APPLICATION OF POLYGONAL FINITE ELEMENTS IN LINEAR ELASTICITY

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1 International Journal of Computational Metod c World Scientific Publiing Company APPLICATION OF POLYGONAL FINITE ELEMENTS IN LINEAR ELASTICITY A. TABARRAEI Department of Civil and Environmental Engineering, Univerity of California, Davi, CA 9566, USA. atabarraei@ucdavi.edu N. SUKUMAR Department of Civil and Environmental Engineering, Univerity of California, Davi, CA 9566, USA. nukumar@ucdavi.edu Received (Day Mont Year) Revied (Day Mont Year) In ti paper, a conforming polygonal finite element metod i applied to problem in linear elaticity. Mefree natural neigbor (Laplace) ape function are ued to contruct conforming interpolating function on any convex polygon. Ti provide greater flexibility to olve partial differential equation on complicated geometrie. Cloed-form expreion for Laplace ape function on pentagonal, exagonal, eptagonal, and octagonal reference element are derived. Numerical example are preented to demontrate te accuracy of te metod in two-dimenional elatotatic. Keyword: barycentric coordinate; mefree metod; natural neigbor; Laplace interpolant.. Introduction Te finite element metod i a powerful numerical tool for olving partial differential equation of matematical pyic. Te ue of triangular and quadrilateral element are well-etablied in conventional finite element metod. However, tere i ignificant bottleneck in generating quality mee uing tree- and four-ided element for complex geometrie. Te ue of element wit a large number of ide will provide greater flexibility and better accuracy to olve problem tat arie in olid mecanic and biomecanic. Since material microtructure in polycrytalline alloy and piezoelectric, and bone can be decribed troug polygonal ub-domain, te ue of polygonal finite element in uc application i a natural coice. Wacpre [Wacpre (975)] propoed te contruction of bai function on convex Correponding autor. Department of Civil and Environmental Engineering, Univerity of California, One Sield Avenue, Davi, CA 9566, USA.

2 A. Tabarraei and N. Sukumar polygon (n-gon) wit any number of ide. Wacpre rational bai function are defined a polynomial of n divided by polynomial of n 3. Recently, ignificant advance ave been acieved in te contruction of barycentric coordinate on irregular polygon [Meyer et al. (00); Dagupta (003); Floater (003); Malc and Dagupta (004); Sukumar and Tabarraei (004)]. A implified expreion for Wacpre bai function i preented by Meyer et al. [00], and Floater [003] derived barycentric coordinate in wic a vertex in a planar triangulation i expreed a a convex combination of it neigboring vertice. In Sukumar and Tabarraei [004], natural neigbor (Laplace) interpolation [Crit et al. (98)] i ued to contruct C 0 (Ω) ape function on polygonal element and ti procedure a been ued for -adaptive refinement on quadtree mee [Tabarraei and Sukumar (005)]. Te contruction of polygonal approximant uing te principle of maximum entropy i decribed in Sukumar [004]. In ti paper, Laplace interpolant are contructed on reference polygonal element, and troug an ioparametric map, ape function on pyical element are contructed. A own in Sukumar et al. [00], Laplace interpolant are linear on element boundarie and can exactly reproduce contant and linear field. Ti permit te direct impoition of eential boundary condition in polygonal finite element [Sukumar and Tabarraei (004)]. Te outline of ti paper i a follow. Te contruction of Laplace interpolant on irregular convex polygon i preented in Section. Te model problem i decribed in Section 3, and numerical example (patc tet, beam bending, edge crack) are preented in Section 4. Finally, ome concluding remark are mentioned in Section 5.. Conforming Polygonal Sape Function In ti ection, we review te contruction of conforming interpolant on irregular polygon. Te intereted reader can ee Sukumar and Tabarraei [004] for a detailed decription of te metod. Conider a domain Ω tat i dicretized by a et of node. If a point p i inerted inide te domain, te Voronoi cell of point p i te locu of all point tat are cloer to p tan to any oter node inide te domain. Te Delaunay teellation can be obtained by connecting toe node tat ave a common edge. Te Voronoi cell, Delaunay triangle and Delaunay circumcircle are own in Fig.. If te point p lie inide te circumcircle of a Delaunay triangulation, all toe node tat form te Delaunay triangle are conidered a te natural neigbor of point p [Sibon (980)]. For example in Fig. b, node,, 5 and 6 are natural neigbor of p. A in claical finite element, to contruct an interpolant on te pyical element, firt te interpolant i defined on a reference element. Polygonal reference element (pentagon, exagon, eptagon, and octagon) in te ξ-coordinate ytem are depicted in Fig.. From Fig., we oberve tat all vertex node lie on te circumcircle of te reference element. Ti property enure tat all te node of te reference element are natural neigbor of any point witin te polygon. Referring

