A simplified calculation method of wall plates of dry gas holders under gas pressure

hr. J. ores. 0 I998 Elsevier PlI:SO308-0161(97)00062-S

ELSEVIER

i'es. & Pipq 74 (1997) 13-18 Science Ltd. All rights reserved Printed in Northern Ireland 0308-0161/97/$17.00

A simplified calculation method of wall plates of dry gas holders under gas pressure L. H. Youa, J. J. Zhangb, H. B. Wu’ & R. B. SunC “1st Department of Mechanical Engineering, Chongqing UniversiQ, Chongqing City 630044, People’s Republic of China bThe National Cerztre for Computer Animation, Bournemouth University Bournemouth, Dorset BH12 5R3, UK ‘Department of Civil Engineering, Chongqing University of Architecture, Chongqing City 63004.5, People’s Repubiic of China

(Received30 December1996;accepted28 July 1997) In this paper,a setof basicequationsfor stiffenedwall platesof dry gasholdersare developedandtheclosedform solutionsof simplifiedequationsarederivedaccording to the deformationbehavioursof the plates under gas pressure.The analytical formulaefor calculatingthe deformations,stresses andinternal forcesare obtained which give the resultsin good agreementwith thoseobtainedby finite element analysisof dry gasholdersundergaspressure.0 1998ElsevierScienceLtd.

NOMENCLATURE

TXY

gas pressureper unit area shearforces per unit length moment of area A about the middle surface of skins displacementsin the x, y and z directions, w is also called lateral deflection distances,coordinates half central angle of a wall plate shearstrain at ordinate z normal strains in the x and y directions at ordinate z, respectively Poisson’sratio normal stressesof skinsin the x andy directions, respectively normal stressesof ribs in the x and y directions, respectively shearstressof skins shearstressof ribs

V2

Laplace operator V’ = $+

P

QmQ, area of a rib b width of a wall plate C spacing between ribs in the x direction c I, c2’ ’ ‘cc integral constants Eh’ D flexural rigidity, D = 12(1 - p2)

A

Dw D” E i.2

h I k L Mm M, Mm N,; N, NXV

s u, v, w x, Y, z CY

effective flexural rigidity of wall plates defined in eqn (9) effective extensional rigidity of wall plates defined in eqn (12) Young’s modulus E

equivalent Young’s modulus, E= -for 1 -/i= skins, and I? = E for ribs thickness of skins moment of inertia of area A about the middle surface of skins Eh extensional rigidity of skins, k = -1- $ distance between two opposite wall plates bending momentsper unit length in the x and y directions, respectively twisting moment per unit length in-plane normal forces per unit length in the x and y directions, respectively in-plane tangential force per unit length

TX>, 6x9 Ey P 0, I I uyf ox ” , uy ” TXY

I I,

2

2

$

1 INTRODUCTION Dry gasholders, mainly consistingof columns, wall plates, corridors, a piston structure, a top structure and a bottom plate (see Fig. l), usually take a large amount of steel to

14

L. H. You et al. strain-displacement

relationships.

au a*w %= z-zyg

2 TX,=

Fig. 1. The geometry of a dry gas holder. 1, columns; 2, wall plates; 3, corridors; 4, piston structure; 5, top structure; 6, bottom plate. build. For example, it will take over 2000 tons of steel to build a dry gas holder with a volume of 120 000 m3. Under gas pressure, wall plates are the main load-carrying members of dry gas holders. About 40% of steel used in building a dry gas holder is for its wall plates. Development of accurate and efficient computing methods is thus very important for the design of wall plates in order to both reduce their steel consumption and increase the reliability. Wall plates are a kind of stiffened plate. A wall plate consists of a skin and some ribs, the ribs being placed on one side of the skin (see Fig. 2). For stiffened plates under simple boundary conditions and loads, research progress has been made on their bending, stability and vibration.le4 However, for the calculation of the wall plates of dry gas holders, to our knowledge, little research has been reported. The detailed analysis of gas holders including wall plates under gas pressure and/or wind load may be performed using the finite element method or series method as done previously by some of the authors of this paper.s-7 However, these methods are not very suitable for the analysis of a preliminary design. In contrast, the simplified calculating method and formulae introduced in this paper, owing to the accuracy and computational efficiency, have proved to be a more effective and feasible alternative.

2 DERIVATION PLATES

OF BASIC EQUATIONS

d”+

!I-&!t

ay

ax

(1)

axay

The relationships of the stress and strain for the skin are

E 7 ‘xy - 2(1 +J”y

(2)

and for the ribs are I, 0, = Txy” = 0 cyt’ = Ecy

(3)

The in-plane internal forces of wall plates are defined as N,

=

J

h/2

J

NY =

- h/2

oyrdz+;,Acgu=k(;+p$f)

OF WALL 1112

For the wall plates of dry gas holders shown in Fig. 2, taking in-plane strains into account and considering the LoveKirchhoff’s assumption, we may obtain the following

Nxy

=

J

-h,2rxy’ dz= +(

$+

2)

h/2

X

M,=

J

-h,20xXzdz=

-D

,$zdz= L

Fig.

