František Kardoš
email: frantisek.kardos@upjs.sk

Fulleroids with Dihedral Symmetry

Keywords: Fullerene graph, Fulleroid, Symmetry group, Dihedral symmetry

Fulleroids are cubic convex polyhedra with faces of size 5 or greater. They are suitable as models of hypothetical all-carbon molecules. In this paper sufficient and necessary conditions for existence of fulleroids of dihedral symmetry types and with pentagonal and n-gonal faces only depending on number n are presented. Either infinite series of examples are found to prove existence, or nonexistence is proved using symmetry invariants.

Introduction

Fullerenes have been objects of interest and study in the past two decades. A fullerene is a 3-valent carbon molecule, where atoms are arranged in pentagons and hexagons. It can be seen as a convex polyhedron, where vertices represent atoms and edges represent bonds between atoms. Fullerenes can also be represented by graphs. In fact, a fullerene graph is a planar, cubic (i.e. 3-regular) and 3-connected graph, twelve of whose faces are pentagons and the remaining faces are hexagons.

The concept of fullerenes can be generalized in several ways. Fowler [START_REF] Fowler | An Atlas of Fullerenes[END_REF] asked whether a fullerene-like structure consisting of pentagons and heptagons only and exhibiting an icosahedral symmetry exists. The answer was given by Dress and Brinkmann [START_REF] Brinkmann | Phantasmagorical Fulleroids[END_REF]. Motivated by these examples Delgado Friedrichs and Deza [START_REF] Delgado Friedrichs | More Icosahedral Fulleroids[END_REF] introduced the following definition: Definition 1 A fulleroid is a convex polyhedron such that all its vertices have degree 3 while all its faces have degree 5 or larger. A Γ-fulleroid is a fulleroid on which the group Γ acts as a group of symmetries. A given Γ-fulleroid is of type (a, b) or a Γ(a, b)-fulleroid if all its faces are either a-gonal or b-gonal.

The set of all Γ(a, b)-fulleroids will be denoted simply by Γ(a, b).

There is a list of groups, that can act as a symmetry group of a convex polyhedron [START_REF] Cromwell | Polyhedra[END_REF]. According to the system of rotational symmetry axes they can be divided into icosahedral, octahedral, tetrahedral, dihedral, cyclic and others.

Symmetry of fullerenes has been studied deeply. The possible symmetry groups Γ for fullerenes were shown to be limited to a total of 28 point groups [START_REF] Fowler | Possible Symmetries of Fullerene Structures[END_REF]. Babić, Klein and Sah [START_REF] Babić | Symmetry of fullerenes[END_REF] divided all fullerenes with up to 70 vertices according to the symmetry group. Fowler and Manolopoulos [START_REF] Fowler | An Atlas of Fullerenes[END_REF] found symmetry groups of all fullerenes with up to 100 vertices. For each symmetry group Γ they found the smallest Γ-fullerene and the smallest Γ-fullerene obeying IPR (isolated pentagon rule). They described how to create a new fullerene with the same symmetry group having more hexagonal faces and obeying IPR once a fullerene is given. Graver [START_REF] Graver | Catalog of All Fullerenes with Ten or More Symmetries[END_REF] published a catalogue of all fullerenes with ten or more symmetries.

Symmetry of fulleroids has also been an object of research in recent years. Among all possible symmetry types, icosahedral symmetry groups were studied first. Let the full symmetry group of a regular icosahedron be denoted by I h ; its subgroup of rotational symmetries by I . Delgado Friedrichs and Deza [START_REF] Delgado Friedrichs | More Icosahedral Fulleroids[END_REF] found I h (5, n)-fulleroids for n = 8, 9, 10, 12, 14 and 15 and asked several questions concerning I (5, n)-fulleroids. Most of their questions were answered by Jendrol' and Trenkler [START_REF] Jendrol | More icosahedral fulleroids[END_REF], who found infinite series of examples of I (5, n)-fulleroids for all n ≥ 8.

Jendrol' and Kardoš [START_REF] Jendrol | On octahedral fulleroids[END_REF] found a necessary and sufficient condition for the existence of O h (5, n)-fulleroids, where O h denotes the full symmetry group of a regular octahedron. Kardoš [START_REF] Kardoš | Tetrahedral Fulleroids[END_REF] characterized Γ(5, n)fulleroids, where Γ is the group of all symmetries of a regular tetrahedron T d , or the group of rotational symmetries of a regular tetrahedron T , or the group T h , which is a subgroup of the group O h with four 3-fold rotational symmetry axes, three 2-fold rotational symmetry axes, and a point of inversion.

In this paper we solve the question of the existence of fulleroids with a symmetry group of dihedral type. In particular, we investigate fulleroids with the symmetry group D m , D md , and D mh , where m ≥ 2.

The symmetry groups D m , D md and D mh have one specific property among all possible symmetry groups of convex polyhedra: all rotational symmetry axes with the exception of one axis lie in one plane (called main, horizontal plane) and the only other axis (called main, vertical axis) is perpendicular to this plane.

The group D m is the group of all rotational symmetries of a regular m-sided prism. It is isomorphic to the triangle group T (2, m) = x, y : x 2 = y m = (xy) 2 = 1 . The group D md is the full symmetry group of a regular m-sided antiprism. It is isomorphic to the triangle group T (2, 2m), where the generator of order 2m is the rotation-reflection. The group D mh is the full symmetry group of a regular m-sided prism. It is isomorphic to the full triangle group x, y, z :

x 2 = y 2 = x 2 = 1, (xy) m = (xz) 2 = (yz) 2 = 1 .
The group D m is a subgroup of index 2 of both D md and D mh . There is another subgroup of index 2 in the group D md . It is denoted by S 2m and it is generated by the rotation-reflection. The relations among these four symmetry types can be easily observed in Figure 1, where examples of polyhedra with the symmetry groups D 7h , D 7d , D 7 , and S 14 are depicted. The well-known Steinitz Theorem states that a connected graph G is the graph of a convex polyhedron if and only if it is planar and 3-connected. Thus, in some cases it is useful to study 3-connected planar graphs instead of convex polyhedra and not to distinguish between a convex polyhedron and the corresponding graph. Furthermore, by the theorem of Mani [START_REF] Mani | Automorphismen von polyedrischen Graphen[END_REF] (see also [START_REF] Ziegler | Lectures on Polytopes[END_REF]), for each such graph G there is a convex polyhedron P such that the graph of P is isomorphic to G and the symmetry group of P is isomorphic to the automorphism group of G. Therefore, to give an example of a Γ(5, n)-fulleroid (Γ is either D m , D md , or D mh ), it is sufficient to find a 3-connected cubic planar graph with pentagonal and n-gonal faces whose automorphism group is isomorphic to Γ. Usually, we draw the graph on the surface of regular prism or bipyramid.

Operations used to generate examples

To prove that for some number n and for some group Γ (D m , D md , or D mh ) the set of all Γ(5, n)-fulleroids is infinite, it is sufficient to find an infinite series of corresponding graphs. This can be done by finding one example and a method of creating a new example from the old.

If the size n of some faces should be increased, two operations are used. If two n-gons are connected by an edge, by inserting 10 pentagons they are changed to (n + 5)-gons (see Figure 2). This step can be carried out arbitrarily many times, so the size of these two faces can be increased by any multiple of 5. When this operation is used later in the paper, it is represented by a rectangle with a number inscribed which denotes the number of edges added to the two adjacent n-gons (see e.g. Figure 8), or alternatively its application is indicated only by thickening the edges (see e.g. Figure 11). If two m-gons (pentagons or n-gons) are separated by two faces in a position as in the left-hand side picture in Figure 3, the sizes of those faces can be increased equally and arbitrarily (see the right-hand side picture in Figure 3). As a special case of the second operation we get the following: If the two original (m-gonal) faces are pentagons, we can change them into two n-gons and 2n -8 new pentagons, so the number of n-gonal faces can be increased by two. For n ≥ 8 this step can be repeated as many times as required, because two pentagons in an appropriate position can be found among the new pentagons again. The new configuration can always be chosen in such a way that possible local symmetry (rotation through 180 • ) is not destroyed. In the figures used in this paper, the two pentagons that can be used this way to create infinitely many examples, are shaded and the path of three edges connecting them is doubled. If the operation from Figure 3 has already been used, the pair of pentagons is not emphasized.

For n = 7 we need two additional pentagons if the operation is to be carried out again, see Figure 4. If this operation is used, the configuration of four starting pentagons is shaded.

-→ Fig. 4. The step to create two new heptagons.

D m (5, n)-fulleroids

For this type of symmetry there are no obstructions that could make the existence of particular structures impossible. Theorem 1 For any n ≥ 6 the sets D 2 (5, n), D 3 (5, n), and D 5 (5, n) contain infinitely many elements. Proof. In Figure 5 the graph of a fulleroid with symmetry group D 2 is shown. It has four n-gonal faces and 4(n -3) pentagonal faces. Two edges marked with an asterisk should be identified to make an embedding of the graph into the sphere. If the graph in the left-hand side picture in Figure 6 is inscribed in all three side faces of a regular 3-sided prism and arrowed semiedges are elongated to the vertices in the center of the bases of the prism, the graph of a D 3 (5, 6)-fulleroid is obtained. Analogously, if we use the graph in the right-hand side picture in Figure 6, we get examples of D 3 (5, n)-fulleroids for all numbers n ≥ 7 with six n-gonal and 6(n -4) pentagonal faces. If the graph in the left-hand side picture in Figure 7 is inscribed in all five side faces of a regular 5-sided prism, the graph of a D 5 (5, 7)-fulleroid with 42 pentagonal and 30 heptagonal faces is obtained. Similarly, if we use the graph in the right-hand side picture in Figure 7, we get examples of D 5 (5, n)-fulleroids for all numbers n ≥ 8 with 10 n-gonal and 10(n -5) pentagonal faces. For n = 6, all D 5 (5, 6)-fulleroids are in fact fullerenes with D 5 symmetry. They are classified in [START_REF] Graver | Catalog of All Fullerenes with Ten or More Symmetries[END_REF]. Theorem 2 Let m = 4 or m ≥ 6 and n ≥ 6 be integers. If n is not a multiple of m, then the set D m (5, n) is empty. If n is a multiple of m, then the set D m (5, n) has infinitely many elements. Proof. Let P be a convex cubic polyhedron with pentagonal and n-gonal faces only exhibiting D m symmetry. Then the main axis of m-fold rotational symmetry intersects P in two points that obviously can be neither vertices nor internal points of edges, since m > 3. So the axis intersects P in two faces. Since it is the axis of m-fold rotational symmetry, the faces must have local m-fold rotational symmetry, too. Since m = 5, the faces cannot be pentagons. Faces that are n-gonal can have m-fold rotational symmetry only if n is a multiple of m.

n -3 n -3 n -3 n -3 * *
It is sufficient now to find examples of D m (5, km)-fulleroids where k ≥ 1 is an arbitrary integer. For k = 1 and m = 6 we get the case of D 6 (5, 6)-fullerenes which are classified in [START_REF] Graver | Catalog of All Fullerenes with Ten or More Symmetries[END_REF]. For k = 1 and m ≥ 7 we can use the graphs shown in Figure 7 and inscribe them in the side faces of a regular m-sided prism. For k > 1, appropriate graphs are in Figure 8. To prove the first claim of this theorem we use the following lemma [START_REF] Jendrol | On octahedral fulleroids[END_REF]. Lemma 1 Let n ≡ 0 (mod 5) and P be a convex polyhedron with pentagonal and n-gonal faces only and all vertices of degree 3. Then there exists the homomorphism Ψ : P → D, where D denotes the dodecahedron.

