Limitations of certain curved finite elements when applied to arches

Int. J. mech. Sei. Pergamon Press. 1971. Vol. 13, pp. 133-139. Printed in Great Britain

LIMITATIONS OF CERTAIN CURVED FINITE ELEMENTS W H E N A P P L I E D TO A R C H E S D. G. ASHWELL and A. B. SABIR D e p a r t m e n t of Civil and S t r u c t u r a l Engineering, U n i v e r s i t y College, Cardiff (Received 22 M a y 1970, and i n revised f o r m 14 October 1970) \

S u n u n a r y - - S e v e r a l segmental arch p r o b l e m s were solved b y finite elements using three shape functions, n a m e l y , I, cubic expression for radial, linear for circumferential displacem e n t s ; I I , a f o r m of Cantin and Clough's shape f u n c t i o n for cylindrical shells; I I I , a shallow element f o r m of I I . F o r a m o d e r a t e l y deep arch w i t h a thick rib all shape functions c o n v e r g e d rapidly to correct solutions w i t h increasing n u m b e r of elements. F o r a deep arch w i t h a t h i n rib, no shape f u n c t i o n g a v e satisfactory solutions for a n u m b e r of elements up to 34. F o r i n t e r m e d i a t e eases, I I was superior to I I I which was superior to I. T h e relevance to shell studies is p o i n t e d out and several conclusions drawn. NOTATION a~ shape f u n c t i o n constants, n = 1. . . . . 6. h thickness of arch rib n n u m b e r of elements in complete arch 'R ' m e a n radius of arch rib w radial displacement v circumferential displacement ~y circumferential co-ordinate (Fig. 2) z radial co-ordinate (Fig. 2) h a l f angle s u b t e n d e d b y arch fl h a l f angle s u b t e n d e d b y element A deflexion $,, 8, c o m p o n e n t s of rigid-body displacement (Fig. 2) e circumferential strain rigid-body r o t a t i o n (Fig. 2) K increase in c u r v a t u r e of arch rib ¢ angular co-ordinate (Fig. 2) 1. I N T R O D U C T I O N

THIS paper discusses the use and limitations of certain shape functions applied to the solution of segmental arches, of linearly elastic material, by the method of finite elements. The investigations to be described originated during a study of the solution of cylindrical shells. 1, * Since a uniform segmental arch is a special ease of a cylindrical shell, shape functions which are unsatisfactory for the arch cannot have general application to the shell, and thus the interest of the present work is not confined to arches. Discussing cylindrical shells, Cantin and Clough 3 described the need for the shape functions to allow rigid-body displacements of the element (i.e. displacements without strain), and showed how a shape function attributed to Gallagher 4 could be modified to satisfy this requirement without introducing additional degrees of freedom. However, m a n y problems can apparently b e solved by shape functions not allowing rigid-body displacements, and there 133

134

D. G. ASHWELL a n d A. B. SABIR

has been some discussion about their true necessity. The present investigation attempts to clarify this situation by applying three shape functions to the two-dimensional equivalent to a cylindrical shell, for which a similar problem exists. This case is the segmental arch, which is shown, with dimensions, in Fig. 1.

hl

F I G . 1.

Dimensions

of arch element.

It is believed that the results obtained are of interest in a wider context than merely that of rigid-body displacements, for we also discuss the possibility of making a shallow-shell theory approximation to the shape function of Cantin and Clough, and the effects of the values of R/h and a (see Fig. 1) on the success of the finite element analysis. Our conclusions are presented in Section 8. 2. S H A P E FUNCTIONS INVESTIGATED Fig. 2 s h o w s a n e l e m e n t of t h e a r c h s u b t e n d i n g a n a n g l e 28, t o g e t h e r w i t h c o - o r d i n a t e s 4( = y/R) a n d z, a n d t h e c o r r e s p o n d i n g d i s p l a c e m e n t s v a n d w. T h r e e sets of s h a p e f u n c t i o n s were used, as follows:

w = al+a2y+a3y2+a4yS, t ! v = aa+aey,

I

w = a , sin ¢ + a s cos ¢ + a s R cos f~ sin ¢ + a 5 y2 + a6 yS, t

v

alcos¢-a2sin¢-a3R(1-cosflcos¢)+a,y,

II

J

al y w = - ~ - + a 2 + a s y - t - a s y S + a 6 y s, y

R~2

y2

III

v = at(1-Y---;) --a2 ~--as(-~--+~2-1~) +a, y. I is t h e a p p l i c a t i o n $o a s e g m e n t a l a r c h of t h e c u r v e d shell e l e m e n t a t t r i b u t e d b y C a n t i n a n d Clough s t o G a l l a g h e r ; a i t is o b t a i n e d b y o m i t t i n g t h e a x i a l c o - o r d i n a t e x. I I is s i m i l a r l y d e r i v e d f r o m C a n t i n a n d C l o u g h ' s e x t e n s i o n of G a l l a g h e r ' s s h a p e f u n c t i o n . I f I I is s u b s t i t u t e d i n t h e expressions for c i r c u m f e r e n t i a l s t r a i n e a n d i n c r e a s e in c u r v a t u r e K, g i v e n b y dv w e = ~yy+~, (la) K =

d 2w 1 dv -+ dy2 R d y '

(lb)

c o n s t a n t s a 1, a v a s a r e f o u n d to d i s a p p e a r f r o m t h e expressions for ~ a n d K. T h e c o r r e s p o n d i n g t e r m s in I I t h e r e f o r e r e p r e s e n t d i s p l a c e m e n t s of t h e e l e m e n t w i t h o u t d e f o r m a t i o n - - i . e , rigid-body displacements. T h e s e r i g i d - b o d y d i s p l a c e m e n t s , e x p r e s s e d i n the

L i m i t a t i o n s of certain c u r v e d finite elements w h e n applied to arches

135

t e r m s of t h e three c o m p o n e n t s 8~, $~, 0 shown in Fig. 2, are : w = 8, cos 4 -- 8~ sin 4 -- OR( 1 - cos fl cos 4),

(2a)

v = 8, sin 4 + 8~ cos 4 + 0R cos fl sin 4"

(2b)

Cantin a n d Clough suggest t h a t such rigid-body displacements cannot be represented b y t h e p o l y n o m i a l displacement functions I. To o b t a i n I I there are a d d e d to (2a) t e r m s in y2 a n d ya to represent t h e change in c u r v a t u r e K, a n d to (2b) a t e r m in y to represent the circumferential strain e.

FIG. 2. Co-ordinate system. I n cases in which f] a n d 4 are small, a possible simplification of I I can be m a d e b y supposing t h a t higher orders off~ 2 a n d 42 can be neglected. F o r such eases e q u a t i o n s (1) b e c o m e dv w -- ~yyq- ~ , (3a) 42 w K = - ~dy .

(3b)

On writing t h e series expansions of t h e t r i g o n o m e t r i c a l functions in e q u a t i o n s (2), a n d including only t h e lowest order t e r m s necessary to m a k e e, K -- 0, e q u a t i o n s (2) b e c o m e w

= 8y y + ~z + Oy,

v ----

-8,~--~-

(4a)

(flt~].

(4b)

T h u s I I I is o b t a i n e d instead o f I I . I I I is t h e a r c h e q u i v a l e n t to t h e shallow-shell e q u i v a l e n t to Cantin a n d Clough's shape f u n c t i o n for a cylindrical element. T h e intrinsic (curvilinear) co-ordinates y, z, shown in Fig. 2, allow t h e use of shallow-arch e q u a t i o n s p r o v i d e d only t h a t t h e elements are shallow (fl~ 1); no restriction is placed on t h e d e p t h of t h e arch into which t h e elements are assembled. Nevertheless, it is found t h a t in some cases t h e shallow-arch elements give i n a c c u r a t e results w i t h i n t h e range of e l e m e n t sizes used in this study. 3. A R C H E S AND LOADING CASES CONSIDERED M o d e r a t e l y deep arches, s u b t e n d i n g 2a = 40 °, a n d semi-circular arches, s u b t e n d i n g 2a = 180 °, were t r e a t e d . These are called moderate a n d deep respectively. T w o values of R/h were used, n a m e l y 40 a n d 320, t h e corresponding arches being called thick a n d thin respectively. Three support conditions a t t h e a b u t m e n t s were considered, n a m e l y : (i) fixed against all linear and a n g u l a r d i s p l a c e m e n t ; (ii) fixed against linear b u t n o t angular d i s p l a c e m e n t ; (iii) fixed against circumferential and angular displacement b u t free to slide radially.

