Progressive Collapse Analysis of a Ship' s Hull Girder under Longitudinal Bending considering Local Pressure Loads

Transcription

1 507 Progressive Collapse Analysis of a Ship' s Hull Girder under Longitudinal Bending considering Local Pressure Loads by Tetsuya Yao*, Member Masahiko Fujikubo**, Member Mohammad Reza Khedmati***, Student Member Summary A method of progressive collapse analysis of a ship's hull girder under longitudinal bending developed by the first author is extended in order to take local pressure loads into account. First, the average stress-average strain relationship of the plate elements subjected to combined thrust and lateral pressure is formulated in an analytical manner. Then, based on the equilibrium conditions of forces and bending moments considering the influence of lateral pressure, the average stress-average strain relationship of stiffener elements with attached plating is derived. The proposed analytical procedure is then verified through the comparison of the calculated results with those by the FEM analysis for both plate and stiffener elements. Finally, developed formulations are implemented into the computer code "HULLST", and the modified code is applied to the analysis of collapse behaviour and ultimate hull girder strength of MV "Energy Concentration." 1. Introduction A ship's hull is basically a box girder structure composed of a number of plates and stiffeners, and is in equilibrium in the sea under the action of static and dynamic local pressure loads arising from buoyancy and sea waves, which are neutralising other loads such as hull and cargo weights. These global forces will create the cross-sectional forces such as shear force and bending moment, which produce stresses and strains in the structural elements. Among these sectional forces, the most predominant one is the vertical bending moment about the horizontal neutral axis in longitudinal waves. Because of the high stresses this moment may produce in the deck and the bottom of a ship's hull girder, these regions may suffer failure due to the buckling or yielding. Consequently, the structural response to this load component has received enough attention during recent years to get its own name- Ultimate Longitudinal Strength-. Available methods to evaluate the ultimate longitudinal strength of a ship's hull girder under longitudinal Graduate School of Engineering, Osaka University Faculty of Engineering, Hiroshima University Graduate School of Engineering, Hiroshima University Received 10th July 2000 Read at the Autumn meeting 16, 17th Nov bending can be catergorised into two groups". The methods of the first group are trying to calculate directly the ultimate hull girder strength. Some simple formulae have been proposed based on these methods to evaluate the ultimate hull girder strength. On the other hand, the methods of the second group are based on performing progressive collapse analysis on a ship's hull girder to shed more light on its behaviour during the progress of collapse. Some of them apply incremental finite element method2"). However, because of tremendous computational requirements imposed by the FEM, some of them utilise an alternative method, the so-called Idealized Structural Unit Method (ISUM), originally developed by Ueda et al.4),5) Some simplified methods have also been developed among which the most effective one is the Smith's method6),7). The key idea of this method is to take the strength reduction (load shedding) of structural members after their ultimate strength into account. The only difficult task in applying the Smith's method is the derivation of stress-strain relationships of component elements taking into account of the buckling and yielding. Smith performed elastoplastic large deflection analysis by the FEM. Such analysis however, may require much work especially when the number of different elements is large. The first author and his colleague8),9) proposed an analytical method to derive the average stress-average strain relationships of plates and stiffened plates considering influences of buckling and yielding. Then they prepared a computer code, "HULLST," which is able to perform progressive collapse analysis of a ship's hull

