ANALYSIS THROUGH CONSTRUCTAL DESIGN OF THE INFLUENCE OF SPACING BETWEEN STIFFENERS IN THE DEFLECTION OF PLATES ANÁLISE VIA DESIGN CONSTRUTAL DA INFLUÊNCIA DO ESPAÇAMENTO ENTRE ENRIJECEDORES NA DEFLEXÃO DE PLACAS

Using the Finite Element Method (FEM), stiffened plates arrangements defined by the application of the Constructal Design Method (CDM) were analyzed. So that, through an Exhaustive Search (ES), different spacing between the stiffeners were tested regarding the central and maximum deflections. Starting from a non-stiffened plate with a fixed volume, a portion of its material was completely removed from its thickness and transformed into stiffeners considering the volumetric fraction φ = 0.5. It were established 4 arrangements: P(2,2), P(2,3), P(3,2) and P(3,3), varying for each one, the spacing between the stiffeners, as well as the parameter hs/ts (ratio between height and thickness of stiffeners). The results showed that stiffeners equally spaced in the longitudinal and transverse directions with higher ratios hs/ts are more effective, being able to reduce the central and maximum deflections by more than 95% compared to the non-stiffened reference plate.


INTRODUCTION
Plates are structural components that have a much smaller dimension than the others, i.e., their thickness is very small when compared to the width and length (SZILARD, 2004). Usually the plates work under flexion and/or compression parallel to its plane, being widely used in engineering, such as aerospace, civil, mechanical and naval (BIRMAN, 2011). However, due to its high slenderness it is necessary to insert stiffeners, traditionally fixed in longitudinal and/or transverse directions, whose main function is to increase stiffness.
Analytical solutions for analyzing the mechanical behavior of plates are uncommon, and when exist, usually lead to inaccurate results. Hence, it is interesting to apply numerical methods to solve this type of problem. Rossow and Ibrahimkhail (1978) presented models of stiffened plates with concentric and eccentric stiffeners using the Method of Constraints. Mukhopadhyay and Satsangi (1984) proposed a model based on the Finite Element Method (FEM) for the analysis of stiffened plates employing an isoparametric type finite element. Tanaka and Bercin (1998) used the Boundary Element Method (BEM) to study stiffened plates under bending, considering the effects of bending, torsion, deformation and eccentricity of the stiffeners in the equation. Salomon (2000) explains that depending on the design requirements, the stiffeners can occupy any position on the plate, aiming an effective structural arrangement, which is capable of resist the imposed load, limiting the stresses and deformations to their allowed values.
More recently, Fernandes (2009) presented a numerical model based on the hypotheses of Kirchhoff's theory and applying the BEM to analyze the bending in plates reinforced by beams considering different materials. Benai and Pedatzu (2010) based on orthotropic plate work of Shade (1940,1941,1951) implemented a computer program to analyze stiffened plates under bending. Damanpack et al. (2013) developed the analysis of functionally graded stiffened plates through BEM, i.e., plates with material properties variable along the thickness.
Finally, the application of the CDM in the structural analysis of aircraft also stands out (MARDANPOUR et al., 2019 andIZADPANAHI et al., 2020).
Therefore, the present study analyzed different arrangements of stiffened plates defined by the application of the CDM, numerically solved by applying the FEM using the ANSYS ® software, and compared among each other through the Exhaustive Search (ES) technique. So, considering the variation of spacing between the stiffeners and variation of hs/ts (ratio between height and thickness of stiffeners), it was possible to identify the geometric configuration that minimizes the central and maximum deflections.