By homomorphism Ψ : P → D we mean the mapping of vertices of the polyhedron P onto those of the polyhedron D, respecting the adjacency structure. Since both P and D are cubic convex polyhedra, it immediately follows that if two edges (faces) of P are adjacent, then also their images are adjacent in D.

Proof of Theorem 3. We prove the first part by contradiction. Let n = 5 + 10k and let P be a D 2d (5, n)fulleroid. Since all faces of P are of odd size and all vertices of P are trivalent, the main symmetry axis (rotation by 180 • ) intersects P in two midpoints of edges. The points will be denoted as x and x ′ , the edges e and e ′ , respectively. There are two vertical symmetry planes, perpendicular to each other; their intersection is the main axis. The edge e must lie in a vertical symmetry plane. Let this plane be ρ. Since the symmetry is D 2d and not D 2h , the edge e ′ lies in the other plane, ρ ′ .

Let Ψ : P → D be the mapping given by Lemma 1. Since the size of all faces of P is a multiple of 5, if a point moves from x to x ′ and back to x along intersection of P and ρ, its image can move only on certain circular lines on D, two of them visible in Figure 9. These lines will be denoted as perimeters of D. Let the perimeter containing Ψ(P ∩ ρ) be denoted by p and the perimeter containing Ψ(P ∩ ρ ′ ) be denoted by p ′ . Then Ψ(x) ∈ p ∩ p ′ , moreover, p and p ′ are perpendicular to each other and Ψ(e) ⊂ p. The only possible relative position of p and p ′ is the position of two perimeters in Figure 9. On the other hand, Ψ(e ′ ) ⊂ p ′ and Ψ(x ′ ) ∈ p ∩ p ′ . This implies that p intersects p ′ in the midpoint Ψ(x ′ ) of the edge Ψ(e ′ ), which is impossible. To prove the second part it is sufficient to find examples of D 2d (5, n)-fulleroids for all other values of the number n. Unlike the case of D 2 (5, n)-fulleroids, because of the nonexistence of D 2d (5, 5 + 10k)-fulleroids it is impossible to find a universal example where increasing the size of n-gonal faces by any number would be possible.

In Figure 10 we show examples of D 2d (5, n)-fulleroids for n = 6 and n = 7. In Figure 11 we show graph segments of D 2d (5, n)-fulleroids for n = 7, 8 + 5k, 9 + 5k, 11 + 5k, and 12 + 5k. To obtain a fulleroid, the graph should be inscribed in all four side faces of a regular 4-sided prism (alternating the graph itself and its mirror image), and the topmost and bottommost pairs of edges should be identified. To make this process clear, note that the first graph in Figure 11 is a representation of the fulleroid in the right-hand side picture in Figure 10.

If the left (right) graph shown in Figure 12 is inscribed in all four sides of a regular 4-sided prism (again alternating the graph and its mirror image), the graph of a D 2d (5, n)-fulleroid is obtained, where n = 10+20k (n = 20 + 20k). Theorem 4 For any n ≥ 6 the sets D 3d (5, n), and D 5d (5, n) contain infinitely many elements. Proof. If the graphs shown in Figure 13 are inscribed in all three side faces of a regular 3-sided prism and arrowed semiedges are elongated to the vertices in the center of both bases of the prism, graphs of D 3d (5, n)fulleroid for n = 7, n = 8 and n ≥ 9 are obtained. D 3d (5, 6)-fulleroids are in Graver's catalogue [START_REF] Graver | Catalog of All Fullerenes with Ten or More Symmetries[END_REF] divided into 18 infinite series.

If the graphs shown in Figure 14 are inscribed in all five side faces of a regular 5-sided prism, graphs of D 5d (5, n)-fulleroid for n = 7, 8, 9, 10 and n ≥ 11 are obtained. D 5d (5, 6)-fulleroids are in Graver's catalogue [START_REF] Graver | Catalog of All Fullerenes with Ten or More Symmetries[END_REF] divided into 5 infinite series.

D mh (5, n)-fulleroids

Although the groups D md and D mh are of the same order, they behave in a different manner. Even if the number m is odd and the groups are both isomorphic to T (2, 2m), there are some values of numbers n and m such that D md (5, n)-fulleroids exist and D mh (5, n)-fulleroids do not.

The group D 2h is not a typical dihedral group, because all three symmetry axes and all three reflection planes are equivalent. It can be seen as a subgroup of the group O h , the full symmetry group of the regular Proof. Let n ≡ 5 or 10 (mod 15) and P be an D 3h (5, n)-fulleroid. Let the vertical symmetry planes of P be denoted by ρ 1 , ρ 2 , ρ 3 and the horizontal symmetry plane by ρ 0 . Lemma 1 gives us the homomorphism Ψ : P → D. If a point moves along the intersection of any of the symmetry planes with P , its image can move only along some perimeter of D. Let p 0 , p 1 , p 2 and p 3 be perimeters of D such that Ψ(ρ i ∩ P ) ⊆ p i ; i = 0, 1, 2, 3. Since the main vertical 3-fold rotational symmetry axis intersects P in two vertices, p 1 , p 2 and p 3 are in relative position like in Figure 17. The point where some ρ i (i = 1, 2, 3) intersects ρ 0 and P can be either an internal point of a face (only if n is even) or a midpoint of an edge. In the first case p i = p 0 and in the second case p i and p 0 intersect each other in two midpoints of edges of D. If p 1 = p 0 , then p 0 intersects p 2 and p 3 in vertices of D, which is a contradiction. If p 0 intersects p 1 in two midpoints of edges, it intersects p 2 and p 3 in centers of faces of D (one of two possible relative positions of p 0 is in Figure 17), which is again a contradiction. Thus, for n ≡ 5 or 10 (mod 15) the set D 3h (5, n) is empty. The remaining part of the proof is to show examples of D 3h (5, n)-fulleroids for n ≡ 5 or 10 (mod 15). For n = 6 the fullerenes with D 3h symmetry are in Graver's catalogue [START_REF] Graver | Catalog of All Fullerenes with Ten or More Symmetries[END_REF] divided into 18 infinite series. For n ≥ 7, n ≡ 0 (mod 5) examples of D 3h (5, n)-fulleroids can be obtained if the graph segments shown in Figure 18 are inscribed in all the faces of a regular bipyramid with a hexagonal base respecting the reflection Proof. Let n ≡ 0 (mod 5), n ≡ 0 (mod 25), and P be an D 5h (5, n)-fulleroid. Let the vertical symmetry planes of P be denoted by ρ 1 , . . . , ρ 5 and the horizontal symmetry plane by ρ 0 . Lemma 1 gives us the homomorphism Ψ : P → D. If a point moves along intersection of any of symmetry planes with P , its image can move only along some perimeter of D. Let p 0 , p 1 , . . . , p 5 be perimeters of D such that Ψ(ρ i ∩ P ) ⊆ p i ; i = 0, 1, . . . , 5. The main symmetry axis intersects P in two faces. These faces can be either pentagons or n-gons. Since n ≡ 0 (mod 25), the perimeters ρ 1 , . . . , ρ 5 are pairwise different and intersect each other in a center of a pentagonal face of D.

The point where some ρ i (i = 1, . . . , 5) intersects ρ 0 and P can be either an internal point of a face (only if n is even) or a midpoint of an edge. In the first case p i = p 0 and in the second case p i and p 0 intersect each other in two midpoints of edges of D. If p i = p 0 for some i, then p 0 = p i cannot intersect any other p j in a midpoint of an edge, thus p 0 = p 1 = . . . = p 5 , which is a contradiction. But nor can p 0 intersect all p i in two midpoints of edges.

To complete the proof it is sufficient to show examples of D 5h (5, n)-fulleroids for n ≡ 0 (mod 5) or n ≡ 0

Fig. 1 .

 1 Fig. 1. Examples of polyhedra with D 7h , D 7d , D 7 , and S 14 symmetry, respectively.

Fig. 2 .

 2 Fig. 2. The step to increase the size of two n-gons by 5.

Fig. 3 .

 3 Fig. 3. The step to increase the size of two m-gons arbitrarily.

Fig. 5 . 4 n - 4 Fig. 6 .

 5446 Fig. 5. The graph of a D 2 (5, n)-fulleroid.

Fig. 7 .

 7 Fig. 7. A segment of the graph of a D 5 (5, n)-fulleroid for n = 7 and n ≥ 8.

Fig. 8 .

 8 Fig. 8. Examples of graph segments of Dm(5, n)-fulleroids for n = (2 + 5c)m, n = (3 + 5c)m, n = (4 + 5c)m, n = (5 + 5c)m and n = (6 + 5c)m, respectively.

4 .

 4 D md (5, n)-fulleroids Theorem 3 Let n ≥ 6 be an integer. If n ≡ 5 (mod 10), then the set D 2d (5, n) is empty. If n ≡ 5 (mod 10), then the set D 2d (5, n) has infinitely many elements.

Fig. 9 .

 9 Fig. 9. Two perimeters of the dodecahedron.

Fig. 10 .Fig. 11 .Theorem 5

 10115 Fig. 10. Examples of D 2d (5, n)-fulleroids for n = 6 and n = 7.

Fig. 12 .

 12 Fig. 12. Graph segments of D 2d (5, n)-fulleroids for n = 10 + 20k and 20 + 20k.

Fig. 13 . 6

 136 Fig. 13. Graph segments of D 3d (5, n)-fulleroids for n = 7, n = 8, and n ≥ 9.octahedron, in which 3-fold rotation symmetries were destroyed. To depict a D 2h -fulleroid we draw a certain graph onto all eight faces of regular octahedron in such a way that the reflection symmetries are retained. Theorem 6 For any n ≥ 6 the set D 2h (5, n) contains infinitely many elements. Proof. For n = 6 we get the case of fullerenes with D 2h symmetry. All D 2h -fullerenes can be divided into 21 infinite series, according to the relative positions of the twelve pentagons. For n ≥ 7 examples of D 2h (5, n)fulleroids can be obtained if the graphs shown in Figure16are inscribed in all eight faces of a regular octahedron respecting the reflection symmetries. Theorem 7 Let n ≥ 6 be an integer. If n ≡ 5 or 10 (mod 15), then the set D 3h (5, n) is empty. If n ≡ 5, 10 (mod 15), then the set D 3h (5, n) contains infinitely many elements. Proof. Let n ≡ 5 or 10 (mod 15) and P be an D 3h (5, n)-fulleroid. Let the vertical symmetry planes of P be denoted by ρ 1 , ρ 2 , ρ 3 and the horizontal symmetry plane by ρ 0 . Lemma 1 gives us the homomorphism Ψ : P → D. If a point moves along the intersection of any of the symmetry planes with P , its image can move only along some perimeter of D. Let p 0 , p 1 , p 2 and p 3 be perimeters of D such that Ψ(ρ i ∩ P ) ⊆ p i ; i = 0, 1, 2, 3. Since the main vertical 3-fold rotational symmetry axis intersects P in two vertices, p 1 , p 2 and p 3 are in relative position like in Figure17. The point where some ρ i (i = 1, 2, 3) intersects ρ 0 and P can be either an internal point of a face (only if n is even) or a midpoint of an edge. In the first case p i = p 0 and in the second case p i and p 0 intersect each other in two midpoints of edges of D. If p 1 = p 0 , then p 0 intersects p 2 and p 3 in vertices of D, which is a contradiction. If p 0 intersects p 1 in two midpoints of edges, it intersects p 2 and p 3 in centers of faces of D (one of two possible relative positions of p 0 is in Figure17), which is again a contradiction. Thus, for n ≡ 5 or 10 (mod 15) the set D 3h (5, n) is empty.The remaining part of the proof is to show examples of D 3h (5, n)-fulleroids for n ≡ 5 or 10 (mod 15). For n = 6 the fullerenes with D 3h symmetry are in Graver's catalogue[START_REF] Graver | Catalog of All Fullerenes with Ten or More Symmetries[END_REF] divided into 18 infinite series. For n ≥ 7, n ≡ 0 (mod 5) examples of D 3h (5, n)-fulleroids can be obtained if the graph segments shown in Figure18are inscribed in all the faces of a regular bipyramid with a hexagonal base respecting the reflection

Fig. 14 .Fig. 15 .Theorem 8

 14158 Fig. 14. Graph segments of D 5d (5, n)-fulleroids for n = 7, 8, 9, 10 and n ≥ 11.

Fig. 16 .

