On a finite dynamic element method for free vibration analysis of structures

This paper explores the concept of finite dynamic elements involving higher order dynamic correction terms in the associated stiffness and mass matrices. Such matrices are then developed for a rectangular prestressed membrane element. Next, efficient analysis techniques for the eigenproblem solution of the resulting quadratic matrix equations are described in detail. These are followed by suitable numerical examples which indicate that employment of such dynamic elements in conjunction with an efficient quadratic matric solution technique will result in a most significant economy in the free vibration analysis of structures.     

Rotation free isogeometric thin shell analysis using PHT-splines

This paper presents a novel approach for isogeometric analysis of thin shells using polynomial splines over hierarchical T-meshes (PHT-splines). The method exploits the flexibility of T-meshes for local refinement. The main advantage of the PHT-splines in the context of thin shell theory is that it achieves C¹ continuity, so the Kirchhoff–Love theory can be used in pristine form. No rotational degrees of freedom are needed. Numerical results show the excellent performance of the present method.     

On the use of deep,shallow or flat shell finite elements for the analysis of thin shell structures

This paper is confined to the study of thin shells. The aim is to summarize the different theories used and to examine the assumptions upon which each of them is based. The intention is to show when it is more suitable to use a particular approximation and to indicate the errors it introduces. Beginning with the general deep shell theory, some simplifications are introduced to obtain the shallow shell theories. The special implications of this theory for the finite element method are also examined. Finally the particular case of flat elements is discussed.     

Application of boundary-domain element method to the free vibration problem of plate structures

The boundary-domain element method is applied to the free vibration problem of thin-walled plate structures. The static fundamental solutions are used for the derivation of the integral equations for both in-plane and out-of-plane motions. All the integral equations to be implemented are regularized up to an integrable order and then discretized by means of the boundary-domain element method. The entire system of equations for the plate structures composed of thin elastic plates is obtained by assembling the equations for each plate component satisfying the equilibrium and compatibility conditions on the connected edge as well as the boundary conditions. The algebraic eigenvalue equation is derived from this system of equations and is able to be solved by using the standard solver to obtain eigenfrequencies and eigenmodes. Numerical analysis is carried out for a few example problems and the computational aspects are discussed.     

薄壁圆柱壳的高阶模态振动特性研究

采用解析法研究了不同边界条件下薄壁圆柱壳的高阶模态振动特性.首先基于Love壳体理论,在简支-简支、固支-固支和固支-自由三种边界条件下,通过伽辽金法建立了动力学模型,对模态特性进行求解,得到了高阶固有频率和三维模态振型,并通过文献和有限元法进行了比较.算例结果表明,两端简支边界条件下采用解析法得到的固有频率误差值不超过2%,当周向波数较小时固有频率先减小后增加,在高阶时的固有频率逐渐升高,当轴向半波数增加时固有频率明显增大,通过解析法、文献和有限元法得到的三维模态振型相吻合.     

An efficient facet shell element for corotational nonlinear analysis of thin and moderately thick laminated composite structures

In the present work, an efficient facet shell element for the geometrically nonlinear analysis of laminated composite structures using the corotational approach is developed. The facet element is developed by combining the discrete Kirchhoff-Mindlin triangular bending element (DKMT), and the optimal membrane triangular element (OPT). The membrane-bending coupling effect of composite laminates is incorporated in the formulation, and inconsistent stress stiffness matrix is formulated. Using corotational formulation and the proposed facet element, some example laminated composite structures with geometric nonlinearity are analyzed, and the results are compared with those found using other facet elements.     

Elastic-plastic analysis of shell structures

The elastic-plastic analysis of shell structures is generally restricted to axially symmetric problems. To eliminate such limitation, the authors implemented a family of degenerated tridimensional finite elements, using the theory of conventional plasticity. In this manner, thick and thin shells of any geometry can be analyzed. The numerical results are compared with those obtained using other formulations, and also with experimental results.     

A moving Kriging interpolation-based meshfree method for free vibration analysis of Kirchhoff plates

The present work aims to make a further development of a novel meshfree method for free vibration analysis of classical Kirchhoff’s plates. The deflection of plates is approximated by the moving Kriging interpolation method which possesses the Kronecker’s delta property. This thus makes the proposed method efficient and straightforward in imposing the essential boundary conditions, and no special treatment techniques are required. A standard weak form is adapted to discrete the governing partial differential equations of plates. Numerical examples with different geometric shapes are considered to demonstrate the applicability and the accuracy of the proposed method.     

Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells

A meshfree approach––the harmonic reproducing kernel particle method is proposed for the free vibration analysis of rotating cylindrical shells. The reproducing kernel particle estimation is employed in hybridized form with harmonic functions, to approximate the two-dimensional displacement field. This is the first instance in which a meshless technique has been adopted for rotating shell dynamics. This technique provides ease of enforcing various types of boundary conditions and concurrently is able to capture the traveling modes. The effects of centrifugal and Coriolis forces as well as the initial hoop tension due to rotation are all taken into account in the present formulation. This study examines in detail the effects of different boundary conditions on the frequency characteristics of rotating shells. The present results, wherever possible, are verified by comparison against results available in the open literature. In general, close agreement between the authors' results and those of others has been found. Further, results presented here in selective parametric studies may be used as benchmarks for future related works.     

A non-parametric free-form optimization method for shell structures

This paper presents a numerical shape optimization method for the optimum free-form design of shell structures. It is assumed that the shell is varied in the out-of-plane direction to the surface to determine the optimal free-form. A compliance minimization problem subject to a volume constraint is treated here as an example of free-form design problem of shell structures. This problem is formulated as a distributed-parameter, or non-parametric, shape optimization problem. The shape gradient function and the optimality conditions are theoretically derived using the material derivative formulae, the Lagrange multiplier method and the adjoint variable method. The negative shape gradient function is applied to the shell surface as a fictitious distributed traction force to vary the shell. Mathematically, this method is a gradient method with a Laplacian smoother in the Hilbert space. Therefore, this shape variation makes it possible both to reduce the objective functional and to maintain the mesh regularity simultaneously. With this method, the optimal smooth curvature distribution of a shell structure can be determined without shape parameterization. The calculated results show the effectiveness of the proposed method for the optimum free-form design of shell structures.     

Nonlinear free vibration of fixed off-shore framed structures

The dynamical behavior of fixed off-shore framed structures is studied using the Wittrick-Williams algorithm to solve the nonlinear eigenvalue problem. The effects of shear deformation and rotary inertia as well as axial static loading are considered in this study of nonlinear free vibration.

The members are assumed to be rigidly connected and the added water mass is assumed equal to the mass of the water displaced. The structural modeling is based on a two-dimensional representation of the three-dimensional tower assuming a constant dimension equal to the base length perpendicular to the plane. The distributed masses of the members in the plane of the frame are computed by summing up the structural mass, the mass of the water contained in the tube, and the mass of the water displaced. The member masses in the plane perpendicular to the frame are assumed to be lumped at the horizontal cross-brace levels.

The results of the study indicate that while the first two frequencies obtained from the nonlinear and linear eigenvalue solutions agree closely, the effect of nonlinear eigenvalue solution is significant for the higher frequencies. The results also highlight the significant effects of the axial static force in the dynamic tangent stiffness matrix in the free vibration study of the off-shore structure. Fields for further research include (i) soil-structure interaction studies for gravity off-shore structures, buried pipelines, and (ii) nuclear power plant structures.     

Buckling and vibration analysis of laminated composite plate/shell structures via a smoothed quadrilateral flat shell element with in-plane rotations

This paper presents buckling and free vibration analysis of composite plate/shell structures of various shapes, modulus ratios, span-to-thickness ratios, boundary conditions and lay-up sequences via a novel smoothed quadrilateral flat element. The element is developed by incorporating a strain smoothing technique into a flat shell approach. As a result, the evaluation of membrane, bending and geometric stiffness matrices are based on integration along the boundary of smoothing elements, which leads to accurate numerical solutions even with badly-shaped elements. Numerical examples and comparison with other existing solutions show that the present element is efficient, accurate and free of locking.     

Application of Chebyshev–Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures

Engineering with Computers - This research develops a nonlocal couple stress theory to investigate static stability and free vibration characteristics of functionally graded (FG) nanobeams. The...     

