DISCRETE ENERGY METHOD FOR THE ANALYSIS OF CYLINDRICAL SHELLS

P. K. MISHRA? and S. S. DEY$

~Unive~ity College of Engineering, Burfa, Sambalpur, Orissa, India ~~~~rnent of Civil Engineering, Indian Institute of Technology, Kharagpur 721 302, India

(Received 17 December 1986)

Abstract-This paper presents an investigation into the use of the closeiy associated finite difference technique for the analysis of shell structures as a feasible alternative to the finite element method. The method discretises the total energy of the structure into energy due to extension and bending and thal due to shear and twisting, contributed by two separate sets of rectangular elements formed by a suitable finite difference network. The derivatives in the corresponding energy expressions are replaced by their finite difference forms and the nodal displacements than constitute the undetermined parameters in the variational formulation. The formulation is also extended to a cylindrical shell element of rectangular plan- form. The results obtained by DEM are compared with existing results and they show excellent agreement.

~RODU~ION

Shells are being used on an increasing scale for roofs, canopies, silos, tanks and even for foundations. Analysis of shells by theoretical methods involves solution of higher order differential equations. Even with the use of such equations, any odd loading and complicated support conditions cannot be handled without great difficulty. The discrete energy or dis- crete element method as proposed by Buragohain and Ramesh [I], with the aid of computers, proves to be a very effective method, which heips us to over- come all these di~cu~ti~ without going into the solution of differential equations.

The paper presents the development of a hybrid finite difference-finite element, referred to as the discrete energy method for the analysis of shells and other structures, composed of interconnected plates. The only difference between this method and the finite element method is that, in the finite element method the shape function is usually in the form of polynomials whereas in the discrete energy or discrete element method, it is in the form of truncated Taylor series expansion. The method consists of the discretis- ation of the total potential energy of the system into energies over discrete elements formed by a suitable finite difference network. Instead of finite difference moIecules, element matrices are derived using matrix procedure, which is similar to a finite element technique.

PROCESS OF ANALYSIS

The process involves the following steps.

(i) Discretisation of the structure into two sets of elements, namely, type A and type 3. The first set considers energy due to extension and bending whereas the second set considers energy due to shear and twist.

(ii) Repla~ment of the derivatives in the corre- sponding strain expressions by finite difference operators.

(iii) Formation of element stiffness matrices using strain energy approach and assembly of all the element stiffness matrices, transformed wherever necessary, to form the global stiffness matrix.

(iv) Allocation of the joint loads. (v) The solution of the set of resulting equations

for joint displacements subjected to boundary conditions.

(vi) Evaluation of element displa~ments and forces.

DERIVATION OF EJEMENT SFIFFNESS MATRKES

In the present method, two different types of elements designated as A and B are used. These A and B elements are primarily categorised into two classes, namely, flat plate elements and curved shell elements.

The continuum whether flat or curved is divided into a number of flat plates. Each plate in turn is discretized into a number of type A and type B rectangular elements. For each plate the local co- ordinate system x, y, z, forming a right hand triad, is used for displacements and forces at all nodes lying in the plate, except at nodes at the fold lines where the global co-ordinate system X, Y, Z (Fig. 1) is used in order to satisfy the compatibility and equilibrium conditions. The element matrices are derived using two dimensional elasticity theory for in-plane action and thin plate thoexy for bending action. The dis- cretixation of the structure into etements of type A and type B is shown in Fig. 2.

753

154 P. K. MISHRA and S. S. DEY

2 *r\ Fig. 1. Co-ordinate system.

Constitutive relations. Since the proposed method is meant for the analysis of three dimensional structures such as shells, there will be both in-plane forces (membrane) and out-of-plane forces (bending). The general stress-strain relationship for in-plane forces can be expressed in matrix form as:

- -

= v$, 4 o

0 0 Gxy_

or

{c)fn = v%lkhn~ (1)

For symmetry, it is assumed that vxEy = vyEx and

au ;5;; au

;i-; au av .av+z I (2)

Similar moment+zurvature relationship for an ortho- tropic plate in bending can be stated as:

-- ax2

{XJbC ;; = -$ 4- XXY 2 azW axay

(4)

For an isotropic material, the constants reduce to

E, = E, = E

Et’ Dx=D*=D= 12(1_v2)

v, = vu = v

D, = VD

Et K=-

l-9

E --

G - 2(1 + v)

D,=D(l -v)/2.

