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Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation

Application of generalized equations of finite difference method to computation of bent isotropic... Curved and Layer. Struct. 2022; 9:54–64 Research Article Seydou Youssoufa, Moussa Sali, Abdou Njifenjou, Nkongho Anyi Joseph*, and Ngayihi Abbe Claude Valery Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation https://doi.org/10.1515/cls-2022-0005 Keywords: rectangular plate, elastic foundation, general- Received May 03, 2021; accepted Sep 29, 2021 ized equations of finite difference method, discontinuity Abstract: The computation of bent isotropic plates, stretched and/or compressed, is a topic widely explored in the literature from both experimental and numerical point 1 Introduction of view. We expose in this work an application of the gener- alized equations of Finite difference method to that topic. One can start by recalling that a plate is a structure, which The strength of the proposed method is the ability to recon- thickness is small beside its length and width. What hap- struct the approximate solution with respect of eventual pens when you crumple up a sheet of paper? How does a discontinuities involved in the investigated function as well general raft supporting a building behavior? Can the futur- as its first and second derivatives, including the right-hand istic roofs that adorn our most glorious buildings (shopping side of the equilibrium equation. It is worth mentioning that centers, airport buildings. . . ), resist the wind? In which way by opposition to finite element methods our method needs the carrosseries of a car is distorted in an accident? Here are neither fictitious points nor a special condensation of grid. some of many scenarios of structure behavior that arise in Well-known benchmarks are used in this work to illustrate real life problems. Characterizing the deformations under- the efficiency of our numerical and the high accuracy of gone under certain constraints is our aim in this work. The calculation as well. A comparison of our results with those issues related to this study are diverse: it will sometimes available in the literature also shows good agreement. be a question for the engineer of designing resistant and / or aesthetic plates or shells. Sometimes covertly, the plate specifications must provide the deformations distributed Seydou Youssoufa, Moussa Sali, Ngayihi Abbe Claude Valery: University of Douala, Laboratory of Research in Energy, Material, Modeling and Methods, National high school polytechnic Douala po box 2701 Douala, Cameroon Abdou Njifenjou: University of Douala, Laboratory of Research in Energy, Material, Modeling and Methods, National high school polytechnic Douala po box 2701 Douala, Cameroon; University ofYaoundé I, Laboratory of mathematical engineering and infor- mation system, National high school polytechnic Yaoundé, Po Box 8390 Yaoundé, Cameroon *Corresponding Author: Nkongho Anyi Joseph: University of Douala, Laboratory of Research in Energy, Material, Modeling and Methods, National high school polytechnic Douala po box 2701 Douala, Cameroon; University ofYaoundé I, Laboratory of mathe- matical engineering and information system, National high school polytechnic Yaoundé, Po Box 8390 Yaoundé, Cameroon; University of Buea, Department of mechanical engineering, HTTTC, Po Box 249 Buea Road, Kumba, Cameroon; E-mail: nkongho.anyi@ubuea.cm Open Access. © 2022 Seydou et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation | 55 over its mid surface. The designed structures aim is to ab- isfactory approximate solutions faster than the successive sorb shocks, for example the front part of a vehicle or even approximation method (SAM, for short). submerged radiators. The work is organized as follows: after the introduc- During their exploitations, plate structures are sub- tion which poses the problematic of the subject, we unfold jected to the transversal loads (statics and dynamics). From the methodology of the implementation of the generalized computational point of view, efficient numerical tools are equations of the finite difference method which is broken necessary for modeling sophisticated mechanical behavior down into three points. Subsequently we transform the of such structures, accounting with their specificities. De- new differential equation by the FDM. The last part of this spite the abundance of literature on computations of plate framework will be devoted to the validation of our approach structures [1, 2, 3], several questions remain topical here. through the numerical resolution of test- problems. The behavior of those structures is governed by a lin- th ear partial differential equation (PDE, for short) of 4 order which is not obvious to be solved using analytical meth- 2 Tools and techniques ods [5, 6, 7]. Hence, to call out for numerical methods easy to implement and less onerous from computational point In several works related to the numerical calculation of of view. Among popular numerical methods used for this plates and shells, the authors use various approaches to topic, the finite element method (FEM, for short) is the most solve the partial differential equations which govern these popular [4, 8, 9, 10]. However, the FEM presents a certain structures. As far as we are concerned in this frame work, number of drawbacks as indicated in [9]: (a) The FEM is our resolution methodology is as follows: poorly adapted to a solution of the so-called singular prob- th (a) transformation of the partial derivatives 4 order de- lems like plates with cracks, corner points, discontinuity formation equation of a rectangular plate of variable internal actions, and of problems for unbounded domains. thickness into a system of two differential equations (b) This method requires the use of powerful computers of nd of 2 order partial derivative; considerable speed and storage capacity. (c) The method (b) introduction of new dimensionless parameters in the presents many difficulties associated with problems of C system of equations obtained and in the equation continuity and nonconforming elements in plate (and shell) describing the boundary conditions; bending analysis. Note that the mathematical theory of FEM (c) transformation of new differential equations by the is exponentially increasing. So the above drawbacks could generalized equations of the finite difference method, be addressed in a near future by the FEM. these permits a system of algebraic equation to be More recent numerical methods have been developed obtain; for addressing singular problems [10, 11, 12, 13, 14, 15, 16]. (d) transformation of boundary conditions; Among the large variety of those methods for addressing (e) elaboration of a calculation algorithms; singular problems, the Generalized Differential Quadrature (f) resolution of the system of algebraic equation in or- (GDQ, for short) is proposed to solve different kinds of struc- der to obtain the bending moment and the maximum tural problems. In many applications present in literature displacement. [17, 18, 19, 20], the GDQ method has shown superb accu- racy, efficiency, convenience, and great potential in solving differential equations. The generalized equations of the Fi- 2.1 Equation of deformation of a bent nite Difference Method (GE-FDM, for short) are part of these recent numerical methods [21, 22, 23]. rectangular plate stretched and /or The aim of this work is to show that the generalized compressed with variable rigidity on equation of the finite difference method (GE-FDM, for short) elastic foundation could be used to address the computation of bent, stretched and/or compressed rectangular plates of variable stiffness In the following paragraphs, only bent, stretched and/or under elastic foundations. One of the main features of this compressed plates of variable thickness will be analyzed. method (GE-FDM) is the ability of dealing with finite discon- So it is convenient to express the governing differential tinuities of the investigated solution and that of its first and equation of this plate. Figure 1 illustrates of the equilibrium second derivatives, including discontinuities of the right of a sample plate stretched or compressed element: hand side of the primary PDE. According to [22] and [23, 25], Let us consider an infinitesimal element d , d as indi- X Y the