Ivo Senjanović

Coupled horizontal and torsional vibration of a ship hull with large hatch openings

Abstract The higher-order theory of thin-walled girders, with mode-dependent stiffness parameters, is used to analyse coupled horizontal and torsional vibration of a ship hull as a nonuniform beam. A beam finite element of eight degrees of freedom is developed. Accuracy of the finite element procedure, in the case of a simple supported rectangular tube and channel girder, is checked. Application of the procedure for ship structures, analysing vibration of a pontoon with a large deck opening, is illustrated.

Safety analysis of ship rolling in rough sea

Abstract Non-linear ship rolling and capsizing in irregular waves is analysed by a single degree-of-freedom system. The random wave excitation depends on sea state, ship speed and ship direction to wave propagation. The governing differential equation of motion is integrated by applying the harmonic acceleration method. Ship response is presented in time and frequency domains. A safe basin in the initial value plane is constructed for different values of the three excitation parameters mentioned above in order to determine the probability of ship survival, which is presented in a PC radar map as a function of heading angle.

A higher-order theory of thin-walled girders with application to ship structures

Abstract A new flexural and torsion theory of thin-walled girders is presented by introducing the concept of effective stiffness and mass parameters as modal quantities. For this purpose a rather simple vibration analysis of simple supported girders is performed. Due to sinusoidal modes the problem is reduced to the girder cross-section plane. The stiffness and mass properties are determined from equivalence of girder deformation energy and inertia work, respectively and its beam idealization that results in the same natural frequencies. Therefore, this beam theory is more accurate than the classical one. For practical reasons the finite element formulation is applied, employing orthotropic strip elements that are necessary for the ship structures. The developed computer program is tested in case of a square tube and a channel girder. For illustration, analysis of a container ship midship section is performed.

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.

A higher-order flexural beam theory

Abstract A formulation of the beam theory for flexural vibration of thin-walled or solid girders, valid in the higher-frequency domain, is presented by introducing the concept of effective stiffness and mass parameters. Determination of these parameters is based on equivalence of natural frequencies of a simply supported girder and its beam idealization. The same effective parameters, which are strongly frequency dependent, may be transferred to a girder with different boundary conditions, as well as to the non-prismatic girder of proportional cross-sections. Application of this higher-order beam theory is illustrated in the case of a flat girder.

Harmonic acceleration method for dynamic structural analysis

Abstract The harmonic acceleration is assumed in each time step of integration of the dynamic equilibrium equation of a structure and its modal transformation. As a result of this two different numerical integration methods have been derived. The both methods are unconditionally stable and very accurate comparing to some other commonly used methods.

The bending and shear coefficients of thin-walled girders

Abstract The bending coefficient is introduced and a new formulation of the shear coefficient is given for thin-walled prismatic girders based on equivalence of natural frequencies of simply supported girders and their beam idealizations. Analytical and numerical procedures are developed and their application is illustrated. Formulae for natural frequencies and stiffness coefficients for some girders of simple cross-sections are tabulated.

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.

Investigation of Restoring Stiffness in the Hydroelastic Analysis of Slender Marine Structures

The restoring stiffness, which couples displacements and deformations, plays a very important role in hydroelastic analysis of marine structures. The problem of its formulation is quite complex and is still discussed in relevant literature. In this paper, the recent formulations of restoring stiffness are correlated and analyzed. Due to some common terms of the restoring and geometric stiffness, the unified stiffness is established and compared with the complete restoring stiffness known in relevant literature. It is found out that the new formula deals with more terms and that under some assumptions, it is reduced to the known complete restoring stiffness. The unified stiffness constitution is analyzed through derived analytical formulae for prismatic pontoon. Its consistency is checked for the rigid body displacements. Also, numerical results of the hydroelastic response of segmented barge are correlated with available model test results. Some issues, that are important for practical implementation in the hydroelastic code for flexible structures, are described.