In order to develop the stiffness matrices of type A and type B elements the strains and curvatures are to be replaced by corresponding finite difference operators.

It may be noted that in the case of flat plate elements, the membrane stiffness and bending stiff-

NOT E

o w NODES A u NODES

X v NODES

.z 0

$ Lt.

Fig. 2. Various elements and load contribution areas.

Discrete energy method for analysis of cylindrical shells

w NN

u --lbl- N

W V W V W

ww W 0 E EE

U Y S

bw ss

Fig. 3, Type A eIement.

ness matrices coexist in an uncoupled form at element level, i.e. these two stiffness matrices are inde~ndent of each other. The coupling between them is estab- lished at the global stage through the transformation matrices.

7”pe A element. The nodal displacements associ- ated with the element (Fig. 3) are

The zeros in the displacement vector are introduced in order to facilitate co-ordinate transformations at fold lines. The strain components consists of the membrane strains L, and eu and the curvatures XI, xu which are expressed in finite difference form as

where h, and h,. are the lengths in x and y directions respectively. Thus the relation [BA] is given by

PAI =

related to strain components by

where

r K vK 0 01

In eqn (9), K = extensional rigidity = El/l - v2 and D =flexural rigidity = Et2/12(1 -v2); E is the Young’s modufus of elasticity, Y is the Poisson’s ratio and t is the thickness of the plate.

The element stiffness matrix can be derived by the application of a variational principle such as minimum potential energy which amounts to the integration of the following matrices:

where [K,] = eIement stiffness matrix of type A

element

AS, = area of type A element

= h,h,.

The element stiEness matrix is obtained by substitut- ing eqns (7) and (9) in eqn (10).

i”rpe B element. The nodal displacements associated with this element (Fig. 4) are

(4) = hw, uN~ %NE~b”,

w,,N&,,“, wO,o, wiVN,o, WEE,“lt (11)

1 -l 7;;h, 0 0 000000000

0 0 1 -1

3;;h, 000000000

0 0 0 0 $0 -’ 0 $0 0 0 x h2 X

0 X

0 0 0 0 $0 0 0 0 0 -lo -l Y h2 h2 Y P

(7)

The stress resultants consist of the in-plane forces The strain components consists of the shearing strain N, and ny and the plate moments Kf, and My and are yXy and generalized twisting curvature xzy, g&en by

P. K. MISHRA and S. S. DEY

The relation matrix [B,] is then obtained as

1 -1 1 -1 71J --- h, h, h, 0 0 0 0 0 0

FM = 0 0 0 0 0

(12)

(13)

The stress resultants are the membrane shear force Following the same steps as in the case of type A

NXY and the twisting moment MXY, given by elements,

where

0

D(1 1 - v)/2 .

Fig. 4. Type B element.

(14) [&I = PB~~~DBI[B~I~&, (16)

where

(15)

[&I = element stiffness matrix of type B elements

ASS = area of type B element

Substituting eqns (13) and (15) into eqn (16), the element stiffness matrix is obtained.

Boundary elements. Figure 2 shows that whereas type B elements can cover the entire region, it is necessary to have nine type A elements, depending on the position of the element relative to the boundary. The area AS, for the boundary elements is either one-half or one-quarter of the regular area. In order to express strains in terms of displacements within the region, in-plane displacement u,, and rotation B,,,, (= awlax) are introduced at the nodes on the bound-

Table 1. Expressions for strain components for boundary elements (flat plate)

Type

A2

A3

A4

A5

A6

A7

A8

A9

2bo - d

hx

2b, - uo)

hx

kv - Us) hx

(UN-%-)

hx

.%o - us) 4

wh - uo) hx

2(u, - us) hx

2(u, - 4 hx

(UE - hv)

4 2(uo - hv)

hY 2(u, - 00)

4 2(u, - hv)