Computational Modeling
The FEM is a numerical method widely applied in the analysis of engineering problems, especially when it is not possible to obtain a precise solution through analytical solutions (COOK et al., 2001;SEGERLIND, 1984).
According to Szilard (2004), the FEM consists of the decomposition of the continuous domain into sub-domains of finite size connected by nodal points.
The displacement field of each element is arbitrated according to the displacement of the nodes, replacing the continuous mathematical model by the equilibrium of each finite element, converting the differential equations into algebraic equations in each element. Thus, this global system allows the determination of the approximate solution after the introduction of the loads and boundary conditions (ASSAN, 2003;SORIANO, 2003).
Revista Mundi Engenharia, Tecnologia e Gestão. Paranaguá, PR, v.5, n.5, p. 275-01, 275-21,2020. DOI: 10.21575/25254782rmetg2020vol5n51354 4 In this study, the FEM was used to solve the proposed plate models by means the ANSYS ® software. The SHELL281 finite element was adopted, since it is suitable for modeling thin and moderately thick plates. This two-dimensional element is based on the Reissner-Mindlin hypotheses, having 8 nodes in its quadrilateral version and 6 nodes in its triangular version, with 6 degrees of freedom per node, being 3 rotations and 3 translations in relation to the x, y and z axes (ANSYS, 2019).

Verification and Validation of the Computational Model
For the verification, it was used the case presented in Fig

Constructal Design Method (CDM)
The Constructal Law is the physical phenomenon behind the vast geometric complexity of the flow systems that exist in nature. These systems generate shape and structure over time in order to facilitate flow access that passes through them. Thus, the geometric design of the system is not a random result; it arises in a natural evolutionary attempt to achieve its best performance (BEJAN AND ZANE, 2008;BEJAN AND LORENTE, 2008).
Based on a principle of restrictions and objectives, the CDM is the practical application of the Construct Law. The performance of a system carries inherent restrictions, which may be the space allocated for its evolution, the material available, as well as limit rates of pressure, temperature and stress. When the problem restrictions are defined, the degrees of freedom related to the geometric parameters are modified, searching an arrangement that achieves the better possible performance according to a predetermined performance indicator (REIS, 2006;DOS SANTOS et al. 2017).
It is possible to find several studies applying the CDM on problems of fluid mechanics and heat transfer. However, its use in structural analysis is still few where Vs is the material volume of the stiffeners and Vr is the material volume of

RESULTS
To determine the appropriate size of the finite elements used in the discretization of the numerical models analyzed in this study, a mesh convergence test was previously performed, using the plate P(2,2) with Sl = 222.22 mm, St = 111.11 and hs/ts = 20.84. The size of the finite elements was reduced consecutively with reference to the plate width b = 1000 mm, according to the mesh definition criteria presented in Troina et al. (2020). The result can be seen in Fig. 7.

Figure 7 -Mesh convergence test
According to Fig. 7, from the M4 mesh with finite element size of 12.5 mm the values converged for the central deflection of the plate Uz. So, this is the size of the finite element determined to discretize all other cases (see Fig. 6).
In a first analysis, Figs. 8 to 11, respectively, for P(2,2), P(2,3) P(3,2) and P(3,3), show the results of central and maximum deflection for each spacing situation (see Fig. 6) as a function of the variation in the hs/ts ratio.

CONCLUSIONS
Applying the CDM, different arrangements of stiffened plates were proposed, being solved by the application of FEM. After that, through the ES it was evaluated the influence of the variation of the spacing between the stiffeners in terms of central and maximum deflections occurred in the plates.
First of all, it can be concluded that transforming a portion of material deducted entirely from a non-stiffened plate into stiffeners, keeping the total material volume constant, improves considerably the mechanical behavior of the plates, being able to achieve reductions greater than 95% in the central and maximum deflections. Still as expected, the raise in the hs/ts ratio caused a significant stiffness improvement in all analyzed plates.
It was also possible to notice that the smallest central deflections were achieved when there was a decrease in the St and Sl spacing. Thus, the optimized configuration that best minimized the central deflection was the plate P(3,3) for the ratio hs/ts = 35.00 for the situation of spacing St = 333.33 mm and Sl = 666.67 mm.
However, concerning the maximum deflection, the lowest values found occurred where the stiffeners were equally spaced in both directions (transverse and horizontal). The best performance achieved in terms of maximum deflection was for plate P(3,3) with ratio hs/ts = 35.00 and St = 500.00 mm and Sl = 1000.00 mm.
In future works it would be interesting to analyze other volumetric fractions , as well as other arrangements with different amounts of stiffeners and spacing.