Dynamic Stationary Response of Reinforced Plates by the Boundary Element Method

A direct version of the boundary element method (BEM) is developed to model the stationary dynamic response of reinforced plate structures, such as reinforced panels in buildings, automobiles, and airplanes. The dynamic stationary fundamental solutions of thin plates and plane stress state are used to transform the governing partial differential equations into boundary integral equations (BIEs). Two sets of uncoupled BIEs are formulated, respectively, for the in-plane state (membrane) and for the out-of-plane state (bending). These uncoupled systems are joined to form a macro-element, in which membrane and bending effects are present. The association of these macro-elements is able to simulate thin-walled structures, including reinforced plate structures. In the present formulation, the BIE is discretized by continuous and/or discontinuous linear elements. Four displacement integral equations are written for every boundary node. Modal data, that is, natural frequencies and the corresponding mode shapes of reinforced plates, are obtained from information contained in the frequency response functions (FRFs). A specific example is presented to illustrate the versatility of the proposed methodology. Different configurations of the reinforcements are used to simulate simply supported and clamped boundary conditions for the plate structures. The procedure is validated by comparison with results determined by the finite element method (FEM).


Introduction
Reinforced panel systems are widely used in buildings, bridges, ships, aircrafts, and machines.These structural systems are efficient, economical, and readily constructed from common materials.The panels are usually built by the association of plates (or shells) with orthogonally displaced beams, which are the reinforcements.The main advantage of applying these structural elements is the increase of structural rigidity without considerable increase in weight.
The static analysis of the reinforced plate systems has been performed using solution strategies such as methodologies based on energy principles [1], semi-analytical methods [2], or the differential quadrature methods [3].Also it is possible to model the behavior of these structures by the finite element method (FEM) [4,5], the boundary element method (BEM) [6][7][8][9][10], or a combination of these numerical methods [11].A rather limited amount of technical literature is available on the dynamic analysis of stiffened plate systems.On the other hand, a significant research effort is under way in both the academia and the industry to improve the numerical models and to develop new modeling methods for the dynamic analysis [12].Finite and boundary elements have some limitations to obtain vibration responses at middle and upper frequency ranges due to the necessity of intense mesh refining.The use of very fine meshes in the finite element analysis results in large algebraic systems.An alternative is posed by the BEM.If formulated with the proper auxiliary state, the BEM only requires boundary discretization, leading to considerable smaller algebraic systems.
Direct boundary element subregion formulations based on Kirchhoff 's plate theory has been applied to the dynamic analysis of thin-walled structures formed by assembling folded plate models using the so-called static fundamental solution [13,14].Assembled plate structures were also analyzed by BEM and comparisons with FEM are given to demonstrate the accuracy of this methodology [15].Another dynamic analysis of elastic plates reinforced with beams takes into account the resulting in-plane forces and deformations in the plate, as well as the axial forces and deformations in the beam, due to combined response of the system [16].The method presented in [16] employs the static solution similar to the models described previously.The consequence of these formulations is that the inertia forces lead to domain integrals.In these previous articles, it was necessary to develop a procedure to deal with the domain integral.An alternative way to derive the governing integral equation for the problem is to use a stationary dynamic fundamental solution [17][18][19].If this fundamental solution is applied, the resulting integral equation requires only the discretization of the boundary of the single-folded plate being analyzed.
The present paper analyzes the dynamic stationary response of reinforced panels subjected to time harmonic loadings using the BEM.In the proposed methodology, the panels are considered as assembled folded-plate structures [20].The formulation is built by coupling BE formulations of plate bending and two-dimensional plane stress elasticity.These uncoupled systems are joined to form a macro-element.The plate structure is divided into several regions, and equilibrium and compatibility equations along the interface boundaries are imposed.The boundaries are discretized by means of linear continuous and discontinuous isoparametric elements.Four displacement integral equations are written for every boundary node.The stationary dynamic responses are characterized by modal quantities, that means by eigenfrequencies and eigenvalues.These quantities are obtained by analyzing the numerically synthesized frequency response functions (FRFs) of the reinforced structures.A harmonic force of constant amplitude excites the structure at a given point and the resulting displacement is measured (calculated) at another point.From the resonances or peaks of the FRFs, the operational eigenfrequencies may be determined.The operational eigemodes (vibration mode shapes) are determined by calculating the folded-plate structure displacement field at the determined operational eigenfrequencies.In the present article, an example is presented to illustrate the proposed methodology where different configurations of the reinforcements are used to simulate simply supported and clamped boundary conditions.The implementation is validated by comparison with numerical results determined by FE solutions.The results obtained by the present BEM are shown to be in good agreement with those obtained by the FEM.The proposed scheme may be seen as an accurate methodology to analyze free and forced stationary vibrations of structures assembled by folded plates, like plate structures and reinforced panels.This methodology may be regarded as an extension of the previous article that analyzed the stationary dynamic behavior of frame structures by the BEM [21].

