OPTIMIZATION OF CORRUGATED SHELL PLATING FOR MARINE STRUCTURES

Transcription

1 Proceedings of the ASME th International Conference on Ocean, Offshore and Arctic Engineering OMAE011 June 19-4, 011, Rotterdam, The Netherlands OMAE OPTIMIZATION OF CORRUGATED SHELL PLATING FOR MARINE STRUCTURES Jonas W Ringsberg Dept of Shipping and Marine Technology Chalmers University of Technology SE Gothenburg, Sweden Jonas.Ringsberg@chalmers.se Hüseyin Sağlam Dept of Shipping and Marine Technology Chalmers University of Technology SE Gothenburg, Sweden saglm@hotmail.com Md Asaduzzaman Sarder Dept of Shipping and Marine Technology Chalmers University of Technology SE Gothenburg, Sweden russelasad003@hotmail.com Anders Ulfvarson Dept of Shipping and Marine Technology Chalmers University of Technology SE Gothenburg, Sweden Anders@Ulfvarson.se ABSTRACT The structural weight of a shell-plated structure can be reduced in numerous ways. The current investigation presents an example of innovative lightweight design of the pontoon of an offshore platform by utilization of corrugated structure. The corrugated shell plating is compared with a conventional stiffened panel with respect to strength, weight and cost. For this purpose, an optimization methodology is developed for shell-plated marine structures. The procedure enables the analysis/comparison of various (structural) solutions with regard to strength characteristics, weight and cost. Here, strength characteristics include ultimate tensile strength, buckling stability and fatigue life analyses. Linear elastic finite element analyses are carried out to generate input to the structural characteristics studies, which involve several design criteria according to classification rules. The results show that in competition with the traditional stiffened panel, the corrugated shell-plated structure can be used as the more lightweight design solution. It can be manufactured and installed at a lower cost. Finally, the structural strength characteristics analyses show that, when designed properly, classification rules are fulfilled without compromising with the safety margins. Keywords: optimization, corrugated panel, fatigue, buckling, ultimate limit state, lightweight design, cost, finite element analysis, parametric study. 1. INTRODUCTION Corrugated panels are often used in the bulkhead design to save weight, ease the construction and reduce maintenance costs. There are, however, still many parts of marine structures which could benefit from being substituted to corrugated panels instead of using the traditional stiffened panel structures. There are several examples in the literature where corrugated panels in marine applications have been studied from various aspects. Rahman [1] presents a panel form optimization useful for improvement of ship structures. In the study, two different types of stiffened panels are compared with a corrugated panel with respect to strength and production cost. Kippenes et al. [] present a study on ultimate strength of open corrugated bulkheads. Three different types of ships with corrugated bulkheads were modelled and compared with regard to ultimate strength. Caldwell [3] presents analyses of the strength of corrugated plating for bulkheads in ships. In Paik et al. [4], a theoretical and experimental study on the ultimate strength of corrugated bulkheads is presented where the objective was to obtain a theoretical background of strength validation of corrugated shell plates. Recent trends in the offshore industry show that the size, deck load capacity and structure weight of new offshore platforms are increasing. The increase in weight of the structure, however, is a design parameter that has a negative impact on, for example, the profit from running the platform. Hence, lightweight design and innovative design solutions which reduce the weight are sought for. The current 1 Copyright 011 by ASME

2 investigation is a contribution to a lightweight design solution applicable on shell-plated marine structures. The idea is to utilize corrugated shell plating in several parts of offshore marine structures. The current study focuses on the description of a procedure developed for optimization of corrugated shell plating for marine structures. It is presented and demonstrated here using the pontoon structure of a semi-submersible structure. A thorough description of the optimization procedure and all steps in its development is found in Sağlam and Sarder [5].. INVESTIGATION METHOD The investigation utilizes interactively two types of tools: the finite element method (FEM) and semi-empirical closed form expressions from the classification rules. The FEM is utilized for the transfer of load effects from the global level to the detailed level of structure, i.e. from the total semi-submersible to individual panels, while the semi-empirical expressions give well-defined constraints for the utilization of individual structural elements. These constraints are based on defined limits like bifurcation, yielding and ultimate limit state of panels, generalized with interaction formulas for considering stresses in different directions. The original structure composed of stiffened flat shell plates is challenged by the corrugation alternative. For both types, the optimum is searched for in a stepwise procedure. In the end, the best solutions found are verified with a standard procedure that identifies the most critical parts for final judgement. 