Interval Finite Element Analysis of Thin Plates

Transcription

1 Interval Finite Element Analsis of hin Plates M. V. Rama Rao ), R. L. Muhanna ) and R. L. Mullen ) ) Vasavi College of Engineering, Hderabad - 5 India, dr.mvrr@gmail.com ) School of Civil and Environmental Engineering, Georgia Institute of echnolog Atlanta, GA -55, USA, rafi.muhanna@gtsav.gatech.edu ) School of Civil and Environmental Engineering, Universit of South Carolina, GA -55, USA, rlm@cec.sc.edu Abstract: his paper focuses on the analsis of thin plates with uncertain structural parameters modelled as intervals. he plate is assumed to be orthotropic. Interval uncertaint is associated with the Young s modulus of the plate and also with the applied load. Interval Finite Element (IFEM) developed in the earlier work for line elements of the authors for truss and frame structures (Rama Rao et al., ) is applied to the case of thin plates in the present work. his method is capable of obtaining bounds for interval forces and moments with the same level of sharpness as displacements and rotations. Eample problems pertaining to various edge conditions of the thin plate are solved to demonstrate that the present method is capable of obtaining sharp bounds. Results are compared to the values of displacements and forces obtained using combinatorial and Monte Carlo solutions. Kewords: interval, interval finite elements, uncertaint, thin plates, structural analsis. Introduction Plates pla a major role in several important structures viz. ships, pressure vessels, and other structural components. hus it is important to understand their structural behaviour and possible conditions of failure especiall under conditions of uncertaint. he structural behaviour of thin plates in bending depends on several important factors including load, stiffness characteristics of plate and support conditions. he problem of plate bending is one of the oldest in the theor of elasticit and is discussed in several tetbooks (imoshenko and Krieger, 959; Szilard, 4; Redd, 7). Lim et.al. (7) derived eact analtical solutions to bending of rectangular thin plates b emploing the Hamiltonian principle with Legendre s transformation. he solution obtained b them for eample problems of plates with selected boundar conditions shows ecellent agreement with the classical solutions. Batista () used the Fourier series to compute the analtical solutions of uniforml loaded rectangular thin plates with smmetrical boundar conditions. On the other hand, structural analsis without considering uncertaint in loading or material properties leads to an incomplete understanding of the structural performance. Structural analsis using interval variables has been used b several researchers to incorporate uncertaint into structural analsis (Koluoglu, Cakmak and Nielson, 995; Muhanna and Mullen, 995; Nakagiri and Yoshikawa, 996; Rao and Sawer, 995; Rao and 6 b authors. Printed in German REC 6 - M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

2 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen Berke, 997; Rao and Chen, 998; Muhanna and Mullen, ; Pownuk, 4; Neumaier and Pownuk, 7). o the authors knowledge, applications of interval methods for the analsis of plates with uncertaint of load and material properties do not eist anwhere in literature. In view of this, we present an initial investigation into the application of interval finite element methods to problems of bending of thin plates. Usuall, derived quantities in Interval Finite Element (IFEM) such as stresses and strains have additional overestimation in comparison with primar quantities such as displacements. his issue has plagued displacement-based IFEM for quite some time. he recent development of mied/hbrid IFEM formulation b the authors (Rama Rao et al., ) is capable of simultaneous calculation of interval strains and displacements with the same accurac. his work presents the application of interval finite element methods to the analsis of thin plates. Uncertaint is considered in both the applied load and Young s modulus as eplained in section. Eamples are finall presented and discussed. In the present stud a rectangular plate is analsed different tpe of edge conditions of such as clamped and simpl supported edge conditions and the deformations are obtained.. Linear Interval Finite Element Finite element method is one of the most common numerical methods for solving differential and partial differential equations with enormous applications in different fields of science and engineering. Interval finite element methods have been developed to handle the analsis of sstems for which uncertain parameters are described as intervals. A variet of solution techniques have been developed for IFEM. A comprehensive review of these techniques can be found in (Zhang, 5; Muhanna et al., 7; Rama Rao et al., ). Interval analsis concerns the numerical computations involving interval numbers. All interval quantities will be introduced in non-italic boldface font. he four elementar operations of real arithmetic, namel addition (+), subtraction ( ), multiplication ( ) and division ( ) can be etended to intervals. Operations {,,, } over interval numbers and are defined b the general rule (Moore, 966; Moore, 979; Moore et al., 9; Neumaier, 99) [min( ),ma( )] for {,,, }, in which and denote generic elements and. Software and hardware support for interval computation are available such as (Sun microsstems, ; Knüppel, 994; INLAB, 999). For a realvalued function f (,..., n ), the interval etension of is obtained b replacing each real variable i b an interval variable i and each real operation b its corresponding interval arithmetic operation. From the fundamental propert of inclusion isotonicit (Moore, 966), the range of the function f (,..., n ) can be rigorousl bounded b its interval etension function f ( ) () f (,.., n) { f (,.., n),.., n n} () REC 6 - M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