%

w, - uo)

h, 2@, - uo)

h, 2(u, - UY)

4

- W,$ - wo + wss) hi

- ~kv, -wo-h&J

hf - (%w - 2wo + wss)

hf

- (wnx - 2wo + %s)

ht

- 2(h,0,, - w. + wSS)

hf

-(lw,- 2wo f %vw)

hi -(WEE -2~o+Tvw)

hi

- wh, - wo - V,) ht

_2w,,+2w,+28, h; h; h, >

- 2(h,0,,, - w, + w&

hl

- 2(w,, - wo - h&0, h:

Discrete energy method for analysis of cylindrical shells 157

ary x = constant and in-plane displacement u, and rotation 8, (=~~/~y) are introduced at the nodes on the boundary y = constant. The displacement vector {dA} and the relation matrix fs,] are modified accordingly. Table 1 gives the expressions for strain components modified for the boundary elements A2 to A9.

transformed element matrix is thus given by

RI = ~~I~I~il~~l, (19)

where i stands for any of the elements A 1 to A9 and B.

The overall stiffness matrix [ST] is then obtained as Load uector. The load corresponding to a particu-

lar displacement can be obtained directly. For uni- formly distributed in-plane loads P, and Py in X- and y-directions respectively, the contributing areas are shown in Fig. 2. The total in-plane load in the x-direction is lumped equally at the neighbouring nodes uN and uNEE and the total in-plane load in the y-direction is lumped equally at the neigh~u~ng nodes vE and v,,,. For uniformly distributed trans- verse load P, in the z-direction, the contributing areas are the same as type A elemental areas and the total load is lumped at the central pivotal node ‘0’.

Co-ordinate transformation and assembly. Trans- formations are carried out only for those elements which are associated with displacements at fold line nodes. As an illustration, if element A4 (Fig. 5) lies on a free edge, the displacement vector is written as

KVJ = CKi,

in which the summation extends over all the element matrices including both type A and type B elements (transformed wherever necessary).

Cyhdrical shell element

VA41 = [u,,, us, 00, uw, wsu,,

0, WNNV 0, w,,, 0, Tvwt 0, fJ,l’. (17)

If A4 lies on a fold line nodes w, (vo, w. and 0, are at the same point), w,, and wSS are on the fold line (Fig. 5). The transformed displacement vector

G&4) = i+t u,, to, vwt tto, t,w

VNN$ Cw ‘IW 0, %tw~ 0, a,oI’, (18)

in which c and q are the displacements in the positive Y and 2 directions respectively (Fig. 1).

The required transformation matrix [7’,J is given

by

The geometry of a shell structure is idealised as a folded plate one instead of a shell even if a fine mesh spacing of flat plates is used. A shell structure should be logically approached through elements having curved surface to provide a close fit and hence a curved shell element of rectangular planform is devel- oped. The basic central difference equations for these elements are similar to those for the flat plate ele- ments. The following changes are incorporated: hy is replaced by tre( = rH), 8y is replaced by rd8 and c?y2 by r2802.

Stra~n~~pIacement relations. The strains at any point in a cylindrical shell with constant radius [2] can be expressed as:

ldv w Lo=----

rlJ0 r

av 1 au l%=-&+;;iii

VA =

1

1

cos 8 -sin 0

1

sin 8 cos 9

cos e -sin 0

sin e cos 8

cos e -sin 0

sin e c0se 1

1

1

(only nonzero elements are shown)

(21)

(221

1 J

in which B = the angle measured cIockwise from the positive y-axis of the plate to the positive Y-axis. The

CAS 27/6-E

PW xx= -ax’ (23)

P. K. MISHRA and S. S. DEY

Wl~~

U

ex,v,w

U

W

NN

U 0 A2 ur’- -- s

WI SS

w NN

Fig. 5. Various A elements with location of nodes.

xxs = - - r aeax +z

W

I NN

wml w”w

W EV EWE U QY

i ab I au xs= ------_fz

r2 a82 (24)

2 azw 2 au ___.

The above relations are used to develop the element stiffness matrices for shell elements.