2 508 Journal of The Society of Naval Architects of Japan, Vol. 188 girder cross-section under longitudinal bending. Different ship cross-sections have been analysed using this code and its effectiveness has been demonstrated10),11). So far, very few of the available simplified methods to evaluate the ultimate hull girder strength, have considered the effects of local pressure loads on the structural elements. Therefore, it is necessary to develop a method of progressive collapse analysis which is based on the simplified methods taking into account of lateral pressure on the individual elements of a ship's hull cross -section in a rational manner. In this paper, the average stress-average strain relationships for plate and stiffener elements subjected to combined thrust and lateral pressure are derived in an analytical manner. Then the developed formulations are implemented into the code, "HULLST," in order to enable the code to perform progressive collapse analysis of a ship's hull girder under longitudinal bending considering local pressure loads. Finally, the modified "HULLST" is applied to progressive collapse analysis on the hull girder of MV "Energy Concentration." 2. 1 Basic assumptions 2. Analysis of Plates A rectangular panel partitioned by longitudinal and transverse stiffeners is considered. Fundamental equations are derived for the analysis of this rectangular plate subjected to combined longitudinal thrust and uniformly distributed lateral pressure. As illustrated in Fig. 1, the panel is assumed to be accompanied by welding residual stresses in a rectangular distribution and initial deflection of a hungry-horse mode expressed as : ( 1 ) The coefficient Aomi is based on the measurement by Yao et al.12),13) The maximum magnitude of initial deflection, womax, is taken as : ( 2 ) where ƒà = b/t E ãƒðy/e is the slenderness parameter of the plate, and b and t are the breadth and the thickness of the panel, and ay and E are the yielding stress and the Young's modulus of the material. The value of womax given by Eq. ( 2 ) is the average magnitude of measured initial deflection of ship plates14)15). To consider the plate continuity, it is assumed that the plate is simply-supported along its four edges which remain straight while subjected to in-plane movements. The influence of welding residual stresses is considered according to the formulations in Ref. 16). The fundamental procedure to derive the average stressaverage strain relationship for plate elements is the same as that described in Ref. 8) except the introduction of the influence of lateral pressure Elastic large deflection analysis (ELDA) When a plate is subjected to combined thrust and lateral pressure, the sequence of loading affects very little on the final collapse mode and strength17). So, in the present paper, lateral pressure is first applied to a specified value, and then longitudinal thrust is exerted. It is known that a continuous plate subjected to uniformly distributed lateral pressure deflects in a clamped mode in each panel partitioned by stiffeners. This mode is approximated by a single component as : ( 3 ) This mode is similar to that of a hungry horse mode, and for the analysis of plate behaviour under lateral pressure, initial deflection expressed by Eq. ( 1 ) is approximated as : where ( 4 ) ( 5 ) Following the theory of elastic large deflection analysis, the relationship between lateral pressure, q, and total deflection, B, in a clamped mode is obtained as follows : ( 6 ) Solving the above equation giving the specified value of q, the maximum value of deflection coefficient, Bch under lateral pressure is obtained. The maximum deflection for this state is expressed as wq=4b,. Then, in the following elastic large deflection analysis of a plate subjected to longitudinal thrust, the plate is assumed to have initial deflection of a hungry-horse mode of which magnitude is 13,, and is simply supported along its four edges. When the compressive load acts in the direction of the longer side of the plate, only one term of sinusoidal deflection mode is amplified and becomes stable above the buckling load12). This stable mode does not necessarily coincide with buckling mode, but is usually one or two higher-order mode than the buckling mode. From these points of view, it is enough to consider only the stable deflection component in the initial deflection as : Fig. 1 Plate model for analysis

3 Progressive Collapse Analysis of a Ship's Hull Girder under Longitudinal Bending considering Local Pressure Loads 509 where in is the number of half waves of a stable mode. Total deflection of the same mode is assumed and elastic large deflection analysis is performed. Details are described in Refs. 8) and 16). The influence of plate-stiffener interaction and lateral pressure on the elastic buckling strength is considered by introducing magnification factors to the buckling strength14),18) for simply supported condition Rigid-plastic mechanism analysis (RPMA) With increase in the applied load, yielding starts and the ultimate compressive strength is attained. Beyond the ultimate strength, the load carrying capacity decreases with the increase in the end-shortening displacement. This behaviour can be well simulated