Progressive Collapse Analysis of a Ship's Hull under Longitudinal Bending

Transcription

1 (Read at the Autumn Meeting of The Society of Naval Architects of Japan, Nov. 1991) 449 Progressive Collapse Analysis of a Ship's Hull under Longitudinal Bending by Tetsuya Yao*, Member Plamenn Ivanov Nikolov**, Member Summary It is very important to estimate the load carrying capacity of a ship's hull as awhole from the viewpoints of safety and economy. For this purpose, a simple method was proposed to simulate the progressive collapse behaviour of a ship's hull girder subjected to longitudinal bending. In this method, the cross section of a hull girder is divided into small elements composed of a stiffener and attatched plating. For each stiffener element, the average stress-strain relationship under axial load is derived based on the equilibrium conditions of forces and moments. The buckling and yielding in both stiffener and plate are taken into account. Then, a step-by-step increase of vertical curvature is applied to the hull girder assuming that the plane cross section remains plane. At each incremental step, the tangential flexural rigidity of the cross section is evaluated using the tangential slope of the average stress-strain curves of the elements as well as the incremental bending moment due to the curvature increment. Performing sample calculations on existing girder models tested under pure bending, the rationality of the proposed method was examined. Then, the analysis was performed on an existing bulk carrier, and the progressive collapse behaviour of the cross section under bending load was discussed. It was found that the full plastic bending moment can not be attained due to buckling of the deck plates under sagging condition and that of the bottom and inner bottom plates under hogging condition. It was also found that the maximum bending moment carried by the cross section under sagging condition is 20% lower than that under hogging condition. 1. Introduction A ship's hull is a typical box girder structure, and is subjected to longitudinal bending loads such as distributed weights, buoyancy forces and wave loads. To design a ship's hull withstanding these load, some allowable stresses have been introduced in a structural design so that the deck and/or bottom platings do not undergo buckling/plastic collapse. From this point of view, many research works have been performed on buckling/plastic collapse strength of isolated plates and stiffened plates both theoretically and experimentally, and useful results have been obtained. At the same time, it is very important to know the load carrying capacity of a ship's hull as a whole from the viewpoints of safety and economy. Such information may fundamentally be obtained by performing Finite Element Analysis on a ship's hull, considering both geometrical and material nonlinearities1,2). However, such analysis requires large amounts of effort, money, computational time and computer memory. So, * Faculty of Engineering, Hiroshima University ** Structural Engineering Course, Graduate School of Hiroshima University some simplified method is required. As far as the maximum bending moment carried at a girder cross section is concerned, simple methods were proposed by Nishihara3) and Endo et. al.4). In their methods, however, the reduction in load carrying capacity of each structural component after the ultimate strength is not considered, and thereby the progressive collapse behaviour was not simulated, although the ultimate bending moment was evaluated with good accuracy. On the other hand, Ueda, Rashed and Paik5) developed efficient elements for plates and stiffened plates under bi-axial tensile and/or compressive and shear loads. In their method, a stiffended panel surrounded by longitudinal girders is considered as one structural unit, and the stiffness matrix is derived for this unit considering the influences of buckling and plastic behaviour of the element. This work was done in the frame work of the Idealized Structural Unit Methods6). Applying the improved unit, Paik7)8) simulated the collapse behaviour of ship's hull girders subjected to longitudinal bending moment. Smith et. al.9-13) proposed another simple but efficient method to analyze the collapse behaviour of box girder structures under longitudinal bending load. Smith's method is fundamentally the development of that