computation of these plates with GE-FDM leads to sat- cated in Figure 1 and projections of membrane forces along 56 | Seydou et al. 2 2 ∂ W ∂ W M ⎪ + = − 2 2 ⎪ D ∂X ∂Y (︁ )︁ 2 2 2 2 2 ∂ M ∂ M ∂ W ∂ W ∂ W + = F + N + N − 2N − RW + 2 2 2 2 Z X Y XY ∂X ∂Y ∂X ∂Y ∂X∂Y (︁ )︁ 2 2 2 2 2 2 ⎩ ∂ D ∂ W ∂ D ∂ W ∂ D ∂ W +(1 − µ ) + − 2 2 2 2 2 ∂X ∂Y ∂Y ∂X ∂X∂Y ∂X∂Y (4) where 2 2 2 2 ∂ W ∂ W ∂ W ∂ W M = + µ ; M = + µ (5) X Y 2 2 2 2 ∂X ∂Y ∂Y ∂X M + M X Y M = (6) 1 + µ M denotes the resultant moment, so M and M are X Y Figure 1: Element of a bent plate stretched and /or compressed. bending moment following X and Y directions, respectively. the Z - axis. N , N and N are respectively the horizon- X Y XY 2.2 Introductions to dimensionless tal components of the normal and shear forces which are parameters exerted on the various facets. The normal unit vector to the facets is tangent to the neutral plane in each direction Rewriting the Eq. (4) using dimensionless parameters [1], (X,Y). By neglecting the forces of volume along the X and [14]: Y directions, we obtain according to Z- axis: Y X F M WD Z 0 η = ; ξ = ; F = ; m = ; v = ; 2 2 2 2 3 3 2 4 ∂ D ∂ W ∂ D ∂ W ∂D ∂ W ∂D ∂ W l l F F l F l 0 0 0 + µ + 2 + 2 + (7) 2 2 2 2 3 2 ∂X ∂X ∂X ∂Y ∂Y ∂X ∂X ∂X∂Y M M D(X; Y ) (ξ ) (η) X Y m = ; m = ; g = 3 2 2 2 2 2 2 2 2 ∂D ∂ W ∂ D ∂ W ∂ D ∂ W ∂ D ∂ W F l F l D 0 0 0 2 + 2 − 2µ + 2 2 2 ∂Y ∂X ∂Y ∂X∂Y ∂X∂Y ∂X∂Y ∂X∂Y ∂Y ∂Y 2 2 2 2 2 Nl N N −2N X Y XY ∂ D ∂ W ∂ W ∂ W ∂ W ¯ l = max (|l | , |l |) ; k = ; α = ; 𝛾 ¯ = ; β = X Y + µ + D∆∆W = N + N − 2N X Y XY D N N N 2 2 2 2 0 ∂X∂Y ∂Y ∂X ∂X ∂X (8) − RW + F where N = max (|N | , |N | , |N |) ;−1 ≤ α ≤ 1, −1 ≤ X Y XY (1) β ≤ 1, −1 ≤ 𝛾 ≤ 1; (η ; ξ) are Cartesian coordinates without with units; F is Load factor; m is Moment coefficient; µ is the co- 2 2 2 2 2 2 ∂ D ∂ W ∂ D ∂ W ∂ D ∂ W efficients of deflection; D denotes the cylindrical stiffness D∆∆W = − +2 (2) 2 2 2 2 ∂Y ∂X ∂Y ∂Y ∂X∂Y ∂X∂Y of any section of the slab; g is the stiffness coefficient; l is plate length; α , β, 𝛾 and k are coefficients without unit. where EH Introducing the parameter in Eqs. (7) and (8) into the D = (3) 12(1 − µ ) system of Eq. (4), we obtain: Eq. (1) is called the deformation equation of a bent rect- 2 2 ∂ v ∂ v m angular plate stretched and /or compressed with variable + = − 2 2 ⎪ g ∂ξ ∂η ⎨ (︁ )︁ 2 2 2 2 2 rigidity on elastic foundation; ∂ m ∂ m ∂ v ∂ v ∂ v + − k α ¯ + 𝛾 ¯ + β + (9) 2 2 2 2 ∂ξ ∂η ∂ξ ∂η ∂ξ∂η ⎪ (︁ )︁ where W = W (X, Y ) is the transversal dis- ⎪ 2 2 2 2 2 2 ⎩ ∂ g ∂ g ∂ g ∂ v ∂ v ∂ v (1 − µ ) + − 2 − λv = F 2 2 2 2 ∂ξ ∂η ∂η ∂ξ ∂ξ∂η ∂ξ∂η placement of the plate (searched function); D = D(X, Y ) stiffness of a variable p late; µ is Poisson’s coef- Rl where λ = . ficient; N , N are normal membrane forces; N denotes X Y XY The Eq. (9) can be written as follows: membrane shear forces ; R is the ground stiffness in N/m ; F represents the volume force along the Z - axis, while H Z 2 2 ∂ v ∂ v m ⎪ + = − 2 2 ⎪ ∂ξ ∂η g =H(X, Y ) is the variable thickness of the plate. (︁ )︁ 2 2 2 2 ∂ m ∂ m ∂ g ∂ v nd + + −kα ¯ + (1 − µ ) + The Eq. (1) can be transformed as a system of 2 order 2 2 2 2 ∂ξ ∂η ∂η ∂ξ ⎪ (︁ )︁ (︁ )︁ ⎪ 2 2 2 2 ⎩ ∂ g ∂ v ∂ g ∂ v partial derivative equations (pde): ¯ ¯ + −k𝛾 ¯ + (1 − µ ) + −kβ − 2(1 − µ ) = F + λv 2 2 ∂ξ∂η ∂ξ∂η ∂ξ ∂η (10) Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation | 57 Eq. (10) is called the generalized algebraic equation of 3 Formulation of the Generalized the finite difference method which replaces Eq. (4). Note that v and its partial derivative can be discontinuous when Equation of the Finite Difference the plate has ball points, while m will be discontinuous if Method (GE- FDM) external point bending moments are applied in one of the directions of the coordinate axes. The technique exposed here for obtaining the generalized equations of the finite difference has been first introduced in [14]. 2.3 Boundary conditions Several boundary conditions are discussed in this work 3.1 Finite difference mesh over the structure in accordance with practical and site works constraints in foundations design. We are going to emphasize on displace- A square mesh of size h is defined over the structure as ment and moments conditions on the borders. indicated in Figure 2 where the roman numerals are used for mesh element numbers and (i,j) represents mesh nodes 2.3.1 Articulated supports If the articulated edge is parallel to the X - axis, in other words, Y = 0, then: W = 0; M =0. If the above conditions are imposed on the edge of the plate: W =W (X) and M =M X , therefore from formu- ( ) 0 Y Y las (10), we obtain: ∂ W (X) Y 0 M = M (X) − D(1 − µ ) (11) ∂X If the edge is parallel to the Y - axis, in other words, X = 0, then: W= 0; M = 0. (12) Figure 2: Square mesh for GE-FDM. 2.3.2 Embedded edges Taking into consideration Eq. (6) established in [14] If the embedded edge is parallel to the Y - axis, then: and identifying Eq. (10) with Eq. (6), we obtained: (︂ )︂ ∂W (a) ; (W ) = 0 (13) y=a ∂Y y=a ij V + V + V + V − 4v = −h (15) i,j−1 i−1,j i+1,j i,j+1 i,j If on the other hand it is parallel to the X - axis, then: ij (︂ )︂ ∂W with P = − ω = v ; α = 𝛾 = 1 ; δ = β = σ = o; also W = 0; = 0 (14) ( ) x=a ∂x considering that m and vare continuous, as well as their x=a first and second order derivatives All the ingredients are gathered to find out the general- ized equations of the finite difference method. 58 | Seydou et al. (b) tion is solved taking into consideration the transformed boundary conditions. m + m + m + m − 4m + i,j−1 i−1,j i+1,j i,j+1 i,j (︁ )︁ ξ ξ η η I−II III−IV I−III II−IV ∆ m + ∆ m + ∆ m + ∆ m + ij ij ij ij [︃ ]︃ (︀ )︀ 3.2 Transformation of the boundary k β v − v − v + v + i−1,j−1 i−1,j+1 i+1,j−1 i+1,j+1 − (︀ )︀ (︀ )︀ + 4 4α ¯ v + v + 4𝛾 ¯ v + v − 8(α ¯ + 𝛾 ¯ )v conditions by the generalized equation i−1,j i+1,j i,j−1 i,j+1 i,j [︃ ]︃ (︀ )︀ ξη of the finite difference method − g v − v − v − v + i−1,j−1 i+1,j+1 i+1,j−1 i−1,j+1 2h + (1 − µ ) (︀ )︀ (︀ )︀ 1 ηη 1 ξξ g v + v − 2v + g v + v − 2v i−1,j i+1,j i,j i,j−1 i,j+1 i,j 2 2 h h In this section, we establish equations called boundary = −h P i,j conditions, which are combined to the Eq. (17) lve the differ- (16) 2 2 ∂ g ∂ g ηη ξξ ξη ential Eq. (2). The said boundary conditions are presented where: P = F + λv . ; g = ; g = and g = i,j i,j i,j 2 2 ∂η ∂ξ 2 below. ∂ g ∂ξ∂η ∂ g With: ω = m ; α = −kα + (1 − µ ) ; 𝛾 = −k𝛾 + 1 2 1 ∂η 2 2 ∂ g ∂ g 3.2.1 Articulated sides (1 − µ ) ; β = −kβ + (1 − µ ) and σ = δ = 0, 2 1 1 1 ∂ξ ∂ξ∂η then also consider that P is constant within inside of each If all the edges of the plate are articulated, it is enough to element but can vary abruptly from one element to another. solve the Eqs. (15) and (16) Those equations are written for Eqs. (15) and (16) are called the generalized equations each point inside the field. Thus, we have: of the finite difference method for a bent rectangular plate stretched and /or compressed with variable stiffness on elastic foundation, which substitute’s Eq. (10), where, = 2, (ξ ) (η) ξξ m = 0 ; v = v (ξ ); m = m (ξ ) − (1 − )v (ξ ); 0 0 3. .. n-1; = 2, 3, ... ,-1. (18) (i) At the end, F+λv will be replaced by F + λv . ξξ (η) i,j i,j where v (ξ ) ; m (ξ ); v (ξ ) are all known. 