The need for quick and easy deflection calculations of various prefabricated slabs causes simplified procedures and numerical tools to be used more often. Modelling of full 3D finite element (FE) geometry of such plates is not only uneconomical but often requires the use of complex software and advanced numerical knowledge. Therefore, numerical homogenization is an excellent tool, which can be easily employed to simplify a model, especially when accurate modelling is not necessary. Homogenization allows for simplifying a computational model and replacing a complicated composite structure with a homogeneous plate. Here, a numerical homogenization method based on strain energy equivalence is derived. Based on the method proposed, the structure of the prefabricated concrete slabs reinforced with steel spatial trusses is homogenized to a single plate element with an effective stiffness. There is a complete equivalence between the full 3D FE model built with solid elements combined with truss structural elements and the simplified homogenized plate FE model. The method allows for the correct homogenization of any complex composite structures made of both solid and structural elements, without the need to perform advanced numerical analyses. The only requirement is a correctly formulated stiffness matrix of a representative volume element (RVE) and appropriate formulation of the transformation between kinematic constrains on the RVE boundary and generalized strains.
Abstract. The article presents the results of validation of static calculations carried out for a monolithic rectangular tank with walls of trapezoidal cross-section. Static calculations were made with the use of software based on the finite element method (FEM) and the finite difference method (FDM) in terms of energy (including spatial static work of the tank). Validation of the results was conducted on a concrete tank model using an innovative measurement tool, i.e. a coordinate measuring arm with a touch probe.
The paper presents the effect of considering the substrate under the floor—insulation in the form of closed-cell polyurethane spray foam, which is used for insulating surfaces particularly exposed to mechanical impact. The layer of thermal insulation was made by spraying, which prevents the occurrence of thermal bridges due to tight filling of the insulated space. It seems extremely important to adopt the appropriate material characteristics of an insulating layer. The basic thermophysical properties of polyurethane foam justifying its choice as an insulation material were the values of its thermal conductivity coefficient (0.022 W/(mK)) and density (36 kg/m3). However, what was the most important for the calculations provided in the work was to determine the stiffness of the foam subgrade so as to assess its impact on the floor load capacity. The paper includes calculations for a floor slab characterized by a static diagram, with all edges free (unfixed), loaded in strips circumferentially. The reinforced concrete slab was 6 × 6 m long, 0.25 m thick, and made of C20/25 concrete resting on an elastic substrate. Calculations were made for two variants taking into consideration two values of subgrade stiffness. The first variant concerned the subgrade stiffness for sprayed polyurethane foam insulation. On the basis of laboratory tests in situ made according to the standard procedure, its average value was assumed as K = 32,000 kN/m3. The second, comparative, computational variant included the subgrade stiffness equal to K = 50,000 kN/m3. A variation approach to the finite difference method was used for static calculations, adopting the condition for the minimum energy of elastic deformation while undergoing bending that was accumulated in the slab resting on a Winkler elastic substrate. Static calculations resulted in obtaining the values of deflections at each point of the discretization grid adopted for the slab. The obtained results have proved the necessity of calculating the floor as a layer element. For the reference substrate with the subgrade stiffness K = 50,000 kN/m3 that was adopted in the work, the value of the bending moment was 17% lower than when taking into account that there was thermal insulation under the floor slab, causing an increase in the deflection of the slab and an increase in its bending moment. If a design does not include the actual subgrade stiffness of the layer under the floor slab, it results in an understatement of the values of the bending moments on the basis of which the slab reinforcement is designed. Adherence of insufficient concrete slab reinforcement may cause subsequent damage to floor slabs.
The method to calculate rectangular tanks as a system of bi-directionally bent plates with the use of separated plates methodology is a widely known and currently most often used approach to verify numerical calculations obtained from computer software supporting the design process, in which spatial operation of tanks is taken into account. In these calculations, due to their static scheme, it is possible to distinguish supported plates, plates fixed on four edges and plates with one edge free and three fixed. The subject literature contains publications on plates or tanks with walls of a constant thickness, however, there are very few references on plates or tanks with walls of linearly variable thickness. The wall plate of a tank is subject to hydrostatic load or soil pressure and might be exposed to thermal load in the case of, i.e. filling it with hot liquid or during climate action. The article presents the results of static calculations for rectangular plates with a linearly variable thickness, a trapezoidal cross-section, three fixed edges and one edge free, subjected to permanent and thermal loads. Trapezoidal cross-section walls are optimal when used in structures where load distribution is triangular in shape (hydrostatic load). For tanks recessed in the ground, the load on walls increases along with the depth of foundations and obtains the highest value in the bottom part of the wall. Trapezoidal or triangular load distribution causes that the highest values of bending moments in the vertical cross-section occur at the point where a wall connects to the bottom, while the upper free edge of the tank takes zero value. The above statements lead to the conclusion that structural and economic considerations should determine the choice of walls with a thickness increasing along with the tank depth, since it is more economical in terms of material usage. The impact of thermal load is often neglected in the design process, which may cause operational problems and even pose a threat to the safety of use. In addition to the numerical analysis, the article presents the results of model tests for a plate with a linearly variable thickness made of resin, subjected to thermal load. The convergence of the obtained results proves the correctness of calculations and tests performed. This also contributes to the recognition of statics in rectangular plates of a trapezoidal cross-section.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.