2 450 Journal of The Society of Naval Architects of Japan, Vol. 170 proposed by Caldwell"), but it takes into account of the progressive loss in stiffness of a cross section due to buckling and yielding of structural components. Smith described the procedure of his method as follows9). ( 1 ) The hull girder cross section is divided into small elements composed of plates and stiffeners. ( 2 ) Vertical curvature of the hull girder is assumed to occur incrementally. The corresponding incremental element strains are calculated on the assumption that the plane section remains plane and that the bending occurs about the instantaneous neutral axis of the cross section. ( 3 ) Incremental element stresses are derived from incremental strains using the slopes of stress-strain curves. ( 4 ) Incremental element stresses are integrated over the cross section to obtain the bending moment increments. Incremental curvatures and bending moments are summed to provide cumulative values. His method may be regarded as a rather simple method comparing to that proposed by Ueda et. al.5-8), but easily gives solutions accurate enough under a simple loading condition of pure bending. The only difficulty in the Smith's method is the derivation of the stress-strain relationships of component elements taking into account of the buckling and yielding. To obtain such stress-strain relationships, Smith suggested to perform elasto-plastic large deflection analysis by the Finite Element Method. Such analysis, however, may require much works especially when the number of elements is large. In the present paper, such stress-srain relationships were derived in an analytical manner. Then, according to the Smith's method, a computer program "HULLST" was developed to analyze the progressive collapse behaviour of the cross section of a box girder type structure subjected to longitudinal bending load. After performing some sample calculations on the existing box girder models to examine the rationality of the proposed method, collapse behaviour of the cross section of an existing bulk carrier of 60,000 tons DWT was analyzed both under sagging and hogging conditions. 2. Method of Analysis 2. 1 General Assumptions The longitudinal stiffening system is usually employed in larger ships at their midlength parts, and the deck, the bottom and the side plating are stiffened by a number of longitudinal stiffeners and girders. If an extreme longitudinal bending load acts on a hull girder, the possible collapse modes of its cross section may be : ( 1 ) local collapse of plates between stiffeners ; ( 2 ) local collapse of stiffeners ; ( 3 ) overall collapse of longitudinally stiffened plate between longitudinal girders and transverse frames ; ( 4 ) overall collapse of orthogonally stiffened plates between transverse bulkheads. For the existing scantling of ship structures, however, the most probable collapse mode may be the overall collapse of stiffened plate after local collapse of the individual plates between stiffeners. In the following, assuming such collapse mode, a simple method is proposed to simulate progressive collapse behaviour of a ship's hull girder under an extreme load. Firstly, the average stress-strain relationship of an isolated plate is derived based on the results of elastic large deflection analysis and plastic mechanism analysis. In the elastic analysis, analytical method was applied taking into account of the influences of initial deflection and welding residual stresses. Then, the collapse of a stiffened plate is treated as that of stiffeners with flanges corresponding to the plating between stiffeners. Their average stress-strain relationships are derived considering the equilibrium conditions of forces and moments. In this derivation, the average stressstrain relationships obtained for plates previously are used to represent the characteristics of the attached flanges corresponding to the plating. After evaluating the average stress-strain relationships of all the component elements in the cross section, the progressive collapse behaviour of a box girder under pure bending is simulated fundamentally following the Smith's method Average Stress-Strain Relationships of Isolated Plates Initial imperfections due to welding and deflection mode under thrust When a longitudinal stiffening system is employed, the dominant compressive and/or tensile load act on the plate in the direction of its longer edge as indicated in Fig. 1. All edges are assumed to be simply supported and remain straight while subjected to inplane movements. The plates in a ship structure usually contain welding residual stresses and initial deflection associated by fillet welding of stiffeners to the plating. In this paper, a rectangular distribution of welding residual stresses is assumed considering the continuity condition of plates. Considering the self equilibrium condition of the residual stresses shown in Fig. 1, the following equation is obtained. ( 1 ) The initial deflection in general may be expressed in Fig. 1 Rectangular plate with initial imperfections due to welding