3 Application of Polygonal Finite Element in Linear Elaticity 3 (a) (b) Fig.. (a)voronoi cell and Delaunay triangle; and (b) Delaunay triangle and Delaunay circumcircle. to Fig. 3, te Laplace ape function at point p are defined a φ i (ξ) = α i(ξ) n, α j (ξ) = j(ξ) α j (ξ) j (ξ), ξ Ω 0, () j= were α i (ξ) i te Laplace weigt function, i (ξ) i te lengt of te common edge between Voronoi cell of point p and node i, and i (ξ) i te Euclidean ditance between point p and node i. Te Laplace ape function atify te following propertie [Crit et al. (98)]: () Non-negative, interpolate, and form a partition of unity: n 0 φ i (ξ), φ i (ξ j ) = δ ij, φ i (ξ) =, () were δ ij i te Kronecker-delta. () Linear completene [Huge (987)]: n x(ξ) = φ i (ξ)x i, (3) i= wic in conjunction wit Eq. () implie tat te Laplace interpolant can exactly reproduce arbitrary contant and linear function. To obtain te polygonal ape function and teir derivative on te pyical element, we ue te ioparametric tranformation given in Eq. (3). Since te Laplace interpolant i linear on any edge of te reference element (only two ape function are non-zero), te ioparametric mapping preerve ti property on te pyical i=

4 4 A. Tabarraei and N. Sukumar ξ ξ Ω 0 ξ Ω 0 ξ (a) (b) ξ ξ Ω 0 ξ Ω 0 ξ (c) (d) Fig.. Reference element. (a) Pentagon; (b) Hexagon; (c) Heptagon; and (d) Octagon. element, wic lead to C 0 (Ω) conformity of te interpolant (Fig. 4). It wa own in Sukumar et al. [00] tat Laplace ape function are identical to barycentric (area coordinate) ape function on te triangle and to bilinear finite element ape function on te bi-unit quare. Hence, te extenion of exiting finite element program to polygonal finite element code i traigtforward. Te Laplace interpolant reproduce Wacpre interpolant on regular polygon, and te Laplace weigt function, α i (ξ), at any point inide te reference element i given by [Sukumar and Tabarraei (004)]: α i (ξ) = ( ξ ξ ) in 3 π n co π n A i A i. (4) In te above equation, A i i te area of te triangle [p, p i, p i+ ] and A i i te area of triangle [p, p i, p i ] (Fig. 5). On uing Eq. (4), cloed-form expreion for ape

5 Application of Polygonal Finite Element in Linear Elaticity p p (a) (b) Fig. 3. Voronoi cell of point p and contruction of Laplace ape function. (a) Pentagonal reference element; and (b) Hexagonal reference element. ξ φ Ω 0 ξ Ω e x Fig. 4. Ioparametric mapping. x function on regular n-gon can be obtained by employing a ymbolic program uc a Matematica TM. Te ape function for a regular pentagon, exagon, eptagon, and octagon are preented in te Appendix. Extenion of numerical quadrature rule from triangular and quadrilateral element to convex polygon wit more tan four edge i a non-trivial tak. No general quadrature rule i available for te purpoe of integrating function (polynomial or rational) on polygonal domain. In ti paper, te Gauian quadrature rule for triangular element i ued to evaluate weak form integral on polygonal element. For te purpoe of numerical integration, te polygonal reference element i partitioned

6 6 A. Tabarraei and N. Sukumar p i+ A i p p i A i- p i- Fig. 5. Equivalence of Laplace and Wacpre ape function on a regular n-gon (n = 8). into n iocele ub-triangle. An affine map i ued from te triangular reference element to te ub-triangle of te polygonal reference element to find te location of Gau point inide te reference element. To find te location of te Gau point in te pyical element, te ioparametric map from te reference element to te pyical element i ued. Te contribution from eac ub-triangle are added up to obtain te integral value over te polygonal element. Ti procedure i illutrated in Fig. 6 and can be expreed a [Sukumar and Tabarraei (004)] f dω = Ω e Ω 0 f J dω = n j= Ω j 0 f J dω = n j= 0 ξ 0 f J j J dξdη. (5) 3. Governing Equation and Weak Form in Linear Elaticity Te equilibrium equation of linear elatotatic in te abence of body force i: wit boundary condition σ = 0 in Ω (6) u = ū on Γ u (diplacement boundary) (7a) σ n = t on Γ t (traction boundary) (7b) were Γ = Γ t Γ u and Γ u Γ t =. In Eq. (7), u i te diplacement vector and σ i te tre tenor. In te finite element metod, diplacement are approximated by u = i N i u i = Nu, (8)

7 Application of Polygonal Finite Element in Linear Elaticity 7 η p ξ ξ N FEM φ Ω 0 p 0 ξ p x Ωe x Fig. 6. Numerical integration ceme baed on te partition of te canonical element. were N i i te finite element ape function for node i and u i i te diplacement at te it node. Te finite element approximation for te train tenor i defined a ε = Bu, (9) were B i te element train-diplacement operator. Uing te contitutive law, te tre matrix can be defined a σ = Dε = (DB)u, (0) were D i te contitutive matrix. Te variational form (principle of virtual work) of te equilibrium equation i: δε T σ δu T tdγ = 0 δu i H0 (Ω), () Γ t Ω were δ denote te variation operator, and H 0 (Ω) i te Sobolev pace of function wit quare-integrable derivative up to order one and vaniing value on Γ u. On uing a tandard Galerkin procedure, we obtain te dicrete equation: Ku = f, K = B T DBdΩ, Ω f = N T tdγ. Γ t (a) (b) (c)

8 8 A. Tabarraei and N. Sukumar For an iotropic material in plane tre condition, te tre-train relationip i defined a σ x σ y = E ν 0 ε x ν σ ν 0 xy 0 0 ν ε y, (3) ε xy were ν i te Poion ratio and E i te Young modulu. Te contitutive equation of plane train cae can be obtained from plane tre equation by replacing E E by ν and ν by ν ν. 4. Numerical Reult We preent four example to ae te performance of te polygonal finite element metod. For te purpoe of error etimation and convergence tudie, te L norm and energy norm of te diplacement error are employed. In our analye, te L norm and energy norm of te diplacement error are defined a ( u u [ L = (u u ) (u u ) ] ) dω, (4a) u u E(Ω) = Ω a(u u, u u ), a(u, u) = Ω ε T Dε dv, (4b) were u and u are te exact and numerical olution, repectively. To perform numerical integration on n-gon wit n > 4, te procedure decribed in Section i employed wit 5 Gau point in eac ub-triangle. On tree-noded triangular element, one point quadrature i adopted and on four-noded bilinear element, Gau-Legendre quadrature rule i ued. 4.. Diplacement patc tet Firt, te ability of polygonal finite element to repreent linear diplacement field i tudied. Te domain i te unit quare and two different et of boundary condition are conidered. For te firt tet, u = x + x i applied on te boundary of te domain; te econd tet i performed by applying u = x and u = x on te boundary of te domain. In Fig. 7, tree different mee tat are conidered in ti analyi are own. Te L norm and energy norm of diplacement error i own in Table, and te reult reveal tat te patc tet i paed to O(0 8 ) and O(0 7 ) accuracy in L and energy norm, repectively. 4.. Equilibrium patc tet Te ability to repreent a uniaxial plane tre field i verified by te equilibrium patc tet. Conider a uniaxial tre σ = pi (plane tre condition are aumed)

9 Application of Polygonal Finite Element in Linear Elaticity 9 Table. Relative error in te L norm and energy norm for te diplacement patc tet. Relative error in te L norm Relative error in te energy norm Mee Number u = x + x u = x u = x + x u = x of node u = x + x u = x u = x + x u = x a b c d (a) (b) (c) (d) Fig. 7. Patc tet on polygonal mee. (a) Me a ( node); (b) Me b (0 node); (c) Me c (00 node) and (d) Me d (5 node). in te x -direction acting on te top edge of te unit quare own in (Fig. 8). Te diplacement of te bottom edge i fixed in te x direction. Te eential boundary condition are indicated in Fig. 8. Te exact diplacement olution i [Sukumar et al. (998)]: u (x, x ) = ν E ( x ), (5a) u (x, x ) = x E. (5b) Te mee own in Fig. 7 are conidered in te analye. Te relative error in te L norm and te energy norm are preented in Table. Relative error of O(0 5 ) in L norm and energy norm are obtained for ti problem Cantilever beam In Fig. 9, a cantilever beam ubjected to a parabolic end load i illutrated. Te beam a lengt L, eigt D, and unit widt. Te top and bottom edge of te beam are traction free. Te left edge i fixed in te x and x direction. Te diplacement

