# مدل های شبه تحلیلی سه بعدی برای واکنش های استاتیک و تجزیه و تحلیل حساسیت کامپوزیت ورق های چند لایه با عیوب سطحی سخت

کد مقاله | سال انتشار | مقاله انگلیسی | ترجمه فارسی | تعداد کلمات |
---|---|---|---|---|

26584 | 2012 | 16 صفحه PDF | سفارش دهید | 10855 کلمه |

**Publisher :** Elsevier - Science Direct (الزویر - ساینس دایرکت)

**Journal :** Composite Structures, Volume 94, Issue 6, May 2012, Pages 1943–1958

#### چکیده انگلیسی

For the stiffened composite laminated plates with interfacial imperfections, the problem of static response and sensitivity analysis was investigated in Hamilton system. Firstly, the meshfree formulation of Hamilton canonical equation for the composite laminated plate with interfacial imperfections was deduced by the linear spring-layer and the state-vector equation theory. And then, based on the equation of plates and stiffeners, governing equation of the composite stiffened laminated plate was assembled by using the spring-layer model again to ensure the compatibility of stresses and the discontinuity of displacements at the interface between plate and stiffeners. At last, a three-dimensional hybrid governing equation was developed for the static response analysis and sensitivity analysis. To demonstrate the excellent predictive capability of this three-dimensional semi-analytical model, several numerical examples were carried out to assess the static responses and static sensitivity coefficients. Good agreement had been achieved between the predictions and the results of finite element code MSC.Nastran. Extensive numerical results were presented showing the effects of the interfacial stiffness on the response quantities and their sensitivity coefficients. Furthermore, distributions along the thickness direction were also presented for the static responses and the static sensitivity coefficients with respect to the material properties and the shape parameters.

#### مقدمه انگلیسی

The stiffened plates/shells structural system is composed of plate or shell elements, reinforced by a series of stiffeners (or ribs, beams, stringers, etc.) that are attached in longitudinal and sometimes in orthogonal directions. Stiffening of the plate/shell is used to increase its load carrying capacity and prevent buckling especially in the case of in-plane loading. The primary advantage of stiffened constructions lies in the structural efficiency of the system, since great savings or conservation of weight can be attained with no sacrifice in strength or serviceability. Thus, there are wide applications of stiffened plate/shell components in a variety of engineering structures. In aerospace industry, they are used in the construction of aircraft fuselage and wings. Meanwhile, the stiffened plates/shells are also utilized in the construction of bridges, buildings, storage tanks, off-shore structures, and recently in petrochemical processing facilities. Research into stiffened structures has been a subject of interest for many years. The vibration and stability of stiffened structures is of great interest, since it generally controls the optimum design of the structures in which they are deployed. Also, only a comprehensive and accurate stress analysis can lead a designer to an appropriate selection of the plate thickness distribution and the stiffener dimensions. As a result, a large number of studies on stress analysis, buckling, and vibration of stiffened panels are available in the literature. Based on energy principles, Kukreti and Cheraghi [1] have presented a procedure to analyze a stiffened plate system supported by a network of steel girders. The beams are assumed to be rigidly connected to the plate. A semi-analytical method for the analysis of bare plates has been extended to the static analysis of stiffened plates by Mukhopadhyay [2]. Both concentric and eccentric stiffeners have been considered. Siddiqi and Kukreti [3] have developed a differential quadrature solution for the flexural analysis of eccentrically stiffened plates subjected to transverse uniform loads. The axial stiffness of the plate and the interaction between the beams and the plate due to the eccentricity are taken into consideration during the analysis of the in-plane forces in the plate. Deb and Booton [4] and Palani et al. [5] have presented respectively linear finite element models based on Mindlin’s shear distortion theory and two isoparametric finite element models (the eight-noded QS8S1 and the nine-noded QL9S1) for static and vibration analysis of plates/shells with stiffeners. The boundary element method (BEM) is also developed to model the static and dynamic response of reinforced plate structures [6], [7], [8], [9] and [10]. Ng et al. [11] have studied the suitability of a new method combining the advantages of both the BEM and the finite element method (FEM) to analyze the more complicated problems of slabs and slab-on-girder bridges. This new method is first applied to investigate the conventional plate bending problems. Due to the complexity of the problem and the many parameters involved, extensive research efforts were devoted over the past years by many researchers to investigate a variety of aspects. Reinforcing the plate/shell with the stiffener elements complicates the analysis, and several assumptions must be made in order to facilitate a solution if the stiffeners are not identical or unequally spaced. And the complication would further increase when analyzing composite laminated plate/shell structures. Currently, three main approaches can be employed for analysis of laminated structures: equivalent single layer theory (classical laminate theory and shear deformation laminated plate theories) [12], [13], [14], [15] and [16], three-dimensional elasticity theory (traditional 3-D elasticity formulations and layerwise theory) [17] and [18] and multiple model methods [19] and [20]. Since the above theories are established on some hypothesis, the most important of which is the neglect of transverse shear deformations and rotatory inertia, only partial fundamental equations can be satisfied and some of the elastic constants cannot be taken into account. Therefore, the errors will increase as the thickness of plates increases and the stress at interface cannot be exactly calculated. To overcome this difficulty, some refined formulations have been established to take into account, e.g., transverse shear deformations and rotatory inertia [21], [22] and [23]. In recent years, the state-vector equation in Hamilton system, which is employed in the analysis of control systems of current significance, has attracted the attention of a number of investigators who are interested in the problems of laminated structures [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37] and [38]. In the state-vector equation, two types of variables (i.e., the transverse stresses and displacements) are synchronously considered in the control equation. And the thick plates/shells or the laminated plates/shells problems can be treated without any assumptions regarding displacements and stresses. Due to the transfer matrix technique being employed, the solution provides exact continuous transverse stresses and displacement field across the thickness of laminated structure. Another significant difference from the classical layer-wise methods is that the scale of the final governing equation system is independent of the thickness and the number of layers of a structure. As a result, Qing et al. [39] have developed a novel mathematical model for free vibration analysis of stiffened laminated plates based on the semi-analytical solution of the state space method. The method accounts for the compatibility of displacements and stresses on the interface between the plate and stiffeners, the transverse shear deformation, and naturally the rotary inertia of the plate and stiffeners. Meanwhile, there is no restriction on the thickness of plate and the height of stiffeners. Since the stiffeners are assumed to be rigidly connected to the plate, the bonding imperfection which may be introduced into the interface between the plate and stiffeners cannot be taken into account in this mathematical model. However, various interlaminar debonding like microcracks, inhomogeneities and cavities may be introduced into the bond in the process of manufacture or service. During the service lifetime, these tiny flaws can get significant. To avoid the local failure of bond or the whole collapse of structure, therefore, the effect of imperfect interfaces on the structural behavior should be accurately evaluated. In recently years, Chen et al. [40], [41], [42] and [43] have used the analytical methods and numerical methods to research the problem of interfacial imperfection for composite laminated plates in Hamilton system. In present work, the static response analysis and static sensitivity analysis of composite stiffened laminated plates with interfacial imperfections are investigated by the spring-layer model, meshfree method and the state-vector equation theory. A hybrid governing set equation of the composite stiffened laminated plates with bonding imperfection is deduced for the response and sensitivity quantities. One of the main advantages of the hybrid governing set equation is that the discontinuity of displacements and the compatibility of stresses on the interface between the plate and stiffeners are accounted. The present three-dimensional semi-analytical model with no initial assumptions regarding displacement and stress accounts for the transverse shear deformation and rotary in the governing equations. Furthermore, by using this hybrid governing equation in the response analysis and sensitivity analysis, the convoluted algorithm can be avoided in sensitivity analysis, and the response quantities and the sensitivity coefficients can be obtained simultaneously.