The procedure followed to derive the element stiffness matrix, up to the formation of the global stiffness matrix, is same as that for flat plate elements. Here it can be noted that no transformation is required and the coupling between membrane and bending stiffness matrices exists even at the element level.

Imposition of boundary conditions. To satisfy force boundary conditions in conventional finite difference method is difficult, since it involves higher order derivatives,’ but these are automatically satisfied in the case of the discrete energy method, an advantage inherent with the energy approaches. The various displacements associated with any node are U, v, w, 0, and 0,. The specified restraints on displacements for different types of supports are as follows:

(I) Free edge-no restraint on any displacement, (2) Simple support: w = 0; u = 0.

W V U

ww w Q w,v,e*

0 A6 -7 U

s

W

SS

Wl~~

W V ww w 0 +I

if4

w,v, 0x

0 AL

U 5

Iw ss

W

d NN

U

0

N A9

w”w WV 0 w,v,ex

(3) Hinged support: w = 0; u = 0; u = 0. (4) Built-in support: w = 0; u = 0; v = 0; 8, = 0;

e, = 0.

These zero deflections in each case are eliminated from the set of equations right at the formation stage by assigning zero NCODE values or made zero by multiplying the corresponding diagonal terms by a large number after the formation of the global matrix. The former procedure leads to a smaller set of equations.

Computation of stress resultants. The assembly of the overall stiffness matrix is carried out in the computer in half band form and the matrix solved for the unknown displacements. After the evaluation of the displacements, the displacement vector dA for each type A element is formed in local coordinates and the stress resultants N,, NY, M, and MY are

computed at the pivotal nodes of type A elements using eqn (8). Similarly, using eqn (14) the stress resultants NXY and Mx, are computed at the centres of type B elements.

In the first four illustrated examples of cylindrical shell analysed by DEM, the shell surface is first divided into a number of flat plates. These plates in turn are discretised into a number of type A and type

Discrctc cncrgy method for analysis of cylindrical shells 759

la) DISCRETIZATION OF SHELL SURFACE BY CURVED ELEMENTS

w/ ss

(b) TYPE ‘A’ ELEMENT WITH LOCATION

OF NODES

1-E EEW

(cl TYPE ‘B’ ELEMENT WITH

LOCATION OF NODES

Fig. 6. Various types of elements in cylindrical shell.

B elements. Although flat plate representation is an In the fifth example, the shell is discretised into approximation to the actual shell surface, a reason- shell elements. ably good fit can be achieved by increasing the The shell parameters such as radius, length, central number of plates. Also the plates can fit into any type angle, thickness, etc. are given in Fig. 7. Taking of directrix, e.g. parabolic, elliptic, etc. Care is taken - _ symmetry into consideration, only one-quarter of the to maintain compatibility at fold lines. shell is discretised.

SHELL ROOF CONFIGURATION WITH CIRCULAR DIRECTRIX

PARABOLIC DIRECTRIX

EXAMPLE 1

[WITH AND WITHOUT

RIGID EDGE BEAM Rs25.0

L= 50.0’

t = 0.25’

SHELL WEIGHT = 90 psf er40*

EXAMPLE 3

PARABOLIC DIRECTRIX L162.0’

t 5 0.317’

SELF WEIGHT= L7’6 psf

LINE LOAD 2200 Ib/ft

(AT CROWN1

FXAMPLE 2

IWITH AND WITHOUT

ELASTIC EDGE BEAM1

R = 33’ 25’

L = 62’0’ t s 0.33

SELF WEIGHT = 50 psf

SNOW LOAD =25 psf

e= 3o”

EXAMPLE 4 ELLIPTIC DIRECTRIX

L = 62’0’

t = 0.317’ SELF WEIGHT=L7’6 psf

LINE LOAD8200 lb/f t

IAT CROWN)

ELLIPTIC DIRECTRIX

Fig. 7. Shell roof problems.

760 P. K. MWRA and S. S. DEY

Table 2. Numerical results for cylindrical shell roof (example 1, example 5)

Number of IO-‘N,, Method

MXC elements 1OU” w, IOU, low, lb/in.

M, lb in/in. lb in/in.

