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Journal of Computational and Applied Mechanics, Vol. 5., No. 1., (2004), pp. 165179

OPTIMUM DESIGN OF STIFFENED PLATES FORDIFFERENT LOADS AND SHAPES OF RIBS

Zoltn VirgDepartment of Equipments for Geotechnics, University of Miskolc

3515 Miskolc Egyetemvros, [email protected]

[Received: December 1, 2003]

Dedicated to Professor Jzsef FARKAS on the occasion of his seventy- fth birthday

Abstract. In this overview of loaded sti ff ened plates various plate types, loadings, andstiff ener shapes are investigated. Mikami [1] and API [2] methods are used for the optimumdesign and comparison of the two methods and uniaxially compressed plates sti ff ened byribs of various shapes. Both methods consider the e ff ect of initial imperfection and residualwelding stresses, but their empirical formulae are di ff erent. The elastic secondary de ectiondue to compression and lateral pressure is calculated using the Paiks solution [3] of thediff erential equation for orthotropic plates, and the self-weight is also taken into account.Besides this de ection some more deformations are caused by lateral pressure and the shrink-age of longitudinal welds. The unknowns are the thickness of the base plate as well as thedimensions and number of sti ff eners. The cost function to be minimized includes two kindsof material and three kinds of welding costs.

Mathematical Subject Classi cation: 74K20,74P10Keywords : stiff ened plate, welded structures, stability, residual welding distortion, structuraloptimization, minimum cost design

1. Introduction

Stiff ened welded plates are widely used in various load-carrying structures, e.g. ships,bridges, bunkers, tank roofs, o ff shore structures, vehicles, etc. They are subject tovarious loadings, e.g. compression, bending, shear or combined load. The shape of plates can be square, rectangular, circular, trapezoidal, etc. They can be sti ff ened inone or two directions by sti ff eners of at, L, trapezoidal or other shape.

Various plate types, loadings and sti ff ener shapes have been investigated. In thispaper two kinds of loads are investigated [6], [7]. These are uniaxial compression andlateral pressure. Structural optimization of sti ff ened plates has been worked out byFarkas [8], Farkas and Jrmai [9], and applied to uniaxially compressed plates withstiff eners of various shapes [10], biaxially compressed plates [11].

This paper contains the minimum cost design of longitudinally sti ff ened plates us-ing the strength calculation methods. De ections due to lateral pressure, compressionstress and shrinkage of longitudinal welds are taken into account in the stress con-straint. The self-weight is added to the lateral pressure. The local buckling constraint

c 2004 Miskolc University Press

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166 Z. Virg

of the base plate strips is formulated as well. The cost function includes two kindsof material and three kinds of welding costs. The unknowns are the thickness of thebase plate as well as the dimensions and number of sti ff eners.

2. Geometric characteristics

The sti ff ened plates are shown in Figures 1 and 2. The plates are simply supported atfour edges. Geometrical parameters of plates with at, L- and trapezoidal sti ff enerscan be seen in Figures 3-5.

Figure 1. Longitudinally sti ff ened plate loaded by uniaxial compression

Figure 2. Longitudinally sti ff ened plate loaded by uniaxial compres-sion and lateral pressure

Figure 3. Dimensions of a at sti ff ener

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Optimum design of sti ff ened plates for di ff erent loads and shapes of ribs 167

The geometrical parameters of the at sti ff ener are calculated as follows

AS = hS tS , (2.1)

hS = 14tS , (2.2)

=

q 235/f y , (2.3)

yG = hS + tF 2 S

1 + S , (2.4)

S = AS btF

, (2.5)

I x = bt3F

12 + btF y

2G +

h 3S tS 12

+ hS tS hS 2 yG2

, (2.6)

I S = h3S

tS 3

, (2.7)

I t = hS t 3S

3 . (2.8)

Figure 4. Dimensions of an L-sti ff ener

The calculations of geometrical parameters of the L-sti ff ener are

AS = ( b1 + b2 ) tS (2.9)

b1 = 30tS , (2.10)b2 = 12.5tS , (2.11)

yG =

b1 tS b1 + tF

2

+ b2 tS

b1 +

tF

2 btF + AS , (2.12)

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I x = bt3F

12 + btF y2G +

b31 tS 12

+

+ b1 tS b12 yG2

+ b2 tS (b1 yG )2

,(2.13)

I S = b31 tS

3 + b21 b2 tS , (2.14)

I t = b31 tS 3

+ b32 tS 3

. (2.15)