Nonlinear free vibration analysis of composite conical shell panel with cluster of delamination in hygrothermal environment

Engineering with Computers - The nonlinear eigenvalue responses of conical composite shell structure with cluster of multiple delaminations are investigated numerically using the displacement-type...     

Finite element analysis of shell structures

Summary A survey of effective finite element formulations for the analysis of shell structures is presented. First, the basic requirements for shell elements are discussed, in which it is emphasized that generality and reliability are most important items. A general displacement-based formulation is then briefly reviewed. This formulation is not effective, but it is used as a starting point for developing a general and effective approach using the mixed interpolation of the tensorial components. The formulation of various MITC elements (that is, elements based on Mixed Interpolation of Tensorial Components) are presented. Theoretical results (applicable to plate analysis) and various numerical results of analyses of plates and shells are summarized. These illustrate some current capabilities and the potential for further finite element developments.     

Large displacement extension of consistent shell element for static and dynamic analysis

The superior performance of the consistent shell element in the small deflection range has encouraged the authors to extend the formulation to large displacement static and dynamic analyses. The nonlinear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear stiffness matrix and the unbalanced load vector for the consistent shell element is presented in this study. Meanwhile, a simplified method for coding the nonlinear formulation is provided by relating the components for the nonlinear B-matrices to those of the linear B-matrix. The consistent mass matrix for the shell element is also derived and then incorporated with the stiffness matrix to perform large displacement dynamic and free vibration analyses of shell structures. Newmark's method is used for time integration and the Newton-Raphson method is employed for iterating within each increment until equilibrium is achieved. Numerical testing of the nonlinear model through static and dynamic analyses of different plate and shell problems indicates excellent performance of the consistent shell element in the nonlinear range.     

A method for shell contact analysis

A relatively general and computationally efficient method of shell contact analysis using the discrete Fourier transform is developed for linear and certain types of nonlinear problems. The method predicts the contact boundary and the interfacial pressure distribution. It is illustrated by calculating the road contact pressure predicted by a finite element toroidal shell model of a pneumatic tire.     

Modeling techniques for adina analysis of stiffened shell structures

The ADINA beam element is inadequate for transient analysis of eccentrically stiffened shell structures, particularly when the lateral stability of the stiffener is of concern. As an alternative to modeling stiffened shells with a large number of continuum or transition elements, a stiffener modeling technique based on the ADINA multi-point constraint option is presented. This technique leads to significant reductions in the number of elements and solution degrees of freedom needed for accurate stiffener modeling, yet allows effects of out-of-plane web distortion, longitudinal warping and torsion to be included in the analysis. Stiffeners with various cross section geometries and boundary conditions have been modeled and predicted response correlates well with experimental data. The approach is of practical significance for large stiffened shell problems, especially for nonlinear analysis.     

A geometrically nonlinear shell finite element for tire vibration analysis

An axisymmetric finite element is developed which includes such features as orthotropic material properties, doubly curved geometry, and both the first and second order nonlinear stiffness terms. This element can be used to predict the equilibrium state of an axisymmetric shell structure with geometrically nonlinear large displacements. Small amplitude vibration analysis can then be performed based on this equilibrium state. The nonlinear path is predicted by using the self-correcting incremental procedure and any point on the path can be checked by using the Newton-Raphson iterative scheme. The present formulation and solution procedure are evaluated by analyzing a series of examples with results compared with alternative known solutions. Examples include: free vibration of an isotropic cylindrical shell, a conical frustum, and an orthotropic cylindrical shell; buckling of a cylindrical shell; large deflection of a clamped disk, a spherical cap, and a steel belted radial tire. The final example is a free vibration analysis of the inflated tire and the natural frequencies obtained compared well with published experimental data.     

Superelastic shell structures modelling: Part I: Element formulation

A C⁰ three-node shell finite element belonging to the assumed shear strain elements family is extended to account for large strains when a rotating frame formulation is adopted to describe the material behaviour. Within an incremental method associated with the Newton iterative scheme, a strain measure is defined and interpolated in an intermediate configuration assuming a linear interpolation of the incremental geometric transformation. This strain measure allows the definition, in a rotated configuration, of a constitutive incremental strain obtained from a material cumulated tensorial strain. The proposed approach is validated herein considering several elastic finite strains examples.