0 0 We write the equation for a regular mesh: h = τ = h = τ i i i i+1 3.2.2 Embedded edges hence, by combining Eqs. (15) and (16) we obtain for We remind that if an edge is embedded, the deflection and Eq. (10): rotation on that edge are zero. Edges parallel to ξ- axis: ij v + v − 4v + v + v = −h ⎪ i−1,j i,j−1 i,j i+1,j i,j+1 ⎪ if the edge η =0 is embedded, then: ij ⎪ ∂v (v ) = 0 either v = v = v = 0 and ( ) = m + m + m + m − 4m + η ⎪ i,j i−1,j i+1,j i,j η=0 i,j−1 i−1,j i+1,j i,j+1 i,j ⎪ ∂y (︁ )︁ η ξ ξ η η 0. Hence, v = 0. By substituting the first equation of the ⎪ I−II III−IV I−III II−IV i,j ⎪ ∆ m + ∆ m + ∆ m + ∆ m + ij ij ij ij )︁ system (17) with the generalized Eq. (3) giving in [14] and [︃ ]︃ k β(v − v − v + v + i−1,j−1 i−1,j+1 i+1,j−1 i+1,j+1 by noting that α = 𝛾 = 1 ; δ = β = σ = o we obtain: ⎪ − (︀ )︀ (︀ )︀ + 4 η m ⎪ 4α ¯ v + v + 4𝛾 ¯ v + v − 8(α ¯ + 𝛾 ¯ )v 1 2 ij i−1,j i+1,j i,j−1 i,j+1 i,j 2v = 0 = (v +v +v +v − 4v −h ), ⎪ 2 i,j−1 i−1,j i+1,j i,j+1 i,j ⎪ [︃ ]︃ i,j h g (︀ )︀ ij ⎪ ξη − g v − v − v − v + ξ 2 i−1,j−1 i+1,j+1 i+1,j−1 i−1,j+1 2 2h ⎪ (︀ )︀ (︀ )︀ + (1 − µ ) either m = v from where: ⎪ 2 i,j+1 1 ηη 1 ξξ i,j h ⎪ g v + v − 2v + g v + v − 2v i−1,j i+1,j i,j i,j−1 i,j+1 i,j 2 2 ⎪ h h {︃ = −h P i,j v = 0 i,j (19) (17) ξ −2g i,j m = V ξ ξ ξ ξ 2 i,2 i,j ξ ξ h I−II I II III−IV III IV where: ∆ m = m − m ; ∆ m = m − m ; i,j i,j i,j i,j i,j i,j where i = 2, 3, ....., n − 1 ; j = 2. ξ ξ ξ ξ ξ ξ I−III I III II−IV II IV ∆ m = m − m ; ∆ m = m − m , i = i,j i,j i,j i,j i,j i,j if the edge η = is embedded then: 2, 3, ....., n − 1; j = 2, 3, ....., n − 1. by proceeding in the same way as previously and by h - Mesh spacing; i - measuring along the Axis; j - mea- considering the pair of elements (a-c), and according to suring along the Axis [14], we obtain: Eq. (17) is called the generalized algebraic equation of {︃ the finite difference method for a bent rectangular plate v = 0 i,η stretched and /or compressed with variable stiffness on i = 2, 3, . . . . . . , n − 1 (20) ξ −2g i,η m = v 2 i,η−1 i,η elastic foundation, which substitute Eq. (10). This equa- h Edges parallel to η- axis: Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation | 59 If the edge ξ = is embedded: 4.1 Example 1: Case of bending combined By considering the pair of element (c-d) we have: with unidirectional compression For this first example, it is a square plate subjected to bend- ξ ij 2v = + v + v + v − 4v − h i,j−1 i−1,j i+1,j i,j+1 i,j i,j ing combined with compression. The four sides of which ij are articulated. It is also subjected to the action of a load and noticing that in [14], α = 𝛾 = 1 ; δ = β = σ = o we uniformly distributed over its entire surface. Moreover, the η 2g 1,j obtain: m = V , 2 i+1,j i,j h compression loads are applied uniformly and parallel to - from where: axis. A mesh of such a plate is shown in the Figure 3. {︃ v = 0 1j , j = 1, 2, . . . .n − 1 (21) η 2g 1,j m = V 2,j 1,j h If edge ξ = is embedded: here, by considering the pair of elements (c-d), and according to [14], we obtain: {︃ v = 0 η,j , j = 1, 2, . . . .n − 1 (22) η 2g η,j m = V 2 η−1,j η,j h Figure 3: Square plate of variable thickness hinged on all edges. 4 Numerical results and discussion Section one: square plate of variable thickness articu- In this first example, we look at two cases: lated on all edges (a) Case where the plate is subjected to a constant thick- In this first section, the case of a square plate of variable ness: thickness articulated on all edges and on the unit side is For g = const, the differential equations of bending examined. It’s a benchmark widely use in literature to test plates of variable stiffness are specials cases of the equa- numerical models. Some examples of calculation of said tions for plates of constant stiffness. plate subjected to simple bending, then to bending com- Using those equations, a computer program was com- bined with compressed and /or traction using the boundary piled for calculating plates with stiffness continuously con- conditions, are presented. The calculations consist in de- stant according to an arbitrary law for the action of breaking termining the maximum values of the coefficients of the static and dynamic loads. The program takes into consid- deflection and the moment of the plate, according to differ- eration all types of boundary conditions; it has been intro- ent meshes. duced into the practice of engineering calculations. Table1 Since the plate is in contact with the ground, then we gives bending momentum and deflection coefficients of the choose for (those cases) the stiffness of the soil, (according plate resulting from this calculation: to geotechnical reconnaissance of the state of Cameroon). In order to evaluate the convergence of the solutions or The plate will have a variable stiffness: to check the error, we determine the speed of convergence ⃒ ⃒ GE−FDM 2 2 ⃒ exp−v ||⃒ max g (η; ξ ) = a η +b ξ +a η +b ξ +c ηξ +d i,j 01 01 02 i,j 02 i,j 01 01 i,j i,j in the form: e = αh v , with r>0, α >0 and max ⃒ ⃒ max where a ; b ; b ; c ; d are known constants. exp 01 01 02 01 01 where r is the order of convergence, α a constant, v and max The Young’s modulus is E = 4 × 10 MPa GE−FDM v denote respectively the maximum value of deflec- max These values will be introduced into the Eq. (17) to ob- tion obtained in [1], [12] and by the Generalized equations tain a new system corresponding to each request. The other of the finite difference method. Eq. (23) can be written as: y parameter will be defined according to the type of stress. = rx + ρ where y = Loge ; x = Logh and ρ = Logα max We will define r and ρ by the least squares’ method. For that, we introduce the function Φ defined by: ∑︀ [︀ ]︀ Φ(r, ρ) = y + (rx + ρ . The least square prob- i i ̂︁ ̂︀ lem consists in finding (r, ρ) such that: 60 | Seydou et al. Table 1: Moment and deflection (cm) coeflcients for bending combined with compression for constant thickness (BCCT). Other researchers Casel:BCCT GE-FDM α ¯ = −1; β = 1; 𝛾 ¯ = 0; k = 1; µ = 0, 16; N < 0; F = −1; and R = 0 [12] [1] [11] Mesh 4 x 4 8 x 8 16 x 16 20 x 20 24 x 24 32 x 32 va PD 0,00411 0,00413 0,00415 0,00416 0,00418 0,00418 0,00417 0,00417 0,00490 max m (a P) 0,07238 0,07486 0,07551 0,07581 0,07582 0,07582 0,07580 0,07580 / max Table 2: Values of the relative error of v . max exp GE−FDM h v v e i x y max max max 4 x 4 0,00411 0,00417 0,00006 1 -0,60206 -4,22184 8 x 8 0,00413 0,00417 0,00004 2 -0,90309 -4,39794 16 x 16 0,00415 0,00417 0,00002 3 -1,20412 -4,69897 20 x 20 0,00416 0,00417 0,00001 4 -1,30103 -5,0000 24 x 24 0,00418 0,00417 0,00001 5 - 1,38021 -5,0000 32 x 32 0,00418 0,00417 0,00001 6 - -1,50515 -5,0000 MinΦ(r, ρ) = Φ(̂︀r,̂︀ρ) ∈ R × R consider half of the plate. The resultant of external loads This leads us to the system: applied to this portion is equal to 1. {︃ {︃ ∑︀ ∑︀ ∑︀ ̂︀ ̂︀ Φ(r,ρ) 6 6 6 = 0 x r) + x )ρ = x y i i i ∂r i i i i that is to say: ∑︀ ∑︀ ̂︀ ̂︀ Φ(r,ρ) 6 6 = 0 x )r + 6ρ = y i i i i ∂ρ The resolution of this system leads to: r = 1, 01 and ρ = 3, 572 ; hence the regression line of y as a function of x is given by: y = 1, 01x − 3, 572; with this error obtained we can say that the conver- gence of the results towards those obtained in is estab- lished. (b) Case where the plate is subjected to a variable thick- ness and under elastic foundation: Using Eq. (17), a computer program was compiled for calculating plates with stiffness continuously varying ac- Figure 4: Square plate of variable stiffness with loading. cording to an arbitrary law (η; ξ ) for the action of breaking static and dynamic loads. The program takes into consid- eration all types of boundary conditions; it has been intro- duced into the practice of engineering calculations. Table 3 gives the corresponding values of maximum moment and 4.2 Example 2: Simple bending plate deflection (coefficients) of the plate resulting from this com- puter program for this case: In this example, the plate is subjected to uniformly dis- Since solutions for this problem do not exist anywhere, tributed load over its entire surface as shown in Figure 4. we checked the error of the results obtained by using the Moreover, the compression loads are applied uniformly principle of static equilibrium of the plate (see Table 2). In and parallel to the axis. The value of the compression loads this view we have determined the sum of the projections of is much lower than the critical value. The dimensionless all the reactions on the axis perpendicular to the average value of distributed load is P = 1. The thickness of the plane of the plate. Under the symmetry property, we can plate varies along η and ξ as shown in the Figure 4. Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation | 61 Table 3: Moment and deflection coeflcients for bending combined with compression for variable thickness (BCVT). Case2: BCVT GE-FDM α = −1; β = 1; 𝛾 ¯ = 0; k = 1; µ = 0, 16; F = −1; and R = 0 Meshes 4 x 4 8 x 8 16 x 16 20 x 20 24 x 24 32 x 32 va PD 0,001961 0,001966 0,001967 0,001968 0,001968 0,001969 max m (a P) 0,07316 0,07365 0,07382 0,07390 0,07395 0,07399 max Table 4 also illustrates the convergence of the numeri- cal solution. While proceeding as in the case of BCCT, the speed of convergence is an order of convergence equal to r = 0, 99and the regression line as a function of x is given by: y = 0, 99x − 2, 64. So the convergence of the results towards those obtained in [24] is established. The g fi ures 5(a) and 5(b) illustrate perfectly this convergence. 