Construction and Calculation of Reinforced Concrete Overlap with A High Spatial Work Effect

The article proposes the construction of a prefabricated monolithic reinforced concrete overlap consisting of beams hollow triangular section. It is shown that in such overlap the effect of spatial work is much higher than the analogous effect in traditional overlap that consists of U-shaped or T-beams and slabs. The technique of determining the forces of interaction of individual beams in the composition of the overlap is given. The technique is based on a discrete-continual method developed by the author, which is adapted to the calculation of overlaps that consist of considered beams. The technique of determining the effort between the shelf and the ribs of a beam during its bending is presented. It is based on the theory of compound rods. The algorithm of calculation taking into account the spatial work is presented as well as the principles of constructing overlaps consisting of beams hollow triangular section, taking into account the change in their rigidity as a result of cracks formation. An approach to the determination of the rigidity of beams with normal torsion fractures is given, based on the approximation of numerical experimental data.

On a hyperbolic 3-manifold with some special properties

We present a closed hyperbolic 3-manifold M with some surprising properties. The universal covering group of M is a normal torsion-free subgroup of minimal index in one of the nine Coxeter groups G, generated by the reflections in the faces of one of the nine Lannér-tetrahedra (bounded tetrahedra in hyperbolic 3-space all of whose dihedral angles are of the form π/n with n ∈ ℕ see [1] or [3]). The corresponding Coxeter group G splits as a semidirect product G = π1M⋉A, where A is a finite subgroup of G, and G is the only one of the nine Coxeter groups associated to the Lannér-tetrahedra which admits such a splitting (this follows using results in [4]). We derive a presentation of π1M and show that the first homology group H1(M) of M is isomorphic to ℚ11. This is in sharp contrast to other torsion-free (non-normal) subgroups of finite index in Coxeter groups constructed in [1] which all have finite first homology (though it is known that they are all virtually ℚ-representable (see [5], p. 434). It follows from our computations that the Heegaard genus of M is 11, and that there exists a Heegaard splitting of M of genus 11 invariant under the action of the group I+(M) ≌ S5 ⊕ ℚ2 of orientation-preserving isometries of M (we compute this group in [4]), so that the Heegaard genus of M is equal to the equivariant Heegaard genus of the action of I+(M) on M. Moreover M is maximally symmetric in the sense of [4, 6]: the order 120 of the subgroup of index 2 in I+(M) which preserves both handle-bodies of the Heegaard splitting is the maximal possible order of a group of orientation-preserving diffeomorphisms of a handle-body of genus 11. (This maximal order is 12(g—1) for a handle-body of genus g; see [7].) By taking the coverings Mq of M corresponding to the surjections π1M→H1(M) ≌ ℚ11→(ℚq)11 for q ∈ ℕ, we obtain explicitly an infinite series of maximally symmetric hyperbolic 3-manifolds.