Effect of interfacial imperfection on bending behavior of composites and sandwich laminates by an efficient C0 FE model

The bending behavior of composites and sandwich plates having imperfections at the layer interfaces is investigated by a refined higher order shear deformation plate theory (RHSDT) and a Least Square Error (LSE) method. In this theory, the in-plane displacement field is obtained by superposing a globally varying cubic displacement field on a zig-zag linearly varying displacement field. This plate theory represents parabolic through thickness variation of transverse shear stresses which satisfy the inter-laminar continuity condition at the layer interfaces and zero transverse shear stress condition at the top and bottom of the plate. In this plate model, the interfacial imperfection is represented by a liner spring-layer model. Finite element method is adopted and an efficient C⁰ continuous 2D finite element (FE) model is developed based on the above mentioned plate theory for the static analysis of composites and sandwich laminates having imperfections at the layer interfaces. In this model, the first derivatives of transverse displacement have been treated as independent variables to circumvent the problem of C¹ continuity associated with the above plate theory (RHSDT). The LSE method is applied to the 3D equilibrium equations of the plate problem at the post-processing stage, after in-plane stresses are calculated by using the above FE model based on RHSDT. The proposed model is implemented to analyze the laminated composites and sandwich plates having interfacial imperfection. Many new results are also presented which should be useful for the future research.     

Frame mode method for thin-walled structures

The Rayleigh-Ritz method is extended to thin-walled structures on continuous supports by means of the combinations of frame modes transversely and continuous beam modes longitudinally. Effects of warping, distortion and shear-lag are considered simultaneously. The thin-walled structure problem is reduced to simpler plane frame and continuous beam problems. Numerical results are compared to the finite element and finite strip methods. The present method is advantageous over both the finite element and finite strip methods in the reduced number of generalised coordinates and its ability to use the existing frame programmes to analyse thin-walled structures. No new elements are required to be generated.     

A general finite strip for the static and dynamic analyses of folded plates

In the conventional semi-analytical finite strip analysis of folded plates, the boundary conditions and the intermediate support conditions must be satisfied a priori. The admissible functions used as the longitudinal part of the displacement functions are sometimes difficult to find, and they are valid for specific conditions only. In this paper, a general finite strip is developed for the static and vibration analyses of folded plate structures. The geometric constraints of the folded plates, such as the conditions at the end and intermediate supports, are modelled by very stiff translational and rotational springs as appropriate. The complete Fourier series including the constant term are chosen as the longitudinal approximating functions for each of the displacements. As these displacement functions are more general in nature and independent of one another, they are capable of giving more accurate solutions. The potential problem of ill-conditioned matrices is investigated and the appropriate choice of the very stiff springs is also suggested. The formulation is done in such a way to obtain a unified approach, taking full advantage of the power of modern computers. A few numerical examples are presented for comparison with numerical results from published solutions or solutions obtained from the finite element method. The results show that this kind of strips is versatile, efficient and accurate for the static and vibration analyses of folded plates.     

Inelastic local buckling of curved plates with or without thickness-tapered sections using finite strip method

This paper addresses the inelastic local buckling of the curved plates using finite strip method in which buckling modes and displacements of the curved plate are calculated using sinusoidal shape functions in the longitudinal direction and polynomial functions in the transverse direction. A virtual work formulation is employed to establish the stiffness and stability matrices of the curved plate whilst the governing equations are then solved using a matrix eigenvalue problem. The accuracy and efficiency of the proposed finite strip model is verified with finite element model using ABAQUS as well as the results reported elsewhere while a good agreement is achieved. In order to illustrate the proposed model, a comprehensive parametric study is performed on the steel and aluminium curved plates in which the effects of curvature, the length of the curved plate as well as circumferential boundary conditions on the critical buckling stress are investigated. The developed finite strip method is also used to determine the buckling loads of the curved plates with thickness-tapered sections as well as critical stresses of the aluminium cylindrical sectors that are subjected to uniform longitudinal stresses.     

The structural behavior of homogeneous and hybrid stub columns under dynamic loading conditions

This paper presents a new finite strip method for the analysis of deep beams and shear walls. The essence of the method lies in the adoption of displacement functions possessing the right amount of continuity at the ends as well as at locations of abrupt changes of thickness. The concept of periodic extension in Fourier series is utilized to improve the accuracy of the stresses at the strip ends. The equilibrium conditions at locations of abrupt changes of thickness are taken into account by the incorporation of piecewise linear correction functions. As these displacement functions are built up from harmonic functions with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as appropriate order of continuity. Numerical results also show that the method is versatile, efficient and accurate.     

Analysis of deep beams and shear walls by finite strip method with C continuous displacement functions

This paper presents a new finite strip method for the analysis of deep beams and shear walls. The essence of the method lies in the adoption of displacement functions possessing the right amount of continuity at the ends as well as at locations of abrupt changes of thickness. The concept of periodic extension in Fourier series is utilized to improve the accuracy of the stresses at the strip ends. The equilibrium conditions at locations of abrupt changes of thickness are taken into account by the incorporation of piecewise linear correction functions. As these displacement functions are built up from harmonic functions with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as appropriate order of continuity. Numerical results also show that the method is versatile, efficient and accurate.     

Buckling optimization of variable thickness prismatic folded plates

This paper deals with the structural shape optimization of prismatic folded plates under buckling load consideration. Buckling loads are determined using linear, quadratic and cubic, variable thickness, C(0) continuity, Mindlin-Reissner finite strips. The whole structural optimization process is carried out by integrating finite strip analysis, cubic spline shape and thickness definition, semi analytical sensitivity analysis and mathematical programming algorithm. The objective is either the maximization of the critical buckling load or minimization of the cross-section of the prismatic folded plate with constraints on the volume and buckling loads. Several examples are included to illustrate various features of the optimization algorithm, including plates and stiffened panels.     

Static and dynamic analysis of thin plates in bending— A novel finite element model

The static and dynamic behavior of thin flat plates in bending have been studied by means of a recently developed¹ doubly curved triangular shell element. The element's formulation is based on a modified mixed variational principle, wherein the primal variable σ (vector of shell stress resultants) and (boundary displacement vector) are required to satisfy a priori: (1) the complete shallow shell equilibrium equations, and (2) interelement C¹ displacement continuity. Several well-selected plate structures have been analyzed and the numerical results obtained indicate that the new element scheme competes most favorably with recently developed as well as with well-established elements included in commercial general-purpose finite element codes.     

Dynamic instability of laminated sandwich plates using an efficient finite element model

For the first time, the dynamic instability of laminated sandwich plates subjected to in-plane edge loading is studied using an efficient finite element plate model, which is developed recently by the authors. The plate model is based on a refined higher order shear deformation plate theory. In this theory, the transverse shear stresses are continuous at the layer interfaces with stress free conditions at plate top and bottom surfaces. It is interesting to note that the plate model having all these refined features requires unknowns at the reference plane only. However, this theory requires C¹ continuity of transverse displacement at the element edges, 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.     

Buckling mode decomposition of single-branched open cross-section members via finite strip method: Derivation

This paper provides the first detailed presentation of the derivation for a newly proposed method which can be used for the decomposition of the stability buckling modes of a single-branched, open cross-section, thin-walled member into pure buckling modes. Thin-walled members are generally thought to have three pure buckling modes (or types): global, distortional, and local. However, in an analysis the member may have hundreds or even thousands of buckling modes, as general purpose models employing shell or plate elements in a finite element or finite strip model require large numbers of degrees of freedom, and result in large numbers of buckling modes. Decomposition of these numerous buckling modes into the three buckling types is typically done by visual inspection of the mode shapes, an arbitrary and inefficient process at best. Classification into the buckling types is important, not only for better understanding the behavior of thin-walled members, but also for design, as the different buckling types have different post-buckling and collapse responses. The recently developed generalized beam theory provides an alternative method from general purpose finite element and finite strip analyses that includes a means to focus on buckling modes which are consistent with the commonly understood buckling types. In this paper, the fundamental mechanical assumptions of the generalized beam theory are identified and then used to constrain a general purpose finite strip analysis to specific buckling types, in this case global and distortional buckling. The constrained finite strip model provides a means to perform both modal identification relevant to the buckling types, and model reduction as the number of degrees of freedom required in the problem can be reduced extensively. Application and examples of the derivation presented here are provided in a companion paper.     

