On-line version ISSN 1679-7825
Lat. Am. j. solids struct. vol.7 no.4 Rio de Janeiro 2010
Anupam ChakrabartiI,*; Abdul Hamid SheikhII
IDepartment of Civil Engineering, Indian Institute of Technology, Roorkee-247667, Ph. +91(1332)285844, Fax. +91(1332)275568 India
IISchool of Civil, Environment and Mining Engineering (Office: EM111), University of Adelaide, North Terrace, Adelaide, SA 5005 Australia
Dynamic instability of laminated sandwich plates having inter-laminar imperfections with in-plane partial edge loading is studied for the first time using an efficient finite element plate model. The plate model is based on a refined higher order shear deformation plate theory, where the transverse shear stresses are continuous at the layer interfaces with stress free conditions at plate top and bottom surfaces. A linear spring-layer model is used to model the inter-laminar imperfection by considering in-plane displacement jumps at the interfaces. Interestingly the plate model having all these refined features requires unknowns at the reference plane only. However, this theory requires C1 continuity of transverse displacement (w) i.e., w and its derivatives should be continuous at the common edges between two elements, which is difficult to satisfy arbitrarily in any existing finite element. To deal with this, a new triangular element developed by the authors is used in the present paper.
Keywords: dynamic instability, imperfection, partial edge loading, sandwich plate, refined plate theory, Finite Element.
The loads acting on parts of turbines, electric machines and parts of aircraft or ships due to aerodynamic and hydrodynamic effects are some typical examples of in-plane partial edge loads, which may induce dynamic instability. Dynamic instability of laminated sandwich plate subjected to in-plane partial edge loading is of considerable importance in mechanical, aerospace and many other engineering fields. The most important feature of laminated composite plates is that they are weak in shear compared to that of extensional rigidity. Due to this, the effect of shear deformation becomes very significant consideration for the analysis of such laminated structures. Moreover, the problem becomes much more complex if some inter-laminar imperfection is found in the form of weak bonding or otherwise. All these aspects have been discussed in detail in some earlier paper by the authors i.e., Chakrabarti and Sheikh .
Plates subjected to in-plane loads may lead to parametric resonance due to certain combination of the load parameters. The instability may occur below the critical load of the structure under in-plane loads over a range of excitation frequencies. The well-known Hill's method of infinite determinants is used for solving a system of Mathieu-type equation in the present problem to predict the stability properties. Dynamic instability of plates under different in-plane loads has been investigated by a number of different investigators in case of perfect interface. The dynamic stability of rectangular isotropic plates under various in-plane forces has been studied by Bolotin , Jagdish  and Yamaki and Nagai . Hutt and Salam  and Deolasi and Datta  used finite element method based on first order shear deformation theory (FSDT) to study the parametric instability characteristics of thin isotropic plates. The dynamic stability of rectangular laminated composite plate due to periodic in-plane load is studied by Srinivasan and Chellapandi  using finite strip method (FSDT). Dynamic stability of laminated composite plates due to periodic in-plane loads is investigated by Chen and Yang  using FSDT. Dynamic instability of composite laminates has been studied by Kwon  (finite element method) and Chattopadhay and Radu  (analytical method) using Higher order shear deformation theory (HSDT). Lee  studied the finite element dynamic stability of laminated composite skew plates containing cutouts based on HSDT. Dynamic instability analysis of composite laminated thin walled structures has been carried out by Fazilati and Ovesy  by using two versions of FSM. Patel et al.  have done the parametric study on dynamic instability behavior of laminated composite stiffened plate by using the FSDT. Dynamic instability behavior of composite and sandwich laminates with interfacial slips has been studied by Chakrabarti and Sheikh  by using RHSDT (refined higher order shear deformation theory).
However, no studies based on RHSDT are found in the literature in case of imperfect laminated sandwich plates having in-plane partial edge loading. In this paper attempt has been made for the first time to study the dynamic instability of imperfect laminated sandwich plates with in-plane partial edge load using a finite element plate model recently developed by the authors based on RHSDT in combination with linear spring layer model. The problem is solved by finite element technique in order to have generality in the analysis and also to generate new results.
