NONLINEAR FINITE ELEMENT ANALYSIS AND ADINA

Proceedings o f the 4th A D I N A Conference

Massachusetts Institute of Technology

15-17 June 1983

Guest Editor

K. J. BATHE Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

PERGAMON PRESS

OXFORD NEW YORK TORONTO SYDNEY PARIS FRANKFURT

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Copyright 1983 Pergamon Press Ltd.

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Published as a special issue of the journal Computers & Structures, Vol. 17, Numbers 5-6 and supplied to subscribers as part of their normal subscription. Also available to non-subscribers.

Computers Structures Vol. 17, No. 5 6, p. (1983). Pergamon Press Ltd. Printed in Great Britain.

PREFACE

The thoughts I presented in the preface of the proceedings of the 3rd ADINA Conference (see Computers and Structures, Vol. 13, No. 5-6, 1981) are also quite applicable to this 4th Conference Nonlinear Finite Element Analysis and ADINA. However, there are some thoughts that, at this opportunity, I would like to mention again and discuss further.

It is well-accepted that finite element methods will be employed to an increasing extent in engineering practice, and that the appropriate application of these analysis methods requires a considerable amount of education and experience.

The effective use of finite element computer programs is based on a thorough understanding of the basic principles of mechanics and materials and a knowledge of the finite element procedures and assumptions used when operating on these principles. Much of the basic knowledge to properly employ finite element methods must be taught at the Universities in the undergraduate and graduate education, and we must expect a considerable evolutionsome of which has taken place alreadyof the University curricula in this respect. This educational goal is clearly not fulfilled by simple use of some finite element computer programs in the courses taught, but there is a deep intellectual challenge to change some of the traditional teaching of mechanics, for both analysis and design.

However, another important area of professional education is the Continuing Education Study so important to all of us in engineering practice. With the methods of analysis and design evolving quite rapidly, we are continuously faced with many new developments to learn and absorb. Also, considering the currently available very impressive capabilities for analysis using finite element methods, the art of good engineering analysis is a very exciting and rewarding field with much room for learning and engineering ingenuity.

The objective of the ADINA Conferences is to contribute to this continuing education process. In these conferences a number of valuable papers are presented on the usage of the ADINA system that provide the focal point for in-depth discussions of current state-of-the-art modeling and solution of complex problems. The reason for concentrating on the use of one finite element program system is to give a common basis of interest and understanding to the conference participants. However, many experiences discussed are quite general and should be of interest to most finite element researchers and practitioners. This volume contains the papers presented at the 4th ADINA Conference.

I am once again personally very pleased that these proceedings are published in Computers and Structures and would like to express my appreciation to Professor H. Liebowitz for this help in my research and educational goals. I am also very thankful to the authors of the papers for their efforts and cooperation, and to Ms. C. Simonsen of the Center for Advanced Engineering Study of M.I.T. and Ms. T. Nolan, my secretary, for their help in organizing this conference. Finally, I would like to thank the ADINA users group whose lasting support is making this complete endeavor possible.

Massachusetts Institute of Technology K . J . B A T H E Cambridge, MA 02139 U.S.A.

V

Computers Structures Vol. 17, No. 5 6, pp. 625 633, 1983 Printed in Great Britain.

0045-7949/83 $3.00+ .00 Pergamon Press Ltd.

THE USE OF ADINA FOR ANALYSIS OF MINES WITH EXPLOSIVE FILLS

F R E D E R I C K H . G R E G O R Y a n d A A R O N D . G U P T A

U.S. Army Ballistic Research Laboratory, U.S. Army Armament Research and Development Command , Aberdeen Proving Ground , M D 21005, U.S.A.

A b s t r a c t T h e structural response of a U.S. M - 1 5 and a Soviet T M - 4 6 land mine with explosive fills subjected to an externally applied pressure wave has been analyzed with the A D I N A finite element code. The finite element model of the two-dimensional axisymmetric configurations predicted response out to 2 msec of real time. Failure of the mine case was predicted, based on a comparison of the value of the three-dimensional second invariant of plastic strain with that of the one-dimensional value obtained from tensile tests.

1. INTRODUCTION

T h i s p a p e r d e s c r i b e s t h e r e s p o n s e o f a n t i t a n k m i n e s o f

t w o different c o n f i g u r a t i o n s t o a t r a n s i e n t b l a s t l o a d .

T h e r a t i o n a l e for th i s a n a l y s i s is t h e n e e d t o d e v e l o p a

r e m o t e , e x p e d i t i o u s m e a n s o f c l e a r i n g a p a t h t h r o u g h

a n e n e m y m i n e field. A t e c h n i q u e o f de l i ve r i ng re la -

t ively l a rge t r a n s i e n t p r e s s u r e t o t h e su r f ace o f t h e

e a r t h by m e a n s o f exp los ives is u n d e r d e v e l o p m e n t .

T h e ob jec t o f th i s s t u d y is t o d e t e r m i n e t h e e x t e n t o f

s t r u c t u r a l d a m a g e t o m i n e b o d i e s f r o m a g iven level o f

b l a s t w a v e a m p l i t u d e a n d s h a p e . T h e p r i n c i p l e d a m -

age m e c h a n i s m is t o b e a s e r i o u s d i s t o r t i o n o r r u p t u r e

of t h e m i n e b o d y r a t h e r t h a n fuze i n i t i a t i o n o r p r e s s u r e

p l a t e r e m o v a l s ince t h e a c t i v a t i o n m e c h a n i s m s c o u l d

be c h a n g e d eas i ly f r o m o n e t y p e o f m i n e t o a n o t h e r

a n d a fa i lu re c o u l d n o t b e g u a r a n t e e d b a s e d o n a

p a r t i c u l a r m o d e o f a c t u a t i o n .

T h e m i n e s i n v e s t i g a t e d r e p r e s e n t typ ica l a n t i t a n k

m i n e s , b o t h fore ign a n d U . S . m a n u f a c t u r e , w h i c h c o n -

sist bas i ca l ly o f r o u n d t h i n m e t a l b o d i e s filled w i t h

exp los ives . T h e s e t ypes o f a n t i t a n k m i n e c o n s t i t u t e a

l a rge p a r t o f t h e i n v e n t o r y o f U . S . a n d fore ign m i n e s .

T h e c o m p o n e n t s m o s t d i s t i nc t i ve a r e t h e fuze m e c h -

a n i s m s . T h e r e a r e a va r i e ty o f r ad i ca l l y di f ferent fuzes

for t hese m i n e s , d i f ferent b o t h in m e c h a n i c a l d e s i g n s

a n d m e t h o d o f a c t i v a t i o n . T h e r e f o r e t h e n u m e r i c a l

m o d e l s a d a p t e d for t h e t w o m i n e s a r e r e p r e s e n t a t i v e o f

a l a rge c lass o f b o t h fo re ign a n d U . S . m i n e s .

T h e p a p e r h a s fou r m a j o r a r e a s a s fo l lows: (a) p r o b -

lem def in i t ion , (b) d e t e r m i n a t i o n o f m a t e r i a l p r o p e r -

ties a n d se lec t ion o f fa i lure c r i t e r i a , (c) finite e l e m e n t

m o d e l d e s c r i p t i o n a n d c a l c u l a t i o n s , a n d (d) d y n a m i c

r e s p o n s e p r e d i c t i o n o f t h e s t r u c t u r a l a s s e m b l y .

2. PROBLEM DEFINITION

2.1 TM-46 antitank mine description

T h e T M - 4 6 l a n d m i n e h a s a cy l ind r i ca l steel b o d y

wi th a p r i m a r y fuze well in t h e c e n t e r o f t h e t o p a n d

o n e o n t h e b o t t o m , p r e s u m a b l y for ant i l i f t o r b o o b y

t r a p p i n g p u r p o s e s . In a d d i t i o n , it h a s a s e c o n d a r y fuze

well in t he s idewal l u n d e r n e a t h t h e c a r r y i n g h a n d l e . A

sec t iona l d r a w i n g o f t h e m i n e is s h o w n in F i g . 1. T h e

m i n e h a s a m o n i n a l d i a m e t e r o f 29 .7 c m , h e i g h t o f

7.3 c m , a n d we ighs 8.7 k g w i th a m a i n c h a r g e o f 5.7 k g

T N T .

T h e m i n e b o d y is m a d e o f t h r e e p ieces o f shee t steel

wh ich a r e j o i n e d a t t h e u p p e r p e r i p h e r y by a 360

c r i m p . T h e t o p of t h e m i n e b o d y is o n l y 0 .635 m m

th i ck a n d h a s t h r e e s t e p s . T h i s c o v e r c o n n e c t s t o a

c e n t r a l c i r c u l a r p l a t e f o r m e d by s p o t - w e l d i n g o f a

t h i ck p l a t e t o t h e t h i n c o v e r sec t ion . T h e i n t e r m e d i a t e

wal l is f o r m e d f rom 0.94 m m th ick steel shee t t o w h i c h

a h o l l o w cy l indr i ca l p iece 0 .56 m m th ick is a t t a c h e d t o

f o r m t h e c e n t r a l l y l o c a t e d t o p fuze well . T h e fuze well

c o n t a i n s a 4 0 g te t ry l b o o s t e r c h a r g e for a c t i v a t i o n .

T h e l o w e r p a r t o f t h e m i n e b o d y is f o r m e d by a d e e p

d r a w i n g o p e r a t i o n w h i c h resu l t s in very in-

h o m o g e n e o u s m a t e r i a l p r o p e r t i e s . T h e c e n t r a l cav i ty

in t h e m a i n b o d y o f t h e m i n e is filled w i t h a c h a r g e o f

5.7 k g T N T exp los ive . T h e c a v i t y b e t w e e n t h e t o p a n d

i n t e r m e d i a t e wa l l s is unfi l led. H o w e v e r c o m p r e s s i o n

o f a i r in th i s r e g i o n c a n c o n t r i b u t e t o a l t e r a t i o n of t h e

r e s p o n s e b e h a v i o r o f t h e m i n e a n d s u b s e q u e n t u n -

c r i m p i n g o f t h e j o i n t .

T h e n o r m a l m e t h o d o f a c t i v a t i o n o f t h e fuze is b y

m e a n s o f force a p p l i e d t o t h e p r e s s u r e c a p d e p r e s s i n g

t h e fuze a n d r e l eas ing t h e s t r i k e r t o s t r i ke t h e b o o s t e r

c h a r g e in t h e fuze wel l . T h i s a c t i v a t e s t h e t e t ry l b o o s t e r

w h i c h in t u r n d e t o n a t e s t h e p r i m a r y T N T c h a r g e . T h e

s e c o n d a r y fuze well o n t h e T M - 4 6 m i n e gives it a n

a n t i - d i s t u r b a n c e c a p a b i l i t y .

2.2 M-15 antitank mine description

T h e M - 1 5 m i n e h a s a cy l ind r i ca l b o d y s imi l a r t o

t h e T M - 4 6 m i n e . H o w e v e r t h e r e is n o i n t e r m e d i a t e

wa l l o r unfi l led s p a c e in t h e U . S . m i n e . T h e m i n e h a s

a n o m i n a l d i a m e t e r o f 32 .13 c m , h e i g h t o f 9.88 c m ,

a n d w e i g h s 14.3 kg . T h e c e n t e r o f t h e t o p of t h e m i n e

h a s a d e p r e s s e d a r e a w h i c h h o u s e s t h e p r e s s u r e p l a t e

a s s e m b l y . I s o m e t r i c a n d s ide v iews o f t h e m i n e a r e

s h o w n in F ig . 2.

T h e m i n e is m a d e essen t i a l ly o f t w o pieces o f

W D - 1 0 1 0 steel w h i c h a r e j o i n e d a t t h e l o w e r p e r i p h -

e ry by a 360 c r i m p . T h e u p p e r p a r t o f t h e m i n e b o d y

is f o r m e d by a d e e p d r a w i n g o p e r a t i o n w h i c h resu l t s

in ve ry i n h o m o g e n e o u s m a t e r i a l s p r o p e r t i e s as is t h e

III

Fig. 1. Soviet anti tank mine.

625

626 F. H. GREGORY and A . D. G U P T A

A R M I N G P L U G IN PRESSURE

Fig. 2. U.S. M - 1 5 anti tank mine.

case w i th t h e Sovie t T M - 4 6 m i n e . T h e c e n t r a l c av i ty

in t h e l o w e r ha l f of F i g . 2 is filled w i t h 10 k g o f

c o m p o s i t i o n exp los ive .