3. DESCRIPTION OF THE REFERENCE STRUCTURE The pontoon structure studied is part of an ongoing design of a new semi-submersible production unit with the following main particulars: Deck load capacity: 33,000 metric tonnes Length (overall): 146 m Width (overall): 137 m Length of the pontoons: 105 m Draught: 41.5 m The initial (reference) design of the global pontoon structure is made of stiffened flat steel plates. For the purpose of the current study, it is only necessary to consider a small part of the pontoon structure in the numerical analyses. Hence, five sections in the middle of the pontoon structure are used in the model-making and analysis procedure to demonstrate the lightweight potential of using corrugated shell plating; see Sağlam and Sarder [5] for details. Figure 1 shows the geometry model of the reference structure with five sections. All structural elements that contribute to global stiffness in a structure analysis are incorporated in this reference structure model. Table 1 presents the material properties used in the numerical analyses together with the dimensions of the reference structure and some of its structural members. Thickness (mm): Fig. 1: Geometry model of the reference stiffened structure. Table 1. Material properties and reference structure dimensions used in the numerical analyses. Property Value Young s modulus, E [GPa] 06 Poisson s ratio, υ 0.3 Tensile yield stress, σ y0. [MPa] 355 Shear yield stress, k s [MPa] 05 Density, ρ [kg/m 3 ] 7850 Length [m] 1.8 Width [m] 17.3 Height [m] 11.5 Depth [m] 30.0 Stiffener spacing [mm] 640 Stiffeners (bulb type) [mm] Shell plate thickness [mm] between 19 and 3 4. INTRODUCTION TO STRENGTH ANALYSIS The strength analysis of the panel structures presented in this investigation is based on linear-elastic theory. The ultimate strength capacity of the reference and corrugated structures are calculated and used for comparison as a measure of merit in the lightweight design optimization. Ultimate strength is one of the design criteria of shellplated structures. Buckling and yielding are the dominating phenomena for attaining the ultimate strength limit when compressive stresses are significant. The ultimate strength limit is normally reached when yielding is spread to an extent with or without the occurrence of buckling. In the current investigation, buckling and yielding are incorporated as part of the design and optimization procedure. The methods used here to quantify the parameters related to buckling and yielding are obtained from the references ABS [6] and DNV [7]. The safety margin against yielding is checked by calculation of the von Mises effective stress, which is compared with the material s yield stress. The buckling strength assessment procedure should ultimately consider the following five explicit collapse scenarios: Overall collapse of corrugations as a unit, in which case the corrugations are not strong enough as a beam-column and buckle together with the plating as a unit; see Fig.. Panel failure by yielding along the edge under biaxial compression without any corrugation failure; see Fig. 3. As a secondary effect, panel collapse will occur Copyright 011 by ASME

3 Beam-column type collapse in which the ultimate strength is reached by yielding of the corrugation, first at mid-span and then at the supports; see Fig. 4. Local buckling of the corrugation in the web or flange where ultimate strength is reached when the web/flange buckles are subjected to local compressive stress; see Fig. 5. Gross yielding might take place when the panel is subjected to axial tensile load predominantly and the panel cross section yields before the global or local buckling of the panel takes place; see Fig. 6. Buckling occurs when the load effect of external compressive loads reaches a critical level. The corrugated shell-plated panel will be designed in such a way that the largest loads act in the longitudinal (corrugation) direction of the panel. The reason is that corrugated panels have less stiffness/resistance to buckling in the transverse (perpendicular to the corrugations) direction. In the current study, the design of the corrugated panel considers buckling due to compression, shear and lateral pressure loads or combinations of all three types of loads. Note that lateral loading and deflection reduce the in-plane rigidity of the structure, which, in turn, induces in-plane (membrane) stresses. Lateral loads also give rise to bending and in-plane stresses. Fig. : Overall collapse of corrugations as a unit. Fig. 3: Panel failure by yielding. Fig. 4: Beam-column type collapse. Fig. 5: Local buckling of the corrugation web or flange. Fig. 6: Gross yielding. It is described in Sections 5 to 10 how the buckling stresses are calculated by means of a parametric optimization approach. Several external operational load cases are superimposed to cover realistic loading and operation conditions of the semi-submersible s pontoon structure. Corrugation profiles are varied and evaluated together with a slenderness and panel aspect