by performing rigid-plastic mechanism analysis. Detail of this analysis is described in Ref. 8) Derivation of average stress-average strain relationship Typical average stress-deflection and average stressaverage strain relationships are shown in Fig. 2( a) and ( b ), respectively. The procedure to get the average stress-deflection and average stress-average strain relationships of the plate under longitudinal thrust can be described as follows : 1) From the origin (point 0) to the initial yielding point (A), the solution of elastic large deflection is used. 2) The ultimate strength of the plate (point B) under combined thrust and lateral pressure is accurately predicted using proposed formulae in Ref. 19). 3) Between points A and B, and also between points B and C, appropriate curve-fittings are performed. 4) After point C, the solution of rigid-plastic mechanism analysis is used. 5) After obtaining average stress-deflection relationship, the average stress-average strain relationship is derived by using strain-deflection relationship considering geometrical nonlinearity. The details are described in Refs. 8) and 20). 2.5 Comparison of average stress-average strain relationship with FEM results Example calculations are performed on panels indicated below. The initial deflection of a hungry horse mode expressed by Eq. ( 1 ) is given, and the pressure (7) load is ranged (a) Average stress-deflection relationship (b) Average stress-average strain relationship Fig. 2 Prediction of plate behaviour based on analytical method ( a ) Average stress-deflection from 0 m to 45 m in water head. The calculated results are shown in Fig. 3 ( a ) through (c) together with the calculated results applying the FEM19). In the case of very thin plates ( t =10 mm), under small to moderate levels of lateral pressure, say h< 20 m, FEM results show larger inplane rigidity than that of the analytical method. The reason is that the thin plates having a large slenderness parameter, Ĉ, possess a large value of maximum initial deflection (in an idealised hungry-horse mode), even when there exists no lateral pressure. In such a case, the present model considering an uni-modal initial deflection corresponding to the stable-mode component results in some error in inplane rigidity. relationship ( b ) Average stress-average strain relationship The ultimate strength is accurately predicted by the present model, but the post-ultimate strength capacity tends to be overestimated compared to the FEM results. This is mainly because a localization of plastic defor-

4 510 Journal of The Society of Naval Architects of Japan, Vol. 188 (a) t=10 mm (b) t=14 mm (c) t=20 mm Fig. 3 Comparison of calculated and FEM results for plates mation takes place in the FEM analysis considering the initial deflection of a hungry-horse mode, while the plastic deformation of a roof mode is assumed over the plate in the analytical model. However, the predicted curves are generally in good correlation with those obtained by the FEM analysis. 3. Analysis of Stiffened Plates 3. 1 Basic assumptions A typical cross-section of a stiffened plate is shown in Fig. 4 ( a). The shaded part corresponds to the crosssection of a stiffener element composed of a stiffener and attached plating. The stiffener between points 1 and 2 in Fig. 4 ( b) is considered as a stiffener element. (a) Cross-section (b) Initial deflection Fig. 4 Stiffened plate model for analysis ( a ) Cross section ( b ) Initial deflection The initial deflection of a sinusoidal mode shown in Fig. 4 ( b) is assumed for this element. This element is subjected to axial load, P, and uniformly distributed lateral pressure, go. To derive the average stress-average strain relationship of the stiffener element, three basic assumptions are made as : 1) Attached strips of plating behave according to the average stress-average strain relationship obtained in Chapter 2. 2) Stiffener material is assumed to follow an elastic -perfectly plastic stress-strain relationship. 3) Plane cross-section remains plane and the strain varies linearly over it. The fundamental procedure to derive the average stress-average strain relationship for stiffener elements is also the same as that described in Ref. 8) except the introduction of the influence of lateral pressure. 3.2 Derivation of average stress-average strain relationship In Ref. 21), a thorough investigation was performed on the collapse behaviour of stiffened plates subjected to combined thrust and lateral pressure. It was found that under small to moderate levels of lateral pressure, the stiffened plate collapses in an Eulerian buckling mode, as shown in Fig. 5( a ). On the other hand, under high lateral pressure levels, collapse of clamped mode is observed as indicated in Fig. 5 ( b). The elastic deflection at midspan of a beam-column subjected to combined thrust, P, and lateral pressure, q, is approximated as : where (8) (9) (10) and ƒ =P/Pcr and u =(ƒî/2) ãƒ. The first term cone-