3 Progressive Collapse Analysis of a Ship's Hull under Longitudinal Bendin 451 the form") : ( 2 ) and the total deflection under inplane compressive load as : ( 3 ) However, it is known that only a single deflection component is amplified as the compressive load increases above the buckling load while other components become negligibly small until secondary buckling takes place"). As a result, the behaviour of the plate may be approximately represented by taking single deflection modes, Aom and Am in Eqs. ( 2 ) and ( 3 ), respectively. m is equal to or greater than the number of half waves in buckling mode, and is taken as : m=1 for a/b< 1.3 m=n for n-0.7 alb< n+0.3 where a/b is the aspect ratio of the plate, and n is an integer greater than 1. Hereafter, A0m and Am are denoted as A0 and A, respectively Relationship between average stress and deflection Performing elastic large deflection analysis, the following relationship is derived between average compressive stress and deflection"). two sets of plastic mechanism may exist as illustrated in Fig. 2. For each mechanism, the following relationships are derived"). (7) ( 8 ) ( 9 ) The average stress-deflection relationship terminates from that expressed by Eq. ( 4 ) to that by Eq. ( 7 ) or Eq. ( 8 ) at their intersecting point. The intersecting point gives the compresive ultimate strength Relationship between average stress and average strain The average stress-strain relationship according to the elastic large deflection analysis is expressed as") : (10) On the other hand, that according to the plastic mechanism analysis is expresed as : (11) (12) where a= a/bm and ( 4 ) ( 5 ) ( 6 ) The relationship between average compressive stress and deflection is derived also according to the plastic mechanism analysis assuming rigid-perfectly plastic material. Depending on the aspect ratio a/b of the plate, Until the compressive ultimate strength is attained, the average stress-strain relationship follows Eq. (10). At the ultimate strength, the strain is assumed to change from the value evaluated by Eq. (10) to that by Eq. (11) or Eq. (12). After this, the average stress-strain relationship follows Eq. (11) or Eq. (12). Here, when a plate is thick and undergoes plastic buckling, the plastic inplane strain until buckling is not considered, since the first term in Eq. (10), (11) or (12) represents the elastic inplane strain, and the second term in these equations the strain produced by lateral deflection. In this case, the constant strain increment evaluated as the difference between the elastically evaluated buckling strain and the yield strain is added to the strain after the ultimate strength has been attained. In the tension range, the stress-strain relationship of an elastic-perfetly plastic material was assumed as the average stress-strain relationship. When there exist welding residual stresses as illustrated in Fig. 1, the compressive load in not carried at the central part of the plate after this part has been yielded in compression under applied compressive load. In this case, the compressive load is carried only at the edge parts where residual stresses are in tension, and the tangential inplane rigidity may be expressed as Fg. 2 Plastic mechanism of plate under thrust 2btE/b. When the ultimate strength is greater than ƒðƒá the tangential inplane rigidity where average

4 452 Journal of The Society of Naval Architects of Japan, Vol. 170 compressive stress is greater than ƒðƒá - ƒðc is set equal to 2btE/b. On the other hand, when tensile load is applied, the inplane rigidity is expressed as : (13) The influences of residual stresses both in tensile and compressive ranges are schematically show in Fig Accuracy of the proposed stress-strain relationship To examine the accuracy of the average stress-strain relationships of plates proposed here, a series of elastoplastic large deflection analysis by the Finite Element Method was performed. The calculated relationships for a plate of a/b =0.8 are plotted in Fig. 4 ( a ) together with that evaluated by Eq. (7). It is seen that the deflection evaluated by the plastic mechanism analysis is smaller than those by the FEM analysis. This may be partly because the elastic deformation is ignored in the plastic mechanism analysis. Although more precise discussion is necessary about this, the deflection obtained by Eq. ( 7 ) or ( 8 ) was modified by dividing it by ƒð to get better agreement with that by the FEM analysis. The modified deflection is plotted by a dashed line with in Fig. 4 ( a ). The stress-strain relationships are also compared in Fig. 3 Influences of residual stresses on plate behaviour under tensile and compressive inplane loads Fig. 4 ( b ). When strain was evaluated by Eq. ( 11 ), the modified deflection mentioned above was used. Good correlations are observed between results of the FEM analysis and the present method Average Stress-Strain Relationships of Stiffeners with Plating General assumptions In the present method, a stiffened plate is treated as an assemblage of stiffeners with attached strips representing the plating. A typical cross section of the stiffener element is shown in Fig. 5( a ). When this element is subjected to axial compression, it may deflect as illustrated in Fig. 5( b). In this case, three types of modeling may be posible, which are two single span models, AB and BC, and a double span model, 12. Among them, the