AP05_4.vp 1 Introduction The concept design methodology for monotonous, ta- pered, thin-walled structures (wing /fuselage /ship /bridge) is presented. The problem solution is based on the OCTOPUS program [1, 2]. It contains: (A) response and feasibility anal- ysis modules (FIN-CREST), (B) decision making-synthesis modules (DeMak) and (C) interaction /visualization programs (MAESTRO MM/MG and DeVIEW) that irerate in the design cycle. The modules are summarized in Table 1 as modules 1a–8c. (A) The analytical (CREST) modules and methods are fully described in [2]. Module-1a INDAT is used for data genera- tion, combined with the MAESTRO FEM MODELER [8]. Module-2 LOAD is used for design load generation. Mod- ule-1b MIND is used for determining the minimal scantlings based on prescribed rules (LR, DnV, ABS, etc.) Module-3a LTOR is used for direct calculation of the primary strength (shear flow and corrected stresses in bending and warp- ing torsion). They are calculated using an original extended beam theory[5, 6]. Module-3b TOKV is used for the trans- verse strength calculation (newly developed 8-node stiffened panel macro-elements are used for modeling the transverse structural response). Module-4 is the PANEL library of struc- tural serviceability and ultimate strength criteria for the structural adequacy calculation [4], using the response fields generated in modules 3a and 3b. Module-8a is VB-SHELL for the designer-model interaction. Module-8b MAESTRO GRAPHIC [8] is used for presenting the model (loading, response, adequacy, etc.) (B) Synthesis (DeMak) modules and methods are documented in [1, 3]. Local variables for substructures (s � 1, …, NS), are denoted xs � {xi} s � {tplating, nstiffeners, hweb …}s. Substructure areas X � {Xs}are intermediate (global) variables, where Xs � Xs(x s). Project k is defined as Pk � {x1, …, xNS, xfixed} k. Design criteria (attributes, objectives, con- straints) are formulated as a library of mathematical func- tions/procedures for driving the optimization process or feasibility check. OCTOPUS metamodeling of failure sur- faces is based on the most unsatisfied constraint from each local problem. They are added to the set of global constraints. The value function for a global level is a multicriterion combi- nation of normalized attribute functions. The solution strat- egy involves generation of designs, using (a) a Random Num- ber Generator in the first cycles of design space exploration, and (b) Fractional Factorial Experiments for subsequent cy- cles. Coordination is performed by modifying v(xs) respective to its divergence from globally optimal substructure area Xs . Special provisions: � generation of promising designs using 27 designs obtained from Orthogonal Array L27. � extensive usage of tables of optimized profiles to speed up the generation process. Modules 6 and 7c (GAZ) are used for calculating the sensi- tivity of the structural response with respect to the design variables, based on the global strength module FIN/LTOR [7]. Module-7a GLO is used for global level MODM optimiza- tion (level 1) of the cross-section. Module-7b LOC is used for local coordinated MADM decision making via sequen- tial application of stochastic search methods and Theory of experiments. (C) Module-8c DeVIEW is used for the designer-model in- teraction with graphic presentation of designs in the design and attribute spaces. The stratified distances from the target (or the ideal) design, calculated by Lp metric are used as a means of visualizing the multidimensional space of the design attributes and/or free variables. Visualization is the most pow- erful tool for the designer’s understanding of the Decision Support Problem. It generates expert knowledge about the problem for all participants involved, and helps the designer to identify advantageous combinations of variables, feasible options and clusters of non-dominated designs (Pareto fron- tier, thus enabling realistic decision support to the head and structural designer). 