Plates are structural elements of many engineering structures, like bridges, ships, aircrafts, etc. Plates are classified into three categories depending on thickness-span ratio: thin plates, h/L<0.01, moderately thick plates, 0.01<h/L<0.2, and thick plates, h/L>0.2. Structural analysis (strength, vibration, buckling) of thin plates has been performed within the well-known Kirchhoff plate theory , while moderately thick plates are analysed by the Mindlin plate theory as a 2D problem , . Thick plates are considered to be an elastic body analysed by 3D theory of elasticity.
Dynamic behaviour of moderately thick plates is a more complex problem than that of thin plates, since influences of shear and rotary inertia are taken into account. The Mindlin theory deals with a system of three differential equations of motion in terms of three independent variables, i.e. deflection and two angles of cross-section rotation. A large number of papers has been published on this challenging problem and a comprehensive literature survey up to 1994 can be found in .
Generally speaking, there are two approaches to analysis of structural problems of moderately thick plates, i.e. analytical methods for solving differential equations of motion and numerical procedures based on the Rayleigh-Ritz energy method as well as the finite element method (FEM). Different analytical methods have been developed depending on which independent variables are selected as fundamental ones in the reduction of the system of differential equations of motion. Some methods operate with three, two or even one variable, as shown in , , , respectively. Developed analytical methods for vibration analysis of simply supported plates are relatively simple, as well as those for plates with simply supported two opposite edges. For vibration analysis of plates with any combination of simply supported and clamped edges a sophisticated closed-form solution is presented in .
The Rayleigh-Ritz method is widely used for vibration analysis of plates with arbitrary boundary conditions (simply supported, clamped and free) as well as with elastically supported edges. The achieved level of accuracy and convergence of solution depend on the chosen set of coordinate functions for definition of natural modes. Usually, two dimensional polynomials , or static deflection functions of the Timoshenko beam, , are used. An efficient solution is also achieved by applying the assumed mode method , .
The finite element method is a universal numerical tool for structural analysis of complex engineering structures concerning both the topology and material properties. A few triangular, rectangular and quadrilateral finite elements with different number of nodes have been developed for Mindlin plate and incorporated in the library of commercial FEM software. Generally, the elements deal with three independent displacement fields, i.e. deflection and two cross-section rotations. They are prescribed by polynomials of the same order and in transition from thick to thin plate it is not possible to capture pure bending modes and zero shear strain constraints. In order to overcome this shear-locking problem in the FEM analysis, a few procedures have been developed, which are referred in : reduced integration for shear terms , , which is commonly used in commercial software; mixed formulation of hybrid finite elements , , ; Assumed Natural Strain , , ; and Discrete Shear Gap (DSG) . Recently, a new shear-locking-free finite element formulation for static analysis of moderately thick plates has been proposed, based on an extension of the Kirchhoff thin plate theory .
In order to overcome the above problems, the Mindlin theory has been modified . The system of three governing differential equations of motion is reduced to a single equation with bending deflection as a potential function for determining shear deflection and cross-section rotation angles. By employing the modified Mindlin theory a new finite element formulation for moderately thick plates is presented in , . Four-node rectangular finite element and three-node triangular finite element are worked out utilizing polynomial shape (interpolation) functions. The stiffness matrix, consisting of the bending stiffness matrix and the shear stiffness matrix, as well as the mass matrix, are developed. In this formulation the shear stiffness matrix of a thin plate becomes negligible compared to the bending stiffness matrix. Hence, the developed finite elements are shear-locking-free.
The above two finite elements are non-conforming since compatibility conditions of displacements are satisfied only in the nodes of the adjacent finite elements. As a result, convergence of natural frequencies is not monotonous. Therefore, in this paper a conforming four-node rectangular finite element is presented, which satisfies compatibility conditions along all edges of the adjacent finite elements. Shape functions of plate defection are defined as a product of the Timoshenko beam shape functions in longitudinal and transversal direction. The bending and shear stiffness matrices, as well as the transversal and rotary mass matrices are derived in a relatively simple way by employing an ordinary variational formulation . The finite element is shear-locking-free and values of natural frequencies monotonously converge to exact solution from above.
2 Outline of the modified Mindlin theory
Displacements of a thick plate, i.e. total deflection, w, and angles of rotations, ψx , ψy , are shown in Figure 1 in the Cartesian coordinate system. The basic idea of the modified Mindlin thick plate theory is, like in case of the modified Timoshenko beam theory , decomposition of the total deflection into bending deflection and shear deflection
Rotations of the plate cross-sections are caused only by bending, and one can write for rotation angles
Bending moments and twist moments are results of plate curvature and warping, respectively
is the plate flexural rigidity, and h, E and ν is plate thickness, Young’s modulus of elasticity and Poisson’s ratio, respectively.
Shear strain is defined as summation of the plate generatrix rotation and cross-section rotation, i.e.