double span model may be most ( a ) Average stress-deflection relationships rational and only this model is considered here. To derive the average stress-strain relationship of a stiffener element, the following assumptions were made. ( 1 ) Attached strips behave according to the average stress-strain relationships derived in Section ( 2 ) Plane cross section remains plane and the strain varies linearly over it. ( 3 ) The material is assumed to be elastic-perfectly plastic. ( 4 ) The deformation in torsional buckling mode is not considered. Firstly, the average stress-strain relationship in the compression range is derived. Then, that in tension range is considered Assumed deflection mode The element is assumed to be initially deflected along its span in the following mode : ( b ) Average stress-strain relationships Fig. 4 Comparison of the calculated results by the present method and FEM (Plate)

5 Progressive Collapse Analysis of a Ship's Hull under Longitudinal Bending 453 ( a ) Cross section ( b ) Deflection mode under axial compression (22) Deflection and curvature in a single span range are shown in Fig. 5( c ) Axial force and bending moment in crosssection On the other hand, the strain and stress distributions in the cross section at both ends of the element take one of the patterns indicated in Fig. 6. For each cross section, the axial force and bending moment are evaluated by integrating the stresses in Fig. 6. If unloading is detected in the plate, the stress-strain relationship is assumed to follow the unloading path Equilibrium conditions Neglecting the influences of reaction forces at the supporting point, the following equilibrium conditions of forces and moments have to be satisfied : (23) (24) where P1,_P2,M1,--2and ĉ are indicated in Fig. 7. P1,M Here, if the average stress-strain relationships derived at Section 2.2 are used for flanges, the effective flexural rigidities at Cross Sections 1 and 2 differ with each other, since the strains in the flange at each cross section are different. Consequently, the point where the bending moment becomes zero moves along the span from the original supporting point with the increase in an axial force. Denoting the distance between Cross ( c ) Elastic and plastic components of deflection and curvature (14) The total deflection under axial compression is expressed as the sum of the elastic and the plastic components' 8). (15) The elastic component is assumed to be in the same mode with initial deflection as follows : (16) According to Ref. 19), the elastic component is evaluated as : (17) On the other hand, the plastic component is expressed as") : Fig. 5 Stiffener element (18) (19) From the assumed deflection modes, the total curvature K and it's elastic and plastic components at the mid-span point are expressed as : (20) (21) Section 1 and the zero bending moment point as a1/2, and that between Cross Section 2 and the zero bending moment point as a2/2, the elastic buckling loads for each span are expressed as follows. (25) where EL and Ell are the flexural rigidities at Cross Sections 1 and 2, respectively. In deriving Eq. (25), it is assumed that effective flexural rigidities at Cross Sections 1 and 2 are uniformly distributed along the corresponding spans. In evaluating EL and E12, the reduction in rigidity of plates is taken into acount. The lengths of spans a1 and a2 are evaluated from the following conditions : and are given as : where (26) (27) (28) The deflection along each span and the curvatures at Cross Sections 1 and 2 are evaluated according to Eqs. (14) through (19) substituting al or a2 into a. When both spans are elastic, the ratio of the curvature at Cross Sectins 1 and 2 is expressed as : (29) Relationship between average stress and average strain After the equilibrium conditions are satisfied, the axial compressive strain is evaluated by the following equation.

6 454 Journal of The Society of Naval Architects of Japan, Vol. 170 CROSS SECTION 1 CROSS SECTION 2 Fig. 6 Possible distributions of strain and stress at cross sections 1 and 2 Fig. 7 Forces and moments acting on both ends of stiffener element where (30) F represents the total sectional area, e is the position of original neutral axes and dz is the current zero stress point and the subscript i shows the corresponding span or cross section. In the tensile range, the average stress-strain relationship is assumed to follow that of the elastic-perfectly plastic material Procedure to derive average stress-strain relationship The average stress-strain relationship is derived according to the flow in Fig. 8, which may be described as follows. ( 1 ) At the first step, a1 and a2 are set equal to a. ( 2 ) The curvature K1 at Cross Section 1 is firstly given increasing it incrementally step by step. Fig. 8 Flow chart to derive average stress-strain relationship of stiffener element

