The boundary element analysis of stress around the bottom outlet

The boundary element analysis of stress around the bottom outlet JUSHENG

YANG

AND

JUNFA

ZANG

Shaanxl Instttute o f Mechamcal Engmeermg, P R C

In this paper, a F O R T R A N program of BEM is established by using the 'direct' technique, and some g o o d results are obtained in stress analysis of bottom outlets, which belong to a high dam of a large hydropower station situated in P.R.C. In comparison with traditional photoelastic experiment results and calculated results of FEM, it seems to us that this method is an effective technique in solving similar problems. Therefore it can be used as an important reference in engineering design. Since this method possesses advantages of saving time, high computing speed and low expense as compared with FEM, it can be provided for engineering designers to solve 2D problems of elasticity. Key words Bottom outlet, photoelastlc experiment, 'd~rect' technique, superposltlOn theorem, tangential, normal stress, horizontal axial thrust

basic expression can be written as

I

bku,~ dfl + ft

f

~r~ j uk d9 = -

"

~2

akPf dE

f

r~ /5*u~ dE

+ f

UkP~k dF

where

fi I'l

- the searching domain of the solution, - boundary of known displacement,

- boundary of known surface forces, -- virtual displacement, fi~ = 0 if It satisfies the homogeneous boundary condition at I't, if u~ is understood as a weighting function, that condition Is unsatisfied, Pk = yjk -- surface forces corresponding to u~', body forces, bk stress component defined by Einstein crjk appointment

1"2 HA

The fundamental solution u~ and p~ can be found by equlhbrlum equations

~rjk j + A, = 0 1. I N T R O D U C T I O N There 1s a dam with two 5 × 8 bottom outlets, standing side by side, in a certain large hydropower station The bottom outlets weaken the dam segment very much Generally the dam takes the shape of an arc w~thm a plane and could stand for horizontal axml forces Conslderlng the dam as a whole, we suppose that this problem is a stress analysis problem of a hole in an infinite domain According to our experience with FEM, the numbers of elements and nodes are much less, the number of freedom is about 1]10, the calculating workload decreases a lot and a high computing accuracy is provided It can be expected 6'7 that the BEM would have a wide application progressively in stress analysis of hydro-structures lO ~ 12

2. T H E O U T L I N E OF C A L C U L A T I N G T H E O R Y

AND ITS PROGRAM The 'direct' techmque based upon Green's theorem can be explained as a weighted residual method 4,8 The

A c c e p t e d J u n e 1988 Discussion closes J a n u a r y 1989

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Adv Eng Software, 1988, Vol 10, No 4

(1)

(2)

Then, apply equation (1) to the boundary, if the tth point, with unit forces applied in three directions, is considered, we have

u[+ f

r

ukP~ d F = f

r

Pku,*k d F + f

~2

bkn~ 69

(3)

This is a basic equation of BEM for the lsotroplc elasticity For slmplloty only the 2D case wtll be stud~ed here and its boundary will be divided into n elements Discretlzing equation (3), it will be

C'u'+ ~ I£I,juj = ~ GuPJ d=l

(4)

J=l

The whole set of equations for n nodes can now be lsoparametnc element, the finite element model has 259 elements and 305 nodes and IS dealt with as infinite domain Calculated assumptions are as follows (1) Take account of the whole action of the dam but the horizontal forces are transmitted from connguous segments of dams The hole stress is a mulnply connected plane-strain problem in infinite domain (2) On the horizontal sections, which are hu above the top of the bottom outlet and hd under the base, the

,c, 1988 Computanonal Mechamcs Pubhcatlons

expressed in matrix f o r m as

HU = GP

(5)

Note that there are k n o w n nl displacements on F~, k n o w n n2 forces on 1'2 (1' = I'~ + 1"2, and 2n = n~ + n2), hence there should be 2n u n k n o w n in (5) Reordering the equations in such a way that all the u n k n o w n s are on the left h a n d side, one can write

AX = F

(6)

where X is the vector o f u n k n o w n s u and P Once the values o f U and P on the whole b o u n d a r y are k n o w n one can calculate the value o f the displacements and stresses at any interior point using the following expressions (The b o d y force are not considered)

lll=~=l (Ir~ U* dF) Pk-k~=~ (IrA

P* dF)u, (7)

°q=k~=n1

( fr Du dF) Pk-k~=t ( fr S's dF) u*

where

Dts= [DI,j

L D21j],

S,j= [S~,j S2,s],

P=

[Pl

u = [u~

P2] T

u2]T

(8)