 16 Fig. 16. Graph segments of D 2h (5, n)-fulleroids for n = 7, 8, 9, 10 and n ≥ 11.

1 Fig. 17

 117 Fig. 17. The perimeters of the dodecahedron.

  Fig. 17. The perimeters of the dodecahedron.

Fig. 18 .

 18 Fig. 18. Graph segments of D 3h (5, n)-fulleroids for n = 7, 8 + 10k, 9 + 10k, 11 + 10k, 12 + 10k, 13 + 10k, 14 + 10k, 16 + 10k, and 17 + 10k.

Fig. 19 .

 19 Fig. 19. Graph segments that can be used for construction of examples of D mh (5, n)-fulleroids for various m and n. See the text for precise values of the parameters 5k, p, q, and r.

Fig. 20 .Theorem 9

 209 Fig. 20. Graph segments of D 5h (5, n)-fulleroids for n = 7, 8 + 10k, 9 + 10k, 11 + 10k, 12 + 10k, 13 + 10k, 14 + 10k, 16 + 10k, and 17 + 10k.

5(c + ck - 2 )

 2 Fig. 21. Graph segment of a D mh (5, n)-fulleroid for m = 6 + 5k, k ≥ 0, and m = c(6 + 5k), c ≥ 2.

⋆ This work was supported by the Slovak VEGA Grant 1/3004/06 and AVPT-20-004104 Grant.


Chapter 1

Symmetry of Fulleroids

One of the important features of the structure of molecules is the presence of symmetry elements. Symmetry arguments can e.g. tell us whether the molecule is chiral and so it could be optically active. Structure of the symmetry group of a molecule affects several spectroscopic aspects and vice-versa. Thus, it is clearly important to know the possible symmetries of fullerenes and similar structures.

In this chapter, we study symmetry groups of fulleroids. In the most general manner, a fulleroid can be viewed as a cubic (i.e. 3-valent) convex polytope with all faces of size at least five. This definition is a natural generalization of the notion of fullerenes. Fulleroids can also be represented by planar graphs. In the first section, we recall the basic definitions and properties of convex polyhedra and planar graphs. In the second section, we mention connections between symmetries of convex polyhedra and automorphisms of planar graphs, which allow us not to distinguish between a polyhedron with its symmetry group and a graph with its automorphism group.

In the third section, all groups that can act as symmetry groups of convex polyhedra are listed and described. For each such group, its symmetry elements are listed in a table, and a few more information is then mentioned in the text. We also provide examples and constructions of polyhedra and/or their graphs with particular symmetry groups.

The fourth section gives basic relations between rotational symmetries and faces sizes of fulleroids, which imply some easy to see necessary conditions on face sizes for the existence of fulleroids with particular symmetry groups.

The fifth section brings a brief overview of results concerning symmetries of fullerenes. One can find a list of all possible fullerene symmetry groups and counts of vertices of the smallest representatives of them.

In the next two sections, we study fulleroids with icosahedral symmetry and fulleroids with the symmetry group being a subgroup of an icosahedral group. We provide construc-tions of fulleroids with icosahedral symmetry group with pentagonal and n-gonal faces only for each n ≥ 7. We also mention constructions, which assure that there are infinitely many such structures for each n ≥ 7. It turns out that for subgroups of the icosahedral group I h , one can find examples of fulleroids with any faces sizes.

In the next section, we study symmetry of fulleroids with multi-pentagonal faces. We use tools from both algebra and geometry to prove nonexistence of fulleroids for the seven groups. Therefore, for the remaining seven groups without local constrains, there are values of integer n such that the corresponding fulleroids with pentagonal and n-gonal faces do not exist.

In the ninth section, we complete the characterization of symmetry of fulleroids by listing the results concerning fulleroids with octahedral, prismatic, or pyramidal symmetry.

In the last section of this chapter, we focus on fulleroids with pentagonal and heptagonal faces and their symmetry. We list all possible symmetry groups and for each of them, we provide examples with minimal number of vertices.

Convex polyhedra and planar graphs

Fullerene-like molecules are often represented by convex polyhedra, where the atoms are placed in the vertices of a polyhedron, whereas the bonds among atoms are realized along the edges of the polyhedron. Combinatorial structure of convex polyhedra can be represented by planar graphs. A graph is a pair G = (V, E) of finite sets such that the elements of E are 2-element subsets of V . The elements of V are the vertices of the graph G, the elements of E are its edges.

A graph G is planar, if it can be represented in the 2-dimensional plane R 2 in such a way that vertices of G are distinct points in R 2 , edges of G are arcs between the vertices, such that different edges have different sets of endpoints and the interior of an edge contains no vertex and no point of any other edge. Such representation is called drawing or planar embedding of G. For every drawing of a graph G, the regions of R 2 \ G are the faces of G. One of the faces is always unbounded -the outer face; the other faces are its inner faces.

Given a convex polyhedron P , the vertices and the edges of P form a graph, called the graph of P and denoted by G(P ). One can find a planar drawing of G(P ) by projecting P into a face f of P . Such drawing of the graph G(P ) of a convex polyhedron P is also called the Schlegel diagram of P . Any face of P can be chosen to be the outer face in the diagram.

A graph G is polyhedral if it is (isomorphic to) a graph of some convex polyhedron. For every convex polyhedron P , the graph G(P ) is planar and 3-connected. On the other hand, these conditions are also sufficient for the graph to be polyhedral, by Steinitz' theorem.

Theorem 1.1 (Steinitz) [START_REF] Cromwell | Polyhedra[END_REF]) A graph G is polyhedral if and only if G is planar and 3-connected.

Convex polyhedra P 1 and P 2 are equivalent, if the corresponding graphs G(P 1 ) and G(P 2 ) are isomorphic. In this case, we say P 1 and P 2 are of the same type.

An arbitrary planar graph can have several different drawings, with the faces of different size. Another important property of 3-connected planar graphs is given by Whitney's theorem.

Theorem 1.2 (Whitney) [START_REF] Diestel | Graph Theory[END_REF] Any two drawings of a 3-connected graph are equivalent.

It means that the faces (their boundaries and sizes) of a drawing of G are uniquely determined by the graph itself. Therefore, we can speak about the faces of a 3-connected planar graph without specifying its drawing. Whitney's theorem also says that each 3connected planar graph corresponds to precisely one type of convex polyhedra. Hence, we can identify a convex polyhedron P with its graph G(P ); we represent the fulleroids by 3-connected planar graphs with all faces of size at least five.

To find more information about convex polyhedra, we refer the reader to the books [START_REF] Cromwell | Polyhedra[END_REF]) and [START_REF] Grünbaum | Convex Polytopes[END_REF]. The recommended introductory book for the graph theory is the book of Chartrand and Lesniak (1972).

Polyhedral symmetries and graph automorphisms

An automorphism of a graph G is an isomorphism of G onto itself. The set of all automorphisms of a graph G together with the composition operation forms the automorphism group of G and is denoted by Aut(G). For every graph G the group Aut(G) is nonempty and finite.

Symmetry is a property of a convex polyhedron which causes it to remain invariant under certain classes of transformations (such as rotation, reflection, inversion, or more abstract operations). Strictly speaking, a symmetry of a convex polyhedron P is an isometry of the 3-dimensional space R 3 , under which the polyhedron P remains invariant. The set of all symmetries of a convex polyhedron P together with composition operation forms the symmetry group of P and is denoted by Γ(P ). For every convex polyhedron P the group Γ(P ) is nonempty and finite.

If φ is a symmetry of a convex polyhedron P , then φ restricted to the vertices and edges of P is an automorphism of the graph G(P ). Moreover, the group Γ(P ) is (isomorphic to) a subgroup of the group Aut(G(P )). Stronger connection between automorphisms of polyhedral graph and symmetries of corresponding polyhedra is given by the theorem of Mani:

Theorem 1.3 (Mani) [START_REF] Mani | Automorphismen von polyedrischen graphen[END_REF] Let G be 3-connected planar graph. Then there is a convex polyhedron P whose graph is isomorphic to G and Γ(P ) is isomorphic to Aut(G).

This theorem allows us not to distinguish between a polyhedron P with the group Γ(P ), and its graph G with the group Aut(G). Therefore, to give examples of convex polyhedra with the symmetry group Γ it is sufficient to find polyhedral graphs such that Aut(G) ∼ = Γ.

Point symmetry groups

As mentioned above, the symmetry groups of convex polyhedra are nonempty and finite. But not all finite groups can be symmetry groups of convex polyhedra. The characterization of all the groups that act as a symmetry group of a convex polyhedron is known. The list of all such groups, their structure and properties, can be found in [START_REF] Cromwell | Polyhedra[END_REF]. Another source of general information on finite groups is the book of [START_REF] Coxeter | Generators and Relations for Discrete Groups[END_REF].

Every symmetry of a convex polyhedron P has at least one fixpoint (its centre of gravity), and this point is common for all symmetries of P . The group Γ(P ) is thus called a point group.

According to the number of rotational symmetry axes and their relative position all point groups can be divided into icosahedral, octahedral, tetrahedral, dihedral, skewed, pyramidal and others. The list of all groups that can act as symmetry groups of convex polyhedra is in Table 1.1.

First seven point groups (I h , I , O h , O, T h , T d , and T ) can be interpreted as the symmetry groups of platonic solids or their subgroups. They contain more than one axis of at least 3-fold rotation.

i) The group I h is the full symmetry group of a regular icosahedron (and a regular dodecahedron). It is also the symmetry group of the most famous fullerene, the bucky-ball C 60 . It is isomorphic to A 5 × Z 2 , where A 5 is the group of even permutations of 5 elements.

I h ∼ = x, y, z | x 2 = y 2 = z 2 = (xy) 2 = (xz) 3 = (yz) 5 = 1 .
ii) The group I is the group of rotations of a regular icosahedron. It is a subgroup of I h of index 2 and it is isomorphic to A 5 .

I ∼ = a, b | a 2 = b 3 = (ab) 5 = 1 .

