#### Abstract

For a system obtained by placing more than two elastic plates side by side, the transmission conditions are obtained at the common boundaries. Finite difference equations are developed for the problem of plates with internal hinges and applied for determination of the response of a system assembled from three different plates with different mechanical constraints between adjacent plates in this study. An algorithm is written to find out how long the size of the plates should be in order to obtain the desired amount of bending against the force affecting the system under different boundary conditions. The bisection and multigrid methods are used for this. These two methods are compared based on the obtained data.

#### 1. Introduction

Since laminated composite structures have high strength and stiffness, the use of such structures is gradually increasing in implementations in the field of engineering. Timoshenko mathematically explains the theory of elasticity and plasticity in a previous study [1]. The deformation problem of elastoplastic plates whose mathematical model is expressed by biharmonic equations is defined by the boundary value problem. The deformation problems of thin plates modelled by the classical Kirchhoff theory were considered priorly by Reddy [2]. Using the monotone potential operator theory, Hasanov developed the variational approach theory for nonlinear biharmonic equations related to bending of elastoplastic plates [3]. Likewise, the plate theory for multilayered and sandwich plates was developed in different studies [4, 5]. Recently, the first-order shear deformation theory for vibration of rectangular plates with an internal line hinge was investigated. The literature appears to contain a limited number of studies related to plates with internal hinges. Previous studies have analytically solved the buckling problem using the Levy method. The first known numerical solutions were introduced by some researchers [6]. Using the Ritz method, the vibration and buckling analyses of plates with an internal hinge were studied [7, 8]. Moreover, using the Levy-type solution, the exact solutions of natural vibration of rectangular plates were also presented [9]. The vibration studies of plates with point and line supports were also investigated by several researchers [10, 11]. The problem of bending of a rectangular plate given by symmetrical boundary conditions along its edges under a load was also investigated. Quintana and Grossi studied the free vibration of plates with different geometrical shapes with internal line hinges [12, 13]. The vibration of Timoshenko beams with an internal hinge was studied in a previous study [14]. Additionally, a numerical solution was obtained using the finite element method [15]. Grossi conducted a study on anisotropic plates with an arbitrarily located internal line hinge with elastic supports [16]. Grossi and Raffo extended the model for several arbitrarily located internal line hinges [17]. Internal hinges were also investigated using the Modified Stiffness Matrix Method [18]. Furthermore, some researchers studied the variational approach to vibrations of plates with line hinges [19]. Different possible combinations of simply supported, clamped, or free boundary conditions may be given along each plate edge as the boundary conditions. To calculate the deformation of thin plates under quasistatic axial loading, the three-hinge-line method was presented [20]. The transmission conditions on common boundaries were obtained by the functional approximation method in a previous study [21]. In a previous study of ours, we considered the deformation problem of a plate system (formed side by side) composed of multistructure plates.

In this study, the transmission conditions obtained on the common border of the plates that constituted the system in our previous study [22] are extended for a system consisting of plates with different mechanical properties, and the finite difference expressions of these conditions are given. It is the aim of this study to determine how long the sizes of the plates should be for obtaining the desired amount of bending against the force affecting the system under different boundary conditions. For this purpose, we created an algorithm, and the bisection and multigrid methods were utilized. These two methods were compared based on the obtained data. In relation to the bending of the system obtained by applying three elastic plates, the numerical results are presented in tables and plots.

#### 2. Problem Formulation

Let us consider inhomogeneous elastoplastic plates with different properties. The plates make up a system formed side by side. For instance, let us consider the system-filled regions, whose dimensions are , such that , (Figure 1). Here, is the number of plates that form the system, and are the dimensions for each . Moreover, the common boundary of the and regions is considered as .

The mathematical model of the problem of deformation of the multistructure plate system may be written as follows [21]: where is the bending of the system at any and is the force applied vertically onto the -th plate. The cylindrical stiffness coefficients of the plates of the system are . Moreover, the , , and values are Young’s modulus, Poisson constants, and thicknesses for each plate, respectively.

