Crack Paths 2006

L x x O T G 4 2 ˜

(4)

Substituting Eq. (4) into Eq. (3), the relationship between )(OP and )(GV b becomes

P

tL

³

) ( b

) ( 4

O

d

0 2

˜ ˜ ) ( 4 0 a W L

G G G V

O O

(5)

It is difficult, if not impossible, to solve Eq. (5) analytically to determine the

bridging stress distribution )(GVb. Therefore we propose anumerical inverse analysis

method to solve Eq. (5) for )(GVb based on experimental data of )(OP in the bridging

section.

I N V E R SAEN A L Y S IMS E T H O D

O at the loading

From bending test, we can obtain the loads )(OP and the displacement

point. Using this information, we can obtain the bridging stress from Eq. (5)

numerically. In what follows, the numerical inverse analysis used to determine the

bridging stress is descrived.

As shown in Fig. 3, bridging stress characteristics is first approximated by a

polygonal line function. The total element number is m and the opening displacement

C G G . The magnitude of bridging stress at the point of 1 G ,2 G , ...,

is zero when

m G are

1s ,

2 s , ...,

m s , which are the unknowns to be obtained.

Polygonal lines as shown in Fig. 3 (a) are expressed by superposition of m

number of trianglar function in Fig. 3 (b). Each triangular function is multiplication of

magnitude of bridging stress at bending point and unit weight functions given by (see

Fig. 4)

C C G G G

, ) 2 (

­

» ¼ º « ¬ ª d d i m i m

0

°

C

°

G G G

G

G G

(6)

» ¼ º « ¬ ª d d ) 1 ( ) 2 ( i m i m i m i m C C

f

° ®

i

°

°

) (

» ¼ º « ¬ ª d d ) 1 ( i m i m

G G

2

° ¯

Hence the load )(OP in the experiment can be expressed apploximately as

follows:

a W

4

O

im

) (

˜

˜ ˜

tL

L

l)(Pnn , f s

) (

¦ ³ 1 O O Let )(11lP , )(22lP, ..., P 0 2 4 ) (

G G G ) (mt nbe the measured data of )(OP . From Eq. d (7)

i i

0

(7), one obtains

¦ m

¦

¦ m

i n i n

im i i

i R s p R s p 1 , 1 , 1 1 ) ( , , ) ( , ) ( O O O " (8) i i

R s p

1 , 2 2 2

i

where

ij R , is given by