Boundary integral formulations
The dynamic equilibrium equations for plane stress and thin plate theory will be presented next with Latin indices taking values {1, 2, and 3} and Greek indices assuming the range {1, 2}.In the plane macro-element formulation, the membrane displacements u 1 and u 2 are in the x 1 -x 2 plane.The thin plate transversal displacement w is in the x 3 directions.The equilibrium equations for the dynamic plane stress problem in the domain Ω is given by where σ i j represents the stresses components, ρ is the mass density, F α (α = 1,2) are the body forces components in the x 1 -x 2 plane and dots over the quantities indicate differentiation with respect to time.
The equilibrium equations for an infinitesimal thin plate element under a dynamical transverse loading g and in absence of a body forces are given by where ρh is the mass density per unit area, h is the thickness, q α (q 1 , q 2 ) represent the shear forces, m αα (m 11 ,m 22 ) represent the bending moments and m αβ (m 12 ,m 21 ) represent the twisting moments.Now consider the plane element occupying the area Ω, bounded by the contour Γ, in the plane x 1 -x 2 .The displacement boundary integral equation for the plane stressproblem (membrane) and smooth boundaries is given by where δ αβ is the Kronecker delta; dΓ and dΩ denote boundary and domain differentials, respectively; u β (Q) and t β (Q) are displacement and traction boundary values associated with a boundary point Q, respectively.The term U * αβ (Q,P) represents a displacement fundamental solution and may be interpreted as the displacement at point Q in the direction α due to a harmonic unit point force applied at the point P in the direction β.Analogosly, the term T * αβ (Q,P) represents the traction fundamental solution and may also be interpreted as the traction at point Q in the direction α due to a harmonic unit point load applied at P in the direction β.
Considering that all variables are undergoing a time harmonic displacement with circular frequency ω, the displacement and traction fundamental solutions are given, respectively, by the expressions [22] where where δ αβ is again the Kronecker delta, n is the normal vector, K 0 and K 1 are the zero and first-order modified Bessel function of second kind, r is the distance between load and displacement point, Additionally, the integral equation for the thin plate theory is employed to describe the bending action: where δ(P) is equal to Kronecker delta for a smooth boundary, w is the out-of-plane displacement, w ,n is the rotation in the direction of outward normal to the boundary Γ, V n is the equivalent shear, M n is the bending moment, and R c is the corner reaction.The classical theory makes use of the equivalent shear (V n ) in boundary integrals and a corner reaction (R c ) at each corner when polygonal plates are considered, ) where Q n is the shear in the direction of outward normal and M ns is the twisting moment in the direction normal and tangential to the boundary Γ.The expression (2.8) presents the corner reaction (R c ) at corner k as the difference between the twisting moments at the corner neighborhood on the forward side (M F ns ) and the backward side (M B ns ).Considering again that all variables are undergoing a time harmonic displacement, u(t) = uexp(iωt) with circular frequency ω.Under this circumstance, load g and deflections w will also vary harmonically and the fundamental solution for (2.6) has the form [23,24] with (2.10) In (2.9) to (2.10), the flexural rigidity D is equal to ], E is the Young Modulus and ν is the Poisson ratio.The variables J 0 and Y 0 are the zero-order Bessel functions of the first and second kind, respectively, K 0 is the zero-order modified Bessel function of the second kind.Explicit expressions for derivatives of fundamentals solutions, rotations w * ,n , moments M * n , and shear forces V * n are as follows [23]: where J 1 and Y 1 are the first-order Bessel functions of the first and second kind, respectively; K 1 is the first-order modified Bessel function of the second kind and β the angle formed between r and n.