4.3 Example 3: Case of flexion combined with unidirectional traction (a) Graphical representation of the coefficients of moment. Figure 6 shows a square slab of length 1 pivotally supported along its contour, the stiffness and distributed load of which in the direction y vary linearly. The results of the compu- tation when half of the plate is loaded make it possible to obtain a solution in the case of loading the entire plate with the same load. The values of the largest bending moments and deflec- tions are obtained by us on a 36x36 square bit. (b) Graphical representations of the coefficients of deflection. Figure 5 The goal is to compare our results to the reference val- ues available in literature for the same element in order to better quantify the influence of the flexible foundation, the variable stiffness and the influence of the flexible founda- tion and the influence of membrane forces. The computation results on various meshes for a square Figure 6: Square plate under elastic foundation. slab hinged along the contour, the rigidity of which changes in two directions, on the action of the load evenly dis- This makes it possible to note the good behavior of tributed over the entire area in Figure 4 are compared with the method. It should be noted that the algorithm was de- the numerical solution of [24]. veloped with the aim of writing a code of calculation on the basis of generalized equations of the finite difference 62 | Seydou et al. Table 4: Maximum moment and deflection coeflcients for simple bending (SB). Case3: SB GE-FDM α = 0; β = 0; 𝛾 ¯ = 0; k = 0; N = 0; µ = 0, 16; F = −1; and R = 0 [24] Meshes 4 X 4 8x8 16 X 16 20 X 20 24 x 24 32 X 32 v2 0,0904 0,0915 0,0916 0,0916 0,0918 0,0918 0,0917 0 0 max m(η) 0,0457 0,0469 0,0475 0,0475 0,0476 0,0476 0,0477 X max m(η) 0,0456 0,0472 0,0478 0,0478 0,0480 0,0482 0,0481 Y max Table 5: Moment and deflection coeflcients for bending combined with unidirectional traction (BCT). Case 4: BCT GE-FDM α ¯ = 0; β = 0; 𝛾 ¯ = 1; k = 1; µ = 0, 16; F = −1; (Normal forces tend to lengthen the plate: N>0); R = 22754.0 Meshes 4 X 4 8 X 8 16x16 20 X 20 24 X 24 32 X 32 va PD 0,001986 0,001992 0,001993 0,001994 0,001994 0,001994 max m (a P) 0,07418 0,07467 0,07484 0,07492 0,07497 0,07501 max Table 6: Deflection (cm) and moment of the plate with variable thicknesses under a uniformly distributed load (BCVT). Case 1: BCVT GE-FDM α = −1; β = −1; 𝛾 ¯ = 0; k = 1; N < 0; F = −1 and R=0 [11] FEM EXPERIMENTAL Meshes 4 x 4 8x8 16 X 16 20 X 20 24x24 32 X 32 va PD 0,0049 0,0054 0,0058 0,0062 0,0063 0,0063 0,0069 0,0070 0,0059 max m (a P) 0,07238 0,07486 0,07551 0,07581 0,07582 0,07582 / / / max method. Thus, we can say that with a mesh course, the Section two: rectangular plate of variable thickness generalized equations give good results. The refinement of freely supported at two opposites edges and the other the mesh makes it possible to observe the convergence of two edges fixed the results. Tables 1 to 5 above illustrate the Convergence Figure 8 shows a rectangular plate with variable thick- well. nesses, freely supported at two opposite edges y =0, y = b to evaluate the impact which the application of the and two fixed edges x 0, x =a. It should be noted that here normal forces of membranes causes on a bent, tended and the plate is not under elastic foundation (R=0). or compressed plate of variable rigidity, we will refer to In order to solve the Eq. (17) and to obtain the numerical Tables 1 to 4 then the curves of Figures 7a and 7b 7; hence : results of deflection and moment, a computer program was a) In all cases when the mesh is increased, the mo- used. The results are presented by the values of deflection ments and arrows increase and are almost monotonic from for the case of a plate with variable thicknesses (h = 6 mm, a certain mesh pitch. h = 8 mm, Figure 8) loaded by the uniformly distributed b) The decrease in bending forces caused by the elastic load. This choice is in order to compare our results with foundation with the normal forces of membranes acting in the experiments ones obtain by [11], using a tensile test traction in one direction is equal to the increase in forces machine with additional equipment. caused in the same conditions compared to those acting in The total load is equal to 24 kPa for the uniformly dis- compression in the same direction. The combined effects tributed load. The calculations were made for plates with of these two forces cancel each other out. dimensions of 180 mm in width and 400 mm in length One can deduct from these interpretations that the de- loaded by the uniformly distributed load. The steel grade flection of a plate of variable rigidity on a flexible founda- NVA with yield stress 235 N/mm is used. tion and subjected to bending combined with traction or compression is less important or even negligible when the two stresses are combined simultaneously. Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation | 63 (a) Figure 8: Rectangular plate with variable thicknesses with two opposite edges freely supported and two xed fi edges [11]. that the GE – FDM has displayed ability to yield accurate solutions on relatively coarse grid, with an order of conver- gence equal to 0, 99. This shows as well the stability of the method. Acknowledgement: The authors are very grateful to Dr Amba Chills, Dr Yakada S. and Dr Mezoue Cyrille for their useful comments on methodology and writing of this paper. Funding information: The authors state no funding in- volved. (b) Author contributions: All authors have accepted responsi- bility for the entire content of this manuscript and approved Figure 7: a) Graphical representation of the moment coeflcients; b) its submission. Ggraphical representations of coeflcients of deflection. Conflict of interest: The authors state no conflict of inter- est. 5 Conclusion In this work we have exposed a mathematical technique th that transforms the PDE of 4 order governing the deforma- References tion equation of a rectangular plate into a system of PDE of nd [1] TimoshenkoI S, Woinowsky-Krieger S. Theory of Plates and 2 order. An analyst process leading to a dimensionless pa- Shells. Ed. McGraw-Hill, New York. 1966;591. rameters and unknown functions have been implemented. [2] eontiv NN, Leontiv AN, Sobolev DN, Anohin NN. Fundamentals of Then after generalized equations of finite difference method the theory of beams and plates on a deformed base. Moscow: nd are derived from a dimensionless system of 2 order PDE Ed. MISI. 1982;119. previously obtained. The resolution takes into account the [3] Nzengwa R, Tagne BH. A two-dimensional model for linear elastic boundary conditions which was done using the iterative thick shells. Int J Solids Struct. 1999;34(36):5141-5176. [4] Nkongho AJ, Nzengwa R, Amba JC, Ngayihi CVA. Approximation method of Gauss-Seidel. of linear elastic shell by curved triangular finite element base on This numerical approach has been tested on bench- elastic thick shells theory. Math Probl Eng. 2016. mark problems found in the literature. The provided nu- [5] Blassov BZ, Leontiv NN. Plates and shells on anelastic base. merical solutions are satisfactory and are in accordance Moscow: Ed. GIFML; 1960;208. with those found in the literature. It is worth mentioning 64 | Seydou et al. [6] Leontiv NN, Leontiv ANm Sobolev DN, Anohin NN. Fundamentals [17] Tornabene F, Fantuzzi N, Ubertini F, Viola E. Strong formulation of the theory of beams and plates on a deformed base. Moscow: finite element method based on differential quadrature: a survey. Ed. MISI. 1982;119. Appl Mech Rev. 2015;67(2):20801. [7] Koreneva EB. Analytical methods for calculating variable thick- [18] Fazzolari FA, Viscoti M, Dimitri R, Tornabene F. 1D – Hierarchical ness plates and their practical applications. Ed. ABS publisher, Ritz and 2D-GDQ Formulations for free vibration analysis of cir- Moscow. 2009. cular/elliptical cylindrical Shells and beam structures. Compos [8] Sharma S, Gupta US, And Singhal P. Vibration Analysis of Struct. 2021;158:113338. Non-Homogeneous Orthotropic Rectangular Plates of Variable [19] Tornabene F, Viscoti M, Dimitri R, Reddy JN. Higher order theories Thickness Resting on Winkler Foundation. J Appl Sci Eng. for the vibration study of doubly – curved anisotropic shells 2012;15(3):291-300. with a variable thickness and isogeometric mapped geometry. [9] Ventsel E, Krauthammer T. Thin Plates and Shells Theory, Anal- Compos Struct. 