3 Interval Finite Element Analsis of hin Plates f (,..., n contains the range of f (,..., n ) Eq. () indicates that ) for all. A natural idea to implement interval FEM is to appl the interval etension to the deterministic FE formulation. Unfortunatel, such a naïve use of interval analsis in FEM ields meaningless and overl wide results (Muhanna and Mullen, ; Dessombz et al., ). he reason is that in interval arithmetic each occurrence of an interval variable is treated as a different, independent variable. It is critical to the formulation of the interval FEM that one identifies the dependence between the interval variables and prevents the overestimation of the interval width of the results. In this paper, an element-b-element (EBE) technique is utilized for element assembl (Muhanna and Mullen, ; Zhang, 5). he elements are detached so that there are no connections between elements, avoiding element coupling. he Lagrange multiplier method is then emploed to impose constraints to ensure the compatibilit. hen a mied/hbrid formulation is incorporated to simultaneousl calculate the interval strains and displacements (Rama Rao, Mullen and Muhanna, ). his linear formulation results in the interval linear sstem of equations that has the following structure: (, () with interval quantities in D and b K BDA) u a Fb ( K A onl. he term BD ) represents the interval structural stiffness matri and the a + F b term, the structural loading. An interval solver can be used to solve Eq. (), however, the following iterative scheme that is developed b Neumaier and Pownuk (Neumaier and Pownuk, 7) is superior for large uncertaint, defining: i C : ( K BD A) (4) where D is chosen in a manner that ensures its invertabilit (often D is selected as the midpoint of D), the solution u can be written as: u ( Ca) ( CF) b ( CB) d (5) o obtain a solution with tight interval enclosure we define two auiliar interval quantities, v Au d ( D D) v, which, given an initial estimate for u, we iterate as follows: v k { ACa ) ( ACF ) b ( ACB) d } v, {( D D ) v k k k k k c c until the enclosures converge, from which the desired solution u can be obtained in a straightforward manner. In this paper the above mentioned iterative enclosure has been used for the solution of the linear interval sstem of Eq. (). he solution includes displacements, strains, and forces simultaneousl with the same high level of accurac.. Finite Element Model of the Plate hin plates are characterized b a structure that is bounded b upper and lower surface planes that are separated b a distance h as shown in Figure. he - coordinate aes are located on the neutral plane of the plate (the "in-plane" directions) and the z-ais is normal to the - plane. In the absence of in-plane d d, i (6) (7) REC 6 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

4 4 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen loading, the neutral plane is at the midpoint through the thickness. In the present work, it will be assumed that the thickness of plate h is a constant. Consequentl, the location of the - aes will lie at the mid-surface plane (z=). In most plate applications, the eternal loading includes distributed load normal to the plate (z direction), concentrated loads normal to the plate, or in-plane tensile, bending or shear loads applied to the edge of the plate. Such loading will produce deformations of the plate in the coordinate directions which in general can be characterized b displacements, and in the, and z directions, respectivel. u(,, z) v(,, z),, z w(,, z) Figure. Geometr of thin plate. he plate is discretized into rectangular ACM (Adini-Clough-Melosh) plate elements. he ACM element is a non-conforming element with degrees of freedom ( degrees of freedom at each of the four nodes). Degrees of freedom at each node (i) are the transverse displacement and normal rotation about each ais, w, i w i and i i, as illustrated in Figure. Note that is a vector in the negative direction. Node "" is selected at the lower left comer of the plate (=-a, =-b) and that the nodes are numbered,,,4 in counterclockwise direction around the plate. We assume that the plate dimensions are given b a and b as shown in Figure and that the - coordinate sstem is located at the center of plate. he degrees of freedom are arranged in the vector of generalized nodal displacementsd as: wi w w w w (8) d i REC 6 - M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