T h e fuze is a c t i v a t e d by m e a n s o f force a p p l i e d t o

t h e p r e s s u r e p l a t e ( 1 2 5 0 - 2 0 0 0 n e w t o n s ) w h i c h in t u r n

is t r a n s f e r r e d t o t h e bel levil le s p r i n g s . A t a c e r t a i n

def lec t ion , t h e bellevil le s p r i n g s s n a p t h r o u g h , d r i v i n g

t h e firing p in i n t o t h e d e t o n a t o r . T h e e x p l o s i o n o f t h e

d e t o n a t o r a c t i v a t e s t h e t e t ry l b o o s t e r w h i c h in t u r n

d e t o n a t e s t h e p r i m a r y c o m p o s i t i o n c h a r g e . T h e r e

a r e t w o a u x i l i a r y fuze wel ls o n t h e M - 1 5 m i n e t o

a l l ow a n t i - d i s t u r b a n c e c a p a b i l i t y s imi l a r t o t h e Sov ie t

m i n e .

2.3 Guidelines for the numerical model In k e e p i n g w i t h t h e p h i l o s o p h y o f iden t i fy ing a

gene ra l fa i lure m e c h a n i s m i n d e p e n d e n t of s o m e specific de s ign f ea tu r e , all p r e s s u r e c a p s o r p l a t e s , fuzes a n d s p r i n g s w e r e o m i t t e d f r o m t h e finite ele-m e n t m o d e l o f b o t h m i n e s . T h i s w a s d o n e in a c c o r d -a n c e w i t h t h e p r e v i o u s l y s t a t e d gu ide l i ne o f n o t iden t i fy ing fa i lure o f t h e fuze c o m p o n e n t s . T h e m o d -els s h o w n d o n o t i n c l u d e s e c o n d a r y fuzes a n d filling ho l e s . H o w e v e r t h e s e c o n d a r y te t ry l b o o s t e r c h a r g e is i n c l u d e d in t h e Sov ie t m i n e t o fac i l i ta te a s s e s s m e n t of t h e inf luence of t r a p p e d a i r in t h e unfi l led s p a c e b e l o w t h e t o p wa l l .

T h e a u x i l i a r y fuze wells w e r e n o t c o n s i d e r e d in t h e c u r r e n t i n v e s t i g a t i o n s ince t hey m a k e t h e m i n e b o d i e s h igh ly s u s c e p t i b l e t o d a m a g e d u e t o s t ress c o n c e n -t r a t i o n s n e a r t h e j u n c t i o n b e t w e e n t h e b o d y a n d t h e fuze. T h u s , t h e s impli f ied m o d e l is c o n s e r v a t i v e in t e r m s o f b l a s t l o a d r e q u i r e d for m i n e d e a c t i v a t i o n .

A l s o , i nc lu s ion o f these u n s y m m e t r i c a l l y l o c a t e d

s t r u c t u r e s w o u l d h a v e n e c e s s i t a t e d t h e use of a t h r e e -

d i m e n s i o n a l ( 3 - D ) finite e l e m e n t m o d e l r e s u l t i n g in

s igni f icant i n c r e a s e in c o m p u t i n g t i m e a n d c o s t s . T h e

d i m p l e s a t t h e b a s e of b o t h m i n e s w e r e e l i m i n a t e d for

t h e s a m e r e a s o n s . B e c a u s e of t h e s e s impl i f i ca t ions ,

t h e 2 - D a x i s y m m e t r i c m o d e l s w e r e a d e q u a t e for

d y n a m i c r e s p o n s e e v a l u a t i o n .

2.4 Base support and surface loading

D u r i n g field e m p l a c e m e n t , t h e m i n e s m a y b e p l a c e d

o n t h e su r f ace a n d c o v e r e d w i t h g r a s s o r o t h e r

m a t e r i a l s for c o n c e a l m e n t . In o t h e r cases , t h e m i n e s

m a y b e s h a l l o w b u r i e d . I n e i t he r c a se , t h e m i n e s will

e x p e r i e n c e t r a n s i e n t p r e s s u r e l o a d i n g o n t h e t o p

su r f ace d u e t o d e t o n a t i o n o f a c o u n t e r - m i n e exp los ive

in t h e v ic in i ty . T h e b a s e a n d s ide b o u n d a r y c o n d i -

t i o n s w e r e t r e a t e d in t w o di f ferent w a y s in t h e M - 1 5

m i n e s t u d y . It is e x p e c t e d t h a t t yp ica l field b o u n d a r y

s u p p o r t c o n d i t i o n s w o u l d b e b r a c k e t e d by t h e t w o

e x t r e m e c o n d i t i o n s s i m u l a t e d . In o n e case , t h e b a s e

w a s s u p p o r t e d o n n o n - l i n e a r s p r i n g s s i m u l a t i n g soi l .

In th i s c a s e , t h e m i n e w a s s i m u l a t e d a s b e i n g b u r i e d

in soil u p t o i ts t o p su r f ace b y a l l o w i n g d o w n w a r d

a c c e l e r a t i o n / m o v e m e n t o f t h e m i n e b a s e d o n d y -

n a m i c p r o p e r t i e s of t h e soil m e d i u m as d e s c r i b e d in

[1]

T h e o t h e r s u p p o r t c o n d i t i o n u s e d for t h e M - 1 5 a n d T M - 4 6 m i n e s w a s a r igid s u p p o r t w h i c h c lose ly m o d e l e d t h e e x p e r i m e n t a l c o n d i t i o n s d e s c r i b e d in [2]. A ro l le r s u p p o r t c o n d i t i o n w a s used a l l o w i n g l a t e r a l , b u t n o ve r t i ca l , m o t i o n . T h e i nd i r ec t l o a d i n g o f t h e m i n e t h r o u g h s h o c k w a v e s p a s s i n g t h r o u g h t h e soil

The use of A D I N A for analysis of mines with explosive fills 627

T i m e ( m s )

Fig. 3. Shock loading function for ant i tank mines.

m e d i u m w a s n o t m o d e l e d . I n th i s r ig id s u p p o r t c o n d i t i o n , t h e i n p u t s h o c k l o a d is a p p l i e d t o t h e t o p a n d s ides o f t h e m i n e ; w h e r e a s , in t h e s p r i n g s u p p o r t c o n d i t i o n , o n l y t h e t o p o f t h e m i n e w a s l o a d e d d i rec t ly .

F o r s t r u c t u r a l l o a d i n g t h e p r e s s u r e p u l s e u s e d in th is p a p e r s i m u l a t e d p e a k p r e s s u r e a n d i m p u l s e m e a -s u r e d f r o m e x p e r i m e n t s c o n d u c t e d w i t h m i n e c lea r -a n c e t y p e s o f exp lo s ive s in [2]. T h e p e a k p r e s s u r e w a s 13.8 M P a a n d t h e i m p u l s e de l i ve red w a s 6.5 k P a - s e c . A d e c a y i n g e x p o n e n t i a l f u n c t i o n w a s fitted t o t h e s e p a r a m e t e r s r e s u l t i n g in t h e f o l l o w i n g e q u a t i o n

/>(*) = 13.76 e "2 1 1 7

' . (1)

A c u r v e o f th i s f u n c t i o n v a r y i n g in t i m e is s h o w n in

F i g . 3 .

3. MATERIAL PROPERTIES A N D FAILURE CRITERIA

M a t e r i a l p r o p e r t i e s w e r e r e q u i r e d for t h e steel

j a c k e t s , t h e e x p l o s i v e filler m a t e r i a l s , t h e t r a p p e d a i r ,

a n d t h e soil in w h i c h t h e m i n e is e m p l a c e d . M e c h a n -

ical p r o p e r t i e s w e r e m e a s u r e d for t h e steel j a c k e t s b y

e m p l o y i n g u n i a x i a l t ens i le t e s t s . T h e d a t a for t h e

exp los ive a n d soil w e r e t a k e n f r o m a v a i l a b l e p u b l i c a -

t i o n s . F a i l u r e c r i t e r i a u s e d for t h e s teel j a c k e t s a n d

t h e filler m a t e r i a l s w e r e s imi l a r t o t h e f o r m u l a t i o n s in

[3]

TOP

S I D E W A L L

( a ) L O C A T I O N O F S P E C I M E N S , M - 1 5 M I N E

S I D E W A L L '

( b ) L O C A T I O N O F S P E C I M E N S , T M - 4 6 M I N E

60i,12R 2.03 i . 2 5J

( c ) P R E P A R A T I O N O F S P E C I M E N D I M E N S I O N S ( c m )

Fig. 4. Details of tensile specimen sampling and prepara-tion.

3.1 Steel casing T h e M - 1 5 j a c k e t is m a d e o f a m e d i u m s t r e n g t h

steel a l l oy w i t h a d e n s i t y o f 7 .80 g / c m3 a n d a t h i c k n e s s

o f 0 .94 m m . T h e T M - 4 6 j a c k e t is m a d e o f a l o w c a r b o n soft m a g n e t i c s teel e q u i v a l e n t t o m i l d s teel . T h e l o w e r p a r t o f t h e c a s i n g w a s d e e p d r a w n , b u t it r e t a i n e d a n e q u i a x e d g r a i n m i c r o s t r u c t u r e w i t h i so -t r o p i c p r o p e r t i e s . T w o tens i le s p e c i m e n s w e r e c u t f r o m e a c h o f t h e s igni f icant su r f aces o f t h e m i n e b o d y . L o c a t i o n s o f t h e s e s p e c i m e n s a r e s h o w n in F i g . 4 (a , b ) . T h e s p e c i m e n s w e r e m a c h i n e d w i t h a l a rge r a d i u s o n t h e tes t s ec t i on a s s h o w n in F i g . 4 (c) . A n e x t e n s o m e t e r a n d a b i ax i a l s t r a i n g a g e w e r e a t t a c h e d a t t h e l o c a t i o n o f t h e m i n i m u m w i d t h a n d t h e spec i -m e n s w e r e t e s t ed in a n I n s t r o n T e s t i n g M a c h i n e . T y p i c a l s t r e s s - s t r a in c u r v e s for t h e U . S . a n d t h e Sov ie t m i n e b o d y a r e s h o w n in F i g s . 5 a n d 6 re -spec t ive ly . E v i d e n c e o f w o r k h a r d e n i n g a n d r e s i d u a l s t ress w a s s igni f icant in t h e Sov ie t m i n e d u e t o t h e f o r m i n g o p e r a t i o n .

B i l inea r a p p r o x i m a t i o n s t o t h e s t r e s s - s t r a i n c u r v e s

Pressure plate well tensile tests 5 0 0

Q_ T e s t 2

B i l i n e a r

a p p r o x i m a t i o n

T e s t I

6 . 0 9 . 0

T r u e s t r a i n ( % )

Fig. 5. Stress-strain curves for the pressure plate well specimens for the M - 1 5 mine.

628 F . H . GREGORY and A . D . GUPTA

? 100

5 10 E N G I N E E R I N G S T R A I N (X)

Fig. 6. Stress-strain curves for the top cover plate speci-mens for the T M - 4 6 mine.

o b t a i n e d by a v e r a g i n g t h e d a t a for t h e i n d i v i d u a l

s p e c i m e n s a r e s h o w n s u p e r i m p o s e d in F ig s . 5 a n d 6.

T h e A D I N A [4, 5] finite e l e m e n t c o d e u s e d in th i s

ana lys i s h a s a b i l i nea r , e l a s t i c - p l a s t i c , v o n M i s e s yield

c o n d i t i o n , k i n e m a t i c h a r d e n i n g , a x i s y m m e t r i c 2 - D

e l e m e n t for t h e steel j a c k e t .

T h e c r i t e r i o n se lec ted t o p r e d i c t fa i lu re o f t h e steel

c a s ing m a t e r i a l w a s d e s c r i b e d in [3] a s t h e v a l u e o f t h e

s e c o n d i n v a r i a n t o f p l a s t i c d e v i a t o r i c s t r a i n a t fa i lu re ,

Ijjie1*), def ined a s

(2)

w h e r e t h e s t r a i n s i n d i c a t e d a r e t o b e t h e s t r a i n s a t

fai lure. In t h e u n i a x i a l t e n s i o n test w h e r e t h e l o a d is

a p p l i e d in t h e Z - d i r e c t i o n , w e h a v e ,

= 3 / 4 ( z)2. (3)

3.2 Characterization of explosives-

T h e r e a r e t w o t y p e s of exp los ives e m p l o y e d in t h e

T M - 4 6 m i n e , i.e. T N T as t h e m a i n c h a r g e a n d te t ry l

a s t he fuze well c h a r g e . F o r t h e U . S . M - 1 5 m i n e ,

c o m p o s i t i o n B-3 exp los ive c o n s i s t i n g o f 6 0 % R D X

a n d 4 0 % T N T is u sed in ca s t f o r m a s t h e m a i n c h a r g e .