Figure 5. Dimensions of a trapezoidal sti ff ener

The calculations of geometrical parameters of the trapezoidal sti ff ener are

AS = ( a 1 + 2 a 2 ) tS , (2.16)

a 1 = 90 [mm], a3 = 300 [mm], thus

hS = a22 105

2

1 / 2

, (2.17)

sin2 = 1 105a2 2

, (2.18)

yG = a1 tS (hS + tF / 2) + 2 a 2 tS (hS + tF ) / 2btF + AS (2.19)

I x = bt3F

12 + btF y

2G + a 1 tS hS + tF 2 yG

2

+

+ 16

a 32 tS sin2 + 2 a 2 tS hS + tF 2 yG

2

, (2.20)

I S = a1 h3S tS +

23

a32 tS sin2 , (2.21)

I t = 4A2P

Pbi /t i

, (2.22)

AP = hS a 1 + a 3

2 = 195hS . (2.23)

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3. Design constraints in case of uniaxial compression

3.1. Global buckling of the sti ff ened plate. According to Mikami [1] the e ff ect of initial imperfections and residual welding stresses is considered by de ning bucklingcurves for a reduced slenderness

= ( f y / cr )1 / 2 . (3.1)

The classical critical buckling stress for a uniaxially compressed longitudinally sti ff -ened plate is

cr = 2 DhB 2 1 + S 2R + 2 + 2R for R = L/B < R 0 = (1 + S )1 / 4 , (3.2)

cr = 2 2 D

hB 2 h1 + (1 + S )1 / 2i for R R 0 . (3.3)

Figure 6. Global buckling curve considering the e ff ect of initial im-perfections and residual welding stresses

When the reduced slenderness is known the actual global buckling stress can becalculated according to Mikami [1] as follows

U /f y = 1 for 0.3, (3.4)U /f y = 1 0.63 ( 0.3) for 0.3 1, (3.5)

U /f y = 1 / 0.8 + 2

for > 1. (3.6)The global buckling constraint is de ned byN A U

P + S 1 + S

, (3.7)

in which S is given by Equation 2.5,

A = Bt F + (1)AS , (3.8)

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and the factor is

p = 1 if UP > U , (3.9)P = UP /f y if UP U . (3.10)

Figure 7. Global buckling curve according to Mikami and APIAccording to API [2]

U /f y = 1 if 0.5, (3.11)U /f y = 1 .5 if 0.5 1, (3.12)

U /f y = 0 .5/ if > 1. (3.13)

The global buckling constraint can be written as followsN A U . (3.14)

3.2. Single panel buckling. This constraint eliminates the local buckling of thebase plate parts between the sti ff eners. From the classical buckling formula for asimply supported panel uniformly compressed in one direction

crP = 4 2 E 10.92 tF b

2

, (3.15)

the reduced slenderness is

P = 42 E

10.92f y 1 / 2 b

tF =

b/t F 56.8

; = 235f y 1 / 2

(3.16)

and the actual local buckling stress considering the initial imperfections and residualwelding stresses is

UP /f y = 1 for P 0.526, (3.17)UP f y =

0.526P

0 .7

for P > 0.526. (3.18)

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The single panel buckling constraint is

N A UP . (3.19)

3.3. Local and torsional buckling of sti ff eners. These instability phenomena

depend on the shape of stiff

eners and will be treated separately for L stiff

ener.The torsional buckling constraint for open section sti ff eners is

N A UT . (3.20)

The classical torsional buckling stress is

crT = GI T

I P +

EI L2 I P

, (3.21)

where G = E / 2.6 is the shear modulus, I T is the torsional moment of inertia, I P isthe polar moment of inertia and I is the warping constant. The actual torsional

buckling stress can be calculated as a function of the reduced slendernessT = ( f y / crT )

1 / 2 , (3.22)

UT /f y = 1 for T 0.45, (3.23)UT /f y = 1 0.53 ( 0.45) for 0.45 1.41, (3.24)

UT /f y = 1/2 for > 1.41. (3.25)

Figure 8. Limiting curves for local plate buckling ( P

) and torsionalbuckling of open section ribs ( T )

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172 Z. Virg

4. Design constraints in case of uniaxial compression and lateral pressure

4.1. Calculation the de ection due to compression and lateral pressure.Paik et al. [3] used the di ff erential equations of large de ection orthotropic platetheory and the Galerkin method to derive the following cubic equation for the elasticde ection Am of a stiff ened plate loaded by uniaxial compression and lateral pressure

C 1 A3

m + C 2 A2

m + C 3 Am + C 4 = 0 , (4.1)where

C 1 = 2

16 E x m4 B

L3 + E

LB 3 ; C 2 = 3

2 Aom16 E x m

4 BL3

+ E LB 3 ,

C 3 = 2 A2om

8 E x m4 B

L3 + E

LB 3 + m

2 BL

xav + 2

tF Dx m4 B

L3 + 2H

m 2

LB + D

LB 3 ,

(4.2)