5 5 Interval Finite Element Analsis of hin Plates Z Y w4 Z Y w ϴ 4 a a ϴ4 4 ϴ b X w X b ϴ ϴ Figure. Rectangular element with degrees of freedom. ϴ ϴ.. SIFFNESS MARIX AND FORCE VECOR OF HE ACM PLAE ELEMEN We assume that w(, ) is some function over the plate geometr as follows: w(, ) a a a a4 a5 a6 a7 a4 a9 a a a (9) w(, ) () which is represented as w(, ) N(, ) () w(, ) () Substituting the values of and coordinates of nodes,,,4 of the plate in the Eq. will ield: where d () d and element respectivel. Substituting from Eq. () in Eq. (), we obtain are vector of unknown coefficients and vector of generalized nodal displacements for the d w(, ) N(, ) (4) REC 6 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

6 6 he curvatures of the plate element viz., differentiation of w.r.t and as follows: w(, ) w w ; ; M. V. Rama Rao, R. L. Muhanna and R. L. Mullen and are obtained from the second order partial w (5) hus the vector of curvatures (e) for the plate element can be epressed as d (6) (7) ( e) B d From the moment-curvature relationship { M} D we obtain where M M M D D B d Eh D (9) ( ) ( ) otal potential energ stored in the plate due to bending is given b (Gallagher, 975; Bathe, 996) a b a b ( ) e ( e M dd d P d B D B dd d d P ) () ab ab Using the first variation of, we obtain, a b ( e ) ( e) P B DBdd d K d () ab where the stiffness matri K (e) is given as a b ( e) K B D B dd () ab (8) REC 6 - M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

7 7 and the nodal force vector for the plate element a b ( e) P p z N(, ) dd ab Interval Finite Element Analsis of hin Plates P (e) is given as: () 4. Interval Finite Element Model of the Plate An element-b-element (EBE) technique is utilized for element assembl as outlined in section. Interval uncertaint is considered in pressure and Young s modulus of the plate E. Accordingl, the stiffness matri and the force vector of the plate element are rewritten, denoting interval quantities in boldface, as follows: and a b K B (e) D B dd ab a b p N(,) dd z ab B is denoted as a b K (e) B DB dd p z P (e) (5) For convenience, he ab B. hus Equation (4) can be rewritten as (6) D matri appearing in the above equation is an interval matri owing to the uncertaint of Young s modulus E. It can be epressed as follows: Eh Eh ( ) ( ) Eh Eh Eh D (7) ( ) ( ) ( ) ( ) Eh 4( ) where E is the interval Young s modulus and is the Poisson s ratio. Following the work of Xiao et.al., (Xiao, Fedele and Muhanna, ), thed matri is decomposed as follows: D A diag ( Λ α ) A (8) k k k k (4) REC 6 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

8 8 α E k M. V. Rama Rao, R. L. Muhanna and R. L. Mullen h h h where ; Λ k ; A k ( ) 4( ) (9) Appling numerical integration to Eq. (6), then M N (e) ~ K w w B (, ) D(, ) B (, () i j that leads to K (e) i j i i i i i i) M N i j Eq. () can be rewritten as B, ) Ak B (, ) Ak... w w B (, ) A diag ( Λ α i j ) A B (, i i k k k k i i) diag ( Λ kαk (, )) Ak B (, ) K (e) ( diag ( Λk α (, )) Ak B (, ) () k So the decomposition for the element stiffness matri ( e) ( e) K A diag ( α) A (e) K (e) is epressed as () he stiffness matri of the structure K is obtained from the element stiffness matrices described in Eq. () as follows: (e) ( e) K diag ( α ) A (e) ( e) K ( ) ( e) ( e) ( e) diag α... A K A A A (4) (e) ( ) e K diag ( ) α A his can be denoted as K A D A (5) When each plate element is subjected to an interval pressure z (e) K () p, the corresponding interval force vector P described in Eq. () for the structure can be defined using the M matri approach outlined b the authors (Muhanna and Mullen, 999) as follows: (e) P (e) P n M δ... P n m m (e) P (6) REC 6 - M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

