Nikola Vladimir

A new finite element formulation for vibration analysis of thick plates

ABSTRACT A new procedure for determining properties of thick plate finite elements, based on the modified Mindlin theory for moderately thick plate, is presented. Bending deflection is used as a potential function for the definition of total (bending and shear) deflection and angles of cross-section rotations. As a result of the introduced interdependence among displacements, the shear locking problem, present and solved in known finite element formulations, is avoided. Natural vibration analysis of rectangular plate, utilizing the proposed four-node quadrilateral finite element, shows higher accuracy than the sophisticated finite elements incorporated in some commercial software. In addition, the relation between thick and thin finite element properties is established, and compared with those in relevant literature.

Some aspects of structural modelling and restoring stiffness in hydroelastic analysis of large container ships

The increase in world trade has largely contributed to the expansion of sea traffic. As a result, the market demand is leading to ultra-large container ships (ULCS), with expected capacity up to 18,000 TEU (twenty-foot equivalent unit) and length about 400 m, without changes in the operational requirements (speed up to 27 knots). The particular structural design of the container ships leads to open midship sections, resulting in increased sensitivity to torsional and horizontal bending loads that is much more complex to model numerically. At the same time, due to their large dimensions, the structural natural frequencies of ULCS become significantly lower so that the global hydroelastic structural responses (springing and whipping) can become a critical issue in the ship design and should be properly modelled by the simulation tools since the present classification rules do not cover described operating stages completely. There are several research projects worldwide aiming at solving this problem, and one of them is the EU FP7 project TULCS (tools for ultra-large container ships) for development of the integrated design tools, based on numerical procedures, model tests and full-scale measurements. This paper is based on research activities and results of the project, with particular emphasis on the part that deals with global hydroelastic loading and response. Special attention is paid to beam structural model based on the advanced beam theory. It includes shear influence on bending and torsion, contribution of transverse bulkheads to hull stiffness and an appropriate modelling of relatively short engine room structure of ULCS. Along with that, a hydrodynamic model is presented in a condensed form. Further on, a fully consistent formulation of restoring stiffness, which plays an important role in the hydrostatic model, is described. Theoretical contributions are illustrated within the numerical example, which includes a complete hydroelastic analysis of an 11,400 TEU container ship. In this case, validation of the one-dimensional (1D) finite-element method (FEM) model is done by a correlation analysis with the vibration response of the fine three-dimensional (3D) FEM model. The procedure related to determination of engine room effective stiffness is checked by a 3D FEM analysis of a ship-like pontoon, which has been made according to the 7800 TEU container ship properties. The obtained results confirm that the sophisticated beam model is a very useful numerical tool for the designer and represents a reasonable choice for determining wave load effects on ULCS, in preliminary design stage.

Natural vibration analysis of stiffened panels with arbitrary edge constraints using the assumed mode method

Natural vibration analysis of stiffened panels represents an important issue in different kinds of engineering applications. In this article, a procedure for the vibration analysis of stiffened panels with arbitrary edge constraints is presented. It is based on the assumed mode method, where natural frequencies and modes are determined by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equations of motion. The Mindlin thick plate theory is applied for a plate, while the effect of stiffeners having the properties of Timoshenko beams is accounted for by adding their strain and kinetic energies to the corresponding plate energies. The accuracy of the proposed procedure is justified by several numerical examples which include the natural vibration analysis of stiffened panels with different framing sizes, their lengths and orientations, plate thicknesses and different combinations of boundary conditions. A comparison of results with those obtained by the finite element method is provided, and good agreement is achieved.

Approximate natural vibration analysis of rectangular plates with openings using assumed mode method

ABSTRACT Natural vibration analysis of plates with openings of different shape represents an important issue in naval architecture and ocean engineering applications. In this paper, a procedure for vibration analysis of plates with openings and arbitrary edge constraints is presented. It is based on the assumed mode method, where natural frequencies and modes are determined by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagranges equations of motion. The presented solution represents an extension of a procedure for natural vibration analysis of rectangular plates without openings, which has been recently presented in the literature. The effect of an opening is taken into account in an intuitive way, i.e. by subtracting its energy from the total plate energy without opening. Illustrative numerical examples include dynamic analysis of rectangular plates with rectangular, elliptic, circular as well as oval openings with various plate thicknesses and different combinations of boundary conditions. The results are compared with those obtained by the finite element method (FEM) as well as those available in the relevant literature, and very good agreement is achieved.

Analytical solution for free vibrations of a moderately thick rectangular plate

In the present thick plate vibration theory, governing equations of force-displacement relations and equilibrium of forces are reduced to the system of three partial differential equations of motion with total deflection, which consists of bending and shear contribution, and angles of rotation as the basic unknown functions. The system is starting one for the application of any analytical or numerical method. Most of the analytical methods deal with those three equations, some of them with two (total and bending deflection), and recently a solution based on one equation related to total deflection has been proposed. In this paper, a system of three equations is reduced to one equation with bending deflection acting as a potential function. Method of separation of variables is applied and analytical solution of differential equation is obtained in closed form. Any combination of boundary conditions can be considered. However, the exact solution of boundary value problem is achieved for a plate with two opposite simply supported edges, while for mixed boundary conditions, an approximate solution is derived. Numerical results of illustrative examples are compared with those known in the literature, and very good agreement is achieved.

