Recibido: 22 de julio de 2020 - Aprobado: 02 de diciembre de 2020
C. V. S. Araújo,J. P. M. Xavier & R. L. Pereira, “Determination of stresses in to-way slabs of concrete through the finite difference method”, Revista Ingenio, 18(1), pp. 25-32 ,2021.
From the implementation of analytical solutions for thin rectangular slabs using the Finite Difference Method, the present paper was developed with the purpose of comparing the determined stresses in a concrete plate with those that would be obtained using the tables originated from the plate theory found in the literature. For this, the Python programming lan-guage was used, taking as a case study, a slab with dimensions of 5 m long by 3 m wide and thickness equal to 0.10 m. In the analysis of the stresses, several support conditions were considered, in which the internal stresses and displacements obtained by the Finite Difference Method were compared with the results obtained through the tables proposed in the litera-ture. It was verified that the Finite Difference Method constitutes a good alternative for the resolution of thin solid plates, since the results were similar to the classic solution proposed in the literature. The implemented program allows the visual-ization of the stresses through of spectrums of zone which facilitates the understanding of the distribution of the stresses along the slab, differing slightly from the uniform distribution adopted in the tables consulted.
Palabras clave:Concrete, Elastic method, Finite differences method, Two-way Slabs.
A partir de la implementación de soluciones analíticas para losas rectangulares delgadas utilizando el Método de Diferencias Finitas, el presente artículo tiene como objetivo comparar los esfuerzos determinados sobre una placa de hormigón con los que se encontrarían utilizando tablas originadas a partir de la teoría de las placas de los autores consagrado en la literatura. Para ello, se utilizó el lenguaje de programación Python, tomando como caso de estudio, una losa con dimensiones de 5 m de largo por 3 m de ancho y grosor igual a 0.10 m. En el análisis de esfuerzos se consideraron varias condiciones de apoyo y se compararon los esfuerzos y desplazamientos internos encontrados por el Método de Diferencias Finitas con los resultados obtenidos a través de las Tablas propuestas en la literatura. Se encontró que el Método de Diferencias Finitas es una buena alternativa para la resolución de losas delgadas, ya que los resultados fueron similares a la solución clásica propuesta en la literatura. El programa implementado permite visualizar los esfuerzos a través de los espectros de rango que facilitó la comprensión de su distribución a lo largo de la losa, diferenciándose de la distribución uniforme adoptada en las tablas consultadas.
Palabras clave:Hormigón, Método elástico, Método de diferencias finitas, Losas bidireccionales.
There are several processes for determining the stresses, among which we can highlight: finite difference, finite elements, equivalent grid and use of series to represent the load value in each coordinate of the plane (p (x, y)).
Thus, the objective of this work is to present a comparative study between the stresses deter-mined by the finite difference method and the tables proposed by . For this, an implementation of an algorithm in Python programming language, analyzing a slab with different boundary conditions, was made.
According to item 14.7.3 of Brazilian code for design of concrete structures, NBR 6118 , methods based on the theory of elasticity can be used in plate structures, with Poisson’s ratio v equal to 0.2, as long as the conditions dealing with linear analysis, stiffness values, redistributing moments, and ductility are met.
In order to use this method, it is necessary to admit some hypotheses, that is: homogeneous, elastic, isotropic material, physically linear and small displacements. It is also admitted that the section remains flat after deformation and the elements representation can be done through its average plane.
In order to facilitate the use of boundary conditions in solving the stress determination prob-lem, most of the time, other considerations are made, such as: the action of the plates on the contour beams occurs only by means of vertical forces, with no transmission of torsion or moments to the beams; the actions of the plates on the contour beams are uniformly distributed, and there is no load transmission directly to the columns; the contour beams cannot be displaced in the vertical direction; and the rotation of the plates in the contour is free (simple support) or totally prevented (embedded).
Solving the fundamental equation (1), we obtain the expression for the elastic surface w=(x,y), and with its derivatives, the moments mx and my in directions x and y, are pre-sented in respectively equations:
The fundamental equation was obtained for moments acting per unit of length, that is, if acting a total moment Mx (adopts as Mx the moment that must be resisted in the direction x) on the plate, of width ly, the moment ly in a unit width (Figure 1) is shown below:
So, let Δx>0 and consider the following Taylor series of a function around a point x0, respectively to the right and left:
Denoting Δxn = (Δx)n and isolating the first de-rivative, we have:
Immediately two possible approximations are obtained for the first derivative of u in x0 , as shown below:
Equation (9) is called a progressive finite difference and equation (10) is called Finite Difference Regression.
The derivatives of the polynomial are consid-ered approximately equal to those of the unknown function, whose partial derivatives are intended to be replaced by finite differences, such as:
Also, according to  apud , these expres-sions are applied to several points, allowing the solution of the problem to be made, generally, through a system of linear equations. In order to facilitate the understanding of the use of the Fi-nite Difference Method in solving thin plate dif-ferential equations, the approach performed by  is presented. As shown in Figure 2, one can observe the nodal points that form the finite dif-ference mesh. Thus, it is expected that the greater the number of points, the better the approximation.
From equation (1), the finite difference equation can be obtained:
Figure 3 shows the numbering of the nodes cor-responding to the displacements (Wi) of equation (12). The ai coefficients are presented in the following set of equations:
Using the coefficients presented in the set of equations (13) and in the sum presented in equation (12), and referencing the elements in the global numbering, we obtain the system of linear equations presented in equation (11), with the number of unknowns equal to the number of displacements to be determined:
The system presented in equation (14) has as a solution the displacements of all analysed nodal points. Once the displacements are obtained, the requesting stresses can be found, substituting in the differential equations that correlate the stresses and the displacements (Figure 4). In this way, the stresses are obtained in a plate element.
