Optimum design of welded stiffened plates loaded by hydrostatic normal pressure*

Transcription

1 Please reffer to the following article: Farkas J., Jarmai K. Optimum design of welded stiffened plates loaded b hdrostatic normal pressure STRUCTURAL OPTIMIZATION (ISSN: ) 0: pp (000) Optimum design of welded stiffened plates loaded b hdrostatic normal pressure* J.Farkas and K.Jármai Universit of Miskolc, Hungar, H-515 Miskolc, Hungar, altfar@gold.unimiskolc.hu Abstract The optimal positions of horizontal stiffeners are computed considering the condition that the base plate parts, having equal thicknesses and loaded b bending moments, should be stressed to ield strength. The trapezoidal stiffeners are designed for bending using the stress and local buckling constraints. The optimal number of stiffeners is determined on the basis of material and fabrication cost calculations. It is shown b a numerical example that the non-equidistant stiffener arrangement gives 6% weight and 19-1% cost savings. Kewords Stiffened plates, minimum cost design, welded structures, hdrostatic pressure 1 Introduction Stiffened plates can be loaded b hdrostatic pressure in several structures, for instance in bunkers and gates (Fig.1). In the book of Wickert and Schmausser (1971) a solution is given to obtain the optimal distances of horizontal stiffeners using the condition that all the stiffeners should be equall loaded. This solution disregards the dimensioning of the base plate, thus, it does not give a realistic optimum. McGrattan (1985) has worked out a weight and cost optimization for equall spaced longitudinal and orthogonal stiffeners. This stud does not consider the non-equidistant arrangement of stiffeners. Kravanja et al. (1998) have used a mixed-integer nonlinear programming method for cost optimization of hdraulic steel gate structures. In their article the problem is described generall for non-equidistant horizontal stiffeners and the numerical example of the orthogonall stiffened Intake Gate (Aswan, Egpt) structure is solved showing 8% cost savings compared with the actual solution. In the present stud the stiffened plate consists of a square base plate of constant thickness and horizontal stiffeners. Vertical stiffeners are also treated for cost comparison. This arrangement is selected for its relative simplicit regarding the fabrication. When the horizontal stiffeners are equall spaced onl the lowest part of the base plate can be stressed to ield strength, thus, these solutions are not optimal.

2 The optimum position of stiffeners can be calculated on the basis of the condition that all parts of the base plate, having the same thickness, should be stressed to ield strength. *This stud was partl presented at WCSMO- held in Buffalo New York For the calculation of the maximum bending moments in simpl supported base plate parts due to hdrostatic pressure the values given in tabulated form in (Vanberg 1970) are approximated b nonlinear functions of the ratio of side dimensions b/a (Fig.). The optimum stiffener positions characterized b xj (Fig.) are calculated from a set of nonlinear equations using the MathCAD software. Trapezoidal stiffeners are considered (Fig.4) and designed for bending considering also the local buckling constraint. Figure 1. A bunker constructed from stiffened plates The cost function consists of material as well as fabrication costs and is calculated in the function of number of stiffeners for equidistant and non-equidistant (optimal)

3 stiffener positions. It is shown that, using optimall spaced stiffeners, significant mass and cost savings can be achieved. Optimum position of horizontal stiffeners The thickness of each base plate part can be calculated from the stress constraint 6M t F f (1) where M is the factored bending moment, tf is the base plate thickness, f is the ield stress. Figure. Points where the maximum bending moment arises The maximum bending moment in a simpl supported rectangular plate due to hdrostatic pressure can be obtained using tables given in the book of Vanberg (1970) in the form of M max = k1 pa () where k1 is given b tabulated values depending on the ratio b/a. In a plate stiffened b horizontal stiffeners the upper plate part is loaded b a distributed load of triangular shape, in which the maximum bending moment arises at point I in distance of a/8 from the center point O (Fig.). This bending moment is given b k1. Tabulated values of k1 can be approximated in function of x0 = b/a in the following form c1 10 k = 1 1+ d exp( f x ) ; c1 = 6.877, d1 = 1.974, f1 =.1664 () 1 1 0