Af te r s u r v e y i n g t h e a v a i l a b l e m a t e r i a l p r o p e r t i e s o f

exp los ives a n d t h e v a r i o u s 2 - D a x i s y m m e t r i c m a t e r i -

als m o d e l s in t h e A D I N A c o d e , it w a s d e c i d e d t h a t

t h e c u r v e d e s c r i p t i o n m a t e r i a l m o d e l (see [4] S e c t i o n

X I I , p p . 17 -22 ) w a s t h e a p p r o p r i a t e m o d e l t o use .

T h i s m o d e l r e q u i r e s t a b l e s o f l o a d i n g a n d u n l o a d i n g

bu lk m o d u l i a n d s h e a r m o d u l i v e r s u s v o l u m e s t r a i n .

A r e l a t i o n s h i p b e t w e e n t h e v o l u m e s t r a i n a n d t h e

bu lk m o d u l u s o b t a i n e d f r o m t h e M i e - G r n e i s e n

e q u a t i o n of s t a t e [ 1 , 6 ] is g iven as

( + 1 ) ( 2 + V +

The use of A D I N A for analysis of mines with explosive fills 629

B U L K M O D U L U S

/

S H E A R M O D U L U S

[I F A I L U R E I N T E N

I f AT v =-0

0 2 4 6

V O L U M E S T R A I N (%)

s I I 2.0

- BULK MODULUS

G - SHEAR MODULUS

3 - I

.

0.0 0.02 0.04 0.06 0.08 0.10

v - VOLUMETRIC STRAIN

Fig. 8. Bulk and shear moduli vs volume strain for T N T explosive.

- B U L K MODULUS , ^ 2 0 . 0

S

3 5

0.0 0.02 0.04 0.06 0.08 0.10

v - VOLUMETRIC STRAIN

Fig. 7. Bulk and shear moduli vs volume strain for com- Fig. 9. Bulk and shear moduli vs volume strain for tetryl position B-3 explosive of the M - 1 5 mine. explosive.

Table 2. A D I N A Input values for bulk and shear moduli for filler materials

COMPOSITION B - 3 E X P L O S I V E

P o i n t No. , . n K

G V I u

(%) ( G P a ) ( G P a ) ( G P a )

1 0 1 3 . 52 13 .52 6 .60

2 1.0 14. 00 14.00 6 .84

3 2 . 5 14 .91 14.91 7 .28

4 3.75 15 .83 15 .83 7 .73

5 5 .0 16 92 16.92 8.26

6 10 .0 23 36 23.36 11.41

TNT E X P L O S I V E

1 0 21 72 21 .72 10.62

2 1.0 23 03 23 .03 11.24

3 3.0 25 65 25 .65 12 .55

4 5 .0 28 68 28 .68 14.01

5 9 .0 35 85 35 .85 17.51

6 11 .0 40 20 40 .20 19 .65

TETRYL F I L L E R

1 0 10 5 10 .5 4 . 0 3

2 1 .0 11 15 11 .15 4 .27

3 3 .0 12 59 12.59 4 . 8 3

4 5 .0 14 .24 14.24 5.46

5 8 . 0 17 .2 17 .2 6 .60

6 10.0 19 .56 19.56 7.50

630 F . H . GREGORY and A . D . G U PT A

a p p l i e d a t a n e l e m e n t i n t e g r a t i o n p o i n t is g iven for a n

e l e m e n t , j , by

(6)

w h e r e pe is t h e d e n s i t y of t h e o v e r b u r d e n ; is t h e

s h a p e func t ion for n o d e / o f e l e m e n t j ; a n d Zi} is t h e

ver t ica l c o o r d i n a t e for n o d e / in e l e m e n t / T h e

p o s i t i o n of t he s y s t e m ver t i ca l c o o r d i n a t e c a n be

o b t a i n e d f rom t h e e q u a t i o n ,

gPe

w h e r e 0 is t he ini t ia l b u l k l o a d i n g m o d u l u s ; ej is t h e

v o l u m e t r i c fai lure s t r a i n , n e g a t i v e in t e n s i o n ; a n d g is

t h e a c c e l e r a t i o n d u e t o g r a v i t y .

3.3 Soil simulation

F o r t h e s t r u c t u r a l r e s p o n s e c a l c u l a t i o n s o f t h e

s h a l l o w b u r i e d M - 1 5 m i n e , o n l y t h e t o p o f t h e m i n e

w a s e x p o s e d t o b l a s t p r e s s u r e whi le t h e r e m a i n d e r

w a s a s s u m e d t o b e e m b e d d e d in soi l . A n impl ic i t

m o d e l i n g t e c h n i q u e w a s e m p l o y e d w h e r e b y n o d a l t ie

e l e m e n t s w e r e u s e d t o m o d e l t h e b a s e s u p p o r t a s

n o n l i n e a r s p r i n g s . N o s i m u l a t i o n o f t h e soil w a s

neces sa ry for t h e r igid s u p p o r t c a l c u l a t i o n s .

T h r e e di f ferent t ypes o f n o d a l t ie e l e m e n t s w e r e

ava i l ab l e in t h e Bal l is t ic R e s e a r c h L a b o r a t o r y v e r s i o n

of t he A D I N A c o d e . T h e p a r t i c u l a r t y p e c h o s e n is t h e

b o u n d a r y t y p e e l e m e n t def ined b y o n e n o d e o n l y a n d

is c a p a b l e o f t h r e e t r a n s l a t i o n a l a n d t h r e e r o t a t i o n a l

deg ree s of f r e e d o m . In t h e M - 1 5 m i n e , t h e e l e m e n t s

a l o n g t h e b a s e o f t h e m i n e w e r e u s e d t o t r a n s m i t a

ver t ica l force ( F z) , wh i l e t h o s e a l o n g t h e s ide e x e r t e d

a h o r i z o n t a l force (FY).

D u e t o t h e l a rge va r i e ty o f soils in w h i c h m i n e s

w o u l d b e e m p l a c e d , it is p o s s i b l e o n l y t o select a soil

s i m u l a t i o n m o d e l w h i c h w o u l d b e r e p r e s e n t a t i v e o f

s o m e s u b c l a s s o f soi ls . T h u s , a typ ica l l o a d def lec t ion

c u r v e [7] w a s se lec ted t o def ine t h e n o d a l t ie e l e m e n t

p r o p e r t i e s . T h e a v e r a g e l oad -de f l ec t i on for s lowly

v a r y i n g l o a d s in t h e e las t ic l o a d i n g r a n g e f r o m [7] is

0 .0815 M P a / c m . T o a c c o u n t for t h e d y n a m i c re -

s p o n s e o f soil a t t h e b a s e o f t h e m i n e , a n o n l i n e a r

q u a d r a t i c c o m p o n e n t w a s a d d e d t o t h e force

def lec t ion p r o p e r t y .

F o r t h e s u p p o r t a l o n g the ver t i ca l s ides o f t h e m i n e ,

a l inear s p r i n g force w a s u s e d d u e t o c o n s i d e r a t i o n of

smal l l a t e ra l m o v e m e n t . T h e l inea r n o d a l t ie e l e m e n t

stiffness v a l u e s a l o n g t h e ver t ica l s ide a r e p r o p o r -

t i ona l t o t h e h e i g h t o f t h e p a r t i c u l a r e l e m e n t o n t o

w h i c h t h e n o d a l tie b o u n d a r y e l e m e n t is a t t a c h e d .

S imi la r ly , t h e n o n l i n e a r stiffness v a l u e s for s p r i n g s a t

t h e b a s e in t h e A D I N A i n p u t a r e a d j u s t e d by a f a c to r

p r o p o r t i o n a l t o t h e a n n u l a r s ec to r o f pi r a d i a n s a n d

a r ad i a l e x t e n t a p p r o p r i a t e for t h e p a r t i c u l a r n o d a l t ie

e l e m e n t . F o r soil m o d e l i n g o f t h e T M ^ 6 m i n e in t h e

A D I N A c o d e , a n expl ic i t t e c h n i q u e u s i n g t w o l ayers

o f c o m p r e s s i b l e soil e l e m e n t s s u r r o u n d i n g t h e m i n e

will be e m p l o y e d .

3.4 Simulation of void in - 4 6 mine

T h e T M - 4 6 m i n e h a s a cav i ty b e t w e e n t h e u p p e r

p r e s s u r e p l a t e a n d t h e m i d d l e p l a t e c o v e r i n g t h e

p r i m a r y c h a r g e . T h i s cav i ty h a s a i r in it w h i c h w o u l d

t r ans f e r s o m e l o a d t o t h e m i d d l e p l a t e as t h e v o l u m e

of t h e cav i ty is d e c r e a s e d . A n a t t e m p t w a s m a d e t o

m o d e l t h e a i r w i t h 2 - D a x i s y m m e t r i c fluid e l e m e n t s

c o m p o s e d of a n invisc id l i nea r c o m p r e s s i b l e m a t e r i a l .

A c o n s t a n t b u l k m o d u l u s w a s u s e d in l ieu o f a

p r e s s u r e d e p e n d e n t b u l k m o d u l u s d u e t o lack o f

a v a i l a b l e d a t a for a i r . H o w e v e r , t h e p r i m a r y difficulty

w i t h th is m o d e l w a s t h a t t h e r e w a s n o t h i n g in t h e

m o d e l t o p r e v e n t t h e u p p e r p l a t e f rom p e n e t r a t i n g t h e

m i d d l e p l a t e as t h e d e f o r m a t i o n p r o g r e s s e d .

S ince t h e a i r w a s j u d g e d t o a p p l y o n l y a m i n i m a l

r e s t r a i n t o n t h e m o t i o n o f t he u p p e r p l a t e a n d d u e t o

t h e n e e d t o p r e v e n t t h e t w o p la t e s f r o m p a s s i n g

t h r o u g h o n e a n o t h e r , a different m o d e l h a s b e e n

a d o p t e d . T h e m o d e l c o n s i s t s o f ax ia l t r u s s e l e m e n t s

c o n n e c t i n g t h e t w o c i r c u l a r p l a t e s . T h e m a t e r i a l

m o d e l for t h e t ru s ses is n o n l i n e a r a n d d e v e l o p s on ly

a sma l l force u p un t i l t h e ax ia l s t r a i n in t ru s ses

a p p r o a c h e s 1 . A t th i s s t r a i n , a l a rge stiffness is

specified t o s i m u l a t e c o n t a c t b e t w e e n t h e t w o p l a t e s .

C o n s t r a i n t s a r e a p p l i e d t o t h e u p p e r e n d of t h e

t ru s se s t o i n s u r e t h a t i ts r a d i a l c o o r d i n a t e is t h e s a m e

as t h e r a d i a l c o o r d i n a t e of its l o w e r e n d . A l s o , t h e

ax ia l c o o r d i n a t e o f u p p e r e n d is c o n s t r a i n e d t o

t r a n s l a t e w i t h t h e u p p e r p l a t e .

4. FINITE ELEMENT M O D E L DESCRIPTION AND

CALCULATIONS

T h e t w o m i n e s w e r e m o d e l e d as a x i s y m m e t r i c 2 - D

s t r u c t u r e s u s i n g t h e A D I N A finite e l e m e n t c o d e . T h e

steel c o m p o n e n t s w e r e m o d e l e d w i t h s i x - n o d e ele-

m e n t s i n c l u d i n g m i d - s i d e n o d e s o n t h e p l a t e su r face .

T h e exp los ive c o m p o n e n t s w e r e m o d e l e d w i t h four -

n o d e Q U A D e l e m e n t s e x c e p t w h e r e t hey i n t e r f aced

t h e steel j a c k e t , in w h i c h c a s e a m i d - s i d e n o d e w a s

i n c l u d e d o n t h e in t e r f ace e d g e .