136

D . G . ASHWnLL a n d A. B. SABIR

T h e s e s u p p o r t s are called fixed, pinned a n d sliding respectively. L o a d i n g cases c o n s i d e r e d were a c e n t r a l c o n c e n t r a t e d r a d i a l load, a u n i f o r m l y distrib u t e d r a d i a l l o a d o v e r t h e whole a r c h a n d a series of e q u i d i s t a n t r a d i a l c o n c e n t r a t e d loads. Some c a l c u l a t i o n s were also m a d e for u n s y m m e t r i c a l h o r i z o n t a l loads (wind t y p e loads), b u t t ~ e r e s u l t s were n o t significantly different f r o m t h o s e o b t a i n e d w i t h r a d i a l loads. 4. P R E S E N T A T I O N OF RESULTS T h e r e s u l t s a r e p r e s e n t e d b y p l o t t i n g t h e c a l c u l a t e d deflexion A, for a g i v e n c o n s t a n t v a l u e of t h e load a n d a g i v e n s h a p e f u n c t i o n , a g a i n s t t h e n u m b e r of e l e m e n t s n i n t o w h i c h t h e (whole) a r c h w a s d i v i d e d . A b r o k e n h o r i z o n t a l line i n d i c a t e s a s o l u t i o n o b t a i n e d b y e x a c t analysis. I n t h i s way, t h e effectiveness of e a c h s h a p e f u n c t i o n , i n c o n v e r g i n g to t h e c o r r e c t solution, c a n b e r e a d i l y o b s e r v e d . Since t h e n u m e r i c a l v a l u e s c h o s e n for loads, elastic c o n s t a n t s a n d d i m e n s i o n s (except for a a n d B/h) axe n o t significant, n o n u m b e r s are a t t a c h e d t o t h e v e r t i c a l (deflexion) axes, b u t t h e y a r e p l o t t e d w i t h l i n e a r scales h a v i n g t h e i r zeros a t t h e origins of t h e g r a p h s . As is t o b e e x p e c t e d w h e n u s i n g l i n e a r e l e m e n t s (which are, of course, c o n f o r m a b l e ) , c o n v e r g e n c e to t h e correct deflexion is a l w a y s f r o m below. 5. F I X E D SUPPORTS AND CENTRAL CONCENTRATED LOAD Fig. 3 gives results for a r c h e s w i t h fixed s u p p o r t s a n d c e n t r a l c o n c e n t r a t e d loads.

I Z

<3

\/

r 0

0

i

L

4.

8

i

i

12

i

i

le zo

i

~

:'4 28

Number of" elements

J

~2 3e

0

n

i

i

4

8

i

i

I

i

I

i

t

12 I~ 20 24 28 32 56

Number of e l e m e n t s n

(c)

(d) -----

~E

¢-

T

9 ~5

r .0

r~

o

T

0

4

$

2

I

20 24. '2S 32 ~

Number o~ ¢lement"s

n

0

4

8

12

I

20 24 28

Numb=r c~ etcmcn'ts

52 ~vo n

FI¢. 3: C o n v e r g e n c e c u r v e s for a r c h e s w i t h fixed s u p p o r t s a n d c e n t r a l load.

Limitations of certain curved finite elements when applied to arches

137

For the moderate-thick case (Fig. 3a) all shape functions converge satisfactorily to the correct solution, I I slightly more rapidly t h a n I I I , and I I I t h a n I. For the moderate-thin case (Fig. 3b) I is unsatisfactory throughout the range of n considered, while both I I and I I I give solutions for n > c. 12. On considering deep arches (Figs. 3c, d) the differences between the shape functions are found to be much greater, only I I giving a satisfactory solution for the deep-thick arch, and no solution being obtained from a n y shape function for the deep-thin arch. Of particular interest are the apparent convergence of I I I to a solution 13 per cent too low for the deep-thick case, and the failure of I to provide a solution even of the right order of magnitude for the deep-thin case. 6. P I N N E D

AND SLIDING SUPPORTS AND CENTRAL CONCENTRATED LOAD Fig. 4 gives similar results for arches with sliding supports and central concentrated loads. Their behaviour is essentially similar to that of the curves for fixed supports, except for a slight increase in the differences between the shape functions. Curves for arches with pinned supports are not given; their behaviour lies generally between that of the curves for fixed and sliding supports.