4 The maximum bending moment in the plate parts loaded b a distributed load of trapezoidal shape, given b p1 and p, can be calculated adding two moments as follows: one moment is due to uniform loading p1 and the second is due to triangular loading p - p1. Calculation shows that the resulting moment is larger for point O than that for point I, thus we use tabulated values for point O. M = k p1a + k( p p1 ) a (4) where k and k can be approximated b functions as follows Figure. The distances of stiffeners and the pressures for the calculation of bending moments acting on stiffeners k k = c 1+ d exp( fx0) ; c = 1.044, d = 1.775, f =.1695 (5) = c 1+ d exp( f x ) ; c = , d = , f =.1709 (6) 0 In the case of n stiffeners the position of the jth stiffener is characterized b xj (j = 1,...,n) (Fig.) and the maximum bending moments in the base plate parts can be given as follows: M = k px / ; in k1 x0 = b/x1 (7) a M j = k px ( x x ) / a + k p( x x ) / ; in k and k x = b /( x x ) (8) j 1 j j 1 j j 1 a 0 j j 1 Mn+ 1 = k pxn ( a xn) / a + k p( a xn) / a; in k and k x0 = b /( a xn ) (9) Using these formulae n equations can be written expressing that these bending moments should be equal to each other, since the thickness of the base plate parts is the same. The set of equations for unknowns xj is the following:

5 k1 x1 = kx1 ( x x1 ) + k( x x1 ) (10). k 1 x1 = kx j 1( x j x j 1) + k( x j x j 1) ) ; in k and k x 0 = b /( x j x j 1) (11). k x = k x a x ) + k ( a x (1) 1 1 n( n n) This set of equations is solved b MathCAD software. Design of stiffeners Having obtained the optimal stiffener positions, the required dimensions of trapezoidal stiffeners can be calculated. The maximum bending moment in the jth stiffener is given b (Fig.) pb M x x x + x x (1) ( )( ) Sj = j+ 1 j 1 j+ 1 j + j 1 64a Considering trapezoidal stiffeners according to Stahlbau Handbuch (1985) with given dimensions a1 = 90, a = 00 mm, appling the local buckling constraint for the inclined webs according to Eurocode a 8t S ; = 5/ f (14) Taking equation (14) as equalit, the section characteristics of a stiffener can be expressed b the unknown thickness ts. The cross-sectional area is (Fig.4) A = ( t S ) t S (15) the height of the stiffener and the angle of the web is Figure 4. Dimensions of a trapezoidal stiffener ( 8 ) / h = ; t S The distance of the gravit center G is sin 105 = 1 8 t S (16)

6 z G a1 + a = hts (17) a t + a + a ) t 4 F ( 1 S and the moment of inertia is I a t sin h ( 6 S = a4tf zg + a1t S h zg ) + + ats zg (18) The stress constraint for a stiffener is expressed b M( h zg ) / I f / M1 ; M1 = 1.1 (19) from which the required stiffener thickness ts can be calculated. 4 The cost function We have developed on the basis of COSTCOMP (1990) software (Bodt 1990) a cost function containing the material and fabrication cost (Farkas-Jármai 1997, Jármai- Farkas 1999) K k f 0 V C ( V ). 5 = + 1d +1. T. (0) km km where is the material densit, kf and km are the fabrication and material cost factors, respectivel, is the number of structural elements to be assembled, V is the volume of the structure, d is the difficult factor expressing the complexit of a structure (planar or spatial, using simple plates or profiles), the coefficient for the preparation time is C1 = 1.0 min/kg 0.5. To give internationall usable results, the ratio of kf/km is varied in a wide range. For steel km = kg, for fabrication including overheads kf= 0-60$/manhour = 0-1$/min, thus, the ratio ma var in the range of 0 - kg/min, the value of 0 corresponds to the minimum weight design. The welding time is given b n T = CiaWiLWi (1) i the factor of 1. expresses that the additional time for chipping, deslagging and n changing the electrode is approximated b T = 0.T.. The formulae for C a W are given for various welding technologies and weld tpes. aw is the weld size, LW is the weld length. 5 Numerical example To illustrate the optimization procedure the following data are taken: a = b = 6 m, the intensit of the factored hdrostatic load is p = p0 = 1. 5x0. 06 = N/mm, f = 5 MPa. It is assumed that the base plate is butt-welded from plate strips of width

7 1500 mm. A stiffener is welded to the base plate with fillet welds of size aw = 0.7tS. The welding times are calculated with the following data: use GMAW-M welding technolog (Gas Metal Arc Welding with mixgas). For (1) the following formulae are used: aw = mm X butt welds CaW n = 0.14aW aw= 4-15 mm V butt welds CaW n = aW aw= 0-15 mm fillet welds CaW n = 0.58aW and LW is calculated in mm. The difficult factor is chosen for n = 0 (base plate without stiffeners) d =, and for n>0 d =. The results of computations are summarized in Tables 1, and and Fig. 5. Table 1. Results of the numerical example. Optimum stiffener positions, thicknesses and costs for different numbers of stiffeners n xj (m) ts (mm) tf (mm) K/km(kg) for kf/km=0 K/km(kg) for kf/km=1 K/km(kg) for kf/km= Table. Results of the numerical example (continuation of Table 1.) n xj (m) ts (mm) tf (mm) K/km(kg) for kf/km=0 K/km(kg) for kf/km=1 K/km(kg) for kf/km=