96 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 45 No. 4/2005 Czech Technical University in Prague Primary Response Assessment Method for Concept Design of Monotonous Thin-Walled Structures V. Zanic, P. Prebeg A concept design methodology for monotonous, tapered thin-walled structures (wing/fuselage/ship/bridge) is presented including modules for: model generation; loads; primary (longitudinal) and secondary (transverse) strength calculations; structural feasibility (buckling/fa- tigue/ultimate strength criteria); design optimization modules based on ES/GA/FFE; graphics. A method for primary strength calculation is presented in detail. It provides the dominant response field for design feasibility assessment. Bending and torsion of the structure are mod- elled with the accuracy required for concept design. A ‘2.5D-FEM’ model is developed by coupling a 1D-FEM model along the ‘monotonity’ axis and a 2D-FEM model(s) transverse to it. The shear flow and stiffness characteristics of the cross-section for bending and pure/restrained torsion are given, based upon the warping field of the cross-section. Examples: aircraft wing and ship hull. Keywords: thin-walled structures, shear flow, FEM, concept design. 2 Modeling philosophy for primary response in concept design Classical FE modeling, giving good insight into stresses and deformations, is not capable of giving the efficient and fast answers regarding feasibility criteria (buckling, fatigue, yield) required by the Rules. However, structural feasibility and compliance with the Rule requirements are of primary in- terest, not stresses or deformations. Most of the local failure criteria, e.g. various buckling fail- ure modes of stiffened panels, require specified force and dis- placement boundary conditions. They are available only if logical structural parts, such as complete stiffened panels between girders and frames, are modeled (macro-elements). For the concept design structural evaluation of the primary response (longitudinal strength, torsional strength), the beam idealization of a wing /ship /bridge is often used. A primary strength calculation provides the dominant re- sponse field (Demand) for design feasibility assessment. The evaluation is based on extended beam theory, which needs cross-sectional characteristics. These are obtained using analytical methods, which can be very complicated for real combinations of open and closed cross-sections. Application of energy based numerical methods gives an opportunity for an alternative approach to the given prob- lems. The method is based on decomposing a cross-section into the line finite elements between nodes i and j with coordinates (yi, zi), (yj, zj); element thickness t e; material char- acteristics (Young’s modulus E /shear modulus G); material © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 97 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 4/2005 MODULE OCTOPUS (1a) STRUCTURAL MODEL MAESTRO files generated by program MM and used in OCTOPUS (s/r INDAT) (2) LOAD MODEL Rule Loads + designer given loads generated automatically by OCTOPUS s/ r LOAD (1b) MINIMAL DIMENSIONS Minimal dimensions by OCTOPUS s/r MIND (3a) RESPONSE CALCULATIONS - - PRIMARY STRENGTH (u - displ.; stresses �x, �) Extended beam theory (cross section warping fields in bending and torsion, normal stresse s, respective shear flows) program LTOR (3b) RESPONSE CALCULATION - -TRANSVERSE STRENGTH (displacements v, w, �x stresses �y) FEM calculation using beam element with or without rigid ends and stiffened panel macroelements program TOKV (4) FEASIBILITY CALCULATION ii ii i DC DC g � � � (Normalized Safety Factor) Calculation of macroelement feasibility using library of safety criteria in program PANEL (C – capability; D – Demand from 3a and 3b) (5) RELIABILITY CALCULATION (not used) FORM approach to panel reliability. Upper Dietlevsen bound as design attribute (6) DECISION SUPPORT PROBLEM DEFINITION (interactive) Constraints: User given Minimal dimensions Library of criteria (see 4) Objectives: minimal weight, minimal cost, maximal safety (7a, b, c) OPTIMIZATION METHOD Decision making procedure using a) Global MODM optimization program GLO b) Local MADM optimization module LOC c) Coordination module GAZ (8a, b, c) PRESENTATION OF RESULTS a) VB Environment, b) Program MG, c) DeVIEW graphic tool Table 1: Summary of OCTOPUS modules � x px+ x x qy q(x) x x qy q(x) x Fig. 1: Transverse strip (S1-S2) with external loading p, warping fields u and 1D / 2D FEM idealization efficiency RN and RS (due to cutouts, lightening holes, etc.) with respect to normal /shear stresses. Using the FEM approach, a procedure is developed for calculating the set of cross-sectional geometric and stiffness characteristics at position x denoted Gx with the following elements: � Cross-section area A � Center of gravity YCG, ZCG, � Shear /torsion center YCT, ZCT. � Moments of inertia with respect to the center of gravity: IY , IZ, Iyz, Ip ; principal: I1, I2, �0-angle of axis-1 w.r.t. Z-axis, � Horizontal and vertical bending: � Flexural stiffness EIZ , EIY � Shear stiffness GAV, GAH, � Cross-section axial stiffness EA � Torsional stiffness GIT � Warping stiffness EIW. Standard stiffness matrices (alternatively with geometri- cal nonlinearity [4]), for axial (ku) and flexural (kv, kw) and torsional response modes are given as functions of the geo- metric set Gx: k k u v � � � � � � � � � � � AE L EI L Symm L LZ Y Y 1 1 1 1 1 12 6 4 3 2 ; ( ) ( ) � � � � � � � � � � � � � � � � 12 6 12 6 2 6 4 12 2 2 L L L L L EI Y Y Y z ( ) ( ) ; � � � L GAY 2 , kw is obtained similarily. Stiffness matrices for free torsion (kT) and restrained warping (k�) read: k T T� � � � � � GI L Symm L L L L L L L 6 5 10 2 15 6 5 10 6 5 10 2 30 10 2 15 2 2 2 � � � � � � , k 4 � � � � � � � � � � � � � EI L Symm L L L L L L L W 3 2 2 2 12 6 12 6 12 6 2 6 4 . Global stifness matrix K1D is obtained by the combination of modal stiffnesses corrected for centroid and shear centre position relative to the position of the origin of global C.S. a u v w z L y z x � � � � � � � � � � � � � � � � � � � � � � � � T T T T CG � � � � � � , 1 0 0 0 � � � � � � � � � � y z y z y CG CG CG CS CS 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 � � � y z U V W y z CS CS0 0 0 0 0 0 0 0 0 0 1 � � � � � � � � � � � � � � � � , x � � � � � � � � � � � � � � � � � � � � � � � t aG f f f f f k k k k k L u v w u v w T 0 0 0 0 0 0 0 0 0 0 0 0 � � � � � � � � � � � � � � � � � � �+ � � � � � � � � � � � � � � � � � � � � � � � � � � � � a a a a k a u vB wB L L � e where a T a T t tL G � � � � � � �, 0 0 . Finally, the global stiffness matrix K1D is obtained as the sum of the element stiffness matrices k T k TG L e T e� with ap- propriate node numbering. The system K1D a � F can now be solved for unknown displacements a which in turn enable de- termination of element parameters aL. From these parame- 98 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 45 No. 4/2005 Czech Technical University in Prague Fig. 2: OCTOPUS/MAESTRO GRAPHIC shear stress fields in wing and bulk-carrier transverse strip ters the element axial, bending and torsion parameter distri- butions, based on the applied shape functions, can be derived (e.g. �(x), �, x(x), �, xx(x), �, xxx(x) for torsion). The key element for calculation of the response of the complex thinwalled structure is therefore determination of elements of Gx. A sim- ple and elegant FEM procedure for such a calculation is pre- sented in the sequel. 3 Calculation of response for a transverse strip with a complex cross section The shear flow and geometrical characteristics of the cross section in bending and torsion is usually calculated using analytical methods. Such calculations become rather com- plicated for multiple-connected cross section graphs with a combination of open and closed (cell) contours. Application of numerical methods based on the energy approach offers an elegant alternative. The procedure is based on section decomposition into finite elements, as first introduced by Herman and Kawai. In the sequel, the method of calcula- tion as described