7 Progressive Collapse Analysis of a Ship's Hull under Longitudinal Bendin 455 ( 3 ) At Cross Section 1, the zero stress point d1 is given, and the axial force Pt and the bending moment M1 are evaluated against the prescribed K1 and d1. ( 4 ) The curvature at Cross Section 2 is evaluated according to Eq. (29). ( 5 ) At Cross Section 2, the zero stress point d2 is given incrementally until the axial force P2 becomes equal to P1. ( 6 ) If the continuity condition of the slope at the supporting point is not satisfied, the curvature at Cross Section 2 is modified from that by Eq. (29). Then return to step ( 4 ). If it is satisfied move to the next step. where Di and F, are effective tangential modulus and sectional area of the i-th element ;yi and zi are horizontal and vertical coordinates of the i-th element with respect to the reference coordinate system. Di is evaluated from the slope of the stress-strain curve for a specified strain. When unloading occures, that of the unloading path is used. yg and zg are the horizontal and the vertical coordinates of the instantaneous neutral axes for horizontal and vertical bending, respectively. They are calculated by the following equations. (33) ( 7 ) The elastic and plastic components of deflection at Cross Sections 1 and 2 are evaluated by Eqs. (17) and (19). ( 8 ) The equilibrium condition of moment is examined. If the condition is not satisfied return to step In the analysis, the strains in the elements are firstly calculated for the specified curvature. Then, the corresponding effective tangential moduli of the elements are ( 3 ). If it is satisfied, proceed to the next step. ( 8 ) The axial strain is evaluated by Eq. (30). ( 9 ) Ĉ and the span lengths a1 and a2 are evaluated as well as the lengths of the yielded zone ap1 and ap2 at each span, and return to step ( 2 ). To evaluate the length of the yielded zone, the same method was applied as that in Ref. 18) Accuracy of the proposed average stressstrain relationships Elasto-plastic large deflection analyses were performed to examine the accuracy of the average stressstrain relationships by the proposed method. The shaded part in Fig. 9 was analyzed with the dimensions and material properties indicated in the same figure. The plate part was divided into 10 ~ 30 elements in the transvere and longitudinal directions, respectively. The calculated average stress-strain relationships are compared with those by the proposed method in Fig. 10. Fig. 9 Stiffened plate for elasto-plastic large deflection analysis Although there exist some differences between both results near the ultimate strength in case of a thin plate, it may be concluded that the predicted average stressstrain relationship is accurate enough Progressive Collapse Analysis of Cross Section Subjected to Pure Bending According to the Smith's paper", the equilibrium equation of the cross section of a girder subjected to bi-axial longitudinal bending is expressed in the following form. (31) DH, D, and DHV are the tangential rigidities of the cross section, and are evaluated by the following equations. ( a ) Thin stiffened plate ( b ) Thick stiffened plate (32) Fig. 10 Comparison of the calculated results by the present method and FEM (Stiffened plate)

8 456 Journal of The Society of Naval Architects of Japan, Vol. 170 evaluated, and the instantaneous neutral axes are obtained by Eqs. (33). After this, the flexural rigidity of the cross section are calculated by Eqs. (32). Here, prescribing the increments of horizontal and vertical curvatures, the increments of corresponding bending moments are calculated by Eq. (31). The stress increments are also evaluated using the stress-strain relationships of the elements. Then, adding the increments to the previous values, the cumulative values of curvatures and bending moments of the cross-section are evaluated as well as those of stresses and strains of all the elements. The influences of both buckling and yielding are taken into account automatically by employing the stressstrain relationship derived in Section The computer program "HULLST" is developed to perform such analysis. 3. Numerical Calculations and Discussions 3. 1 Calculations on Box Girder Models To examine the rationality of the proposed method, a series of analysis was performed on existing box girder models subjected to longitudinal bending load. Four specimen are chosen for analysis. They are Model 23 by Reckling"), Model 4 by Dowling et. al.") and Models MST-3, MSD-S and MSD-H by Nishihara3). The former two models were analyzed also in Dow's paper"). The principal dimensions and the material properties are summarized in Table 1 together with the initial imperfections used in the analysis.those with * are the measured results Reckling's Model 23 Reckline carried out collapse test on seven steel girder models under pure bending. Model 23 has the largest number of stiffeners, and is most similar to ship's hull girder. This is the reason why Dow chose this model for analysis. From the same reason, Model 23 was chosen also in