2021;267:113829. ysis and Applications. The Pennsylvania State University and [20] Dimitri R, Fantuzzi N, Tornabene F, Zavarise G. Innovative nu- University Park, Pennsylvania. 2001;651. merical methods based on SFEM and IGA for computing stress [10] Gabbassov RF, Filatov VV, Ovarova NB, Mansour AM. Dissection concentrations in isotropic plates with discontinuities. Int J Mech Method Applications for complex shaped membranes and plates. Sci. 2016;118:166-187. Procedia Eng. 2016;153:444–9. [21] Gabbasov RF. Difference equations in problems of strength and [11] Aryassov G, Gornostajev D, and Penkov I. Calculation method for stability of plates. Soviet Appl Mech. 1982;18:820-824. plates with discrete variable thickness under uniform loading or [22] Uvarova N, Gabbasov RF. Calculation of plates in a geometrically hydrostatic pressure Int J of Appl Mech Eng. 2018;23(4):835-853. nonlinear setting with the use of generalized equations of finite [12] Gabbassov RF, Gabbassov AR, Filatov VV. Numerical construction difference method, Theoretical foundation of civil engineering. of discontinuous solutions to problems of structural mechanics. Web Conf. 2018;196:01024. Moscow: Ed. ABC. 2008;280. [23] Gabbasov RF. Generalized equations of the finite difference [13] Komlev AA, Makaev SA. The calculation of rectangular plate on method in polar coordinates on problems with discontinuous elastic foundation: the finite difference method. J Phys Conf Ser. solutions. Resistance of materials and theory of structures. Kiev: 2018;944:012056. Budivelnik. 1984;45:55-58. [14] Gabbassov RF, Moussa S. Generalized Equations of Finite Differ- [24] Smirnov VA. Calculation of plates of complex shape. M. Stroyiz- ence Method and their Application for Calculation of Variable dat; 1978;300. Stiffness Curved Plates. Ed. News of Higher Educational Institu- [25] Nkongho AJ, Amba JC, Essola D, Ngayihi Abbe VC, Bodol Momha tions Construction. 2004;(17–22):5. M, Nzengwa R. Generalised assumed strain curved shell finite [15] Moussa S, Lontsi F, Hamandjoda O, Raidandi D. Calculation of elements (CSFE-sh) with shifted-Lagrange and applications on plates on elastic foundation by the generalized equations of N-T’s shells theory. Curved Layar Struct. 2020;7(1):125-138. finite difference method. Int J Eng Sci. 2018;7(8):32- 38. [16] Moussa S, Njifenjou A, Youssoufa S. Équations généralisées de la méthode des différences finies pour le calcul des plaques minces isotropes d’épaisseur constant soumises à une flexion, compression et ou/traction. AfriqueSCIENCE. 2019;15(3):49 - 63. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Curved and Layered Structures de Gruyter

Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation

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Curved and Layer. Struct. 2022; 9:54–64 Research Article Seydou Youssoufa, Moussa Sali, Abdou Njifenjou, Nkongho Anyi Joseph*, and Ngayihi Abbe Claude Valery Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation https://doi.org/10.1515/cls-2022-0005 Keywords: rectangular plate, elastic foundation, general- Received May 03, 2021; accepted Sep 29, 2021 ized equations of finite difference method, discontinuity Abstract: The computation of bent isotropic plates, stretched and/or compressed, is a topic widely explored in the literature from both experimental and numerical point 1 Introduction of view. We expose in this work an application of the gener- alized equations of Finite difference method to that topic. One can start by recalling that a plate is a structure, which The strength of the proposed method is the ability to recon- thickness is small beside its length and width. What hap- struct the approximate solution with respect of eventual pens when you crumple up a sheet of paper? How does a discontinuities involved in the investigated function as well general raft supporting a building behavior? Can the futur- as its first and second derivatives, including the right-hand istic roofs that adorn our most glorious buildings (shopping side of the equilibrium equation. It is worth mentioning that centers, airport buildings. . . ), resist the wind? In which way by opposition to finite element methods our method needs the carrosseries of a car is distorted in an accident? Here are neither fictitious points nor a special condensation of grid. some of many scenarios of structure behavior that arise in Well-known benchmarks are used in this work to illustrate real life problems. Characterizing the deformations under- the efficiency of our numerical and the high accuracy of gone under certain constraints is our aim in this work. The calculation as well. A comparison of our results with those issues related to this study are diverse: it will sometimes available in the literature also shows good agreement. be a question for the engineer of designing resistant and / or aesthetic plates or shells. Sometimes covertly, the plate specifications must provide the deformations distributed Seydou Youssoufa, Moussa Sali, Ngayihi Abbe Claude Valery: University of Douala, Laboratory of Research in Energy, Material, Modeling and Methods, National high school polytechnic Douala po box 2701 Douala, Cameroon Abdou Njifenjou: University of Douala, Laboratory of Research in Energy, Material, Modeling and Methods, National high school polytechnic Douala po box 2701 Douala, Cameroon; University ofYaoundé I, Laboratory of mathematical engineering and infor- mation system, National high school polytechnic Yaoundé, Po Box 8390 Yaoundé, Cameroon *Corresponding Author: Nkongho Anyi Joseph: University of Douala, Laboratory of Research in Energy, Material, Modeling and Methods, National high school polytechnic Douala po box 2701 Douala, Cameroon; University ofYaoundé I, Laboratory of mathe- matical engineering and information system, National high school polytechnic Yaoundé, Po Box 8390 Yaoundé, Cameroon; University of Buea, Department of mechanical engineering, HTTTC, Po Box 249 Buea Road, Kumba, Cameroon; E-mail: nkongho.anyi@ubuea.cm Open Access. © 2022 Seydou et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation | 55 over its mid surface. The designed structures aim is to ab- isfactory approximate solutions faster than the successive sorb shocks, for example the front part of a vehicle or even approximation method (SAM, for short). submerged radiators. The work is organized as follows: after the introduc- During their exploitations, plate structures are sub- tion which poses the problematic of the subject, we unfold jected to the transversal loads (statics and dynamics). From the methodology of the implementation of the generalized computational point of view, efficient numerical tools are equations of the finite difference method which is broken necessary for modeling sophisticated mechanical behavior down into three points. Subsequently we transform the of such structures, accounting with their specificities. De- new differential equation by the FDM. The last part of this spite the abundance of literature on computations of plate framework will be devoted to the validation of our approach structures [1, 2, 3], several questions remain topical here. through the numerical resolution of test- problems. The behavior of those structures is governed by a lin- th ear partial differential equation (PDE, for short) of 4 order which is not obvious to be solved using analytical meth- 2 Tools and techniques ods [5, 6, 7]. Hence, to call out for numerical methods easy to implement and less onerous from computational point In several works related to the numerical calculation of of view. Among popular numerical methods used for this plates and shells, the authors use various approaches to topic, the finite element method (FEM, for short) is the most solve the partial differential equations which govern these popular [4, 8, 9, 10]. However, the FEM presents a certain structures. As far as we are concerned in this frame work, number of drawbacks as indicated in [9]: (a) The FEM is our resolution methodology is as follows: poorly adapted to a solution of the so-called singular prob- th (a) transformation of the partial derivatives 4 order de- lems like plates with cracks, corner points, discontinuity formation equation of a rectangular plate of variable internal actions, and of problems for unbounded domains. thickness into a system of two differential equations (b) This method requires the use of powerful computers of nd of 2 order partial derivative; considerable speed and