Following the concept of refined higher order plate theory (RHSDT) and linear spring layer model discussed earlier, the through thickness variation of in-plane displacements (Fig. 1) may be expressed as follows.
where H(z-zi ) and H(-z+zi+1 ) are the unit step functions.
The transverse displacement is assumed to be constant over the plate thickness i.e.,
The stress-strain relationship of a lamina (say k-th lamina) in structural axes system (x-y) may be expressed as
where the rigidity matrix can be formed with the material properties and fiber orientation of the k-th lamina following the usual techniques of laminated composites.
The imperfection at the k-th interface is characterized by the displacement jumps Δuk (Fig. 1) and Δvk , which may be expressed in terms of inter-laminar shear stresses at that interface utilizing the concept of linear spring-layer model as
where are the compliance coefficients of the idealized linear spring layer at the k-th interface whereas and are the transverse shear stresses on that interface. Taking an adjacent layer of k-th interface, and may be expressed in terms of γxz and γyz (transverse shear strains) of that layer at this interface with the help of Eq. (4). Again Eqs. (1)-(3) may be used to express where Δuk and Δvk will not fortunately appear and it will help to express Δuk and Δvk in terms of other terms easily. Now the condition of zero transverse shear stress/ strain at the top and bottom surfaces of the plate is utilized to express βx, βy, ηx and ηy as
Finally, the condition of transverse shear stress continuity at the interfaces between the layers is imposed to express and in terms of the quantities at the reference plane as
where γx ( = w,x - θx = w,x + ) and γy ( = w,y - θy = w,y + ) are the transverse shear strains at the reference plane. The constants (axx, axy, bxx, byy, ...) found in the above equation are dependent on the material properties of the two layers adjacent to the i-th interface.
The FE formulation of the element has been described in detail in Reference . The element developed may have any arbitrary triangular shape and orientation as shown in Fig. 2(a). It is mapped in a different plane (ζ-η) to have a regular shape as shown in Fig. 2(b).
The stiffness matrix [k], geometric stiffness matrix [kg] and mass matrix [m] are evaluated for all the elements and assembled together to form the overall stiffness matrix [K], geometric stiffness matrix [KG] and mass matrix [M] of the whole structure and these matrices are stored in a single array following the skyline storage technique. With these matrices, the equation of equilibrium for an elastic system undergoing small displacements at the instant of buckling may be written as
In the above equation, the in-plane load factor P is periodic and may be expressed in the form
where PS is the static portion of P, Pt is the amplitude of the dynamic portion of P with Ω as the frequency of excitation. The buckling load Pcr may be used to express PS and Pt as follows:
where α and β are static and dynamic load factors respectively. Using Eqs. (10-11) the equation of motion (9) may be expressed as
Eq. (12) represents a system of second order differential equation with periodic coefficients of Mathieu-Hill type. One of the most interesting characteristics of the equation is that, for certain relationships between its coefficients, it has solution which is unbounded. The regions that correspond to the regions of dynamic instability is the physical problem under consideration in the present study. The boundaries of dynamic instability are formed by the periodic solution of period T and 2T, where T = 2ρ/Ω. The boundaries of the primary instability region with period of 2T are of practical importance and solution can be achieved in the form of trigonometric series:
After substitution of the above equation into Eq. (12), if the first term of the series is considered, it leads to a series of algebraic equations for the determination of instability regions. Principal instability region, which is of practical importance, corresponds to k = 1 and for this case the dynamic instability equation leads to
The two conditions under plus and minus signs correspond to two boundaries (upper and lower) of the dynamic instability region. Eq. (14) is solved by the simultaneous iteration technique proposed by Corr and Jennings . The above eigenvalue solution give the value of Ω, which are the bounding frequencies of the instability regions for the given values of α and β. Before solving the above equations, the stiffness matrix [K] is modified through imposition of boundary conditions. The boundary conditions used are same as discussed in some earlier studies .