Algebraic formulation of the macro-elements
In this session, the plane macro-element will be assembled by superposition of the membrane and thin plate effects.The plane stress boundary integral equation (2.3) representing the membrane may be discretized leading to the following algebraic system of equations: Analogosly, the BIE (2.6) describing the out-of-plane bending effect (thin plate) may be discretized as follows: in-plane membrane displacements associated with the in-plane tractions t 1 and t 2 .The plate displacement normal to the x 1 -x 2 plane is w and its derivative with respect to the boundary normal n is w ,n .The corresponding generalized forces are the shear forces V n and the bending moment M n .Equations (3.1) and (3.2) may be superposed to form the plane macro-element in which membrane and bending mechanisms are uncoupled: The interface boundaries between macro-elements must be parallel to a single axis.In a global coordinate system, this axis is called x 3 , as shown in Figure 3.1.Figure 3.1 also shows a plate with reinforcements.It can be noticed that the reinforcements are all aligned parallel to the x 3 axis.
The plane macro-element given by (3.3) is written in terms of a local coordinate system.To perform the coupling of distinct macro-elements, as the ones shown in Figure 3.1, it is necessary to transform (3.3) from a local to a global coordinate system.This is done by means of an intermediate coordinate system and a set of two coordinate transformation matrices.After the macro-element equations have been written in terms of the global coordinate system, the assemblage may take place.The vector of generalized displacements and forces may now be subdivided into ones belonging or not to a common interface.For the case of two macro-elements, the individual equations for every macroelement may be written as The coupling of the macro-elements is performed by considering kinematic compatibility and equilibrium at the interface nodes.Considering T the vector of external loads 8 Mathematical Problems in Engineering applied at the elements interface, compatibility and equilibrium is given by (3.5)After (3.5) has been applied to (3.4), the basic system of equation for two coupled macro-elements is given by In (3.6), U 1 and U 2 are generalized displacement vectors (bending and stretching) related to subregions Ω 1 and Ω 2 , respectively.T 1 and T 2 are the corresponding generalized forces.The displacement vector U i and the corresponding forces vector T i stand for the values at the interface; T 1 i and T 2 i represent forces vectors at the interfaces for each one of the macro-elements.

BEM formulations
In this paper, the macro-elements coupled by (3.6) were discretized by rectilinear boundary elements described by linear shape functions.Considering B 1 and B 2 the initial and final coordinates of the elements, the element geometry may be expressed in terms of intrinsic coordinates, σ: This same interpolation is used for the field variables of the boundary elements possessing no corners, leading to an isoparametrical formulation.For elements with corners, the field variables were discretized by discontinuous elements.The corner nodes were displaced towards the interior by one-fourth of the element length (0.25 L e ).Four integral equations were written for every boundary node.The collocation points were placed outside the plane element (macro-element) domains.When collocation point P is placed outside the plate domain (P / ∈ Ω), the integration free-term disappears, δ(P) = 0.Moreover, the corner reactions R ck can be written in terms of neighbor node rotations using a finite difference scheme.Although this is the correct way to treat corner reactions, in the present implementation these terms were neglected.
A final algebraic system [A]{X} = {B} is obtained once the equations are assembled and the prescribed boundary conditions applied.The solution of this system, the vector X, contains all unknown boundary quantities.The system matrix [A(ω)] contains frequency dependent terms.After the vector X is determined, the displacement at the assembled folded plate domains may be readily obtained by the nonsingular integrations indicated in (2.3) and (2.6).