in [5, 6] is presented. It has been successfully used in practical calculations since its development for [10]. The simplest decomposition of thin-walled cross-section (symmetric or not) into line finite elements (segments) is shown in Figs. 1 and 2. These elements form stiffened panel macro-elements for the feasibility evaluation. The methodology is based on applying the principle of minimum total potential energy (�) with respect to parame- ters which define the displacement fields of the structure. The primary displacement field (following classical beam theory) is defined via displacements and rotations of the cross section as a whole. Secondary displacement field u u2( , , ) ( , , )x y z x y z� represents warping (deplanation) of the cross section. For piecewise-linear FEM idealization of the cross-section, di- vided into n elements, with shape functions N in the element coordinate system (x, s), the warping field reads: u s s l s l u ux x e e e e i j ( ) � � � � � � � � � � � � � � � � 0 1N uT . Element strain and stress fields � and � are obtained from the strain-displacement and stress-strain relations: � � � � � � � ��� � � � � � � � � � � xs e e e i j u s l l u u B uT 1 1 , and � � �� � � xs e xs e eG G B uT . The total potential energy of the �x -long transverse strip of the beam, with the cross-section divided into n elements, reads: � � � � � � � � � � � � �� � � T V Sn V V p x s u s S V F e e e d d dT ( , ) ( ) ( 1 2 B B s u s S x Se e e e e e e e ) ( ) , d T T �� � � � � � � � � �� � � �� � 1 2 u K u u F where p x s( , ) is the external loading on two cross sections (S1 and S2) of the strip. Minimization of � leads to the classical FEM matrix relation K2D u2D � F2D (shortened to K u � F). The element stiffness matrix for the proposed linear displace- ment distribution along the line element (the same for bend- ing and torsion) reads: K e e e e e G t RS l � � � � � � � � 1 1 1 1 , where RS is the prescribed shear efficiency. 4 Cross-sectional shear stress distribution due to bending In the case of bending, the net external load (due to bend- ing moments M(x+�x) and M(x)) is the normal stresses: p x s x x s x s x x s x Q x s I S S C ( , ) ( , ) ( , ) ( , ) ( ) ( ) � � � � � � � � � � � � �2 1 ( )x x� , where � C s( ) is distance from the point to N. A. The load vector for a nonsymmetrical cross-section in, e.g., bending about the z axis reads: F Fz e y z e e y e e Y Z YZ Y x Q x x E Q x t RN E I I I I ( ) ( ) ( ) ( ) ( ) � � � 2 � � � � � � � � � � � � � � � � � y l l y l l i e e e i e e e C C 2 6 2 3 2 2 sin sin � � I z l l z l l YZ i e e e i e e e � � � � � � � � � � � � � � C C 2 6 2 3 2 2 sin sin � � � � � � � � � � � � . For bending around the Y and Z axes, the matrix relations K u � F with u u� Q x( ) can be converted into expressions for the warping due to unit load F. For node warping ui(x), unit warping u( x) must be multiplied by Q(x). This enables the assessment of shear stresses �Y e or �Z e from the expression � e j i eG u u l� �( )2 2 in each element e between nodes i and j. If necessary, it is possible to calculate shear stress distribution �xs e s( ) more accurately, from the mean stress �xs ke obtained from FEM, and the contribution to each element calculated analyt- ically � �� � � �xse u xs ke u u ke us s( ) ( ) ( )� � �1 1 u � y or z from expres- sion (for symmetrical section): � ��xse z z e T y e e e Y e i y s Q G E RN EI l z z ( ) � � � � �� � B u 3 1 2C C ! " # � � � � � ! " # # � � � � z s z z s l i j i eC C C( ) . 2 2 In this case, the sectional characteristics and shear center are easily obtained. The shear /torsion center position reads: Y t d sSC Z e e e l n Q e Z � � � � � � ��� � � C d 0 1 , Z t d sSC Y e e e l n Q e Y � � � � � � ��� � � C d 0 1 , where deC is the normal distance from the centroid to e. The shear stiffness for bending about the Y and Z axes, GAV , GAH reads: © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 99 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 4/2005 � � GA G t s RS xs e y Q e e l e eY e V d� � � � � � � � � � � � �� ( )� 1 2 0 � � � � �1 ; � � GA G t s RS xs e z Q e e l e eZ e H d� � � � � � � � � � � � �� ( )� 1 2 0 � � � � �1 . 