Research on Transverse Ultimate Strength for High Speed Trimaran＊

XU Shuangxi WU Weiguo LIU Weiqin WU Haochen

（School of Transportation，Wuhan University of Technology，Wuhan 430063）1）

（Key Laboratory of High Speed Ship Engineering，（Wuhan University of Technology），Ministry of Education Wuhan 430063）2）

（Systems Innovation Department，the Graduate School of Engineering，the University of Tokyo，Tokyo，Japan.113－8656）3）

0 Introduction

High speed trimaran has a wide application prospects because of its excellent sporting performance，high speed trimaran is suitable for being research ships，maritime shipping，maritime transport，exploration and rescue boats，etc.The structure of high speed trimaran is complex，it has side hulls and cross structures.The motion and structural response of trimaran are more complex than monolithic ship and catama－ran，the ultimate bearing capacity of trimaran has been the concern for ship mechanical workers，it＇s a hot spot of studying on the ultimate strength of ship structure and design method.According to sole rule for trimaran in the world（Rules for the classification of trimaran 2006），All designing external loads include a overall longitudinal still water bending moment，wave bending moment，hogging splitting moment，sagging splitting moment，longitudinal torsional moment，transverse torsional moment，trans－verse bending moment horizontal bending moment so on.In which，the transverse loads including transverse torsional moment and transverse bending moment.Therefore，under these transverse external load，cross brigde of high speed trimaran may be damaged in ultimate condition.It is very necessary that the transverse ultimate strength of trimaran is researched.Chen etc.proposed the ultimate bearing of a ship the finite element method firstly，applied to any load type and structure model［1］.Kutt used this method to study longitudinal ultimate strength of four different ship types by a variety of loading conditions，different finite element model were calculated and analyzed.In each analysis are included in the buckling，post buckling and plasticity effects［2］.Bin Liu had applied nonlinear finite element method to study ultimate strength of a SWATH ship，ultimate bending capacity was predicted to the actual ship，and the impact of SWATH ship＇s the typical structure to transverse ultimate strength was analyzed［2－4］.Weiguo Wu，Weiqin Liu and others have taken methods of nonlinear finite element and model test to study longitudinal ultimate strength of high speed trimaran［5－6］.The failure rule and mode of different ships is not the same，the ultimate strength of trimaran and its external load is more complex，its facture mode is much more different from general ship greatly，but transverse ultimate strength of high speed trimaran have been not researched by ship mechanical researchers at present.Therefore，in this paper，the respective ultimate strength was calculated under the transverse bending condition and transverse torsional condition，then the typical components was thickened locally and their transverse ultimate loads were calculated in two conditions.Two groups of load－displacement curves were obtained，the transverse ultimate bearing capacity was calculated as well，the influence of typical components to the transverse ultimate strength was analyzed，and the trimaran failure mode and rule could be illustrated in two ultimate conditions.This study might provide a useful reference to the design of the same kind of high speed trimaran.

1 Arc－length Method Principle

Nonlinear finite element method is an important method to calculate the ultimate strength of ship structure.With regard to large general－purpose finite element software，As long as boundary conditions，the nonlinear properties of materials，reasonable element types and mesh size，and various factors influenced to ultimate strength of real ship structure are considered reasonably，large finite element programs can obtain more accurate ultimate load of the hull structure.The geometric nonlinearity and material nonlinearity are included in ultimate strength problem. Using incremental arc－length method to track non－linear buckling／instability path which include the geometric nonlinearity and material nonlinearity influence，it＇s the most direct method.In this chapter，Crisfield arc－length method is illustrated，its basic principle is：the load factoris an λindependent variable，in each increment，equilibrium iteration is carried out（self－correction method），during the equilibrium iteration process，value of load factor is controlled automatically，spherical surface constraint is applied as displacement constraint equations.That is assumed：during the equilibrium iteration process，the calculated load and displacement results of them thincremental step are amand λm，in the（m＋1）thincremental step，modified balance iteration is carried out again.

There is a balance equation

Owing to increase of unknown，so a constraint equation is added.Fig.1shows the illustration of single freedom problem.

Fig.1 Constraint equations of crisfield arc－length method derivation radius vector

The constraints of incremental arc－length method is，in each iteration step the radius vector remain the same，the following relationship is established.

In iterative process，the point of i，i＋1step is carried out along an arc，Therefore，the method is arc length method.The main problem of this method is how to identifyand

Because of equation（5），

use（4a）to substitute（6），so

Coefficients are

According to equation（7）and（8），is obtained.Substituteinto equation（3），displacement increment and load increment factor are get during i＋1iterative process，be iterated until convergence solution is received.