T h e t i m e s t ep u s e d for t h e c a l c u l a t i o n s w a s d e t e r -

m i n e d f r o m t h e C o u r a n t s t ab i l i ty c o n d i t i o n

At = (7)

w h e r e Atcnt is t h e m i n i m u m C o u r a n t s t ab i l i t y t i m e

s t ep ; / is t h e d i s t a n c e b e t w e e n the t w o c loses t n o d e s

in t h e s y s t e m ; is t h e Y o u n g ' s m o d u l u s for t h e

stiffest m a t e r i a l ; is t h e d e n s i t y of t h e m a t e r i a l ; a n d

is t h e n u m b e r o f t i m e s t eps w h i c h we wish t o

r e p r e s e n t t h e s h o c k w a v e in p a s s i n g t h r o u g h t h e

d i s t a n c e / . T h e v a l u e o f Ar c r it w a s a p p r o x i m a t e l y 200

n a n o s e c o n d s , for b o t h t h e T M - 4 6 a n d M - 1 5 m i n e s .

A v a l u e o f o f f o u r w a s u sed , so t h a t t h e t ime s t ep

for t h e c e n t r a l d i f ference expl ic i t t ime i n t e g r a t i o n

m e t h o d w a s 50 n a n o s e c o n d s .

4.1 M - 1 5 mine calculations A s i n d i c a t e d p r e v i o u s l y in Sec t ion 2 .4 , t w o

different b o u n d a r y c o n d i t i o n s were u sed in m o d e l i n g t h e M - 1 5 m i n e . T h e p r i m a r y dif ference b e t w e e n t h e t w o c a l c u l a t i o n s w a s in t h e b a s e s u p p o r t c o n d i t i o n . O n e u sed a n o n l i n e a r s p r i n g s u p p o r t a n d t h e o t h e r used a r igid ver t ica l b a s e s u p p o r t . T h e m e s h c o n f i g u r a t i o n for b o t h M - 1 5 m o d e l s is s h o w n in F ig . 10.

E i g e n f r e q u e n c i e s w e r e g e n e r a t e d a n d t h e a s soc i -a t e d m o d e s h a p e s w e r e p l o t t e d via t h e A D I N A p o s t - p r o c e s s o r , P L O T 3 D [ 8 ] . T h e n a t u r a l f r equenc i e s

The use of A D I N A for analysis of mines with explosive fills 631

Fig. 10. Finite element mesh for the M - 1 5 mine.

a r e i m p o r t a n t for e s t i m a t i n g t h e r a t e o f r e s p o n s e o f

a s t r u c t u r e . F o r s imi l a r l o a d i n g s , t h e h i g h e r t h e

n a t u r a l f r equenc ie s o f a s t r u c t u r e , t h e fas te r t h e

s t r u c t u r e will r e s p o n d . I n a d d i t i o n , r a p i d r e s p o n s e s

c a u s e h i g h e r s t r a i n r a t e s t o b e effected. T h i s is

s ignif icant for s t r a i n r a t e sens i t ive m a t e r i a l s s u c h a s

mi ld steel w h i c h b o t h o f t h e sub jec t m i n e s e m b o d y .

H o w e v e r , s t r a i n r a t e sens i t iv i ty w a s n o t m o d e l e d in

these c a l c u l a t i o n s . T h e m o d e s h a p e s a s s o c i a t e d w i t h

t he l o w e r e i g e n f r e q u e n c i e s o f t en give a g o o d i n d i -

c a t i o n o f t h e d e f o r m e d s h a p e w h i c h will r e su l t f r o m

t h e a p p l i c a t i o n o f t yp i ca l l o a d s . T h i s w a s espec ia l ly

e v i d e n t in t h e d e f o r m a t i o n o f t h e T M - 4 6 m i n e . T h e

lower e i g e n f r e q u e n c i e s a n d p e r i o d s for t h e M - 1 5

m i n e a r e g iven in T a b l e 3 .

4.2 TM-46 mine calculations

T h e A D I N A c a l c u l a t i o n s for t h e T M - 4 6 m i n e

h a v e n o t b e e n c o m p l e t e d . H o w e v e r , s o m e o f t h e

sa l ien t f e a t u r e s o f t h e m o d e l h a v e b e e n d e v e l o p e d

f r o m p r o g r e s s m a d e in s t u d i e s o f t h e m i n e t h u s far .

A d r a w i n g o f t h e c u r r e n t m e s h c o n f i g u r a t i o n is

s h o w n in F i g . 11 .

F r o m o u r e x p e r i e n c e w i t h t h e M - 1 5 m i n e , w e

e x p e c t e d s igni f icant ly di f ferent m a t e r i a l s p r o p e r t i e s in

the o u t e r steel j a c k e t o f t h e T M - 4 6 m i n e . M e a -

s u r e m e n t s s h o w e d t h a t t h i s w a s , i n d e e d , t h e c a s e .

Severa l d i f ferent se ts o f m a t e r i a l s p r o p e r t i e s w e r e

used t o m o d e l t h e v a r i o u s steel c o m p o n e n t s o f t h e

m i n e .

It w a s e v i d e n t f r o m t h e first t h a t t w o p a r t i c u l a r

difficulties w o u l d b e e n c o u n t e r e d in m o d e l i n g t h e

TM-46 m i n e . F i r s t , t h e di f ference in stiffness b e t w e e n

t h e steel p l a t e s a n d t h e a i r filled r e g i o n l e ads t o

n u m e r i c a l p r o b l e m s . T h e c o l l a p s e o f t h e a i r filled

r eg ion l e ads t o t h e i m p a c t o f t h e u p p e r p l a t e o n t h e

m i d d l e p l a t e . T h i s p h e n o m e n o n n e e d s t o b e m o d e l e d

r a t h e r ca re fu l ly . S e c o n d , t h e t h i n s t e p p e d t o p c o v e r

s h o w n in t h e u p p e r r i gh t p a r t o f F i g . 11 l e a d s t o a

very inefficient l o a d t r a n s f e r f r o m t h e t o p c o v e r t o t h e

m a i n m i n e b o d y . O n t h e o t h e r h a n d , a n y v iab le

fa i lure m e c h a n i s m for t h e m i n e m u s t i n e v i t a b l y in-

vo lve a fa i lu re o f t h e m a i n m i n e b o d y .

S ince t h e A D I N A 1981 c o d e d o e s n o t h a v e a

c o n t a c t e l e m e n t t o sense w h e n t h e t o p c o v e r p l a t e a n d

m i d d l e p l a t e i m p a c t , w e h a v e u s e d n o n l i n e a r t r u s s

e l e m e n t s t o a p p r o x i m a t e t h e i n t e r a c t i o n o f t h e t w o

p l a t e s . T h i s a p p r o a c h w a s d e s c r i b e d in S e c t i o n 3.4.

E i g e n f r e q u e n c i e s a n d m o d e s h a p e s w e r e a l s o o b -

t a i n e d for t h e T M - 4 6 m i n e m o d e l . T h e e igen-

f r equenc ie s a n d a s s o c i a t e d p e r i o d s for t h e l o w e r

m o d e s a r e g iven in T a b l e 4 .

Al l c a l c u l a t i o n s d e s c r i b e d h e r e i n u s e d t h e t o t a l

L a g r a n g i a n f o r m u l a t i o n w i t h a l u m p e d m a s s m a t r i x

w i t h t h e e x c e p t i o n o f t h e n o d a l t ie a n d t r u s s e l e m e n t s .

T h e f o r m u l a t i o n s u s e d for t h e s e w e r e m a t e r i a l n o n -

l inea r i ty o n l y a n d u p d a t e d L a g r a n g i a n a n a l y s i s p r o -

c e d u r e , respec t ive ly .

5. DYNAMIC R E S P O N S E PREDICTIONS

Severa l m o d i f i c a t i o n s t o t h e A D I N A p r o g r a m w e r e

m a d e t o ass is t u s in i n t e r p r e t i n g t h e r e s p o n s e p r e d i c -

t i o n s . T h e s e a r e d e s c r i b e d fully in [1]. A s u m m a r y o f

t h e s e m o d i f i c a t i o n s will b e g iven h e r e . D u e t o t h e ve ry

l a r g e a m o u n t o f s t r e s s - s t r a i n d a t a a v a i l a b l e f r o m t h e

A D I N A resu l t s , s o m e m e a n s o f select ively e x t r a c t i n g

s igni f icant p a r t s o f t h e r e su l t s w a s d e s i r e d . S ince t h e

c o m p o n e n t w h i c h i n v o l v e d t h e m o s t c r e d i b l e fa i lure

m e c h a n i s m s w a s t h e steel j a c k e t , w e focused o u r a t t e n -

t i o n o n it . T h e m o d i f i c a t i o n s w e r e m a d e in t w o

dif ferent a r e a s . F i r s t , r o u t i n e s w e r e w r i t t e n t o m o n i t o r

t h e e x t r e m e ( m a x i m u m / m i n i m u m ) s t resses a n d s t r a i n s

in t h e steel c o m p o n e n t s . I n f o r m a t i o n o n t h e l o c a t i o n ,

t i m e , a n d v a l u e o f t h e s e e x t r e m e s t resses a n d s t r a i n s

w e r e s a v e d a n d p r i n t e d a t i n t e r v a l s d u r i n g t h e ca l cu -

l a t i o n . S e c o n d , r o u t i n e s w e r e w r i t t e n t o c a l c u l a t e a n d

m o n i t o r t h e s e c o n d i n v a r i a n t o f p l a s t i c s t r a i n . T h e

v a l u e o f th i s q u a n t i t y w a s c o m p a r e d t o a n i n p u t v a l u e

in o r d e r t o p r e d i c t fa i lu re o f t h e s teel j a c k e t . T a b l e s o f

t h e m a x i m u m v a l u e o f th i s q u a n t i t y w e r e s t o r e d a n d

p r i n t e d a t p r e se l ec t ed i n t e r v a l s .

In a d d i t i o n t o t h e a b o v e m o d i f i c a t i o n s t o A D I N A ,

o n e f u r t h e r m o d i f i c a t i o n w a s n e c e s s a r y t o successful ly

o b t a i n t h e s o l u t i o n t o s u c h l o n g r e s p o n s e t i m e s u s i n g

t h e expl ic i t t i m e i n t e g r a t i o n s c h e m e . I n t h e s t a n d a r d

A D I N A p r o g r a m , (1977 v e r s i o n ) w h e n e v e r p las t i c i ty

o c c u r s in t h e k i n e m a t i c h a r d e n i n g m o d e l for a so l id

e l e m e n t , a l i nea r i zed c o r r e c t i o n is a p p l i e d t o b r i n g t h e

s t ress t e n s o r b a c k t o t h e v o n M i s e s yield su r f ace .

B e c a u s e t h e l inea r i zed c o r r e c t i o n leaves t h e s t ress a t a

p o s i t i o n in s t ress s p a c e a l o n g a t a n g e n t t o t h e c o n v e x

Table 3. Eigenfrequencies and periods for the M - 1 5 mine

S p r i n g S u p p o r t e d Mine R i g i d l y S u p p o r t e d Mine

F r e q u e n c y P e r i o d F r e q u e n c y P e r i o d

( c p s ) ( s e c ) ( c p s ) ( s e c )

36* 2.744 1 0 "?

6426 1.556 1 0 "u

3636 2.750 1 0 "u

7899 1.266 1 0 "u

6710 1.490 ]0~k

9685 1.032 ^0~^

8531 1 .172 10"14

12186 8 .205 1 0 "5

* R i g i d body mode.

632 F . H . GREGORY and A . D . G U P T A

Deflection Magnification = 15X

t = 0 . 6 7 ms

Fig. 11. Finite element mesh for the T M - 4 6 mine.

i

...

Table 4. Eigenfrequencies and periods for the TM-46 mine

Frequency

( c p s )

P e r i o d

( s e c )

3041

10466

17068

31071

3.288 10"

9 .555 10"

5.859 10"

3.218 10"

yield sur face , t h e r e su l t i ng s t ress will b e s l ight ly o u t -

s ide t he yield su r f ace . A n a c c u m u l a t i o n o f e r r o r r e su l t s

f rom th is l i n e a r i z a t i o n a n d af ter m a n y t i m e s t eps

causes i m a g i n a r y r o o t s t o b e o b t a i n e d in s o l v i n g for

t he s t ress c o r r e c t i o n . It w a s n e c e s s a r y t o i n c l u d e t h e

q u a d r a t i c s t ress c o r r e c t i o n t o a v o i d th i s p r o b l e m . T h e

de ta i l s o f th i s m o d i f i c a t i o n a r e d e s c r i b e d in [1 , 9] .

