Mathematical modeling of planar physically nonlinear inhomogeneous plates with rectangular cuts in the three-dimensional formulation

Mathematical models of planar physically nonlinear inhomogeneous plates with rectangular cuts are constructed based on the three-dimensional (3D) theory of elasticity, the Mises plasticity criterion, and Birger’s method of variable parameters. The theory is developed for arbitrary deformation diagrams, boundary conditions, transverse loads, and material inhomogeneities. Additionally, inhomogeneities in the form of holes of any size and shape are considered. The finite element method is employed to solve the problem, and the convergence of this method is examined. Finally, based on numerical experiments, the influence of various inhomogeneities in the plates on their stress–strain states under the action of static mechanical loads is presented and discussed. Results show that these imbalances existing with the plate’s structure lead to increased plastic deformation.

A Numerical Approach for the Design of RC Beams Subjected to Axial and Transverse Loads

The present contribution deals with a numerical approach for the design of RC beams subjected to axial and transverse loads. It is based on the finite-element implementation of the kinematic approach of the yield design (or limit analysis) theory combined with a “mixed modelling” where the concrete material is regarded as a classical two-dimensional continuum while the longitudinal reinforcements are modelled as one-dimensional elements working in tension-compression only. For the beams reinforced in shear, stirrups are incorporated in the analysis through a homogenization procedure. An optimization problem is formulated, then solved using conic quadratic optimization method. As a result, an upper bound estimate to the yield strength domain of RC beams may be drawn in the plane of axial and transverse loads. For illustrative purpose, calculations are conducted on typical RC beams with different longitudinal and transverse reinforcement degrees. Furthermore, it is shown that such numerical predictions prove to be in good agreement with the results derived from other numerical simulations of the same problem using a finite element-based limit analysis commercial software. In order to assess their practical validity, these predictions are also compared to some available experimental results published in the literature.

On the Solution to Pre- Buckling of a Thin Rectangular Plate Subjected to a Thick Strip in-plane Loading

The current paper deals with the problem of the simply supported thin rectangular plate subjected to the intermediate strip in-plane loading. Based on the strain energy method (Fourier ansatz), the critical (minimum value) of buckling stress occurrence was determined in a general form dependent only on the strip thickness, strip location, plate width and stress magnitude. Compatible with the classical columns Euler method it was found that the plate stability is decreased with the increasing of the plate width due to larger induced stresses. Also, strip location relative to the support region was found to influence the buckling (same analogy to the Euler buckling theory; consider the strip as a both sides pressed rod). Additionally, the strip width parameter increase is likely to cause larger buckling stress. Moreover, expressions that includes both axial and transverse loads for different extended cases configurations were also derived and examined based on the strain energy method alongside explanation for possible applications (thin aluminum plate welding). In a general view, it was found that the cases of combined axial and perpendicular loading action are less stabilized than cases where only one kind of loading configuration is participated. Finally, the buckling stress was found to agree qualitatively with the cited literature.

Mathematical modeling of physically nonlinear 3D beams and plates made of multimodulus materials

In this work, mathematical models of physically nonlinear plates and beams made from multimodulus materials are constructed. Our considerations are based on the 3D deformation theory of plasticity, the von Mises plasticity criterion and the method of variable parameters of the theory of elasticity developed by Birger. The proposed theory and computational algorithm enable for solving problems of three types of boundary conditions, edge conditions and arbitrary lateral load distribution. The problem is solved by the finite element method (FEM), and its convergence and the reliability of the results are investigated. Based on numerical experiments, the influence of multimodulus characteristics of the material of the beam and the plate on their stress–strain states under the action of transverse loads is illustrated and discussed.

Static Response of Double-Layered Pipes via a Perturbation Approach

A double-layered pipe under the effect of static transverse loads is considered here. The mechanical model, taken from the literature and constituted by a nonlinear beam-like structure, is constituted by an underlying Timoshenko beam, enriched with further kinematic descriptors which account for local effects, namely, ovalization of the cross-section, warping and possible relative sliding of the layers under bending. The nonlinear equilibrium equations are addressed via a perturbation method, with the aim of obtaining a closed-form solution. The perturbation scheme, tailored for the specific load conditions, requires different scaling of the variables and proceeds up to the fourth order. For two load cases, namely, distributed and tip forces, the solution is compared to that obtained via a pure numeric approach and the finite element method.

Longitudinal-transverse bending of a multilayer bar made of a physically nonlinear material

В работе рассматриваются многослойные бетонные стержни постоянного поперечного сечения. Закон деформирования каждого слоя стержня принят в виде аппроксимации полиномом третьего порядка. Предполагается, что на защемленный стержень действуют квазистатические продольные и поперечные нагрузки и сила тяжести. Рассматриваемые задачи решаются методом Бубнова-Галеркина. The paper considers multilayer concrete rods of constant cross-section. The deformation law for each layer of the bar is adopted as an approximation by a third-order polynomial. It is assumed that quasi-static longitudinal and transverse loads and gravity act on the restrained rod. The problems under consideration are solved by the Bubnov-Galerkin method.