this paper, and the results of analysis are shown in Figs. 11 ( a) and ( b ), where average stress-strain relationships for typical elements and the moment-curvature relationships of the girder are plotted, respectively. In Fig. 11 ( a ), the solid and the chain lines represent the results by the present method and by Dow, respectively, and the dashed line by the elasto- plastic large deflection analysis by the FEM similar to that in Section In the Dow's calculation, the average stress-strain relationship was derived by performing elasto-plastic large deflection analysis by FEM, but plate part was treated as a single fiber of the beamcolumn element of which stress-strain relationship follows that of a plate derived before this analysis. This must be the main reason why there exists some differences between the results of the two FEM analyses. The average stress-strain relationship obtained by the present method shows good correlation with that by the present FEM analysis. In Fig. 11 ( b ), the solid and chain lines represent the calculated results by the present method and by Dow, ( a ) Element stress-strain relationships ( b ) Moment-curvature relationships Fig. 11 Results of analysis for Reckling's Model 23 and the dashed line the experimental result. Experimentally, this specimen failed by compressive flange buckling. The final collapse occurred by nearly simultaneous local buckling of the compression flange plating between the stiffeners and overall buckling of the whole compression flange. The calculated collapse behaviour was almost the same as that of the test model, although some differences are observed in the measured and calculated rigidities. In the present and Dow's analyses, one stiffener element located at each corner was treated as a hard corner element of which stress-strain relationship follows that of a elastic-perfectly plastic material both in tensile and compressive ranges. Dow performed an alternative analysis considering two stiffeners at each corner as hard corner elements, and almost the same ultimate strength was obtained with the measured one Dowling's Model 4 Dowling et. al." carried out collapse test on six steel girder models with three spans, in which three were tested under three point bending and the other three under four point bending. The analysis was performed on Model 4 tested under pure bending having the largest number of stiffeners, and the results are plotted in Figs. 12 ( a) and ( b ) In this analysis, the welding residual stresses and the initial deflections shown in Table 1 were assumed. The observed failure mode was the compression flange buckling. Also this model was analyzed by Dow. In Fig. 12 ( a ), the average stress-strain relationships are plotted. There exist large differences between stress-strain relationships by the present method and by Dow. According to Fig. 12 ( a ), the tension flange is much stronger than the compression flange. However, looking over the dimensions and the material properties in Table 1, there can not be so much differences. The stress-strain relationships by the present method seems to be more rational. Nevertheless, the

9 Progressive Collapse Analysis of a Ship's Hull under Longitudinal Bendin 457 Table 1 Principal dimensions, material properties and welding imperfections of girder models moment-curvature relationships by the present method and by Dow's method plotted in Fg. 12 ( b) do not show so much differences. In the present analysis, two cases are considered regarding the hard corner effect. In one case, hard corner element was not introduced, and in another case, the four corners were regarded as hard corner elements. By introducing hard corner elements, the rigidity and ultimate strength increased. Dow also performed an alternative analysis considering two stiffener elements from each corner as hard corner elements, and got a higher ultimate strength than measured result. ( a ) Element stress-strain relationships ( b ) Moment-curvature relationships Fig. 12 Results of analysis for Dowling's Model Nishihara's Models MST-3, MSD-S and MSD -H Nishihara3) also carried out collapse test on nine steel girder models with one span under four point bending. The analysis was performed on Models MST-3, MSD- S and MSD-H. He described in his paper that the welding residual stresses were not removed after fabricating the models. However, no measurement was performed on welding residual stresses and initial deflection. In the present analysis, the initial imperfection shown in Table 1 were used. An alternative analysis was also performed without considering welding residual stresses. Figure 13 ( a) shows the average stress-strain relationship of the element, where solid and chain lines represent those with and without considerig residual sresses. On the other hand, the dashed line represents the stress-strain relationship without residual stresses obtained by the FEM analysis. Good correlations are observed between the results by the present method and the FEM. It is also known that the ultimate strength is reduced by 10% of the yield stress due to the assumed welding residual stresses. Figure 13 ( b) shows the measured and calculated moment-deflection relationships. It is known that the assumed welding residual stresses reduce the ultimate bending moment about 4% under sagging condition. With the welding residual stresses of 2bt/b=0.2, the ultimate bending moment is well predicted. Contrary to this, the accuracy of deflection is not so good. This may be attributed to the difference in number of spans. The test was carried out on a single span model, whereas the proposed model is a double span model applicable to a

