Crack Paths 2006

respectively, in equilibrium with the imposed external stress ^`xyV.

^`xyH

Total strain

can be evaluated by the addition of the strain concrete between cracks ^`'cxyH

and that of

^ ` 1cr cxy

H :

the concrete in the crack

^`^`^`^`1crxy'cxycxyxyH HH

(4)

^`

^` > @ ^` )23( 1 11cr122 11crxy3 1 T u\ u u H H

T

1m1 1 m 1 1 c r 1 2 av aw ¿¾½ ¯ ® H

where

and

where w1 and v1 are the opening

Wxy

and sliding of crack surfaces, respectively (Fig.2b).

W

b

1

W

y

bmax

V y

1 1

W=W bmax (s/s 1 )

y

2 1

W yx

\

v 1

x

W xy

w 1

1

2

1

1

T n

\ 1

Vx

W

f

(b)

a m 1

(d)

A sn

s

s

s

0

s

1

3

2

A si

Slip

1

T i

s i

a

m /2

a m /2

V x

w

s n

W

W b

b

t

x

(a)

W yx

V y

H

s,av

(c)

H

s,min

Hs,crHs,max

Figure 2. (a) R/C membrane element in the singly cracked stage: geometry and

notation, (b) kinematical parameters of crack, (c) adopted bond relationship, (d)

stabilized cracking stage: shear bond stress, tensile stress and strain of steel bar.

At the crack the stress field is transmitted by axial stiffness and dowel action of the

steel bars crossing the crack, in addition to the bridging and interlock actions of

concrete aggregate. R/C between cracks transmits the stress field by the concrete and by

axial resistance of the steel bars embedded in the concrete. The strain of the steel in concrete between cracks ^`'sxyH is assumed equal to the global average strain of the steel

^`xyH

bar evaluated for the whole element.

Therefore, it follows:

in the crack : ^`^`^`^`>@>@^`>@^`1crc1crxy1crc1crsxy1crcxy1crsxy1crcxyxy1crxyDDH V VV

(5)

in the concrete between cracks: