A method for optimal design of large-scale framework structures is presented. The approach is based on a reanalysis technique in combination with suboptimization for each member group in the structure. The suboptimization is performed using a nonlinear programming algorithm based on sequential quadratic programming. Given the topology and material quality, the cross-sectional dimensions of wide-flange and hollow-section members are treated as design variables and the minimum weight or cost solution is sought. The structural problem formulated using the matrix displacement method. Large three-dimensional rigidly jointed frames subjected to multiple loading cases are analyzed. Constraints can be imposed on maximum Von Mises stresses, beam stability, and the fatigue damage at nodal points. The approach is general, robust, and efficient in that it avoids numerous global structure analyses. The practical engineering utility of the method is emphasized. Case studies of an offshore deck module and a platform jacket structure are included. Future work should focus on a refined formulation of the fatigue constraint. INTRODUCTION The process of detailed design of large three-dimensional framed structures subjected to multiple load cases is quite involved. The structural analysis is normally carried out using the matrix displacement method with beam elements. After an initial study of topology and material, the designer will have to carry out several analyses searching for the optimum cross-sectional dimensions giving the minimum weight or cost design. The work could be both challenging and cumbersome, and the advantage of automation using a mathematical optimization technique has been recognized. In recent years the development of hardware facilities and numerical solution algorithms has made it possible to automate this optimization process even in the practical design of large-scale structures. A general overview of different approaches and literature on the subject is given by Levy and Lev (1987). For large-scale structures, Khan (1984) has used an optimality criterion technique for optimizing frames. The examples are limited to plane frame analysis. Kirsch (1982) has introduced a procedure based on scaling and explicit approximations of the response variables in terms of design variables to avoid multiple analyses. The present approach is based on a reanalysis technique combined with suboptimization. The method takes advantage of the fact that large-framework structures often have a rather limited number of member groups, i.e., groups with beams of same cross-sectional dimensions. Searching for an optimum design for these frames, one can conduct a suboptimization for each member group individually. The suboptimization is based on the idea that only one constraint and one element within each member group are critical at any stage of the process. Although there is no theoretical basis for this assumption, experience with the process has indicated that it is robust in that it consistently provides optimum solutions at a low computer cost. The subproblems are treated as uncoupled problems with fixed internal beam forces and unchanged critical constraint during the iteration process. The constraint gradient computation is carried out by a simple code check for a single beam without the time-consuming frame analysis. When the subproblems have converged, the coupling between the subproblems is taken care of by a complete frame reanalysis. The internal end forces of the beams are then updated before returning to a new suboptimization. The final solution will have at least one active constraint for at least one element in each member group. If the number of member groups equals the total number of beams (one single beam per group), the solution may coincide with the solution obtained by the use of an optimality criterion-based method. One important difference, however, is that the present method incorporates a suboptimization based on mathematical programming for each beam in the structure. Thus, for beams with several independent design variables, a cross-sectional shape optimization is included. The examples presented demonstrate that the method can efficiently handle rather complex and realistic design problems. The optimal designs obtained by the present simplified algorithm compare favorably to those obtained by more conventional methods. In the normal case when the number of beams is much larger than the number of member groups, the suboptimization results in an appreciable reduction in computational efforts. The suboptimization is performed using a nonlinear programming algorithm based on sequential quadratic programming (NLPQL). The algorithm was developed by Schittkowski (1981a, 1981b) and has proved to be very efficient compared to other algorithms available today. The scope of the present work is to analyze three-dimensional framework structures with more than one design variable per member and with general cross-sectional relationships. This will make the method suitable