Accurate dynamic response of laminated composites and sandwich shells using higher order zigzag theory

Forced vibration response of laminated composite and sandwich shell is studied by using a 2D FE (finite element) model based on higher order zigzag theory (HOZT). This is the first finite element implementation of the HOZT to solve the forced vibration problem of shells incorporating all three radii of curvatures including the effect of cross curvature in the formulation using Sanders' approximations. The proposed finite element model satisfies the inter-laminar shear stress continuity at each layer interface in addition to higher order theory features, hence most suitable to model sandwich shells along with composite shells. The C₀ finite element formulation has been done to overcome the problem of C₁ continuity associated with the HOZT. The present model can also analyze shells with cross curvature like hypar shells besides normal curvature shells like cylindrical, spherical shells etc. The numerical studies show that the present 2D FE model is more accurate than existing FE models based on first and higher order theories for predicting results close to those obtained by 3D elasticity solutions for laminated composite and sandwich shallow shells. Many new results are presented by varying different parameters which should be useful for future research.     

Vibration and buckling of thin-Walled structures by a new finite strip

This paper presents the application of a new finite strip to the analysis of folded-plate structures. The displacement function of a flat shell finite strip is made up of two parts, namely, the two in-plane displacement interpolations and the out-of-plane displacement interpolation. Each of the three displacement components is interpolated by a set of computed shape functions in the longitudinal direction and, as usual, one-dimensional shape functions in the transverse direction. Only standard beam shape functions are involved in the longitudinal computed shape functions. When compared with other finite strips, the present finite strip is relatively simple in dealing with boundary and internal support conditions. In addition, the method can be easily implemented by incorporating a standard finite strip program with a continuous-beam program. The computation of the stiffness matrix involves no numerical integration. To verify the accuracy and efficiency of the new finite strip, a few numerical experiments are conducted in which the present finite strip results are compared with those using other finite strips and/or finite elements for the vibration and buckling of folded-plate structures with varying complexity.     

Robust control of plate vibration via active constrained layer damping

In this paper, the theoretical modeling of a plate partially treated with active constrained layer damping (ACLD) treatments and its vibration control in an H_∞ approach is discussed. Vibration of the flat plate is controlled with patches of ACLD treatments, each consisting of a viscoelastic damping layer which is sandwiched between the piezo-electric constrained layer and the host plate. The piezo-electric constrained layer acts as an actuator to actively control the shear deformation of the viscoelastic damping layer according to the vibration response of the plate excited by external disturbances. In the first part of this paper, the Mindlin–Reissner plate theory is adopted to express the shear deformation characteristics of the viscoelastic damping layer, meanwhile GHM (Golla–Hughes–McTavish) model of viscoelastic damping material and FEM (finite element model) are incorporated to describe the dynamics of the plate partially treated with ACLD treatment. In the second part, particular emphasis is placed on the vibration control of the first four modes of the treated plate using H_∞ robust control method. For this purpose, an H_∞ robust controller is designed to accommodate uncertainties of the ACLD parameters, particularly those of the viscoelastic damping core which arise from the variation of the operation temperature and frequency. Disturbances and measurement noise are rejected in the closed loop by H_∞ robust controller. In the experimental validation, external disturbances of different types are employed to excite the treated plate. The results of the experimental clearly demonstrate that the proposed modeling method is correct and the ACLD treatments are very effective in fast damping out the structural vibration as compared to the conventional passive constrained layer damping (PCLD).     

On torsional and warping stiffness of thin-walled girders

The torsional and warping stiffness of the beam idealization of thin-walled girders is defined from equivalence of their natural frequencies for torsional vibration of sinusoidal modes. The membrane and plate strip theory are employed for the girder structural members. The stiffness parameters are determined analytically for a few girders of simple cross-section, while for those of complex cross-section a numerical procedure is formulated.     

Application of spline finite strip method in the analysis of space structures

The spline finite strip based on the thin plate theory has been demonstrated to be a versatile tool for the analysis of plates and shells. Applications of this method to the analysis of prismatic space structures have also been reported. This paper attempts to extend the method to the analysis of non-prismatic space structures. In the analysis, the plates that form the space structures are treated as flat shells. The Mindlin plate theory is adopted to model the bending action of the plate, and the in-plane action is modelled in the usual manner. The present formulation, which requires only C° continuity for the displacement interpolation functions, allows greater flexibility in the geometry of the structures. The accuracy and versatility of the method are also demonstrated by numerical examples.     