5 Corrected normal stresses due to the influence of shear (shear lag) The normal stress must be corrected for (a) stress arising from a longitudinal change of the warping field and (b) nor- mal stress due to correcting bending moment (Mc), compen- sating for the loss of cross section equilibrium: (a) ( ) ( )� � � x c y i e y i e z y i y c eE u x E p u u RN� � � � ! " # � � � E � ; (b) � �M s z s t syc xc y c e l e e � �� � ( ) ( ) d 0 . The total normal stress correction in node i reads: ( ) ) (� x cT y i e Z i YZ i Y Z YZ Y c e e zRN I z I y I I I M E E p u� � � �C C E( 2 y i y cu� � � � � �) . The approximate value of normal stress for simultaneous bending about axes y and z for node I reads: � �x i e Z Z e i x cT z i e Y Y e i RN M EI E y RN M EI E z � � � � � � � � � C C ( ) (� x cT y i) . � � � � � 6 Calculation of warping and primary shear stresses due to pure torsion A transverse strip of a thin-walled beam of length x is subjected to torsional loading. The displacement field of the middle line of thin walled elements can be expressed using the warping function u s t( ) �0, rotation v xs t( ) �0 around the centre of twist, twist rate (�x,x) and angle (�x) of the twist reads: u x s u s xt x,x( , ) ( ) ( )� �0 0� , v x s t v x d xs t t x( , , ) ( ) ( )� � � �0 0 0T � , where dT is the normal distance from the element to the cen- ter of torsion. The strain (with s $ 0) and stress fields read: � � � � � � � � � �� � ! " # � � � �� � � � �� � � � � � x xs x xx x x u u s d , ,T , � �� � �� � ! " # � � � �� � � � �� E E u G u s d x xx x x � � � � , ,T . The total potential energy of a section is given by the stan- dard expression: � � � � ��U W V W V 1 2 � � T d . After summation of all elements and transformation of lo- cal element displacements u N ue e� T and loads F e into global displacements u and loads F we get: U x RS d l t Gx x e e e e e e� � � � � ! " # #�� � , 2 21 2 1 2 u Ku u FT T T . Where K BBe e e e l RS G t s e � � Td 0 , and F Be e e e e l RS G t d s e � � �T d 0 . Minimization of total potential energy leads to two sets of equations: (1) � �� % � 0 (1D beam torsion) and (2) �� u � 0 (2D cross-section warping). A second set of equations, � �� u uU� � � �0 Ku F , en- ables determination of the unit warping field. The primary shear stresses on the elements which are parts of closed contours (cc) and open sections (os) can now be calculated as functions of 1D twist rates �, x(x) (to be obtained from the first relation for 1D beam torsion): � �xs ke e x x e e i j eG l l u u d ( ) , cc T� � � � � � � � � � � � � � � � � ! " 1 1 # # and � �xs ke e x x e G t max , (os) � 2 . 7 Calculation of torsional and warping stiffness of thin-walled structures To solve the equation for 1D beam free torsion, the tor- sional stiffness of elements which are parts of the open eo and closed cells ec can now be calculated using the known unit warping field u: GI G l t RSe e e e e To o � � 3 3 , GI G l t u u l d RSe e e j i e e e e Tc T c � � � � � ! " ## � 2 and GI GI GIT To Tc� � . Warping stiffness is calculated using the expression: EI E l t u u u u RNe e e i i j i e e W � � � � 3 2 2( ) . Using GIT and EIW, the matrix K1D for the 1D beam problem can be formed and relevant parameter distributions �(x), �, x(x), �, xx(x), �, xxx(x) can be determined for use in shear stress calculations. 