10 0 A. Tabarraei and N. Sukumar Table. Relative error in te L norm and energy norm for te equilibrium patc tet. Mee Number Relative error in Relative error in of Node te L norm te energy norm a b c d x σ = pi in. in. x Fig. 8. Equilibrium (uniaxial tenion) patc tet [Sukumar et al. (998)]. vector olution i given by [Timoenko and Goodier (987)] u (x, x ) = P x 6ĒI u (x, x ) = P 6ĒI [ (6L 3x )x + ( + ν) )] (x D, (6a) 4 ] [ 3 νx (L x ) + (4 + 5 ν) D x + (3L x )x 4, (6b)

11 Application of Polygonal Finite Element in Linear Elaticity x Γ u P D x L Fig. 9. Cantilever beam model [Sukumar et al. (998)]. were E (plane tre), Ē = E ν (plane train), ν (plane tre), ν = ν (plane train). ν (7a) (7b) Te tree are given by σ (x, x ) = P (L x )x, I (8a) σ (x, x ) = 0, (8b) σ (x, x ) = P I (D 4 x ), (8c) were I i te moment of inertia, wic for a beam wit rectangular cro-ection and unit tickne i: I = D3. (8d) In te finite element computation, te following parameter are ued: P = 000 lb, D = in., L = 8 in., E = 0 5 pi, ν = 0.3 and plane train condition are aumed. To compare te reult, we ue quadrilateral and polygonal mee to olve te problem. Te polygonal mee are own in Fig. 0. Te quadrilateral mee are compoed of 80 (30 6) and 890 (05 8) element. Te exact end deflection u (L, 0) = in. In Table 3 and 4, te numerically computed end deflection i compared wit tat obtained from engineering beam teory. Te numerical reult uing polygonal mee are in good agreement wit beam teory prediction.

12 A. Tabarraei and N. Sukumar (a) (b) Fig. 0. Polygonal mee of cantilever beam. (a) Me a (0 node); and (b) Me b (00 node) Stre intenity factor computation for an edge crack Stre intenity factor (SIF) computation for an edge cracked pecimen are preented. To compute te tre intenity factor, te domain form of te interaction integral i ued [Yau et al. (980); Moran and Si (987)]. Two et of boundary condition are conidered. A te firt problem, a crack of lengt a in a emi-infinite elatic plate ubjected to uniform loading i conidered. Te loading can be decom- Table 3. Deflection of te cantilever beam uing polygonal element. Mee Number of Number of Normalized end element node deflection a b Table 4. Deflection of te cantilever beam uing quadrilateral element. Mee Number of Number of Normalized end element node deflection a b

13 Application of Polygonal Finite Element in Linear Elaticity 3 Near tip diplacement field a = W r Crack W (a) (b) (c) Fig.. Te emi-infinite edge crack problem. (a) Interaction integral domain and boundary condition; (b) Me a (00 node); and (c) Me b (895 node). poed into an opening (K I ) and a earing mode (K II ). To calculate te tre intenity factor, a mall region near te crack tip i conidered. Te exact near-tip diplacement field i impoed on te boundary, wit K I = and K II =. To verify te domain independence of te computation, te SIF are evaluated for tree different contour. Te contour interaction integral domain and te boundary condition are own in Fig. a, and polygonal dicretization of te domain are own in Fig. b and c. Te problem parameter are: W = 7.0 in. and = 8.0 in.. Te numerical reult of tre intenity factor are own in Table 5. Te SIF reult are domain independent and accurate. For te econd problem, a finite-dimenional plate ubjected to uniform tenion on te top urface under plane tre i conidered. Te geometry and boundary condition of te problem are own in Fig.. Te following parameter are coen: W = 7.0 in. and = 6.0 in.. Due to ymmetry, alf of te domain i analyzed. For te boundary condition own in Fig., te exact normalized tre intenity factor K ref I =.864 [Tada et al. (000)]. σ πa Te polygonal mee of Fig. are ued in ti analyi, and te computed tre intenity factor are preented in Table 6. Te numerical reult are witin two percent of te reference olution. 5. Concluion In ti paper, we reviewed te contruction of Laplace ape function on irregular convex polygon. We owed tat te Laplace interpolant i a generalization of tree- and four-noded element to convex polygon wit any number of vertice. Due to te appealing propertie of te Laplace interpolant, te extenion of claical finite element code to polygonal finite element i imple and direct. Cloed-form expreion for ape function on pentagonal, exagonal, eptagonal, and octag-

14 4 A. Tabarraei and N. Sukumar σ = pi a = W σ = pi a = W r W Crack W (a) (b) Fig.. Te finite dimenional edge crack problem. (a) Domain and boundary condition; and (b) Interaction integral domain and model ued in te numerical computation. onal reference element were preented. In te numerical integration of te weak form, a iger-order Gauian quadrature rule wa ued. Te polygonal finite element metod paed te diplacement patc tet to O(0 8 ) in te L norm and O(0 7 ) in energy norm. Error in te L norm and energy norm of O(0 5 ) wa obtained for te equilibrium patc tet. Macine preciion accuracy i not realized in te patc tet computation due to te error introduced in te numerical integration of te weak form integral. Accurate numerical reult for te cantilever Table 5. Normalized SIF: Semi-infinite edge crack problem. Me a Me b r K I K II K I K II 0.a a a

15 Application of Polygonal Finite Element in Linear Elaticity 5 beam and edge cracked plate problem were obtained. 6. Acknowledgment Te financial upport of ti work by te National Science Foundation troug reearc award CMS to te Univerity of California, Davi, i gratefully acknowledged. Reference Crit, N. H., Friedberg, R. and Lee, T. D. (98). Weigt of link and plaquette in a random lattice. Nuclear Pyic B., 0: Dagupta, G. (003). Interpolant witin convex polygon: Wacpre ape function. Journal of Aeropace Engineering, 6: 8. Floater, M. S. (003). Mean value coordinate. Computer Aided Geometric Deign, 0: 9 7. Huge, T. J. R. (987). Te Finite Element Metod, Prentice-Hall, Prentice-Hall. Malc, E. A. and Dagupta, G. (004). Interpolation for temperature ditribution: A metod for all non-concave polygon. International Journal of Solid and Structure, 4: Meyer, M., Lee H., Barr, A. H. and Debrun, M. (00). Generalized barycentric coordinate for irregular polygon. Journal of Grapic Tool, 7: 3. Moran, B. and Si, C. F. (987). Crack tip and aociated domain integral from momentum and energy balance. Engineering Fracture Mecanic, 7: Sibon, R. (980). A vector identity for te Diriclet teelation. Matematical Proceeding of te Cambridge Piloopical Society, 87: Sukumar, N., Moran, B., Semenov, A. Yu and Belikov, V. V. (00). Natural neigbor Galerkin metod. International Journal for Numerical Metod in Engineering, 50: 7. Sukumar, N. (004). Contruction of polygonal interpolant: A maximum entropy approac. International Journal for Numerical Metod in Engineering, : Sukumar, N., Moran, B. and Belytcko, T. (998). Te natural element metod in olid mecanic. International Journal for Numerical Metod in Engineering, 43: Sukumar, N. and Tabarraei, A. (004). Conforming polygonal finite element. International Journal for Numerical Metod in Engineering, 6: Tabarraei, A. and Sukumar, N. (005). Adaptive computation on conforming quadtree mee. Finite element in Analyi and Deign, 4: Table 6. Normalized SIF: Finite-dimenional edge cracked plate under tenion. r Me a ( ) K ref I σ %Error πa Me b ( ) K ref I σ %Error πa 0.a a a

16 6 A. Tabarraei and N. Sukumar Tada H., Pari, P. C. and Irwin, G. R. (000). Te Stre Analyi of Crack Handbook, ASME Pre, New York, N.Y. Timoenko, S. P. and Goodier, J. N. (987). Teory of Elaticity, McGraw-Hill, New York, N.Y. Wacpre, E. L. (975). A Rational Finite Element Bai, Academic Pre, New York. Yau J., Wang, S. and Corten, H. (980). A mixed-mode crack analyi of iotropic olid uing conervation law of elaticity. Journal of Applied Mecanic, 47: Appendix A. Appendice Sape function on regular convex n-gon are derived. Te nodal coordinate of a regular n-gon are: ξ m = co πm n, ξm = in πm n (m =,,..., n). A.. Pentagonal reference element (n = 5) N i (ξ) = a i(ξ) b(ξ) (i =,,..., 5), were b(ξ) = ξ.7004 ξ, and a (ξ) = ( ξ ) ( ξ.76393ξ ) ( ξ ξ ), a (ξ) = ( ξ.76393ξ ) ( ξ.76393ξ ) ( ξ ξ ), a 3 (ξ) = ( ξ ξ ) ( ξ.76393ξ ) ( ξ.76393ξ ), a 4 (ξ) = ( ξ ) ( ξ ξ ) ( ξ.76393ξ ), a 5 (ξ) = ( ξ ) ( ξ ξ ) ( ξ ξ ).