for application to offshore steel structures where large wide-flange beams are often built-up by welding. Typical examples are deck modules. A minimum-weight design of these structures will give favorable material and fabrication costs and will improve the platform stability and dynamic behavior due to reduced topside weight. FORMULATION OF DESIGN PROBLEM A framework structure can be described by a set of quantities defining topology, material properties, configuration, and cross-sectional dimensions. In practical frame design the topology and configuration are preassigned, while member sizes are treated as continuous design variables. Often-used cross sections are wide-flange (WF), circular hollow-section (CHS), and rectangular hollow-section (RHS) as shown in Fig. 1. The cross section for each member type may be described by only one independent variable, the other dimensions being treated as dependent variables under different linking assumptions according to international standards. For WF cross sections the web height is the independent variable, while the other dimensions are chosen according to Euronorm 53-62 HE-B. Some modified linking is possible if one wishes to keep total web height or flange widths within specified limits. For the tubular sections, the outer diameter is chosen as the independent variable while wall thickness is linked according to different International Standard Organization (ISO) specifications. Another possibility is to specify nonstandard cross sections and let the plate thicknesses be dependent variables based on recommendations from different structural building codes to avoid local plate buckling. Under this assumption, WF beams will have both the web height and the flange width as free variables. The mathematical formulation of the design problem reads min W(x) = |summation~|A.sub.i~|l.sub.i~ = pV (1) subject to |g.sub.j~(x) |is greater than or equal to~ 0, j = 1, |m.sub.t~, |x.sub.1~ |is less than or equal to~ x |is less than or equal to~ |x.sub.u~ (2) The objective function is here chosen to be the primary weight of the structure. A more sophisticated function taking into account the fabrication costs is W(x) = |K.sub.m~V + |K.sub.f~(|T.sub.1~ + |T.sub.2~ + |T.sub.3~) (3) where |K.sub.m~ and |K.sub.f~ = material and fabrication costs; and |T.sub.i~ = times related to preparation, welding, and control. Farkas (1989) found that the material costs are the most important part and the analysis carried out in this report is based on (1). Only the most critical constraint function within one member group is specified |g.sub.j~(x) = 1 - |D.sub.max~ j = 1, 2 ... |m.sub.t~ (4) The usage factor D is in general defined as the ratio of response over allowable response (capacity). Thus, D will approach unity when the corresponding constraint becomes active. |D.sub.max~ is defined as the highest usage factor within one member group for any of the actual dimension criteria. For the yield criterion based on Von Mises stresses, D reads |D.sub.y~ = ||sigma~.sub.emax~/||sigma~.sub.a~ (5) For the beam stability criterion according to the Norwegian Building Codes (1984) (minor axis) D reads |Mathematical Expression Omitted~ There is a corresponding interaction formula for buckling with respect to the major cross-sectional axis. For tubular members the axes referred to are defined by in-plane and out-of-plane buckling. For a fluctuating load condition, it is mandatory to consider the critical fatigue constraints at nodal joints. A common approach is to assume a linear relationship in the log-log scale between the constant stress range and the number of cycles to failure lgN = lga - mlg|delta~|sigma~ The graph of this equation is the S-N curve given in the building codes (S = |delta~|sigma~). With constant stress range, the usage factor for the fatigue criterion then reads |D.sub.f~ = mlg|delta~|sigma~/lga - lgN (8) where m and lga = the parameters in the S-N equation found in the design building codes; and N = the design fatigue life. The hot spot stress ranges |delta~|sigma~ are calculated from frame analysis results. The stresses in the member at its end are multiplied by the appropriate stress concentration factor (SCF) for the actual nodal joint. For structures subjected to variable amplitude loading, an allowable cumulative damage design format could be considered. For offshore structures subjected to random sea-wave loading, the long-term stress range distribution could be approximated by a simple two-parameter Weibull distribution. The details are given by Almar-Naess (1985). The fatigue criterion based on a continuous integration of Miners rule then reads |Mathematical Expression Omitted~ If the structure is subjected to wave loading one will have to carry out a quasi-static frame analysis for an extreme sea-state condition simulating maximum wave height and wave period. This will give a reasonably good approximation of the maximum stress range |delta~||sigma~.sub.1~. DESIGN PROCEDURE The basic idea for the solution algorithm is the suboptimization for each member group while the internal forces are kept constant. Because this suboptimization only works with the most critical member and constraint within one member group, the computation is fast and only a limited number of global iterations with updating of displacements and internal forces are needed. After having decided the frame configuration, the method can be summarized in three steps: 1. Choose any initial value for the design variables. 