2 Alculation Transverse Ultimate Strength

2.1 Transverse Bending Ultimate Strength

When trimaran voyages in the transverse wave，the fluid load，own weight，and inertia loads of the main hull and side hulls are different，the main hull and side hulls structure appear the vertical downward trend，so that cross bridge structures need to afford transverse bending moment，vertical shear and axial force.Especially when the encountered wavelength is the same to vertical distance of two side hulls center line，if the main hull is located on the peak，two films are located on the trough of wave（transverse hogging），or the main structure is located on the trough of wave，two side hulls are located on the peak of wave（transverse sagging），the vertical wave bending moment of the cross bridge structure is maximum.Owing the ultimate bearing capacity of transverse sagging is bigger than transverse hogging；the ultimate bearing capacity of transverse sagging is analyzed when the trimaran is considered its ultimate transverse strength.Fig.2is figure of wave distribution of transverse sagging.Boundary conditions of finite element model of trimaran are selected to simulate the boundary conditions of transverse bending primarily.Finite element model is half of trimaran，symmetry constraints are used on the longitudinal section，appropriate nodes are selected to constrain vertical displacement，and rigid body displacement of trimaran is restricted as a result.Five point forces in 5calculated sections are exerted，the maximum force of each node is 400 kN.The inertial load is exerted as gravity.Fig.3is von－Mises stress contour when trimaran is in transverse ultimate state.Get hold of transverse bending load－deformation curve for trimaran，The load factorμ＝0.432 5.

Fig.2 Wave distribution of transverse sagging

Fig.3 von－Mises stress contour in transverse bending ultimate state

In equation：Ftbis the transverse bending ultimate load；Fnis the maximum value of every point force；Nis the number of load force node；μis the load factor.

2.2 Transverse Torsional Ultimate Strength

When trimaran voyages in the oblique wave，main hull and side hulls are put in different location，with regard to main hull and side hulls，the drafts aren＇t equivalent in stern and bow.Owing to the superposition between weight and Hydrodynamic，relative vertical displacement trends are caused in stern and bow.Therefore，cross structures need to afford torsional moment，vertical shear and axial force.Trimaran has wide deck，structural response of three－hulls portion is considered in transverse torsional strength problem.So transverse torsional ultimate strength is considered in three－hulls portion. At present，the transverse torsional strength calculation of trimaran is carried out by Lloyd’s rules for the classification of trimaran.From Fig.4，antisymmetric load distribution is adopted on trimaran’s deck.The whole trimaran model is choosed in nonlinear FEA model，symmetry constraints are used on the longitudinal section，and appropriate constraints are selected to constrain rigid body displacement.Fig.5is von－Mises stress contour when trimaran is in transverse torsional ultimate state.Get hold of transverse torsional load－deformation curve for trimaran，The load factorμ＝0.443 3.

Fig.4 Transverse torsional load distribution

Fig.5 von－Mises stress contour in transverse torsional ultimate state

In equation ：Fttis the transverse bending ultimate load；Fnis the maximum value of every point force；Nis the number of all point force；μ is the load factor.

3 Sensitive Components Determine the Improvement Transverse Ultimate Strength

Search for the sensitive components affecting the transverse ultimate strength.To improve transverse ultimate strength of trimaran greatly，its weight should be increased at least as far as possible.In initial finite element model of trimaran，the typical structural components are thickened 1mm locally，stiffeners are not changed.Two parameters are compared and analyzed，percentage of weight increasing and percentage of ultimate strength increasing，overall ultimate loads of high speed trimaran and a number of curves of load－displacement are obtained，the influence of typical structural components on overall ultimate strength is analyzed.The chief typical structural components are：（1）main deck；（2）wet deck and（3）transverse bulkheads.

There are three nonlinear FEA model to analyze improvement relation on transverse bending ultimate strength and transverse torsional ultimate Strength，they are the thickened main deck model，thickened wet deck and thickened transverse bulkhead.Ultimate strengths load factors of four models including the initial one listed in the Tab.1.From the Tab.1we find that thickening of the transverse bulkheads results a significant increase of the load factors of transverse ultimate strengths.

Tab.1 Load factors

Tab.2give the percentages that added weight with a thickening of 1mm to weight of initial structures of Main deck／Wet deck and Transverse bulkhead respectively，and percentages of change values to the initial model values of Transverse bending ultimate load and Transverse torsional ultimate load.Through comparison，it turn out to be that Transverse bulkhead have the minimum increase of weight，Transverse bending ultimate load and Transverse torsional ultimate load have the maximum amount of increase.Therefore，thickening of Transverse bulkhead works best to improve the transverse ultimate strength of trimaran，that is to say，Transverse bulkhead is the most sensitive components of trimaran about transverse ultimate strength.

Tab.2 Percentage of weight increasing and ultimate load increasing item

4 Conclusion

The nonlinear finite element numerical method is adopted to calculate the transverse ultimate strength in this paper，ultimate bearing capabilities are obtained in two ultimate conditions.The sensitivity influence of typical structural components on transverse ultimate strength is analyzed.Some collusion is obtained.

1）Various mechanics analysis is carried out in two conditions to base on wave transverse distribution of trimaran.Some effective calculated methods of transverse ultimate strength of trimaran are obtained.

2）Failure modes are evaluated in two transverse ultimate conditions，the failure mode of trimaran is transverse bulkheads are damaged in transverse bending ultimate condition，main deck and transverse bulkheads are damaged in transverse torsional ultimate conditions.

3）Thickened transverse bulkheads have a greatest contribution to improvement of transverse bending ultimate strength，Main deck and Wet deck just has a little effect at transverse ultimate strength.Transverse bulkhead is the most sensitive component of trimaran about transverse ultimate strength.

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