ratio. Johnson-Ostenfeld correction is applied to incorporate the influence of yielding on buckling behaviour and limit state. The calculated elastic buckling stress at bifurcation is thus converted to critical buckling stress. The procedure proposed here for strength analysis of corrugated panel structures is limited to three of the five collapse modes depicted in the Figs to 6; see Sağlam and Sarder [5] for details. Hence, the buckling and failure modes of the corrugated shell-plated structure presented in the current investigation are as follows: Local face/web plate buckling: in this buckling mode, different plate strips of the corrugation profile are checked for out-of-plane buckling. Column buckling of a unit corrugation: buckling of any unit corrugation of the corrugated panel is checked. Entire corrugation buckling (global buckling): check for global buckling, i.e. out-of-plane buckling of the entire panel. 5. THEORETICAL ASSESSMENT BY CLASS RULES: BUCKLING, ULTIMATE LIMIT STATE AND FATIGUE STRENGTH This section gives a very brief overview of the criteria used to check the strength capacity of conventional stiffened and corrugated shell-plated panel structures. The criteria are only mentioned briefly, without detailed explanation and motivations. Instead, interested readers are directed to ABS [6] or to Sağlam and Sarder [5]. The notation of the variables used in the equations follows the notations defined in ABS [6]; see also the notation list. Although the DNV rules are not 3 Copyright 011 by ASME

4 directly employed in the analysis, their design criteria for plates and corrugated bulkheads are considered. Local buckling assessment The buckling strength of the flange and web plate subjected to in-plane loads is assessed by Eq. (1). In Section 6, a discussion concerning strength utilization factors, η, is presented. x,max Cx y,max Cy C 1 Beam-column buckling assessment Unit corrugation is treated as beam-column where Eq. () describes the limit state that must be satisfied. Equation () describes the unit corrugation as a beam and it is evaluated with regard to resistance to beam-column buckling influenced by lateral loads, which is developed from the Perry- Robertsson formula. Note that flange efficiency is considered fully effective in the calculations. a Ca CB C m b 1 1 /( ( ) ) a E C Global buckling assessment Global buckling strength of the corrugated panel is assessed with respect to biaxial compression and edge shear, see Eq. (3); note that the influence from lateral loading conditions is disregarded. For the current reference structure and corrugation shell-plated structure, this buckling mode is not critical. x Gx y Gy G 1 Ultimate strength assessment Paik and Thayamballi [8] and offshore practice according to ABS [6] give clear descriptions how an ultimate strength check should be carried out for plate panels. The ultimate strength for a plate between stiffeners subjected to a combination of biaxial compression and edge shear during plane stress conditions has to satisfy Eq. (4). The criterion considers the ultimate strength in all directions of loading together. The slenderness ratio and length of the short plate edge have here been the critical parameters for the ultimate strength of the plate areas. x,max Ux y...,max Uy x,max Ux U y,max Uy 1... (1) () (3) (4) For design of a panel structure it is desired that the plates buckle before the ultimate strength of the panel is reached. This was checked for each of the loading directions separately, i.e. for longitudinal, transverse and shear loadings. In Eqs (5) to (7), an example for the longitudinal direction is presented. The maximum allowed compressive stress in this direction should be less or equal to the ultimate stress in the longitudinal direction multiplied by a strength utilization factor, η, see Table in Section 6: x, max Ux (5) The local buckling criterion in Eq. (1) gives: x, max Cx (6) The criterion for ultimate strength design requires that the ultimate stress in any direction must be larger than the corresponding buckling stress taking account also of the slenderness ratio, C x, see Eq. (7). Eqs (5) to (7) give the final expression in Eq. (8), which is the design criterion for buckling taking place before ultimate strength is reached in the longitudinal direction. Cx Ux C x 0 (7) x, max C x 0 (8) In the ultimate strength assessment procedure, yielding of the plates is checked when subjected to lateral loads and/or inplane loads. The ultimate strength of a plate subjected to uniform lateral pressure combined with in-plane stresses has to satisfy the relationship in Eq. (9). It incorporates the influence from lateral pressure and also the in-plane stresses as the von Mises effective stress, σ e according to Eq. (10). Note that the spacing of the corrugation, s, and the length of the short plate edge, t, have a significant influence on this limit state. t 0 a 1 1 e q u 4 (9) s 0 Yielding strength assessment The effective stress is compared with the von Mises yield criterion. The von Mises effective stress must be below the allowed ULS limit. In the current investigation, the influence from stress concentration is only considered in the fatigue analysis. For a plated structure, the von Mises effective stress is defined as: e ( x,max ) x,max y,max ( y,max ) 3 (10) The entire structure is assessed by calculation of effective stress in Eq. (10). The design criterion used here is that all 4 Copyright 011 by ASME

