Neven Hadžić

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.

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.

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.

Nonlocal vibration of a carbon nanotube embedded in an elastic medium due to moving nanoparticle analysed by modified Timoshenko beam theory – parametric excitation and spectral response

Abstract The Timoshenko beam theory, which deals with the deflection and rotation in two partial differential equations of motion, is transformed into a single partial differential equation with pure bending deflection as a potential function for the determination of the total deflection, rotation angle, and sectional forces. Inclusion of a nonlocal stress parameter results in the extension of the governing differential equation from the 4th to the 6th order. A simply supported nanotube is considered, and the governing differential equation is decomposed into a system of ordinary differential equations by employing the modal superposition method, separation of variables, and the Galerkin method. Both moving nanoparticle gravity and inertia force are consistently taken into account, resulting in ordinary and parametric excitation, respectively. As a novelty, the parameters are split into a constant and a time-dependent part. The former is added to the ordinary system of equations, which is solved analytically in the frequency domain by the harmonic balance method, while the system with variable coefficients is solved in the time domain by the perturbation method. The effects of slenderness ratio, nonlocal parameter, stiffness of elastic medium, nanoparticle gravity and inertia force, and velocity on the free and forced nanotube response are also investigated. Special attention is paid to the influence of damping on resonance. Performed parametric analysis is physically transparent due to the obtained semi-analytical solution. Some analytical results of illustrative examples are compared with numerical ones from the relevant literature, and notable differences are discussed.

Analytical Solution of the serial Bernoulli production line steady-state performance and its application in the shipbuilding process

The modern shipbuilding industry is faced with numerous challenges urging abiding improvement of the shipbuilding process and its management. Such a demanding problem is usually approached using complex production management models involving large data basis handling. The same problem can be solved using simpler, intuitive and mathematically transparent models developed within production system engineering. For these purposes, the analytical solution of the serial Bernoulli production line steady-state performance involving an arbitrary number of machines and buffers of arbitrary occupancy is developed based on Markov chain approach and eigenvalue problem including formulation of the generalised transition matrix as a key novelty. Performance measures in case of general serial Bernoulli production line are developed and the theory is validated using serial Bernoulli lines composed of two, three, four and five machines. The obtained results are compared to those determined using a semi-analytical approach. In order to further demonstrate the applicability of the developed theory, performance measure analysis is performed in cases of longer lines composed of 6, 7, 8, 9 and 10 machines with non-equal buffer capacities. Application of the developed procedure in the analysis of the shipbuilding process is illustrated in a case of the plate prefabrication line usually placed in each shipyard.

Vibration analysis of thin circular plates with multiple openings by the assumed mode method

Circular plates with openings are used in many engineering structures, especially as swash bulkheads in cargo tanks of liquefied gas carriers. In this article, an approximate procedure is worked out for the vibration analysis of circular plates with different boundary conditions and a different number and size of openings with an arbitrary arrangement within the plate area. The assumed mode method, recently used in the vibration analysis of rectangular plates, is applied in this case. Following the basic idea, the effect of the openings is accounted for by subtracting the potential and kinetic energy of the cut-out plate parts corresponding to the total plate energies without openings. The procedure is illustrated with numerical examples related to the vibration analysis of annular plates and circular plates with openings. The results are evaluated through their comparison with an analytical and finite element method solution.

An analytical solution to free rectangular plate natural vibrations by beam modes – ordinary and missing plate modes

Relatively simple analytical procedures for the estimation of natural frequencies of free thin rectangular plates, based on the Rayleigh quotient and the Rayleigh-Ritz method, are presented. First, natural modes are assumed in the usual form as products of beam natural modes in the longitudinal and transverse directions, satisfying the grillage boundary conditions. Based on a detailed FEM analysis, the missing of some natural modes, defined as a sum and a difference of the cross products of beam modes, is noted. The frequencies of these modes are very similar and identical in some special cases, manifesting in such a way a double frequency phenomenon. These families of natural mode shapes form a complete natural frequency spectrum of a free rectangular plate as a novelty. The reliable approximation of natural modes enables the application of the Rayleigh quotient for the estimation of higher natural frequencies. The application of the developed procedure is illustrated by the case of a free thin square plate. The obtained results are compared with those determined by FEM and also with rigorous ones from the relevant literature based on the Rayleigh-Ritz method. The achieved accuracy is acceptable from the engineering point of view. Furthermore, the same problem is solved by the Rayleigh-Ritz method using approximate natural modes as mathematical ones. Direct and iterative procedures are presented. A small number of mathematical modes and iteration steps are sufficient to achieve reliable results.

Conforming shear-locking-free four-node rectangular finite element of moderately thick plate

Abstract An outline of the modified Mindlin plate theory, which deals with bending deflection as a single variable, is presented. Shear deflection and cross-section rotation angles are functions of bending deflection. A new four-node rectangular finite element of moderately thick plate is formulated by utilizing the modified Mindlin theory. Shape functions of total (bending+shear) deflections are defined as a product of the Timshenko beam shape functions in the plate longitudinal and transversal direction. The bending and shear stiffness matrices, and translational and rotary mass matrices are specified. In this way conforming and shear-locking-free finite element is obtained. Numerical examples of plate vibration analysis, performed for various combinations of boundary conditions, show high level of accuracy and monotonic convergence of natural frequencies to analytical values. The new finite element is superior to some sophisticated finite elements incorporated in commercial software.

Some Aspects of Mega-Floating Airport Design and Production

Mega-Floating Airports (MFA) are unique and complex offshore transport system components that emerged as a consequence of tremendous land price increase in the vicinity of very large coastal cities. An overview of MFAs design and production aspects is presented within this paper including design concept, model tests and full scale measurement, air transport analysis, infrastructure, main particulars and structure, wave breaker, hydroelastic analysis due to wave load and airplane moving mass, mooring analysis, production technology and environmental aspects. MFA dynamic response due to airplane load is emphasized as the most challenging problem. Theoretical outline as well as a realistic illustrative numerical example are presented.

Analytical and numerical methods for vibration analysis of thick rectangular plates by modified Mindlin theory

Total deflection and angles of rotations in the Mindlin plate theory are decomposed into bending and transverse shear deflection, bending rotations and in-plane shear angles. Single differential equation of flexural vibrations is derived in terms of bending deflection as potential function for determination of all displacements and sectional forces. The equation is solved analytically for different combinations of boundary conditions. Shear locking-free rectangular finite element is formulated. Illustrative examples are solved analytically and numerically, and the obtained results are compared with the ones available in the relevant literature.