irection along the entire span. he material properties for each element are given in able, while the cross sectional imensions are in. in m 0.054m. ll elements have a moulus of elasticit of 0 ksi 0.84 GPa. Solution is obtaine b following the proceure outlines in the earlier section. For ouble integral, two integration points are use along the length of the each element an two integration points across the height. For reuce integration using single integral, one integration point is chosen at the level of the neutral ais of the beam along the irection of span. linear constitutive moel is use to analze this eample. w Figure. Fie-Fie beam subjecte to istribute loaing In orer to valiate the present formulation, the beam is subjecte to an incremental loaing from 0 to 0 lb/in. in steps lb/in. Consiering zero loa uncertaint, the vertical isplacements of the mi-span of the beam are compare to those given b Re 00 an are plotte in Figure. It is clearl observe from the Figure that the vertical isplacements compute b the present approach closel match the results obtaine b Re. REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao

9 Non-inear nalsis of Beams with arge Deflections n Interval Finite Element pproach Figure Fie-Fie beam- vertical isplacements at mi-span at various levels of loaing Figure. Fie-Fie beam- vertical isplacements at mi-span at various levels of uncertaint with 0 lb/in. loaing REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao

10 R.. Mullen, R.. Muhanna an M.V. Rama Rao Figure 4. Fie-Fie beam- bening moment at mi-span at various levels of uncertaint with 0 lb/in. loaing Figure shows the plot of vertical isplacement of mi-span for a loa of 0 lb/in. acting all over the span using the present approach, combinatorial approach an Monte Carlo simulation approach 00 realization at various levels of uncertaint. It is observe from Figure, that the present solution encloses the combinatorial solution sharpl an Monte Carlo simulation provies sharp inner bouns to the combinatorial solution at various levels of uncertaint. Figure 4 shows a plot of the mi-span bening moment for the same case. It is also observe from Figure 4 that the combinatorial solution is enclose sharpl from insie an outsie b the Monte Carlo solution an the present interval solution respectivel. he secon eample chosen is a beam with pinne-pinne supports at both ens as shown in the Figure 5. It has the same geometric an material properties an same loaing as that of the fie beam cite in the first eample. lb/ inch m Figure 5. Pinne-Pinne beam subjecte to istribute loa 00 inch m Figure shows the plot of vertical isplacement of mi-span for a loa of lb/in. acting all over the span using the present approach, combinatorial approach an Monte Carlo simulation approach at various levels of uncertaint. It is observe from Figure, that the present solution encloses the combinatorial solution REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao

11 Non-inear nalsis of Beams with arge Deflections n Interval Finite Element pproach sharpl an Monte Carlo simulation provies sharp inner bouns to the combinatorial solution at various levels of uncertaint. Figure 7 shows a plot of the mi-span bening moment for the same case. It is also observe from Figure 7 that the combinatorial solution is enclose sharpl from insie an outsie b the Monte Carlo solution an the present interval solution respectivel. However, it is observe that the present formulation is not able to able to provie an enclosure to combinatorial solution at higher loas in case of pinne beam. Efforts are on to investigate this issue. Figure. Pinne-Pinne Beam - Vertical isplacement at mi-span at various levels of uncertaint with lb/inch loaing Figure 7. Pinne-Pinne beam- Bening moment at mi-span at various levels of uncertaint with lb/in. loaing REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao

12 R.. Mullen, R.. Muhanna an M.V. Rama Rao 5. Conclusions new formulation for interval analsis of geometricall nonlinear beam is presente. he present approach is valiate b comparing it with the results obtaine earlier b Re. Solutions are compute using the present approach, combinatorial approach an Monte Carlo simulation. It is observe that the present solution encloses the combinatorial solution at various levels of uncertaint.. References lefel, G. an Herzberger, J.98. Introuction to Interval Computations, caemic Press, New York, NY. Bathe, K. 99. Finite Element Proceures, Printice Hall, Englewoo Cliffs, New Jerse 07, New Jerse. Corliss,G., Fole,G., an Kearfott, R.B Formulation for Reliable nalsis of Structural Frames, Muhanna, R.. an Mullen, R.. es, Proc. NSF Workshop on Reliable Engineering Computing, Savannah. Gallagher, R. H Finite Element nalsis Funamentals, Printice Hall, Englewoo Cliffs,US. Henzol, Georg, Hanbook of Geometrical olerancing, Design, Munufacturing an Inspection. John Wile & Sons t., Englan, 995. IS , "Inian Stanar olerances for fabrication of steel CED 7: Structural Engineering an structural sections", Sith Reprint, Bureau of Inian Stanars, New Delhi, Inia ISO/IEC International Organisation for Stanarization008. "Uncertaint of Measurement Part-III- Guie to the Epression of Uncertaint in Measurements, GUM-995", First Eition, Printe in Switzerlan. Koluoglu, U. an Elishakoff, I " Comparison of Stochastic an Interval Finite Elements pplie to Shear Frames with Uncertain Stiffness Properties, " Computers an Structures, Vol. 7, No. -, pp Köluoglu, U., Cakmak,.S., an Nielsen, S. R. K "Interval lgebra to Deal with Pattern oaing an Structural Uncertainties, " Journal of EngineeringMechanics, November, Kölüoglu, U., Cakmak, S., hmet, N., an Soren, R. K. Interval lgebra to Deal with Pattern oaing an Structural Uncertaint. Journal of Engineering Mechanics, : 49-57, 995. Muhanna, R.. an Mullen, R "Development of Interval Base Methos for Fuzziness in Continuum Mechanics, " Proc., ISUM-NFIPS 95, September 7-0, Mullen, R.. an Muhanna, R..99. "Structural nalsis with Fuzz-Base oa Uncertaint, " Proc, 7th SCE EMD/SD Joint Specialt Conference on Probabilistic Mechanics an Structural Reliabilit, WPI, M, ugust 7-9, 0-. Muhanna, R.. an Mullen, R "Formulation of Fuzz Finite Element Methos for Mechanics Problems, " Computer- ie Civil an Infrastructure Engineering previousl Microcomputers in Civil Engineering, Vol.4, pp Mullen, R.. an Muhanna, R "Bouns of Structural Response for ll Possible oaings, " Journal of Structural Engineering, SCE, Vol. 5, No., pp Muhanna, R.. an Mullen, R..: Uncertaint in Mechanics Problems Interval-Base pproach, J. Engrg. Mech. 7, pp , 00. Muhanna,R.., Erolen., an Mullen,R.. 00, "Geometric Uncertaint in russ Sstems: n Interval pproach ",NSF Workshop on Moeling Errors an Uncertaint in Engineering ComputationsREC-00, Center for Reliable Engineering Computing, Georgia ech, Savannah, US. Muhanna, R.., Zhang, H., & Mullen, R Interval finite element as a basis for generalize moels of uncertaint in engineering mechanics. Reliable Computing,, Nakagiri, S. an Suzuki, K "Finite Element Interval analsis of eternal loas ientifie b isplacement input uncertaint, " Comput. Methos ppl. Mech. Engrg. 8, pp. -7. Nakagiri, S. an Yoshikawa, N. 99. "Finite Element Interval Estimation b Conve Moel, " Proc., 7th SCE EMD/SD Joint Specialt Conference on Probabilistic Mechanics an Structural Reliabilit, WPI, M,, ugust 7-9, Neumaier, "Overestimation in inear Interval Equations, " SIM Journal on Numerical nalsis, Vol. 4,, pp REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao

13 Non-inear nalsis of Beams with arge Deflections n Interval Finite Element pproach Neumaier,. an Pownuk,. inear Sstems with arge Uncertainties, with pplications to russ Structures. Reliable Computing, : 49-7, 007. Popova, E., R. Iankov, Z. Bonev, 00. Bouning the Response of Mechanical Structures with Uncertainties in all the Parameters. In R..Muhannah, R..Mullen Es: Proceeings of the NSF Workshop on Reliable Engineering Computing REC, Savannah, Georgia US, Feb. -4, pp Rama Rao, M. V., Mullen, R.., Muhanna, R.., 00, Primar an Derive Variables with the same accurac in Interval Finite Elements, Fourth International Workshop on Reliable Engineering Computing- Robust Design, Coping with Hazars, Risk an Reliabilit, -5 March, National Universit of Singapore, Singapore. Rama Rao, M. V., Mullen, R.., Muhanna, R.., 0, New Interval Finite Element Formulation with the Same ccurac in Primar an Derive Variables, International Journal of Reliabilit an Safet, Vol. 5, Nos. /4. Rao, S. S., Berke, "nalsis of Uncertain Structural Sstems Using Interval nalsis, " I Journal, Vol. 5, No. 4, pp Rao, S. S., i Chen, 998. "Numerical Solution of Fuzz inear Equations In Engineering nalsis, " Int. J. Numer. Meth. Engng. Vol. 4, pp Rao, S. S., Sawer, P "Fuzz Finite Element pproach for nalsis of Imprecisel Define Sstems, " I Journal, Vol., No., pp Re, J. N. 00. n Introuction to nonlinear finite element analsis. Ofor Universit Press. Zhang, H Noneterministic inear Static Finite Element nalsis: n Interval pproach. Ph.D. Dissertation, Georgia Institute of echnolog, School of Civil an Environmental Engineering. Zienkiewicz,O.C. an alor,r. 000, he Finite Element Metho, Butterworth Heinemann, Ofor, UK. REC 04 - R.. Mullen, R.. Muhanna, an M. V. Rama Rao