storage capacity. (c) The method (b) introduction of new dimensionless parameters in the presents many difficulties associated with problems of C system of equations obtained and in the equation continuity and nonconforming elements in plate (and shell) describing the boundary conditions; bending analysis. Note that the mathematical theory of FEM (c) transformation of new differential equations by the is exponentially increasing. So the above drawbacks could generalized equations of the finite difference method, be addressed in a near future by the FEM. these permits a system of algebraic equation to be More recent numerical methods have been developed obtain; for addressing singular problems [10, 11, 12, 13, 14, 15, 16]. (d) transformation of boundary conditions; Among the large variety of those methods for addressing (e) elaboration of a calculation algorithms; singular problems, the Generalized Differential Quadrature (f) resolution of the system of algebraic equation in or- (GDQ, for short) is proposed to solve different kinds of struc- der to obtain the bending moment and the maximum tural problems. In many applications present in literature displacement. [17, 18, 19, 20], the GDQ method has shown superb accu- racy, efficiency, convenience, and great potential in solving differential equations. The generalized equations of the Fi- 2.1 Equation of deformation of a bent nite Difference Method (GE-FDM, for short) are part of these recent numerical methods [21, 22, 23]. rectangular plate stretched and /or The aim of this work is to show that the generalized compressed with variable rigidity on equation of the finite difference method (GE-FDM, for short) elastic foundation could be used to address the computation of bent, stretched and/or compressed rectangular plates of variable stiffness In the following paragraphs, only bent, stretched and/or under elastic foundations. One of the main features of this compressed plates of variable thickness will be analyzed. method (GE-FDM) is the ability of dealing with finite discon- So it is convenient to express the governing differential tinuities of the investigated solution and that of its first and equation of this plate. Figure 1 illustrates of the equilibrium second derivatives, including discontinuities of the right of a sample plate stretched or compressed element: hand side of the primary PDE. According to [22] and [23, 25], Let us consider an infinitesimal element d , d as indi- X Y the computation of these plates with GE-FDM leads to sat- cated in Figure 1 and projections of membrane forces along 56 | Seydou et al. 2 2 ∂ W ∂ W M ⎪ + = − 2 2 ⎪ D ∂X ∂Y (︁ )︁ 2 2 2 2 2 ∂ M ∂ M ∂ W ∂ W ∂ W + = F + N + N − 2N − RW + 2 2 2 2 Z X Y XY ∂X ∂Y ∂X ∂Y ∂X∂Y (︁ )︁ 2 2 2 2 2 2 ⎩ ∂ D ∂ W ∂ D ∂ W ∂ D ∂ W +(1 − µ ) + − 2 2 2 2 2 ∂X ∂Y ∂Y ∂X ∂X∂Y ∂X∂Y (4) where 2 2 2 2 ∂ W ∂ W ∂ W ∂ W M = + µ ; M = + µ (5) X Y 2 2 2 2 ∂X ∂Y ∂Y ∂X M + M X Y M = (6) 1 + µ M denotes the resultant moment, so M and M are X Y Figure 1: Element of a bent plate stretched and /or compressed. bending moment following X and Y directions, respectively. the Z - axis. N , N and N are respectively the horizon- X Y XY 2.2 Introductions to dimensionless tal components of the normal and shear forces which are parameters exerted on the various facets. The normal unit vector to the facets is tangent to the neutral plane in each direction Rewriting the Eq. (4) using dimensionless parameters [1], (X,Y). By neglecting the forces of volume along the X and [14]: Y directions, we obtain according to Z- axis: Y X F M WD Z 0 η = ; ξ = ; F = ; m = ; v = ; 2 2 2 2 3 3 2 4 ∂ D ∂ W ∂ D ∂ W ∂D ∂ W ∂D ∂ W l l F F l F l 0 0 0 + µ + 2 + 2 + (7) 2 2 2 2 3 2 ∂X ∂X ∂X ∂Y ∂Y ∂X ∂X ∂X∂Y M M D(X; Y ) (ξ ) (η) X Y m = ; m = ; g = 3 2 2 2 2 2 2 2 2 ∂D ∂ W ∂ D ∂ W ∂ D ∂ W ∂ D ∂ W F l F l D 0 0 0 2 + 2 − 2µ + 2 2 2 ∂Y ∂X ∂Y ∂X∂Y ∂X∂Y ∂X∂Y ∂X∂Y ∂Y ∂Y 2 2 2 2 2 Nl N N −2N X Y XY ∂ D ∂ W ∂ W ∂ W ∂ W ¯ l = max (|l | , |l |) ; k = ; α = ; 𝛾 ¯ = ; β = X Y + µ + D∆∆W = N + N − 2N X Y XY D N N N 2 2 2 2 0 ∂X∂Y ∂Y ∂X ∂X ∂X (8) − RW + F where N = max (|N | , |N | , |N |) ;−1 ≤ α ≤ 1, −1 ≤ X Y XY (1) β ≤ 1, −1 ≤ 𝛾 ≤ 1; (η ; ξ) are Cartesian coordinates without with units; F is Load factor; m is Moment coefficient; µ is the co- 2 2 2 2 2 2 ∂ D ∂ W ∂ D ∂ W ∂ D ∂ W efficients of deflection; D denotes the cylindrical stiffness D∆∆W = − +2 (2) 2 2 2 2 ∂Y ∂X ∂Y ∂Y ∂X∂Y ∂X∂Y of any section of the slab; g is the stiffness coefficient; l is plate length; α , β, 𝛾 and k are coefficients without unit. where EH Introducing the parameter in Eqs. (7) and (8) into the D = (3) 12(1 − µ ) system of Eq. (4), we obtain: Eq. (1) is called the deformation equation of a bent rect- 2 2 ∂ v ∂ v m angular plate stretched and /or compressed with variable + = − 2 2 ⎪ g ∂ξ ∂η ⎨ (︁ )︁ 2 2 2 2 2 rigidity on elastic foundation; ∂ m ∂ m ∂ v ∂ v ∂ v + − k α ¯ + 𝛾 ¯ + β + (9) 2 2 2 2 ∂ξ ∂η ∂ξ ∂η ∂ξ∂η ⎪ (︁ )︁ where W = W (X, Y ) is the transversal dis- ⎪ 2 2 2 2 2 2 ⎩ ∂ g ∂ g ∂ g ∂ v ∂ v ∂ v (1 − µ ) + − 2 − λv = F 2 2 2 2 ∂ξ ∂η ∂η ∂ξ ∂ξ∂η ∂ξ∂η placement of the plate (searched function); D = D(X, Y ) stiffness of a variable p late; µ is Poisson’s coef- Rl where λ = . ficient; N , N are normal membrane forces; N denotes X Y XY The Eq. (9) can be written as follows: membrane shear forces ; R is the ground stiffness in N/m ; F represents the volume force along the Z - axis, while H Z 2 2 ∂ v ∂ v m ⎪ + = − 2 2 ⎪ ∂ξ ∂η g =H(X, Y ) is the variable thickness of the plate. (︁ )︁ 2 2 2 2 ∂ m ∂ m ∂ g ∂ v nd + + −kα ¯ + (1 − µ ) + The Eq. (1) can be transformed as a system of 2 order 2 2 2 2 ∂ξ ∂η ∂η ∂ξ ⎪ (︁ )︁ (︁ )︁ ⎪ 2 2 2 2 ⎩ ∂ g ∂ v ∂ g ∂ v partial derivative equations (pde): ¯ ¯ + −k𝛾 ¯ + (1 − µ ) + −kβ − 2(1 − µ ) = F + λv 2 2 ∂ξ∂η ∂ξ∂η ∂ξ ∂η (10) Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation | 57 Eq. (10) is called the generalized algebraic equation of 3 Formulation of the Generalized the finite difference method which replaces Eq. (4). Note that v and its partial derivative can be discontinuous when Equation of the Finite Difference the plate has ball points, while m will be discontinuous if Method (GE- FDM) external point bending moments are applied in one of the directions of the coordinate axes. The technique exposed here for obtaining the generalized equations of the finite difference has been first introduced in [14]. 2.3 Boundary conditions Several boundary conditions are discussed in this work 3.1 Finite difference mesh over the structure in accordance with practical and site works constraints in foundations design. We are going to emphasize on displace- A square mesh of size h is defined over the structure as ment and moments conditions on the borders. indicated in Figure 2 where the roman numerals are used for mesh element numbers and (i,j) represents mesh nodes 2.3.1 Articulated supports If the articulated edge is parallel to the X - axis, in other words, Y = 0, then: W = 0; M =0. If the above conditions are imposed on the edge of the plate: W =W (X) and M =M X , therefore from formu- ( ) 0 Y Y las (10), we obtain: ∂ W (X) Y 0 M = M (X) − D(1 − µ ) (11) ∂X If the edge is parallel to the Y - axis, in other words, X = 0, then: W= 0; M = 0. (12) Figure 2: Square mesh for GE-FDM. 2.3.2 Embedded edges Taking into consideration Eq. (6) established in [14] If the embedded edge is parallel to the Y - axis, then: and identifying Eq. (10) with Eq. (6), we obtained: (︂ )︂ ∂W (a) ; (W ) = 0 (13) y=a ∂Y y=a ij V + V + V + V − 4v = −h (15) i,j−1 i−1,j i+1,j i,j+1 i,j If on the other hand it is parallel to the X - axis, then: ij (︂ )︂ ∂W with P = − ω = v ; α = 𝛾 = 1 ; δ = β = σ = o; also W = 0; = 0 (14) ( ) x=a ∂x considering that m and vare continuous, as well as their x=a first and second order derivatives All the ingredients are gathered to find out the general- ized equations of the finite difference method. 