The c o m p o n e n t s o f 3-order tensor Dk,s and Sk,j are obtained f r o m physical equations correlated with material characteristics 4 According to the above expression, the p r o g r a m is established with F O R T R A N and could be run on I B M - P C , F I L E X and M-150 c o m p u t e r A n y n u m b e r o f boundaries, elements and interior points could be used in this p r o g r a m

P2 Figure 1

PI

3. LOADS A N D A S S U M P T I O N S FOR CALCULATION In 3D photoelasticlty analysis o f the above problem, the two sides o f the calculated section are free G o o d results have been obtained already In order to check the results o f BEM directly, every sections are considered as multiply connected models o f finite d o m a i n and are calculated by B E M (Fig 1) At the same time, we can c o m p a r e the results with F E M o f quadrilateral stress distribution is u n i f o r m Therefore, the infinite d o m a i n mentioned above can be simplified as an infinite strip (Fig 2) (3) Distributed forces P~ and P2 in Fig 2 are assumed to be u n i f o r m and are a d o p t e d as the same with the normal stress try at a distance 9 8 m f r o m side o f holes in photoelastlClty analysis The experimental results o f photoelastlclty analysis show that outside a distance 6 5 m f r o m sides o f holes try tends to be constant

?"

hU h

q ho

I k L I bd b

llll!

P, Figure 2 sections We have

o~)=0 4. THE SKETCH FOR C A L C U L A T I O N A N D THE MODEL OF DISCRETIZATION The applied forces o f the calculated section can be divided into two cases in accordance with the superpositlon t h e o r e m and the uniqueness o f solution s (Fig 3) Case 1 There are two distributive forces P t , P2 and the equivalent weight The stress is u n i f o r m on horizontal

Oy~1) = - (Pi + Y'Y),

(9)

r~ ) = 0 Case 2 In the b o t t o m outlet, the inner water pressure is executed The sketch o f case 2 is for calculation o f BEM, however, the upper and lower horizontal section are no longer finite but Infinite The pressure on the upper surface o f the outlet is qu, on the lower surface

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211

P,

@

® r'

=j-

q2

qll

I

II

-]

I I

II II

I I

t

Jl__!

) +

/ q2(y)

llllll P,

P2

Ftgure 3 IS qd a n d on two side surfaces is q2 q * '==ql q l - qq~ '== q l q~ ( P -l +PI Y ' h- ""y' ) (hu + h) 1 qo 1 q2 = ql + -~ (q3 + ql )(y - h,)

e q u a t i o n for elasticity written as e, -

(10)

h,,, hd are shown as in Fig 3, y is the distance f r o m the u p p e r surface t o w a r d s the lower surface v~2), Ov(2),z(z), are stresses for case 2 T h e real stresses are o b t a i n e d by using the superp o s i t i o n t h e o r e m where b o t h two cases are c o n s i d e r e d

5. C A L C U L A T E D

u , ( p ) = ~,, q,j~tjk- ~ h,jAuj~ (11) tl

A..m-

where, the coefficients, g, h, A , B can be f o u n d in Refs 4, 6 a n d 12 In B E M , the stresses are d i s c o n t i n u o u s on the b o u n d ary, so that the c a l c u l a t e d results have a certain a m o u n t o f e r r o r within a small range c o n c e r n e d with the element size In o r d e r to find out a m o r e accurate d i s t r l b u n o n o f the t a n g e n t i a l stress vt on the b o u n d a r y , we c a l c u l a t e d it by using finite difference a p p r o m m a t ] o n s a c c o r d i n g to the d i s p l a c e m e n t s on the b o u n d a r y T h e a b o v e p r o cedure c o u l d be easily e s t a b l i s h e d b y using the physical

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Adv Eng Software. 1988. Vol 10. No 4

1 (o, - # o . ) E

(12)

where, t indicates the t a n g e n t i a l d~recnon, n indicates the n o r m a l direction, ll ~s the distance between the midpoints o f two a d j a c e n t elements F o r the case o f plane s t r a t a , u a n d E are replaced respectively by u[(l - u ) a n d El(1 - u 2) m a b o v e form u l a , a n d then the t a n g e n t i a l stress on the b o u n d a r y can be o b t a i n e d 2 G A ut /~ ot = - ~1-~t Al 1-~

