Structural analysis of open deck ship hulls subjected to bending, shear and torsional loadings

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1 Structural analysis of open deck ship hulls subjected to bending, shear and torsional loadings Sebastião José Ferraz de Oliveira Soeiro de Carvalho Instituto Superior Técnico, Lisboa, Portugal Abstract The objective of the present work is to study the structural behavior of open deck ship hull structures subjected to bending, shearing and torsion by using analytic and finite element solutions. Two different sets of simplified finite element models are developed as a double shell box girder composed of plates with equivalent thicknesses that accommodate the stiffeners both in longitudinal and transverse directions (in such way that the original sectional properties are respected). The first one, a cargo-hold length model, used to study the behavior when subjected to bending and shear design loads, and the second one, a full pontoon-like ship hull model, used to study the structure subjected to torsional design load. This second set of FEM models is composed by an open deck model, a partially-closed model and a closed model, in order to quantify the influence of the degree of opening on the structural behavior. The values of the stress obtained by the FEM analysis are compared with the simple procedures from the beam theory and from the thin-walled girder theory, demonstrating that the global stresses are generally successfully predicted by these simple methods. The thin-walled girder theory application used here, lies in considering the ship cross-section as a channel section with lumped areas under different boundary conditions, subjected to torsional load, analyzed according to the bi-moment method. Keywords: bending moments, shear loads, torsional moment, open deck structures, finite element method, beam theory, thin-wall girder theory, bi-moment method 1. Introduction Ship structure design and analysis has always been a very important and active field of scientific research, in an effort to make those structures more reliable and cost effective. Much of this work was initially aiming to develop methods to determine the hull girder strength, and although these early methods gave adequately safe designs for common ship structures it has been shown by full-scale tests that the mechanisms of failure where frequently different from the predictions of those methods [1]. The major cause for that discrepancy is the nonlinear behavior of the individual components and subsequently the entire system. These observations led to an increasing concern with the local phenomena as opposed to the global phenomena. A great amount of research was devoted then to the ultimate strength and behavior of individual ship structural components such as individual plates, stiffened plates and grillages. Based on the knowledge of these individual components, various methods were developed in an attempt to determine the ultimate strength for the entire ship hull []. From a variety of methods, one of the most exhaustive is the one developed by Ostapenko [ 5], where the behavior and ultimate strength of longitudinal stiffened ship hull girder segments of rectangular single-cell cross-section, subjected to bending, shear and torque, were analyzed analytically and tested experimentally. This method produces accurate results for the bending and shear load cases, but not so acceptable results when torsion is considered (up to 4% of overestimation). Despite of this research work, the torsion problem kept understudied since it was not the reported cause of accidents. However, torsion induced buckling damage in deck structures of ships with large deck openings were not uncommon, see for an example Hong et al. [6], where a large ore carrier under rough weather was subjected to deck damages due to excessive warping. Besides, most of the Classification Society criteria for the ship structural design are based on the first yield of hull structures together with buckling checks for structural compo- 1

