Residual Ultimate Strength Evaluation Method of Cracked Hull Structure under Combined Bending Moment

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(1.School of Naval Architecture,Faculty of Vehicle Engineering and Mechanics,Dalian University of Technology,Dalian 116024,China;2.State Key Laboratory of Structural Analysis for Industrial Equipment,Dalian 116024,China)

Abstract:Crack damage is inevitable for ship structures and can weaken the ultimate strength of the structures.Therefore,it is of great significance to study the residual ultimate strength of ship structures with crack damage.Existing research mainly focuses on the residual ultimate strength of cracked hulls under vertical bending moment,and there is still a lack of research under combined bending moment.In this paper,the residual ultimate strength of a cracked midship section under combined action of vertical moment and horizontal moment is studied by using nonlinear finite element(FE)analysis.An evaluation formula is proposed to calculate the residual ultimate strength of the cracked midship section under the combined bending moment.Undetermined coefficients in the formula are obtained by fitting FE calculation results of cracked box girders.The results indicate that the proposed method can be used for rapid prediction of the residual ultimate strength of cracked hull structures under combined bending moment.

Key words:residual ultimate strength;crack damage;midship section;combined bending moment

0 Introduction

Crack is one of the most common forms of damage on ship structures,which could be caused by local corrosion or alternating loads.The existence of cracks will cause local stress concentration,reduce the ultimate strength of the ship structures,and even cause disastrous consequences under extreme conditions.Therefore,when cracks appear in ship structures,they should be detected and repaired as soon as possible.However,it is difficult to find the existence of cracks and repair them during daily voyage.Generally,related repair operations are carried out in docking maintenance.Therefore,it is necessary to know the change rule of the ultimate strength of cracked hull structures.On one hand,cracks found during the voyage can provide a basis for judging whether to berth in situ waiting for rescue or continue the voyage.On the other hand,it can be guaranteed that the hull structure will not be destroyed because of existing cracks during a detection cycle,even if the cracks are not found.

In recent years,the residual ultimate strength of cracked hull structures has been studied.Paik et al[1-3]carried out a series of tensile and compression tests on plates with pre-crack damage.A simplified model based on the reduction of cross-sectional area related to crack damage was proposed to predict the ultimate strength of a center or edge cracked plate under axial tension and compression loads.Based on the research by Paik,Shi et al[4]proposed a similar formula to calculate the residual ultimate strength of box girders with transverse cracks under torsional loads.Gao et al[5]used nonlinear finite element(FE)method and idealized structural element method to study the residual ultimate strength of hull structures under vertical bending moment with cracks or corrosion damage.

Studies have indicated that the worst-load case could be assumed to be a vertical bending moment,and the horizontal bending moment could be assumed to be negligible.This method may be valid for ships in intact condition.However,when a ship is damaged,its floating condition could be changed obviously during the voyage.If the ship has a roll,the combined effect of the vertical and the horizontal bending moment could be more serious.Therefore,the effect of the horizontal bending moment should be considered along with vertical hull girder strength to assess the structural safety of damaged ships.Gordo et al[6]used the progressive collapse analysis method to study the ultimate strength of intact midship sections under combined bending moment,and proposed an interaction formula to account for the combination of the load effects based on the results of the collapse calculations.Khan et al[7]studied the ultimate strength of the midship section of a tanker and two bulk carriers under combined vertical and horizontal bending moments using an approximate method in collision and grounding scenarios.However,there are currently few studies on the residual ultimate strength of cracked hull structures under combined vertical and horizontal bending moments.

In this paper,based on nonlinear FE analysis of cracked box girders,an evaluation formula for calculating the residual ultimate strength of cracked midship sections under combined bending moment is proposed.Undetermined coefficients in the function are obtained by fitting the FE results of the cracked box girders.The accuracy of the proposed formula for evaluating the residual ultimate strength is also verified.