Natural vibration analysis of vertical rectangular plates and stiffened panels in contact with fluid on one side

This article presents a simple and efficient procedure for the natural vibration analysis of rectangular plates and stiffened panels in contact with fluid on one side. The assumed mode method is applied, where the natural frequencies and mode shapes are obtained by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equation of motion. The Mindlin thick plate theory is applied for a plate, and in the case of stiffened panels, the effect of framing is taken into account by adding its strain and kinetic energies to the corresponding plate energies. Potential flow theory assumptions are adopted for the fluid, and free surface waves are ignored. The fluid velocity potential is derived from the boundary conditions for the fluid and structure and is utilized for the calculation of added mass using the assumed modes. The applicability and accuracy of the developed procedure are illustrated with several numerical examples using a developed in-house code. A comparison of the results with those obtained by general purpose finite element analysis software is provided, where very good agreement is achieved.

Global hydroelastic analysis of ultra large container ships by improved beam structural model

ABSTRACT Some results on the hydroelasticity of ultra large container ships related to the beam structural model and restoring stiffness achieved within EU FP7 Project TULCS are summarized. An advanced thin-walled girder theory based on the modified Timoshenko beam theory for flexural vibrations with analogical extension to the torsional problem, is used for formulation of the beam finite element for analysis of coupled horizontal and torsional ship hull vibrations. Special attention is paid to the contribution of transverse bulkheads to the open hull stiffness, as well as to the reduced stiffness of the relatively short engine room structure. In addition two definitions of the restoring stiffness are considered: consistent one, which includes hydrostatic and gravity properties, and unified one with geometric stiffness as structural contribution via calm water stress field. Both formulations are worked out by employing the finite element concept. Complete hydroelastic response of a ULCS is performed by coupling 1D structural model and 3D hydrodynamic model as well as for 3D structural and 3D hydrodynamic model. Also, fatigue of structural elements exposed to high stress concentration is considered.

Numerical procedure for the vibration analysis of arbitrarily constrained stiffened panels with openings

Abstract A simple and efficient vibration analysis procedure for stiffened panels with openings and arbitrary boundary conditions based on the assumed mode method is presented. Natural frequencies and modes are determined by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagranges equations of motion, where Mindlin theory is applied for plate and Timoshenko beam theory for stiffeners. The effect of stiffeners on vibration response is taken into account by adding their strain and kinetic energies to the corresponding plate energies whereas the strain and kinetic energies of openings are subtracted from the plate energies. Different stiffened panels with various opening shapes and dispositions for several combinations of boundary conditions are analyzed and the results show good agreement with those obtained by the finite element analysis. Hence, the proposed procedure is especially appropriate for use in the preliminary design stage of stiffened panels with openings.

The contribution of the engine room structure to the hull stiffness of large container ships

Very large container ships are rather flexible due to their large deck openings. Hydroelastic stress analysis is therefore required as a base for reliable structural design. In the early design stage, the coupling of the beam model with a 3D hydrodynamic model is rational and preferable. The calculation is performed utilizing the modal superposition method, so natural hull modes have to be determined in an appropriate way. Consequently, the advanced thin-walled girder theory, which takes the influence of shear on both bending and torsion into account, is applied to calculate the hull flexural and torsional stiffness properties. A characteristic of very large container ships is the quite short engine room, whose closed structure behaves as an open hold structure with a shear centre outside the cross-section, very close to that of the open section. As a result, torsionally induced horizontal bending is negligible, while the distortion of the cross-sections appears as a new problem. The task is solved by an energy balance approach that enables the use of effective stiffness. Hence, the effect of interior decks is taken into account by increasing the torsional stiffness of the open cross-section within the engine room domain. The procedure is checked by the 3D FEM analysis of a ship-like pontoon. Such a modified beam model of the engine room structure can be included in the general beam model of a ship hull for the need of hydroelastic analysis, where only a few first dry natural frequencies and mode shapes are required. For practical use in the preliminary design of ship structures, the simplicity of the beam model presents an advantage over 3D FEM models.

Simplified procedure for the free vibration analysis of rectangular plate structures with holes and stiffeners

Abstract Thin and thick plates, plates with holes, stiffened panels and stiffened panels with holes are primary structural members in almost all fields of engineering: civil, mechanical, aerospace, naval, ocean etc. In this paper, a simple and efficient procedure for the free vibration analysis of such elements is presented. It is based on the assumed mode method and can handle different plate thickness, various shapes and sizes of holes, different framing sizes and types as well as different combinations of boundary conditions. Natural frequencies and modes are determined by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equations. Mindlin theory is applied for a plate and Timoshenko beam theory for stiffeners. The applicability of the method in the design procedure is illustrated with several numerical examples obtained by the in-house developed code VAPS. Very good agreement with standard commercial finite element software is achieved.