Structural Optimization 8, 262-267 © Springer-Verlag 1994

Minimum cost design of laterally loaded welded rectangular cellular plates* J. Farkas and K. dfirmai

University of Miskolc, H-3515 Miskolc, Hungary Abstract Material and fabrication costs are included in the cost function. The fabrication cost is calculated by three formulae relating to the preparation, welding and additional costs. The design constraints are related to bending stresses, the local buckling of ribs due to bending and shear and to the limitation of the plate thicknesses. The local buckling of the compressed face plate elements is considered by an effective width calculation. In the numerical examples, the variables are the plate dimensions and the numbers of ribs in two directions. The optimization is carried out for steel Fe 360 and Fe 510 and for various values of the fabrication cost factor. The computations are performed by using the backtrack discrete combinatorial method, Rosenbrock's Hillclimb method and the FSQP method developed by Zhou and Tits (1992), and the results are compared with each other. illlttt

1

|

l

~1.

Introduction

A cellular plate consists of two parallel face sheets welded to an orthogonal grid of ribs sandwiched between them (Fig. 1). This type of welded sandwich plate has the following advantages over plates stiffened on one side: (i) the torsional stiffness is much larger, (ii) the height of ribs can be much smaller, (iii) fabricational imperfections due to the shrinkage of welds are much smaller because of the symmetry of the structure, and (iv) the planar surface can be more easily protected against corrosion. Cellular plates may be applied in ships, bridges, dock, gates, light-weight roofs, elements of machine structures, etc. Their disadvantage is that, if the rib height is smaller than approximately 800 mm, the face plates cannot be welded to ribs from inside. Then these connections can be made externally by arc-spot-welding, electron beam welding, slot or plug welds. A brief survey of selected literature was given by Farkas (1992). Research has been carried out predominantly in the field of double bottoms of ships. The present paper is a generalization of the investigation of the minimum cost design of square cellular plates (Farkas 1992). Our aim is to show how to optimize the plate dimensions, mainly the numbers of ribs. The effect of fabrication costs and the yield stress of steel on the optimal number of ribs will be shown by numerical examples. *Partly presented at the international conference "Structural Optimization'93", Rio de Janeiro, August 2-6, 1993.

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if', F

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Fig, 1, Details of a welded rectangular cellular plate 2

T h e cost f u n c t i o n

It is assumed that the fabrication has the following steps. First the grid of ribs is welded from cold-formed channels or from welded/-beams. The grid nodes should be completely welded to be able to carry bending moments and shear forces. Then the elements of the upper and lower cover plates are welded to the ribs from outside with fillet welds (Fig. 1). This method is selected since one cannot find cost data for other welding methods. The ribs are continuous in the y-direction and intermittent in the z-direction. The cross-sectional area of a rib is approximately 2htr where h is the height and tr is the thickness. The integer numbers of rib distances are tax and ~y, respectively (Fig. 1). The number of ribs in the xand y-directions is ~x + 1 and ~y q-1, respectively. Assuming that all ribs have the same cross-sectional area, the whole volume of the cellular plate is

V = 2bxbytf q- 2bxhtrx(~y -4- 1) q- 2byhtry(~x "4- 1), (1) where t f is the thickness of face plates. The total cost consists of the material and fabrication costs

g = Km + I<~ = kmpY + As ~ r ~ , or in another form

(2)

263

K/km = pV + k f / k m ( T 1 + T 2 + T3),

(3)

where km ($/kg) and kf ($/min) are the material and fabrication cost factors, respectively, p is the material density, and T i are the fabrication times in minutes. In order to give internationally usable solutions, the following ranges of km and kf may be considered. For steel Fe 360 km = 0.5-1.2 $/kg, for fabrication including overheads kf =15-45 $/manhour = 0.25-0.75 $/min. Thus the ratio kf,/km may vary in the range 0-1.5 kg/min. The value k f / k m = 0 in (3) corresponds to the minimum volume design. For the calculation of times T i we use the method proposed by Pahl and Beelich (1982).