1 Ultimate strength under combined bending moment

In this paper,a two-dimensional vector space is used to describe the vertical bending momentMVand the horizontal bending momentMHof ships.The combined bending momentMcan be expressed as

where

As shown in Fig.1,the ratio of the vertical bending moment to the horizontal bending moment will change as the value ofθchanges.For example,in the case thatθ=0°,the horizontal bending momentMHbecomes 0,and the hull is only subjected to the vertical bending momentMV.The ultimate bending moment with different values ofθcan be calculated to form a critical failure surface.If the critical failure surface can be expressed by an equation,the ultimate strength of a hull girder under combined bending moment can be assessed quickly and expediently.

Fig.1 Combined bending moment of hull girder

Gordo et al[6]studied the ultimate strength of intact midship sections under combined bending moment and proposed a governing equation as

whereMV0is the ultimate vertical bending moment of the intact structure,MH0is the ultimate horizontal bending moment and the power coefficientαcan be obtained by fitting numerical results.

Paik[3]proposed a formula for calculating the residual ultimate strength of cracked structures based on the assumption of effective cross-sectional area as follows

whereσuis the residual ultimate strength of the cracked structure,σu0is the ultimate strength of the intact structure,Ais the cross-sectional area of the cracked structure,andA0is the original crosssectional area.

From Eq.(4),we can see that the effect of the crack on the ultimate strength can be characterized by the cross-sectional area of the cracked structure,which is a function of the crack length.Based on Eq.(4),the right side of Eq.(3)can be changed into a corresponding reduction factor,and the equation for calculating the residual ultimate strength of the cracked midship section under combined bending moment can be given as

wheref(c)is the reduction factor and a function of crack lengthc.f(c)andαcan be obtained by fitting numerical results.

When calculating the ultimate strength of the hull structure,it is necessary to select an appropriate failure criterion.For cracked hull structures,theKcriterion or other fracture criteria can be used as the failure criterion.However,for materials with a high fracture toughness,the general yield of the hull structure can be used as the failure criterion by assuming that the crack does not propagate,which avoids the difficulty of calculating fracture parameters accurately and the results are also acceptable[8].

2 Finite element model of midship section

In this paper,the midship section is simplified as a box girder with stiffeners.It is assumed that the through-wall crack does not propagate and the two crack surfaces do not contact before the structure is destroyed.

As shown in Fig.2,the longitudinal extent of the midship FE model covers 1/2+1+1/2 cargo holds for reducing the adverse effects of the boundary conditions to a minimum.The FE model of the box girder is 600 mm long,500 mm wide and 400 mm high.The heights of both the transverse frame and the longitudinal one are 40 mm.The longitudinal spacing is 100 mm.All plates are 3 mm thick.

Fig.2 Schematic diagram of the box girder structure

For a box girder with longitudinals,the longitudinals are the primary load-bearing members,and the stress levels near the longitudinals are generally higher than those at other locations[1,4].In addition,when the box girder is under bending moment,the location farther away from the neutral axis has a higher stress level,so in the following numerical calculation the crack is placed at the edge of the box girder,as shown in Fig.2.The width of the crack is given as 3 mm.

It is assumed that the material is elastic-perfectly plastic,and the strain hardening of the material is neglected.The yield stress of the materialσyis 313.6 MPa.The Young’s modulusEand Poisson’s ratioνare 210 GPa and 0.3 respectively.

Initial deformation usually exists in hull structures due to improper welding operation or loading,which will affect the ultimate strength of the structures[9].In this paper,the initial deformation of the cracked box girder is considered and assumed to be the first-order buckling mode of the structure[4].The amplitude of the initial deformation is determined by the following empirical formula proposed by Smith et al[10]:

whereA0is the amplitude of initial deformation,bis the short edge length of plate element,tis the thickness of plate,σyis the yield limit andEis the elastic modulus.

In this paper,the FE analysis is carried out using ANSYS software,in which SHELL181,a four-node element with six degrees of freedom at each node,is selected for the numerical calculation.The SHELL181 element is suitable for analyzing thin to moderate-thick shell structures.