M - 1 5 mine dynamic response

T h e c a l c u l a t i o n s for b o t h b a s e s u p p o r t c o n d i t i o n s

were r u n t o 2.0 m s e c o f r e s p o n s e t i m e . T h i s a m o u n t o f

r e s p o n s e t i m e c o r r e s p o n d s t o seven a n d twe lve t i m e s

t he p e r i o d of t h e f u n d a m e n t a l d i s t o r t i o n a l e i g e n m o d e

for t h e s p r i n g s u p p o r t e d a n d rigidly s u p p o r t e d m i n e ,

respect ive ly (see T a b l e 3).

T h e first p r e d i c t e d fa i lure o f t h e s p r i n g s u p p o r t e d

m i n e ( s i m u l a t i n g a m i n e b u r i e d in soi l ) o c c u r r e d a t

0.67 msec . T h e fa i lures o c c u r r e d in t h e fuze well in t h e

cen t e r of t h e m i n e as s h o w n in F i g . 12. O t h e r a r e a s o f

t h e m i n e c a s i n g h a d severe p l a s t i c flow as i n d i c a t e d in

F i g . 12; h o w e v e r , t h e s e c o n d i n v a r i a n t o f p l a s t i c s t r a i n

d id n o t r e a c h t h e fa i lure v a l u e . E x p e r i m e n t s d e s c r i b e d

in [2] s h o w e d a s imi l a r b e h a v i o r . F u z e wells w e r e t o r n

f r o m the steel j a c k e t a n d exp los ive m a t e r i a l w a s ejec-

ted f rom t h e ins ide cav i ty . T h e d e f o r m e d s h a p e o f t h e

m i n e a t t h e t i m e o f t h e first fa i lure o f t h e c a s i n g is

K

34

Predicted t i m e s of fai lure

1 @ 0 . 6 7 ms

2 @ 0 .97 ms

3 (a> 1.03 ms

4 (a> 1.88 m s

A r e a s of s ign i f icant , but lesser p last ic f low

t

Fig. 12. Predicted failures of the M-15 mine with nonlinear spring supported base.

Fig. 13. Deformed shape of the M-15 mine on nonlinear spring support at the time of the first predicted failure.

s h o w n in F i g . 13. In m a k i n g th i s p l o t , r ig id b o d y

m o t i o n o f t h e m i n e o n t h e s p r i n g s u p p o r t w a s s u b -

t r a c t e d a n d t h e r e s u l t i n g d i s p l a c e m e n t s w e r e

magn i f i ed by 15 t o m a k e t h e d e f o r m a t i o n p a t t e r n

m o r e g r a p h i c . T h e d o t t e d l ines in F i g . 13 i n d i c a t e t h e

u n d e f o r m e d s h a p e o f t h e m e t a l c a s i n g a n d t h e c ro s se s

i n d i c a t e t h e o r i g i n a l n o d a l p o s i t i o n . C o n t o u r p l o t s o f

r a d i a l a n d h o o p s t ress s h o w e d h i g h s t ress g r a d i e n t s in

t h e n e i g h b o r h o o d o f t h e p r e d i c t e d r u p t u r e p o i n t s .

E x t e n s i v e c r a c k i n g o f t h e exp los ive filler m a t e r i a l o c -

c u r r e d in t he se c a l c u l a t i o n s a c c o r d i n g t o t h e 0 . 1 %

tens i le s t r a i n fa i lure c r i t e r i o n . It w o u l d be r e a s o n a b l e

t o a s s u m e t h a t s o m e o f t h e c r u s h e d exp los ive filler

m a t e r i a l w o u l d b e expe l l ed t h r o u g h a n y r u p t u r e s

w h i c h o c c u r r e d in t h e c a s i n g .

T h e finite e l e m e n t c a l c u l a t i o n s o f t h e M - 1 5 m i n e o n

r igid ro l l e r s u p p o r t w e r e a l so c a r r i e d o u t t o 2.0 m s e c .

T h i s c o n f i g u r a t i o n is a m u c h m o r e h igh ly c o n s t r a i n e d

s t r u c t u r e t h a n w a s t h e p r e v i o u s case . T h i s fact is

ref lected in t h e h i g h e r n a t u r a l f requenc ies ( T a b l e s 3

a n d 4 ) . T h e fa i lures p r e d i c t e d for th is c o n f i g u r a t i o n

o c c u r r e d in t h e s a m e g e n e r a l a r e a , t h e c e n t r a l fuze

well . H o w e v e r , t h e t i m e s r e q u i r e d for fa i lure t o o c c u r

w e r e m u c h s h o r t e r t h a n t h o s e for t h e s p r i n g s u p p o r t e d

m i n e as o n e m i g h t excep t . T h e fa i lure o f th i s m i n e w a s

p r e d i c t e d a t t w o l o c a t i o n s a t t i m e s of 0 .255 a n d

0 .609 m s e c . T h e first fa i lure o f t h e fuze well o c c u r r e d

in t h e c e n t e r o n t h e in i t ia l d o w n w a r d c o m p r e s s i o n

p h a s e . In t h e s p r i n g s u p p o r t e d r e s p o n s e , t h e first fail-

u r e o c c u r r e d in a r e b o u n d m o t i o n o f t h e fuze well .

T h e s e r e s p o n s e s a r e d e s c r i b e d m o r e fully in [1].

6. C O N C L U S I O N S

T h e expl ic i t t i m e i n t e g r a t i o n m e t h o d g a v e t h e m o s t a c c u r a t e r e su l t s for t h e s h o c k l o a d e d m i n e s . T h i s s t a t e -m e n t is b a s e d o n t h e s m o o t h n e s s o f t h e s t resses a n d s t r a i n s a s a f u n c t i o n o f t i m e . W e f o u n d t h a t s e c o n d o r d e r c o r r e c t i o n s t o a s s u r e t h a t t h e s t ress s t a t e is o n t h e yield su r f ace d u r i n g p l a s t i c flow a r e r e q u i r e d t o k e e p t h e c a l c u l a t i o n a l p r o c e d u r e f rom fai l ing.

T h e p a r t s o f t h e o u t e r steel j a c k e t o f t h e M - 1 5 a n d m i n e s w h i c h a r e w o r k h a r d e n e d in t h e d e e p

d r a w i n g m e t a l f o r m i n g o p e r a t i o n h a v e s igni f icant ly v a r y i n g m a t e r i a l s p r o p e r t i e s . T h e s e v a r i a t i o n s in s t r e s s - s t r a i n r e l a t i o n s m u s t be m e a s u r e d a n d m o d e l e d carefu l ly s ince t h e y h a v e a d i r ec t in f luence o n m i n e fa i lure u n d e r b l a s t l o a d s .

T h e soil m e d i u m s u p p o r t i n g t h e m i n e a n d t h e n a -t u r e o f t h e l o a d i n g o f t h e s idewal l h a v e a s igni f icant in f luence o n t h e r e s u l t i n g r e s p o n s e . I t is r e c o m m e n d e d t h a t t h e soil m e d i u m b e i n c l u d e d expl ic i t ly in a n y

The use of A D I N A for analysis of mines with explosive fills 633

fu tu re s t ud i e s . A t t e n u a t i o n of t h e s h o c k in t h e soil in

t h e n e i g h b o r h o o d o f t h e m i n e s idewal l s h o u l d b e in-

ves t iga t ed .

F a i l u r e o f t h e M - 1 5 m i n e o c c u r r e d in t h e a r e a o f t h e

c e n t r a l fuze cav i ty w h e n s u b j e c t e d t o a 13.8 M P a p e a k

p r e s s u r e , 6.5 k P a - s e c i m p u l s e level b l a s t l o a d in b o t h

t he r igid s u p p o r t a n d soil s u p p o r t s i m u l a t i o n s . T h i s

ag ree s wi th e x p e r i m e n t a l t es t s in w h i c h c a t a s t r o p h i c

fa i lure o f t h e m e t a l c a s i n g o c c u r r e d , a s well a s e jec t ion

of s e c o n d a r y fuze wel ls .

W e h o p e t o be a b l e t o r e p o r t o n t h e a n a l y s i s o f t h e

T M - 4 6 m i n e in a f u tu r e c o m m u n i c a t i o n .

REFERENCES

1. F. H. Gregory, Failure of the M-15 anti tank mine due to blast loads. BRLTR-02420, Oct. 1982.

2. A. J. Tulis, R. Remaly, M. Nusbaum (HT Research Institute), D. C. Heberlein, I. A. Berg and D. Stefanye ( M E R A D C O M ) , Improved fuel air explosives. U.S. Army Mobility and Equipment R & D C o m m a n d Rep. 2222, Sept. 1977.

3. F. H. Gregory, Finite element modeling of the vulner-ability of an M - 1 5 land mine using an explicit integration scheme. Proc. 1981 Army Num. Anal. Comput. Conf. A R O Rep. 81-3 , Aug. 1981.

4. A D I N A , A finite element program for automatic dy-namic incremental nonlinear analysis. A D I N A En-gineering, Inc., Water town, MA, Rep. AE81-1, Sept. 1981.

5. K. J. Bathe, Static and dynamic geometric and material nonlinear analysis using A D I N A , MIT-82448-2, May 1977.

6. M. S. Chawla and R. B. Frey, A numerical study of projectile impact on explosives. BRLMR-2741 , Apr. 1977.

7. C. A. Hogentogler, Engineering Properties of Soil. 1st Edn., p . 223. McGraw Hill, New York (1937).

8. J. E. Crawford, Private communicat ion. Aerospace Cor-porat ion, El Segundo, CA, March 1982.

9. A. D. Gupta , J. M. Santiago and H. L. Wisniewski, An improved strain hardening characterization in the A D -INA code using the mechanical sublayer concept. Proc. \st Chautauqua on Finite Element Modeling, Har-wichport, MA, 15-17 Sept. 1980.

Computers & Structures Vol. 17, No. 5 6, pp. 635-642, 1983 Printed in Great Britain.

0045-7949/83 $3.00 + .00 Pergamon Press Ltd.

USE OF ADINA IN SOIL MECHANICS WITH CASE STUDIES FOR EXCAVATIONS

M I C H E L D Y S L I

Soil Mechanics Laboratory of the Swiss Federal Institute of Technology EPFL-Ecublens , CH-1015 Lausanne, Switzerland

AbstractThe reliability and the variety of mathematical procedures related to incremental solution strategies for the nonlinear analysis as well as the general availability of the A D I N A code make this latter particularly well adapted for soil and rock mechanics. These materials, especially the soils, have indeed a marked nonlinear strain-stress behaviour. These qualities are somewhat counterbalanced by the relative lack of constitutive laws applicable to soil however.

The first part of this paper deals with the general and practical use of A D I N A in soil and rock mechanics.

The second presents, as case study, an A D I N A use for the evaluation of the deformations around excavations in clayey soils. With the classical methods of geomechanics, it is not possible to determine these deformations correctly. Only the use of the F E M with nonlinear constitutive laws for ground performance and possibly considering also the pore pressure allows for the good evaluation of these deformations. Two very large excavations have been analysed in this way and it was possible to compare the results of the computat ion with many in situ observations.

1. INTRODUCTION

T h i s p a p e r i n t e n d s t o s h o w t h e p r a c t i c a l poss ib i l i t i e s

of u s i n g t h e A D I N A c o d e in soil a n d r o c k m e c h a n i c s .

It is t h e resu l t o f severa l y e a r s o f c lose c o l l a b o r a t i o n

b e t w e e n e n g i n e e r i n g offices a n d a un ive r s i t y .

T h e F E M a p p l i e d t o t h e e v a l u a t i o n o f s t resses a n d

s t r a in s is b a s e d o n c o n t i n u u m m e c h a n i c s . T h e soi l ,

h o w e v e r , is . a typ ica l d i s c o n t i n u u m w i t h severa l

p h a s e s : a sol id p h a s e , a p o r e fluid p h a s e , a n d a p o r e

gaz p h a s e w h e n t h e soil is u n s a t u r a t e d . T h e in t e r -

a c t i o n s b e t w e e n t h e p o r e p h a s e a n d t h e so l id p h a s e

i n d u c e i m p o r t a n t s t resses a n d s t r a i n s w h i c h a r e q u i t e

of ten difficult t o s i m u l a t e w i t h a o n e p h a s e m o d e l

ba sed u p o n c o n t i n u u m m e c h a n i c s .