Numerical analysis
This session applies the previously described strategy to analyze reinforced panels.The strategy is simple.Consider two joined rectangular plates, simply supported (S) in two edges z = 0 and z = L z , and freely supported (F) at the remaining edges, x = 0 and x = L x , as shown in Figure 5.1.The plates are excited by a concentrated force applied at the point with coordinates x p and z p .The frequency of excitation is continuously changed within a preestablished range.The displacement response at some point of the plates to this frequency dependent excitation is the so-called frequency response function (FRF).
In the sequence, reinforcements are placed at the boundaries x = 0 and x = L x .If the reinforcement is very thin but very high in the y-direction, then the bending effect of the support is very small compared to the membrane effect.This should simulate a simply supported (S) boundary condition.On the other hand, if the plate thickness is increased, then the clamped (C) boundary condition should be simulated.The validation strategy is composed of these three steps.In the first step, the FRF of S-F-S-F plate is determined and the natural frequencies of the present methodology compared to results from a finite element (FE) analysis.In the second step, high but thin reinforcements are placed at the originally free boundaries, giving rise to a model that simulates completely simply supported plates S-S-S-S.Again the operational eigenfrequencies are obtained from the FRF.Comparisons are also made with the FE solution.Finally, thicker reinforcements are placed at the free boundaries, simulating the clamped (C) boundary condition.The operational eigenfrequencies for this C-S-C-S plate are compared to the FE results.This strategy is sketched in Figure 5.2.
Take initially the two plates loaded by a unit harmonic normal excitation on the interface between the two plates at distances x 1 = 304 mm and x 3 = 160 mm (X-Z plane), as shown in Figure 5.1.The two plates are assembled and are simply supported (S) at their edges z = 0 and z = L z , andfree (F) at their boundaries x = 0 and x = L x .Each  assembled plate is made of same constitutive properties with Young's modulus E = 6.9 × 10 10 kN/m 2 , thickness h = 4 mm, density ρ = 2700 Kg/m 3 , length L x = 704 mm, width L z = 400 mm, and Poisson ratio ν = 0.3.
Computations by the BEM are carried out for the following boundary discretization (two macro-elements) using linear micro-elements: Mesh 1 : 18 and 20 boundary elements per macro-element (plate 1 and 2, resp.) and Mesh 2 : 28 and 30 boundary elements per macro-element (plate 1 and 2, resp.).An example of the discretization of boundary is shown in Figure 5.3.
Figure 5.4 shows the FRF 18-18 for the first BEM mesh.The FRF 18-18 is obtained by exciting node 18 (see Figure 5.3) and measuring the response at the same node.In this FRF, the resonances and antiresonances can be clearly recognized.The system operational eigenfrequencies (natural frequencies) are determined from the frequencies at which resonances in the FRF occur.
The values of the first six eigenfrequencies taken from the FRF of the two assembled plates given in similar discretization of the FEM by ANSYS using SHELL63 elements.The discretization of de FEM by ANSYS consisted of the 18 × 30 finite elements.Now the reinforcements are included in the originally freely supported boundary conditions (F).The resulting reinforced panel structure is shown in Figure 5.5.In the reinforcements, only the central nodes of the sides are submitted to simply supported boundary conditions.The remaining nodes are free.
To simulate a simply supported boundary condition (S) as shown in Figure 5.6, the reinforcement is a thin and high macro-element.In this case, the plates and reinforcements are made of same material properties described in the previous example.
The reinforced panel is discretized with 4 macro-elements, 2 elements for the plates and 2 elements for the reinforcements (see Figure 5.7).Two BE meshes are used to perform the calculations.In the first mesh, all 4 macro-elements are discretized with 18 × 20  Let us now consider the plate structure with its entire contour under clamped (C) boundary conditions (Figure 5.9(a)).The intention is to simulate a clamped boundary condition of the two plates on the boundaries z = 0 and z = L z using a thick reinforcement.The idea was to introduce thick reinforcements to increase the bending rigidity.This strategy is illustrated in Figure 5.9(b).In the reinforcement, only the central nodes of the sides are considered clamped.The others are considered free.In this case, the plates are made of same material properties and geometry described in the previous example.However, the thickness of the two reinforcements is increased to 40 mm.This thickness is increased by a factor 100, compared to the previous case.
The FRF 18-18 , of the reinforced (campled) panel structure discretized by BE is shown in Figure 5.10.The discretization utilized is the same as the previous case (mesh 1).The values of the first six (operational) eigenfrequencies are reproduced in Table 5.3.The eigenvalues obtained by the commercial FEM code ANSYS using 18 × 30 SHELL63 elements are also given in Table 5.3.
The other modal quantity necessary to characterize the stationary dynamic behavior of the reinforced panel structure is given by the eigenmodes or the natural modes of vibration.For the last case (clamped bc), the operational eigenmodes are obtained by calculating the displacement field at boundary of the structure at each resonance frequency present in the FRF.
Luiz Carlos Facundo Sanches et al. 15 Figure 5.11 shows the boundary displacements corresponding to the first four eigenmodes of the excited structure.In this case, all external boundary nodes of the structure are campled.It should be noticed that the second and third modes present the same boundary displacement.

Concluding remarks
An implementation of the direct version of the boundary element method has been presented to analyze the stationary dynamic behavior of the reinforced panel structures.The dynamic stationary fundamental solution has been used to transform the differential equations governing the thin plate and membrane behavior into boundary-only integral equations.The proposed scheme is used, exemplarily, to obtain modal data, that is, operational eigenfrequencies and eigenmodes of the assembled plates and reinforced panel structures with different boundary conditions.The formulation was shown to be capable of modelling plates subjected to varied boundary conditions and out-of-plane loadings.Frequency response functions may be determined for every boundary or domain point of the structure.In the reported examples, the FRF of a node on an interface boundary is used to recover eigenfrequencies.The eigenfrequencies are determined from the resonances of the FRF.At these resonance frequencies, the displacement fields of the structure furnish the operational eigenmodes.The presented results agree well with numerical solutions obtained by a FEM commercial code.The proposed scheme may be seen as an accurate methodology to analyze free and forced stationary vibrations of structures assembled by folded plates, plate structures, and also reinforced panels which only require the discretization of the folded plate boundary.The simplicity of the BE mesh generation presents some advantages over other domain methods.

Table 5 .
1. First six natural frequencies of the two SFSF plates [Hz].

Table 5 .
2. First six natural frequencies of the SSSS structure[Hz].The values of the first six eigenfrequencies of the reinforced panel structure are reproduced in Table5.2.These values are compared with results obtained by the FEM commercial code ANSYS using 18 × 30 SHELL63 elements.

Table 5 .
3. First six natural frequencies of the CCCC structure [Hz].