Ultimate strength assessment of a container ship accounting for the ...

Maritime Technology and Engineering – Guedes Soares & Santos (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02727-5

Ultimate strength assessment of a container ship accounting for the effect of neutral axis movement M. Tekgoz, Y. Garbatov & C. Guedes Soares

Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

ABSTRACT:  The objective of this work is to analyse the effect of the neutral axis movement, translation and rotation, of the mid-ship section on the ultimate load carrying capacity of a container ship subjected to asymmetrical bending moment for a non-damaged ship hull. The ultimate bending moment assessments are performed by the solution stipulated by the Common Structural Rules, finite element method and MARS2000 software. The achieved results are compared and discussed. 1  Introduction The asymmetrical bending moment is the most common structural load when ships operate under extreme sea conditions. The non-uniform ship deterioration and damage can also be the reason for the ship asymmetrical bending moment loading. The application of asymmetrical bending to the ship hull results in a neutral axis translation and rotation, which needs to be undertaken in the strength assessment unlike the pure symmetrical bending moment, in which case the hull stays strongly vertical with a central line identifying the transverse symmetry of the ship hull. In the event of not considering a complete movement of the neutral axis, when ship hull is subjected to asymmetrical bending moment, it may lead to a misestimate of the hull strength. Several other studies of ultimate limit state assessment were reported by Gordo & Guedes Soares (2000), Wang et  al. (2002), Ziha & Pedisic (2002), Smith & Pegg (2003), Yoshikawa et  al. (2008), Hussein & Guedes Soares (2009) wherein all studies are considered only one convergence criterion, force equilibrium condition, to search for new positions of the neutral axis plane as the curvature increases in a progressive manner and the position of the neutral axis is defined only by a translation without rotation from its initial position. Joonmo et al. (2012) presented two convergence criteria to find translational and rotational locations of the neutral axis plane for intact and damaged vessels. Definition of three types of asymmetries of a ship section was proposed, including material, load, and geometry-induced asymmetries. Tekgoz et al. (2014a) studied the intact ship shaped structures under asymmetrical bending moment and have proposed a new equilibrium criteria to account

for the neutral axis rotational influence on the ultimate strength of the ship hull and in addition to this, they proposed a new bending moment interaction model to relate the vertical and horizontal bending moments as a function of the heeling angle. Tekgoz et  al. (2014b) investigated the residual strength of ship hulls under asymmetrical bending moment. They have shown that, the damage case has great influence on the strength reduction, the maximum bending moment shifts as a function of the heeling angle and the damage gives rise to the neutral axis rotations, which were derived based on the results from the finite element method. The objective of this work is to analyse the effect of the neutral axis movement on the ultimate load carrying capacity of a container ship subjected to asymmetrical bending moment of an intact ship hull structure. For the ship hull ultimate strength assessment, the solutions provided by the New Common Structural Rules and implemented in the home made software, MARS2000 (2011) software and a finite element analysis performed by commercial software, ANSYS (2012) are compared and discussed. 2  ASYMMETRICAL BENDING MOMENT Analysis The mid-ship section of the container ship studied here is presented in Figure  1. The stiffener types and their specifications are shown in Table 1. The asymmetrical bending moment occurs when the external bending moment acts on the two crosssectional principle planes namely the vertical and horizontal planes. Depending of the asymmetry of loading, the neutral axis not only translates but also rotates

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Figure 2.  Asymmetrical bending moment.

Figure 1.  Container ship. Table 1.  Container ship stiffener specifications. Stiff. No

Dimensions (mm)

Type

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

350 × 100 × 12/17 400 × 100 × 11.5/16 300 × 90 × 13/17 300 × 90 × 11/16 250 × 90 × 12/16 250 × 90 × 10/15 300 × 28 300 × 38 300 × 90 × 13/17 230 × 10 150 × 10 300 × 90 × 11/16 150 × 90 × 9/9 150 × 12 250 × 90 × 12/15 150 × 90 × 12/12

Angle-bar Angle-bar Angle-bar Angle-bar Angle-bar Angle-bar At-bar At-bar Angle-bar At-bar At-bar Angle-bar Angle-bar At-bar Angle-bar Angle-bar

Figure  3.  Asymmetrical bending moment interaction curve layout.

due to the curvature difference in between the principle planes (see Fig. 2). A more detail analysis can be found in a study performed by Tekgoz et al. (2014a), Tekgoz et al. (2014b). Figure 3 shows the vertical and horizontal bending moment curve layout as a function of ship heeling, which varies between 0 and 180 degrees of heeling under sagging and hogging conditions. Mh stands for the horizontal bending moment and Mv stands for the vertical bending moment.