As a result the shear forces read
where S=ks Gh is shear rigidity and ks is shear correction coefficient.
Plate natural vibrations are performed under action of distributed inertia force and bending moments
where m̅=ρh is the plate mass per unit area, J=ρI=ρh3/12 is the mass moment of inertia of the cross-section per unit breadth, h is the plate thickness and ρ is mass density.
Consideration of the equilibrium of vertical forces and moments around the x and y axis, by applying the above relations, leads to a single differential equation of motion in terms of bending deflection
where is the Laplace differential operator and qe is distributed excitation load. Once bending deflection wb is determined the total deflection (1) is obtained by the following formula 
Final plate deformation depends on boundary conditions.
3 Formulation of shape functions
The four-node rectangular finite element with three degrees of freedom (d.o.f.) per node is considered, Figure 2. The dimensionless coordinates ξ=x/a and η=y/b are introduced due to reason of simplicity. The shape (interpolation) functions of plate deflection are assumed in the form of products of thick beam shape functions, based on the modified Timoshenko beam theory , Xi (ξ) and Yj (η), i, j=1, 2, 3, 4, in x and y direction respectively
The beam shape functions, which take both shear stiffness and rotary inertia into account, are specified in Appendix A.
Each of 12 shape functions Φk (ξ, η)=Xi (ξ)Yj (η), where indexes i and j appear in combinations (10), can be expanded into beam bending and shear terms, and mixed bending and shear terms
The beam bending and shear shape functions are also given in Appendix A. Hence, each of plate shape functions consists of four displacement fields, i.e. bending in both x and y direction, Xib Yjb ; bending in x and shear in y direction, Xib Yjs , and vice versa, Xis Yjb ; and shear in both directions, Xis Yjs .
In order to formulate finite element bending and shear stiffness matrix it is necessary to decompose bending and shear shape functions (11), as in the case of non-conforming finite element , . Unfortunately, this is not possible in the considered case. However, cross-section rotation angles and shear angles, which are caused by bending and shear respectively, can be extracted from (11)
Hence, for derivation of all finite element properties it is necessary to operate with the known shape functions of total deflection (11) and rotation and shear angles (12) and (13), respectively.
The first two deflection shape functions of the present conforming and non-conforming finite element of aspect ratio a/b=1 and thickness-span ratio h/b=1.6, Appendix B, are shown in Figure 3. High value of h/b is a result of plate thickness-span ratio h/B=0.2 and the finite element mesh 8×8. The first shape function of non-conforming finite element is very close to that of conforming element, while the difference between the second shape functions are relatively large.
4 Bending and shear stiffness matrix
For derivation of bending and shear stiffness matrix the ordinary finite element technique is used. Finite element deflection is expressed as product of deflection shape functions and nodal displacements
where according to Figure 2
In a similar way one can write for rotation angles
and shear angles
According to (3) the finite element bending curvature and warping can be presented as vector
By substituting (12) into (16), and taking (18) into account, yields
By using a general formulation of stiffness matrix from the finite element method based on variational principle , one can write for bending stiffness matrix
Elements of matrix [K]b , after multiplication of the subintegral matrices, can be presented in the form
where, according to (20)
Related to the shear stiffness the shear strain vector according to (5) reads
By substituting (13) into (17), and then into (25), yields
Analogously to (21) one can write for the shear stiffness matrix
is matrix of the plate shear rigidity. Since [D]s includes unit matrix, Eq. (28) is reduced to the form
5 Translational and rotary mass matrix
According to the general formulation , mass matrix depends on deflection shape function Φk , Eq. (11), and one can write for the translational mass matrix
The beam shape functions Xi and Yj are specified in Appendix A.
Mass matrix due to mass rotation is derived in the following way. According to (2) vector of cross-section rotation reads
and taking into account (12) and (16), yields
Rotary mass matrix is specified similarly to the shear stiffness matrix, Eq. (26)
Hence, one can write
Finally, the finite element equation for natural vibration analysis of thick plate reads
where ω is natural frequency. Elements of shear stiffness matrix [K]s , according to (27) and (A3), includes parameters α and β, Eqs. (A4) and (A5), respectively. Their values are rapidly reduced for thin plate due to small thickness-span ratio h/a and h/b. In case of very thin plate shear stiffness matrix [K]s becomes negligible compared to bending stiffness matrix in (38). This fact indicates that the presented finite element is shear-locking-free. Assembling of the finite element Eqs. (38) is performed in ordinary way of the finite element technique , .