Group Order Rotations Reflections Inversion

I h 120 6C 5 , 10C 3 , 15C 2 15 yes I 60 6C 5 , 10C 3 , 15C 2 - - O h 48 3C 4 , 4C 3 , 6C 2 9 yes O 24 3C 4 , 4C 3 , 6C 2 - - T d 24 4C 3 , 3C 2 6 - T h 24 4C 3 , 3C 2 3 yes T 12 4C 3 , 3C 2 - - D nh 4n C n , nC 2 n + 1 if n even D nd 4n C n , nC 2 n if n odd D n 2n C n , nC 2 - - S 2n 2n C n - if n odd C nh 2n C n 1 if n even C nv 2n C n n - C n n C n - - C s 2 - 1 - C i 2 - - yes C 1 1 - - -
Table 1.1: Point symmetry groups. Each row of the table lists one such group, its order, the number and sort of rotational symmetry axes, the number of mirror symmetry planes and presence of point inversion in the group. iii) The group O h is the full symmetry group of a regular octahedron (and of a cube). It is isomorphic to S 4 × Z 2 , where S 4 is the group of all permutations of 4 elements.

O h ∼ = x, y, z | x 2 = y 2 = z 2 = (xy) 2 = (xz) 3 = (yz) 4 = 1 . O h O T h T d T Figure 1.2: The groups O h , O, T h , T d a
T can be represented as groups of symmetries of a cube respecting certain patterns drawn on it.

iv) The group O is the group of rotations of a regular octahedron. It is a subgroup of O h of index 2 and it is isomorphic to S 4 . v) The group T h is the full symmetry group of a volleyball ball, or of a pyritohedron. It is a subgroup of both O h and I h and is a supergroup of T . It is isomorphic to A 4 × Z 2 , where A 4 is the group of even permutations of 4 elements.

O ∼ = a, b | a 2 = b 3 = (ab) 4 = 1 . O h O
T h ∼ = a, b | a 2 = b 3 = (abab -1 ) 2 = 1 .
vi) The group T d is the full symmetry group of a regular tetrahedron. It is a subgroup of O h and it is isomorphic to S 4 (and O indeed).

T d ∼ = x, y, z | x 2 = y 2 = z 2 = (xy) 2 = (xz) 3 = (yz) 3 = 1 .
vii) The group T is the group of rotations of a regular tetrahedron. It is a subgroup of index 2 of both T h and T d and it is isomorphic to A 4 . Next seven infinite series of groups (D nh , D nd , D n , S 2n , C nh , C nv , and C n ) can be interpreted as symmetry groups of regular prisms or antiprisms or their subgroups. i) D mh -full symmetry group of a regular m-sided prism. For m even it is isomorphic to vii) C m -group of rotations of a regular m-sided pyramid. It is isomorphic to Z m and is a subgroup of all groups listed above.

T ∼ = a, b | a 2 = b 3 = (ab) 3 = 1 . T h T d T
D m × Z 2 ,
The remaining three groups are the groups of low symmetry.

i) The group C s contains a mirror reflection and an identity.

ii) The group C i contains a point inversion and an identity.

iii) The group C 1 contains only an identity.

Local restrictions

In general, the presence of rotational symmetry axes in the symmetry group implies a local rotational symmetry of the sites where the axes intersects the polyhedron:

Lemma 1.1 Let P be a fulleroid. If an axis of an m-fold rotation intersects a face of size n, then m divides n. If an axis of an m-fold rotation intersects a vertex of P , then m = 3.

If an axis of an m-fold rotation intersects an edge of P , then m = 2.

These connections, together with the fact that every fulleroid has at least 12 faces of degree 5, give us necessary conditions for the face sizes of fulleroids with given symmetry group. If the symmetry group of a fulleroid P contains an m-fold rotational axis, where m = 2, 3, 5, then P must have faces of size n, where m divides n.

Symmetry of fullerenes

In the special case of fullerenes, there are only pentagonal and hexagonal faces. Therefore, by Lemma 1.1 the fullerene symmetry group can contain axes of m-fold rotation only for m = 2, 3, 5 and 6. This makes the list of possible symmetry groups be finite; it consist of 36 groups:

I h , I , T h , T d , T , D 6h , D 6d , D 6 , S 12 , C 6h , C 6v , C 6 , D 5h , D 5d , D 5 , S 10 , C 5h , C 5v , C 5 , D 3h , D 3d , D 3 , S 6 , C 3h , C 3v , C 3 , D 2h , D 2d , D 2 , S 4 , C 2h , C 2v , C 2 , C s , C i , C 1 .
Using more detailed consideration it was proved that whenever a 5-fold or 6-fold rotational axis is present, the structure of the fullerene implies a perpendicular 2-fold rotational axis [START_REF] Fowler | Possible symmetries of fullerene structures[END_REF]. This means that the groups S 12 , C 6h , C 6v , and C 6 only occur as subgroups of dihedral groups D 6h , D 6d , and D 6 ; as the groups S 10 , C 5h , C 5v , and C 5 only occur as subgroups of isosahedral or dihedral groups I h , I , D 5h , D 5d , and D 5 . The final list of 28 fullerene symmetry groups is therefore:

I h , I , T h , T d , T , D 6h , D 6d , D 6 , D 5h , D 5d , D 5 , D 3h , D 3d , D 3 , S 6 , C 3h , C 3v , C 3 , D 2h , D 2d , D 2 , S 4 , C 2h , C 2v , C 2 , C s , C i , C 1 .
As we will see in the next sections, this is the only case where realizability of a symmetry group is not transferred onto its subgroups.

For each of the 28 fullerene symmetry groups, the smallest examples are known [START_REF] Fowler | An Atlas of Fullerenes[END_REF]. The number of vertices of them is listed in Table 1.2.

I h I T h T d T D 6h D 6d D 6 D 5h D 5d D 5 D 3h D 3d D
S 6 C 3h C 3v C 3 D 2h D 2d D 2 S 4 C 2h C 2v C 2 C s C i C

Icosahedral fulleroids

It was Fowler who first asked whether there is fullerene-like structure consisting of pentagons and heptagons only and exhibiting icosahedral symmetry. The answer came immediately, when [START_REF] Brinkmann | Phantasmagorical fulleroids[END_REF] found two such structures with a minimal possible number of vertices and proved that there are only two of them. They also defined Γ(a, b)-fulleroid to be a fulleroid with only a-gonal and b-gonal faces, on which the group Γ acts as a group of symmetries. Delgado Friedrichs and [START_REF] Delgado Friedrichs | More icosahedral fulleroids[END_REF] found more icosahedral fulleroids with pentagonal and n-gonal faces for n = 8, 9, 10, 12, 14, and 15. [START_REF] Jendrol | More icosahedral fulleroids[END_REF] constructed I (5, n)-fulleroids for all n ≥ 8.

The properties and structure of two-faces maps (including fulleroids) are studied in the book of [START_REF] Deza | Geometry of Chemical Graphs[END_REF].

The results [START_REF] Brinkmann | Phantasmagorical fulleroids[END_REF][START_REF] Delgado Friedrichs | More icosahedral fulleroids[END_REF][START_REF] Jendrol | More icosahedral fulleroids[END_REF]) are all concerned with the group I . But the largest fullerene symmetry group is I h , hence we focus now on I h (5, n)-fulleroids.

Theorem 1.4 For any n ≥ 6 there are infinitely many I h (5, n)-fulleroids.

Proof. For n = 6 we get the case of I h -fullerenes, which are completely characterized in the catalogue of [START_REF] Graver | Catalog of all fullerenes with ten or more symmetries[END_REF]. They can be divided into two infinite series.

To depict an I h (5, n)-fulleroid, it is not necessary to draw whole its graph. If we cut the regular dodecahedron D along all its mirror planes, it falls into 120 congruent triangular pieces, called flags. The flags have the shape of a right triangle ABC with |BC| < |AB| < |AC|, where A (B, resp. C) represents the point where the 5-fold (2-fold, resp. 3-fold) rotational axis intersects the polyhedron -the face centre, the egde midpoint, and the vertex of D.

If a graph is drawn onto one flag and it is copied onto all other flags, altogether we get a graph with the automorphism group isomorphic to I h . If this graph is cubic, planar, and 3-connected, with all faces of size at least 5, we get an example of a graph of an I h -fulleroid.

For n = 7, examples of I h (5, 7)-fulleroids are obtained, if the graphs from Figure 1.8 are used this way.

To prove there are infinitely many I h (5, 7)-fulleroids, one can search for a configuration of four pentagons (see Figure 1.9, left) and change them into two heptagons and six pentagons (in every flag). Since there is such a configuration again, the operation can be carried out arbitrarily many times.

Examples of I h (5, n)-fulleroids for n = 8, 9 . . . 17 are obtained, if the graphs from Figure 1.10 and Figure 1.11 are drawn into all the flags of a regular dodecahedron. To prove that for some number n the set of all I h (5, n)-fulleroids is infinite it is sufficient to find an infinite series of corresponding graphs. We start with an example of fulleroid with pentagonal and m-gonal faces, where m ∈ {8, 9, . . . 17} such that 10 divides n -m, e.g. some of those in Figures 1.10 and 1.11. We first change all the m-gonal faces to n-gons and then increase the number of them.

If the size m of some faces should be increased, two operations can be used. If two m-gons are connected by an edge, by inserting 10 pentagons they are changed to (m + 5)-gons (see Figure 1.12 for illustration). This step can be carried out arbitrarily many times, so the size of those two faces can be increased by any multiple of 5. In the Figures 1.10 and 1.11, the application of this operation is indicated by thickening the edges. As a special case of the second operation we get the following: If original two faces are pentagons, we can change them into two n-gons and 2n -8 new pentagons, so the number of n-gonal faces can be increased by two. For n ≥ 8 this step can be repeated as many times as required, because two pentagons in an apropriate position can be found among the new pentagons again. The two pentagons, that can be used this way to create infinitely many examples, are in Figures 1.10 and 1.11 shaded grey and the edges connecting them are doubled.

It is worth to note that given integers 5 < n 1 < n 2 < • • • < n k , we can make an I hfulleroid with faces of sizes 5, n 1 , n 2 , . . . , n k if we start with graph segments from Figures 1.8, 1.10, and 1.11 and use the operations from Figures 1.9, 1.12, and 1.13.

Subgroups of I h

In the previous section we proved that there are infinitely many I h (5, n)-fulleroids for any n ≥ 7. It is natural to ask for which groups Γ there are infinitely many Γ(5, n)-fulleroids for any n ≥ 7.

It is clear that such groups can only contain m-fold rotational axis for m = 2, 3, and 5. Therefore, the list of candidates is finite and it is possible to check them one by one. In particular, we get the following 29 groups:

I h , I , T h , T d , T , D 5h , D 5d , D 5 , S 10 , C 5h , C 5v , C 5 , D 3h , D 3d , D 3 , S 6 , C 3h , C 3v , C 3 , D 2h , D 2d , D 2 , S 4 , C 2h , C 2v , C 2 , C s , C i , C 1 .
Proposition 1.5 Let Γ be a symmetry group that is a subgroup of the group I h . Then there are infinitely many Γ(5, n)-fulleroids for any n ≥ 7.

Proof. Since Γ is a subgroup of I h , it has an action on the set of flags of regular dodecahedron D. Under this action, all 120 flags of D split into 120/|Γ| orbits, each containing |Γ| flags. (Here |Γ| denotes the order, i.e. the number of elements of the group Γ.) If we start with an I h -fulleroid and change the graph in the flags forming one orbit under Γ in such a way that no new symmetries arise, we obtain a Γ-fulleroid. We can use the examples of I h (5, n)fulleroids described in the proof of Theorem 1.4 to obtain infinite series of Γ(5, n)-fulleroids for all n ≥ 7 and all subgroups Γ of I h .