10 458 Journal of The Society of Naval Architects of Japan, Vol. 170 ( a ) Element stress-strain relationships ( b ) Moment-curvature relationships Fig. 13 Results of Analysis for Nishihara's Models MST-3, MSD-S and MSD-H hull girder of an existing bulk carrier of 60,000 DWT. The principal dimensions are shown in Fig. 14. HT36 steel was used at the deck plate and the upper part of the side shell, and HT32 steel in the remaining part. In the analysis, the welding residual stresses of 2bt/b =0.1 was assumed in the plates between stiffeners, and the initial deflections of A0/t =0.01 and ĉ0/a =0.002 in the plates and stiffeners, respectively. At the corner parts of the cross section, hard corner elements were introduced. For some typical elements, the average stress-strain relationships are shown in Fig. 15 ( a ). Both hogging and sagging moments were applied, and the results are plotted in Fig. 15 ( b ). The stress distribution in the cross section before, at and after the ultimate strength are also shown in Figs. 16 ( a) and ( b) under hogging and sagging conditions, respectively. In Fig. 15 ( b), the dashed line represent the results when all elements are assumed to follow the stressstrain relationship of an elastic-perfectly plastic material. In this case, the same results are obtained under Fig. 14 Principal dimensions of cross section of bulk carrier of 60,000DWT multi-span girder. The differences between the actual and the assumed initial imperfections may be the other factors. Through the results of analyses on five girder models, it may be concluded that the present method of analysis is applicable to the progressive collapse analysis of a box girder under pure bending Progressive Collapse Analysis on Existing Bulk-Carrier Progressive collapse analysis was performed on the sagging and hogging conditions, and the ultimate strength is equal to the full plastic bending moment of the cross section. On the other hand, solid and chain lines represent the results under hogging and sagging conditions using the average stress-strain relationships in Fig. 15 ( a) obtained by the present method. In case of a hogging condition, the local collapse firstly took place at the deck plate by yielding in tension. Then, after all the panels composing a top side tank had been yielded, the bottom plate buckled. With further increase in applied curvature, the buckling took place also at the inner bottom plate, and the ultimate strength was attained. The stress distribution at this state is plotted by solid line in Fig. 16 ( a ). It is seen that

11 Progressive Collapse Analysis of a Ship's Hull under Longitudinal Bendin 459 ( a ) Element stress-strain relationships ( a ) Hogging condition ( b ) Moment-curvature relationships Fig. 15 Progressive collapse behaviour of cross section of bulk carrier under longitudinal bending the neutral axis of the cross section moved a little downward. This is because the effectiveness of the upper part of the cross section decreased due to yielding. However, after the ultimate strength had been attained, the load carrying capacities at the buckled bottom and inner bottom plates decreased, and the neutral axis moved upward as indicated by the dashed line in Fig. 16 ( a ). In case of a sagging condition, the local buckling firstly took place at the upper part of the bottom of a top side tank, and the buckling of the deck plate near the hatch side followed. After this, buckled part spreads at both the deck and the bottom of a top side tank from the center to the side. When buckling took place at all the deck plate, half of the bottom of the top side tank and upper part of the side shell, the ultimate strength was attained. The reserve strength after the first local buckling was very small comparing to the case of ( b ) Sagging condition Fig. 16 Change in stress distributions during progressive collapse hogging condition. This is because the stresses at the buckled part in compression decrease whereas that of yielded part in tension do not decrease. With furher increase in applied curvature, the neutral axis rapidly moved downward, and the middle part of the side shell came to the compression side of bending. The dashed line in Fig. 16 ( b) shows the stress distribution after buckling had occured at this part. The yielding did not take place at the bottom plate in tension side within the curvature applied here. 4. Conclusions A simple method was proposed to analyze the pro-