5 Progressive Collapse Analysis of a Ship's Hull Girder under Longitudinal Bending considering Local Pressure Loads 511 curvatures at both ends for the next step is determined using the present curvature expressed as : (a) Eulerian buckling mode where icp is the curvature produced by plastic deflection (14) component. Ď= -1 when wa+wq>0, whereas Ď = 1 when wa + wq < 0. The former case corresponds to the collapse of a simply supported mode, and the latter to (b) Clamped mode Fig. 5 Possible collapse modes ( a ) Eulerian buckling mode ( b ) Clamped mode sponds to the deflection in Fig. 5( a ), and the second term to that in Fig. 5( b ). The elastic deflection along the span is assumed as : (11) In the present method, lateral pressure is first applied to a specified level, and then the thrust load is applied. Depending on the relationship among ao, q and P, the stiffener element may deflect in the same directions or in the opposite directions in the adjacent spans. After yielding starts, plastic deflection, wp, which gives constant curvature at the mid-span region is introduced, and the total deflection is expressed as a sum of the elastic and plastic components as8) : (12) The direction of the plastic deflection is assumed to be the same as that of the elastic component. The forces and bending moments acting on the stiffener element are shown in Fig. 6. The equilibrium conditions can be expressed as : that of a clamped mode Comparison of average stress-average strain relationships with those by the FEM To check the validity of the proposed method in this chapter, the stiffened plates in Ref. 21) are analised. Comparisons are made for some typical cases in Fig. 7 between the average stress-average strain relationships obtained by the proposed method. Both results show relatively good correlations when the lateral pressure is below 20 metres water head. Above this pressure level, the proposed method overestimate the ultimate compressive strength. This is first because of the overestimation of compressive capacity of plate elements which were shown in Fig. 3 in Chapter in Eq. (12), the plastic deformation is concentrated at the midspan. However, under large lateral pressure, yielding is spreads in the neibourhood of the end supports also. Ignoring such a plastic deformation leads to the underestimation of the average compressive strain. (13) To derive the average stress-average strain relationship, curvatures are imposed at both ends of the element, and determine the elastoplastic stress distributions which produces P1, P2, M1, M2, WI and W2 which satisfy Eq. (13). This procedure is carried out step by step increasing the imposed curvature. The ratio of the (a) t=13 mm (angle-bar) (b) t=20 mm (angle-bar) (c) t=13 mm (tee-bar) (d) t=20 mm (tee-bar) Fig. 6 Equilibrium of loads, forces and bending moments Fig. 7 Comparison of calculated and FEM results for stiffened plates ( a ) t =13 mm (angle bar) ( b ) t =20 mm (angle bar) ( c ) t =13 mm (tee bar) ( d ) t = 20 mm (tee bar)

6 512 Journal of The Society of Naval Architects of Japan, Vol. 188 From these viewpoints, further improvement is necessary for the case with high lateral pressure. Table 1 Longitudinals of "Energy Concentration" 4. Analysis of Hull Girder 4. 1 Smith's Method The procedure of the Smith's method for progressive collapse analysis of a hull girder can be summarised as follows6),7) 1) The cross-section is subdivided into small elements composed of plates and stiffened plates. 2) Their average stress-average strain relationships are derived by the analytical method explained in Sections 2 and 3. 3) The curvatures of the cross section in two perpendicular directions are given incrementally, assuming that the plane cross-section remains plane and that bending takes place with respect to the instantaneous neutral axes. 4) At a certain incremental step, the in-plane rigidity of the elements are obtained using the slope of average stress-average strain curve at each specified strain. 5) Using the in-plane rigidities of individual elements, locations of the instantaneous neutral axes are determined, and the flexural rigidities of the cross-section is calculated. 6) Increments of bending moments corresponding to the applied increments of curvatures are then obtained. At the same time, the increments of strains and stresses in individual elements are calculated. 7) At the end of each step, increments of curvatures and bending moments are summed as well as those of strains and stresses to provide their cumulative values. 