This paper proposes a numerical solution to deep beams using the layerwise displacement theory. Most of the methods for performing structural analyses of deep beams have geometric and boundary conditions limitations, as well as modeling inconveniences. This paper provides a finite element solution for deep beams based on a layerwise displacement field considering the full stress/strain tensors. In this formulation, the cross section is discretized in a pre-defined number of independent virtual layers, with linear interpolation within the thickness direction. To validate the model developed, two numerical examples are analyzed. The first is a reinforced concrete deep beam with two supports, loaded over the top face, validated by finite element analysis based on solid-element ABAQUS TM software. Next, an isotropic deep beam with both ends cantilevered is analyzed and the outcome is compared to the literature. The results of both numerical examples are accurate and can estimate the complete state of stress over all domains of the element. Moreover, the layerwise formulation does not suffer from shear and membrane locking, and it may use fewer computational resources than equivalent 3D finite element analyses.

The Brasselet number of a function f hnonisolated singularities describes numerically the topological information of its generalized Milnor fibre. In this work, using the Brasselet number, we present several formulas for germs f : (X , 0) → (C, 0) and g : (X , 0) → (C, 0) in the case where g has a one-dimensional critical locus. We also give applications when f has isolated singularities and when it is a generic linear form.

In this work, we investigate the topological information captured by the Euler obstruction of a map-germ, [Formula: see text], where [Formula: see text] denotes a germ of a complex [Formula: see text]-equidimensional singular space, with [Formula: see text], and its relation with the local Euler obstruction of the coordinate functions and consequently, with the Brasselet number. Moreover, under some technical conditions on the domain, we relate the Chern number of a special collection related to the map-germ [Formula: see text] at the origin with the number of cusps of a generic perturbation of [Formula: see text] on a stabilization of [Formula: see text].

Let X ⊂ P r be a projective d-variety with isolated determinantal singularities and ω be a 1-form on X with a finite number of singularities (in the stratified sense). Under some technical conditions on r we use two generalization of Poincaré-Hopf index with the goal of proving a Poincaré-Hopf Type Theorem for X.