2. Carry out a complete frame analysis and code check for all loading conditions, elements, and constraint types. Select the element in each member group that for a given loading condition has the most critical constraint value. The most critical loading condition, element, and constraint type are defined. The internal beam forces for these elements are saved. 3. A subproblem is formulated for each member group individually. During the suboptimization the internal beam forces are kept constant and only the most critical constraint is computed. This in, in fact, a single beam element optimization with only a few design variables and one constraint. A simple code check and weight computation is carried out in each iteration. The coupling between the subproblems is taken care of by means of one complete reanalysis when convergence of the subproblem is reached. The beam internal forces are updated. The most critical loading condition, element, and constraint in each member group are redefined. Often there will be no changes for any of them. The algorithm then returns to suboptimization. The convergence criterion in the subproblem is given by a limit on the sum of constraint violations and on the Kuhn-Tucker optimality criterion. Both the sum of violations and the norm of the Kuhn-Tucker vector is limited to |10.sup.-3~ in the present work. This will give approximately three correct digits in the final solution. The convergence for the major loop with the total reanalysis is controlled by setting d limit of 0.5% to the relative decrease in the object function. Because of the simplification with the uncoupled subproblems, one cannot be sure that the obtained solution is an optimum one. To verify the solution it is possible to omit the suboptimization step described previously. The algorithm will then treat all of the variables simultaneously and carry out a complete frame reanalysis when computing the constraint gradients. This is time-consuming and was performed only to verify that the suboptimization technique yielded a result close to the optimum one derived by a direct mathematical approach. This was the case for both the examples presented here. For further verification of convergence, the NLPQL method was replaced with the widely used sequential unconstrained minimization technique (SUMT) with the use of a penalty function along with the Powell's search scheme. Both methods gave almost the same optimal designs, but the SUMT technique demanded considerably longer central processing unit (CPU) time. A similar comparison was carried out by Khan (1984). The program system used in the following practical applications was developed by Lassen (1989). The suboptimization is carried out by using the NLPQL algorithm. This algorithm is based on successive solutions of quadratic programming subproblems and a subsequent one-dimensional line search. The mathematical details and some convergence results are given by Schittkowski (1981a, 1981b). When the algorithm is called in the present procedure a member group subproblem has been defined and the algorithm will work with one or two variables and one inequality constraint. DECK MODULE EXAMPLE Offshore deck modules are often constructed as framework structures with large WF members. The present structure is a mud and cementing treatment system module that is analyzed for a static in-place loading condition. The main loading is caused by the weight of the large mud and cement tanks and different process equipment. In the present study the material properties, structure topology, and geometric layout are preassigned. The choice of transverse frame spacing was based on a simple initial study of the longitudinal stiffeners and deck-plate panel contribution to the secondary steel weight. The aim is to carry out comparative studies by varying the number of member groups, and by using standard contra nonstandard (welded) profiles, etc. The minimum-weight solution was sought for the following specifications. * Topology * Two main upper and lower decks * Two movable mezzanine decks * Seven transverse frames with a spacing of 5 m * Diagonals in side walls * Material * St. 52-111 yield stress |f.sub.y~ = 360 MPa * Allowable stress ||sigma~.sub.a~ = 360/1.15 = 313 MPa (constant) * Loading * Total static loading on all decks is P = 45.5 MN (4,550 t) * Loading from the messanine decks given as nodal point forces on top of vertical columns * Large bulk tanks cause uneven loading on bottom deck * Loading multiplication factor 1.3 * Functional requirements on dimensions * Maximum web height bottom deck = 1.6 m * Maximum web height sides/upper deck = 0.8 m * Minimum web height 0.2 m * Cross sections--Due to the fabrication requirements, the number of member groups is set to 4 with the following member group definition * Group 1 = all bottom girders (element Nos. 1-6, 19-31) * Group 2 = side diagonals (element nos. 41-56) * Group 3 = side columns (element Nos. 57-70) * Group 4 = top deck girders (element nos. 33-40) Euronorm HEB standard was chosen for the smaller girders (groups 2 and 4), while welded profiles were chosen for the larger girders (groups 1 and 3). The frame model is shown in Fig. 3. Only the two main decks and side walls are part of the model. The model consists of 30 nodes and 70 WF beam elements. Definition of the loading condition is given in Fig. 4, and details are listed in Table 1. Constraints are imposed on member sizes and on the stress and stability criteria. Design constraints are given in accordance to the functional requirements. Two different assumptions were made during optimization. In the first model, all design constraints were specified as in the original design concept. In the second model the design constraint on the web heights was ignored to see how much weight increase this restriction causes. Both models have a total of six design variables with maximum two design variables per member group. The results from the optimization process are shown in Tables 2, 3, and 4. The first suboptimization required 66 function evaluations and 58 gradient evaluations before convergence for all member groups was reached. The gradients were computed numerically by the use of forward differences. For model 1, which satisfies all the given constraints, the weight decreases from an arbitrary chosen initial design of 388 |10.sup.3~ kg to an optimum solution of 325 |10.sup.3~ kg. As seen from the optimum dimensions, the web height of the large bottom girders (group 1) causes the only active design constraint H = 1.59 m. These girders also have active stress constraints due to equivalent stresses in the longitudinal girder in the cross section adjacent to the deck support (element 2, ||sigma~.sub.eq~ = 313 MPa). The critical elements for groups 2 and 3 (diagonals and columns) are elements 47 and 62 due to stability interaction about the minor axis. This is the reason why the flange widths of the columns become relatively large B = 0.5 m. The nearest standard profiles for element groups 2 and 4 are HEB 500 and HEB 750, respectively. Model 2 does not comply with all the design restrictions and is analyzed as an alternative concept. lt is demonstrated that if one allows for a web height of h = 1.9 m in the large bottom girders, the weight will further decrease to 312 |10.sup.3~ kg. This solution coincides with the principle of leading the heavy loads directly to the nearest supports, thus avoiding large influence on the rest of the structure. When setting up the functional design constraints, one should be aware of such consequences on the steel weight. The final solution is to some extent sensitive to the chosen initial design. It is an advantage to choose the initial height of the girders close to the new design restriction, recognizing that weight reduction can be gained in the rest of the structure by increased stiffness of the deep bottom girders. For different arbitrary chosen initial designs the derived solutions are in the range of 312-320 |10.sup.3~ kg, i.e., still close to the optimum design. The analysis was carried out with and without the use of suboptimization. With the use of suboptimization the number of global iterations dropped from 50 to four complete reanalysis (each reanalysis required 15 s CPU time) (PRIME 2455 computer). The direct global optimization for model 2 gave a final steel weight of 309 |10.sup.3~ kg, i.e., 1% reduction compared with the solution derived by the suboptimization scheme. The SUMT algorithm was used as a reference and yielded the same optimal solution but with over six times in applied CPU. Other models of interest could allow for nonstandard profiles for element groups 2 and 4 and for an increase in the number of member groups because many elements have a low usage factor in the present configurations due TABULAR DATA OMITTED to uneven loading on the main bottom deck. These alternative solutions will decrease material costs while fabrication costs will increase. These comparative studies of alternative design concepts are useful in the predesign stage when assessing topside steel weight and costs. TABLE 2. Design History for Deck Module under Different Constraints Number of Number of Iteration Model 1 Mo del 2 function gradient number (kg) (kg) evaluations eva luations (1) (2) (3) (4) (5) Initial 388,000 40,600 -- -- 1 329,000 321,000 66 58 2 326,000 313,000 48 42 Optimum 325,000 3 12,000 34 38 DESIGN OF JACKET STRUCTURE Offshore jacket structures are fabricated as truss frameworks in which tubular members constitute the structural elements. The tubular joints represent TABULAR DATA OMITTED structural discontinuities