2 nents (i.e. not for the whole hull structure), which proved themselves to be effective for intact vessels in normal seas and loading conditions, however they fail to do so after accidental situations such as collisions or groundings [7]. Later, Paik [7 9] also worked on the ultimate strength under torsion based on the thin-walled theory and finite element analyses, suggesting a multisegment model between two neighboring transverse bulkheads or a single-segment model between two neighboring transverse frames as sufficient models to study the warping effects. Sun and Soares [1] conducted a model experiment, as well as a nonlinear finite element analysis, of a large deck opening ship-type structure to investigate the ultimate strength and collapse mode under a torsional moment. The elastic behavior of ship structures and its stiffness parameters, under torsion, were a subject of study by Pedersen [11, 1], Pavazza [13, 14] and Senjanović [15 3]. All of these authors developed methods based on the thin-walled girder torsional theory (since ships are a good example of a thinwalled structural application), but while the first two authors modeled the contribution of transverse bulkheads, deck strips and engine rooms as axial elastic foundations, the latter considered them as short closed cross-sections with an increased torsional stiffness and used a finite element analysis and the energy approach to estimate this increment. Iijima et al. [4] developed a practical method for torsional strength assessment, including a wave load estimation method and a proposal of design loads by a dominant regular wave condition. The topic of thin-walled girders under torsion is also studied in other engineering fields such as civil engineering, where the methods are not fundamentally different, as for an example, the investigation of Sapountzakis and Mokos [5, 6]. On a more particular level, Villavicencio et al. [7], developed a method, based on finite element analysis, to estimate the normal warping stress fluctuation in the presence of transverse deck strips in large container ships. The St. Venant component accounts for the torsion effects assuming that there are no in-plane deformations, i.e. a plane cross-section remains plane during the twist. This kind of torsion is only verified in circular closed cross-sections, while in the remaining cases the warping component must be considered, since the cross-sections no longer remain in-plane during the twist. These warping deformations vary with the rate of twist and as a function of the position across the cross-section. The differential equation that describes this nonuniform torsion, equivalent to Eqn. 1, is given by: GJ t dφ dx EJ(ω)d3 φ dx 3 = T () where φ is the twist angle, G is the shear modulus, J t is the torsional modulus of the section, E is the Young modulus, J(ω) is the sectorial moment of inertia (or warping constant) of the section and T is the sectional torque applied to the structure. Once Eqn. is solved, under the appropriate load and boundary conditions, the bi-moment method is ready to be applied in order to find the shear and flexural warping stresses verified at the structure (Fig. 1). τ(ω) = T (ω) S(ω) J(ω)t σ(ω) = ω M(ω) J(ω) (3) (4) where S(ω) is the sectorial static moment, ω is the sectorial coordinate, t is the thickness of a given segment. T (ω) is the warping torsional moment and M(ω) is the bi-moment applied on the structure and determined according to Eqns. 5 and 6: T (ω) = EJ(ω) d3 φ dx 3 (5) M(ω) = EJ(ω) d φ dx (6). Background The torsional moment T is divided in two components: St. Venant torsional moment T S and warping torsional moment T ω, hence: T = T S + T ω (1) (a) Figure 1: Channel section cross-sectional distribution of shear (a) and normal (b) stresses due to warping. (b)

3 3. Implementation The midship section used for this study is a simplification of the one taken from [8]. The ship main dimensions are given in Tab. 1. Table 1: Ship main dimensions. Main dimensions [m] Length, L bp 15.9 Beam, B 6. Depth, D 16. Draft, T 1.8 For all the computations where the block coefficient C B is required, it is assumed a typical block coefficient for a container ship, as C B =.65. Table : FE model section properties. Section properties Area, A[m ].37 1 st moment of area, S[m 3 ] Neutral axis, z NA [m] 5.67 Shear center, z e [m] -8.7 nd moment of area, I yy [m 4 ] nd moment of area, I NA [m 4 ] nd moment of area, I zz [m 4 ] Section modulus, Z deck [m 3 ] 7.4 Section modulus, Z bottom [m 3 ] Section modulus, Z side [m 3 ] Required section modulus, W min [m 3 ] 7.36 Table 3: Simplified section properties. Section properties Torsion constant, J t [m 4 ] 4.6 nd moment of area, I Y [m 4 ] 5.97 Shear center position, e Y [m] Warping constant, J(ω)[m 6 ] Sectorial static moment, S(ω)[m 4 ]. second case they are constrained from warping deformation (Eqn. 7b). In both of these scenarios the torsion is constrained (Eqn. 7c) at the extremities of the beam. Figure : Midship section layout. Fig. presents the original midship section layout in dashed line (that is subjected to some modifications regarding the thicknesses of some segments in order to fulfill the minimum section modulus, according to the DNV - Det Norske Veritas classification rules [9]), and in bold line the simplification that is used in the bi-moment method analysis. For the finite element model built based on this structure, the curved bilge is replaced by a rectangular bilge. Tab. presents the midship cross-sectional properties after all the supra mentioned simplifications, as well as an estimation of the shear center position according to Paik et al. [7], while Tab. 3 presents the properties of the simplified section used in the bi-moment method analysis Analytical models For the torsional loading case, two different boundary condition scenarios, regarding the warping constraining, are considered. In the first case the extremities are free to warp (Eqn. 7a) and in the d φ = free warping, (7a) dx dφ = constrained warping. (7b) dx φ = constrained torsion, (7c) The chosen load is a concentrated sectional torque applied amidship, as shown in Fig. 3. Figure 3: model. Torsion loading applied on the beam This load is then expressed as: { T / for the left half, (8a) T = T / for the right half. (8b) 3