8) The same procedure is repeated from the step ( 3 ). Such a procedure is implemented into the computer code "HULLST" Analysis on VLCC "Energy Concentration" The structural failure of the MV Energy Concentration, which broke its back in July 1980 during discharging of cargo at Rotterdam, provides a case study of ultimate strength of ships. This casualty allows the authors to calibrate the computer code "HULLST" with an unintentional, unfortunate, and almost a full scale test. A comprehensive review of the ship and its conditions before and at the time of casualty is given in Ref. 22). The sizes, types, and material of the longitudinals are indicated in Table 1. The dimensions of the crosssection and some structural details are shown in Fig. 8. Initial deflection of the plates is assumed to be in an idealised hungry-horse mode with a maximum magnitude given by Eq. ( 2 ). The initial imperfections of the stiffeners consist of a buckling-mode initial deflection with the maximum magnitude of a/1000, Fig. 8 Cross section of MV "Energy Concentration" and an angular distortion 00 which is taken as a is the length of stiffener span, and h. the height o] stiffener web. Welding residual stresses are considerec here. Corrosion damage is assumed as22) plating :1 mm reduction in thickness stiffener web : 1 mm reduction in thickness stiffener flange : 2 mm reduction in thickness When the accident took place, the ship draught approximately 12 in. So, three cases are analysed here no pressure loads, draught, d, of 12 metres at the casu alty and that of 20 metres for comparison purpose. ThE obtained moment-curvature relationships for each casc under hogging condition have been shown in Fig. 9 ( b). In Fig. 9 ( b), the dashed line represents the results when

7 Progressive Collapse Analysis of a Ship's Hull Girder under Longitudinal Bending considering Local Pressure Loads 513 point a (initial collapse) point b (ultimate strength) (a) Stress-strain relationships of selected elements point c point d Fig. 10 Stress distribution (draught =12 m) ( b ). After the ultimate strength, buckling spreads over the side shell and longitudinal bulkhead in the compression side of the bending, while unloading takes place in the tension side of the bending, as is observed in stress distributions at the points (c) and ( d ). This is because the load carried at the compression side of bending decreases and the neutral axis moves upward. The same features are observed for the other two cases. (b) Moment-curvature relationships Fig. 9 Results of analysis for "Energy Concentration" ( a ) Stress-strain relationships of selected elements ( b ) Moment-curvature relationships all elements are assumed to follow the stress-strain relationship of an elastic-perfectly plastic material. The ultimate strength in this case is equal to the fully plastic bending moment of the cross-section. On the other hand, other lines represent the moment-curvature curves when elements behave following the average stress-average strain curves shown in Fig. 9 ( a ). It is known that a fully plastic bending moment can not be sustained in the cross-section because of the buckling. The stress distributions corresponding to the selected points on the moment-curvature relationship of the case with the draught of 12 metres has been shown in Fig. 10. At point ( a ), the initial local collapse takes place in the deck by yielding in tension. With further increase in the applied curvature, yielding spreads all over the deck and at some elements in the upper part of side shell, longitudinal bulkhead, and deck longitudinal girders. At the same time, bottom elements and some elements in the lower part of longitudinal bulkhead and bottom longitudinal girders buckle, and the ultimate strength of the whole cross section is attained at point A numerical comparison of the results is given in Table 2. The value of ultimate bending moment (MuH) for the case of casualty draught, is about 1.5% less than that for the case when no pressure loads act. This is because the bottom plating is relatively thick and the ultimate longitudinal strength of bottom stiffened plates does not decrease significantly under the action of lateral pressure. A similar difference of about 1.9% is observed between the ultimate bending strengths of the two cases ; when no pressure loads is considered and when the draught is assumed to be 20 metres. Estimated applied still water bending moment at collapse in hogging, is equal to ~ 103 MN Em22), which differs about 7% from the calculated ultimate strength for the case of draught of 12 metres. Significance of the influence of local pressure loads on the ultimate bending moment of the whole cross-section Table 2 Numerical comparison of the results