that give rise to very high stress concentrations in the intersection areas. To achieve a cost-optimal design it is necessary to take into account the nodal fatigue constraints. The present example is a small well-protector platform in an offshore gas field. It is constructed as a three-pile jacket, and the outer legs form an equilateral triangle. The triangle side length at mud line is 15 m and 10 m at top bracing. The height from mud line to top bracing is 17.8 m, and the water depth is 15 m. There are three intermediate bracing levels as shown on the model in Fig. 5. The simplified loading condition due to static deck loading and an extreme-wave loading condition is specified in Table 5. The topology and configuration satisfy the fabrication and installation constraints and are kept fixed during the optimization of the member sizes. It was decided to use three member groups and the outer tube diameters are chosen as continuous free variables while the wall thicknesses are dependent variables in accordance with cross-section class No. 1 in the building code (fully developed plastic hinges without local plate buckling). The member groups are: (1) Outer legs; (2) lower side K-braces; and (3) the rest of the structure. The structure is subjected to two load cases: 1. Static loading condition including permanent deck loads and wave loads corresponding to maximum wave height and period at an extreme sea state (100 years). 2. Long-term stress distribution characterized by the maximum stress range |delta~||sigma.sub.1~ during the fatigue design life (20 years). The distribution is derived from the maximum wave height analysis and reasonable assumptions for the Weibull long-term shape parameter p. TABLE 5. Simplified Loading on Jacket Structure due to Static Deck Loading and Extreme-Wave Loading Condition. All Loads in One Bracing Level Are Equally Distributed on Nodes in Plane Bracing level First triangle Second triangle Deck support (kN) (z = 2.0 m) (z = 10.0 m) (z = 17.8 m) (1) (2) (3) (4) |P.sub.x~ 300 600 750 |P.sub.y~ 300 600 750 |P.sub.z~ -- -- 2,600 The stress and global member buckling constraints are imposed on loading condition 1, while the fatigue constraints are used in condition 2. Punching shear constraints are not considered. The simplified fatigue analysis contains four major steps. * Determine the wave height, period, and direction that correspond to the most extreme sea state characterizing the wave climate. Calculate hydrodynamic forces according to Morison's equation. * Carry out global frame analysis in order to obtain nominal member stresses at the member ends. * Calculate SCFs at joints by recommended parametric formulas and find the hot-spot stress ranges. * Choose a suitable Weibull shape parameter based on experience from similar structures, and calculate fatigue damage usage factor according to (9). The analysis is based on the following data: * Material St. 52 with yield stress 360 MPa * S-N T-curve lga = 12.16 and m = 3 * Fatigue design life N = |10.sup.8~ cycles (20 years) * Extreme sea state (100 years) H = 18 m, |T.sub.p~ = 12 s * Weibull shape parameter p = 0.8 * Assumed average SCF = 3.0 Based on the aforementioned data, the allowable nominal fatigue stress for the extreme design wave is ||sigma~.sub.a~ = 140 MPa. The results of the optimization are shown in Tables 6, 7, and 8. The minimum weight is 49 |10.sup.3~ kg with an outer-leg diameter of 0.86 m. All the member sizes are governed by the fatigue constraints only (nodes 5, 6, and 12). The stress and buckling usage factors are in the range D = 0.3-0.7. The plate thicknesses are relatively thin and one can improve the final design if the tube wall thickness is treated as a free variable with the section class 1 thickness as a lower limit. If the fatigue constraints are neglected, the minimum weight will decrease further to 32 |10.sup.3~ kg (outer-leg diameter 0.67 m) with equivalent stress constraint active in outer legs (element 2) and buckling constraints active in diagonal braces (elements 13 and 31). TABLE 6. Design History for Jacket Structure Number of Number of Iteration Weight function gradient number (kg) evaluations evaluations (1) (2) (3) (4) Initial 62,000 -- -- 1 53,000 42 34 2 50,000 20 18 Optimum 49,000 16 16 TABULAR DATA OMITTED TABULAR DATA OMITTED A final design could be quite close to this solution if the fatigue constraints are dealt with by favorable local geometry in the joints. The first suboptimization required 42 function evaluations and 34 gradient evaluations, demanding 3.5 s CPU time. Each frame analysis required 9 s CPU time. If the number of design variables is increased from the chosen three to six variables the corresponding function and gradient evaluations are 72 and 56, respectively. A further increase to nine variables yields 98 and 90 function-gradient evaluations. This means that there is almost a linear relationship between the number of evaluations and the number of design variables, as long as the number of reanalysis is not increased. In