Determination of geometric reliability index of piles at site-specific scale: case studies

Abstract

An application of geometric reliability techniques on various pile types based on 67 load–displacement curves obtained from pertinent literature is presented in this paper. These static loading tests were performed at local scale (even building-specific sites) under essentially identical geotechnical conditions. A power-law function with two parameters was used to fit the measured load–settlement curves. For each site, the means and coefficients of variation for the power-law parameters were obtained. Since the number of tests conducted at each site is usually small, it is extremely difficult to identify a certain distribution type for these regression parameters. Thus, for simplicity, a bivariate normal distribution was assumed to represent the set of regression pairs. This joint distribution was incorporated into a geometric reliability method, which offers an estimation of the bearing capacity of piles at the serviceability limit state. Examples are provided to illustrate the application of the proposed reliability method to interpret the visually determined reliability index of the bearing capacity of piles. These case studies showed that the proposed method is a useful and comprehensive tool for capturing the load–displacement response and for evaluating the bearing capacity of piles by considering uncertainties in their load–displacement behaviour.

Solving the geometric reliability index for a case involving multivariate random variables in the original physical space

“…In the geometric reliability algorithm proposed previously, 4,5 it is necessary to find the design point on the critical probability density contour and the pseudo‐design point on the single standard deviation configuration, whereby the reliability index β is defined by a distance ratio, as follows: $$$$\begin{equation}\beta = {{{L}_{\mathrm{D}}} \mathord{\left/ {\vphantom {{{L}_{\mathrm{D}}} {{L}_{\mathrm{C}}}}} \right. \kern-\nulldelimiterspace} {{L}_{\mathrm{C}}}}\end{equation}$$$$where L D is the distance from the design point on the critical probability density contour to the mean‐value point, and L C is the distance from the pseudo‐design point on the single standard deviation hyper‐ellipsoid to the mean‐value point.…”

“…2. Take the intermediate value 𝛽 mid between 𝛽 min and 𝛽 max , substituting it into Equation ( 4) and combining Equations (5) to (10) in order to derive the PDP coordinates associated with the reliability index of 𝛽 mid . These coordinates are substituted into the limit state equation to determine whether there is a point outside the critical surface (𝑔 ≤ 0).…”

“…In the geometric reliability algorithm proposed previously, 4,5 it is necessary to find the design point on the critical probability density contour and the pseudo-design point on the single standard deviation configuration, whereby the reliability index 𝛽 is defined by a distance ratio, as follows:…”

“…Based on the geometric reliability algorithm proposed by the author, 4,5 the critical PDPs and the limit state curve intersect at a specific point, which is defined as the design point and is denoted by “× .” The pseudo‐design point (represented by “+”) 17 is an intersection of the line between the mean point and the design point and the single standard deviation configuration. The computed regression pairs are represented by “*,” and their mean value is represented by “○.” For comparison, the probability density iso‐contour configuration with $$\beta = 1$$ is illustrated in Figure 7, which is close to the single standard deviation ellipse, but the two do not completely coincide due to the non‐normal characteristics of the joint distribution of the two random variables.…”

“…Such a physical space is also denoted as "generalised space" as it holds a clear physical meaning for all variables. An interactive and visual solution of the reliability index under fewer random variables is achieved in several areas 5,6 using the R platform. 7,8 However, a solution in the case of multiple random variables needs to be further studied.…”