Reference [S] -1.51 3.70 -8.73 -5.22 6.50 -2087.1 - 128.4 Reference [6] - 3.36 - 4.92 6.00 - 1900.0 -90.0 Exact [3] -1.51 3.696 -8.76 -5.24 6.41 - 2056.0 -92.7 DEM (flat plate) 200(334 DOF) - 1.374 3.51 -6.8 -5.21 5.93 - 1960.47 -92.5 DEM (shell element) 137(266 DOF) - 3.52 - -5.22 - -2095.3 - 112.74

Example 1. A cylindrical shell with a circular direc- This problem has been solved by Scordelis [3] trix, shown in Fig. 7, is loaded by its self weight and based on the theory of Gibson [4]. This solution may is supported by diaphragms at the ends but is free be classified as the exact solution. The same problem along the sides. The diaphragms are considered also has been tackled by Cowper et al. [5] using a infinitely rigid in their own plane but perfectly flexible shallow shell finite element of triangular shape. in the direction normal to their own plane. Using The same shell with two rigid edge beams is symmetry, only one-quarter of the shell is analysed. analysed by DEM, asigning zero displacement code A 10 x 6 grid is used with 334 degrees of freedom. numbers to all the displacements along the free edge.

CROWN

-r 0

-l!k EDGE 0

0.8 0.5 0 -0.L 10.8

[aI M, lklp ft /ft 1 AT MID SPAN

REF. 18) DEM.

CROWN

EDGE I I I

-2’0 -1’0 0 1’0

lb) My tkip ft / f t ) AT MID SPAN

CROWN CROWN

- REF.181

0 DEM.

THOUT EDGE

ICI N, lkip /f t 1 AT MID SPAN (d) Nxy lkip/ftl AT SUPPORT

- REF. 191

o DEM.

SUPPORT MIDSPAN

::ip]

le) VERTICAL DISPL. (11) ALONG

FREE EDGE

tf 1 VERTICAL DISPL. lft) ALONG RIDGE

Fig. 8. Results of cylindrical shell roof.

Discrete energy method for analysis of cylindrical shells 761

Table 3. Results at midspan of cylindrical shell roof with elastic edge beam (example 2)

M, (lb ft/ft)

Reference [8] ASCE 181

0.0 277.83 460.28 461.91 331.64 i 16.27

- 141.29 -403.48 -824.90 -943.01 -983.52

25EO 420:10

2oz3 409:62

433.12 411.41 313.23 287.45 107.23 88.98

- 142.00 - 1 IS.75 - 397.20 -343.53 - 809.07 -751.36 -924.78 - 845.95 -964.43 -878.30

DEM Reference [S] ASCE [81

0.0 0.0 - 6999.62 -7031.92 -7729.21 - 7732.42 -7823.56 -7820.83 -7445.32 -7440.17 -6732.31 -6726.81 -5796.14 -5791.67 -4723.33 -3573.91 -2393.33 -2392.38 -1198.19 - 1197.83

0.0 0.0

DEM

0.0 Edge -7361.33 -8217.40 -8310.81 - 7742.54 - 7069.34 -6170.63 -3747.50 -2533.00 - 1266.23

0.0 Crown

The results obtained are compared (Table 2) with those furnished by Tilak and Siddhaye [6] using Cantin and Clough’s cylindrical shell element [7j and are presented in Fig. 8.

Example 2. A cylindrical shell with parameters given in Fig. 7 is analysed with and without elastic edge beams. The edge beam is considered as a part of the shell with change in thickness for the analysis by DEN while the conventional edge beam theory is adopted for the analysis by the ASCE exact theory. Variations in some of the important results are given in Table 3.

Example 3. This example consists of a cylindrical shell with a parabolic directrix (Fig. 7). It is analysed for its self weight and a line load acting at its crown separately. The values of M,. obtained using an 8 x 6 grid with 270 degrees of freedom are compared with those reported by Chakrabarti [8] and presented in Table 4.

Exumple 4. This example consists of cylindrica1 shell with an eIliptic dire&x (Fig. 7). It is analysed for a line load acting on its crown. The values of M,, obtained using an 8 x 6 grid with 270 degrees of freedom are compared with those reported by Charkrabarti [8], who has solved it by using flat plate elements in the form of strips. The results are presented in Table 4.