Based on the input data (which consists of the plate geometry, material property, loading and boundary conditions), the programming starts to promote the generation of the mesh to be used in the determination of nodal displacements considering the points of intersection between lines that were configured for 0.20 m spacing. It is im-portant to highlight that the characteristics of the steel bars were not modelled in the algorithm, considering that only a simplified analysis was performed, that is, the influence of the reinforcement was not verified through the equivalent elasticity module.
For the application of the finite difference method, the generated mesh was expanded in two points to the outer sides of the slab support limits and the Lagrange equation was applied to each mesh node. The boundary conditions varied between simple supported, fixed or free edges, semi-fixed edges were not considered in this example. The developed algorithm identifies the boundary condition inserted as input data and for the condition of the simple support type, a unit is added to the pivot (central element coefficient, for example, the coefficient in Figure 3), for the fixed support one unit is subtracted from the pivot and when the informed side is free board, the values of the Lagrange equation are modified at the edge and at the internal border.
After this procedure, the weight matrix is assembled and the program calculates the displacement at each point using the finite difference method. The displacement calculation is followed by the plotting of the values found in the form of iso-bands. The bending moment and shear forces are determined from the displacements and for each of these steps, the diagrams representative of the isovalues found are also plotted.
The total load considered was the live load plus the permanent load due to the weight of a 0.10 m thick reinforced concrete slab (specific weight equal to 25.0 kN/m³). All slabs were 3 m wide according to the “x” axis and 5 m long according to the “y” axis. The slabs behave like reinforced plates in two directions since the ratio between the longest side and the shortest side was less than two (λ = 1.67). Figure 7 shows the five cases of boundary conditions studied, related to the interaction between the plate and its support.
Figure 7(a) shows the four edges simply supported. In Figure 7(b), only the left side of 5.0 m was fixed and the others remained supported. In Figure 7(c), the two 5.0 m sides are fixed, and the two 3.0 m sides are simply sup-ported. The model shown in Figure 7(d), present all edges fixed. Figure 7(e) corresponds to a slab with three sides fixed and a 3.0 m edge (free) without support.
Displacements and stresses are shown in table 3 for case “b”.
Displacements and stresses are shown in table 4 for case “c”.
Table 5 shows the results for case “d”.
Table 6 shows the results for case “e”.
The data of the shear stresses could not be determined for the “e” case (three fixed supports and a free edge) because the Bares’s tables used do not provide the support reactions for this situ-ation.5.2 Analysis and discussions
Figure 8(a) shows the graphs of positive and negative bending moments isovalues for the “y” direction of case “b”. It is observed by the isovalue bands indicated in the graph that in the region of the connection corner between the fixed and the simple support a negative moment develops due to the influence that the fixed support has on the beginning of the simply supported section. Such behavior could also be seen in the isovalue graph for case “c”, shown in Figure 8(b).
The shear efforts were the output data of the processing that showed the biggest percentage differences. It should be noted that the shear forces on the slab that had a free edge were not evaluated. Despite the divergence of the values deter-mined by the two methods, it was possible to perceive symmetry when the opposite edges present the same boundary conditions.
Figure 9 shows the results of shear stresses in the “y” direction for the “c” case (two opposite sides fixed). Despite corresponding to the greatest differences between the stresses deter-mined according to the two methods (47.72%), it is possible to observe the expected symmetry between the values from the existing support conditions.
As the Bares’s tables show an average value between the different stresses along the beam length and the results obtained by the finite difference method were the highest values observed, the marked difference is attributed to the simplifications that guided the making of the tables for dimensioning of reinforced concrete slabs.
The simplification adopted in the tables (which use the theory of elasticity applied to the plates) in which an average value is considered over the entire length of the support. In some cases, it can generate a detail of inadequate reinforcement positioning, mainly when considering the distribution of stresses in the corner regions (support region). Using the finite difference method, it is possible to visualize a more real distribution of these stresses, thus leading to a better reinforcement arrangement and eventual savings in the structural design.
Thus, it was found that depending on the boundary conditions, some results have a better refinement through the method of finite differ-ences compared to the process of classical literature (Bares’s tables), which can be a mathematical tool that can be used by software due to its ease of implementation.7. Acknowledgments
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 Associação Brasileira de Normas Técnicas - ABNT: NBR 6118 “Projeto de estruturas de con-creto – Procedimento”. Rio de Janeiro, 2014.
 J. D. Hoffman and S. Frankel. Numerical meth-ods for engineers and scientists. CRC Press, Boca Raton, 2001.
 G. D. Smith. Numerical solution of partial differ-ential equations: finite difference methods. 3rd ed. Oxford University Press, 1985.
 J. Biezuner. Class notes. (2016, November 24) [Online]. Available: http://www.mat.ufm g.br/~rodney/notas_de_aula/df.pdf.
 J. M. Araújo. Curso de concreto armado. 4th ed. v. 2. Rio Grande, Dunas, 2014.
[ D. A. O. Martinelli, I. Montanari and W. Savassi. “Placas Elásticas”. EESC-USP, São Carlos, Bra-zil, Publicação 018/93, 1986.
 S. D. C. Júnior. “Elaboração de um programa computacional para o cálculo e dimensionamento de lajes maciças”. Final year project, Faculdade de Engenharia, Universidade Federal de Juiz de Fora, Juiz de Fora, 2008
 J. V. Guttag. Introduction to Computation and Programming Using Python: With Application to Understanding Data. 2nd ed. Cambridge, MIT Press, 2016.
 Associação Brasileira de Normas Técnicas - ABNT: NBR 6120 “Cargas para o cálculo de es-truturas de edificações”. Rio de Janei-ro, 1980.
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