(a) Preparation, assembly and tacking T1 = C15vf~vfa,

C1 = 1.0 min/kg 0'5 ,

(4)

where 5 is a difficulty factor, and i¢ is the number of structural elements to be assembled. The number of continuous ribs in the y-direction is !az + 1, the number of rib elements in the x-direction (internal intermittent, peripheral continuous, Fig. 1) is !ax(~Oy- 1) + 2, and the number of face plate elements is 2~z!Oy. Thus the number of all structural elements is

= 3(~x~ay + 1) and T 1 = 3 y / ~ / 3 ( ! O z ~ y + 1).

(5)

(b) Welding T2 = E C2iawl'hLwi'

3

T h e design c o n s t r a i n t s

Constraints on compressive elastic stresses in the central upper face plate element are as follows: o'z. max + ~zf. max -< Cradm ,

(10)

ay. max + O'yf. max ~ (Tadm, (11) where Cradm is the admissible stress, az.max and ~y. max are caused by the bending of the entire plate, (~xf max and ayf. max are normal stresses due to the local bending of the face plate element. It can be verified, similarly to the case of a square cellular plate (Farkas 1985), that because of the large torsional cell stiffness, the whole rectangular cellular plate can be calculated as an isotropic one, crx. max = Mx maxElel/B ; ay. max = My maxEle 1 / B . (12) According to the isotropic plate theory (Timoshenko and Woinowsky-Krieger 1959)

i x max = cmzp b2 and My max = CmyPb2 , where p = 1.1p0 , the selfweight is considered with the factor of 1.1, and crux and Cmy are given by Timoshenko and Woinowsky-Krieger (1959) for simply supported edges. The bending stiffness B is calculated considering the effective width of the compressed face plate element (Fig. 2a), ~x

(6)

"////~//////

where C~ = 0.8×10 - 3 min/(mml'5× mm) for manual arcwelding C~I = 0.hxl0 - 3 min/(mml"hx mm) for CO 2welding, and aw and Lw are the size and length of welds in mm, respectively. It is assumed that the grid nodes are joined by manual-arc-welding with fillet welds and the face plate elements are connected to the grid by CO2-welded fillet welds.

(c) Electrode changing, weld deslagging and chipping T 3 = v ~ E C3iawi1.5 Lwi,

C ~ = C'2,

C~' = C~'.

(7)

/////1////, /

"~J-c,,po4

Fig. 2. (a) Effective cross-section for calculation of the bending stiffness; (b) local bending of an upper plate element due to the uniformly distributed lateral load

The number of perpendicular joints of ribs is 2(~x + 1) + 2~x(~y - 1) = 2(~x~y + 1).

B = ElI/a = E:lh2tf¢/(1 + ¢ ) ,

It is assumed that the rib webs are welded with fillet welds of size aw = 0.7try in the length of 2h, and the flanges of ribs of length h are welded with welds of aw :-try (Fig. 1). Thus the T2 + T 3 times for manual-arc-welded nodes are (in rain)

where E, is the modulus of elasticity and v is Poisson's ratio. We use here the effective width formula proposed by Usami and Fukumoto (1982)

+

= (1 + v

)0.8 x 10 - a ×

+

(8)

The total length of fillet welds for face plate elements is

y(2a

+ 2ay) = 4

y(bx/v

or

¢ = 1.426/~p;

all sizes are in mm.

~p = a trY

+ by/Vy),

af~_ax E "

(14)

Substitution of (13) into (12) yields

max = cm pb /(tfh ey),

T~' + T~' = (1 + v/5)0.5 x 10-3(0.5t f) 1'5 ×

+ by/ y),

(13)

a /12(1 v2)~rmax = tf V 7-

and the fillet weld size is taken as awf = 0.5 t f, thus

4V y(b /v

E1 = E/(1 - ¢2),

/

+

2h(O.7try)l"5] .

