Marko Tomić

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.

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.

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.

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.

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.

Beam Structural Modelling in Hydroelastic Analysis of Ultra Large Container Ships

Ultra large container ships are very sensitive to the wave load of quartering seas due to considerably reduced torsional stiffness caused by large deck openings. As a result, their natural frequencies can fall into the range of encounter frequencies in an ordinary sea spectrum. Therefore, the wave induced hydroelastic response of large container ships becomes an important issue in structural design. Mathematical hydroelastic model incorporates structural, hydrostatic and hydrodynamic parts (Senjanovic et al. 2007, 2008a, 2009b, 2010b). Beam structural model is preferable in the early design stage and for determining global response, while for more detailed analyses 3D FEM model has to be used. The hydroelastic analysis is performed by the modal superposition method, which requires dry natural vibrations of the structure to be determined. For each mode dynamic coefficients (added mass and damping) and wave load are calculated based on velocity potential. The governing equation of ship motion in rough sea specified for the impulsive (slamming) load as a transient problem is solved in time domain. The motion equation is also given for the case of harmonic wave excitation (springing), which is solved in the frequency domain. In the chapter, methodology of the ship hydroelastic analysis is described, and position and role of the beam structural model is explained. Beam finite element for coupled horizontal and torsional vibrations, that includes warping of ship cross-section, is constructed. Shear influence on both bending and torsion is taken into account. The strip element method is used for determination of normal and shear stress flows, and stiffness moduli, i.e. shear area, torsional modulus, shear inertia modulus (as a novelty), and warping modulus. In the modelling of large container ships it is important to appropriately account for the contribution of transverse bulkheads to hull stiffness and the behavior of relatively short engine room structure. In the former case, the equivalent torsional modulus is determined by increasing ordinary (St. Venant) value, depending on the ratio of the strain energy of a bulkhead and corresponding hull portion. Equivalent torsional modulus of the engine room structure is also determined utilizing the energy approach. It is assumed that a short closed structure behaves as an open one with the contribution of decks. Application of the beam structural model for ship hydroelastic analysis is illustrated in case of a very large container ship. Correlation of dry natural vibrations analysis results for the beam model with those for 3D FEM model shows very good agreement. Hydroelastic analysis emphasizes peak values of transfer functions of displacements and sectional forces

Improvements of beam structural modelling in hydroelasticity of Ultra Large Container Ships

Increase in global ship transport induces building of Ultra Large Container Ships (ULCS), which have a capacity up to 14000 TEU with length up to 400 m, without changes of the operational requirements (speed around 27 knots). Natural frequencies of such ships can fall into the range of encounter frequencies in an ordinary sea spectrum. Present Classification Rules for ship design and construction don’t cover such conditions completely and hydroelastic analysis of ULCS seems to be the appropriate solution for analysis of their response in waves. This paper deals with numerical procedure for ship hydroelastic analysis with particular emphasis on improvements of the present beam structural model. The structural model represents a constitutive part of hydroelastic mathematical model and generally it can be formulated either as 1D FEM or 3D FEM model. For the preliminary design stage hydroelastic model derived by coupling 1D FEM structural model and 3D BEM hydrodynamic one seems to be an appropriate choice. Within the paper the importance of hydroelastic approach and methodology of hydroelastic analysis are elaborated. Further on, structural model based on advanced beam theory is described in details. The improvements include taking into account shear influence on torsion, contribution of bulkheads to hull stiffness as well as determination of effective stiffness of engine room structure. Along with that, hydrodynamic and hydrostatic models are presented in a condensed form. Numerical example, which includes complete hydroelastic analysis of a large container ship, is also added. In this case, validation of 1D FEM model is checked by correlation analysis with the vibration response of the fine 3D FEM model. The procedure related to determination of engine room effective stiffness is checked by 3D FEM analysis of ship-like pontoon which has been made according to the considered ship characteristics.Copyright

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.