Crack Paths 2012

A Ta: |19zi7r @ 0y : i i : ’i i: 5]67,1" 2 U 19:i1r

(31)

<7 0

W W Wn20

w h e r e

1 [211+3(,u2(h1+1)—,u1(h2+l]))_

211+3

en : —

(12,, $111

1r

(32)

00

2

#2 + NW2

2

It is worthwhile noting that the en coefficients vanish in the homogeneouscase.This is the

m a i ndifference betweenthe twocases.

Substituting Eqs(26),(27) and (28) in Eqs(7) and (8) gives

2n+3 2 k 2 + 3 w : 2 y-T[ M a m ] Sin n T (33)

11:0

'u/l'ua

2

M 2 ( k 1 k 2 — :|

_

2714-3

6 : 1" 2

— i n sin

1t

a (34)

g i

Mk2 I #2 2

2

According to the well established literature on the mechanical behaviour of concrete joints

(see Cervenka et al. [7]),softening only depends on we”: \/w2 + 62:

2 + l ' j / 2 k l ) 2

— 1 2 ( k I k Z ) 2 ) S i n 1 / 2 2

2n+3

2

(#1112)2

1” ,u%h%—,u% 2”

2

11:0

U ) n + 1i1: —eff :2r22_3dn

wefrc n20

;, _ 2 2(MW 2)1/2S,n2n+3

1t

(36)

TL

_

(#1112)2

1” u%h%—,u§ 2”

2

Let us consider the truncated N + 1 terms of 11) (36), and denote do : d0,dn : dn/do

(11 > 1)

N

1) = d01~%(1+ 2 dnr")

(37)

11:1

F r o mthis relation, w e can obtain

=<—.”>

2 1) 3 T 2 (+2111 1 : N ( i-

(2 1)

w w (2' 1)

(2T1)

1':

e

3

1I

3

C

( 2 ' I 1 )

( 2 ' I 1 ) ( 2 ' I 1 )

M

1) 3

=d0 3 1" 2 (nzemrn), (MEN)

(39)

11:1

in which:

N

I

_ time)

_

.

”

a. - m ,

fro - (1+ Zdnr )

(40)

11:1

674