5 parts of the structure must have a stress level which is lower than 90% of the material s yield stress. This criterion is applied for the severe storm loading case, which is the most extreme loading condition with regard to ultimate limit strength; see Sections 6 and 9. Fatigue strength assessment Wave-load induced fatigue stress ranges are considered in a simplified fatigue analysis which follows the ABS rules [9]. A safety factor based on 3 30 = 90 years of operation was considered in the fatigue assessment. The numerical analysis presented accounts for mesh dependency in the fatigue assessment procedure, i.e. with regard to material properties, and the Palmgren-Miner damage accumulation rule is used. The fatigue calculation is based on the design wave approach which is described in ABS [9]. In total, 33 environmental load cases were identified and used in the analysis. The response from a 100-year winter storm is recalculated to a 30-year response period using a Weibull distribution as defined ABS [9]. In addition, the fatigue strength of the panel structure is characterized in terms of a maximum allowable stress range based on a fatigue design factor (FDF). If Δ = 1/FDF is the maximum allowable accumulated fatigue damage at the probability level corresponding to N R cycles, the stress range value that exceeds on average once every N cycles is: S R FDF N T R ln( N R ) m / m r 1, z 0 1, z rm A C (11) The allowable stress range according to the fatigue assessment must fulfil the SR SR criterion. Since δ is a function of S R, an iterative procedure is necessary in order to find the appropriate value of S using the 33 load cases. R 6. LIMIT STATES Limit states can be used as guidelines for how to calculate safety factors for loads and material parameters. In the current investigation, the design of the corrugated panel structure is governed by the ultimate limit state (ULS) and the fatigue limit state (FLS). The primary loading conditions considered here, which give input to design load levels and factors, are: Normal operation conditions service limit state (SLS): this condition defines the stresses due to environmental loading combined with dead and maximum real loads relevant to the function and operations of the structure [6]. Severe storm ultimate limit state (ULS): this condition defines the same stresses as in normal operation conditions but during a design environmental condition using the 100- year extreme storm condition wave spectrum [6]. In Table, the strength factors for different loading conditions are presented. They were calculated as follows: a finite element analysis (FEA) of the reference structure was carried out using the DNV software SESAM [10]. The loading conditions representative for a severe storm were used in the analyses. The weakest part of this structure was searched for and identified. This part of the structure was then used for the analysis in the proceeding development and optimization procedure of the corrugated panel structure, see Sağlam and Sarder for details [5]. The FEA clearly showed that the weakest part with the largest stresses of the reference structure was a part of the top shell plating. Hence, only this part of the pontoon structure was employed to challenge with the corrugation shell-plated panel structure. Table. Load condition and buckling/yield utilization factors. Instead of using a safety factor against a maximum strength, a maximum strength utilization factor, η, was employed, which was defined as the inverse of the safety factor. The value of η depends on the loading condition, type of structural component and the consequence from failure [6]. Furthermore, in the parametric study presented in Section 10, different values on η are used for different loading situations. For normal operation conditions, η = 0.60ψ, while for severe storms, η = 0.80ψ, see [6]. Here, ψ is an adjustment factor which depends on the type of loads applied on the structure (tension, compression) and structural member under consideration. The design loads considered here for the offshore structure are based on a 100-year return period design philosophy. Thus, the allowable utilization factors are always less than STRENGTH ASSESSMENT PROCEDURE This section gives a brief overview of the strength assessment procedure used to optimize the corrugated panel structure, see Fig. 7. The numerical analyses are carried out using the DNV software SESAM [10]. A global model of the semi-submersible under consideration was used in hydrodynamic simulations to generate load histories necessary for the structural analysis of the pontoon structure. In order to lower the computational cost in the detailed structure analysis of the pontoon structure, only a smaller part of it was modelled with all detail structures. Thus, a section far off from the pontoon ends (boundaries) was chosen and it was coupled to the global model using submodelling tools available in SESAM [10]. This part of the structure was selected because it is has the general structural characteristics of the pontoon, and hence, it was considered 5 Copyright 011 by ASME