the present case this was not necessary. The analysis highlights the importance of nodal joint fatigue constraints on a feasible and cost-effective design of jacket structures. It must be emphasized that the assumptions made regarding the fatigue analysis are quite simple and only applicable at an early design stage. A more comprehensive analysis should consider several sea states and wave directions in order to calculate the long-term stress range distributions with more accuracy, see Lotsberg et al. (1988). The SCF was chosen based on d rough estimate, because the parametric formulas are limited to simple geometry of uniplane nodes only. A more detailed stress analysis would have to incorporate a finite element node model within the global frame analysis, as suggested by Gibstein (1989). In future work it will be interesting to study to what extent these refinements could be taken into account in the optimization procedure without increasing the computational effort to an unacceptable level. CONCLUSIONS The present study confirms that numerical structural optimization has progressed far enough to equip the design engineer with a practical design tool. A design algorithm based on suboptimization within each member group under the most critical constraints by using the cost-effective NLPQL method seems to have a great potential as an engineering utility for the optimization of large-framework structures. Realistic case studies of an offshore deck module and a jacket platform have been carried out. The optimum design gives the member sizes minimizing the weight function and satisfying the constraints. For offshore structures where the fatigue damage in member joints is critical, a suggestion for future work is to study the possibilities of optimization with refined fatigue analysis. APPENDIX I. REFERENCES Farkas, J. (1989). 'Minimum cost design of tubular trusses considering buckling and fatigue constraints.' Tubular structures, E. Niemi, ed., Elsevier Appl. Sci., New York, N.Y., 451-495. Fatigue Handbook. A. Almar-Naess (ed.) (1985). Tapir, Trondheim, Norway, 405-422. Gibstein, M. (1989). 'Refined fatigue analysis approach and its application to the Veslefrikk jacket.' Tubular structures, E. Niemi, ed., Elsevier Appl. Sci., New York, N.Y., 321-332. Khan, M.R. (1984). 'Optimality criterion technique applied to frames having general cross-sectional relationships.' AIAA J., 22(5), 669-676. Kirsch, U. (1982). 'Optimal design based on approximation scaling.' J. Struct. Engrg., ASCE, 108(4), 888-909. Lassen, T. (1989). ODA-theoretical manual. Agder Coll. of Engrg., Grimstad, Norway. Levy, R., and Lev, O.E. (1987). 'Recent development in structural optimization.' J. Struct. Engrg., ASCE, 113(9), 1939-1962. Lotsberg, I., Taarnes, A., and Simensen, P.A. (1988). 'Design of Karin deepwater platform with respect to fatigue.' J. Struct. Engrg., ASCE, 114(60), 1211-1229. Norwegian building codes, steel structures NS3472. (1984). Norwegian Standardization Society, Oslo, Norway. Schittkowski, K. (1981a). 'The nonlinear programming method of Wilson, Han, and Powell with an augmented Lagrangian type line search function. Part 1: Convergence analysis.' Numer. Math., 38(4), 83-114. Schittkowski, K. (1981b). 'The nonlinear programming method of Wilson, Han, and Powell with an augmented Lagrangian type line search function. Part 2: An efficient implementation with linear least square subproblems.' Numer. Math., 38(4), 115-127. APPENDIX II. NOTATION The following symbols are used in this paper: A = area of cross section; B = flange width; |c.sub.z~ = stability interaction coefficients; D = usage factor; DY = outer diameter; |g.sub.j~(x) = constraint inequality; H = web height; |I.sub.x~ = torsional stiffness; |I.sub.y~ = moment of inertia about y-axis; |I.sub.z~ = moment of inertia about z-axis; |K.sub.E~ = stability interaction coefficient; l = element length; lga = parameter in S-N curve equation; |M.sub.vd~ = lateral buckling moment capacity; |M.sub.y~ = applied moment about y-axis; |M.sub.z~ = applied moment about z-axis; |M.sub.zd~ = yield state capacity z-axis; m = slope of S-N curve; |m.sub.t~ = total number of constraints; N = number of stress cycles; |N.sub.kzd~ = buckling force capacity z-axis; |P.sub.x~ = applied axial force; p, q = parameters in Weibull long-term distribution; T = plate thickness; |T.sub.p~ = wave period; V = structure volume; x = (|x.sub.1~, |x.sub.2~, ... |x.sub.n~) = design variables; |x.sub.1~ = lower-bound design variables; |x.sub.u~ = upper-bound design variables; (x, y, z) = local beam coordinates; W(x) = object function; |gamma~ = gamma function; ||gamma~.sub.m~ = material coefficient; ||gamma.sub.f~ = load coefficient; |delta~|sigma~ = stress range; |delta~||sigma~.sub.0~ = fatigue limit stress range; |delta~||sigma~.sub.1~ = maximum stress range; |rho~ = material density; ||sigma~.sub.a~ = allowable stress; and ||sigma~.sub.e~ = Von Mises equivalent stresses. Tom Lassen, Assoc. Prof., Dept. of Struct. Engrg., Agder Coll. of Engrg., 4890 Grimstad, Norway.