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
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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