3 NUMERICAL EXAMPLES
In this section some numerical examples are presented for imperfect laminated sandwich plates subjected to in-plane partial edge loading. Two different types of loading are considered. In loading type (I), the loaded length in the plate edge is near the corners (see Fig. 3(a)) while in loading type (II), the loaded length is in the middle of the edge (see Fig. 3(b)). In case of uni-axial loading the in-plane edge load is considered to be acting along the x-direction (Fig. 3). As there is no published result on dynamic instability of imperfect as well as perfect laminated sandwich plates subjected to partial edge loading, problem of a perfect isotropic plate subjected to uniform in-plane edge loading is presented for validation. The present results are found to be matching well in this case with the standard results. Subsequently in some cases the present results are compared with those obtained by using the FE package Abaqus (version 6.8). A number of new results are presented for imperfect laminated sandwich plates subjected to partial edge compression, which should be useful in future research.
3.1 Square isotropic plate simply supported at the four edges
To study the convergence and to validate the present results a square isotropic (ν = 0.3) plate simply supported at all the edges having uni-axial in-plane uniform edge loading is analyzed by the present finite element model considering different mesh divisions (4x4, 6x6, 8x8, 12x12, 16x16 and 20x20). It is observed that the convergence in case of thin plate (h/a = 0.01) is obtained for mesh division 16x16. Hence all subsequent analyses are made taking mesh division: 16x16 for the higher thickness ratio (i.e., h/a = 0.05 and 0.20). The static load factor (α) and the dynamic load factor (β) are varied to identify the lower and upper boundaries of the excitation frequency. The values of the excitation frequency parameters, Ω = wa2√(ρh/D) obtained by using the proposed finite element plate model are presented in Table 1. The present results corresponding to thin plate (h/a = 0.01) are compared with the analytical results of Hutt and Salam ; and finite element (FSDT) results of Srivastava et al. . The results are found to be matching well. Unfortunately there is no published result on the present problem for the thicker plates (h/a ≥ 0.05). It may be observed that for a given a the difference between upper and lower excitation frequencies of the system increases with increase in the values of the dynamic load factor (i.e., β). This behavior is more prominent in case of thin plates.
3.2 Cross-ply square laminate simply supported at the four edges
The problem of a simply supported cross-ply (0/90/90/0) square laminate (Fig. 3, a = b) subjected to uni-axial in-plane partial edge loading is studied in this example. The analysis is carried out by the proposed element using mesh sizes (full plate) of 10x10 taking h/a = 0.10. In this problem, all the layers are of same thickness and material properties (E1 = 40E, E2 = E, G12 = G13 = 0.6E, G23 = 0.5E and ν12 = 0.25). The imperfections at the layer interfaces are defined by the parameters: = Rh/E and where the non-dimensional parameter R is taken as 0.0, 0.5 and 1.0 (R = 0.0 represents perfect interface). In this paper some of the reported experimental values of imperfection parameters (R) are used, which generally varies from 0 to 1.0 or slightly higher. The loaded edge length (%) for both the load types are considered as 100 (i.e., fully loaded edge), 60 and 20 respectively, while α (0.0 to 0.6) and β (0.25 to 0.75) are varied to identify the lower and upper boundaries of the excitation frequency. The results obtained are presented in the form of excitation frequency parameter, in Table 2. Some of the present results (R = 0.0 and 100% in-plane edge loading) are compared with the results obtained by using the FE software package Abaqus (version 6.8). The results obtained from the two sources compared well. In this case it is specifically observed that the excitation frequencies reduce rapidly in all the cases with increase in the values of imperfection parameters (R). This is the expected behavior because the stiffness of the plate reduces with increase in the values or R. Also the values of upper excitation frequencies obtained in case of Type: I loading are more compared to Type: II loading for α = 0.0 while this is not so for higher values of α. The values of the lower excitation frequencies are always lesser in case of Type: II loading.