58 | Seydou et al. (b) tion is solved taking into consideration the transformed boundary conditions. m + m + m + m − 4m + i,j−1 i−1,j i+1,j i,j+1 i,j (︁ )︁ ξ ξ η η I−II III−IV I−III II−IV ∆ m + ∆ m + ∆ m + ∆ m + ij ij ij ij [︃ ]︃ (︀ )︀ 3.2 Transformation of the boundary k β v − v − v + v + i−1,j−1 i−1,j+1 i+1,j−1 i+1,j+1 − (︀ )︀ (︀ )︀ + 4 4α ¯ v + v + 4𝛾 ¯ v + v − 8(α ¯ + 𝛾 ¯ )v conditions by the generalized equation i−1,j i+1,j i,j−1 i,j+1 i,j [︃ ]︃ (︀ )︀ ξη of the finite difference method − g v − v − v − v + i−1,j−1 i+1,j+1 i+1,j−1 i−1,j+1 2h + (1 − µ ) (︀ )︀ (︀ )︀ 1 ηη 1 ξξ g v + v − 2v + g v + v − 2v i−1,j i+1,j i,j i,j−1 i,j+1 i,j 2 2 h h In this section, we establish equations called boundary = −h P i,j conditions, which are combined to the Eq. (17) lve the differ- (16) 2 2 ∂ g ∂ g ηη ξξ ξη ential Eq. (2). The said boundary conditions are presented where: P = F + λv . ; g = ; g = and g = i,j i,j i,j 2 2 ∂η ∂ξ 2 below. ∂ g ∂ξ∂η ∂ g With: ω = m ; α = −kα + (1 − µ ) ; 𝛾 = −k𝛾 + 1 2 1 ∂η 2 2 ∂ g ∂ g 3.2.1 Articulated sides (1 − µ ) ; β = −kβ + (1 − µ ) and σ = δ = 0, 2 1 1 1 ∂ξ ∂ξ∂η then also consider that P is constant within inside of each If all the edges of the plate are articulated, it is enough to element but can vary abruptly from one element to another. solve the Eqs. (15) and (16) Those equations are written for Eqs. (15) and (16) are called the generalized equations each point inside the field. Thus, we have: of the finite difference method for a bent rectangular plate stretched and /or compressed with variable stiffness on elastic foundation, which substitute’s Eq. (10), where, = 2, (ξ ) (η) ξξ m = 0 ; v = v (ξ ); m = m (ξ ) − (1 − )v (ξ ); 0 0 3. .. n-1; = 2, 3, ... ,-1. (18) (i) At the end, F+λv will be replaced by F + λv . ξξ (η) i,j i,j where v (ξ ) ; m (ξ ); v (ξ ) are all known. 0 0 We write the equation for a regular mesh: h = τ = h = τ i i i i+1 3.2.2 Embedded edges hence, by combining Eqs. (15) and (16) we obtain for We remind that if an edge is embedded, the deflection and Eq. (10): rotation on that edge are zero. Edges parallel to ξ- axis: ij v + v − 4v + v + v = −h ⎪ i−1,j i,j−1 i,j i+1,j i,j+1 ⎪ if the edge η =0 is embedded, then: ij ⎪ ∂v (v ) = 0 either v = v = v = 0 and ( ) = m + m + m + m − 4m + η ⎪ i,j i−1,j i+1,j i,j η=0 i,j−1 i−1,j i+1,j i,j+1 i,j ⎪ ∂y (︁ )︁ η ξ ξ η η 0. Hence, v = 0. By substituting the first equation of the ⎪ I−II III−IV I−III II−IV i,j ⎪ ∆ m + ∆ m + ∆ m + ∆ m + ij ij ij ij )︁ system (17) with the generalized Eq. (3) giving in [14] and [︃ ]︃ k β(v − v − v + v + i−1,j−1 i−1,j+1 i+1,j−1 i+1,j+1 by noting that α = 𝛾 = 1 ; δ = β = σ = o we obtain: ⎪ − (︀ )︀ (︀ )︀ + 4 η m ⎪ 4α ¯ v + v + 4𝛾 ¯ v + v − 8(α ¯ + 𝛾 ¯ )v 1 2 ij i−1,j i+1,j i,j−1 i,j+1 i,j 2v = 0 = (v +v +v +v − 4v −h ), ⎪ 2 i,j−1 i−1,j i+1,j i,j+1 i,j ⎪ [︃ ]︃ i,j h g (︀ )︀ ij ⎪ ξη − g v − v − v − v + ξ 2 i−1,j−1 i+1,j+1 i+1,j−1 i−1,j+1 2 2h ⎪ (︀ )︀ (︀ )︀ + (1 − µ ) either m = v from where: ⎪ 2 i,j+1 1 ηη 1 ξξ i,j h ⎪ g v + v − 2v + g v + v − 2v i−1,j i+1,j i,j i,j−1 i,j+1 i,j 2 2 ⎪ h h {︃ = −h P i,j v = 0 i,j (19) (17) ξ −2g i,j m = V ξ ξ ξ ξ 2 i,2 i,j ξ ξ h I−II I II III−IV III IV where: ∆ m = m − m ; ∆ m = m − m ; i,j i,j i,j i,j i,j i,j where i = 2, 3, ....., n − 1 ; j = 2. ξ ξ ξ ξ ξ ξ I−III I III II−IV II IV ∆ m = m − m ; ∆ m = m − m , i = i,j i,j i,j i,j i,j i,j if the edge η = is embedded then: 2, 3, ....., n − 1; j = 2, 3, ....., n − 1. by proceeding in the same way as previously and by h - Mesh spacing; i - measuring along the Axis; j - mea- considering the pair of elements (a-c), and according to suring along the Axis [14], we obtain: Eq. (17) is called the generalized algebraic equation of {︃ the finite difference method for a bent rectangular plate v = 0 i,η stretched and /or compressed with variable stiffness on i = 2, 3, . . . . . . , n − 1 (20) ξ −2g i,η m = v 2 i,η−1 i,η elastic foundation, which substitute Eq. (10). This equa- h Edges parallel to η- axis: Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation | 59 If the edge ξ = is embedded: 4.1 Example 1: Case of bending combined By considering the pair of element (c-d) we have: with unidirectional compression For this first example, it is a square plate subjected to bend- ξ ij 2v = + v + v + v − 4v − h i,j−1 i−1,j i+1,j i,j+1 i,j i,j ing combined with compression. The four sides of which ij are articulated. It is also subjected to the action of a load and noticing that in [14], α = 𝛾 = 1 ; δ = β = σ = o we uniformly distributed over its entire surface. Moreover, the η 2g 1,j obtain: m = V , 2 i+1,j i,j h compression loads are applied uniformly and parallel to - from where: axis. A mesh of such a plate is shown in the Figure 3. {︃ v = 0 1j , j = 1, 2, . . . .n − 1 (21) η 2g 1,j m = V 2,j 1,j h If edge ξ = is embedded: here, by considering the pair of elements (c-d), and according to [14], we obtain: {︃ v = 0 η,j , j = 1, 2, . . . .n − 1 (22) η 2g η,j m = V 2 η−1,j η,j h Figure 3: Square plate of variable thickness hinged on all edges. 4 Numerical results and discussion Section one: square plate of variable thickness articu- In this first example, we look at two cases: lated on all edges (a) Case where the plate is subjected to a constant thick- In this first section, the case of a square plate of variable ness: thickness articulated on all edges and on the unit side is For g = const, the differential equations of bending examined. It’s a benchmark widely use in literature to test plates of variable stiffness are specials cases of the equa- numerical models. Some examples of calculation of said tions for plates of constant stiffness. plate subjected to simple bending, then to bending com- Using those equations, a computer program was com- bined with compressed and /or traction using the boundary piled for calculating plates with stiffness continuously con- conditions, are presented. The calculations consist in de- stant according to an arbitrary law for the action of breaking termining the maximum values of the coefficients of the static and dynamic loads. The program takes into consid- deflection and the moment of the plate, according to differ- eration all types of boundary conditions; it has been intro- ent meshes. duced into the practice of engineering calculations. Table1 Since the plate is in contact with the ground, then we gives bending momentum and deflection coefficients of the choose for (those cases) the stiffness of the soil, (according plate resulting from this calculation: to geotechnical reconnaissance of the state of Cameroon). In order to evaluate the convergence of the solutions or The plate will have a variable stiffness: to check the error, we determine the speed of convergence ⃒ ⃒ GE−FDM 2 2 ⃒ exp−v ||⃒ max g (η; ξ ) = a η +b ξ +a η +b ξ +c ηξ +d i,j 01 01 02 i,j 02 i,j 01 01 i,j i,j in the form: e = αh v , with r>0, α >0 and max ⃒ ⃒ max where a ; b ; b ; c ; d are known constants. exp 01 01 02 01 01 where r is the order of convergence, α a constant, v and max The Young’s modulus is E = 4 × 10 MPa GE−FDM v denote respectively the maximum value of deflec- max These values will be introduced into the Eq. (17) to ob- tion obtained in [1], [12] and by the Generalized equations tain a new system corresponding to each request. The other of the finite difference method. Eq. (23) can be written as: y parameter will be defined according to the type of stress. = rx + ρ where y = Loge ; x = Logh and ρ = Logα max We will define r and ρ by the least squares’ method. For that, we introduce the function Φ defined by: ∑︀ [︀ ]︀ Φ(r, ρ) = y + (rx + ρ . The least square prob- i i ̂︁ ̂︀ lem consists in finding (r, ρ) such that: 60 | Seydou et al. Table 1: Moment and deflection (cm) coeflcients for bending combined with compression for constant thickness (BCCT). Other researchers Casel:BCCT GE-FDM α ¯ = −1; β = 1; 𝛾 ¯ = 0; k = 1; µ = 0, 16; N < 0; F = −1; and R = 0 [12] [1] [11] Mesh 4 x 4 8 x 8 16 x 16 20 x 20 24 x 24 32 x 32 va PD 0,00411 0,00413 0,00415 0,00416 0,00418 0,00418 0,00417 0,00417 0,00490 max m (a P) 0,07238 0,07486 0,07551 0,07581 0,07582 0,07582 0,07580 0,07580 / max Table 2: Values of the relative error of v . max exp GE−FDM h v v e i x y max max max 4 x 4 0,00411 0,00417 0,00006 1 -0,60206 -4,22184 8 x 8 0,00413 0,00417 0,00004 2 -0,90309 -4,39794 16 x 16 0,00415 0,00417 0,00002 3 -1,20412 -4,69897 20 x 20 0,00416 0,00417 0,00001 4 -1,30103 -5,0000 24 x 24 0,00418 0,00417 0,00001 5 - 1,38021 -5,0000 32 x 32 0,00418 0,00417 0,00001 6 - -1,50515 -5,0000 MinΦ(r, ρ) = Φ(̂︀r,̂︀ρ) ∈ R × R consider half of the plate. The resultant of external loads This leads us to the system: applied to this portion is equal to 1. {︃ {︃ ∑︀ ∑︀ ∑︀ ̂︀ ̂︀ Φ(r,ρ) 6 6 6 = 0 x r) + x )ρ = x y i i i ∂r i i i i that is to say: ∑︀ ∑︀ ̂︀ ̂︀ Φ(r,ρ) 6 6 = 0 x )r + 6ρ = y i i i i ∂ρ The resolution of this system leads to: r = 1, 01 and ρ = 3, 572 ; hence the regression line of y as a function of x is given by: y = 1, 01x − 3, 572; with this error obtained we can say that