In soil m e c h a n i c s , d i s t i n c t i o n is m a d e b e t w e e n t h e

effective s t ress (g r a in t o g r a i n ) a n d t h e t o t a l s t ress

a c t i n g o n a n i so l a t ed p o r t i o n o f t h e s o l i d - f l u i d - g a z :

wi th : u = p o r e p r e s s u r e = / ( d e r / d t ) ; a n d

= K r o n e c k e r d e l t a .

In t h e a b o v e e q u a t i o n , t h e soil m e c h a n i c s c o n -

v e n t i o n of n e g a t i v e tens i le s t ress is u sed , a l t h o u g h in

A D I N A , t h e r eve r se is t h e ca se .

W h e n us ing a c o d e s u c h as A D I N A , t h e q u e s t i o n

is w h e t h e r o n e w o r k s in effective o r in t o t a l s t ress a n d ,

in t he first c a se , h o w t o i n t r o d u c e t h e p o r e p r e s s u r e .

A fu r the r ' i m p o r t a n t p r o b l e m ar i ses w h e n t h e d i -

l a t a n c y h a s t o be s i m u l a t e d . In c o n t i n u u m m e c h a n i c s ,

th is p h e n o m e n o n c o r r e s p o n d s t o a n i n c r e a s e o f t h e

s t r a in ene rgy w h i c h is phys i ca l l y a b e r r a n t a n d c a n

lead t o e l e m e n t s w i t h n e g a t i v e stiffness t e r m s o n t h e

d i a g o n a l a n d t h e n t o a n ove ra l l stiffness m a t r i x n o t

pos i t ive def ini te .

T h e use o f c o n t i n u u m m e c h a n i c s in soil m e c h a n i c s

d o e s , h o w e v e r , n o t d a t e f r o m t h e i n t r o d u c t i o n o f t h e

F E M . I n d e e d , t h e t w o h i s to r i ca l m o d e l s o f m e c h a n i c s

a r e t h e e las t ic l inea r a n d t h e r ig id -pe r fec t ly p l a s t i c

m o d e l s w i th t h e V o n M i s e s a n d t h e M o h r - C o u l o m b

yield c r i t e r i a .

2. S O M E C O M M E N T S ON THE USE OF ADINA IN SOIL MECHANICS

2.1 Constitutive laws

2.1.1 Linear elastic A l t h o u g h t h e soil is t o a h igh

d e g r e e a p l a s t i c m a t e r i a l , t h e l inea r e las t ic m o d e l s

i s o t r o p i c a n d o r t h o t r o p i c m a y , in p r a c t i c e , a d v a n -

t a g e o u s l y be u sed for t h e s t a t i c c o m p u t a t i o n of

s e t t l e m e n t s a n d for s o i l - s t r u c t u r e i n t e r a c t i o n ana ly s i s ,

p r o v i d i n g t h a t t h e l o a d s a r e n o t t o o i m p o r t a n t a n d

t h a t t hey a l m o s t a l w a y s a r e l o a d i n g . T h e o r t h o t r o p i c

l inea r e las t ic l aw is espec ia l ly useful b e c a u s e t h e soil

e x h i b i t s n e a r l y a l w a y s a n i s o t r o p y .

2 .1.2 Curve description. F o r t h e p r a c t i t i o n e r o n l y

l i t t le a c q u a i n t e d w i t h t h e t h e o r y o f p las t i c i ty , th i s

c o n s t i t u t i v e l aw s e e m s a t first s igh t q u i t e i n t e r e s t i n g .

T h e d i r ec t i n t r o d u c t i o n o f t r i ax ia l test r e su l t s a s

p a r a m e t e r s o f a c o n s t i t u t i v e l aw is i n d e e d t e m p t i n g .

Still , th i s a p p e a r a n c e is m i s l e a d i n g a n d th i s m o d e l

is f r equen t ly n o t ve ry useful in soil a n d r o c k m e c h a n -

ics. a n d G = F(ev) a r e n o t i n t r i n s i c r e l a t i o n s of t h e

soil (K = b u l k m o d u l u s , G = s h e a r m o d u l u s a n d

ev = v o l u m e s t r a i n ) ; t h e y s t r o n g l y d e p e n d o n the

b o u n d a r y c o n d i t i o n s a n d s t ress p a t h s , a n d such re-

l a t i o n s b a s e d u p o n a n a x i s y m m e t r i c t r i ax ia l test a r e

n o t a p p l i c a b l e t o a n y o t h e r b o u n d a r y c o n d i t i o n s

( p l a n e s t r a i n , p l a n e s t ress , 3 D ) . F u r t h e r m o r e , th is

m o d e l is very sens i t ive t o t h e in i t ia l m o d u l i a n d these

m o d u l i a r e r a t h e r difficult t o d e t e r m i n e in p r a c t i c e .

O n t h e o t h e r h a n d , t h e poss ib i l i t y of m a t e r i a l

w e a k e n i n g u n d e r tens i le s t ress is m o s t i n t e r e s t i ng : e.g.

in r o c k m e c h a n i c s for t h e a n a l y s i s o f c r a c k i n i t i a t i on

a n d g r o w t h , a n d in soil m e c h a n i c s for t h e in-

v e s t i g a t i o n o f t h e c o n t a c t s b e t w e e n t h e soil a n d a

f o u n d a t i o n e l e m e n t ( l a t e ra l ly l o a d e d pi le , for e x a m -

ple) .

C A S Vol . 17, No . 5/6 635

636 MICHEL DYSLI

2.1.3 Drucker-Prager. T h u s , t h e c o n s t i t u t i v e l aws

of V o n M i s e s a n d M o h r - C o u l o m b a r e t h e t w o

h i s to r ica l l aws o f soil m e c h a n i c s . B o t h of t h e m

invo lve t h e cr i t ica l s t a t e c o n c e p t w h i c h is t h e bes t

a d a p t e d for t h e ana lys i s o f soil b e h a v i o u r .

T h e D r u c k e r - P r a g e r m o d e l , in s o m e w a y a t r i -

d i m e n s i o n a l g e n e r a l i z a t i o n o f t h e M o h r - C o u l o m b

m o d e l , is t h e r e f o r e m o s t useful for t h e soil e n g i n e e r .

M o r e o v e r , t h e i n t r o d u c t i o n o f a c a p h a r d e n i n g in t h e

A D I N A c o d e in 1978 h a s b r o u g h t f u r t he r i m -

p r o v e m e n t . H o w e v e r , th i s h a r d e n i n g c a p , i.e. a p l a n e

p e r p e n d i c u l a r t o t h e s t ress s p a c e d i a g o n a l , is q u i t e

a p p r o x i m a t e a n d it d o e s n o t a g r e e w i t h fine soi ls .

H o w e v e r , th is law c a n easi ly be a l t e r e d in t h e A D I N A

c o d e .

2.1.4 Von Mises. T h i s c o n s t i t u t i v e l aw w i t h i so -

t r o p i c h a r d e n i n g is very useful for soil m e c h a n i c s

ana lys i s . It is efficient for t he ana ly s i s in t o t a l s t ress

as well as in effective s t ress . In t h e l a t t e r ca se , t h e

inc rease of t h e soil s t r e n g t h , in f u n c t i o n o f its c o n s o l i -

d a t i o n r a t i o , m u s t be s i m u l a t e d by t h e i n t r o d u c t i o n

of several soil l ayers h a v i n g e a c h a di f ferent yield

s t ress o r w h i c h h a v e been p r e v i o u s l y c o n s o l i d a t e d b y

a p p r o p r i a t e l o a d i n g s . It h a s t o b e p o i n t e d o u t t h a t t h e

d i s p l a c e m e n t s o b t a i n e d by t h e c o n s o l i d a t i o n s t ep s

m u s t be d e d u c e d f r o m t h e s u b s e q u e n t d i s p l a c e m e n t s .

As this c a n n o t b e d o n e by t h e A D I N A c o d e , w e c a r r y

it o u t by a g r a p h i c a l p o s t - p r o c e s s o r d e v e l o p e d in o u r

ins t i tu te .

In s o m e p a r t i c u l a r cases , t h e k i n e m a t i c h a r d e n i n g

m a y be useful for t he s i m u l a t i o n o f t h e hys te res i s o f

a soil subjec t t o cyclic l o a d s .

2.1.5 Concrete material model. W e h a v e n e v e r a p -

pl ied this m o d e l t o t h e A D I N A c o d e . H o w e v e r , tes ts

wi th t he o r ig ina l m o d e l of H u s s a i n K h a n a n d

S a u g y [ l ] w i th a different c o d e r evea led i n t e r e s t i n g

poss ibi l i t ies for r o c k m e c h a n i c s .

2.1.6 Creep laws. A s t h e soil is a t w o o r t h r e e p h a s e

m a t e r i a l , a n y m o d i f i c a t i o n o f t h e s t ress t e n s o r i n d u c e s

a flow in t h e p o r e p h a s e . T h u s , m o s t soi ls s h o w s o m e

viscos i ty , a n d m a n y of t h e m s h o w a g r e a t d e a l . S u c h

c r e e p b e h a v i o u r c a n be s i m u l a t e d w i t h a c o u p l e d

m o d e l o r w i th a c r e e p c o n s t i t u t i v e l aw.

If a cr i t ica l s t a t e c o n c e p t is a s s u m e d , th i s v i scos i ty

o c c u r s w h e n t h e s t ress t e n s o r r e a c h e s t h e yield su r -

face. It is t h u s a v i s cop la s t i c b e h a v i o u r .

T h e t h e r m o - e l a s t i c - p l a s t i c a n d c r e e p m o d e l is h o w -

ever o n l y useful in soil m e c h a n i c s for c e r t a i n t i m e -

s e t t l e m e n t p r e d i c t i o n s .

2.2 Dynamic analysis

In soil a n d r o c k m e c h a n i c s , t h e d y n a m i c ana lys i s is

a b o v e all nece s sa ry for t h e des ign o f m a c h i n e f o u n d a -

t i o n s a n d for t h e se i smic des ign o f s t r u c t u r e s e m -

b e d d e d in soi l .

T h e soil is a n e m i n e n t l y n o n l i n e a r m a t e r i a l , in

s t e a d y s t a t e o r d y n a m i c c o n d i t i o n s . T h e d a m p i n g a n d

t h e d y n a m i c s h e a r m o d u l i t h e r e f o r e s t r o n g l y d e p e n d

u p o n t h e s h e a r s t r a i n . F i g u r e 1 s h o w s t h e r e l a t i o n s o f

t h e s h e a r m o d u l u s a n d d a m p i n g vs s h e a r s t r a i n for a

m a r l y r o c k a n d t w o soi ls ; F i g . 2 s h o w s t h e effect o f

t hese n o n l i n e a r r e l a t i o n s d u r i n g a m e d i u m e a r t h -

q u a k e .

T h e n o n l i n e a r r e l a t i o n s h i p b e t w e e n t h e s h e a r

m o d u l u s a n d s t r a i n c a n b e i n t r o d u c e d in t h e A D I N A

c o d e , t h e d a m p i n g u n f o r t u n a t e l y n o t , un less it is

m o d e l e d by t h e n o n l i n e a r m a t e r i a l b e h a v i o r . A D I N A

is t h e r e f o r e h a r d l y eve r u s e d b y us for d y n a m i c

ana lys i s o f e a r t h o r e m b e d d e d s t r u c t u r e s .

H o w e v e r , A D I N A c a n b e q u i t e a d v a n t a g e o u s l y

a p p l i e d in c o n j u n c t i o n w i t h a p r o g r a m m e specia l ly

d e v e l o p e d for t h e ana ly s i s o f t h e s o i l - s t r u c t u r e , l ike

t h e p r o g r a m m e F L U S H for e x a m p l e [ 2 ] . In th is ca se ,

t h e s o i l - s t r u c t u r e i n t e r a c t i o n is a c h i e v e d by a r o u g h

g e o m e t r i c a l s i m u l a t i o n o f t h e s t r u c t u r e , e.g. o f c o n -

c re te . N e x t , t h e r e s p o n s e o f th i s s t r u c t u r e is i n t r o -

d u c e d i n t o a m u c h m o r e ref ined g e o m e t r i c a l m o d e l

w h i c h c a n t h e n be t r e a t e d by A D I N A .

