## Khk FIGURE 12.2 Limit state concept. (Adopted from Probability, Reliability and Statistical Methods in Engineering Design, by Haldar and Mahadevan, 2000, with permission from John Wiley & Sons, Inc.) Note: A number in parenthesis indicates iteration number.

FIGURE 12.2 Limit state concept. (Adopted from Probability, Reliability and Statistical Methods in Engineering Design, by Haldar and Mahadevan, 2000, with permission from John Wiley & Sons, Inc.) Note: A number in parenthesis indicates iteration number.

The limit state equation for the strength limit state in pure bending can be expressed as

where Mu is the unfactored applied moment and Mn is the nominal flexural strength of the member.

Using the AlSC's LRFD design criteria, the limit state equation for the strength limit state for a beam-column element can be expressed as g (X) = 1.0 -g (X) = 1.0 -

Pu 8 Mu

Pn Mu

where Pu is the unfactored tensile and compressive load effects, Pn is the nominal tensile and compressive strength, Mu is the unfactored flexural load effect, and Mn is the nominal flexural strength. Limit state equations can similarly be defined for other strength design criteria like tension, compression, shear, etc. All the strength-related parameters in Equations 12.8 to 12.10 are functions of the geometry, material, and cross-sectional properties of the members under consideration. The code also suggests analytical procedures to evaluate the load effects in some cases. The important point is that the loads effects in Equations 12.8 to 12.10 are to be unfactored in evaluating the reliability index.

For the serviceability limit state, the midspan deflection of beams under live load and the side sway (interstory drift and lateral deflection) of frames are commonly used. They can be represented as g (X) = 1.0 -g (X) = 1.0 -

d dlimit u u limit

where d is the vertical deflection of a beam under unfactored live load, diimit is the allowable or prescribed vertical deflection, u is the lateral deflection under unfactored loads, and ulimit is the allowable or prescribed lateral deflection. The deflection limits diimit and uiimitare selected by the designer based on the performance requirements of the structure. They have to be preselected to evaluate the reliability index.

Incorporating the concept of limit state or performance function, the basic reliability evaluation formulation represented by Equation 12.2 can be rewritten as

P(failure) =/•••/ fX(x1, x2, ..., xn) dx1 dx2 ••• dxn (12.13)

in which fX(x1, x2,..., xn) is the joint PDF of the basic random variables, and the integration is performed over the failure region. If the random variables are statistically independent, then the joint PDF can be replaced by the product of the individual PDFs in Equation 12.13. Reliability evaluation using Equation 12.13 is known as the full distributional approach.

Initially, all the basic random variables are considered to be statistically independent; they can have different types of distribution, and the limit state equation can be linear or nonlinear. FORM is an iterative technique. The final products of this technique are the reliability index b, the corresponding coordinates of the design or checking point or the most probable failure point (xJ, xJ,..., xn), and the sensitivity indexes indicating the influence of the individual random variables on the reliability index. In the context of FROM, the reliability index b has a physical interpretation. It is the shortest distance from the origin to the limit state function at the checking point in the reduced standard normal variable space as shown in Figure 12.2 and Figure 12.3. As will be discussed further later, an optimization technique is used to estimate it iteratively. Once the information on b is available, the probability of failure can be obtained as

where F is the CDF of the standard normal variable. If b is large, the probability of failure will be small. The coordinates on the limit state surface where the iteration converges represents the worst combination of the random variables that would cause failure and is appropriately named the design point or the most probable failure point (xJ, xJ,..., xn). All these aspects of FORM are discussed in the following section.

The estimation of probability of failure using Equation 12.13 is expected to be complicated. As mentioned earlier, Rackwitz  suggested a solution strategy that can be used to evaluate the

Design point

Design point 0 0