Issue 51

F. Jafari et alii, Frattura ed Integrità Strutturale, 51 (2020) 136-150; DOI: 10.3221/IGF-ESIS.51.11

0 2 * * * ,  y t f T A A T T   r s



T

(3)

dead load

live load

s

r M ,

r V and r

T are the bending resistance, the shear resistance, and the torsion resistance of the cross

In Eqns. (1) to (3),

, , , ,  s y c A f f b d refer to areas of longitudinal steel, the yield strength

sections, respectively. Furthermore, the parameters of

of steel, 28-day compressive strength of concrete, width and effective depth of the cross-sections, respectively. The nominal values of the steel are calculated according to ACI 318M-14.2015 standard[16]. For L and T shaped sections, the proposed equations are obtained [12]. To express the shear and torsion limit state functions, all sections of L, T, and rectangular shapes of concrete beams are considered. The factors considered in the ACI standard are: concrete strength reduction factor ( 0.90 c   ), dead load increase factor   1.20 D   and live load resistance factor( 1.60) L   . In addition, the steel strength reduction factor of steel is 0.90 s   . Tab. 1 shows the statistical information (mean, covariance and probability density function(pdf)) which are used to obtain the safety indexes.

Standard deviation 0.18-20

Mean

pdf

Value of parameters

Random parameters

19.3

Normal

21

c f (MPa) 

0.12

472.5

Normal

420

y f (MPa )

Dimension (mm)

b/10 h/17 d/15

b h d

Normal

b h d

0.03-0.05 0.03-0.05 0.03-0.05

Normal

A A A

A A A

2 Area(mm )

s

s

v

v

t

t

Normal Gamble

D L

0.1 0.40-0.25

1.05D L

Loading

Table1 : Statistical values of the used parameters for design the sections of reinforced concrete [17-20].

Limit function and safety index equation Eqn. 4 shows the limit state function which is used to the safety design of concrete beams under the simultaneous effects of bending, shear, and torsion.

G R S  

      ,    M M V V T T      ,    

(4)

 ;   {{     

   }}

s

r

s

r

s

r

After employing Eqn. (4), in order to predict the safety index ( )  , the values of R (resistance surface) and S (load surface) are calculated for the three mentioned states (bending, shear and torsion) using Hasofer-Lind equation [17-18]. Eqn. 5 shows the value of the safety index [18]. In this study, the safety index is calculated for current limit states functions using Monte- Carlo simulation

R 2 R s mean mean σ σ   2

(5)

S



eta index 

Safety surface of shear-torsion Based on Eqn. (4), by limiting the left side of Eqn. (6) to a specified value (

 

  

v

2

'

* 3 c

  

f

)  the combination of

c

w b d

simultaneous effects of shear and torsion can be achieved for different cross-sections.

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