Limit state function

Explicit integration of Eqn. (40) is intractable due to the form of the limit state function which defines the integration domain ([Pg.380]

DuraCrete R9, Statistical quantifications of the variables in the limit state functions . Document BE95-1347/R9, January 2000, The European Union - Brite EuRam III, DuraCrete - Probabilistic Performance based Durability Design of Concrete Structures. 

These transformations are in general nonlinear and are obtained by applying Rosenblatt s or Nataf s transformations, respectively (Huang Du 2006). They are linear only if the random vector x is jointly Gaussian distributed. By transformation 0 = Txg (x), also the Performance Function (PF) or Limit State Function (LSF) gx -) defined in the physical space (Section 1) can be transformed into g ( ) in the standard normal space ... 

Artificial neural networks (ANNs) are most often used for function approximation and for an object classification even if only incomplete and noisy data are available (Rafiq et al. 2001). In structural rehability analysis the role of ANN as a universal tool for function approximation is utilized when the limit state function under consideration is complicated and computer-time consuming, cf Hurtado Alvarez (2001), Gomes Awruch (2004). Typical examples are nonlinear problems, e.g. the assessment of post-budding strength of plates or shells. The inevitable FEM calculations of strength are carried out for suitably chosen sets of training and validation patterns. A subsequent reliability analysis then can be performed by the obtained ANN approximation of the strength function. 

The limit state function forthe slab exposed to bending may be written as follows ... 

Bucher (1990) proposes an algorithm to locate a new centre of sample points A, closer to the true limit state function than the mean value by a hnear interpolation given by ... 

MCS requires extensive computing times as it requires generating N sets of sample values of A to evaluate the limit state function g(X) for each value. The failure probability is then estimated as the ratio of themunberofeventswithg(A) < 0 to the total number... 

The limit state function g(.) is expressed in terms of the limit value of the crack width wijm and the random... 

Theoretical probabilistic safety assessment relies on identifying the structure parameters and the physical model which governs the failure (Elegbede 2005). This physical model is generally described by a function named limit state function in structural reliability theory (Madsen et al. 2006). Since the physical parameters of the structures are generally covered with uncertainties, there are generally modeled by random variables. Thus, to assess failure probability of... 

Equation 1 expresses the fact that the failure domain D is measured by means of probability measure. It is not easy to calculate Pf using Equation 1, therefore many techniques are developed in the literature. The well known approaches are the FORM/SORM (respectively, First Order Reliability Methods and Second Order Reliability Methods) that consists in using a transformation to change variables into an appropriate space where vector U = T X) is a Gaussian vector with uncorrelated components. In this space, the design point, , is determined. Around this point, Taylor expansion of the limit state function is performed at first order or second order respectively for FORM or SORM method (Madsen et al). In the case of FORM, the structure reliability index is calculated as ... 

We apply the presented maximiun entropy concept within this example for the modeling of the soil parameters and evaluate the influence of the parameter imcertainties on the failure probability. The limit state function of the bearing failure of a simple strip foimdation with pure vertical loading can be derived based on Terzaghi (1943) and Meyerhof (1953) as... 

In (Burgazzi 2003, Burgazzi 2007) the quantification of the probability of failure of the system is accom-phshed assuming a fixed and unchanging distribution (with constant mean and standard deviation values over a time interval) for both the threshold and actual values of the characteristic parameter, thus implying the time independence of the characteristic parameter. As stressed before, here we treat the parameters as time dependent upon time, starting from expression (1), which defines the time dependent limit state function. 

Time dependent factor does not always complicate the analysis. In many cases it is possible to transform a time dependent failure mode into a corresponding time independent mode. If the limit state function is monotonically decreasing with time, we simply have ... 

Another case could he overload failure of a time independent resistance R and a time varying action S. The limit state function for a reference period T can be written as ... 

Z t) limit state function W random parameter (as mass flow-rate)... 

The limit state function(s) (LSF), defined as g(0) = 0, is(are) linear with respect to the random variables (and these variables are Gaussian distributed). 

The limit state function does not deviate significantly from a hyperplane, i.e. a weakly non linear behavior of g( ) with respect to 0 is expected, and the number of random variables involved in the problem is low, e.g. less than 20. 

A common means to perform such mapping is to use an approximate method like the Nataf s model (Liu and Der Kiureghian 1986). Once the random variables involved in Equation 1 have been expressed into the standard normal space, it is possible to define the design point (x ) using a geometrical or probabilistic interpretation (see, e.g. (Freudenthal 1956)). In the geometrical interpretation, the design point is defined as the realization in the standard normal space which lies on the limit state function... 

The first order reliability method (FORM) is an approximate method for assessing the reliability of a structural system. Its basic assumption is to approximate the limit state function (g(Tj,g(x)) = 0) of the structural reliability problem by means of a hyperplane which is orthogonal to the design point vector note that this approximation is constructed in the standard normal space. Thus, the failure probability can be estimated using the Euclidean norm of X, i.e. 

Equation 5 provides an exact estimate of the failure probability provided that the limit state function is linear with respect to the Gaussian distributed vector of uncertain parameters. Under more general conditions, Equation 5 yields only approximate results. Moreover, it should be noted that EORM does not produce any measure of the error introduced by the linearization assumption. 

The generic expression for the performance or limit state function G(X) is... 

EVT is progressively utilized to compute limit state equations, which correspond to different failure modes of coastal and offshore structures. The input parameters of the limit state function are the stochastic load and strength parameters... 

The limit state function for calculating the failure probability of the l-th section, after rp i sections have failed, can be expressed as ... 

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