9 9 Interval Finite Element Analsis of hin Plates where n is the number of degrees of freedom for the structure and m is the number of elements. Each column matri contains the contribution of deterministic pressure on each plate element and the interval M of vector δ contains interval multipliers corresponding to pressures acting on each of the m elements of the structure. In addition, the vector of point loads acting on the nodes of the structure can be represented. as P c he current interval formulation is based on the Element-B-Element (EBE) finite element technique (Muhanna and Mullen, ; Rama Rao, Mullen and Muhanna, ). In the EBE method, each element has its own set of nodes, but the set of elements is disassembled, so that a node belongs to a single element. A set of additional constraints is introduced to force unknowns associated with coincident nodes to have identical values. hus, the constraint equation takes the form CU V C U where the constraint matri C is a deterministic one (fied point matri). Eq. (7) can be rewritten to represent the interval form of strain-curvature relationship as κ (e) B d (8) d where is the vector of nodal displacements for the element. At the global level, this relation can be epressed as B U κ (9) where B is the strain-curvature matri and is the vector of interval curvatures for the structure. Eq. κ (9) can be used as an additional constraint in addition to Eq. (7). hus the modified potential energ can be epressed as * U K UU P ( C U- V ) ( B U- κ ) (4) Invoking the stationarit of and *, that is *, and considering Eq. (4), we obtain U PC C B A M λ C D A δ λ (4) B I I where interval displacements U (primar unknowns) as well as interval values ofλ,λ unknowns) with the same level of sharpness (Rama Rao, Mullen and Muhanna,). It is further observed λ have zero value. are vectors of Lagrange Multipliers. he solution of Eq. (4) will provide the values of from Eq. (4) that the Lagrange multipliers p z (7) * and κ (secondar REC 6 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

10 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen REC 6 - M. V. Rama Rao, R. L. Muhanna and R. L. Mullen δ P λ λ U K C M I I B C B C (4) Eq. (4) is now similar to Eq. () and thus can be solved using the Neumaier s approach outlined in section. he vector of interval moments M can be obtained from the vector of interval curvatures κ as h κ κ κ E κ κ κ D M M M ) ( ) ( (4) he applicabilit of the procedure outlined above is illustrated b solving numerical eamples in the net section. 5. Eample Problems A thin rectangular plate with clamped edges is chosen to illustrate the applicabilit of the present approach to handle uncertaint in load and material properties in case of thin plate problems. hese eamples are chosen to demonstrate the abilit of the current approach to obtain sharp bounds to the displacements and forces even in the presence of large number of interval variables. he two eample problems are solved for various levels of interval widths of the loads centered at their nominal values. All interval variables are assumed to var independentl. Solution procedure outlined in the previous sections is used to perform the linear interval finite element analsis. he material and geometric properties of the plate are given in able below. he discretization scheme adopted is shown in Figure. able. Properties of rectangular plate and discretization scheme. Length L. m Width L. m hickness.5 m Young s modulus GPa Poisson s ratio. Applied Pressure p z 4. Pa Number of divisions along -ais n Number of divisions along -ais n Notation for discretization scheme n n