3.3 Simply supported square sandwich plate having three orthotropic layers
The dynamic instability of a simply supported three layered square plate (h/a = 0.1) subjected to in-plane uni-axial partial edge loading (Fig. 3) is studied here for both types of loadings (i.e., I and II, Fig. 3). The central orthotropic layer has a thickness of 0.8h while each of the face layers is 0.1h thick. The material properties of the orthotropic face layers are taken as multiple (Kt) of those of the central layer/core where the value of Kt is taken as 5.0. The material properties used for the core are E22/E11 = 0.543, G12/E11 = 0.2629, G13/E11 = 0.1599, G23/E11 = 0.2668, ν12 = 0.3. The imperfections at the layer interfaces are defined by the parameters: = Rh/E11 and where the non-dimensional parameter R is varied from 0.0 to 1.2 (R = 0.0 represents perfect interface). The uni-axial in-plane partial edge loading is considered for both type I and type II loadings, while α and β are varied as before. The loaded edge length (%) for both the load types are taken as 100 (i.e., fully loaded edge), 60 and 20 respectively. The same loadings are also considered for all subsequent examples. The results obtained for the excitation frequency parameters, are presented in Table 3. Some of the present results (R = 0.0 and 100% in-plane edge loading) are compared here also with the results obtained by using the FE software package Abaqus (version 6.8). The results are quite close to each other in all such cases. In the present case of sandwich plate, it is also observed that the values of both the upper and lower excitation frequencies reduce with increase in the values of imperfection parameter, R. But the rate of reduction is not as high as observed in case of the previous example of laminated composite plate. It is also found that with the increase in the values of loaded edge length the upper excitation frequencies always increase.
3.4 Square sandwich plate with laminated face sheets simply supported at the four edges
The problem of a square simply supported sandwich plate (0/90/C/90/0) having laminated face sheets subjected to in-plane partial edge loading (Fig. 3) is studied in this example for both types of loading (i.e., I and II, Fig. 3). The thickness of the core is 0.8h while that of each ply in the laminated face sheets is 0.05h. Taking thickness ratio (h/a) of 0.20, the problem is solved for α= 0.0, 0.3 and 0.6 while β is varied from 0.2 to 0.75. The non-dimensional excitation frequency parameters Ω = 100wa√(pc/E11 obtained for upper and lower boundary are presented in Table 4. The results show a consistent trend in their variation. The material properties of a ply in the laminated face sheets and those of core are as follows.
Face: E11/E = 40.0, E22/E = 1.0, G12/E = G13/E = G23/E = 1.0, ν12 = 0.25, ρf = ρ
Core: E/Ec = 11.945, Gc12/Ec = Gc13/Ec = 1.173/6.279, Gc23/Ec = 2.415/6.279, νc12 = 0.0025 and ρ/ρc = 0.6818. The imperfections at the layer interfaces are defined by the parameters: = Rh/Ec and where the non-dimensional parameter R is varied from 0.0 to 1.0.
In this example of a thick plate (h/a = 0.2) it is observed that the difference between the lower and upper excitation frequencies are not so much as observed in the previous examples.
3.5 Sandwich plate under bi-axial in-plane partial edge loading
A square laminated sandwich plate (-45/45/C/45/-45) subjected to bi-axial in-plane partial edge loading (Fig. 3) is analyzed with boundary conditions SCSC i.e., two opposite edges (parallel to y: Fig. 3) simply supported and the other two edges clamped. The thickness and material properties of the core and the plies in the face sheet; and imperfection defined at the layer interfaces are same as those used in the previous example. Taking the thickness ratio (h/a) = 0.10, static load factor (α) and dynamic load factor (β) are varied as in the previous examples. The excitation frequency parameters Ω = 100wa√(pc/E11 obtained by the proposed FE model are presented in Table 5. In this example of bi-axial loading it may be observed that the values of the lower and upper excitation frequencies are closer compared to the other examples of uni-axial loading.
3.6 Simply supported double core rectangular sandwich plate with laminated stiff sheets
The problem of a simply supported rectangular sandwich plate having double core with three laminated stiff sheets (0/90/90/0/C/0/90/90/0C/0/90/90/0) is studied in this example taking uni-axial in-plane partial edge loading (Type I). The thickness of each core is 0.425h while that of each ply in the stiff laminated sheets is 0.0125h. Their material properties and the imperfections at the layer interfaces are same as those used in the previous example. In this example, the plate subjected to uni-axial in-plane partial edge loading is analyzed with the proposed FE model with aspect ratio (a/b) = 2.0 and thickness ratio (h/a) = 0.20 for different values of imperfection parameter (R). In this example by varying α and β from 0 to higher values the full range of dynamic instability is studied. For all these cases, the excitation frequency parameter Ω = 100wa√(pc/E11 obtained are presented in Figures 4-12. The figures (4-12) show the locations of the tips (i.e., β = 0.0) of the whole dynamic stability regions, which also shows the difference between the stability and instability regions It is observed from the results shown in the figures (4-12) that for higher values of α and β the uniform trend is disturbed in some cases.