100 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 45 No. 4/2005 Czech Technical University in Prague 8 Normal and secondary shear stresses due to restrained warping Restrained warping of a thin-walled beam will induce (a) normal stresses in a cross-section and (b) secondary shear stresses which will balance the longitudinally non-uniform distribution of normal stresses. This additional mechanism will influence the strain energy and the work of expression, so an iterative solution may be needed for greater accuracy. Let u x s u s xx x( , ) ( ) ( ),� � be the warping field in the cross- -section calculated from the case of free torsion. Normal stresses are caused by restraining the warping, and vary along the x axis. They are given by: � �� � � �x x x xE E x u s xw � � ( ) ( ), or � �x i e x xx i eE u RNw � , . Let u2(x, s) be the secondary displacement field containing a displacement correction due to restrained warping. The to- tal potential energy of a transverse strip consists of the inter- nal energy generated from the fields �2 and �2 (based on u2) and the additional work done by the strip axial load px on the secondary displacements u2. If the change of u2 along strip length � x is neglected, the total potential energy reads: � �� � � � ! " # � � � � �e e e V x S G u s V x p x u x s S e e 1 2 2 2 2 2 � � � � d d� ( , ) � � � � � � � e . The net external load �px due to restrained warping reads: � � �p x s x s E u s x RN xx x e x xxx e( , ) ( , ) ( ) ( ),� � � � � �w , and the total potential energy of the element, using the same shape functions as before, reads: � e e e e e l ex t G RS s x e � � � ! " # # # � � 1 2 2 0 2 2 � � u BB u u T Td e x xxx e e e l et E RN s e T Td� , . � � ! " # # # � N N u 0 Minimization of the total potential energy with respect to the unknown displacement field u2 leads to: ��u x2 0 02 2� � � � � � �� ( )Ku F Ku F, where: K is the global stiffness matrix as before, u2 is the global vector of unknown displacements u u2 2� �x xxx, , F is the global load vector F F� �x xxx, . The element load and the secondary shear stresses (constant on element) read: F e x xxx e e e e i j xRN E t l u u � � � � � � � � � � � � �� �, , 1 3 1 6 1 6 1 3 xxx eF ; � � � �xs ke e e j i e e j i e x xxx G u s G u u l G u u l ( ) , 2 2 2 2 2 2� � � � � . The shear stress distribution can be calculated more ac- curately along the element (similar to the bending case) from the known element average stress �xs ke( )2 , the direction of shear stress flow, local element contribution �2( )s and its average �2 ke using expression � � � �xs e xs ke kes s ( ) ( ) ( ) ( ) 2 2 2 2� � � . After rearranging, it reads: �xs e e j i e e e e i j e i s G u u l E RN RS u u l u ( ) ( ) ( ) 2 2 2 1 2 3 � �� � � � � � �� � ! " # # � � � � s u u l s j i e x xxx2 2 � , . 9 Examples The first example is based on reports from the Advanced Subsonic Technology (AST) program. In the course of this program an experimental model of a composite wing box was made and tested (Fig. 3). [11] gives the loads carried by the hydraulic actuators which simulate the in-flight loading conditions. The example shows a way of rapidly modeling and calcu- lating the overall response of a similar metal wing box with linear behavior during the early stages of wing structural design. The analyzed wing box with reference to the AST box was shortened to 9.8 meters and modeled with high strength aluminum alloy 7075. The loads are decreased for the short- ened wing box. The wing box is modeled by OCTOPUS 1D/2D combination. Fig. 4 presents a model of the aluminum wing-box under modified �2.5G loading conditions, and the unit response which needs to be multiplied by the values in parentheses to get the actual values for the considered load. The response components are decoupled to show the influence of each type © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 101 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 4/2005 Fig. 3: MD 90 airplane and the test model of the wing box 102 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 45 No. 4/2005 Czech Technical University in Prague a) Metal wing box with 13 elements b) Normal stress (Bending) c) Normal stress (Restr. Warp.) d) Shear stress (Bending) e) Shear stress (Torsion) f) Shear stress (Restr. Warp.) � � xxxW , 1 �� � �MB 1 � 13 1211 10 9 7 8 6 1 2 3 4 5 � �Q Q 1 � � �xxT ,1 �� � �xxxxW ,1 �� Fig. 4: Wing box model and unit responses of 1D FEM element 10 (between ribs 10 and 11) Stress Accuracy (Strake 5) -200.0 -150.0 -100.0 -50.0 0.0 50.0 1 2 3 4 5 6 7 8 9 10 11 12 13 Element S ig m a /T a u [N /m m 2 ] SIGx M TAUxy M SIGx O