The boundary conditions are generally classified for the equilibrium equation in two manners depending on the physical meanings:

As the system consists of plates with different properties, the equilibrium equation (1) becomes discontinuous at the common boundaries of . Therefore, when the deformation problem of the plate system is solved numerically, it is needed to provide the numerical expressions of the transmission conditions that are suitable for the connection form of the common boundaries. For this, firstly, the uniform and nonuniform meshes in the interval are defined as follows: uniform mesh, non-uniform mesh where . Here, is the number of steps for the mesh created for each plate. Additionally, is a symbolic expression of greatest integer function.

Let the plates that form the system connect to each other with a beam from the common boundary of . Here, the coefficients of stiffness and the torsional stiffness of beams are and , respectively.

In this case, the potential energy of the system is calculated by the following formula:

Moreover, the nonnegative constants and are the stiffness coefficients of the plates connected with hinges and supports provided by the hinge, respectively. For different values of and , there are different physical meanings of the transmission conditions: (i): plates are attached together along their edges using an ideal hinge that is greased in the mechanical sense and moves freely similar to the simply supported condition(ii): it means there is a hinge with a finite stiffness on the common boundary. It behaves as rusted, but it can be hardly moved in the mechanical sense. When the value of is increased, the hinge is going to be rusted more than before, and the plates building the system are caused to start moving together. The movement of the hinge becomes impossible(iii): it means that these plates forming the system behave like a single plate and move together, also bending together(iv): there is no support provided by the hinge on . When the force is applied on the common boundary , the system bends easily(v): there is a support provided by the hinge on the common border, and it has a finite rigidity. When we apply the force on , it is against the force externally applied. It also prevents bending of the system on the common boundary(vi): in this case, there is a support provided which has infinite stiffness by the hinge. There is also no bending on the common boundary

As seen above, the value of is increased, plates building the system start to move together, the value of is increased, and bending on the common boundary becomes stiffer.

The first variation of the full potential energy of the plates denoted functional must equal zero to reach the equilibrium for the loaded elastic body:

After calculating the Gateaux derivative of the functional , the coefficients of the expressions belonging to the common boundary are equal to zero. By adding the discontinuity condition at the common boundary of the plates, the transmission conditions are obtained as follows: .

Here, is called the jump of at .

For the finite difference approximations of the transmission conditions, we use the functional approximation method. Let be the nonuniform and be the uniform meshes on the axes and , respectively. Here, for the each -th plate, is the length of the step on the -axis and is the length of the step on the -axis. Let us define the mesh in the region .

Considering (7)–(9) and , , and , ( is the stiffness coefficients of the beam on the common boundary ), we may denote the finite difference approximation of the potential energy such that

Here, corresponds to the general case, means that there is no torsional case, and means that the stiffness coefficients of beams are the same. Moreover, is an approximation value of the -bending of the system and are the finite difference approximations of the deformation energy functions on the meshes for each -th plate. The finite difference expression of the functional is obtained by the functional approximation method. In order to do this, finite difference approximations are written in the place of derivatives in the energy functional (5). Furthermore, integrals are computed by the numerical integration formula. In the expression (5), integrals containing mixed derivatives are calculated using the rectangle method, while integrals containing other 2^{nd}-order partial derivatives are calculated by the trapezoid method. Then, the finite difference expressions of partial derivatives are written. In order to write the obtained expressions easily, we define the following coefficients [23]:
where , then we may rewrite by using (11)–(13) as follows:
where the function is
and below the sum symbol mean in equation (14) is the mesh contained inner points of mesh .

Because depends on the variables , it is a multivariable function. Considering this, we compute the derivative of the functional and equalize it to zero. Then, we obtain the finite difference expression of equation (1) as follows:

The finite difference expressions of the collected terms that belong to the common boundary are written as from equation (14) obtained as follows:

Using equation (16), we obtain the finite difference approximations of the transmission conditions on the points of the common border and their neighbours, i.e., , , and . Firstly, equation (16) is turned into (18) for the points :

For the points that belong to the common border in , the finite difference expressions of the transmission conditions are

Finally, we reach equation (20) for the points :

Thus, we obtain the finite difference approximations of the transmission conditions on , , and as equations (18)–(20) for the nonuniform mesh.

Consequently, the coefficients of the equations belonging to the points on the common borders and their neighbours of the linear algebraic equations system obtained may be calculated by using (18)–(20).

#### 3. Numerical Example

In relation to the bending of the system obtained by placing three elastic plates side by side, computer experiments were carried out for the analysis of the numerical solution in the case where a clamped boundary condition is provided at the boundaries of the system. Where , let us suppose that there are three thin plates, occupying the rectangular region .

for these meshes, such that the dimensions of the system are cm, thickness is cm, and mesh dimensions are . Each of the plates forming the system is divided into equal steps (uniform mesh) within itself (Figure 2).

When the sizes of the plates composed the system are the same (), the maximal bending corresponding to any force affecting the system is obtained as the solution to the direct problem and denoted by . Since the maximal bending of the plate system takes different values corresponding to different values of and , the initial data is different for each case.

Firstly, a system of plates with the same mechanical properties was considered. To determine the sizes of the middle plate corresponding to the desired bending, the bisection method and the multigrid method are utilized by using the initial data. As the initial data, using the value of maximum bending occurring on the system based on the effect of a known force ( kN/cm^{2} is applied at 5 points on the middle surface of the system), the size of the middle plate of the system may be obtained with the bisection method by considering the mechanical properties of the plate-forming material that are given in Table 1. Then, the approach speeds are investigated.

As a first numerical example, a situation is handled such that Young’s modulus and the Poisson ratios are , respectively. The bending that occurs as a result of the applied force is investigated. In order for the plates forming the system to move together, are obtained as corresponding to different values of the nonnegative constants and in the transmission conditions obtained, and by using and initial data , the size of the middle plate is obtained by the bisection method. The approximate values of and solutions obtained are shown in Table 2.

Based on different values of the nonnegative constants and , after determining the solution interval, with a precision of , the size value of is obtained in 4 steps with error for cases and and and , while the size value of is obtained in 7 steps with error for cases and and and .

Then, let us handle a situation where plates on the left and right sides of the system with the same properties and the middle plate with a different property are given. Similar processes are followed as above when the clamped condition is given on the boundaries of the system for different values of the nonnegative constants and . For this, let us consider iron and copper plates. (i)To do this, firstly, let us consider the two plates on the sides forming the system as copper and the plate in the middle as iron (, , , and ). In this case, we get the middle plate stiffer than the plates on the sides of the system (Figure 3).(ii)Consider the two plates on the left and right sides of the system as iron and the middle plate as copper. Young’s modulus is and for iron and copper, respectively. Additionally, the Poisson ratios for these plates are considered as and , respectively (Figure 4).

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**(d)**

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We are comparing these two situations to our examination (Table 3).

According to different values of the nonnegative constants and , for both Cases 1 and 2 after determining the solution interval, with a precision of , the number of steps, used to obtain the size values of , and approximate error are given in Table 4.

We used the multigrid method for solving the problem. The algorithm of the method that was applied may be written as follows:

Firstly, the sizes of the right and left plates from the plates forming the system are equal, and the size of the middle plate is given by . With the effect of the force, , which is the maximum bending that occurs at the surface of the plate at the size of , is found. Then, considering the bending of the system under the effect of the same force, the maximum bending , which is previously determined, and , which is numerically found, are compared. is the step size of the mesh (I)(a) If , the problem is solved by taking ; (b) if , the problem is solved by taking , and the obtained and are compared(II)The processes in the first step are continued until the bending value is between the maximum bending values and corresponding to the sizes and that are found consecutively. Eventually, one of the conditions or will be met

The following consecutive processes may be performed by using a acceleration parameter to find the value faster: (III)If is defined as , and may be written

Let us suppose and . If is defined, is obtained by the equation for . Since the real value is between and corresponding to the -th and -th approaches, the -th approximation of may be computed as . (IV)The value of found by using value of can be compared to , which was given previously. If the condition is satisfied, the processes are stopped(V)If , then and are taken. If , then and are taken

The steps III-VI are repeated until is satisfied.