RESULTS

The b o u n d a r y c o n d i t i o n in the case we are studying here, is that the surface force is given, so, the solution for the basic e q u a t i o n o f b o u n d a r y element is the element d i s p l a c e m e n t The stress a n d d i s p l a c e m e n t fields in the given d o m a i n can be d e t e r m i n e d by using the superp o s i t i o n t h e o r e m a c c o r d i n g to b o u n d a r y force t, a n d b o u n d a r y d i s p l a c e m e n t u, F o r a n y p o i n t in the d o m a i n , we have

o,l(p) =

Out Aut Ot Al

The tangential s t r a t a m a y be

o.

where, u t = u t , , - u t , - l , IS the difference d i s p l a c e m e n t s for two a d j a c e n t elements

(13) of

the

6. C O N C L U S I O N S W e consider the stress d i s t r i b u t i o n s on sections 2, 5, 8, a n d 11 o f the c a l c u l a t e d section as e x a m p l e s (Fig 5a) The f o u r results that are o b t a i n e d b y 3D p h o t o e l a s t l c l t y test, B E M c a l c u l a n o n with fimte (Fig 4) a n d infinite b o u n d a r y (Fig 5b) respectively a n d 8-node lSOp a r a m e t r i c finite c a l c u l a t i o n are c o m p a r e d in T a b l e l 6 1 On each section (Fig 5a), the B E M stress distributions are in c o r r e s p o n d e n c e with the results o f the p h o t o e l a s t l c l t y test a n d F E M c a l c u l a t i o n s T h e B E M solutions are in c o r r e s p o n d e n c e with given b o u n d a r y c o n d i t i o n s W i t h classical p r o b l e m s , tests o f the t h e o r y o f c a l c u l a t i o n a n d p r o g r a m are all right T h e r e f o r e , the c a l c u l a t e d results are reliable a n d the stress d I s t r l b u n o n s are correct q u a l i t a t i v e l y 6 2 The m a x i m u m tensile stress o f all the sections is 13 54 k g / c m 2 a n d occurs on the u p p e r o r lower surfaces o f the outlet (Fig 5a) one meter f r o m the side surface Its l o c a t i o n is m c o r r e s p o n d e n c e with the results o f the p h o t o e l a s t l c l t y test a n d F E M c a l c u l a t i o n

-g4

~4

.gg

51.

SZ O l ~

44"

9O

t?

4t:

68 4~t

?f~....~91

/0

14

r0

. . . .

IZ

47"

II I0 I

I

!

Z~4

I

I

678

;

98'/ 9

W

t#

Fzgure 5-a

Ftgure 4 6 3 Compared to the results of the photoelastlclty test the maximum relative error of the tangential normal s t r e s s a l o n g t h e s u r f a c e o f o u t l e t is a b o u t 50°7o T h e r e a s o n is t h a t t h e t w o s~des o f t h e m o d e l m t h e p h o t o e l a s t l c l t y t e s t a r e f r e e , s o t h e d e f o r m a t m n s in x direction are free Sides of the calculated model of BEM are infinite, so d e f o r m a t i o n s in x d l r e c t m n a r e constrained In o r d e r to verify the a b o v e c o n c l u s m n , a c a l c u l a t i o n w a s m a d e b a s e d o n t h e size a n d t h e c a l c u l a t e d m o d e l o f t h e p h o t o e l a s t l c m o d e l ( F i g 4) It is s h o w n t h a t t h e e r r o r o f t a n g e n t m l n o r m a l s t r e s s ~s r e d u c e d t o 27 p e r cent at the m i d p o i n t o f the u p p e r s u r f a c e If the quadratic external patching method was used to c a l c u l a t e t h e s t r e s s e s at i n n e r p o i n t , t h e e r r o r c a n b e r e d u c e d t o less t h a n 10 p e r c e n t It is s h o w n t h a t t h e s t r e s s e s c a l c u l a t e d a c c o r d i n g t o i n f i n i t e d o m a i n a r e less

59, I; ; 6J ;;

2J

(

"t

\

'I

;To/ ',/(sf

1# .

.

.

.

.

.