TAUxy O (a) (b) Fig. 5: (a) Stress accuracy across a wing span between Octopus (O) and Maestro (M), (b) cross-sectional properties of 1D-FEM element 10 1D-Element 10 yCG [mm] �14.5818 zCG [mm] �337.763 � [°] 7.14E-02 ySC [mm] -16.4946 zSC[mm] �354.305 A [mm2] 66270 Iy [mm 4] 2.78E+10 Iz [mm 4] 1.66E+09 Iyz [mm 4] 3.26E+07 I1[mm 4] 2.78E+10 I2 [mm 4] 1.66E+09 It [mm 4] 3.96E+09 Iw [mm6] 2.85E+14 Av[mm 2] 2412.777 Ah[mm 2] 31405.07 y z 6,39(0,502)(0,304) (0.09) (0.686) 1.6 (1.39) (1.65) (1,387) 1.66 (0.59) 1.6 �WT[MPa] �B 17 1614 15 12 10 13 9 8 7 6 5 2 1 4 3 11 835 (881) z y 121311.218 .5 4.9 2.2 3.1 3.1 16.6 12.3 16.5 21.9 27.2 10 16 1 0.6 16.7 13 12 16 14 13 10 9 68 4 7 5 3 12 �2WT[kp/cm 2 ] Fig. 6: (a) U-beam shear stress �W, (b) container ship shear stress �W, (c) general cargo ship shear stress �Bvert of load on the response. The cross section geometric charac- teristics obtained from 2D FEM and used in 1D FEM analysis are given in Fig. 5b. The accuracy of the method is demonstrated in Figures 5a and 6. Strake 5 of Fig. 5a is located in the middle of the upper skin. Fig. 6 presents the application to the U-channel and two standard ship structures. It can be seen that the accuracy of the shear stress distribution based on FEM (constant per ele- ment) and analytical formulae (continuous line) in examples 6a and 6c is very good, even without parabolic correction. The verification examples are taken from [5]. 10 Conclusions A simple and practical method for calculating the primary response of monotonous structures (wings, ships, bridges) has been presented. All cross-section parameters are easily deter- mined for complex stiffened thin-walled structures using a special FEM procedure. It could successfully replace classical, often cumbersome, analytical calculations. The method has been in constant use since 1980, applied to many real struc- tures for concept design (in the OCTOPUS system) or as the generator of the force boundary conditions for partial 3D FEM models. References [1] Zanic, V., Das, P. K., Pu, Y., Faulkner, D.: “Multiple Cri- teria Synthesis Techniques Applied To Reliability Based design of SWATH Ship Structure.” (Chapter 18). In: In- tegrity of Offshore Structures 5, (Faulkner, Das, Incecik, Cowling, editors), EMAS Scientific Publications, Glas- gow, 1993, p. 387–415. [2] Zanic, V., Rogulj, A., Jancijev, T., Bralic, S., Hozmec, J.: “Methodology for Evaluation of Ship Structural Safety.” Proc. of 10th Intl. Congress IMAM 2002, Crete, Greece, 2002, p. 54 + CD. [3] Zanic, V., Andric, J., Frank, D.: “Structural Optimization Method for the Concept Design of Ship Structures.” Proceedings of the 8th International Marine Design Conference, Vol. 1; Papanikolau, A. D. (ed.), Athens, Greece, 2003., p. 99–110. [4] Hughes, O. F.: Ship Structural Design. Wiley, 1983, SNAME 1992. [5] Zanic, V.: “Calculation of Shear Flow in Cross-Section of Ship in Bending.” (in Croatian). Proc. of Sorta Con- ference: Theory and practice in Shipbuilding, Split, Croatia, 1982. [6] Zanic, V.: “Determinazione Degli Sforzi Principali -Fles- sione e Torsione-Sollo Scafo di una nave Applicando Particolari Elementi Finiti.” Technica Italiana (Rivista D’Ingegneria), No.3, 1985, p. 105–113. [7] Prebeg, P.: Diploma thesis, University of Zagreb, 2003. [8] MAESTRO Documentation, Proteus Eng. Stevensville, MD, USA, 2003. [9] CREST Documentation, Croatian Register of Shipping, Split, Croatia, 2004. [10] Hughes, O. F., Mistree, F., Zanic, V.: “A Practical Meth- od for the Rational Design of Ship Structures.” Journal of Ship Research, Vol. 24 (1980), No. 2, June 1980, p. 101–113. [11] Karal, M.: AST Composite Wing Program – Executive Sum- mary, NASA CR 2001-210650, 2001. Vedran Zanic phone: +385-1-6168122 fax: +385-1-6168399 e-mail: vedran.zanic@fsb.hr Pero Prebeg e-mail: pero.prebeg@fsb.hr University of Zagreb Faculty of Mechanical Engineering and Naval Architecture I. Lucica 5 10000 Zagreb, Croatia © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 103 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 4/2005