2.3 Geometric nonlinearity

T h e soil b e i n g a m u c h m o r e d e f o r m a b l e m a t e r i a l

t h a n c o n c r e t e o r i r o n , for e x a m p l e , it is q u i t e t e m p t -

ing t o use a f o r m u l a t i o n w h i c h t a k e s g e o m e t r i c

n o n l i n e a r i t i e s s u c h as offered by t h e A D I N A c o d e

i n t o c o n s i d e r a t i o n ; t h e t o t a l L a g r a n g i a n a n d u p d a t e d

L a g r a n g i a n f o r m u l a t i o n . N u m e r o u s c o m p a r i s o n s

h a v e p r o v e d , h o w e v e r , t h a t t hese f o r m u l a t i o n s a r e

u n n e c e s s a r y in m o s t p r a c t i c a l ca ses .

Y e t , t he se f o r m u l a t i o n s a r e r a t h e r useful t o a n a l y s e

p rec i se c o l l a p s e o r w h e n s i m u l a t i n g j o i n t s o r even for

t h e a n a l y s i s o f t h e pi le b u c k l i n g p r o b l e m , for e x a m -

ple .

Fig. 1. Example of damping and shear modulus vs shear strain relations.

Use of A D I N A in soil mechanics with case studies for excavations 637

F R E E F I E L D

S T R A I N COMPATIBLE CONTROL MOTION 10-10* 20-10* 30-10* MOLASSE

D A M P I N G [V . ] S H E A R MODULUS [ k N m - ]

Fig. 2. Effect of the damping and shear modulus non-linearity during an ear thquake.

3. CASE STUDIES: EXCAVATIONS IN CLAYEY SOILS

3.1 Generality

T h e h igh cos t o f l a n d in u r b a n c e n t e r s a n d t h e

t echn ica l m e a n s a v a i l a b l e t o d a y c a u s e u r b a n b u i l d i n g

be f o u n d e d a t g r e a t e r d e p t h . In c o m p r e s s i b l e soi l , t h e

p r o b l e m of e v a l u a t i n g d e f o r m a t i o n s a r o u n d l a rge

e x c a v a t i o n s , u sua l l y s u p p o r t e d b y s lu r ry wa l l s , b e -

c o m e s t h e r e f o r e i nc r ea s ing ly c r i t i ca l . F i g u r e 3 s h o w s

typ ica l m e a s u r e d s e t t l e m e n t s a r o u n d s u c h a n exca -

v a t i o n . T h e d e t e r m i n a t i o n o f t he se d e f o r m a t i o n s by

m e a n s o f t h e c lass ical o e d o m e t r i c s e t t l e m e n t a n a l y s i s

a n d t h e c o m p u t a t i o n o f s t resses b y m e a n s o f a l i nea r

e las t ic l aw l eads , in t h e c a s e o f c o m p r e s s i b l e soi ls , t o

a to t a l ly e r r o n e o u s s e t t l e m e n t p r e d i c t i o n .

T h e F E M e m p l o y i n g n o n l i n e a r c o n s t i t u t i v e l aws is

a m u c h m o r e s a t i s f ac to ry t o o l for t h e e v a l u a t i o n o f

these d e f o r m a t i o n s . H o w e v e r , i ts p r a c t i c a l u se is

de l i ca te d u e t o , in p a r t i c u l a r , t h e p r o b l e m s o f c o n -

ve rgence c o n n e c t e d w i t h t h e s o l u t i o n o f n o n l i n e a r

e q u a t i o n s y s t e m s a n d d u e t o t h e c h o i c e o f t h e c o n s t i -

t u t i ve l aw a n d its p a r a m e t e r s .

T w o l a rge e x c a v a t i o n s a r e e x a m i n e d in these ca se

s tud i e s w h o s e m a i n ob jec t ive is t h e c o m p a r i s o n o f t h e

resu l t s o b t a i n e d by a n o n l i n e a r F E M ana ly s i s , w i th

t h e a c t u a l b e h a v i o u r in situ. T h i s c o u l d b e a c c o m -

p l i shed by u s i n g t h e m a n y o b s e r v a t i o n s m a d e d u r i n g

a n d af te r t h e e x e c u t i o n o f t h e p r o j e c t s . In e a c h o f t h e

t w o e x c a v a t i o n s , t h e s e t t l e m e n t s a r o u n d a n d in t h e

e x c a v a t i o n w e r e m e a s u r e d r e g u l a r l y a t m o r e t h a n 100

p o i n t s ; i n c l i n o m e t e r s p l a c e d in t h e wal l s m a d e it

pos s ib l e t o fo l low prec i se ly t h e h o r i z o n t a l de fo r -

m a t i o n s ; m a n y p o r e p r e s s u r e cells m a d e it p o s s i b l e t o

c o n t r o l t h e d e v e l o p m e n t o f t h e p o r e p r e s s u r e s b e h i n d

t h e wal l s a n d severa l s t ress g a u g e s c o n t i n u o u s l y

m e a s u r e d t h e forces in t h e b r a c i n g e l e m e n t s .

3.2 Presentation of the two cases

T h e t w o l a rge e x c a v a t i o n s h a v e e a c h a h o r i z o n t a l

su r face of o v e r 5000 m2 a n d t h e m a x i m u m d e p t h of

o n e exceeds 20 m . T h e y w e r e b o t h l a te ra l ly s u p p o r t e d

by s lu r ry wal l s .

T h e first s t u d y w a s u n d e r t a k e n a posteriori a n d t h e

s e c o n d a priori o n a n e x c a v a t i o n in p r o g r e s s a t t h e

p r e s e n t t i m e . T h e first s t u d y h a s a l r e a d y b e e n d i s -

cus sed in a p a p e r in F r e n c h by Dys l i et al.[3].

T h e first p ro j ec t , t h e G r a n d C a s i n o , w a s bu i l t f rom

1975 t o 1980; t h e s e c o n d , a t p r e s e n t t i m e u n d e r

c o n s t r u c t i o n , is ca l led C o n f e d e r a t i o n C e n t r e . T h e s e

t w o b u i l d i n g s a r e p l a c e d in a n iden t i ca l geo log ica l

e n v i r o n m e n t , c h a r a c t e r i z e d by fine s a t u r a t e d soi ls o f

h igh c o m p r e s s i b i l i t y a n d g r e a t d e p t h , w h i c h t h e

w u r m i a n g lac ie rs , a t t h e t i m e of the i r r e t r e a t , d e p o s -

i ted o n a m o l a s s i c b e d r o c k .

T h e Grand Casino is bu i l t o n a si te r e c l a i m e d f rom

t h e L a k e of G e n e v a . T h e a v e r a g e d e p t h o f e x c a v a t i o n

r e a c h e s 13.5 m . F i g u r e 4 gives a n o u t l i n e o f th is

e x c a v a t i o n a n d t h e b r a c i n g p l a c e d ; F i g . 5 r e c a p i t u -

la tes , a m o n g o t h e r t h i n g s , t h e g e o t e c h n i c a l c o n d i t i o n s

a n d d e s c r i b e s t h e m a i n e x c a v a t i o n s t ages .

F i g u r e 6 is a s u m m a r y o f t h e m e a s u r e m e n t s c a r r i e d

o u t d u r i n g t h e e x c a v a t i o n ; it c o m p r i s e s t h e wal l

d i s p l a c e m e n t s m e a s u r e d a t t h e c e n t e r of t h e l o n g s ide

af ter t h e t h r e e e x c a v a t i o n s t ages , t h e v a r i a t i o n o f t h e

p o r e p r e s s u r e b e h i n d t h e s lu r ry wal l a n d t h e d e -

v e l o p m e n t o f t h e s e t t l e m e n t s a t a p o i n t s i t u a t e d a t

14 m b e h i n d t h e wal l .

T h e Confederation Centre c o m p l e x s t a n d s a t t h e

foo t o f t h e hill o f t h e O l d C i t y o f G e n e v a . T h e

e x c a v a t e d d e p t h s r a n g e f r o m 23 .5 m ( u p p e r s ide) t o

1 8 . 0 m ( lower s ide) .

Fig. 3. Settlements a round an excavation (Grand Casino).

6 3 8 MICHEL DYSLI

S I T U A T I O N

A POSTERIORI ANALYSIS AREA

" 1

12

GLOTZL C E L L S

* er

- A I

I N C L I N O M E T E R S

C R O S S S E C T I O N A - A

= 3 .

20 AO 60 ,

Fig. 4. Grand Casinoplan and cross section of the excavation.

PORE PRESSURE

CELLS 0.00 = 374.7 m asl E X C A V A T I O N S T A G E S

Fig. 6. Grand Casinowall displacements, pore pressure behind the wall and settlements.

F i g u r e 7 s u m m a r i z e s t h e g e o t e c h n i c a l c o n d i t i o n s

a n d t h e e x c a v a t i o n p r o c e d u r e e m p l o y e d a t t h e C o n -

f ede ra t i on C e n t r e . A dif ference in e x c a v a t e d d e p t h

exists b e t w e e n t h e u p p e r a n d l o w e r p a r t s o f t h e

p ro jec t ; th i s resu l t s in a n a s y m m e t r y o f e a r t h p r e s -

su res w h i c h is o n e o f t h e c h a r a c t e r i s t i c s o f th i s

p ro jec t .

T h e e x c a v a t i o n w a s s u b d i v i d e d i n t o n i n e p i t s d u g

i n d e p e n d e n t l y o f o n e a n o t h e r a n d b u t t r e s s e d by

s t r u t s in t h e t r a d i t i o n a l w a y .

3.3 2D analysis

3.3.1 Mesh and incremental solution strategy. Al l

t h e ana ly s i s c a r r i e d o u t w e r e t i m e i n d e p e n d e n t s t a t i c

ana lys i s .

T h e finite e l e m e n t d i s c r e t i z a t i o n w a s r a t h e r e l a b o -

Fig. 5. Grand Casinogeometric model of the excavation and geotechnical characteristics.

Use of A D I N A in soil mechanics with case studies for excavations 639

a Excavation chronology

Fig. 7. Confederation centreplan, cross section I-I and geotechnical characteristics.

r a t e ; in p a r t i c u l a r e a c h e x c a v a t i o n c o n c r e t i n g a n d

b r a c i n g s t age w a s exac t ly s i m u l a t e d b y t h e b i r t h a n d

d e a t h o f e l e m e n t s a t t h e de s i r ed m o m e n t .

F i g u r e 5 gives s o m e i n f o r m a t i o n a b o u t t h e finite

e l e m e n t m e s h for t h e G r a n d C a s i n o a n d F i g . 10 for

t h e C o n f e d e r a t i o n C e n t r e . T h e in i t ia l s t ress s t a t e w a s

c r e a t e d by t h e p r o g r e s s i v e a p p l i c a t i o n o f g r a v i t y in

a b o u t t en s o l u t i o n s t eps . I n t h e c a s e o f t h e G r a n d

C a s i n o , t h e s i m u l a t i o n as a w h o l e r e q u i r e d 32 s o l u -

t i on s t eps , w h e r e a s for t h e C o n f e d e r a t i o n C e n t r e , 48

s teps w e r e n e c e s s a r y . Af t e r e a c h s o l u t i o n s t e p , a n e w

stiffness m a t r i x w a s f o r m e d , b u t n o e q u i l i b r i u m

i t e r a t i o n w a s p e r f o r m e d in e a c h s t e p ; th i s p r o c e d u r e

w a s d i c t a t e d b y e x p e r i e n c e r a t h e r t h a n b y t h e o r e t i c a l

c o n s i d e r a t i o n s . T h e n u m b e r o f s t eps w a s i n d e e d

sufficiently g r e a t t o a v o i d e q u i l i b r i u m i t e r a t i o n s . Al l

ana lys i s w e r e c o n d u c t e d in p l a n e s t r a i n .

3.3.2 Constitutive laws. I n t h e a posteriori s t u d y

( G r a n d C a s i n o ) , t h r e e di f ferent c o n s t i t u t i v e l a w s w e r e

u s e d w i t h t h e c h a r a c t e r i s t i c s o f t h e soil d r a w n d i r ec t ly

f r o m t h e g e o t e c h n i c a l t es t s . N o n e o f t he se c h a r a c t e r -

istics w a s a d j u s t e d t o fit t h e r e su l t s o f t h e F E M

c a l c u l a t i o n .

F i r s t o f al l , a linear elastic law w a s u s e d t o d e m o n -

s t r a t e t h e a b s u r d i t y o f t h e r e s u l t i n g s e t t l e m e n t s

a r o u n d t h e e x c a v a t i o n (see F i g . 8) . F o r th i s a n a l y s i s ,

Y o u n g ' s m o d u l u s o f e a c h l aye r soil w e r e t a k e n f r o m

t h e swel l ing i n d e x C s a n d P o i s s o n ' s r a t i o w a s fixed a t

0 .35 ( d r a i n e d c o n d i t i o n s ) .