:'! \ Tf 67.~

I d.7 --I : 6.0 Ftgure 5-b

Table 1 Comparison of stress resultsfor calculated sectton

0)

0

-0 05 2 6 0 1 0 1 2 6 0 1 4

26 52 (~) 21 68 1070 095 r~-, - 6 03 ~.d.J- 6 0 3 26 21 17 65 (~) 1042 1 10 -603 ~) - 4 57 000

12 89 895 300 099 -600 - 6 13 12 90 6 28 012 1 05 -60 -608 -525

19 05 1500 680 050 -600 -599 19 21 15 50 680 0 70 -60 -525 000

7 50 4 16 -100 -265 -605 -625 7 O0 0 59 -114 - 2 89 -616 -617 -630

- 5 63 - 5 29 -560 - 1 1 20 -2351 -2724 - 6 53 - 6 47 -735 - 13 88 -4289 -3134 -383

- 5 50 -509 -721 - 1 1 74 -2265 -2261 -650 -679 - 8 25 -1296 - 2 4 69 -2237 -1769

~ - 5 50 - 5 20 -700 - 1 1 40 -2783 -2778 -650 -610 - 7 90 -1300 - 3 9 33 -2939 -1250

FEM

~

- 5 17 -573 -780 -1273 -21 19 -2128 -621 -785 - 8 72 -1360 - 2 2 29 -2105 -1974

-003 1 40 337 008 000 000 -004 -200 - 2 96 -070 0 00 -008 000

-000 020 064 005 000 -001 000 -045 - 0 70 -012 0 00 006 004

~

FEM

000 1 O0 220 0 10 000 000 000 -090 - 2 20 020 0 00 014 000

005 -005 -015 020 001 003 -007 -036 0 50 021 - 0 03 002 -002

A d v Eng Software, 1988, Vol 10, N o 4

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t h a n t h o s e o f a single d a m b l o c k T h e r e f o r e c o n s i d e r i n g t h e d a m as a w h o l e , t h e s t r e n g t h o f t h e o u t l e t ~s improved It ts also s h o w n , ff the axial t h r u s t s a m o n g a d j a c e n t d a m b l o c k s a r e c o n s i d e r e d ( a r c h effect), t h e t a n g e n t m l tensile stress o n t h e u p p e r s u r f a c e c a n be r e d u c e d furt h e r S o t h a t t h e a s s u m p t t o n a b o u t h o r i z o n t a l axial t h r u s t is r e a s o n a b l e a n d t h e d a m t e n d s to be s a f e r 6 4 B E M lS a s i m p l e , t i m e - s a v i n g , c o n v e m e n t , l o w cost, a n d r e h a b l e m e t h o d to c a l c u l a t e t h e o u t l e t stresses f o r finite a n d infinite p l a n e p r o b l e m s C o m p a r e d w i t h F E M , result h s t l n g ~s s~mpler a n d m o r e a c c u r a t e , t h e a m o u n t o f w o r k is r e d u c e d to 1/5, C P U - U m e is r e d u c e d to a b o u t 4•5, a n d the c a l c u l a t e d a c c u r a c y is a b o u t t h e s a m e ( T a b l e 1)

3 4 5 6 7 8 9 10 11

REFERENCES Rlzzo, F J An mtegral equation approach to boundary value problems of classical elastostatlcs, Journ of Appl Math, Apr 1967, xxv, 1 2 Alarcon, E , Martin, A and Pans, F Boundary elements m 1

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potential and elasuc~ty theory, Computer and Structures, Apt 1979, 10, 1/2 Brebbla, C A and Nakaguma, R Boundary elements m stress analysts, Proc of The ASME J of the Eng Mech Dry , Feb 1979, 105, EMI Brebbla, C A The BEM for Engmeers, 1978 Banerjee, P K and Butterfield, R , BEM m Engineering Science, 1981 Croch, S L and Starfild, A M BEM m Sohd Mechamcs, 1983 CIDa, X and BeHao, H Theory, Method and Program of FEM m Sohd Mechamcs (in Chinese), Water Resources and Electric Power Press, 1983 Jusheng, Y Introduction of BEM (m Chinese), The Institute of Mechamcal Engineering, 1984 The report on 3D photoelast~c~ty for the bottom outlet of Ankang Water Power Station (in Chinese), The Research group o| Hydro Engineering, 1983 Jlahn, Z and Junfel, X BEM for Plane Elasticity problems and its Implementation (m Chinese), Proc of Chinese First Conference on BEM m Engineering, 12, 1985 Xmchuan, W The Displacement Dlscontmmty Method and its Apphcation m the Stress Analysis of Hydro-Structure (in Chinese), Proc of Chinese First Conference of BEM m Engineermg, 12, 1985 Llhong, C The BEM Of Elastic Mechamcal and its Application In Engineering (m Chinese), Hehal Umvensty (Master Degree thesis)