8 Figure 5. Costs in the function of number of stiffeners in the case of kf/km=0,1 and Table. Results of the numerical example(continuation of Table.) n xj (m) ts (mm) tf (mm) K/km(kg) for kf/km=0 K/km(kg) for kf/km=1 K/km(kg) for kf/km=

9 It should be noted that, in the case of 8 stiffeners equidistantl arranged, the K/km - values for kf/km = 0, 1 and are as follows: 449, 647 and 8595, respectivel. It means that the non-equidistant arrangement results in savings of 6, 19 and 1%, respectivel. 6 Comparison with vertical stiffeners For another comparison, the vertical arrangement of stiffeners is designed as well (Fig.6). According to Vanberg (1970), the maximum bending moment arises in the point P. When the number of stiffeners is greater than 5, the base plate thickness can be calculated as a simpl supported strip. Thus, from the stress constraint 0.75pb /( n + 1) f () 8t F / 6 M1 the required base plate thickness is Figure 6. Solution with vertical stiffeners 1/ p 0.75b t F () / f M1 n + 1 In the case of our numerical example, for n = 5 and 8 tf = 1 and 8 mm, respectivel. The maximum bending moment in a simpl supported vertical stiffener is b M max = pa (4) n + 1 For this moment the required stiffener thickness can be calculated in the same manner as in the case of horizontal stiffeners. For n=5 and 8 ts=8 and 7 mm, respectivel. The corresponding cost values, in the case of kf/km=0, 1 and kg/min, are as follows: for

10 n=5 4710, 675 and 8794 kg, for n=8 901, 5805 and 7709 kg. This means that, for optimal arranged horizontal stiffeners these values are 4-18% smaller. Figure 7. Values of K/km (kg) for a stiffened steel plate loaded b hdrostatic pressure for kf/km = 1: (a) 647 for 8 stiffeners in equidistant position, (b) 544 for 8 stiffeners in optimized position, (c) 704 for stiffeners in optimized position 7 Conclusions The optimum horizontal stiffener positions can be calculated on the basis of the condition that each base plate part, having the same thickness and loaded b bending, should be stressed to ield strength. This non-equidistant stiffener distribution is more economic than the equidistant or verticall arranged ones. This econom can be verified b cost calculations (Fig.7). It is shown b a illustrative numerical example that the optimum number of stiffeners is 7 or 8 and the non-equidistant stiffener arrangement is 6% lighter and 19-1% cheaper than the equidistant one. The difference between the costs of non-equidistant solutions with or 8 stiffeners for kf/km = 1 is ( )/544x100 = 5%, thus, the search for the optimum number of stiffeners results in significant cost savings. Acknowledgement This stud has been supported b the grants 1900 and 846 of the Hungarian National Fund for Fundamental Research OTKA. References Bodt,H.J.M. (1990). The global approach to welding costs. The Netherlands Institute of Welding. The Hague. COSTCOMP (1990). Programm zur Berechnung der Schweisskosten. Düsseldorf, Deutscher Verlag für Schweisstechnik. Farkas,J., Jármai,K. (1997). Analsis and optimum design of metal structures. Balkema, Rotterdam-Brookfield. Jármai,K., Farkas,J. Cost calculation and optimisation of welded steel structures. J. Constructional Steel Research 50(1999)

11 Kravanja,S., Kravanja,Z. and Bedenik,B.S. (1998). The MINLP optimization approach to structural snthesis. Part III. Snthesis of roller and sliding hdraulic steel gate structures. Int. Journal for Numerical Methods in Engineering 4, McGrattan,R.J. (1985). Weight and cost optimization of hdrostaticall loaded stiffened flat plates. Journal of Pressure Vessel Technolog Transactions of ASME, 107. Febr Stahlbau Handbuch Band. 1985, Stahlbau-Verlag. Vanberg,D.V., Vanberg,E.D. (1970). Calculation of plates (in Russian). Kiev, Budivelnik. Wickert,G., Schmausser.G. (1971). Stahlwasserbau. Springer, Berlin-Heidelberg-New York.