This answers the question of existence of Γ(5, n)-fulleroids for arbitrary n ≥ 7 for 22 subgroups of I h :

I h , I , T h , T , D 5d , D 5 , S 10 , C 5v , C 5 , D 3d , D 3 , S 6 , C 3v , C 3 , D 2h , D 2 , C 2h , C 2v , C 2 , C s , C i , C 1 .
As we will conclude later, the condition of being subgroup of I h is not only sufficient for the existence of Γ(5, n)-fulleroid for arbitrary n ≥ 7, but it is also a necessary condition.

Different constructions of tetrahedral fulleroids can be also found in Kardoš 2007b.

For the remaining 7 groups, T d , D 5h , C 5h , D 3h , C 3h , D 2d , and S 4 , there are values of integer n such that corresponding fulleroids do not exist. All those cases of nonexistence fall within fulleroids with multi-pentagonal faces.

Fulleroids with multi-pentagonal faces

A face is multi-pentagonal, if its size is a multiple of 5. Fulleroids with multi-pentagonal faces play a special role among others, because they can be mapped onto (a graph of) a regular dodecahedron: Lemma 1.2 [START_REF] Jendrol | On octahedral fulleroids[END_REF] Let P be a cubic convex polyhedron such that all faces are multi-pentagons, i.e. the size of each face is a multiple of five. Then there exists an orientation-preserving homomorphism Ψ : P → D, where D denotes a regular dodecahedron.

By orientation-preserving homomorphism Ψ : P → D we mean a map from the set V (P ) of vertices of the polyhedron P into the set V (D), respecting the adjacency structure, which also preserves the order of the vertices incident with any vertex (up to a cyclic permutation) once an orientation has been assigned to both P and D before.

Lemma 1.3 (Kardoš 2007a) Let P be a cubic convex polyhedron such that all its faces are multi-pentagons and let Ψ : P → D be an orientation-preserving homomorphism, where D denotes a regular dodecahedron. If ϕ ∈ Γ(P ) is a symmetry of P , then Ψ • ϕ : P → D is also an orientation-preserving homomorphism, moreover, the symmetry ϕ of P uniquely determines a symmetry Ψ(ϕ) of D once Ψ is fixed.

Theorem 1.6 (Kardoš 2007a) Let P be a cubic convex polyhedron such that all its faces are multi-pentagons. Then there exists a homomorphism Ψ : Γ(P ) → I h , where Γ(P ) is the symmetry group of P and I h denotes the symmetry group of a regular dodecahedron D.

This theorem is a special case of a result of [START_REF] Malnič | Regular homomorphisms and regular maps[END_REF] concerning map homomorphisms into regular maps: Theorem 1.7 [START_REF] Malnič | Regular homomorphisms and regular maps[END_REF] If Ψ : N → M is a map homomorphism with M regular, then Aut(N ) projects.

The homomorphism Ψ : Γ(P ) → I h is a strong tool for proving nonexistence of fulleroids with multi-pentagonal faces and with symmetry groups which are not subgroups of I h . Proposition 1.8 Let P be a cubic convex polyhedron such that the sizes of all its faces are odd multiples of five. Then the symmetry group Γ(P ) does not contain the group S 4 as a subgroup.

Proof. Let P be a convex polyhedron satisfying the premise of the claim and let ϕ be the rotation-reflection that generates S 4 in Γ(P ). Since all the face sizes are odd, the rotation axis of ϕ intersects P in the midpoints of two edges, let one of these two edges be denoted by e. Let e = uv. Then ϕ 2 (u) = v and ϕ 2 (v) = u. It means ϕ 2 maps two adjacent vertices u and v onto each other.

Let Ψ : P → D be the homomorphism given by Lemma 1.2 and Ψ : Γ(P ) → I h be the corresponding homomorphism given by Theorem 1.6. The rotation-reflection ϕ is an element of order 4 in Γ(P ). There are no elements of order 4 in I h , so Ψ(ϕ) can be only of order 2 or 1, what implies Ψ(ϕ 2 ) = Ψ(ϕ) 2 = id. The vertices u and v are adjacent, hence so are the vertices Ψ(u) and Ψ(v). On the other hand,

Ψ(v) = Ψ(ϕ 2 (u)) = Ψ(ϕ 2 )(Ψ(u)) = id(Ψ(u)) = Ψ(u),
what is a contradiction. Corollary 1.9 There is no fulleroid P such that the sizes of all its faces are odd multiples of five with the symmetry group S 4 , D 2d , or T d .

Proof. It immediately follows from Proposition 1.8, since S 4 is a subgroup of both D 2d and T d . Proposition 1.10 Let P be a cubic convex polyhedron such that all its faces are multipentagons and none of the face sizes is divisible by three. Then the symmetry group Γ(P ) does not contain the group C 3h as a subgroup.

Proof. Let P be a convex polyhedron satisfying the premise of the claim and let ρ and θ be the rotation and the reflection that generate C 3h in Γ(P ). Since no face of P has the size divisible by three, the rotation axis of ρ intersects P in two vertices, let one of those vertices be denoted by v. Let the neighbours of v be denoted by v 1 , v 2 , v 3 in such a way that ρ(v 1 ) = v 2 , ρ(v 2 ) = v 3 , and ρ(v 3 ) = v 1 .

Let Ψ : P → D be the homomorphism given by Lemma 1.2 and Ψ : Γ(P ) → I h be the corresponding homomorphism given by Theorem 1.6. The rotation ρ is an element of order 3 in Γ(P ).

If

Ψ(ρ) = id, then Ψ(v 2 ) = Ψ(ρ(v 1 )) = Ψ(ρ)(Ψ(v 1 )) = Ψ(v 1 ). But Ψ : P → D maps distinct neighbours of v onto distinct ones, a contradiction. Hence ord(Ψ(ρ)) = 3, what implies Ψ(ρ) is a rotation by ±120 • .
We know that θ is orientation-reversing symmetry of P and ord(θ) = 2, so Ψ(θ) must be an orientation-reversing element of I h such that ord(Ψ(θ)) ≤ 2. The reflection θ has at least one fixpoint on the surface of P (midpoints of edges intersected by the mirror plane), thus also Ψ(θ) must have at least one fixpoint on the surface of D, indeed it can not be a point inversion. So Ψ(θ) it is a reflection.

In P , the mirror plane corresponding to θ is perpendicular to the rotational axis of ρ. Moreover, ρ • θ = θ • ρ and the subgroup C 3h of Γ(P ) generated by ρ and θ is commutative. What is the relative position of Ψ(ρ) and Ψ(θ)? There is no mirror plane of D perpedicular to any of its 3-fold rotational symmetry axis. Moreover, if a 3-fold rotational axis Ψ(ρ) is chosen, a mirror plane of D together with Ψ(ρ) generate either a non-commutative subgroup of I h with 6 elements (the group C 3v , if the mirror plane contains the axis), or the whole group I h (otherwise), what is a contradiction.

Corollary 1.11 There is no fulleroid P such that all its faces are multi-pentagons, none of the face sizes is divisible by three, and the symmetry group of P is C 3h or D 3h .

Proof. It immediately follows from Proposition 1.10, since C 3h is a subgroup of D 3h . Proposition 1.12 Let P be a cubic convex polyhedron such that all its faces are multipentagons and none of the face sizes is divisible by 25. Then the symmetry group Γ(P ) does not contain the group C 5h as a subgroup.

Proof. Let P be a convex polyhedron satisfying the premise of the claim and let ρ and θ be the rotation and the reflection that generate C 5h in Γ(P ). The rotation axis of ρ intersects P in two faces, let one of those faces be denoted by f . Let v be a vertex incident with f . Let Ψ : P → D be the homomorphism given by Lemma 1.2 and Ψ : Γ(P ) → I h be the corresponding homomorphism given by Theorem 1.6. The rotation ρ is an element of order 5 in Γ(P ). Since no face of P has the size that is a multiple of 25, Ψ(v) and Ψ(ρ(v)) = Ψ(ρ)(Ψ(v)) are two distinct vertices of the face Ψ(f ). Thus, Ψ(ρ) is not an identity, so Ψ(ρ) is an element of order 5 in I h , hence Ψ(ρ) is a rotation by ±72 • .

Using the same arguments like in the proof of Proposition 1.10 we get that Ψ(θ) it is a reflection.

In P , the mirror plane corresponding to θ is perpendicular to the rotational axis of ρ. Moreover, ρ • θ = θ • ρ and the subgroup C 5h of Γ(P ) generated by ρ and θ is commutative. What is the relative position of Ψ(ρ) and Ψ(θ)? There is no mirror plane of D perpedicular to any of its 5-fold rotational symmetry axis. Moreover, if a 5-fold rotational axis Ψ(ρ) is chosen, a mirror plane of D together with Ψ(ρ) generate either a non-commutative subgroup of I h with 10 elements (the group C 5v , if the mirror plane contains the axis), or the whole group I h (otherwise), what is a contradiction.

Corollary 1.13 There is no fulleroid P such that all its faces are multi-pentagons, none of the face sizes is divisible by 25, and the symmetry group of P is C 5h or D 5h .

Proof. It immediately follows from Proposition 1.12, since C 5h is a subgroup of D 5h .

Corollaries 1.9, 1.11, and 1.13 cover all the seven groups, which are not subgroups of I h , even though they do not force any local restrictions. For all seven of them, there are values of n, for which there are no Γ(5, n)-fulleroids. On the other hand, for all other values of n not covered by Corollaries 1.9, 1.11, and 1.13, examples of Γ(5, n)-fulleroids can be found in Kardoš (2007a).

The results mentioned above combined with constructions in sections 1.6 and 1.7 complete the proof of the following characterization:

Theorem 1.14 Let Γ be a point symmetry group. Then there are infinitely many Γ(5, n)fulleroids for any n ≥ 7 if and only if Γ is a subgroup of the group I h .

1.9 Fulleroids with octahedral, prismatic, or pyramidal symmetry There are groups which contain an m-fold rotational symmetry for m = 4 or m ≥ 6. Local conditions (see Lemma 1.1) do not allow existence of (5, n)-fulleroids for arbitrary n -in this case n must be a multiple of m. However this necessary condition is not always also sufficient.

Octahedral symmetry groups O h and O both contain three 4-fold rotational symmetry axes, therefore, the size of some faces must be a multiple of four.

Theorem 1.15 [START_REF] Jendrol | On octahedral fulleroids[END_REF] Let n ≥ 6. There are infinitely many O h (5, n)fulleroids and O(5, n)-fulleroids if and only if (i) n ≡ 0 (mod 60) or (ii) n ≡ 0 (mod 4) and n ≡ 0 (mod 5).

The groups D mh , D md , D m , S 2m , C mh , C mv and C m contain m-fold rotational symmetry, therefore, the size of some faces must be a multiple of m. Theorem 1.17 (Kardoš 2007a) Let m = 4 or m ≥ 6 and n ≥ 7 be integers and let m ≡ 0 (mod 5). Let Γ be D mh or C mh . Then there are infinitely many Γ(5, n)-fulleroids if and only if n is a multiple of m.

Theorem 1.18 (Kardoš 2007a) Let m ≥ 10 and n ≥ 7 be integers and let m ≡ 0 (mod 5). Let Γ be D mh or C mh . Then there are infinitely many Γ(5, n)-fulleroids if and only if n is a multiple of 5m.

Examples and constructions can be found in Kardoš (2007a).

(5, 7)-fulleroids

Unlike the case of fullerenes, i.e. (5, 6)-fulleroids, for (5, 7)-fulleroids, all point symmetry groups that satisfy local restrictions are realizable.

For the 36 groups Γ, which do not contain m-fold rotation unless m = 2, 3, 5, or 7, the numbers of vertices of the smallest examples of Γ(5, 7)-fulleroids are listed in Table 1

Figure 1 . 1 :

 11 Figure 1.1: Examples of fullerenes with symmetry group I h and I , respectively. Pentagonal faces (except for the outer face) are grey.