In this paper the dynamic instability characteristics of imperfect laminated sandwich plate subjected to in-plane partial edge loading is carried out by an efficient finite element plate model. The present finite element model has all the necessary features for an accurate modeling of the present problem while it is computationally as elegant as any single layer plate. As there is no investigation on dynamic instability of imperfect laminated sandwich plate using such a refined plate model, a number of problems are solved including different plate geometry, boundary conditions, stacking sequences, thickness ratio and other aspects. In general it may be observed that the dynamic instability region is more expanded in case of 100% (i.e. full) loaded edge length. Also with the increase in the imperfection parameter (R) the dynamic instability region is contracted and the effect of different loaded edge length is gradually reduced. In this process many results are generated, which should be helpful for future research.
 V. V. Bolotin. The Dynamic stability of Elastic systems. Holden-Day, San Francisco, 1964. [ Links ]
 A. Chakrabarti and A. H. Sheikh. Vibration of composites and sandwich laminates subjected to in-plane partial edge loading. Comp. Sc. Tech., 67:10471057, 2007. [ Links ]
 A. Chakrabarti and A. H. Sheikh. Dynamic instability of composite and sandwich laminates with interfacial slips. International Journal of Structural Stability and Dynamics, 10(2):205224, 2010. [ Links ]
 A. Chattopadhyay and A. G. Radu. Dynamic instability of composite laminates using a higher order theory. Computer and Structure, 77:453460, 2000. [ Links ]
 L. W. Chen and J. Y. Yang. Dynamic stability of laminated composite plates by the finite element method. Computer and Structure, 36:845851, 1990. [ Links ]
 R. B. Corr and A. Jennings. A simultaneous iteration algorithm for symmetric eigenvalue problems. International Journal Numerical Methods Engineering, 10:647663, 1976. [ Links ]
 P. J. Deolasi and P. K. Datta. Parametric instability of rectangular plates subjected to localized edge compressing (compression or tension). Computer and Structure, 54:7382, 1995. [ Links ]
 J. Fazilati and H. R. Ovesy. Dynamic instability analysis of composite laminated thin-walled structures using two versions of FSM. Composite Structures, 92(9):20602065, 2010. [ Links ]
 J. M. Hutt and A. E. Salam. Dynamic stability of plates by finite element method. Journal of Engineering Mechanics ASCE, 3:879899, 1971. [ Links ]
 K. S. Jagdish. The dynamic stability of degenerate systems under parametric excitation. Ingenieur-Archive, 43:240 246, 1974. [ Links ]
 Y. W. Kwon. Finite element analysis of dynamic instability of layered composite plates using a high-order bending theory. Computer and Structure, 38:5762, 1991. [ Links ]
 S. Y. Lee. Finite element dynamic stability analysis of laminated composite skew plates containing cutouts based on HSDT. Composite Science & Technology, 70(8):12491257, 2010. [ Links ]
 S. V. Patel, P. K. Datta, and A. H. Sheikh. Parametric study on the dynamic instability behavior of laminated composite stiffened plate. Journal of Engineering Mechanics ASCE, 135(11):13311341, 2009. [ Links ]
 R. S. Srinivasan and P. Chellapandi. Dynamic stability of rectangular laminated composite plates. Computer and Structure, 24:233238, 1986. [ Links ]
 A. K. L. Srivastava, P. K. Datta, and A. H. Sheikh. Dynamic instability of stiffened plates subjected to non-uniform harmonic edge loading. Journal of Sound and Vibration, 262:11711189, 2003. [ Links ]
 N. Yamaki and K. Nagai. Dynamic stability of rectangular plates under periodic compressive forces. report of the institute of high speed mechanics. Technical report, Tohoku University, Japan, 1975. [ Links ]
Received 22 Jun 2010;
In revised form 22 Oct 2010