8 514 Journal of The Society of Naval Architects of Japan, Vol. 188 generally depends on the magnitude and distribution of such loads over the cross-section, and also on the geometry of the individual elements. Under very high local pressure loads, the compressive ultimate strength of the stiffened plates is more decreased and subsequently the ultimate longitudinal bending strength of the whole cross-section is more reduced. Improvement of the present model to consider larger lateral pressure and the study on the general influence of local pressure loads on the ultimate bending moment of the ship's cross-section remain as future work. 5. Conclusions A method of progressive collapse analysis of a ship's hull girder under longitudinal bending considering the effects of local pressure loads is developed. The average stress-average strain relationships for plate and stiffener elements subjected to combined thrust and lateral pressure is derived in an analytical manner, and then implemented into the computer code "HULLST." The modified code is applied to the progressive collapse analysis of MV "Energy Concentration" which collapsed in a hogging condition. The following can be drawn : 1) The proposed analytical procedure has been verified through the comparison of the calculated results with those obtained by the FEM analysis, for both plate and stiffener elements. 2) The ultimate longitudinal strength of MV "Energy Concentration" calculated by the devel - oped computer code "HULLST" is in good agreement with the estimated applied still water bending moment at her collapse in hogging. 3) The influence of local pressure loads on the ultimate bending moment of the whole crosssection is not significant for the considered ship and the draught condition. In the present study, the effect of transverse thrust arising from the action of pressure loads on the side shell was not considered. Inclusion of this effect remains as future work. References 1) Yao, T.: "Ultimate Longitudinal Strength of Ship Hull Girder Historical Review and State of Art," Int. J. Offshore and Polar Engineering, Vol. 9.1 (1999), pp ) Japan Ministry of Transport : Report on the Investigation of Casualty of Nakhodka, The committee for the Investigation on Causes of the Casualty of Nakhodka (1997). 3) Japan Shipbuilding Research Association : Investigation into Structural Safety of Aged Ships, Report No. 74 Rule and Regulation Committee (2000) (in Japanese). 4) Ueda, "Plate Y., Rashed, and Stiffened S. M. H., and Paik, J. K. : Plate Units of The Idealized Structural Unit Method (1st Report) -Under Inplane Loading-," J. of Soc. Naval Arch. of Japan, Vol (1984), pp (in Japanese). 5) Ueda, Y., and Rashed, S. M. H. : "The Idealized Structural Unit Method and its Application to Deep Girder Structures," Computers and Structures, Vol. 18, No. 2 (1984), pp ) Smith, C. S. : "Influence of Local Compressive Failure on Ultimate Longitudinal Strength of a Ship's Hull," Proc. of PRADS, Tokyo, Japan, Paper No. A-10 (1977), pp ) Smith, C. S. : "Structural Redundancy and Damage Tolerance in Relation to Ultimate Ship Hull Strength," Proc. of Int. Symp. on The Role of Design, Inspection, and Redundancy in Marine structural Reliability, Williamsburg, USA (1983). 8) Yao, T., and Nikolov, P. I. : "Progressive Collapse Analysis of A Ship's Hull under Longitudinal Bending," J. of Soc. Naval Arch. of Japan, Vol. 170 (1991), pp ) Yao, T., and Nikolov, P. I. : "Progressive Collapse Analysis of A Ship's Hull under Longitudinal Bending (2nd Report)," J. of Soc. Naval Arch. of Japan, Vol. 172 (1992), pp ) ISSC, Report of Committee 111.1, Ductile Collapse-, Proc. of the 12th Int. 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9 Progressive Collapse Analysis of a Ship's Hull Girder under Longitudinal Bending considering Local Pressure Loads (in Japanese). 19) Fujikubo, M., Yao, T., and Khedmati, M. R. : Estimation of Ultimate Strength of Ship Bottom Plating Under Combined Biaxial Thrust and Lateral Pressure, Trans. of the West-Japan Soc. Naval Arch., No. 99 (2000), pp ) Yao, T., Fujikubo, M., and Nie, C.: Buckling/ Plastic Behaviour of Plates Under Inplane Cyclic Loading, Proc. of Structural Dynamics-EUR- ODYN' 93, (Edt.) Moan, T. et al. (1993), Balkema, Rotterdam, The Netherlands. 21) Khedmati, M. R., Fujikubo, M., Yanagihara, D., and Yao, T.: Buckling/Plastic Collapse Behaviour and Strength of Continuous Stiffened Plates Subjected to Combined Biaxial Thrust and Lateral Pressure, Trans. of the West-Japan Soc. Naval Arch., No. 100 (2000), in printing. 22) Rutherford, S. E., Caldwell, J. B.: Ultimate Longitudinal Strength of Ships : A Case Study, Trans. of SNAME, Vol. 98 (1990), pp