Figure 1 . 3 :

 13 Figure 1.3: Examples of fulleroids with symmetry group O h and O, respectively, with the same number of vertices and the same number of faces of size 5, 6, and 8. Pentagonal faces are grey. To obtain the graph of the fulleroid, it suffices to draw the corresponding graph segment onto all faces of a cube.

Figure 1 . 4 :

 14 Figure 1.4: Examples of fulleroids with symmetry group T h , T d , and T , respectively, with the same number of vertices and the same number of faces of size 5, 6, and 7. Pentagonal faces are grey. To obtain the graph of the fulleroid, it suffices to draw the corresponding graph segment onto all faces of a cube.

Figure 1 . 7 :

 17 Figure 1.7: Examples of fulleroids with symmetry groups C s , C i , and C 1 , respectively.

Figure 1 . 8 :

 18 Figure 1.8: The segments of the graph of the smallest I h (5, 7)-fulleroid and another I h (5, 7)fulleroid.

Figure 1 .

 1 Figure 1.10: The segments of the graphs of I h (5, n)-fulleroids for n = 8 + 10k, 9 + 10k, 10 + 10k, 11 + 10k, 12 + 10k, and 13 + 10k.

Figure 1 .

 1 Figure 1.11: The segments of the graphs of I h (5, n)-fulleroids for n = 14 + 10k, 15 + 10k, 16 + 10k, and 17 + 10k.

Figure 1

 1 Figure1.12: The step to increase the size of two m-gons by 5.

Figure 1

 1 Figure 1.13: The step to increase the size of two m-gons arbitrarily.

  Theorem 1.16(Kardoš 2007a) Let m = 4 or m ≥ 6 and n ≥ 7 be integers and let Γ be D md , D m , S 2m , C mv or C m . Then there are infinitely many Γ(5, n)-fulleroids if and only if n is a multiple of m.

  for m odd it is isomorphic to D 2m . (Here D k denotes the dihedral group of 2k elements.) ii) D md -full symmetry group of a regular m-sided antiprism. It is isomorphic to D 2m . iii) D m -group of rotations of a regular m-sided prism. It is isomorphic to D m and is a subroup of index 2 of both D mh and D md . iv) S 2m -cyclic group generated by improper rotation (glide) for 360 • 2m about vertical axis. It is isomorphic to Z 2m and is a subgroup of index 2 of D md . v) C mh -group generated by a rotation for 360 • m about vertical axis and a reflexion about Figure 1.6: Examples of fulleroids with symmetry groups D 7h , D 7d , D 7 , S 14 , C 7h , C 7v , and C 7 , respectively. a vertical plane. It is isomorphic to Z m × Z 2 and is a subgroup of index 2 of D mh . vi) C mv -full group of symmetries of a regular m-sided pyramid. It is isomorphic to D n and is a subgroup of index 2 of both D mh and D md .

Figure 1.5: Examples of polyhedra with symmetry groups D 7h , D 7d , D 7 , S 14 , C 7h , C 7v , and C 7 , respectively.

Table 1 .

 1 2: The counts of vertices of the smallest representatives of 28 fullerene symmetry groups.

	1

  .3.The examples with dihedral, skewed, pyramidal, and low symmetry are depicted in Figures 1.14-1.17.I h I T h T d T D 7h D 7d D 7 S 14 C 7h C 7v C7 500 260 116 116 68 84 28 140 196 112 112 140 D 5h D 5d D 5 S 10 C 5h C 5v C 5 D 3h D 3d D 3 S 6 C 3h 60 60 100 140 80 80 100 44 44 44 68 56C 3v C 3 D 2h D 2d D 2 S 4 C 2h C 2v C 2 C s C i C1 68 68 52 36 36 52 44 44 44 44 60 52 Table1.3: The counts of vertices of the smallest Γ(5, 7)-fulleroid for all 36 possible symmetry groups Γ.Figure1.14: The smallest (5, 7)-fulleroids with dihedral symmetry groups.Figure1.15: The smallest (5, 7)-fulleroids with skewed symmetry groups.Figure1.17: The smallest (5, 7)-fulleroids with low symmetry.

	D 7h	D 5h		D 3h	D 2h
	C 7h	C 5h		C 3h	C 2h
	D 7d	D 5d		D 3d	D 2d
	C 7v	C 5v		C 3v	C 2v
	D 7	D 5		D 3	D 2
	C 7	C 5		C 3	C 2
	Figure 1.16: The smallest (5, 7)-fulleroids with pyramidal symmetry groups.
	C i	S 14	C s	S 10 C 1
		S 6		S 4

Acknowledgement

This work was supported by Slovak Research and Development Agency under the contract No. APVV-0007-07.


Samy Guezouli
email: 1samy.guezouli@insa-rennes.fr

Mohammed Hjiaj
email: mohammed.hjiaj@insa-rennes.fr

Nguyen Quang Huy

LOCAL BUCKLING INFLUENCE ON THE MOMENT REDISTRIBU-TION PERCENTAGES FOR COMPOSITE CONTINUOUS BEAMS OF BRIDGES

Keywords: Eurocodes, bridges, FEM, composite beams, local buckling, moment redistribution

The present paper is concerned with the elastic design optimisation of continuous composite beams. This optimisation is based on the analysis of the beam in the inelastic range including the concrete creep and shrinkage, the tension stiffening and temperature difference effects as well as the possible local buckling instability. The finite element program "Pontmixte" (adapted to study continuous beams at real scale with short time computation) is first presented with its different sections: Pre-design (in accordance with Eurocode specifications), Non linear finite element (FE) calculation and Post-processing. In order to valid the proposed model, the numerical results are compared to experimental test ones on the example of a twin-span beam in reduced scale (7.5m length for each span) without taking into account the local buckling phenomenon avoided in the experimental test by using web-stiffeners. After that, special attention is paid to study the influence of the local buckling instability on the moment redistribution percentages from hogging to sagging zones. The application concerns different 3-span beams of bridge at real scale with medium span lengths (40m -60m -40m). The post-buckling behaviour represented by moment-rotation curves (M-θ) is obtained by the simulation of a simplified 3D model using Castem FE code. The hyperbolic decreasing of these curves is computed in "Pontmixte" using a specific tortional spring FE in order to take into account the local buckling phenomenon. The influence of this instability on the moment redistribution percentages calls the Eurocode predictions into question.

Introduction

Steel-concrete composite structures are common practice today in bridges and industrial buildings. The advantages of both materials lead to a very economic alternative especially in terms of high bearing capacity. The Structural Laboratory of INSA, Rennes, France set some experimental tests including one of a twin-span beam that will be used to valid the finite element model "Pontmixte" [START_REF] Guezouli | Pontmixte": a user friendly program for continuous beams of composite bridges, International Colloquium on Stability and Ductility of Steel Structures[END_REF]. Depending on the hogging cross-section class, the Eurocodes give the max moment redistribution percentages allowed in the case of cracked or uncracked elastic global analyses, so the knowledge about the influence of some phenomena in the inelastic range on the proposed values can reduce high rectification costs. The focus is on the relative less resistant classes of cross-sections that require an Elastic Global Analysis (EGA). The local buckling begins generally before reaching the elastic bending resistance for class 4 and between elastic and plastic resistances for class 3. Experimental and finite element studies on the local buckling of steel girders have been described in many papers [START_REF] Skaloud | The ultimate load behaviour of plate girders loaded in shear[END_REF][START_REF] Davies | The collapse behavior of tapered plate girders loaded within the tip[END_REF][START_REF] Shanmugam | Experimental and finite element studies on tapered steel plate girders[END_REF] and the elastic as well as the inelastic behaviour of plate girders having uniform crosssection along the beam is well understood. Without taking into account a specific classification of these cross sections, [START_REF] Skaloud | The ultimate load behaviour of plate girders loaded in shear[END_REF] concluded that the ultimate load carrying capacity is influenced by the flexural rigidity of the flanges for girders having similar proportions to those employed in civil engineering construction and [START_REF] Porter | The collapse behavior of plate girders loaded in shear[END_REF] assumed that the failure will occur when a certain region of the web yields as a result of the combined effect of the inclined tensile membrane stress field and the web buckling stress. So, it appears that the combined rigidity of compressed flange and the web, for a steel panel under negative bending moment, remains the first parameter influencing the load carrying capacity of the cross-section..

The model "Pontmixte"

Fig. 1 shows the main organisation of "Pontmixte" with its different options. The program has its graphical post-processor "Pmixtpost" for plotting needed variable all along the beam or against the increasing load [START_REF] Guezouli | Approche aux éléments finis pour l'étude du comportement des poutres de ponts mixtes à l'échelle réelle[END_REF]. The continuous beam could be pre-designed with constant cross-section along the beam (for buildings) or different flange thicknesses on hogging zone than those in sagging zone (for bridges) [START_REF] Brozzetti | Design development of steelconcrete composite bridges during construction[END_REF]CEN 2004). The algorithm select the most critical loading cases on hogging and sagging zones between the possible ones (example: for a twin-span beam, 4 possible loading cases for asymmetrical beams reduced to 2 cases for symmetrical ones if both distributed and concentrated variable loads could be applied to the beam). In the case of class 3 or 4 cross-sections on hogging zone, the unknown values of bottom flange thicknesses could be found by an iterative process balanced between two critical loading cases:

• critical loading case in sagging zone: maximum moment close to the plastic resistant moment;

• critical loading case on hogging zone: maximum moment close to the elastic resistant moment.

Fig. 1. Main organisation of "Pontmixte"

Different finite element analyses of composite beams have been carried out [START_REF] Guezouli | Approche aux éléments finis pour l'étude du comportement des poutres de ponts mixtes à l'échelle réelle[END_REF][START_REF] Sotelino | Three-dimensional finite element modeling of composite girder bridges[END_REF][START_REF] Nguyen | Nonlinear F.E. analysis of composite beams[END_REF]. The finite element model (Fig. 2) concerned with the program "Pontmixte considers a concrete slab with reinforcing steels connected to a steel girder. The composite beam finite element (node i to node j) has 4 degrees of freedom per node:

{ } { } ( ) ( ) ( ) ( ) = θ θ t c a c a e i i j j i i j j d u u v u u v (1)
For the node "i" for example (Fig. 2), the longitudinal displacements are: u i (c) for the concrete slab and u i (a) for the steel girder applied at each corresponding centre of gravity, the vertical displacement v i and the rotation θ i both applied at the neutral axis of the entire composite cross-section. The stud slip is defined by:

i ) a ( i ) c ( i i d u u θ × + - = γ (2)
where d is the distance between the slab and the girder neutral axis. First numerical integration is performed along the element (2 Gauss points) and the second one concerns each fibre constituting the entire composite crosssection (Fig. 3). Non linear equations are solved using a step-by-step method including a secant algorithm (Fig. 4). The automatic longitudinal mesh of the beam could be in accordance with the connection distribution or not. Along the beam, 5 Gauss points are necessary in the variable part of the beam (slope = 1/4) and 2 Gauss points are enough elsewhere. The calculation remains running until the imposed stopping calculation (such as reaching the elastic moment at intermediate support, or application of the whole defined loads etc.). This generally occurs before anyone of the following material failure criteria is reached:

• max compression in the concrete slab;

• max strain in the steel girder;

• max strain in the reinforcing steel;

• max slip of the stud.