The mesh size of the FE model should be fine enough to accurately simulate the mechanical response of the cracked box girder.Convergence study is carried out by analyzing FE models of the box girder with different global mesh sizes.Fig.3 shows the variation of the ultimate strengthMUfor different numbers of incorporated elements.Based on this figure,the model with a mesh refinement of 10×10 mm(15 905 elements)produces results which tend to be stable and is therefore used as the minimum requirement.FE modeling and analysis of cracked structure involve maximizing the precision associated with the calculation of stresses and displacements near cracks,plus local effects imposed by the cracks on the overall response of the box girders.The denser FE meshing near cracks is used to increase precision.Therefore,the box girders are divided into three zones,namely the“crack tips”,along the“crack sides”,and“away”from the crack.These divisions are suggested for assigning different mesh densities for zones relative to the position of cracks[11].

Fig.3 Ultimate strength under different global mesh sizes

Tab.1 illustrates how the degree of mesh refinement can affect the evaluation of the ultimate strength.It shows that there is no significant difference between the ultimate strengths for different degrees of refinement,and that a higher mesh refinement along the crack is not really necessary.However,no refinement at all can produce relatively large error.So the middle degree of refinement is used in the subsequent FE calculations.

Tab.1 Ultimate strength under different mesh sizes around the crack tip

Fig.4 shows the FE mesh of the cracked box girder.The mesh size is 10×10 mm outside the refinement zones.The singular element is not used at the crack tip,because more attention should be paid to the overall mechanical properties of the structure when using the general yield criterion as the ultimate strength failure criterion.

Fig.4 FE mesh of the cracked box girder

Boundary conditions of the FE model are indicated in Tab.2.Two independent points are set at neutral axis in center-line as shown in Fig.2.The multi-point constraint element MPC184 in ANSYS is used to connect the nodes at the front end of the model to the independent pointAand connect the nodes at the rear end to the independent pointB,so that the constrains of the FE model can be transferred to the independent points.The vertical and horizontal bending moments are also applied to the two independent points.

Tab.2 Loads and boundary conditions applied to the FE model

3 Results and discussion

From Eqs.(1)～(2),it can be seen that the combined bending momentMand the corresponding neutral axis change asθchanges.As shown in Fig.5,whenθchanges from 90° to 180° and from 270° to 360°,the crack location is close to the neutral axis,so that the crack will have little effect on the ultimate strength of the box girder.To simplify the problem,only the ranges ofθfrom 0° to 90° and from 180° to 270° are considered and de fi ned as Quadrant I and Quadrant II respectively.

Fig.5 Schematic diagram of the box girder

When the combined bending moment is located in Quadrant I,the stress perpendicular to the crack surface is tensile stress,thus the crack tends to open.When the combined bending moment is located in Quadrant II,the stress perpendicular to the crack surface is compressive stress,thus the crack tends to close.

Cui et al[11]studied the effect of crack width(1 mm,3 mm and 10 mm)on the ultimate strength.In their research,the ultimate strengths of cracked plates with different widths are similar when the crack does not close.From the FE results,it is observed that the crack surfaces are not closed under the combined bending moment.For actual structures,the crack may be closed under pressure,and the ultimate strength of the structure will increase partly due to the influence of stiffness recovery.In this paper,the crack is assumed not to be closed,and the result tends to be conservative.

In Eq.(5),the reduction factorf()cis a function of the crack lengthc,so five cases of the crack length(c=30 mm,40 mm,50 mm,60 mm,70 mm)are numerically calculated to obtain the residual ultimate strengthMUof the box girder with different values ofθ.The vertical bending momentMVand the horizontal bending momentMHunder different ultimate states can be determined by

The dimensionless valuesMV/MV0andMH/MH0under different ultimate states are calculated by using the vertical ultimate bending momentMV0and horizontal ultimate bending momentMH0of the intact structure under the single bending moment.The least square method is used to fit the hyperelliptic function,based on FE results of the total 10 cases of Quadrant I and Quadrant II with five crack lengths.Fig.6 shows the FE calculation results and the fitted hyperelliptic function in the case thatθchanges from 0° to 90°(Quadrant I)andc=30 mm.The fitted values of the power coefficientαand the reduction coefficientf()

care 1.58 and 0.982 9 respectively.From Fig.6,it can be seen that the hyperelliptic function in Eq.(5)can fit the FE results of the bending moment well.All fitted values of the coefficientαandf(c) with different crack lengths are indicated in Tab.3.

