## VSR SD s

The checking point must satisfy the performance function represented by Equation 12.22; that is g() = 0 = (mR — 0.13 x b x aR x mR) — (144.3 — 18.76 x b x aD) — (54.1 — 20.02 x b x aL) Substituting the direction cosine values in the preceding equation and simplifying will result in mR — 144.3 — 54.1

This is a quadratic equation in terms of the mean value of R. Solving the equation gives mR = 361.7 kN m. To select a section, the nominal value of the resisting bending moment, Rn, is necessary and can be shown to be 0.9 x 361.7 = 325.5 kN m. A W18 x 35 of Grade 50 steel will satisfy this requirement. The same section was obtained using the codified LRFD approach.

On the other hand, suppose the beam needs to be designed for a reliability index of 4, implying that the underlying risk has to be much smaller than before or that the beam needs to be designed more conservatively. Following the same procedure discussed above, it can be shown that a larger size member of W21 x 44 of Grade 50 steel will be required. This result is expected.

This example clearly demonstrates the advantages of the reliability-based design procedure. It will not only suggest a section but also give the underlying risk in selecting the section. Thus, using the reliability-based design procedure, engineers are empowered to design a structure considering an appropriate acceptable risk different than that considered in the codified approach for a particular structure.

12.6 Reliability Evaluation with Nonnormal Correlated Random Variables

The previous section's discussion of reliability evaluation using FORM implicitly assumes that all the random variables in the performance function are uncorrelated. Considering the practical aspect of structural engineering problems, some of the random variables are expected to be correlated. Thus, the reliability evalution of a structure using FORM for correlated random variables is of considerable interest. Although this is considered to be an advanced topic, it is discussed very briefly below. More detailed information can be found elsewhere [14].

The correlation characteristics of random variables are generally presented in the form of the cov-ariance matrix as

cov(X„, X2) cov(X„, X2) The corresponding correlation matrix can be shown to be

rX2,X1

rX1,X2 1

where rXi,Xj is the correlation coefficient of the Xi and Xj variables.

Reliability evaluation for correlated nonnormal variables X requires the original limit state equation to be rewritten in terms of the uncorrelated equivalent normal variables Y. Haldar and Mahadevan [14] showed that this can be done using the following equation:

where mN and sN are the equivalent normal mean and standard deviation of X, respectively, evaluated at the checking point using Equations 12.15 and 12.16, and T is a transformation matrix. Note that the matrix containing the equivalent normal standard deviation in Equation 12.25 is a diagonal matrix. The matrix T can be shown to be

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