The convergence of the iterative process is tested on the norm of the displacements limited to 10 -4 . The symmetrical stiffness matrix of the composite beam element (ij) including a stud at the node "j" for example, is presented in Eq (3) with the following notation: The specific finite element for local buckling instability (Fig. 5) includes longitudinal displacements for the concrete slab and the girder connected to a tortional spring. The (Moment-Rotation) curve of the local buckling will be followed as soon as the point (M v , θ

] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 c c c c c c
k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k K k R + + + + + + + = + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
.

c
R k k R k R k k R k k k k Sym k k R                   +       + +     + +     + +     (3)
( ) 2 1 1 c c u Q Q e -γ = - (5)
The parameters c1 and c2 (Eq (5)) depend on the ductility of the stud and can be easily obtained using a push-out test; usual values are: c1 = 0.7 and c2 = 0.8. In order to take into account the creep effect (Fig. 6d), the elastic modulus of the concrete is reduced to

( ) ( ) / c a E E n φ =
, where E (a) is the usual elastic modulus of structural steel and n φ is the modular ratio (for first steps loading: n φ = ∞, when the concrete slab is dry: n φ = 18 and leads to:

n φ = 6). σ (a) ε (a) ε e (a) µ 1 (a) ε e (a) µ 2 (a) ε e (a)
Atan(E

ε (s) ε e (s) µ 1 (s) ε e (s) µ 2 (s) ε e (s)
Atan(E (s) ) f y

σ (c) ε (c) ε m ε r f cm f ck Arctg(E cm ) Compression Tension 0.4 f cm ε σ f ctm ε creep σ creep

Fig. 6d. Concrete behaviour with creep

The temperature difference effect is considered by a superposition of a tension N sh in the concrete slab and a compression in the neutral axis of the composite crosssection (homogenised with n φ = 12, Fig. 7) with:

( ) ( ) / c a sh sh N A E n ∞ φ = ε and sh sh sh x N M × =
. If the temperature difference between slab and steel girder is about ±5 °C, the total shrinkage including the temperature effect for usual concrete can vary from The tension stiffening effect (Fig. 8a) is limited between two lines:

• uncracked stage (concrete and reinforcing steel resisting together):

1 ) ( ) ( ) , ( ) 1 ( 1 ε ρ + = ε n A E F s s s , (6) with: ( ) ( ) s c E n E = and ( ) ( ) c t s A A ρ = ( ( ) c t
A the area of the slab in tension);

• full cracked stage (reinforcing steel resisting alone):

2 ) ( ) ( ) , ( 2 ε = ε s s s A E F . ( 7
)
Between both precedent limits, the tension stiffening is acting since the beginning of the concrete cracking corresponding to a stress equal to f ctm (Fig. 8b). Contrary to the homogenization in steel equivalence of the reinforced slab in tension used to obtain the Eq (6), the homogenization must be done now in concrete equivalence to obtain the limit of the uncracked stage ε 1,max corresponding to f ctm.

) 1 ( 1 1 ) ( max , 1 - -ρ + = ε n E f c ctm (8) N (s) ε (s) N (s) ε m ε 2 β∆εmax X Fig. 8a. Tension stiffening phenomenon If 1,max 1 ( , ) ( , ) s s F F ε ε >
, the average strain corresponding to the tension stiffening effect is defined with the following relationship : and considering the Eqs ( 6) -( 7), ε m becomes:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ρ + × β - = β - = ε ε ε ε ε ε ε ε n A E F F F A E F F F A E F s s s s s s s s s s s s s m m m 1 max , 1 2 2 2 2 , , , , , , ) ( ) ( , ( ) ( ) ( ) ( ) ( ) ( ) 1,max 2 , , , 1 1 m s s m s s s F F E A n F ε ε ε     ⇔ ε = -β   + ρ   (9) F (s) ε (s) ( ) 1 s, F ε Uncracked stage Tension stiffening ( ) 2 s, F ε ( ) 1,max s, F ε ( ) s y F 1,max ε (s) y ε Fig. 8b. Tension stiffening model

FE model validation

The pre-design algorithm leads for the twin-beam under investigation (Fig. 9) to a hogging cross-section of class 3. In sagging zones, the cross-section is assumed to be of classe 1 because the slab is fully connected to the girder. An uncracked elastic global analysis is used with the following material mechanical characteristics:

• concrete slab Young's modulus: E (c) = 36000 MPa;

• steel girder Young's modulus: E (a) = 190000 MPa;

• reinforcing steel Young's modulus: E (s) = 200000 MPa.

The self weight is taken into account (4.17 kN/m for sagging zones and 4.26 kN/m for hogging ones). Only concentrated loads are applied on the beam, a load P applied at the left mid-span increases proportionally to a load Q applied at the right mid-span.

-For the first critical loading case (the one concerning the sagging zone), only P is applied, -For the second critical loading case (the one concerning the hogging zone), both P and Q are applied proportionally.

Fig. 9. Geometrical characteristics of the twin-beam

It is assumed that the hogging zone concerns 15% of the span length on each side of the intermediate support. For this zone, the pre-design algorithm proposes 15 mm for the bottom flange thickness while for other cross-sections in sagging zones only 10 mm are required. The top flange thickness is equal to the bottom one. After the beam pre-design, a non linear calculation is carried out with the following loading history: the load P applied at the left mid-span increases proportion-ally to the load Q applied at the right mid-span until 550 N. At this load level, Q remains constant and P still increases until reaching one of the failure criteria described above. Mechanical characteristics are summarised in Table 1. Figs. 10 and 11 respectively, show that the comparison between numerical and experimental results is satisfactory both for deflexion (unfortunately the measurements under the load Q have not been done), and for the bending moment under the load P at the intermediate support. Numerical and experimental failures are reached by concrete cracking under the load P for P + Q ≈ 1400 kN giving:

• maximum displacement under P:

+ =
It is noted that the real conditions of the specimen test where considered in this numerical simulation taking into account the tension stiffening, the concrete creep and shrinkage as well as the temperature difference effect. Without these options, the results could not be as close as those obtained. This validation does not include the local buckling phenomenon because the concrete cracking (max compression) occurred before as the failure criterion. For this reason, the simulation of a continuous beam at real scale with hogging cross-sections of class 3 or 4 appears necessary to simulate the local buckling using the specific finite element defined in Fig. 5.

Table 1. Mechanical characteristics

Material

Parameters values

Concrete slab

E cm = 36 000 MPa, f ck = 40 MPa, f cm = 48 MPa, f t = 2 MPa, ε m = 0.0022, ε r (c) = 0.004 Steel girder E (a) = 190 000 MPa, f y (a) = 475 MPa, f u (a)
= 620 MPa, µ 1 (a) = 10, µ 2 (a) = 28

Reinforcing steel

E (s) = 200 000 MPa, f y (s) = 443 MPa, f u (s) = 565 MPa, µ 1 (s) = 1, µ 2 (s) = 32, ε u (s) = µ 2 (s) ε e (s) Stud Q u =

Influence of local buckling on the moment redistribution for a continuous beam at real scale

Firstly, the pre-design of the 3-span beam leads to a hogging cross-section of class 3 and a bottom flange as well as the web of class 3 (Fig. 13). Mechanical characteristics are given in Table 2.

Table 2. Mechanical characteristics

Material

Parameters values

Concrete slab

E cm =
E (s) = 200 000 MPa, f y (s) = 400 MPa, f u (s) = 432 MPa, µ 1 (s) = 1, µ 2 (s) = 25 Stud Q u = 174 900 N, c 1 = 0.7, c 2 = 0.8, γ max = 6 mm
For reminder, the whole cross-section class is the max one between the compressed flange and the web. Table 3 shows different cross-sections provided for by this investigation taking care to be always in the case of a girder cross-section of class 3. The cross-section obtained by the pre-design is noted H5, it represents the less resistant one by comparison to the other ones (H1 to H4) for which the thicknesses were arbitrarily increased or decreased in order to vary the classes of the bottom flange and the web from class 1 to 3. It should be noted that the flange thicknesses in sagging zones remain the same for all the beams, the web thickness is constant all along each beam, the selfweight of the girder becomes different from one beam to another and the critical position of concentrated variable loads Q is the same for all the beam in the case of type A (symmetrical loading case) and supposed the same in the case of type B (asymmetrical loading case). It is pointed out that only the pre-designed beam remains optimized at ULS and SLS. Geometrical characteristics of hogging cross-sections are given in Table 4 with also some arbitrary modifications proposed for the web and the bottom flange giving always a girder cross-section of class 4 on hogging. The traffic loads applied to the bridge (here a twin-steel girder bridge) have values in accordance with the Model 1 of EN 1991-2 Part 2, namely a "U.D.L." of 9 kN/m 2 for the lane 1 and 2.5 kN/m 2 for the lanes 2 and 3, and a Tandem System with 2 axle loads each equal to 300 kN. In the transverse direction the traffic loads are distributed according to a linear influence line. Finally, the numerical characteristic values of traffic loads for the most loaded lane are: q = 31.2 kN/m and Q1 = Q2 = 406 kN (with the distance 1200 mm). The 3-D model developed on Cast3M (2003) (Fig. 13) represents the steel girder as well as the stiffeners meshed by 4-nodes shells, the studs are meshed using 3D beams to ensure the displacements continuity with the shells (same degrees of freedom) and the reinforcing bars are replaced by equivalent shells supposed at the top of the studs. The panel length is equal to twice the web height (h w ) beginning from the cross-section H-H.

Fig. 13. Simplified 3-D model

It is supposed that on hogging zone the concrete slab is totally cracked so it does not need to be modellized. Nevertheless, very stiff springs should be modellized to keep same distance between the top flange of the girder and the reinforcing bars during the loading. It's pointed out that the springs have no influence on the shear behavior of the studs [START_REF] Faella | Steel and concrete beams with flexible shear connection: Exact analytical expression of the stiffness matrix and applications[END_REF]Guezouli, Hjiaj, Nguyen 2008). This simplification requires common mesh nodes between the studs and the reinforcing bars. The comparison of the trans-verse displacements of the web at ultimate limit state (Figs 14a,14b) shows clearly that the addition of very stiff spring elements to the model has a negligible influence on the mechanical behaviour and the studs appear working exclusively in shear. The model is loaded by applying a displacement at its end to avoid possible geometrical element distortion. The vertical displacement Uz equal to 200 mm is applied in 10 steps.

The curves (M-θ) are plotted for different crosssections of class 3 (Fig. 15) and of class 4 (Fig. 16). The rotation capacity of the cross-section is similar for all the cross-sections. The max point of each curve represents the beginning of the local buckling (M v , θ v ). After this point, a hyperbolic decreasing model (Eq (10)) could be adopted with appropriate values for M 0 , M v and θ v (M 0 represents a horizontal asymptotic line). The web thickness has a largest influence on these curves than the bottom flange one (on the initial stiffness and also on the max hogging moment). The decreasing curves after buckling is similar with nearly the same value of buckling rotation (θ v ≈ 0.014 rad). The curve related to Eq (10) plotted for H1 as an example, gives a good prediction for the behaviour after buckling. The buckling rotation is also nearly the same in the case of cross-section of class 4 (θ v ≈ 0.012 rad).

( )

0 0 , v v v M M M M θ = + - ∀θ ≥ θ θ ( 10
)
Remarks:

• Before setting about a non-linear stability calculation, a linear stability one is carried out and the whole displacement field is reduced by a scale of 1/20 in the aim to represent an initial deformation of the specimen.

• There is no significant sense to talk about available rotation capacity at the intermediate support for crosssections of classes 3 and 4 because of the local buckling instability.