Thermal buckling analysis of rectangular composite plates with temperature-dependent properties based on a layerwise theory

Thermal buckling analysis of rectangular composite multilayered plates under uniform temperature rise is investigated using a layerwise plate theory. von Karman strain–displacement equations are employed to account for large deflections occurrence. It is already proven that the layerwise theory results are compatible with the three-dimensional theory of elasticity results. The accuracy of the present results is increased by substituting each layer by many virtual sub-layers. The final governing equations are not simplified or linearized. Material properties are assumed to vary with temperature. Hermitian finite element formulation is used to ensure a C¹ continuity for the lateral deflections. No semi-analytic solution is employed to reduce the problem to an eigenvalue one. Layerwise formulations are usually displacement-based. Therefore, force or moment boundary conditions (e.g. simply supported boundary condition), are approximately satisfied. A FEM algorithm is presented to exactly incorporate the boundary conditions. A proposed numerical scheme and a modified Budiansky instability criterion presented by the author are used to determine the buckling temperature in a computerized solution. Finally, results of the present techniques are compared with the results of the high-order theories presented by some well-known researchers and the influences of various geometric and mechanical properties parameters of the composite plate on the buckling temperature are studied.     

Comprehensive stability design of planar steel members and framing systems via inelastic buckling analysis

This paper presents a comprehensive approach for the design of planar structural steel members and framing systems using a direct computational buckling analysis configured with appropriate column, beam and beam-column inelastic stiffness reduction factors. The stiffness reduction factors are derived from the ANSI/AISC 360-16 Specification column, beam and beam-column strength provisions. The resulting procedure provides a rigorous check of all member in-plane and out-of-plane design resistances accounting for continuity effects across braced points as well as lateral and/or rotational restraint from other framing. The method allows for the consideration of any type and configuration of stability bracing. With this approach, no member effective length (K) or moment gradient and/or load height (C _b ) factors are required. The buckling analysis rigorously captures the stability behavior commonly approximated by these factors. A pre-buckling analysis is conducted using the AISC Direct Analysis Method (the DM) to account for second-order effects on the in-plane internal forces. The buckling analysis is combined with cross-section strength checks based on the AISC Specification resistance equations to fully capture all the member strength limit states. This approach provides a particularly powerful mechanism for the design of frames utilizing general stepped and/or tapered I-section members.     

基于曲率模态小波分析的单塔斜拉桥损伤识别

基于曲率模态小波分析原理及有限元法分析了含有损伤单元的单塔斜拉桥的振动特性;以Mexh小波为母小波,通过对损伤斜拉桥的曲率模态做连续小波变换,由小波系数模极大值位置识别斜拉桥损伤的位置,建立了一种基于曲率模态小波分析识别斜拉桥损伤的方法;采用该方法对单塔斜拉桥的损伤识别进行了计算分析。结果表明:该方法具有有效性,对于各类型桥的损伤诊断具有指导意义。     

Optimal design of composite laminates with and without cutout undergoing free vibration

In this article, the optimal designs of freely vibrating composite laminates with and without cutout undergoing are investigated. A simple higher order shear deformation theory (HSDT) with four unknown displacements (u ₀, v ₀, w_b, w_s ) is employed to obtain the vibration response. A C ¹ continuity shear flexible element based on HSDT using the Hermite cubic polynomial is used for the rectangular element. The optimisation exercise is performed using genetic algorithms to maximise the fundamental frequency of vibration. The aspect ratio of the laminate, fibre orientation, thickness of plies, modulus are treated as design variables. On the basis of the investigation, it is observed that the non-dimensional frequency of the laminate with cutout can be higher or lower than those of the laminate without cutout depending on the size of the cutout. The presence of cutout significantly affects the frequency.     

Free vibration analysis of circular thin plates with stepped thickness by the DSC element method

Novel formulation is presented by using the discrete singular convolution (DSC) for free vibration analysis of circular thin plates with uniform and stepped thickness. Different from the commonly used ones in literature, regularity conditions are not needed at the circular plate center point to avoid singularity. DSC circular and annular thin plate elements are established. For the DSC circular plate element with radius of R₁, the stiffness equation is first formulated in region [−R₁, R₁] with even number of nodes and then reduced to region [0, R₁] by using either symmetric or anti-symmetric conditions. The proposed DSC circular and annular plate elements are used for obtaining frequencies of uniform/stepped circular thin plates or annular thin plates with different boundary conditions. Comparison of the present DSC results to existing analytic and numerical solutions verifies the proposed formulations. The present research extends the DSC method to free vibration of circular thin plates with stepped thicknesses.