2v

e I -- h/(1 -t- ¢);

and

(9)

ay max = cmyP b2/(tfh e x ) .

(15)

Elimination of ¢ from (15) is performed using (14) and (15) and yields

264 2 242 24 2 az. m a x = Cmzp bzay/(1.426 t f E h ), 2

242

24

4

2

C~ymax = emyP bzax/(1.426 t f E h ).

(16)

Furthermore, 2 2 2 2 ~xf. max -- 6cfxpoax/tf, and tryf. max = 6cfypoax/tf, (17) where cfz and Cry are given in Table 1 according to Timoshenko and Woinowsky=Krieger (1959) for a uniformly loaded rectangular isotropic plate with clamped edges (Fig. 2b). Table 1. Coefficients for the calculation of bending moments and shear forces in a simply supported uniformly loaded rectangular isotropic plate

b~/b= 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.9 2.0 3.0 4.0 oo 104cm= 479 104cmy 479 lO~cq= 338 103cq~ 338

554 627 594 755 812 493 5011503 502 498 3601380 397 411 424 347 353 357 361 363

862 492 435 365

908 985 i86 471 144 459 367 369

1017 464 465 370

1189 496 493 372

1235 1250 384 375 498 500 372 372

Table 2. Coefficients for the calculation of maximum stresses in a uniformly loaded rectangular isotropic plate with clamped edges

a=/ay

The optimization procedure

In a numerical example the values of p0 , by, bx, ~radm, E, v, crux, Cmy Cqx, Cqy, t o are given and the unknowns to be optimized for m i n i m u m cost Kmi n are as follows: ~x, ~y, h, t f, trz and try. In the cost function the kf/km ratio is varied in the range of 0-1.5. For comparison purposes we have used here the following three mathematical programming methods. 1~3

~(kg)'

Fe 360

40Fe 510

30-

30 /

_i26

"

~}05

23

20

16

}0

15 10-

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 oo

104clz 513 581 639 687 726 757 780 799 812 822 829 833 104cry 513 538 554 563 568 570 571 5711571 571 571 571

7

8

9

1

11

1)

13 14

15

Since cfz and Cry vary during the optimization procedure, these values are calculated with approximate analytical formulae in the form of a p o l y n o m

Fig. 3. Results of a numerical example: minimal costs for various

cfx = co Jr clay/ax Jr" c2(ay/ax) 2 "Jr c3(ay/ax) 3 • On the contrary, values of cruz, Cmy, Cqx and Cqy are constant during an optimization procedure, because by and bx

103~k (kg)'

are given in a numerical example• Constraints on local buckling of rib webs due to bending are

kl/km-ratios and the ~=.opt values

I

kf//km= I.

40-

}I.o los

30-

az. max < 23"97r2E1 (~_x--) 2 -

127-------~

'

(18a)

0

and (ry. max _<

23"97r2E1 ( ~ _ ~ ) 2 12% '

Fe 360 Fe 510

(18b)

where 3'b is the safety factor for buckling. Constraints on local buckling of rib webs due to shear are

1.0

Qzay cqzpbxay<5.34zr2El(t~) 2 rz = htrz htrx - 127-----~ vz < tad m ,

(19a)

Qyax cqypbzax <5.347r2E1 ( ~ 1 2 Vy = htry htry 127b ry < radm,

(195)

where tad m = tradm/X/3 is the admissible shear stress, Cqz and Cqy are given in Table 1 according to Timoshenko and Woinowsky-Krieger (1959). Size constraints are the thickness limitations

trx >_tO,

try >_t o , tf >_tO, (20) where t o is the minimal thickness considering the welding technology. Note that the deflection constraint is not considered here because of the large stiffness of the whole cellular plate.