Example 5. In all of the previous four problems, the cylindrical shell is discretixed with flat plate elements. This particular example was solved where the cylindrical shell was discretixed with circular shell elements with rectangufar planform developed in this paper. The data for this example are same as those of example 1. Some of the important results are given in Table 2, where it is compared with the results obtained by using flat plate elements and exact analysis. The aspect of convergence of the procedure was studied for a cylindrica1 shell roof with an increasing number of elements and is presented in Table 5.

Table 5. Convergence studies for cylindrical shell

(a) M,. at midspan with flat plate elements

Number of elements DOF

Mr fib ftlf0

DEM Reference [9]

80 142 - 1875.64 160 270 -2194.85 -2376.0 200 334 -2271.28

(b) Deflection at free edge at midspan (example 5) with shell elements

Number of Degection (in.)

elements DOF DEM

:: 149 25 2.98 3.43 137 226 3.52

Exact [5]

3.696

Table 4. M, at midspan for cylindrical shell with parabolic and elliptic direct&s

Due to self weight Due to live load at crown

Parabolic Elliptic Parabolic Elliptic

‘Reference [8] DEM Reference [8] DEM Reference [S] DEM Reference [8] DEM

10:: 0.00 0.06 Edge

- 139:86

8;:

-80137

-23%

-218:i 1

- 204.23 0.23 - 18.34 0.00 - 17.45 0.00 -30.56 -25.41

- 190.57 - 36.57 - 34.93 -62.35 -54.34 -581.06 -403.52 - 192.93 - 168.31 -49.18 -44.95 - 76.23 - 67.37

- 1071.86 - 965.27 - 165.86 - 145.46 -44.07 -49.54 - 56.42 - 50.36 - 1512.12 - 1377.82 - 140.14 - 121.33 -3.65 -7.69 10.32 -4.65 - 1345.56 - 1656.59 - 118.83 - 103.95 92.62 90.48 135.73 121.36 -2018.32 - 1775.39 - 104.21 -91.61 265.15 258.44 330.53 294.93 -2113.75 - 1988.52 -97.65 -83.78 531.42 472.26 605.00 533.54 Crown

162 P. K. MI~H~U and S. S. DEY

CONCLUSION

The discrete energy method presented here pro- vides a powerful and economic procedure based on matrix concepts for efficient computation of deflections and stress in shell structures. The result- ing matrix is symmetrical and the manipulations are similar to those of any other stiffness matrix procedure. Although the finite element approach is capable of numerically modelling the response of such structures to loads, the analysis is usually tedious, expensive and requires very powerful com- puters with large storage facility. The discrete energy method presented here shows significant improve- ment in computer time and storage requirements.

REFERENCES

1. D. N. Buragohain and C. K. Ramesh, A discrete analysis technique for folded plates wtih openings and overhangs. Indian Cow. JI 45, 156161 (1971).

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9.

V. G. Rekach, Static Theory of Thin-walled Space Sfructures. Mir, Moscow (1978). A. C. Scordelis, Analysis of cylindrical shells and folded plates. Proc. Symposium on Concrete Thin Shells, 6th AC1 Annual Convention, New York, pp. 207-236 (1966). J. E. Gibson, The Design of Cylindrical Shell Roe/, 2nd Bdn. Spon, London (1961). G. R. Cowper, G. M. Lindberg and M. D. Olson, A shallow shell finite element of triangular shape. Inf. J. Solidr Struct. 6, 1133-1156 (1970). B. G. Tilak and V. R. Siddhaye, Analysis of thin cylindrical shells by finite element method. J. Struct. Engng, 5, 135-143 (1977). G. Cantin and R. W. Cloueh. A curved cvlindrical shell finite element. AIAA JI 6,-l&7-1062 (1968). S. Chakrabarti, Analysis of thin cylindrical shells by finite element method. J. Insr Engrs (India), 380403 (1967). Ni. Chi-Mou, A quadrilateral finite difference plate element for nonlinear transient analysis of panels. Compuf. Srrucr. 15, l-10 (1982).