6 representative for the analyses presented in the current investigation. Convergence and sensitivity analyses with regard to model size and mesh density are carried out. The top plating of the middle section between the transverse web frames are analyzed more thoroughly with respect to resolution of stresses in the model. First, this was carried out using the reference structure model, see Section 3. Thereafter, a model of the corrugated panel structure was developed for comparison. The geometry of the corrugated panel structure was parameterised to enable a parametric study. This study was only carried out for a part of the top plating of the pontoon. The parameters describing the corrugation geometry were varied systematically with the objective to find the most lightweight solution while satisfying the ABS rules for buckling and ultimate strength of corrugated panels, see Section 5. The structural characteristics and response (stresses) in the corrugated panel structure models, developed during the iterative parametric study, are all evaluated by FEA. The iteration procedure was interrupted when the variations in parameters between iterations show minor changes in weight, buckling and ULS. Subsequently, a fatigue analysis was carried out followed by cost-benefit analysis. 8. FEA OF THE CORRUGATED STRUCTURE The analysis of the corrugated structure comprises four major steps: development of an FE model, FE analysis, postprocessing of stresses, and finally, strength assessment according to the procedure presented in Section 5 according to ABS rules [6, 11]. The geometry of the first (local) model of the corrugated structure, see Fig. 8, was extracted directly from the global model and used as a basis; the position and size of this model was discussed in Section 6. In addition, the selected part of the pontoon was modelled in the SESAM module Prefix [10]. The boundary conditions are assigned as prescribed displacements, see Fig. 9, to couple them to the displacements and motions of the global model s characteristics using SUBMOD in SESAM [10]. The structural plating was represented by 8-node quadratic shell elements, except for a few triangular areas with 6-node elements. Additionally, hydrostatic pressure loads representing a design draught of 30 metres are applied with a normal pressure distribution to all surfaces to match the load pattern acting on the global model. Mesh convergence analysis was carried out, which resulted in a 100 mm mesh size in the top plating and 00 mm in the other parts of the sections; see Sağlam and Sarder [5] for details. The total number of elements that form the FE model was 16,660. Finally, in the parametric study, the results obtained from the FE analysis are taken at a sufficient distance from the boundaries in the midsection, see Fig. 10. Fig. 8: Geometry model of a part of the corrugated structure. Fig. 9: Displacement boundary conditions applied on the boundaries marked by the ellipses. Fig. 7: Outline of optimization procedure for the corrugated panel structure. Fig. 10: Shell panel area used for extracting stress responses. 6 Copyright 011 by ASME

7 9. DESIGN LOADS The design loads acting on the local FE model of the corrugated panel structure are composed by global loads and local lateral loads. The lateral loads are applied manually on the FE model on the shell surfaces and the magnitude was defined by the draught of the semi-submersible. The global loads are applied on the local FE model via SUBMOD in SESAM [10], and they are calculated using the definitions in the ABS MODU Rules [1]: Static loads: These loads are defined as still water loads which cause a global bending moment and shear forces in the hull girder. The external sea pressure and internal tank pressure results in a local response of the plates, stiffeners and girders. Equilibrium in the FE analysis was attained when the sum of the static forces was zero. Static loads are classified as permanent loads (lightweight, mooring lightweight, etc.) and variable loads (ballast water, riser tension, fuel and crude oil). Environmental (dynamic) loads: These loads are caused by wind, current, ice and waves, green water, etc. The equilibrium in the FE analysis was attained when the sum of the dynamic forces was zero. The load components for the static and the environmental loads are combined to achieve the most severe section forces in critical sections of the model. The global bending moments, shear and axial forces are superimposed and maximum values are obtained which are used in the buckling and yield check assessment. Thus, to obtain conservative estimates, the phase angles between external sea pressure and bending moments are disregarded in the analysis. The external sea pressure distribution is conventionally defined as a normal pressure with an annual probability of exceedance equal to 10 - (100- year return period) for ULS. The results from the FE analyses of the different load cases are stress values in each finite element of the model. In the assessment procedure, different types of stress in each element are computed following the criteria in Section 5, for example: For the ultimate and buckling strength