#### نتیجه گیری انگلیسی

The spring-layer model and the state-vector equation theory have been applied to the problems of the static response analysis and static sensitivity analysis for the composite stiffened laminated plates with interfacial imperfections based on the meshfree method. A hybrid governing equation of the composite stiffened laminated plates with bonding imperfection was developed for the response and sensitivity quantities. The semi-analytical method (SA) and the finite different method (FD) were presented for the sensitivity analysis. One of the main advantages of the present three-dimensional semi-analytical model is that the discontinuity of displacements and the compatibility of stresses on the interface between the plate and stiffeners are accounted. The present model with no initial assumption regarding displacement and stress accounts for the transverse shear deformation and rotary in the governing equation of structure. Furthermore, by using this hybrid governing equation in the response analysis and sensitivity analysis, the convoluted algorithm can be avoided in sensitivity analysis, and the response quantities and the sensitivity coefficients can be obtained simultaneously. Numerical results for an eccentrically stiffened plate with double stiffeners were obtained and compared with the solutions of finite element code MSC.Nastran. Good agreement has been achieved between the predictions and the results of finite element code MSC.Nastran. Extensive numerical results were also used to show the effects of interfacial imperfection stiffness on the response quantities and sensitivity coefficients along the thickness direction. Based on results obtained under various cases, observations have been summarized in respective section. These observations lead to certain broad conclusions which are stated as: (1) The stiffness of entire stiffened structure decreases with the increment of the interfacial imperfections stiffness R. And the influence of the stiffness value R on the stresses is considerably larger than that on the displacements. (2) The influence of the stiffness value R on the static response quantities and sensitivity coefficients of the stresses σxz, σyz and σzz is obvious in the surface of the laminated plate with stiffeners and the interface between the laminated plate and stiffeners. (3) The influence of the stiffness value R on the static response quantities and sensitivity coefficients of the displacements u and v is also obvious in the surface of the laminated plate with stiffeners and the interface between plate and stiffeners. (4) The sensitivity values of displacements and stresses with respect to shape parameters, especially the thickness and the width of stiffeners, are considerably larger than those with respect to the material properties. In general, among all the shape parameters and material properties, the sensitivity values with respect to the thickness t (the total thickness of the stiffeners) are considerably largest.