Let us now compare the bisection and multigrid methods that we use to find the size of the middle plate, provided that the sizes of the two plates on the sides are equal, and the plates with the different properties in the system are as the Cases 1 and 2 above. Let us consider and for this situation.

As the Case 1 above, the size values of the middle plate corresponding to initial data cm are given in Table 3, provided that the sizes of the two plates on the sides are equal, in the problem of bending of the system consisting of three plates whose Young’s moduli and and Poisson ratios and are as formed side by side. Then, for Case 1, with a precision of , the size value of is obtained by using the multigrid method, and the size values of corresponding to the bending are calculated by taking the step length in Table 5.

After determining the solution interval with a step length of for the middle plate and precision, using the multigrid method, the size value is calculated in one step with error. Using the bisection method, with precision, the size value corresponding to the same bending of the middle plate was calculated in 7 steps with error as shown in Table 4. If we compare the values of Tables 3 and 5, it is clear that the multigrid method is faster than the bisection method.

As the Case 2 above, the size values of the middle plate corresponding to initial data cm are given in Table 3, provided that the sizes of the two plates on the sides are equal, in the problem of bending of the system consisting of three plates whose Young’s moduli and and Poisson ratios and are as formed side by side.

Now, for both Cases 1 and 2, with precision, the size value of is obtained by using the multigrid method, and the size values of corresponding to the bending are calculated by taking the step lengths of and in Table 6. Using the bisection method, the obtained solutions and size values of corresponding to both Cases 1 and 2 are shown in Table 6.

As seen in Tables 5 and 6, with the step size for the middle plate, when the bending values for and precision are compared, it is seen that the solution is approached with less error when the precision is reduced. Such that, using the bisection method, with precision, the size value was calculated in 7 steps with error, while using the multigrid method, even with the precision of , the size value was calculated in 5 steps with error. After determining the solution interval as seen in Tables 4 and 6, with how many steps and what error the solution is found by using the multigrid and bisection methods for a given precision is shown in Table 7.

*Case 1. *After determining the solution interval with the step lengths and for the middle plate, in Table 6, with precision, using the multigrid method, the size values corresponding to bending cm of the middle plate are calculated in 5 steps with error and in 2 steps with error, respectively.

*Case 2. *After determining the solution interval with the step lengths and for the middle plate, in Table 6, with precision, using the multigrid method, the size value corresponding to bending cm of the middle plate is calculated in 5 steps with error and in 2 steps with error, respectively.

Comparing the values in Table 7, it may be seen that the interval in which the solutions are solved with smaller steps is narrowed more, that is, the smaller the step is, the faster the solution is approached for both Cases 1 and 2, when we compare the values that are obtained corresponding to and .

#### 4. Conclusions

In this study, a finite difference method was developed for the problem of plates with internal hinges and applied to determine the response of a system assembled from three different plates with different mechanical constraints between adjacent plates. In order to obtain the desired amount of bending against the force affecting the system under different boundary conditions, an algorithm was written. The bisection and multigrid methods were used to derive the sizes of the plates forming the system. The results obtained from these methods are compared. Our study is distinctive for this research domain. Because this study dealt with mathematics and many disciplines of engineering, this research may indeed prove to be significant for fellow researchers and scientists working in the same discipline.

#### Appendix

The finite difference expressions, which are used in the finite difference equations obtained for corresponding to any point of the nonuniform mesh, are as follows:

Finite difference expressions of the first-order derivatives, where backward and forward differences are with respect to and , respectively.

Finite difference expressions of the second-order derivatives,

Finite difference expressions of the second-order mixed derivatives, and finite difference expressions of the fourth-order derivatives,

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

#### Authors’ Contributions

All authors read and approved the final manuscript.