C 4 = Aomm 2 B

L xav

16LB 4 tF

p,

E x = E 1 + nAS Bt F ; E y = E . (4.3)Since the self-weight is taken into account, the lateral pressure is modi ed as

p = p0 + V gBL

, (4.4)

where g is the gravitation constant, 9.81 [m/s 2 ].The exural and torsional sti ff nesses of the orthotropic plate are as follows:

D x = Et 3F

121 2xy

+ Et F y2G1

2xy

+ E I x

b ,

D y = Et 3F

121 2xy

,(4.5)

x = 0.86v uuuuuut

E

E x Et 3F 12 + Et F y

2

G +

EI xb

Et 3F 12

EI xb E E x

2 , (4.6)

y = E E x

x ; xy = x y , (4.7)H =

Gxy I tb

; Gxy = E

2 (1 + xy ), (4.8)

Xbit i

= a1 + 2 a 2

tS +

a3tF

. (4.9)

The de ection due to lateral pressure is

Aom = 5qL4

384EI x ; q = pb; b = B/ . (4.10)

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The solution of equation (4.1) is

Am = C 23C 1

+ k1 + k2 , (4.11)

where

k1 = 3

s Y

2 +

r Y 2

4 +

X 3

27 ; k2 =

3

s Y

2 r Y 2

4 +

X 3

27 , (4.12)

X = C 3C 1

C 223C 21

; Y = 2C 3227C 31

C 2 C 33C 21

+ C 4C 1

. (4.13)

4.2. De ection due to shrinkage of longitudinal welds. According to [9] thede ection of the plate due to longitudinal welds is as follows

f max = CL2 / 8, (4.14)

where the curvature for steels is

C = 0 .844x10 3 QT yT /I x , (4.15)

QT is the heat input, I x is the moment of inertia of the cross-section containing astiff ener and the base plate strip of width b, yT is the weld eccentricity

yT = yG tF / 2. (4.16)The heat input for a sti ff ener is

QT = 2 x59.5a2W . (4.17)

4.3. The stress constraint. The stress constraint includes several e ff ects as follows:the average compression stress and the bending stress caused by de ections due tocompression, lateral pressure and the shrinkage of longitudinal welds.

max = xav + M I x

yG UP , (4.18)where

M = xav (A0 m + Am + f max ) + qL2

8 , (4.19)According to [1], the calculation of the local buckling strength of a face plate strip

of widthb1 = max( a3, b a3), (4.20)

is performed taking into account the e ff ects of initial imperfections and residual weld-ing stresses

UP = f y when P 0.526, (4.21)UP = 0.526P

0 .7

when P 0.526, (4.22)where

P = 4 2 E 10.92f y

1 / 2 b1tF =

b1 /t F 56.8 . (4.23)

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5. Cost function

The objective function to be minimized is de ned as the sum of material and fabri-cation costs

K = K m + K f = km V + kf XT i , (5.1)or in another form K

km= V +

kf km

(T 1 + T 2 + T 3 ) , (5.2)

where is the material density, V is the volume of the structure, K m and K f as well askm and kf are the material and fabrication costs as well as cost factors, respectively,T i are the fabrication times as follows:

time for preparation, tacking and assembly

T 1 = dp V , (5.3)where d is a diffi culty factor expressing the complexity of the welded structure, isthe number of structural parts to be assembled;

T 2 is time of welding, and T 3 is time of additional works such as changing of electrode, deslagging and chipping. T 3 0.3T 2 , thus,

T 2 + T 3 = 1 .3XC 2 i anwi Lwi , (5.4)where Lwi is the length of welds, the values of C 2 i anwi can be obtained from formulae ordiagrams constructed using the COSTCOMP [4] software, aw is the weld dimension.

Welding technology aw [mm] 103 C 2 anwSAW 0-15 0.2349a2w

SMAW 0-15 0.7889a2wGMAW-M 0-15 0.3258a2w

Table 1. Welding times versus weld size aw [mm] for longitudinal llet welds, downhand position

6. Optimiztion method

Rosenbrocks hillclimb [5] mathematical method is used to minimize the cost function.This is a direct search mathematical programming method without derivatives. Theiterative algorithm is based on Hooke & Jeeves searching method. It starts with agiven initial value, and takes small steps in the direction of orthogonal coordinatesduring the search. The algorithm is modi ed so that secondary searching is carried outto determine discrete values. The procedure nishes when the convergence criterionis satis ed or the iterative number reaches its limit.