2.

Y

The element of a wall plate.

-D($+p$)

Wall plate calculation

s h/2

Mxy

=

‘zdz=

_ h/2%

-D(l-,)&

(4)

The equilibrium equationsof wall plates can be obtained as follows aN, dNm

15

of dry gas holders

behave like cylindrical bending, as demonstrated by the finite element analysis. Consequently, all the displacements can be thought independentof the x coordinate andthe basic equationsare simplified as d’u -0 dy2 -

,+x=O

$+ z+p=o

Since the first part of eqn (8) has influence only on shear strain yxY, shearstressr,, and in-plane tangential force N,,, it will not be taken into account in the following discussions. Eliminating displacement v from the second and third parts of eqn (S), and denoting D, as (1 - y2)ES2

EI Dw=D+

(5) Substituting the last three parts of eqn (4) into the last two parts of eqn (5) gives

c-

c[ch+

(9)

(1 -/,?)A]

a fourth-order differential equation is obtained as d4w D,--=p dy4

(10)

The closed form solution of eqn (10) is given by Q,=

-D&V%

4 w =

Qy= -D32w+

;$

Sv-I$ (

1

(6)

$&

w

+

LY3

6

+

;c2y2 +

c3y

+

c4

Substituting eqn (11) into the second part of eqn (8) integrating it and denoting D, as

The basic equationsof wall plates are derived asfollows by substituting eqn (6) and the first three parts of eqn (4) into the first three parts of eqn (5)

(12) the closed form solution of in-plane displacement v is derived as v=D.(

&+

;qy2)

+c5y+c6

1 +/.L a”u ~~+[~$+(l+~A)$]v 2 axay l-/L2

4 DETERMINATION OF THE INTEGRAL CONSTANTS AND CALCULATING FORMULAE

a3w

- L.hVayj=O 4

3

DV4+ 7%

w-

=--$=p

(7)

>

Under gas pressure, the deformation of wall plates is symmetrical with respect to the middle point of its width. Therefore the following conditions can be derived. 1. Boundary conditions of bending of wall plates

3 CLOSED FORM SOLUTION EQUATIONS

OF SIMPLIFIED

The dimensions of wall plates between two adjacent corridors are much larger than those between two adjacent columns. The wall plates under gas pressure therefore

y=o

50 4

y=b

d"=() 4

(w!ydl= W,=b

(14)

16

L. H. You et al.

Substituting obtains w=

w of eqn (11) into the above equations, one

&(y’-

2by3+

b2y2)

+

c4

w

(15)

formulae have the forms w= &

- 2by3 + b*y’ + -

b3

ES D, - ~ ck+EA (23)

2. Boundary condition of in-plane displacementof wall plates y=;

v=o

(16)

Substituting v of eqn (13) into the above boundary condition, we obtain (24) 6by2 + b3) + c5

(17)

3. Equilibrium condition between in-plane force and gas pressure Taking the force equilibrium of Fig. 3 into account, we have PL=2(N,)

b y= 2

(18)

NY=&&(k+

F) -

$4y2-4hy+b2)+$ (25)

My =

Substituting w of eqn (15) and v of eqn (17) into the second part of eqn (4), we obtain (26) (19) Taking y = b/2 in eqn (19) and substituting it into eqn (I 8), we may determine c5 as follows:

ay = i? $$4y2

+

(20)

- 4by + b’)

2(ciEA)k-

- $-(6y2 w

g) - 6by + b2)]

(27)

4. Compatibility condition between lateral deflection and in-plane displacementof wall plates Under gas pressure,columns deform along the diagonal lines of the holders asshownin Fig. 4. Therefore, we obtain the following compatibility condition y=O Wtanor= -v

(21)

5 NUMERICAL

RESULTS

The above formulae have been applied to the computation of the wall plates of a dry gas holder with a volume of

Taking y = 0 in eqns(15) and (17), then substituting w and v into eqn (21), we obtain (22) Substituting c4 and c5 into eqns (15), (17) and (19), we obtain the analytical formulae of deflection W, in-plane displacementv and in-plane normal force N,,. Then substituting w and v into the fifth part of eqn (4) we obtain the formulae of bending moment M,. The stressof wall plates cV can be obtained from eqns (l)-(3). These analytical

Fig. 3. The equilibrium

between in-plane force and gas pressure.

