Issue 51

F. Jafari et alii, Frattura ed Integrità Strutturale, 51 (2020) 136-150; DOI: 10.3221/IGF-ESIS.51.11 138 0 2 * * * ,  y r t s dead load live load f T A A T T T s    (3) In Eqns. (1) to (3), r M , r V and r T are the bending resistance, the shear resistance, and the torsion resistance of the cross- sections, respectively. Furthermore, the parameters of , , , ,  s y c A f f b d refer to areas of longitudinal steel, the yield strength 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 Mean pdf Value of parameters Random parameters 0.18-20 19.3 Normal 21 c f (MPa)  0.12 472.5 Normal 420 y f (MPa ) b/10 h/17 d/15 b h d Normal b h d Dimension (mm) 0.03-0.05 0.03-0.05 0.03-0.05 s v t A A A Normal s v t A A A 2 Area(mm ) 0.1 0.40-0.25 1.05D L Normal Gamble D 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.  ;   {{            ,      ,        }} s r s r s r G R S M M V V T T      (4) 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 S 2 2 R s mean mean σ σ eta index     (5) Safety surface of shear-torsion Based on Eqn. (4), by limiting the left side of Eqn. (6) to a specified value ( ' 2 )  * 3 c c w v f b d         the combination of simultaneous effects of shear and torsion can be achieved for different cross-sections.