## Info

Bar 2 | 3

FIGURE 36.6 Three-bar indeterminate truss modeled as a series-parallel system. (Estes and Frangopol 2001b. Reprinted with permission from the Computational Structural Engineering Institute.)

FIGURE 36.6 Three-bar indeterminate truss modeled as a series-parallel system. (Estes and Frangopol 2001b. Reprinted with permission from the Computational Structural Engineering Institute.)

N[20, 4.0], where 20 and 4.0 are the mean and standard deviation (in kN), respectively. The resistances are independent. The limit state equations that describe the components in the series-parallel model are (Estes and Frangopol 2001b)

g(1) = R1(2.0A1A3 + p2(A1A2 + A3A2)) - p2PA1 (cos 6 + sin 6)- 2.0PA2 cos 6 = 0 g(2) = R2(2.0A1A2 + p2(A1A2 + A3A2)) - p2P((A1 - A3) cos6 +(A1 + A3) sin6) = 0 g(3) = R3(v/2A1A3 + A1A2 + A3A2)+ P(A3 sin 6 - A3 cos 6 - v/2A2 cos 6)

g(4) = R2A2 - P(sin 6 + cos 6) + p2R1Ai = 0 g(5) = R3A3 - PV22 cos 6 + R1A1 = 0

g(6) = R1A1 - V2/2(P(sin 6 + cos 6)) + R2A2 = 0 g(7) = R3A3 - V2/2(P(cos 6 - sin 6)) + R2A2 = 0

g(8) = R1A1 - P\fl cos 6 + R3A3 = 0 g(9) = r2a2 - P(sin 6 + cos 6) + v/2R3A3 = 0

### Answer

Using first-order reliability methods the reliability indices for the three individual bars are found to be bbar1 = 3.48, bbar2 = 7.42, and bbar3 = 3.90. The reliability of the series-parallel system bsystem = 4.19. These reductions and computations were performed using RELSYS (RELiability of SYStems) (Estes and Frangopol 1998), a computer program developed at the University of Colorado. Due to the parallel nature of the model, the reliability of the system was greater than the reliability of two of the bars. The most critical link in the simplified series system was the one involving bars 1 and 3 in parallel, which is why the system reliability was so much lower than the reliability of bar 2. Correlation between failure modes improves the reliability of a series system and decreases the reliability of a parallel system. Although the resistances are independent, the failure modes are correlated because the same load source P was common to all three bars. Estes and Frangopol (2001b) examine the performance and maintenance of this truss system over time.

In a life cycle cost analysis, the system reliability index accounts for the degree of redundancy and extra safety in a structure. As seen in AASHTO (1998), the importance, ductility, and redundancy of a structure all affect the allowable safety level of a structure. For system performance purposes, Frangopol et al. (1992) classified structural redundancy based on a system redundancy measure (asystem) and a redundancy range (gsystem), where

«system b system bweakest member g system bstrongest member bweakest member

Based on these probabilistic measures, structural systems are classified as follows: Very redundant systems

Redundant systems g system > «system > 0

Nonredundant systems

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