4 The solution (for the first half of the length) of the differential equation for non-uniform torsion (Eqn. ) when subjected to the boundary conditions and loads expressed above is: γa, free warp. (9a) 1 + e L k φ(x) = γc, const. warp. (9b) 1 + e L k where: A = ke L k γ = e x k T GJ t (1) k = EJ(ω) GJ t (11) L+4x + ke k e x k x e L+x k x (1) Figure 5: Cargo hold model. C = k(e x k e L k ) e x k (k + x) + e L+x k (k x) (13) φ/φmax x/l Figure 6: Open deck model. Figure 4: Torsion function for both the free warping (solid line) and constrained warping (dashed line) scenarios. 3.. Finite element models To perform the analyses under the different loads, four distinct FE models are developed using the commercial software ANSYS. For the bending and shear load cases, a simple ship hull segment equivalent to one cargo hold is modeled (Fig. 5). For the torsional load case, in order to study the different structural responses for different structural configurations, three distinct simplified full hull models (open deck, partially-closed deck and closed deck (Figs. 6, 7 and 8 respectively) are built. The element type used in the different models is an 8-node quadrilateral shell element, allowing six degrees of freedom at each node, as well as large deflection capabilities. This element type is designated as SHELL93 in ANSYS. Regarding the boundary conditions, in the case of the cargo hold model, all its nodes on one extremity Figure 7: Partially-closed deck model. Figure 8: Closed deck model. 4

5 are constrained in every degree of freedom. Additionally, at the other extremity is placed a transverse rigid plate, in order to smoothly transmit the stresses and deformations across the section, acting as boundary condition as well. For the remaining models, all the nodes at both extremities are constrained in every degree of freedom. To model the overall loads, determined according to DNV [9], to which the FE models are complying, it is necessary to transform the loads in equivalent distributed loads or equivalent set of nodal loads and apply them to the specific nodes of the meshed FE models. The bending moment equivalent set of nodal loads (Fig. 9) is obtained according to Eqn. 14 and applied at the free end of the FE model: σ i = M I N.A. d i = F i A i (14) where σ i is the axial stress at a given node, M is the bending moment, I N.A. is the second moment of area about the neutral axis, d i is the distance of the given node to the neutral axis, A i is the nodal area linked to the given node and F i is the load to apply at the given node. Figure 1: Equivalent vertical shear force. M W T = 4 i=1 1 P ia i L i (16) where M W T is the overall torsional moment, P is the maximum load value, a is the load span (the length of the edge) and L is the distance from the shear center to the equivalent force application point. Additionally, a restriction is applied in order to ensure the balance between the four loads. This restriction assures that the four loads produce an equal moment around the shear center according to Eqn. 17: P al i = P al i+1 (17) Figure 9: Equivalent horizontal bending moment. The shear load equivalent set of nodal loads (Fig. 1) is obtained according to Eqn. 15 and applied at the free end of the FE model: τ approx = Q A shearing = F i A i (15) where τ approxi is the approximated shear stress at a given node, Q is the overall shear load and A shearing is the sectional area of all structural components subjected to shear. The equivalent torsional moment around the shear center (Fig. 11) is obtained by four linear distributed loads, acting at each exterior edge of the midship section, according to Eqn. 16: Figure 11: Equivalent torsional moment. 4. Results and discussion 4.1. Bending moments and shear forces Figs. 1 and 13 present the distributions of the axial stresses along the depth at the side shell and along the beam at the bottom respectively, given by FEM and beam theory. At Tab. 4 are presented the axial stress values at deck and bottom under the vertical bending moment and the values at the side shell at port and starboard under the horizontal bending moment, also both by FEM and beam theory. As it can be seen, the axial stress distribution given by the FE analysis gives an almost perfect 5