assessment, the von Mises effective stress was calculated by superposition of the static and dynamic stress histories to represent the extreme storm loading condition. In the fatigue strength assessment, the maximum principal stresses caused by the dynamics loads are scanned for a 30-year fatigue life. 10. PARAMETRIC STUDY, CORRUGATED PLATING A parametric study of the dimensions of the corrugated structure was carried out to reduce the number of FE analyses in the optimization procedure outlined in Fig. 7. Another argument for using this method is that FEA may not be sufficient to perceive the idea of parametric influence because of the complexity of the model and problem. Consequently, the parametric study was carried out to gain an extensive knowledge on how strength is influenced by each of the parameters of the corrugations, boundary conditions, aspect ratio, loading in the x and y directions and different load cases according to the class rules ABS [6,11] and DNV [7]. The ultimate strength of structural members can be sensitive to type and magnitude of initial distortions. This was, however, omitted in the model-making and the results from the FEA. The influence from residual stresses and initial deformations are incorporated explicitly in the ABS formulations. Also, the elasto-plastic buckling (non-linear) was taken care of in a similar way to the Johnson-Ostenfeld correction. Finally, in the buckling strength calculation of the unit corrugation as a beam, the influence of the effective width concept was ignored. Figure 10 presents the variables of the cross section of the corrugated shell-plating. The following list presents the variables included in the parametric study. The two last variables must be expressed in terms of some of the other variables: Width of corrugated panel, B Length of corrugated panel, L Thickness of the plating, t Width of unit corrugation, corrugation spacing, s Upper flange width, a Height of web plating, d Corrugation angle, Φ Width of lower flange, b = b(a, s, c) Width of web plating, c = (Φ, d) Fig. 10: Sketch of cross section of a corrugated structure [6]. The parametric study was carried out for the part of the top plating between two transverse web frames in the midsection of the local FE model; see Fig. 11 and Fig. 1 for the extracted panel section of this model which was used in the parametric study. The panel section in the top plating was selected because it was identified in the initial FEA as the part of the structure which showed the highest overall and local compressive and tensile stresses; see Section 6. Hence, it was considered as the weakest part of the structure. The parametric study involved nine stress unknowns which had to be identified. Instead of calculating the maximum allowable stress for each unknown stress variable, based on the given allowed limit state for the failure mode in question, it was decided to use the FEA of the local corrugated structure model from which the unknowns could be extracted and subsequently used in the parametric study. Consequently, the assessment for different corrugation profiles could be made with the input stresses from the FEA based on appropriate 7 Copyright 011 by ASME

8 design. In Table 5, the results from the strength assessment are presented. As seen in the table, all criteria are satisfied for this model. Beam-column buckling turns out to be the governing design criterion for this type of structure. A further decrease of the corrugation height leads to failure, so the design process was interrupted. The results demonstrate that there is great potential of using a corrugated shell plating design instead of traditional stiffened flat panel structures. The fatigue analysis of the final design showed variation in the principal stress range between 19 and 9 MPa in various positions of the local FE model of the corrugated structure design. The maximum allowable principle stress was calculated to 98 MPa, using FDF = 3 [9], and hence, the proposed corrugated shell-plating design passes the fatigue assessment. loading and boundary conditions. Note, however, that this calls for an iterative procedure, since the input stresses to the parametric study will differ as the corrugation geometry is modified. Fig. 11: Local FE model with 5 sections and the panel section. Table 4. Dimensions of the final corrugated structure design. Fig. 1: Panel section assessed in the parametric study. Table 5. Strength features of the final corrugated structure. Table 3 presents the nine unknown stress components calculated by means of FEA, which are further used in the strength assessment procedure of the corrugated structure; see Fig. 13 for clarification of notations. Table 3. Unknown stresses obtained from FEA. 1. RESULTS MANUFACTURING COST ANALYSIS One of the challenges with the corrugated plating concept is manufacturing. However, there is equipment available in many shipyards today which can produce the geometry proposed in the current study. The objective of this section is to summarize the manufacturing and cost-benefit analysis presented in Sağlam and Sarder [5]. The analysis demonstrates that the proposed corrugated structure can also be beneficial from a manufacturing and cost point of view. In the cost-benefit analysis, the reference and corrugated structures are compared. The following basic assumptions were made: (i) the ship will operate for 5 years without docking; (ii) the process only shuts down at the end of a shift; (iii) all weldable structures were considered for welding type shielded metal arc welding; (iv) the welded joints in the transverse direction of the shell plate panels were ignored; (v) any difference in the steel cutting process was ignored; (vi) the assigned prices were assumed only for the purpose of comparison. Fig. 13: Coordinate system and notations for stress positions. 