2

4

%

Fig. 4. Results of a numerical example: minimal costs for various kf/km-ratios in the function of bu/b= (1) The "backtrack" combinatorial method is advantageous here, since the n u m b e r of variables is only 6, ~z and ~y are integer numbers and the thicknesses should be commercially available, so the series of discrete values to be investigated can easily be defined• The starting point should be feasible. A detailed description of backtrack with more numerical applications is given by Farkas (1984). The flow chart of the technique can be seen in Fig. 5. (2) The "Hillclimb" method was proposed by Rosenbrock. The method of rotating coordinates is a further development of the Hooke and Jeeves method. No derivatives

265

Discrete values of variables Xlmax, Xlmln, step sizes

(

Compute starting function value Fo (Xlmax), i = 1 ..... N

Print results )

I

of this variable

tCompute variable Xim by the halving technique

l

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.extvariabta ~ 1 < = . - 1

~

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nY

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Compute the last variable

I

fromtheobjectivefunctionFo

n

n

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[

I l

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firstv a r i a b l e ~

-"-....~.tha°max,mum~

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_/~The

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I

n

~

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o, thisvariable

F

~ o u s

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new minimal function value

Fig. 5. Flow chart of the backtrack method are required. The starting point should be feasible. We have supplemented this method with a secondary search for finding discrete values after having continuous ones (J£rmai 1982). The flow chart of the technique can be seen in Fig. 6. (3) FSQP - feasible sequential quadratic programming method. CFSQP 1.0 is a set of C subroutines for the

minimization of smooth objective functions subject to general smooth constraints (Zhou and Tits 1992). If the initial guess provided by the user is infeasible for some constraints, CFSQP first generates a feasible point. Nonlinear equality constraints are turned into inequality constraints. The user must provide subroutines that define the objective functions and constraint functions or re-

266

,. Start 1

I

Input: starting point xl InlUal step sizes SI I = 1,2 ..... N

Evaluate objective

I >~ I

1

Increment xl from belt point by a distance Si parallel to axis and evaluate

~,~ ~

feasible ? J /

objectivefunction

V

Acceleration Sl(new) = Alpha * Sl(old) Alpha >= 1

I=1+1 FIJ

function

...

1

/ | /

Deceleration and reversal Si(new) = - Beta * Sl(old) 0< Beta <= 1

'

I J<

I

n~?Ny

n

Y

Rotate axis

Set step size

Find the nearest discrete values for the continuous ones

Fig. 6. Flow chart of the Rosenbrock method quire that CFSQP estimates them by forward finite differences. CFSQP solves a modified optimization problem with only linear constraints and nonlinear inequality constraints. An Armijo-type line search is used to generate an initial feasible point when required. After obtaining feasibility, either an Armijo-type line search may be used

or a nonmonotone line search is made and analysed. The C version is a quite new development and we have worked also with the beta version on PCs. All programs are written in C and run under Borland C + + on PC 486 type computers. These codes are quicker

267 than the Fortran and Basic codes and are more transportable, we were also able to run them on workstation. 5

Numerical examples

Data: the intensity of the uniformly distributed normal load P0 = 5 × 1 0 - 3 N/ram2, P = 1.1×p0 = 5.5 × 10 - 3 N / m m 2, to = 2 mm, E = 2.1×105 MPa, v = 0.3, p = 7850 kg/rn 3, 7b = 2, ~ = 3, bx = 10 m. To show the effect of the yield stress of steel, calculations are made for steel Fe 360 with ~adm = 120 MPa, and for steel Fe 510 with ~adm = 120x355/235 = 181 MPa. To show the effect of fabrication costs calculations are made for k f / k m = 0; 0.5; 1.0 and 1.5. The results are shown in Figs 3 arid 4 and Table 3. Figure 3 shows the curves of the objective function as a function of z in the vicinity of the optimum value. It can be seen that for larger k f / k m values - larger fabrication costs - ~x.opt. is smaller. With the use of higher-strength steel Fe 510, 4-12% cost savings can be achieved. The sensitivity of the objective function is small. In Fig. 4 the minimal K / k m cost values are plotted in function of by/bz for steel Fe 360 and Fe 510. It can be seen that the K/kin-values vary with by/bx approximately linearly. These results are obtained by the backtrack programming method. Table 3. Results of a numerical example with bx = 10 m, by = 14 m, steel Fe 360 obtained by three mathematical programming methods, dimensions in mm Method

