Bearing Capacity and Critical Normal Stress Distribution of Soils by Method of Variational Calculus

ABSTRACT

A ma thema t i c a l t-echnique is h e r e b y advanced f 01: i n v e s t i -

c j ; ~ t - it13 the bearing c a p a c i t y and a s s o c i a t e d normal stress

t l I :-; tLribut.ion il t- C a i l . u r e of s o i l f o u n d a t i o n s .

'I'hc stab1 1 i t.y e q u a t i o n s are o b t a i n e d using the L i m i l:

i,:clu.i l i b r i u m ( 1,E) c o n d i t i o n s . The c o n d i t i o n s of v e r t i c n 1 ,

I I ~ J J - i z o n t aal .n d r o t a t i o n a l e q u i l i b r i a are transformed

n ~ ; ~ t . l ~ e r n a t i c a Iw. l yi t h r e s p e c t t o the s o i l s h e a r i n g s t r e n y t l l ,

leading to t h e d e r i v a t i o n of t h e e q u a t i o n o f t h e J!unctional

y, and two i n t e g r a l c o n s t r a i n t s .

Genera 1 I. y, no c o n s t i t u t i v e law beyond t h e cou.1 (.,i;lll1 s y i e l d

c 1 . i t e r i o n is i - n c o r p o r a t e d i n the f o r m u l a t i o n . Consequent.Ly,

no c o n s t r a i n t s a r e p l a c e d o n t h e c h a r a c t e r o f the c r i t i c a l s

e x c e p t t h e o v c l - a l l e q u i l i b r i u m o f t h e f a i l i n g s o i l secti-on.

The c r j - t i c a l normal. stress d i s t r i b u t i o n , Cmin, and

c o n s e q u e n t l y the load, b i n , determined as a r e s u l t o f t h e

i u i 11i.mization of t h e f u n c t i o n a l are t h e smallest stress and

Ioatl p a r a m e t e r s t h a t can cause f a i l u r e . I n o t h e r words, lor

a s o i l w i t h s t r e n g t h p a r a m e t e r s c, 6, r , hnd f o o t i n g wiLh

qeornetry 1, I i , when stress 6 4 d n ~ i n (C,fl, r, 1, H ) and l o a d

Q 4 Qrnin(c, 6 , r , I., H ) f o u n d a t i o n is s t a b l e . O t n e r w i s e Lhe

s t a h i l i t y would depend on t h e c o n s t i t u t i v e c h a r a c t e r of the

Foundatioli s o i l . I n t h e m a t h e m a t i c a l method e m p l o y x l , the

..; t a l ~ i l i t ya nalysis i s t r a n s c r i b e d a s a mi n imi z a t i 2 n probl.ern

i n t h e C a l c u l u s o f V a r i a t i o n s .

'I'hc r e s u l t of Lhe analysis shows, among others, Cllat Lhe

7'crzaghi':; st~p