11 Interval Finite Element Analsis of hin Plates Y LY X Figure. Discretization scheme of rectangular plate. L First the present interval approach is validated b solving the problem of a rectangular plate with a 44 discretization scheme. Solution is computed using the present interval approach and combinatorial solution. he computation of results for combinatorial solution required computation of results for 6 =65,56 combinations. he results obtained for clamped plate for vertical displacement at center of the plate (at node ), slope at node and slope at node 7 at various levels of uncertaint of Young s modulus (E) are shown in Figure 4, Figure 5 and Figure 6 respectivel. Figure 7 and Figure 8 show the variation of M M and the center of the plate (at node ) at various levels of uncertaint of Young s modulus (E). hese figures show the lower and upper bounds of the present interval solution and the corresponding results of the combinatorial solution for various levels of uncertaint of Young s modulus from percent to percent. It is observed from these figures that the bounds of the present interval solution sharpl enclose the bounds of combinatorial solution at all levels of uncertaint. at REC 6 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

12 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen Figure 4. Clamped plate- variation of vertical displacement of center of plate (at node ) w.r.t. uncertaint of E. Figure 5. Clamped plate- variation of at node w.r.t. uncertaint of E. REC 6 - M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

13 Interval Finite Element Analsis of hin Plates Figure 6. Clamped plate- variation of at node 7 w.r.t. uncertaint of E. Figure 7. Clamped plate- variation of at center of plate (at node ) w.r.t. uncertaint of E. REC 6 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

14 4 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen Figure 8. Clamped plate- variation of at center of plate (at node ) w.r.t. uncertaint of E. Results are computed for the 44 plate for the following cases: A) Load uncertaint of percent (±5 percent variation of load about its mean value) alone is present B) percent uncertaint of E(±.5 percent variation of E about its mean value) alone C) Load uncertaint of percent along with percent uncertaint of E. ables, and 4 present the results of selected displacements and rotations corresponding to cases A, B and C respectivel. Similarl, ables 5, 6 and 7 present the results of moments at the center of the plate corresponding to cases A, B and C respectivel. It is observed from the ables and 5 that the results of the interval solution coincide with those obtained using combinatorial approach and thus provide eact bounds to the combinatorial solution. It is observed from ables and 6 that the interval solution gives sharp bounds to the values of selected displacements and rotations w.r.t the corresponding values obtained using combinatorial solution. As alread mentioned, these solutions required computation of 6 =6556 combinations. ables 4 and 7 show the results computed for percent load uncertaint along with percent uncertaint of Young s modulus E. It is impractical to compute combinatorial solution in this case, as it would require computation of solution for 6 6 =4,94,967,96 combinations. Instead, the bounds of the solution are computed using MCS (Monte- Carlo Simulations). he results computed in ables 4 and 7 are for, simulations. It is to be noted here that the results obtained using MCS provide inner bounds to the combinatorial solution whereas the results obtained using interval solution provide outer bounds to the combinatorial solution. REC 6 - M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

15 5 Interval Finite Element Analsis of hin Plates able. Clamped rectangular plate (44) selected displacements and rotations of the plate for % uncertaint of load (Case-A). w (m) (radians) at node 7 (radians) at node Lower Upper Lower Upper Lower Upper Combinatorial Interval Error% able. Clamped rectangular plate(44)- selected displacements and rotations of the plate for % uncertaint of E (Case-B). w (m) (radians) at node 7 (radians) at node Lower Upper Lower Upper Lower Upper Combinatorial Interval Error% able 4. Clamped rectangular plate(44)- selected displacements and rotations of the plate for % uncertaint of load and % uncertaint of E (Case C). w (m) (radians) at node 7 (radians) at node Lower Upper Lower Upper Lower Upper MCS Interval Error% able 5. Clamped rectangular plate (44)- moments at the center of the plate for % uncertaint of load (Case-A). M (kn) at node M (kn) at node Lower Upper Lower Upper Comb Interval Error%.... able 6. Clamped rectangular plate (44)- moments at the center of the plate for % uncertaint of E (Case-B). M (kn) at node (kn) at node Lower Upper Lower Upper Comb Interval Error% M REC 6 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