12 460 Journal of The Society of Naval Architects of Japan, Vol. 170 gressive collapse behaviour of a ship's hull girder under longitudinal bending load. This method takes into account of the progressive loss in rigidity due to the occurrence of local buckling and yielding. The fundamental idea of this method was firstly proposed by Caldwell14) and followed by Smith9-13). According to Smith, the cross section is divided into small elements composed of plates and stiffeners. In this paper, simple analytical method was proposed to derive the average stress-strain relationships of elements. The proposed method is as follows : ( 1 ) A stiffener with a flange corresponding to plates between stiffeners is considered as one element. Adjacent two mid-span points between transverse frames are taken as the both ends of an element. ( 2 ) The average stress-strain relationship of the attached plate is firstly derived combining the elastic large deflection analysis and plastic mechanism analysis both in analytical forms. ( 3 ) In the elastic range, deflection mode in a sine half wave is assumed. After yielding starts, plastic deflection component is introduced which gives constant curvature in the yielded zone near the mid-span point. ( 4 ) Giving the curvature incrementally, and assuming a linear distribution of strain over the cross section at both ends, deflection and stress distribution satisfying the equilibrium conditions of forces and moments are determined. At the plate part, the average stress-strain relationships of plates are used, and the effective rigidity is evaluated. ( 5 ) Axial strain is evaluated with the axial force and the deflection. This method was implemented into the computer code "HULLST" to analyze the progressive collapse behaviour of a ship's hull girder under longitudinal bending. Sample calculations were performed on five existing girder models, and the rationality of the proposed method was demonstrated. Then, progressive collapse analysis was performed on the cross section of an existing bulk carrier under pure bending. It was shown that the cross section can not carry the full plastic bending moment due to the occurrence of local buckling in the compression side of bending. More decrease was observed in ultimate strength under sagging condition than hogging condition. In the present paper, only the axial force was considered as the force acting on stiffener elements. In the actual case, the influences of lateral force acting at the bottom plate and shear force acting at the side shell have to be also considered. The assumed deflection was only that of the Eulerian buckling mode. The introduction of deflection component in torsional buckling mode is also important. These remain as future works. References 1) Chen, Y. -K., Kutt, L. M., Piaszczyk, C. M. and Bieniek, M. P. : "Ultimate Strength of Ship Structures", Trans. SNAME, Vol. 91 (1983), pp ) Kutt, L. M., Piaszczyk, "Evaluation of the C. M. and Chen, Y. -K. : Longitudinal Ultimate Strength of Various Ship Hull Configurations", Trans. SNAME, Vol. 93 (1985), pp ) Nishihara, S. : "Analysis of Ultimate strength of Stiffened Rectangular Plate (4th Report) On the Ultimate Bending Moment of Ship Hull Girder", J. Soc. Naval Arch. of Japan, Vol. 154, pp (in Japanese). 4) Endo, H., Tanaka, Y., Aoki, G., Inoue, H. and Yamamoto, Y. : "Longitudinal Strength of the Fore Body of Ships Suffering from Slamming", J. Soc. Naval Arch. of Japan, Vol. 163 (1988), pp (in Japanese). 5) Ueda, Y., Rashed, S. M. H. and Paik, J. K. : "Plate and Stiffened Plate Units of The Idealized Structural Unit Method (1st Report) Under Inplane Loading", J. Soc. Naval Arch. of Japan, Vol. 156 (1984), pp (in Japanese). 6) Ueda, Y. and Rashed, S. M. 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13 Progressive Collapse Analysis of a Ship's Hull under Longitudinal Bending 461 Vol. 5, No. 4 (1985), pp ) Yao, T. and Nikolov, P. I.: "Stiffness of Plates after Buckling", Trans. Kansai Soc. Naval Arch., Vol. 215 (1991), pp ) Okada, H., Ohshima, K. and Fukumoto, Y.: "Compressive Strength of Long Rectangular Plates under Hydrostatic Pressure", J. Soc. Naval Arch. of Japan, Vol. 146 (1979), pp (in Japanese). 18) Yao, T., Fujikubo, M., Bai, Y., Nawata, T. and Tamehiro, M.: "Local Buckling of Brancing Members in Semi-submersible Drilling Unit (1st Report)", J. Soc. Naval Arch. of Japan, Vol. 160 (1986), pp (in Japanese). 19) Timoshenko, S. P. and Gere, J. M.: Theory of Elastic Stability, 2nd Ed., McGrow-Hill, ) Reckling, K. A.: "Behaviour of Box Girders under Bending and Shear", Proc. ISSC, Paris (1979), pp ) Dowling, P. J., Moolani, F. M. and Friez, P. A. : "The Effect of Shear Lag on the Ultimate Strength of Box Girder", Proc. Int. Cong. on Steel Plated Structures, London (1976), pp