Fig.6 FE calculation results and the fitted hyperelliptic function of Quadrant I(c=30 mm)

Tab.3 Fitted values of the coefficient α and f()c with different crack lengths

It seems that the crack length does not have obvious effect on the fitted values ofα.In order to study the value range ofα,some other cracked box girders with different shapes of the transverse section are numerically analyzed.One cracked box girder has a circular transverse section,so that the section modulus keeps constant when the neutral axis of the section rotates.Fig.7 shows FE calculation results and fitted hyperelliptic function of the cracked box girder with the circular section.The fitted value ofαis equal to 2,which is the maximum in the value range ofα.The other cracked box girders have a rectangular transverse section(800 mm×400 mm,700 mm×400 mm,600 mm×400 mm,500 mm×600 mm,500 mm×700 mm,500 mm×800 mm).For example,for the box girder with a width of 800 mm and a height of 400 mm,the width of the section is twice the height,so that the box girder looks more flat and the section modulus varies greatly as the neutral axis rotates.Fig.8 shows FE calculation results and fitted hyperelliptic function of the flat box girder.The fitted value ofαis equal to 1.38,which indicates that the value ofαdecreases as the ratio of the maximum section modulus to the minimum section modulus increases.

Fig.7 FE calculation results and the fitted hyperelliptic function of the cracked box girder with circular section

Fig.8 FE calculation results and the fitted hyperelliptic function of the cracked box girder with rectangular section(800 mm×400 mm)

From Eq.5,it can be understood that the value of the power coefficientαshould not be less than 1.Therefore,the value range ofαshould be taken from 1 to 2.Based on the above analysis,αcan be given as a function of the modulus variation of the transverse section.In this paper,a function to calculate the value ofαis proposed as

whereβis an undetermined coefficient,Wmaxis the maximum section modulus,andWminis the minimum section modulus.The value ofβis obtained as 12.29 by fitting the FE results of the eight box girders mentioned above with the least squares method.The values ofαcalculated by Eq.8 are listed in Tab.4.It can be seen that the calculated results are in good agreement with the fitted values from the FE calculation results.

Tab.4 Fitted values of α from FE results and calculated values by Eq.(8)

According to Paik’s assumption of effective cross-sectional area[3],the reduction factor is equal to the ratio of the residual cross-section area of the crack-free part to the complete cross-section area.As the through-wall crack is considered in the box girders,f(c) can be expressed as

whereSiis the cross-sectional area of each members,cis the length of the crack andtis the thickness of the plate at the crack.Fitted values of the reduction factorf()cobtained from the FE analysis and calculated results by Eq.(9)with different crack lengths are indicated in Tab.5.From the relative errors between the calculated results by Eq.(9)and the fitted values,it can be seen thatf(c) can be determined by Eq.(9)with a high accuracy.

Tab.5 Fitted values of f()c from the FE analysis and calculated results by Eq.(9)

By substituting Eq.(9)into Eq.(5),the residual ultimate strength of cracked midship section can be predicted by the following equation:

where the power coefficientαcan be evaluated by Eq.(8).

4 Conclusions

The residual ultimate strength of the cracked midship section under combined bending moment is studied by using the nonlinear FE method.From the numerical analysis of the cracked box girders,the following conclusions can be drawn:

(1)An evaluation formula for quickly predicting the residual ultimate strength of the cracked midship section under combined bending moment is proposed.

(2)The reduction factorf()cin the proposed evaluation formula can be expressed by the effective cross-sectional area coefficient for small through-wall cracks in structures.The power factorαcan be expressed by a function of the ratio of the maximum section modulus to the minimum section modulus,and the value range ofαis from 1 to 2.

(3)The accuracy of the proposed evaluation formula is verified by comparison with the FE results of the cracked box girders.