The ship ultimate strength calculations have been performed by the methodology stipulated by the Common Structural Rules, IACS (2012), MARS2000 (2011) software and the finite element method performed by commercial software ANSYS (2012). In order to account for the neutral axis rotation on the ultimate strength, the so-called bending moment equilibrium criteria, based on the formulation proposed by Tekgoz et  al. (2014a) is adopted when the methodology of the Common Structural Rules is used and homemade software has been created to calculate the bending moment and the curvature relationship as a function of heeling angle based on the New Common Structural Rules. The homemade software and finite element method cover both the neutral axis rotation and translations.

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An asymmetrical bending moment occurs when the external loading acts on the two-cross sectional principle planes and in this case, not only the crosssectional force equilibrium, but also, the bending moment equilibrium must be taken into account in the strength assessment, which has been implemented in the home made software as can be seen in Figure 4 and Figure 5. Figure 5 shows the two step static equilibrium of forces and bending moment. Firstly, the static force

Figure 4.  Neutral axis translation and rotation.

Figure 5.  Ultimate strength assessment.

equilibrium is satisfied by applying the condition 2 2 of Znai +1 − Znai < 0.1 mm, Zna,i = yna ,i + zna ,i , zna,i and yna,i are the vertical and horizontal shifting of the neutral axis respectively. If the force equilibrium is not satisfied, the neutral axis is shifted in a parallel manner in the vertical and horizontal directions during the iteration process. Secondly, the bending moment equilibrium is employed by satisfying |ψi+1-ψi| < 0.1º. It is important to point out here that the ship heeling angle θ, does not change throughout the analysis. Mz and My are the response (internally developed) bending moments. If the bending moment equilibrium is not satisfied, this leads to an increase or decrease of the neutral axis rotation and this new angle of rotation of the neutral axis also will change the shifting of the neutral axis in the new iteration with respect to the previous one identifying the most appropriate position of the neutral axis including the shifting and rotation. The proposed method, which deals with the neutral axis rotation effect, states that the sum of the internally developed bending moment vectors, which act on the horizontal and vertical principle planes, must be in a line with the externally applied load. Another way to make sure that they are in a line is to calculate the slope of both bending moment vectors, tan(θ ) = M h Mv. Due to the fact that an iterative process is involved, a value of 0.1°, which represents the difference between the current and previous neutral axis position has been set as a convergence criteria. As for the finite element method, there are several important parameters that affect the strength assessments and need a clear definition before a finite element analysis is performed as for an example: a finite element size, boundary conditions, initial imperfection etc. The choice of these parameters and their effect to the ultimate strength have been well covered by Tekgoz et al. (2014a) for the container ship. The boundary conditions have been applied in a way that the structure experiences a pure bending moment in the mid-span of the structure where it is supposed to fail (see Fig. 6). The load has been applied in the way as can be seen in Figure 6. In order to keep the cross-section plane, stiff beam elements have been implemented simulating umbrella boundary conditions (see Fig.  7). The boundary conditions are generated by the use of master nodes located at the crossing point between the central line and the initial elastic neutral axis in the fore and aft net sections of the studied segment. The master nodes are connected by stiff beams to the element nodes of the two edges of the segment, preventing any local corrugation and keeping the two net-section planes during the step loading. One of the master nodes has been constrained against a translation in all directions

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shape displacement functions, leads to a less stiff structure and to an underestimate of stresses. For the containership studied herein, 4 elements for the webs, two elements for the flanges and 11 elements for the plates between longitudinal stiffeners have been adopted and 4-node Shell 181 which is based on the Reissner-Mindlin plate theory that accounts for the transverse shearing stresses over the thickness has been used for the entire ship. 3  Ultimate strEnGTH aNalysis The analysis of ultimate strength of the studied container ship has been undertaken considering the neutral axis movement. Firstly, the ultimate bending moment was estimated and comparisons have been carried out among the solutions defined by the Common Structural Rules, IACS (2012), finite element analysis and MARS2000 (2011) software at 0 degrees of heeling in hogging condition. Figure 8 shows the bending moment and curvature relationships of the container ship in hogging condition, where Model 1, 2 and 3 represent the solution defined by the Common Structural Rules, finite element method and Mars2000 respectively. The ultimate bending moments are shown in Table 2.

Figure 6.  Boundary conditions.