the conver- gence of the results towards those obtained in is estab- lished. (b) Case where the plate is subjected to a variable thick- ness and under elastic foundation: Using Eq. (17), a computer program was compiled for calculating plates with stiffness continuously varying ac- Figure 4: Square plate of variable stiffness with loading. cording to an arbitrary law (η; ξ ) for the action of breaking static and dynamic loads. The program takes into consid- eration all types of boundary conditions; it has been intro- duced into the practice of engineering calculations. Table 3 gives the corresponding values of maximum moment and 4.2 Example 2: Simple bending plate deflection (coefficients) of the plate resulting from this com- puter program for this case: In this example, the plate is subjected to uniformly dis- Since solutions for this problem do not exist anywhere, tributed load over its entire surface as shown in Figure 4. we checked the error of the results obtained by using the Moreover, the compression loads are applied uniformly principle of static equilibrium of the plate (see Table 2). In and parallel to the axis. The value of the compression loads this view we have determined the sum of the projections of is much lower than the critical value. The dimensionless all the reactions on the axis perpendicular to the average value of distributed load is P = 1. The thickness of the plane of the plate. Under the symmetry property, we can plate varies along η and ξ as shown in the Figure 4. Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation | 61 Table 3: Moment and deflection coeflcients for bending combined with compression for variable thickness (BCVT). Case2: BCVT GE-FDM α = −1; β = 1; 𝛾 ¯ = 0; k = 1; µ = 0, 16; F = −1; and R = 0 Meshes 4 x 4 8 x 8 16 x 16 20 x 20 24 x 24 32 x 32 va PD 0,001961 0,001966 0,001967 0,001968 0,001968 0,001969 max m (a P) 0,07316 0,07365 0,07382 0,07390 0,07395 0,07399 max Table 4 also illustrates the convergence of the numeri- cal solution. While proceeding as in the case of BCCT, the speed of convergence is an order of convergence equal to r = 0, 99and the regression line as a function of x is given by: y = 0, 99x − 2, 64. So the convergence of the results towards those obtained in [24] is established. The g fi ures 5(a) and 5(b) illustrate perfectly this convergence. 4.3 Example 3: Case of flexion combined with unidirectional traction (a) Graphical representation of the coefficients of moment. Figure 6 shows a square slab of length 1 pivotally supported along its contour, the stiffness and distributed load of which in the direction y vary linearly. The results of the compu- tation when half of the plate is loaded make it possible to obtain a solution in the case of loading the entire plate with the same load. The values of the largest bending moments and deflec- tions are obtained by us on a 36x36 square bit. (b) Graphical representations of the coefficients of deflection. Figure 5 The goal is to compare our results to the reference val- ues available in literature for the same element in order to better quantify the influence of the flexible foundation, the variable stiffness and the influence of the flexible founda- tion and the influence of membrane forces. The computation results on various meshes for a square Figure 6: Square plate under elastic foundation. slab hinged along the contour, the rigidity of which changes in two directions, on the action of the load evenly dis- This makes it possible to note the good behavior of tributed over the entire area in Figure 4 are compared with the method. It should be noted that the algorithm was de- the numerical solution of [24]. veloped with the aim of writing a code of calculation on the basis of generalized equations of the finite difference 62 | Seydou et al. Table 4: Maximum moment and deflection coeflcients for simple bending (SB). Case3: SB GE-FDM α = 0; β = 0; 𝛾 ¯ = 0; k = 0; N = 0; µ = 0, 16; F = −1; and R = 0 [24] Meshes 4 X 4 8x8 16 X 16 20 X 20 24 x 24 32 X 32 v2 0,0904 0,0915 0,0916 0,0916 0,0918 0,0918 0,0917 0 0 max m(η) 0,0457 0,0469 0,0475 0,0475 0,0476 0,0476 0,0477 X max m(η) 0,0456 0,0472 0,0478 0,0478 0,0480 0,0482 0,0481 Y max Table 5: Moment and deflection coeflcients for bending combined with unidirectional traction (BCT). Case 4: BCT GE-FDM α ¯ = 0; β = 0; 𝛾 ¯ = 1; k = 1; µ = 0, 16; F = −1; (Normal forces tend to lengthen the plate: N>0); R = 22754.0 Meshes 4 X 4 8 X 8 16x16 20 X 20 24 X 24 32 X 32 va PD 0,001986 0,001992 0,001993 0,001994 0,001994 0,001994 max m (a P) 0,07418 0,07467 0,07484 0,07492 0,07497 0,07501 max Table 6: Deflection (cm) and moment of the plate with variable thicknesses under a uniformly distributed load (BCVT). Case 1: BCVT GE-FDM α = −1; β = −1; 𝛾 ¯ = 0; k = 1; N < 0; F = −1 and R=0 [11] FEM EXPERIMENTAL Meshes 4 x 4 8x8 16 X 16 20 X 20 24x24 32 X 32 va PD 0,0049 0,0054 0,0058 0,0062 0,0063 0,0063 0,0069 0,0070 0,0059 max m (a P) 0,07238 0,07486 0,07551 0,07581 0,07582 0,07582 / / / max method. Thus, we can say that with a mesh course, the Section two: rectangular plate of variable thickness generalized equations give good results. The refinement of freely supported at two opposites edges and the other the mesh makes it possible to observe the convergence of two edges fixed the results. Tables 1 to 5 above illustrate the Convergence Figure 8 shows a rectangular plate with variable thick- well. nesses, freely supported at two opposite edges y =0, y = b to evaluate the impact which the application of the and two fixed edges x 0, x =a. It should be noted that here normal forces of membranes causes on a bent, tended and the plate is not under elastic foundation (R=0). or compressed plate of variable rigidity, we will refer to In order to solve the Eq. (17) and to obtain the numerical Tables 1 to 4 then the curves of Figures 7a and 7b 7; hence : results of deflection and moment, a computer program was a) In all cases when the mesh is increased, the mo- used. The results are presented by the values of deflection ments and arrows increase and are almost monotonic from for the case of a plate with variable thicknesses (h = 6 mm, a certain mesh pitch. h = 8 mm, Figure 8) loaded by the uniformly distributed b) The decrease in bending forces caused by the elastic load. This choice is in order to compare our results with foundation with the normal forces of membranes acting in the experiments ones obtain by [11], using a tensile test traction in one direction is equal to the increase in forces machine with additional equipment. caused in the same conditions compared to those acting in The total load is equal to 24 kPa for the uniformly dis- compression in the same direction. The combined effects tributed load. The calculations were made for plates with of these two forces cancel each other out. dimensions of 180 mm in width and 400 mm in length One can deduct from these interpretations that the de- loaded by the uniformly distributed load. The steel grade flection of a plate of variable rigidity on a flexible founda- NVA with yield stress 235 N/mm is used. tion and subjected to bending combined with traction or compression is less important or even negligible when the two stresses are combined simultaneously. Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation | 63 (a) Figure 8: Rectangular plate with variable thicknesses with two opposite edges freely supported and two xed fi edges [11]. that the GE – FDM has displayed ability to yield accurate solutions on relatively coarse grid, with an order of conver- gence equal to 0, 99. This shows as well the stability of the method. Acknowledgement: The authors are very grateful to Dr Amba Chills, Dr Yakada S. and Dr Mezoue Cyrille for their useful comments on methodology and writing of this paper. Funding information: The authors state no funding in- volved. (b) Author contributions: All authors have accepted responsi- bility for the entire content of this manuscript and approved Figure 7: a) Graphical representation of the moment coeflcients; b) its submission. Ggraphical representations of coeflcients of deflection. 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Journal

Curved and Layered Structuresde Gruyter

Published: Jan 1, 2022

Keywords: rectangular plate; elastic foundation; generalized equations of finite difference method; discontinuity

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