16 6 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen able 7. Clamped rectangular plate (44)- moments at the center of the plate for % uncertaint of load and % uncertaint of E (Case-C). M (kn) at node (kn) at node Lower Upper Lower Upper MCS Interval Error% M w z along the Figure 9 presents the variation of the lower and upper bounds of the interval displacement length and width of the plate respectivel. Figure presents the variation of the lower and upper bounds of the slope along the width of the plate. Figure presents the variation of the lower and upper bounds of the slope along the length of the plate. Figure 9. Clamped plate- variation of vertical displacement along the length of the plate with % uncertaint of load and % uncertaint of E. REC 6 - M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

17 7 Interval Finite Element Analsis of hin Plates Figure. Clamped plate- variation of along the width of the plate with % uncertaint of load and % uncertaint of E. Figure. Clamped plate- variation of along the length of the plate with % uncertaint of load and % uncertaint of E. REC 6 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

18 8 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen ables 8, 9 and present the values of selected displacements and rotations for a discretization scheme of (4 elements) for cases A, B and C respectivel. Similarl, ables, and present the values of moments at the center of the plate for cases A, B and C respectivel. It is to be noted here that the computation of combinatorial solution for cases A or B would require 4 combinations while it would require computation of 8 combinations for case C. hus it is practicall impossible for to compute the combinatorial solution. Instead, the results of the bounds obtained using Monte Carlo simulations are presented. It is here to be noted that the bounds of the interval solution enclose the corresponding bounds of the combinatorial solution from outside. On the other hand, the bounds of the results computed using Monte Carlo solution enclose the corresponding bounds of combinatorial solution from inside. It is to be further noted that the percentage error reported in ables 5 through can be reduced b increasing the number of simulations. However it is observed that it is computationall time consuming. able 8. Clamped rectangular plate () displacements at the center of the plate for % uncertaint of load (Case-A). w (m) (radians) at node (radians) at node 6 Lower Upper Lower Upper Lower Upper MCS Interval Error% able 9. Clamped rectangular plate () displacements center of the plate for % uncertaint of E (Case-B). w (m) 4 (radians) at node (radians) at node 6 Lower Upper Lower Upper Lower Upper MCS Interval Error% able. Clamped rectangular plate () displacements at the center of the plate for % uncertaint of load and % uncertaint of E (Case-C). w (m) (radians) at node (radians) at node 6 Lower Upper Lower Upper Lower Upper MCS Interval Error% able. Clamped rectangular plate () moments center of the plate for % uncertaint of load (Case-A). M (kn) at node M (kn) at node Lower Upper Lower Upper MCS Interval Error% REC 6 - M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

19 9 Interval Finite Element Analsis of hin Plates able. Clamped rectangular plate () moments at the center of the plate for % uncertaint of E (Case-B). M (kn) at node M (kn) at node Lower Upper Lower Upper MCS Interval Error% able. Clamped rectangular plate () moments at the center of the plate for % uncertaint of load and % uncertaint of E (Case-C). M (kn) at node M (kn) at node Lower Upper Lower Upper MCS Interval Error% Conclusion A linear Interval Finite Element (IFEM) for structural analsis of thin plates is presented. Uncertaint in the applied load and Young s modulus is represented as interval numbers. Results are also computed using combinatorial solution and Monte Carlo simulations as appropriate. Eample problems illustrate the applicabilit of the present approach to the problem of predicting the structural behavior of thin plates in the presence of uncertainties. Acknowledgements he first author would like to gratefull acknowledge the help received via international travel funding b the echnical Education Qualit Improvement Program (EQIP- Phase II) of Government of India and its sanction b the administration of Vasavi College of Engineering, Hderabad, India. References Bathe, K. Finite Element Procedures, Prentice Hall, Englewood Cliffs, New Jerse 76, New Jerse, 996. Batista, M. Uniforml Loaded Rectangular hin Plates with Smmetrical Boundar Conditions. Cornel Universit Librar, USA, Dessombz, O., F. houverez, J.-P. Laîné, and L. Jézéquel. Analsis of mechanical sstems using interval computations applied to finite elements methods. J. Sound. Vib., 8(5): ,. Gallagher, R. H. Finite Element Analsis Fundamentals, Prentice Hall, Englewood Cliffs, USA, 975. Knüppel,O. PROFIL / BIAS A Fast Interval Librar. Computing, 5:77 87, 994. Kölüoglu, U., S. Cakmak, N. Ahmet and R. K. Soren. Interval Algebra to Deal with Pattern Loading and Structural Uncertaint. Journal of Engineering Mechanics (): 49 57, 995. Lim, C.W., S. Cui, W. A. Yao. On new smplectic elasticit approach for eact bending solutions of rectangular thin plates with two opposite sides simpl supported, International Journal of Solids and Structures, Volume 44(6):596-54, 7. Moore, R., E. Interval Analsis, Prentice-Hall, Englewood Cliffs, N.J., 966. Moore, R. E. s and applications of interval analsis, SIAM, Philadelphia, 979. REC 6 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen

20 M. V. Rama Rao, R. L. Muhanna and R. L. Mullen Moore, R. E., R. B. Kearfott, M. J. Cloud. Introduction to Interval analsis, SIAM, 9. Muhanna, R. L. and R. L. Mullen. Development of Interval Based s for Fuzziness in Continuum Mechanics. In Proceedings of ISUMA-NAFIPS 95: 7- September, IEEE, 995. Muhanna, R. L. and R. L. Mullen. Uncertaint in Mechanics Problems Interval-Based Approach, Journal of Engineering Mechanics 7(6): ,. Muhanna, R. L., H. Zhang and R. L. Mullen. Interval finite element as a basis for generalized models of uncertaint in engineering mechanics. Reliable Computing, ():7 94, 7. Nakagiri, S. and N. Yoshikawa. Finite Element Interval Estimation b Conve Model. In Proceedings of 7th ASCE EMD/SD Joint Specialt Conference on Probabilistic Mechanics and Structural Reliabilit, WPI, MA, August, 996. Neumaier, A. Interval methods for sstems of equations, Cambridge Universit Press, 99. Neumaier, A. and A. Pownuk. Linear Sstems with Large Uncertainties, with Applications to russ Structures, Reliable Computing, ():49-7, 7. Pownuk, A. Efficient method of solution of large scale engineering problems with interval parameters." Proc. NSF workshop on reliable engineering computing (REC4), R. L. Muhanna and R. L. Mullen, eds., Savannah, GA, USA, 4. Rama Rao, M. V., R. L. Mullen, R. L. Muhanna. A New Interval Finite Element Formulation with the Same Accurac in Primar and Derived Variables, International Journal of Reliabilit and Safet, Vol. 5, Nos. ¾,. Rao, S. S. and P. Sawer. Fuzz finite element approach for analsis of imprecisel defined sstems. AIAA J., ():64 7, 995. Rao, S. S. and L. Berke. Analsis of uncertain structural sstems using interval analsis. AIAA J., 5(4):77 75, 997. Rao, S. S. and Li Chen. Numerical solution of fuzz linear equations in engineering analsis. International Journal of Numerical s in Engineering.,4:9 48, 998. Redd, J.N. heor and Analsis of Elastic Plates and Shells. Second edition. CRC Press. alor and Francis group, Boca Raton, London and New York, 7. Rump, S.M. INLAB - INerval LABorator. In ibor Csendes, editor, Developments in Reliable Computing, pages Kluwer Academic Publishers, Dordrecht, 999. Sun microsstems. Interval arithmetic in high performance technical computing. Sun microsstems. (A White Paper),. Szilard, R. heories and Applications of Plate Analsis: Classical Numerical and Engineering s, 4. Wile imoshenko, S. and S. Woinowsk-Krieger. heor of Plates and Shells. Second edition. McGraw Hill, 959. Xiao, N., F. Fedele and R. L. Muhanna. Inverse Problems Under Uncertainties-An Interval Solution for the Beam Finite Element, in Deodatis, B. R. Ellingwood, D. M. Frangopol, Icossar, ISBN: ,. Zhang, H. Nondeterministic Linear Static Finite Element Analsis: An Interval Approach. Ph.D. Dissertation, Georgia Institute of echnolog, School of Civil and Environmental Engineering, 5. REC 6 - M. V. Rama Rao, R. L. Muhanna and R. L. Mullen