11. RESULTS STRENGTH AND WEIGHT ANALYSIS Table 4 presents the dimensions of the final design of the corrugated structure. The weight reduction refers to the reduction in weight in contrast to the reference structure 8 Copyright 011 by ASME

9 The direct savings for the corrugated structure that could be anticipated without any numerical calculation were welding, painting and man-hours. Using a corrugated panel reduces welding materials associated with stiffener attachment to the main plate. Only butt welding of end joints is needed in this case. Additionally, the surface area that must be painted is less for the corrugated panel in contrast to the reference structure; stiffeners and structure details of the reference structure contribute largely to the total area that must be painted. Finally, both during manufacturing and painting of both structures, the number of man-hours needed for welding and painting is less for the corrugated structure. A simplistic cost model was used to calculate the production costs with respect to variation in plate thickness and corrugation height. For this purpose, the cost model proposed by Rahman and Caldwell [13] was used but modified to incorporate also a cost-benefit analysis for painting. The assumed values in the model are presented in Table 6. The results from the cost-benefit analysis are presented in Fig. 14. In terms of both cost and weight, it seen that the corrugated structure has a lower weight and cost compared to the reference structure (flat panel stiffened shell plating); see Sağlam and Sarder [5] for details. Table 6. Assumed values in cost-benefit analysis. Issue Cost Steel cost 800 Euro/tonne Operator cost 10 Euro/hour Paint cost 5 Euro/liter Welding consumables Euro/liter Fig. 14: Summary of weight and cost-benefit analysis. 13. DISCUSSION The ULS was calculated for a maximum load effect in an extreme storm condition (100-year hurricane). The serviceability limit state (SLS) was not employed in this study. The difference between these limit state designs is the utilization factors applied for the assessment of buckling and yield strength. The responses for an extreme storm condition are the highest for the focus structure. Sheet thickness is the most effective parameter among all of the parameters with regard to lightweight design of corrugated shell plating. If the column buckling is the governing failure mode, the most effective weight reduction could be achieved by reducing the thickness. In a parametric variation with a locked structural weight, the decrease in the column buckling capacity is less dramatic, if thickness is reduced rather than corrugation height. As long as no other criterion takes over and determines the design, one should reduce plate thickness until local buckling takes over. This was employed between the iterations. The column buckling of the unit corrugation regarded as a beam is expected to be the significant failure mode for this structure when the bending stresses are high due to hydrostatic pressure. However, it was concluded that the local buckling mode and in some cases the ultimate strength of the plate areas has required attention. The span of the transverse web frames is rather short for column buckling to be the dominating failure mode. In addition, the buckling and ultimate strength of structural components are highly influenced by the amplitude and shape of the imperfections introduced during manufacturing, installation and transportation. These imperfections could be misalignments of joined components or initial distortion due to some fabrication-related process. The FE analysis ignored such imperfections, while the parametric study has included the ABS rules in which these imperfections are implicitly formulated. It should be noted that the reference structure presented in Section 3 is an early design proposed in an ongoing project. In Sağlam and Sarder [5], a weight optimization of this structure is carried out following the procedure presented in the current investigation. The results show that the weight of the reference structure in Section 3 can be reduced by 3 percent. Hence, with reference to this revised reference structure, the weight reduction using a corrugated shell-plated structure is reduced to 40 percent. With a significant weight and strength balance, corrugated panels can be favourable in terms of production cost efficiency. Introduction of a bending machine, an automatic welding and painting machine is a good choice for fast and cheap shell plate panel construction. Surveyors may favour the absence of major obstacles such as stiffeners. 