The results

The moment redistribution percentage is calculated with the following Eq: -the hogging moment obtained by a virtual elastic calculation (using the same loading level λ); 1 2 M - + -the hogging moment resulting only from the first and second loading steps corresponding respectively to the selfweight of the girder and the one of the concrete that is still wet [START_REF] Guezouli | Numerical Investigation of moment redistribution in continuous beams of composite bridges[END_REF][START_REF] Guezouli | Local buckling of class 4 crosssection -application to a steel-concrete continuous beam of bridge at real scale[END_REF][START_REF] Guezouli | Numerical Investigation on instabilities of steel-concrete composite cross-sections[END_REF]. During these steps, the cross-section is not considered resisting as a composite cross-section so, this part of bending moment must be subtracted to the ones defined previously. The proposed hyperbolic model is computed in the user friendly program "Pontmixte" and different applications are carried out with appropriate values of the model parameters (M 0 , M v and θ v ). The stopping calculation criterion is either reaching the plastic resistant moment in sagging zone (generally occurs for the critical loading case of type A) or the elastic moment resistant at the intermediate support (generally occurs for the critical loading case of type B).

( ) ( ) 1 2 1 2 . . ( ) 1 2 . 100 uncr num Ed Rd el Rd uncr Ed Rd M M M M R M M - - - - + + - - + - - - = × - (11) where ( ) .

Results for the beams of class 3

Different variables concerned with the Eq (11) without and with taking into account possible local buckling are given respectively in Tables 5,6. The variation of the hogging moment against the cumulative load (Figs 17, 18, 19), show that:

• for the beams H1 and H2, the elastic resistant moment is reached before the buckling characteristic point (M v , θ v ): no influence of local buckling on the moment redi-stribution percentage, the cross-sections seams to be relatively resistant;

• for the beams H3, H4 and H5, without taking into account the local buckling, the rule of 10% redistribution predicted in the Eurocodes seams to be questionable especially for loading case of type A but the occurrence of local buckling decreases the value of the numerical hogging moment increasing consequently the value of the redistribution R. This value becomes finally over than 10% for both loading cases. Because of local buckling, the calculation stops for lower values of λ (λ buckling ≤ λ no buckling ) especially for loading case of type B and necessary the elastic moment ( uncr ) Ed.Rd M - changes and the value of R consequently. In the column of λ (Tables 4,5), the letters correspond to the stopping calculation criterion for each beam [(A) → It must be noted that the beams H1 and H4 have been arbitrary defined on hogging zone stronger than the one obtained by the pre-design optimization (H5); this leads necessary to values of λ ≥ 1.35. For example, the loading case B of the beam H3, the numerical bending moment can be plotted along the beam by comparison to the virtual elastic calculation. Without local buckling (Fig. 20a) the elastic hogging bending is obtained while the lower layer of the girder bottom flange reaches its elastic limit (this occurs for λ = 1.60, value which the curves are plotted for). It is clear that for this critical loading case, the left intermediate support is mostly concerned with the local buckling than the right one (see the dot-line corresponding to buckling moment). If the local buckling is taken into account (Fig. 20b), the same stopping calculation criterion than the one described before is obtained for a lower loading level (λ = 1.41). First finite element begins to buckle on the left intermediate support in accordance with the corresponding buckling model and the equilibrium of the forces along the beam must be carried out. The local buckling is extended on both sides defining the zone subjected to this instability.

Results for the beams of class 4

In the case of class 4, Tables 7,8 summarize only the values of the moment redistribution percentages. Eq (11) has to be calculated with moments under loads corresponding to the specific value of λ. These values are generally less than 1.35 because most of the beams are less resistant than the pre-designed one. The local buckling appears also more sensible for loading case of type B than the one of type A. Very low values of R especially for the loading case A are increasing when the local buckling is taken into account. Sometimes could be negative meaning that the redistribution is inverted and happens from mid-spans to intermediate supports. The influence of local buckling on the mo-ment redistribution percentage appears more important in the case of class 4 than in the case of class 3.

Conclusions

This study attempted to show that it is possible to simulate at a real scale the behaviour in the inelastic range of a steel-concrete composite bridge beam with a relative simple finite element formulation. The comparison of numerical and experimental results in the case of a twinspan beam in reduced scale (1/2) ensures the validation of the proposed model. After that, the main interest concerned the beams with cross-sections of class 3 and 4 on hogging zone that are generally subjected to the local buckling instability. A prior study was concerned with the post-buckling behaviour of a 3D panel on hogging zone to obtain the moment-rotation relationship corresponding to these cross-section classes. A hyperbolic model was proposed depending on 3 parameters: the buckling point (M v , θ v ) and the horizontal asymptotic line M = M 0 . This model was computed in the code "Pontmixte" and several numerical simulations were carried out to show the significant influence of the local buckling instability on the moment redistribution percentage from hogging to sagging zones in the case of a 3-span beam bridge at real scale. This influence appears closely linked to the class combination of the compressed flange and the web of the girder. The local buckling mostly occurs when the web is of a higher class (less resistant) than the compressed flange. In this case, if the local buckling is not taken into account, the rule of 10% moment redistribution proposed by the Eurocodes is questionable especially for the loading case of type A. If the local buckling is taken into account, the calculation of the moment redistribution percentage after buckling until elastic resistant moment is given only for information and it must not be included in the elastic design because specific constructive measures (web-stiffeners) are required to avoid the local buckling instability and consequently assures the possibility of this redistribution. [START_REF] Skaloud | Fatigue behavior of the breathing webs of steel bridge girders[END_REF] paid attention to the post-buckled behaviour and ultimate strength of slender webs; similar approach could be carried out for steel-concrete composite beams and easily computed in the code "Pontmixte". This future prospect could give important information about the influence of the "breathing" web on the moment redistribution percentage.

Fig. 2 .

 2 Fig. 2. Composite beam finite element

Fig. 4 .

 4 Fig. 3. Cross-section integration F

  ij value in the girder stiffness matrix (6×6); R ij : the ij value in the stud stiffness matrix (3×3).

[

Fig. 5 .

 5 Fig. 5. Local buckling element with tortional spring 3. Material behaviours Material behaviour curves are shown on Fig. 6 using following Eqs:

Fig. 6a .

 6a Fig. 6a. Steel girder behaviour

  Fig. 6b. Reinforcing steel behaviour

Fig. 7

 7 Fig. 7 Temperature and shrinkage effect

  due to the concrete between cracks with the parameter β = 0.4 for high adherence bars. Using a hyperbolic model for max ∆ε

Fig. 11 .

 11 Fig. 10 Comparison of deflexions

  35 000 MPa, f ck = 40 MPa, f cm = 48 MPa, f t = 3.5 MPa, ε m = 0.0025, ε r(c)

Fig. 12 .

 12 Fig. 12. Pre-design results and critical loading cases

Fig

  Fig. 14a. Without springs Fig. 14b. With springs

  Fig. 17a. (θ -M) -Case A -Beam H3

Table 3 . Cross-sections under investigation on hogging zone -Class 3

 3

	Cross-section	Web thickness, mm	Web Class	Bottom flange	Bottom flange	Cross-section
				thickness, mm	class	class
	H1	30	1	50	3
	H2	25	2	50	3
	H3	20	3	60	1	3
	H4	20	3	55	2
	H5	20	3	50	3

Table 4 . Cross-sections under investigation on hogging zone -Class 4

 4

	Cross-section	Web thickness, mm	Web Class	Bottom flange	Bottom flange	Cross-section
				thickness, mm	class	class
	K1	30	1	35	4
	K2	25	2	35	4
	K3	20	3	35	4
	K4	15	4	60	1	4
	K5	15	4	55	2
	K6	15	4	50	3
	K7	15	4	35	4

Table 5 . Hogging and sagging bending (kNm) -Related to Table 3 (class 3) All options -Local buckling not included

 5

	Beams	( num ) el.Rd M -	( uncr ) Ed.Rd M -	1 2 M -+	pl.Rd M -	( num) Sd M +	pl.Rd M +	λ	R, %
			Loading case of type A (for reminder see Fig. 13)
	H1 H2 H3 H4 H5	-41706 -39825 -37698 -37632 -37556	-43653 -41166 -39046 -38712 -38340	-17232 -16987 -17090 -16927 -16745	-48958 -47072 -51843 -48487 -45150	37058 33374 29228 29285 29350	37563 33541 29462 29462 29462	1.71(B) 1.51(A) 1.35(A) 1.35(A) 1.35(A)	7 6 6 5 4
			Loading case of type B (for reminder see Fig. 13)
	H1 H2 H3 H4 H5	-41722 -40529 -45685 -42460 -39969	-47264 -45027 -49123 -45793 -43146	-17232 -16987 -17090 -16927 -16745	-48958 -47072 -51843 -48487 -45150	32472 29742 28872 27625 26948	37563 33541 29462 29462 29462	1.54(B) 1.40(B) 1.60(B) 1.42(B) 1.35(B)	18 16 11 11 12

Table 6 . Hogging and sagging bending (kNm) -Related to Table 3 (class 3) All options -Local buckling included

 6

	Beams	( num ) el.Rd M -	( uncr ) Ed.Rd M -	1 2 M -+	pl.Rd M -	( num) Sd M +	pl.Rd M +	λ	R, %
			Loading case of type A (for reminder see Fig. 13)
	H1 H2 H3 H4 H5	-41706 -39825 -36379 -36371 -36362	-43653 -41166 -39046 -38712 -38340	-17232 -16987 -17090 -16927 -16745	-48958 -47072 -51843 -48487 -45150	37058 33374 29229 29285 29350	37563 33541 29462 29462 29462	1.71(A) 1.51(A) 1.35(A) 1.35(A) 1.35(A)	7 6 12 11 9
			Loading case of type B (for reminder see Fig. 13)
	H1 H2 H3 H4 H5	-41722 -40529 -36220 -36578 -36746	-47264 -45027 -46051 -44832 -43146	-17232 -16987 -17090 -16927 -16745	-48958 -47072 -51843 -48487 -45150	32472 29742 28074 27335 26995	37563 33541 29462 29462 29462	1.54(B) 1.40(B) 1.41(B) 1.36(B) 1.35(B)	18 16 34 29 24

Table 7 . Hogging and sagging bending -Related to Table 4 (class 4) All options -Local buckling not included

 7

	Beams	K1	K2	K3	K4	K5	K6	K7
			Loading case of type A (for reminder see Fig. 13)
	λ R, %	1.35(B) 24	1.22(B) 19	1.01(B) 16	1.22(A) 2	1.15(A) 1	1.15(A) -0.0	0.81(B) 12
			Loading case of type B (for reminder see Fig. 13)
	λ	1.15(B)	1.01(B)	0.88(B)	1.46(B)	1.35(B)	1.15(B)	0.74(B)
	R, %	30	28	25	7	8	9	21

Table 8 . Hogging and sagging bending (kNm) -Related to Table 4 (class 4) All options -Local buckling included

 8

	Beams	K1	K2	K3	K4	K5	K6	K7
			Loading case of type A (for reminder see Fig. 13)
	λ	1.35(B)	1.22(B)	1.01(B)	1.15(A)	1.15(A)	1.15(A)	0.81(B)
	R, %	24	19	16	53	52	50	20
			Loading case of type B (for reminder see Fig. 13)
	λ	1.15(B)	1.01(B)	0.88(B)	1.35(B)	1.22(B)	1.08(B)	0.74(B)
	R, %	30	28	25	67	61	55	29