Wall plate calculation

17

of dry gas holders

Fig. 4. The deformation compatibility between a wall pl,ateand a column. 120000 m3 under gas pressure. The cross-section of the holder is a regular polygon of 20 sides. The wall plate of each side is 85.05 m in height and 7 m in width. Each side has a central angle of 18” and the dimensions of the crosssection are shown in Fig. 5. The gas pressure is 7000 N/m2. The Young’s modulus and Poisson’s ratio of the material of the wall plates are E = 2.1 X 10” N/m2 and 1~ = 0.3 respectively. Using eqns (23) and (27), we obtained the curves of deflection and stress in the bottom surface of the ribs of the wall plates, as shown in Figs 7 and 8. The maximum deflection of the wall plates was 5.52 mm occurring at y = b/2, and the maximum stress was 89.9 MPa. The design criteria adopted in the case study were provided by the Chongqing Institute of Steel and Iron, a leading professional institute in the field in China. The allowable deflection is b/1000, i.e. 7 mm, and the allowable stress is that of the steel used. As far as the holder is concerned, the wall plates are made of Q235 steel whose allowable: stress is 215 MPa. Therefore, both the maximum deflection of the wall plates and their maximum stress satisfy the criteria. In order to check the validity of the analytical solution proposed in this paper, the finite element method was used to perform the analysis of the holder under the gas pressure. One truss of the top structure, one column and adjacent two half wall plates were considered according to the symmetry of the holder under gas pressure. The truss was effectively treated as a member. The skin of each half wall plate was divided into 6 X 105 plate elements. Each column was correspondingly divided into 105 beam elements. Each half rib and half corridor were respectively divided into six beam elements corresponding to the plate elements. The member was taken as a beam element. The finite element mesh is shown in Fig. 6. The corresponding nodes between skins and ribs, between skins and corridors, and between skins and columns were treated as main and subordinate nodes respectively, i.e. the main nodes represent the skins, and subordinate nodes represent the ribs, corridors, and columns. The centre of the top structure., node 1,

Fig. 6. Finite element mesh. and the bottom of the column and wall plates were considered to be fixed. Accounting for the symmetric conditions of the wall plates, the displacements along the width of the wall plates and the rotation around axis were set to zero at the nodes of the symmetric planes 2, 15, 28, . . . and 14, 27, 40, . . . , etc. The calculation was performed with the SAP5 Structural Analysis Program and its results are given in Figs 7 and 8 where AS indicates the analytical solution and FEM the finite element solution. Compared with the results of the finite element analysis, the errors of the deflections of the wall plates at y = 0 and y = b/2 are 0.78% and 5.16% respectively, and the error of the maximum stress of wall plates is 10.9% occurring at the same position y = 0.

6 CONCLUSIONS

1. The basic equations and closed form solution of the cylindrical bending for wall plates under gas pressure are developed. The analytical formulae of deflection, in-plane displacement, internal force and stress of wall plates are derived from the conditions of compatibility, static equilibrium and boundary of the holders. mm

W

-AS

i=$ejp% botto< Surface of ribs

-------L-_--L

88

Fig. 5. The sizes of wall plates.

0

1.167

2.333

'Y 3.5

4.667

5.833

Fig. 7. The deflections of wall plates.

7m

18

L. H. You et al. REFERENCES

0

0.583

1.17

1.75

2.33

2.92

3.5

m

Fig. 8. The stresses in the bottom surface of the ribs.

1. Troitsky, M. S., StifSened Plates-Bending, Stability and Vibrations. ElsevierScientificPublishing,New York, 1976. 2. Boot, J. C. and Moore, D. B. Stiffenedplatessubjectedto transverseloading.Int. J. Solids Structures, 1988,24, 89. 3. Holopainen,T. P. Finite elementfree vibration analysisof eccentrically stiffened plates. Computers and Structures,

1995,56, 993. 4. Danielson,D. A., Cricelli, A. S., Frenzen, C. L. and

A numerical example of wall plates of a dry gas holder with a volume of 120000 m3 under gas pressure is given. Compared with the results of finite element analysis, the method given in this paper has demonstratedgood computing accuracy. The formulae derived in this paper have good potential to become an effective tool for the preliminary design of the wall plates under gas pressure.In contrast to a full finite element analysis method, this method will also substantially facilitate the optimization processof the design due to its simplicity and computational efficiency, as optimum design of the wall plates will be necessaryin a increasing number of cases.

ACKNOWLEDGEMENTS LHY would like to gratefully acknowledge the British Council for its financial support to his research in the Imperial College of Science, Technology and Medicine, University of London, UK.

Vasudevan,N. Buckling of stiffened plates under axial compression and lateral pressure.Int. J. Solids Structures, 1993, 30, 545. 5. You, L. H., Wu, H. B., Sun, R. B. and Wang, D. Y. Static

analysisof the body of dry gasholderswith a volume of 120000 m3. Journal of Chongqing University of Architecture, 1991, 13, 7. 6. You, L. H., Wu, H. B., Sun, R. B. and Wang,D. Y. Finite

elementcalculationof body of dry gasholderswith a volume of 120000m3 under internalpressure.Journal of Guizhou Institute of Technology, 1991, 20, 66. 7. You, L. H., Wu, H. B., Sun,R. B. andWang,D. Y., Analytic

Solutionsof body of dry gasholdersunderinternalpressure basedon theoriesof stiffenedplates,Proceedings of International

Conference on Steel and Aluminium

Structures,

Singapore,22-24 May 1991. Elsevier Applied Science Publishers,Oxford.