14. CONCLUSIONS The current investigation presents an example of an innovative lightweight design of the pontoon of a semi-submersible platform through the utilization of a corrugated structure. For this purpose, an optimization methodology is developed for the analysis of strength, weight and manufacturing cost of shell-plated marine structures. A small part of the pontoon structure is selected for demonstration. A reference structure model of this part is designed and optimized with regard to the guidelines for conventional stiffened panel structures. Another model using corrugated shell-plating is designed in order to challenge the conventional stiffened panel structure using the presented methodology. The results show that in competition with the traditional stiffened panel, the corrugated shell-plated structure can be used as the more lightweight design solution. The reduction in weight is almost 60 percent and the total cost reduction is estimated to 49 percent. It is also found from the structural strength characteristics analyses that, when the corrugation structure is designed properly, classification rules are fulfilled without compromising with the safety margins. 9 Copyright 011 by ASME

10 ACKNOWLEDGEMENTS The authors acknowledge the support from Lighthouse Maritime Competence Centre ( Sweden. REFERENCES [1] Rahman, M. K., 1996, Optimization of Panel Forms for Improvement in Ship Structures, Structural and Multidisciplinary Optimization, 11(3-4), pp [] Kippenes, J., Byklum, E. and Steen, E., 007, Ultimate Strength of Open Corrugated Panels, Proc. PRADS 007 Conference, Houston, Texas, U.S.A. [3] Caldwell, J. B., 1955, The Strength of Corrugated Plating for Ships Bulkheads, Trans. RINA, 97, pp [4] Paik, J. P., Thayamballi, A. K. and Chun, M. S., 1997, Theoretical and Experimental Study on Ultimate Strength of Corrugated Bulkheads, Journal of Ship Research, 41(4), pp [5] Sağlam, H. and Sarder, Md. A., 010, Use of Corrugated Shell Plating in Semi-submersible Offshore Platforms, Report No. X-10/51, Dept of Shipping and Marine Technology, Chalmers University of Technology. [6] American Bureau of Shipping (ABS), 004, Guide for Buckling and Ultimate Strength Assessment for Offshore Structures. [7] Det Norske Veritas (DNV), 010, Offshore Standard, Structural Design of Offshore Ships, DNV-OS-C10. [8] Paik, J. P. and Thayamballi, A. K., 003, Ultimate Limit State Design of Steel-Plated Structures, John Wiley & Sons, Inc. [9] American Bureau of Shipping (ABS), 003, Fatigue Assessment of Offshore Structures. [10] Det Norske Veritas (DNV), 007, SESAM User Manual Sestra: Superelement Structure Analysis, Version 8.3. [11] American Bureau of Shipping (ABS), 009, ABS buckling software (StruProg) manual. [1] American Bureau of Shipping (ABS), 008, ABS MODU Rules - Rules for Building and Classing Mobile Offshore Drilling Units. [13] Rahman, M. K. and Caldwell, J. B., 199, Rule-Based Optimization of Midship Structures, Marine Structures, 5(6), NOTATIONS a Width of upper flange of corrugation [mm] b Width of lower flange of corrugation [mm] c Width of web plating of corrugation [mm] d Height of web plating of corrugation [mm] k s Shear yield stress of the material [MPa] m,r Weibull parameters in Eq. (11); see ABS [6] s Corrugation spacing [mm] t Thickness of plating [mm] q u Lateral pressure, first end of the corrugation [kn/m] x,y,z Coordinate system axes, directions of loading A,C Weibull distribution constants in Eq. (11); see ABS [6] B Width of corrugated panel [mm] C m Bending moment factor, simply supported panel [knm] C x Slenderness ratio [-]; see Eq. (6) E Young s modulus [GPa] FDF Fatigue design factor [-]; see ABS [6] L Length of corrugated panel [mm] N R,N R Number of cycles, reference-time period [cycles] N T Number of cycles, target-time period [cycles] S R,S R (Allowable) Stress range [MPa] α Aspect ratio of the panel [-] γ, δ Weibull shape parameters in Eq. (11); see ABS [6] η Strength utilization factor [-] υ Poisson's ratio [-] ρ Density of the material [kg/m 3 ] σ 0 Specified minimum yield point of plate [MPa] σ a Max. compressive stress, corrugation direction [MPa] σ b Max. bending stress along the length [MPa] σ Ca Critical buckling stress [MPa] σ Cx Critical buckling stress, longitudinal direction [MPa] σ Cy Critical buckling stress, transverse direction [MPa] σ CB Critical bending buckling stress [MPa] σ e The von Mises effective stress [MPa] σ E(C) Elastic buckling stress [MPa] σ Gx Critical buckling stress, uniaxial compr. in corrugation dir. [MPa] σ Gy Critical buckling stress, uniaxial compr. in transverse dir. [MPa] σ x Stress in longitudinal direction [MPa] σ x,max Max. compr. stress, longitudinal direction [MPa] σ y Stress in transverse direction [MPa] σ y,max Max. compressive stress, transverse direction [MPa] σ y0. Yield stress of the material [MPa] σ Ux Ultimate strength, uniaxial stress in longitudinal direction [MPa] σ Uy Ultimate strength, uniaxial stress in transverse direction [MPa] τ In-plane shear stress [MPa] τ c Critical buckling stress for edge shear [MPa] τ G Critical buckling stress for shear stress [MPa] τ U Ultimate strength, edge shear [MPa] φ Interaction coefficient between longitudinal and transverse stresses [-]; see Eq. (4) and ABS [6] Φ Corrugation angle [degrees] 10 Copyright 011 by ASME