Figure 7.  BC beam application at the cross-sections.

and it is free to rotate about the y and z axes. The second one is free to translate in the x-direction and free to rotate about the y and z axes. The element size and type are other important aspects of the finite element analysis of structures that has been studied by Garbatov et  al. (2011), Saad-Eldeen et  al. (2012), Tekgoz et  al. (2012). A bigger element size leads to an overestimate of stresses due to the fact that the lack of structural degree of freedom leads to stiffer the structure. Opposite to that is the use of high order finite elements, which employ high order

Figure  8.  Bending ­hogging condition.

moment-curvature

behaviour,

Table 2.  Ultimate bending moment, hogging condition.

Hogging

420

Model 1 (GNm)

Model 2 (GNm)

Model 3 (GNm)

7.28

7.09

6.70

Firstly, the neutral axis rotational effect has been investigated in the solution provided by the Common Structural Rules as labelled, Model 1 here. Figure 9 shows the ultimate bending moment as a function of the heeling angle with and without the neutral axis rotations. Figure  10  shows the difference in the ultimate bending moments between the solutions when the neutral axis rotation is considered or not, Model 1. The ship may experience 10% in the ­ultimate bending moment difference.

The ultimate bending moment as a function of the heeling angle with and without neutral axis translation is presented in Figure 11. It turns out that the neutral axis translation becomes negligible in the hogging conditions due to the fact that it returns back to its original position. Figure 12 shows the ultimate bending moment difference as a function of heeling angle. The difference has been calculated between the solution with a neutral axis rotation and the solution without a neutral axis rotation in the Common

Figure  9.  Asymmetrical bending moment, Model 1, NA-rotation.

Figure  11.  Asymmetrical bending moment, Model 1, NA-translation.

Figure  10.  Ultimate bending moment difference, Model 1, MA-rotation.

Figure  12.  Ultimate bending moment difference, Model 1, NA-translation.

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Structural Rules. The ship may experience as far as 17% strength difference. The bending moment-curvature relationship, along with the corresponding neutral axis translations in sagging conditions at 0 degrees of heeling is given in Figure 13. When the ultimate bending moment is reached, the neutral axis shifts from its origin to about 1.47 m. Figure 14 presents the bending moment and the curvature relationship, along with the corresponding neutral axis translations in hogging conditions at 0 degrees of heeling. When the ultimate bending

moment is reached, the neutral axis returns back close to the origin and shifts to about 0.0011  m from its origin. Figure  15  shows the bending moments and the neutral axis rotations as a function of curvature at 45  degrees of heeling under hogging condition. Model 1 and Model 2 are the Common ­Structural Rules and finite element method solutions respectively. In Model 1, the neutral axis rotation has been predicted by the proposed model by Tekgoz et al. (2014a). It turns out that in the case of the Model 1, the structure is more subject to

Figure 13.  Ultimate bending moment and NA translation, Model 1, sagging condition.

Figure 15.  Bending moment and NA rotation, Model 1 and 2, hogging condition, at 45 degrees of heeling.

Figure 14.  Ultimate bending moment and NA translation, Model 1, hogging condition.

Figure  16.  Asymmetrical bending moment, Model 1 and 2.

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the neutral axis rotations due to the poor structural elements integrity. Figure  16 presents the results of the ultimate bending moment as a function of the heeling angle. Model 1 represents the solution achieved based on the Common Structural Rules, whereas Model 2 represents the solution obtained by the finite element analysis. It is important to point out that both solutions, from Figure 16, cover the neutral axis rotations and translations. The solutions demonstrate pretty close predictions. 4  Conclusions The effect of the neutral axis movement on the ultimate strength of the intact container ship has been studied here. It has been found that, for this particular ship, the neutral axis rotation may lead to as much as 10% strength difference as a function of heeling angle. The proposed model for the neutral axis rotation shows a similar behaviour of the finite element solution. In the hogging condition, the neutral axis translations become negligible whereas it may lead to as far as 17% strength difference in the sagging condition. REFERENCES Ansys 2012. Online Manuals. Release 12. Garbatov, Y., Tekgoz, M. & Guedes Soares, C. 2011. Uncertainty Assessment of the Ultimate Strength of a Stiffened Panel. In: Guedes Soares, C. & Fricke, W. (eds.) Advances in Marine Structures. London, UK: Taylor & Francis Group, 659–668. Gordo, J.M. & Guedes Soares, C. 2000 Residual strength of damaged ship hulls. Proceedings of the 9th International Congress of International Maritime Association of the Mediterranean (IMAM2000), Ischia, Italy. Hussein, A., W. & Guedes Soares, C. 2009. Reliability and Residual Strength of Double Hull Tankers Designed According to the New IACS Common Structural Rules. Ocean Engineering, 36, 1446–1459.

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