Severa l a n a l y s e s w i t h a n elasto-perfectly plastic

law a n d t h e Drucker-Prager yield criterion m a d e it

pos s ib l e t o c h e c k w i t h q u i t e a h i g h level o f a c c u r a c y

t h e s t ab i l i ty ana ly s i s o f t h e e x c a v a t i o n a c c o r d i n g t o

t h e m e t h o d o f slices. T h e y ie lded a r e a o f t h e finite

e l e m e n t m o d e l c o r r e s p o n d s in fact q u i t e well t o t h e

su r f ace o f t h e cr i t ica l c i rc le (see Dys l i et al. [3] for

m o r e de ta i l s ) . H o w e v e r , o w i n g t o loca l ins tab i l i t i e s , it

w a s n o t p o s s i b l e t o o b t a i n a c o r r e c t d e t e r m i n a t i o n of

final d e f o r m a t i o n s .

T h e A D I N A h a s n o w o n e l aw w i t h a s t r a i n

h a r d e n i n g c a p w h i c h c o u l d b e m o r e s u i t a b l e .

T h e l aw w h i c h y ie lded t h e bes t e v a l u a t i o n o f t h e

d e f o r m a t i o n s w a s t h e s i m p l e s t e l a s t o - p l a s t i c l aw . I t is

Von Mises' law with an isotropic strain hardening; it

w a s a p p l i e d in effective s t ress w i t h t h e s h e a r s t r e n g t h s

r e p r e s e n t e d o n F i g . 5 for t h e G r a n d C a s i n o a n d o n

F ig . 7 for t h e C o n f e d e r a t i o n C e n t r e . T h e s e s h e a r

s t r e n g t h s c o r r e s p o n d t o d r a i n e d c o n d i t i o n s , o n t h e

w h o l e . H o w e v e r , t h e y r e m a i n e d c o n s t a n t t h r o u g h o u t

t h e s o l u t i o n s t eps , t h u s n o t t a k i n g t h e v a r i a t i o n s o f

t h e p o r e p r e s s u r e i n t o a c c o u n t . Y o u n g ' s m o d u l i

c o r r e s p o n d i n g t o t h e e las t ic s t a t e w e r e d e t e r m i n e d by

m e a n s o f t h e swel l ing i n d e x C s a n d t h e s t r a i n

h a r d e n i n g m o d u l i b y m e a n s o f t h e c o m p r e s s i o n i n d e x

C c , w h i c h in t u r n w e r e o b t a i n e d f r o m s t a n d a r d

o e d o m e t r i c tes t s .

T h e a priori a n a l y s i s o f t h e e x c a v a t i o n o f t h e

C o n f e d e r a t i o n C e n t r e w a s p e r f o r m e d o n l y w i t h th i s

last c o n s t i t u t i v e l aw.

3.3.3 Effect of the pore pressure variation. I n t h e

ca se o f t h e a posteriori a n a l y s i s ( G r a n d C a s i n o ) , t h e

p o r e p r e s s u r e w a s first o f all c o n s i d e r e d t o b e c o n -

s t a n t ( t r i a n g u l a r d i a g r a m s ) ; t h e d e f o r m a t i o n s t h u s

o b t a i n e d w e r e h a r d l y s a t i s f a c t o r y ( F i g . 8, va r . 36) .

Fig. 8. Grand Casinocomputed and measured deformations after the stage c of the excavation. Var. 11: Linear analysis. Var. 36: Uncoupled nonlinear analysis. Var. 53: Pore pressure coupled nonlinear

analysis.

640 MICHEL DYSLI

T h e prec i se m e a s u r e m e n t o f t h e p o r e p r e s s u r e b e h i n d

the wall m a d e it pos s ib l e t h e n t o i n t r o d u c e its v a r i a -

t ions in t h e s t e p - b y - s t e p p r o c e d u r e ; t h e d e f o r m a t i o n s

o b t a i n e d w e r e t h e n m u c h m o r e s a t i s f ac to ry ( F i g . 8,

va r 53). T h e m o v e m e n t o f t h e wal l p r o d u c e s a

m o d i f i c a t i o n of t h e t en so r i a l s t resses o r o f t h e t en -

sor ia l s t r a i n s w h i c h is t h e c a u s e o f a n i m p o r t a n t fall

in p o r e p r e s s u r e b e h i n d t h e wal l . T h e t r a n s i e n t flow

net p r o d u c e d by th is m o v e m e n t o f t h e wal l is r e p r e -

sen ted o n F ig . 9; t he s t r e a m l ines a n d e q u i p o t e n t i a l

l ines s h o w n by c o n t i n u o u s l ines a r e t a k e n f r o m t h e

on-s i t e m e a s u r e m e n t s a n d t h e b r o k e n l ines a r e d e -

r ived f rom r o u g h c a l c u l a t i o n s .

T h i s fall in p r e s s u r e is t r a n s i e n t b u t in t h e c a s e o f

silty c lay a n d c lay soi ls t h e r e b a l a n c i n g o f t h e p r e s -

sures m a y last a l o n g t i m e as it is s h o w n in F i g . 6.

A n e v a l u a t i o n of t he p o r e p r e s s u r e s af te r e a c h

Fig. 9 . Grand Casinoflow net behind the wall.

e x c a v a t i o n s t age by m e a n s o f S k e m p t o n ' s p o r e p r e s -

su re e q u a t i o n [4] w i t h = 1 a n d A = 0 .5 , s h o w e d t h a t

th is m e t h o d y ie lded a c c e p t a b l e resu l t s .

A c o n s i d e r a t i o n o f p o r e p r e s s u r e in t h e de fo r -

m a t i o n ana ly s i s is t h e r e f o r e n e c e s s a r y t o o b t a i n c o r -

rect d e f o r m a t i o n v a l u e s a r o u n d a n e x c a v a t i o n pe r -

f o r m e d in c l ayey soi ls . U n f o r t u n a t e l y , c o m p u t e r

p r o g r a m s a l l o w i n g s u c h a c o u p l i n g in a d d i t i o n t o all

t h e o t h e r facili t ies offered by s u c h a p r o g r a m as

A D I N A d o n o t exis t . F u r t h e r m o r e , th is c o u p l i n g

a p p r e c i a b l y i nc r ea se s t h e c o m p u t a t i o n t ime .

In t h e a priori ana ly s i s of t h e C o n f e d e r a t i o n C e n t r e

e x c a v a t i o n , th is c o u p l i n g w a s p e r f o r m e d " b y h a n d ' ' :

t h e p o r e p r e s s u r e v a r i a t i o n s as a f unc t i on of t h e

e x c a v a t i o n s t ages w e r e e v a l u a t e d o n t h e bas i s o f t h e

s t resses c o m p u t e d a t t h e t i m e of a first c o m p u t a t i o n

w i th n o v a r i a t i o n in t h e p o r e p r e s s u r e a n d o f

S k e m p t o n ' s e q u a t i o n (Au = 2 /3 Au c o m p u t e d ) . T h i s

r o u g h p r o c e d u r e is, o f c o u r s e , far f r o m sa t i s f ac to ry .

3.3.4 Brief commentary on the results. T h e c o m -

p u t a t i o n o f t h e s e t t l e m e n t s a r o u n d a n e x c a v a t i o n in

c layey soi ls by m e a n s o f a 2 D a n a l y s i s i n d e p e n d e n t

of t h e t i m e t a k e n by t h e F E M w i t h n o n l i n e a r c o n s t i -

tu t ive l aws s u p p l i e s t h e d e s i g n e r w i t h resu l t s of a n

infinitely g r e a t e r v a l u e t h a n t h o s e o b t a i n e d f rom a

t r a d i t i o n a l e las t ic c o m p u t a t i o n . O n t h e o t h e r h a n d ,

t h e s t resses o n t h e wa l l s a n d t h e b r a c i n g , e v a l u a t e d

w i th t h e h e l p o f t h e c lass ica l m e t h o d s de r ived f r o m

R a n k i n e ac t ive a n d pa s s ive s t a t e s c a n b e c loser t o

rea l i ty ; a n o n l i n e a r a n a l y s i s by t h e F E M u n d e r t a k e n

solely t o e v a l u a t e t he se s t resses is, t h e r e f o r e , o f ten n o t

jus t i f ied .

G i v e n t h a t t h e d e f o r m a t i o n m o d u l i i n t r o d u c e d in

t h e m a t h e m a t i c a l m o d e l w e r e o b t a i n e d f rom oe -

d o m e t r i c t es t s , it m a y b e a s s u m e d t h a t t h e c o m p u t e d

d e f o r m a t i o n s c o r r e s p o n d t o t h e e n d o f p r i m a r y c o m -

p r e s s i o n . H o w e v e r , a s t h e c o n t r a c t o r is n o t s u p p o s e d

t o w a i t for t h e e n d of t h e p r i m a r y c o m p r e s s i o n af ter

Fig. 10. Confederation centrecomputed and measured deformations after the lower excavation.

Fig. 10 Confederation Centre -Computed and measured

deformations after the lower excavation

Use of A D I N A in soil mechanics with case studies for excavations 641

each e x c a v a t i o n s t a g e , a n i m p o r t a n t e r r o r m a y t h u s

be i n t r o d u c e d . T h e s e t t l e m e n t c u r v e o n F i g . 6 c lea r ly

s h o w s t h e difficulty o f t h e c h o i c e o f t i m e c o r r e s p o n d -

ing for a c o m p u t a t i o n s t a g e .

A v i scop la s t i c c o n s t i t u t i v e l a w m a y m a k e it p o s -

sible t o o v e r c o m e th i s difficulty; h o w e v e r , a c o n -

s ide rab l e c o m p u t a t i o n effort is n e c e s s a r y . T h e d i s -

c r e p a n c y o b s e r v e d b e t w e e n t h e r ecen t ly m e a s u r e d

s e t t l e m e n t s a n d t h o s e c o m p u t e d a r o u n d t h e C o n -

fede ra t ion C e n t r e e x c a v a t i o n m i g h t b e d u e t o t h e s e

c o n s o l i d a t i o n t i m e p e r i o d s t h a t a r e m u c h l o n g e r t h a n

the d u r a t i o n o f t h e e x c a v a t i o n s t a t e s .

T h e 2 D a n a l y s i s p r e s e n t e d a b o v e d o n o t t a k e i n t o

a c c o u n t t h e effect o f t h e t h i r d d i m e n s i o n w h i c h m a y

be all t h e m o r e i m p o r t a n t t h a n t h e d e p t h / m e a n w i d t h

r a t i o (2h/(b - h i ) ) o f t h e e x c a v a t i o n is t h e g r e a t e r .

T h i s r a t i o is 0 .20 a t t h e G r a n d C a s i n o a n d 0 .80 a t t h e

C o n f e d e r a t i o n C e n t r e . T h e r e f o r e , t h e final d e f o r -

m a t i o n s m e a s u r e d a r o u n d t h e C o n f e d e r a t i o n C e n t r e

e x c a v a t i o n a r e ve ry p r o b a b l y n o t a s c lose t o rea l i ty

as a r o u n d t h e G r a n d C a s i n o e x c a v a t i o n .

It m u s t a l s o b e p o i n t e d o u t t h a t t h e s t u d i e s t h a t

fo rm t h e sub jec t o f th i s p a p e r , i n c l u d e d m a n y o t h e r

c o m p a r i s o n s b e t w e e n t h e c o m p u t e d a n d t h e m e a -

s u r e d v a l u e s ; t he se c o m p a r i s o n s c o n c e r n in p a r t i c u l a r

t he s t resses in t h e wa l l s a n d b r a c i n g a n d t h e e a r t h

p r e s s u r e s . In t h e l imi t ed s p a c e a t o u r d i s p o s a l , it is

n o t p o s s i b l e t o p r e s e n t a d e t a i l e d d i s c u s s i o n o f t he se

i ssues . A s a n e x a m p l e , T a b l e 1 p r e s e n t s t h e c o m -

p a r i s o n b e t w e e n t h e c o m p u t e d a n d t h e m e a s u r e d

forces in t h e c e n t r a l s h o r e o f t h e G r a n d C a s i n o

e x c a v a t i o n .

3.3.5 Cost of analysis. T h e a v e r a g e c o s t p e r a l t e r -

n a t i v e o f 2 D a n a l y s i s ( s u c h a s t h o s e d i s c u s s e d in th i s

p a p e r ) , is a p p r o x i m a t e l y e q u i v a l e n