6 approximation to the theoretical values obtained by beam theory, attaining its maximum differences, around 6% to 9%, at the extreme values. σx[mp a] z[m] Figure 1: Axial stress σ x distribution along the side shell, under vertical bending moment, obtained by beam theory (solid line) and FEM (dashed line). and vertical girders, and at horizontal elements in the immediate proximity of their connection with the vertical elements. The maximum τ zx shear stress values obtained are around 1 MP a, while for the τ yx component they are around 1 MP a. As can be seen in Fig. 14, the maximum of a τ zx shear stress component distribution along the side plating is found at the height of the neutral axis of the cross-section (5.67 m), which is according to the static moment of area definition. τzx[mp a] σx[mp a] z[m] Figure 14: Shear stress τ zx component measured at the model mid-length along the side. 1 1 y[m] Figure 13: Axial stress σ x distribution along the bottom, under horizontal bending moment, obtained by beam theory (solid line) and by FEM (dashed line). Table 4: Summary of the maximum axial stresses. Vertical bending moment FEM BT diff. % σ deck [MP a] σ bottom [MP a] Horizontal bending moment FEM BT diff. % σ side(p S) [MP a] σ side(st BD) [MP a] Similarly to what happens in the vertical shear loading case, with horizontal shear load Q y a development of two shear stress components τ yx and τ zx is seen. Fig. 15 shows the τ yx stress component developed on the bottom plating. τyx[mp a] y[m] Figure 15: Shear stress τ yx component measured at the model mid-length along the bottom. As a result of the vertical shear loading Q z applied on the structure, two shear stress components are developed. Being the τ zx component the more significant and the τ yx almost negligible. The τ zx stress component is developed at the structure vertical elements, as side plates (Fig. 14) Once again the maximum shear stress at the structural elements aligned with the shear load is much more relevant (τ yx around 74 MP a) than the shear stress component developed at the structural elements perpendicular with the same shear load (τ zx around 3 MP a). 6

7 4.. Torsional loading In theoretical models, such as the bi-moment model used here, the maximum warping induced stresses, both axial and shear, take place at the midship section. These values, at the most relevant positions of the said section, are presented in Tab. 5. Table 5: Stresses at midship section (values in P a). free warp. const. warp. diff. [%] τ(ω) max 9.8E6 9.8E6 τ(ω) deck.57e6.57e6 τ(ω) bilge 4.5E6 4.5E6 τ(ω) keel -3.5E6-3.5E6 σ(ω) deck 5.93E7 3.68E7 38 σ(ω) bilge -5.4E7-3.35E7 38 Tab. 5 shows that for the warping induced shear stress, its values at the midship section are independent from the warping constraints at the ends of the model. This is not the case for the warping induced axial stress, where the free warping scenario has a 38% higher stress value. The longitudinal distributions of the maximum warping induced axial and shear stresses (along the sheer strake and at an height symmetrical to the shear center respectively) can be seen in Figs. 18 and 19. The next subject of interest of the present study is to understand and quantify the difference between stress levels on the three different hull configurations, mentioned before, under the same load. In Figs. 16 and 17 are plotted the longitudinal distributions of the axial and shear warping induced stresses for the three said deck configurations. The first observation is that the developed axial stresses (Fig. 16) are almost five times higher for the open structure (45.4 M P a) than the closed structure configuration (9. M P a). Other relevant aspect is the effect of the transverse structural elements (i.e. frames, bulkheads and cross decks) on the same axial stress distributions. The presence of these transverse elements causes stress fluctuations on the open structure configuration, as explained in Villavicencio et al. [7], and has no effect on the closed deck structure. Regarding the shear stresses (Fig. 17), the observations are similar, i.e. the open structure configuration develops considerably higher stresses, approximately two times higher than the closed configuration. The transverse structural elements show, in this case also, a stress fluctuation effect on the open structure. Comparing the longitudinal distributions of axial stress obtained at the sheer strake, from the bimoment method and from FEM for the open model, an almost perfect fit can be observed (Fig. 18). σx[p a] X[m] Figure 16: Axial stress σ x distributions at sheer strake along the length for open-deck (solid line), closed-deck (dashed line) and partially closed-deck (dash-dotted line) cases. τzx[p a] X[m] Figure 17: Shear stress τ zx component distribution along the length for open-deck (solid line), closeddeck (dashed line) and partially closed-deck (dashdotted line) cases. σx[p a] X[m] Figure 18: Comparison between open-deck finite element method (solid line) and free (dashed line) and constrained (dash-dotted line) warping by the bi-moment method for axial stress σ x results along the sheer strake. This fact indicates the justness of the bi-moment method procedure to predict the axial stresses de- 7