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7. Numerical data and optimum results

7.1. Longitudinally sti ff ened plate loaded by uniaxial compression. The givendata are width B = 6000 [mm], length L = 3000 [mm], compression force N =1.974 107 [N], Young modulus E = 2 .1 105 [MPa] and density = 7 .85 10 6

[kg/mm 3 ]. The yield stress is f y = 355 [MPa]. The unknowns the thicknesses of the base plate and the sti ff ener and the number of the ribs - are limited in size. Forwithout fabrication cost the welding cost is not considered, the material minima isnot shown in Tables 4, 5, 6 and 7.

3 tF 40[mm],3 tS 12[mm], (7.1)3 10.

kf /k m tF [mm] tS [mm] K/k m [kg]Mikami 0 22 6 10 5166

1 22 6 10 61522 22 6 10 7138

API 0 19 10 10 52241 21 7 10 62492 21 7 10 7367

Table 2. Optimum dimensions with L- sti ff ener (SAW)kf /k m tF [mm] tS [mm] K/k m [kg]

Mikami 0 9 7 9 34241 12 6 9 49202 17 5 9 6518

API 0 9 7 9 34241 9 7 9 47612 12 6 9 6097

Table 3. Optimum dimensions with trapezoidal sti ff ener (SAW)kf /k m tF [mm] tS [mm] K/k m [kg]

Mikami 1 22 6 10 72322 24 5 10 8846

API 1 21 7 10 75462 21 7 10 9960

Table 4. Optimum dimensions with L- sti ff ener (SMAW)kf /k m tF [mm] tS [mm] K/k m [kg]

Mikami 1 19 4 9 64522 19 4 9 8538

API 1 15 5 9 64442 21 3 10 7955

Table 5. Optimum dimensions with trapezoidal sti ff ener (SMAW)

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kf /k m tF [mm] tS [mm] K/k m [kg]Mikami 1 22 6 10 6329

2 22 6 10 7493API 1 21 7 10 6462

2 21 7 10 7793

Table 6. Optimum dimensions with Lsti ff ener (GMAW-M)kf /k m tF [mm] tS [mm] K/k m [kg]

Mikami 1 11 6 9 49922 16 5 9 6750

API 1 9 7 9 50992 16 5 9 6532

Table 7. Optimum dimensions with trapezoidal sti ff ener (GMAW-M)

7.2. Longitudinally sti ff ened plate loaded by uniaxial compression and lat-eral pressure. The given data are width B = 4000 [mm], length L = 6000 [mm],compression force N = 1 .974 107 [N], Young modulus E = 2 .1 105 [MPa]and density = 7 .85 10 6 [kg/mm 3 ]. There are three values of lateral pressure p0 = 0 .05, 0.1, 0.2 [MPa] and two values of yield stress f y = 255, 355 [MPa]. Theunknowns the thicknesses of the base plate and the sti ff ener and the number of theribs - are limited in size. The results are shown in Tables 8-13. The optimum resultsare given in bold type.

3 tF 40[mm],3 tS 12[mm],3 10.(7.2)

f y p0 tF tS K/k m [kg][MPa] [MPa] [mm] [mm] kf /k m = 0 kf /k m = 1 .5

235 0.1 38 12 10 8014 11758235 0.05 30 12 6 6127 8362355 0.1 28 12 10 6568 10137355 0.05 20 12 9 4825 7914

Table 8. Optimum dimensions with at sti ff ener for kf /k m = 0 , thematerial minima


235 0.1 38 12 10 8014 11758235 0.05 30 12 6 6127 8362355 0.1 28 12 10 6568 10137355 0.05 21 11 8 4852 7312

Table 9. Optimum dimensions with at sti ff ener for kf

/km

= 1 .5,the cost minima

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235 0.2 31 12 5 6993 8933235 0.1 21 12 7 5686 8230235 0.05 20 10 7 4969 6952355 0.2 22 12 7 6107 8641

355 0.1 18 9 10 5036 7389355 0.05 17 7 10 4313 6302

Table 10. Optimum dimensions with L-sti ff ener for kf /k m = 0 , thematerial minima


235 0.2 34 11 4 7132 8584235 0.1 27 10 5 5888 7422235 0.05 24 8 6 5162 6564355 0.2 28 9 6 6528 8149355 0.1 22 8 7 5247 6801355 0.05 19 8 7 4626 6129

Table 11. Optimum dimensions with L-sti ff ener for kf /k m = 1 .5, thecost minima