(b)

(a)

r

¢-

C

.o

2

c~

g~

o

4

S

I

2o 2a 2s ~2 Y,,

o

4.

Number of ¢llmlnls n

S

12 I

~,o 2.4 2S Z,2 ~ ,

Number o~ ~lcm(zn~s n

(c)

~ . . . . . . . . . . .

lr

(d)

11.

g

o

0

~

8

12 16 20 2a 28 52 36)

Number of ¢lemenCs n

0

4-

8

12 I(0 2 0

24-28

32 3(o

Number ~¢lcrncnfs n

FIG. 4 . Convergence curves for arches with sliding s u p p o r t s a n d central concentrated load.

138

D . G . ASHWELL and A. B. SABIR

The sliding s u p p o r t case was included because the deep arch (2a = 180 °) corresponds to t h e pinched cylinder p r o b l e m discussed b y Cantin and Clough 3 a n d o t h e r authors. The dimensions of t h e shell solved b y Cantin and Clough give R / h = 52.5, which is close to t h e v a l u e for the t h i c k arches considered here. Fig. 4(d) suggests t h a t their shape function would be less satisfactory w h e n applied to a t h i n shell. 7. S L I D I N G

SUPPORTS WITH DISTRIBUTED AND EQUIDISTANT LOADS Since shape function I I does n o t contain a t e r m allowing w to be uniform (w = c, say), the case was considered of sliding supports w i t h a u n i f o r m inward radial load. The correct solution for this case is a uniform inward radial displacement. The deflexions calculated were i n d e p e n d e n t of the n u m b e r of elements (for 6~
. . . . . . . . . . . .

(b)

I, ][~ I]I

p P, P, P, p

Cl

4

B

12

~(o 2 0

S

2 ~e

Nurnbcr o~ ¢]¢mcnt$

0

4

n

(e)

s

12

t~o 2 0 2 4 2s 32 3~

Number o{ el=menU's

n

z p

z

<3

<]

-

._~

0

A

l

I

4

s

12 J~ 20

I

I

I

)

i

24 ~8 52

Number of ~l~rncn'~5 n F I G . 5.

i

L

0

4

L

8

12

IG 20 24 2 8

Number of ¢lemqnts

32

3~

r~

Convergence curves for arches with sliding supports and equidistant radial loads.

Limitations of certain curved finite elements when applied to arches

139

The problem shown in Fig. 5 was therefore solved with different numbers of elements, with the results shown. It is evidently necessary to have several elements between loads, if the equidistant load system is to be distinguished from the m~fform load. Again, all shape functions failed for the deep-thin case. 8. CONCLUSIONS The following conclusions can be drawn from the results obtained: (i) The simple shape function I, without terms allowing rigid-body displacements, is satisfactory only for thick arches of moderate or less than moderate depth. For deep, thin arches its solutions are not even of the right order of magnitude. (ii) The addition of the terms representing rigid-body displacements (shape function II) extends the power of the shape function to deal with all types of arch except deep, thin arches, unless a very large number of elements is used. (iii) The use of shallow-arch equations for the element shape functions (III) is satisfactory only for arches of moderate depth or less. For these arches I I I is only slightly inferior to II. (iv) The tendency for shape function I I I apparently to converge to solutions considerably in error (as for the deep-thin arches) contains a warning against using plausible shape functions which have not been thoroughly tested. (v) I t is dangerous to extend the use of shape functions known to be satisfactory for a structure of certain proportions to structures of a similar type but different proportions. (vi) Although the Cantin and Clough shape function I I is much superior to the others considered here, it does not have general application to arches, or, therefore, to shells. In a future paper other shape functions, of greater promise, will be considered, including one based on the work of Bogner et al. 5 REFERENCES A. B. SABIRand D. G. ASHWELL,Int. J. mech. Sci. 11, 269 (1969). A. B. SABIR,Int. J. mech. Sci. 12, 287 (1970). G. CA~TI~and R. W. C~OUGH,A I A A Journal 6, 1057 (1968). R. H. G~LLI~GHE~,The development and evaluation of matrix methods for thin shells structural analysis, Ph.D. Thesis, State University of New York, Buffalo, New York (1966). 5. F. K. BOGNER,R. L. FOX and L. A. S C ~ T , A I A A Journal 5, 745 (1967). 1. 2. 3. 4.