6 Illustrative numerical examples
The developed conforming four-node finite element is validated by three numerical examples of natural vibrations with different boundary conditions: simply supported square plate (SSSS), rectangular plate clumped on transverse edges and simply supported on longitudinal edges (CSCS), and rectangular plate with combined clamped, free, and simply supported boundaries (CFSS). All plates are modelled with 8×8=64 finite elements. Value of plate aspect ratio, a/b, and shear correction factor, ks , as well as formula for dimensionless frequency parameter μ or λ, are given in the title of Tables 1–3 . Thin, moderately thick and thick plate, are considered. Values of frequency parameter determined by the present conforming finite element (FEM-PC), and those by non-conforming finite element (FEM-NC) , , are compared with analytical solution for case SSSS and CSCS, Tables 1 and 2. For boundary conditions CFSS, Table 3, solution obtained by the Rayleigh-Ritz method  is used as the referent one. FEM results agree very well with analytical ones. Between FEM-PC and FEM-NS results small difference can be noticed for thick plate.
The same boundary value problems are solved by commercial software LS-DYNA  and NASTRAN , for comparison. In the former case fully integrated shell element with four nodes, based on the Reissner-Mindlin theory (ELFORM 16) is used, while in the latter case 2D four-node thick plate finite element is applied. The NASTRAN results are considerably lower than the analytical values for all three plate thickness, while those determined by LS-DYNA are somewhat higher for thin plate and lower for thick plate.
The convergence test is carried out for the thick plate and all three cases of boundary conditions, Tables 4–6 . The finite element mesh density is increased from 2×2 to 10×10. It can be observed that values of frequency parameters in all three examples converge monotonically from above as the mesh density is increased. Convergence is faster for lower frequencies, since simple shape of corresponding natural modes can be successfully described by smaller number of finite elements.
An advantage of the modified Mindlin theory, outlined in Section 2, is dealing with only one variable, i.e. bending deflection as a potential function for determining total (bending+shear) deflection, cross-section rotation angles, strains and sectional forces. This enables a new finite element formulation for moderately thick plate by following the ordinary FEM procedure and ensuring in such a way variational consistency. The four-node rectangular finite element is defined by specifying the shape functions of the total deflection as a product of the Timosheko beam shape functions in longitudinal and transversal direction. In this way conformity of the finite elements is ensured, while the application of the Modified Mindlin theory results with shear-locking-free finite element.
As a result of the above two finite element characteristics, the illustrative numerical examples of thin and moderately thick plate with various boundary conditions show high level of accuracy and fast monotonous convergence of natural frequencies to the exact values from above. Moreover, this relatively simple finite element, in the most analysed cases, achieves a higher level of accuracy than the sophisticated finite elements incorporated in the used commercial software.
This investigation was done within the international collaborative project Global Core Research Center for Ships and Offshore Plants (GCRC SOP), established by the South Korean Government (MSIP) through the National Research Foundation of the South Korea (NRF). Therefore, the authors are grateful to both MSIP and NRF for the support under Grant No. 2011-0030013 signed in between NRF and University of Zagreb.
Beam shape functions
Total functions, Xi =Xib +Xis
Bending functions, Xib
Shear functions, Xis
Beam shape functions in y-direction, Yj =Yjb +Yjs , have the same form as those in x-direction, Xi =Xib +Xis . It is only necessary to change argument ξ into η and use parameters β and μ instead of α and λ, where
Shape functions of the non-conforming finite element
The four-node thick plate finite element with compatible nodal displacements, based on the modified Mindlin theory, is presented in , . Here, only shape functions are specified due to comparation to the conforming finite element.
Bending shape functions, shear shape functions and total deflection functions are given in matrix notation
Parameters α and β are defined by formulae (A4) and (A5), respectively.
Matrix [C] is of the following form
Matrix [C] can be inverted in symbolic form by a CAS package. Matrix elements are rather complicated to be presented in the paper.
Formulae for natural frequencies of simply supported plate
Equilibrium of forces
Natural vibrations are harmonic and one can write for bending deflection
where ω is natural frequency. By substituting (C1) into homogeneous differential equation of motion (8), yields
Solution of Eq. (2) can be assumed in the form
Shear deflection, according to (9), is
and total deflection
The above solution satisfy all boundary conditions for simply supported plate.
By substituting (C3) into (C2) one obtains biquadratic equation for determination of natural frequency . Its solution reads
Natural frequencies of simply supported plate can be also determined by the energy approach. Bending strain energy, according to , reads
Substituting (C3) into (C9) one obtains after integration
The shear strain energy is
By taking into account formulae (6) and (C4), one obtains after integration
Kinetic energy due to mass translation is,
Substituting (6) into (C4), yields after integration
Kinetic energy due to mass rotation is
Substituting (C3) into (C18) and after integration one can write
Natural frequency is defined as
Substituting corresponding expressions for strain and kinetic energy, Eqs. (C10), (C13), (C16) and (C19) into (C20) one obtains
Two factors in (C21) can be presented in expanded form
Their values grow rapidly with plate thickness.
Formula (C22) for the first natural frequency (m=n=1) of square plate (a=b) takes quite simple and transparent form
is the first natural frequency of thin plate.
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About the article
Published Online: 2017-05-16
Published in Print: 2016-12-20