235 0.2 28 12 4 6974 8549235 0.1 24 10 4 5723 6975235 0.05 18 10 5 4993 6466355 0.2 21 11 5 6108 7780355 0.1 15 10 6 4944 6635355 0.05 13 8 7 4148 5611

Table 12. Optimum dimensions with trapezoidal sti ff ener for

kf /k m = 0, the material minimaf y p0 tF tS K/k m [kg]

[MPa] [MPa] [mm] [mm] kf /k m = 0 kf /k m = 1 .5235 0.2 35 9 3 7250 8223235 0.1 24 10 4 5723 6975235 0.05 23 8 4 5122 6132355 0.2 28 8 4 6530 7589355 0.1 21 7 5 5111 6284355 0.05 16 7 6 4264 5560

Table 13. Optimum dimensions with trapezoidal sti ff ener forkf /k m = 1.5, the cost minima

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178 Z. Virg

8. Conclusions

The results show that the trapezoidal sti ff ener is the most economic one. Thecost saving can be 69 % compared with various ribs.

In general the Mikami method gives thinner basic plates than those given byAPI.

Materials with higher yield stress give cheaper results. The cost saving canbe 40 % compared with the lower one. Higher strength steel is 10 % moreexpensive.

In most cases the material and cost minima are di ff erent, the number of stiff eners is smaller at cost minima due to welding cost e ff ects. SAW is thecheapest welding process if we do not consider investment cost.

It can be seen from Tables 8 and 9 that there are no solutions for the highestlateral pressure ( p0 = 0 .2 [MPa]) for at sti ff eners due to the size limits.

In case of uniaxially and laterally loaded plate the ratio between materialcost and welding cost ranged from 13 % (for at sti ff ener, higher yield stressand minimum lateral pressure) to 64 % (in case of trapezoidal sti ff ener, loweryield stress and maximum lateral pressure).

For L- and trapezoidal sti ff eners the number of sti ff eners decreases if the

lateral pressure is increased, but it increases if the yield stress of the materialis increased. For at sti ff eners the number of sti ff eners increases if the lateral pressure is

increased and the yield stress of the material is increased.

Acknowledgement. The author wishes to acknowledge the guidance of Prof. Kroly Jr-mai and Prof. Jzsef Farkas. The research work was supported by the Hungarian Scienti cResearch Found grants OTKA T38058 and T37941 projects.

REFERENCES

1. Mikami, I. , and Niwa, K. : Ultimate compressive strength of orthogonally sti ff ened steel

plates. J. Struct. Engng ASCE, 122 (6), (1996), 674682.2. American Petroleum Institute API Bulletin on design of at plate structures . Bulletin

2V. Washington, 1987.3. Paik, J.K., Thayamballi, A.K. and Kim, B. J. : Large de ection orthotropic plate

approach to develop ultimate strength formulations for sti ff ened panels under combinedbiaxial compression/tension and lateral pressure. Thin-Walled Structures , 39 , (2001),215246.

4. COSTCOMP Programm zur Berechnung der Schweisskosten. Deutscher Verlag frSchweisstechnik, Dsseldorf, 1990.

5. Rosenbrock, H. H. : An automatic method for nding the greatest or least value of afunction. Computer Journal , 3 , (1960), 175184.

6. Virg, Z. Minimum cost design of a compressed welded sti ff ened plate using two dif-

ferent buckling constraints, I II. International Conference of PhD. Students, Miskolc,Hungary, 2001, 467474. (ISBN 963 661 482 2)

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7. Virag Z. and Jrmai K. : Parametric studies of uniaxially compressed and laterallyloaded sti ff ened plates for minimum cost, International Conference on Metal Structures(ICMS), Miskolc, Hungary, Millpress, Rotterdam, 2003, 237242. (ISBN 90 77017 75 5)

8. Farkas, J. : Optimum design of metal structures . Budapest, Akadmiai Kiad, Chich-ester, Ellis Horwood, 1984.

9. Farkas, J. and Jrmai, K. : Analysis and optimum design of metal structures , Balkema,Rotterdam-Brook eld, 1997.

10. Farkas, J. and Jrmai, K. : Minimum cost design and comparison of uniaxially com-pressed plates with welded at, L- and trapezoidal sti ff eners. Welding in the World ,44 (3), (2000), 4751.

11. Farkas, J., Simoes, L.M.C. , and Jrmai, K. : Minimum cost design of a weldedstiff ened square plate loaded by biaxial compression. WCSMO-4, 4th World Congressof Structural and Multidisciplinary Optimization, Dalian China, Extended Abstracts,2001, 136137.

12. Farkas, J. , and Jrmai, K .: Economic design of metal structures , Millpress, Rotter-dam. 2003.

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