k f / k m l ~x

~y

tf

trx try

16.0 16.0 8.8 8.0 15.7 9.0 8.0 7.9 15 9 8 8 16 9 8 8

5.1 5.1 6.1 6.0 5.0 6.0 6.0 7.0 5 6 6 7 5 6 6 7

2.2 2.2 3.0 3.1 2.3 3.0 3.2 3.1 3 3 4 4 3 4 4 4

h

K/km

278 285 383 406 299 400 499 374 300 400 450 375 300 400 450 375

14548 23627 31743 38705 14624 24609 31888 39094 15229 25459 33485 40665 16162 26149 33485 40665

(kg) CFSQP without discretization ttillclimb without discretization Hillclimb with discretization

Backtrack

0 0.5 1.0 1.5 0 0.5 1.0 1.5 0 0.5 1.0 1.5 0 0.5 1.0 1.5

13.0 12.8 10;2 9.4 13.1 10.0 10.0 8.0 14 10 10 8 13 10 10 8

2.0 2.0 2.4 2.5 2.0 2.4 2.5 2.6 2 3 3 3 3 3 3 3

In Table 3 the optimal dimensions obtained by three methods are given for a numerical example. It can be seen that the Hillclimb and CFSQP methods resulted in very similar undiscretized optimal values. The results obtained by hillclimb with discretization and by the discrete backtrack are also very similar.

6

Conclusions

Illustrative numerical examples show that, because of the large torsional stiffness of cellular plates, relatively large structures can be realized using thin plates... The optimal number of ribs decreases when the fabrication cost k f / k m increases. The sensitivity of the objective function is small. The use of Fe 510 instead of Fe 360 results in 4-12% cost savings. Active constraints are the normal stress limitation (10) and the constraints on local shear buckling of rib webs (10). The comparison of the three mathematical programming methods shows that the Hillclimb technique is quick hut can result in local minima, the backtrack is suitable for few variables defined by series of discrete values, the CFSQP method is very robust and the starting point can be infeasible. Acknowledgements The authors would like to thank Andre L. Tits and Jian L. Zhou, University of Maryland, for the possiblility of using the CFSQP algorithm. This work received support from the Hungarian Fund for Scientific Research Grants OTKA T-4479 and T-4407 and from the Ministry of Culture and Education Grant No. 167/92. References Farkas, J. 1984: Optimum design o1: metal structures. Budapest: Al~demial; Chichester: Ellis Horwood Farkas, J. 1985: Discussion to "Simplified analysis for cellular structures" by Evans, H.R.; Shanmugam, N.E.J. Struct. Eng. ASCE 111, 2268-2271 Farkas, J. 1992: Cost comparisons of plates stiffened on one side and cellular plates. Welding in the World 30, 132-137 J~rmai, K. 1982: Optimal design of welded frames by the complex programming method. Publ. Techn. Univ. Heavy lnd. Miskolc, Ser. C. 37, 79-85 Pahl, G.; Beelich, K.H. 1982: Kostenwachstumgesetze nach Aehnlichkeitsbeziehungen fiir Schweissverbindungen. VDI-Bericht 457, 129-141 Timoshenko, S.; Woinowsky-Hrieger, S. 1959: Theory of plates and shells (2nd ed). New York: McGraw Hill Usami, T.; Fukumoto, Y. 1982: Local and overall buckling of welded box columns. J. Struct. Div. Proc. ASCE 108, 525-541 Zhou, J.L.; Tits, A. 1992: User's guide for FSQP Version 3.0: a Fortran code for solving optimization problems. Systems Research Center, University of Maryland, Techn. Report SRC-TR-90-60 rlf

Received Aug. 2~, 1993