8 veloped on such a hull structure configuration. τzx[p a] X[m] Figure 19: Comparison between open-deck finite element method (solid line) and free (dashed line) and constrained (dash-dotted line) warping by the bi-moment method for shear stress τ zx component (warping induced plus pure shear stresses) along the length. 5. Conclusions The work presented here dealt with the structural analysis of an open deck pontoon-like ship in its elastic domain of material subjected to vertical and horizontal bending moments and shear forces and also under torsional moment, analyzed using the finite element method and simplified approaches. One of its main focus was to understand if a very simplified thin-wall girder theory application may predict, within an acceptable range, the expected stress levels under torsion on open deck ships. These expected stress levels were estimated by the finite element method. The thin-walled girder application is using an idealization of the original structure through a channel-section with lumped areas, analyzed under the bi-moment method. To create a range of valid stress values, and since there is no simple methodology to determine the degree of warping constrain, the thin-walled girder ends were subjected to two different sets of boundary conditions: torsion constrained and free warping and both torsion and warping constrained. The conclusion of this study is that the simplified thin-walled girder application, with the predefined boundary conditions, provides an almost perfect envelope for the axial warping stresses verified in the FE analysis, but its prediction differs in about 194% for the shear warping stresses comparing to the FE results. The discrepancy between the warping induced shear stress results of the two different methods, and without experimental observations, does not allow to rule any method as either better or worse since both methods are built upon many simplifications and constrains. In any way, since the analytical methods are normally more conservative, the results of the bi-moment method can be regarded as a valid estimation in an initial design stage, fulfilling thus the purpose of being a reliable prediction method. However, this is a point to develop and improve in the future research, adding some complexity to this, otherwise simple method, by considering the effect of transverse bulkheads as already proposed by some authors. References [1] J. Vasta, Lessons learned from full scale structural tests, Transactions of SNAME, vol. 66, p. 165, [] A. Ostapenko and A. Vaucher, Ultimate strength of ship hull girders under moment, shear and torque, tech. rep., Lehigh University, Fritz Laboratory Reports, 198. [3] A. Ostapenko, Strength of ship hull girders under moment, shear and torque, in Proceedings of the SSCSNAME Symposium on Extreme Loads Response, Arlington, USA, pp , [4] A. Ostapenko and Y. Chen, Effect of torsion on strength of ship hulls, tech. rep., Lehigh University, Fritz Engineering Lab, 198. [5] A. Ostapenko and T. R. Moore, Maximum strength of ship hulls subjected to moment, torque and shear, tech. rep., Lehigh University, Fritz Laboratory Reports, 198. [6] D. Hong and O. Kim, Analysis of structural damage of a large ore/coal carrier, in Ship Technology and Research Symposium (STAR), 1th, [7] J. K. Paik, A. K. Thayamballi, P. T. Pedersen, and Y. I. Park, Ultimate strength of ship hulls under torsion, Ocean Engineering, vol. 8, no. 8, pp , 1. [8] J. K. Paik, Ultimate longitudinal strength analysis of a double skin tanker by idealized structural unit method, in Pro. of SSC/SNAME Symposium, vol. 91, [9] J. Paik and A. Thayamballi, A concise introduction to the idealized structural unit method for nonlinear analysis of large plated structures and its application, Thin-walled structures, vol. 41, no. 4, pp , 3. [1] H.-H. Sun and C. G. Soares, An experimental study of ultimate torsional strength of a shiptype hull girder with a large deck opening, 8

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