17 Application of Polygonal Finite Element in Linear Elaticity 7 A.. Hexagonal reference element (n = 6) ( 3ξ ξ 3 ) ( ξ + 3 ) ( 3 ξ + 6ξ ξ + 3 ) N (ξ) = 8 (ξ + ξ 3), ( 3ξ + ξ + 3 ) ( ξ + 3 ) ( 3 ξ 6ξ ξ + 3 ) N (ξ) = 8 (ξ + ξ 3), ( 3ξ N 3 (ξ) = 6ξ ξ + 3 ) ( 4ξ 3 ) 8 (ξ + ξ 3), ( ) ( 3 ξ 3ξ ξ + 3 ) ( 3 ξ 6ξ ξ + 3 ) N 4 (ξ) = 8 (ξ + ξ 3), ( ) ( 3 ξ 3ξ + ξ 3 ) ( 3 ξ + 6ξ ξ + 3 ) N 5 (ξ) = 8 (ξ + ξ 3), ( 3ξ N 6 (ξ) = + 6ξ ξ + 3 ) ( 4ξ 3 ) 8 (ξ + ξ 3). A.3. Heptagonal reference element (n = 7) N (ξ) = ( ξ ξ )( ξ ξ )H, N (ξ) = ( ξ ξ )( ξ ξ )H, N 3 (ξ) = ( ξ )( ξ ξ )H, N 4 (ξ) = ( ξ )( ξ ξ )H, N 5 (ξ) = ( ξ ξ )( ξ ξ )H, N 6 (ξ) = ( ξ ξ )( ξ ξ )H, N 7 (ξ) = ( ξ ξ )( ξ ξ )H, were H = AB CD EF BD.376 EG CG AF,

18 8 A. Tabarraei and N. Sukumar and A = ξ ξ, B = ξ ξ, C = ξ ξ, D = ξ + ξ, E = ξ ξ, F = ξ + ξ, G = ξ. A.4. Octagonal reference element (n = 8) were N i (ξ) = a i(ξ), (i =,,..., 8), b(ξ) b(ξ) = 64 ( + 5ξ 0ξ 4 + 5ξ 0ξ ξ 0ξ 4 + ( ξ + ξ ) ( 3 + 7ξ + 7ξ ) ), and a (ξ) = ( ξ + A) (ξ + B) ( ξ + C) ( ξ + C) ( ξ + D) ( ξ + D), a (ξ) = ( ξ + A) (ξ + A) (ξ + B) ( ξ + C) ( ξ + D) ( ξ + D), a 3 (ξ) = ( ξ + A) ( ξ + B) (ξ + A) (ξ + B) ( ξ + C) ( ξ + D), a 4 (ξ) = ( ξ + A) ( ξ + B) (ξ + A) ( ξ + C) ( ξ + C) ( ξ + D), a 5 (ξ) = ( ξ + B) (ξ + A) ( ξ + C) ( ξ + C) ( ξ + D) ( ξ + D), a 6 (ξ) = ( ξ + B) (ξ + A) (ξ + B) ( ξ + C) ( ξ + C) ( ξ + D), a 7 (ξ) = ( ξ + A) ( ξ + B) (ξ + A) (ξ + B) ( ξ + C) ( ξ + D), a 8 (ξ) = ( ξ + A) ( ξ + B) (ξ + B) ( ξ + C) ( ξ + D) ( ξ + D), were A = ( + ξ ξ ), B = ( + ξ ξ ), C = ( + ξ + ξ ), D = ( + ξ + ξ ).

E. Stanová. Key words: wire rope, strand of a rope, oval strand, geometrical model

E. Stanová. Key words: wire rope, strand of a rope, oval strand, geometrical model Te International Journal of TRANSPORT & LOGISTICS Medzinárodný čaopi DOPRAVA A LOGISTIKA ISSN 45-7X GEOMETRY OF OVAL STRAND CREATED OF n n n WIRES E. Stanová Abtract: Te paper deal wit te matematical geometric