# Reliability of Soil-Structure-Foundation Systems - 4132005

Project Title—ID Number | Reliability of Soil-Structure-Foundation Systems - 4132005 |

Start/End Dates | 10/1/05 – 9/30/06 |

Funding Source | PEER-CA State Transp. Funds |

Project Leader (boldface) and Other Team Members | Joel Conte (UCSD/F), Quan Gu (UCSD/GS), Michele Barbato (UCSD/GS) |

## Project goals and objectives

- - Extension of sensitivity and reliability framework in OpenSees to perform probabilistic response analysis and reliability analysis of Soil-Structure-Foundation (SSF) systems.
- - Implementation in OpenSees of stochastic earthquake ground motion model.
- - Extend current OpenSees capabilities for solving the large-scale nonlinear constrained optimization problems that the search for the design point(s) reduces to.
- - Extend OpenSees framework for response sensitivity and reliability analysis of SSF systems to high-performance computing (to enable treatment of 3-D large-scale problems).
- - Development of "demonstration" applications for building and bridge structures.

## Role of this project in supporting PEER's mission (vision)

This project contributes advanced analytical tools to enable finite element response sensitivity and reliability analysis of soil-foundation-structure- interaction (SFSI) systems based on state-of- the-art computational mechanics models of all system components. Such tools for uncertainty propagation analysis are needed in the PEER PBEE methodology.

## Methodology employed

This project consists of developing and/or integrating analytical tools for stochastic ground motion modeling, finite element response sensitivity analysis, and reliability analysis in order to propagate basic sources of uncertainty related to earthquake loading and material (structural and soil) properties through nonlinear seismic response analysis of SFSI systems.

## Brief Description of previous year's achievements, with emphasis on accomplishments during last year (Year 8)

We performed the following extensions of the finite element response sensitivity analysis (based on the Direct Differentiation Method, DDM) framework in OpenSees to enable response sensitivity analysis (needed for reliability analysis) of 3-D soil-foundation-structure-interaction (SFSI) systems:

- - Extended transformation constraints (useful for connecting foundation elements to the soil domain and modeling rigid diaphragms of building structures) for sensitivity computation to the 3D case.
- - Added nodal and local (i.e., Gauss point level) sensitivity recorders.
- - Extended DispBeamColumn3d (used in modeling 3D bridge and building structures), FiberSection3d, LinearCrdTransf3d (coordinate transformation from element basic to global DOFs), ElasticSection3d, and section Aggregator, for sensitivity computation capabilities.
- - Extended for sensitivity computation the pressure-independent multi-yield surface material model from 2D to 3D.
- - Extended variableTransient integration Analysis for sensitivity computation to enable response sensitivity analysis when using the adaptive time stepping scheme in OpenSees.
- - Extended for sensitivity computation the 3D bbarbrick element (8 Gauss points for deviatoric part and reduced integration for volumetric part) available in OpenSees. We also modified the algorithm for the response computation part of the element to improve computational efficiency (by avoiding duplications in element state determination).

_{y,col}of the longitudinal steel in the columns (red curve in Fig. 2) and the shear strength τ

_{4}of the fourth (bottom) soil layer (dashed black curve in Fig. 2) are the most important parameters for this problem.

We investigated the effects of using (1) smooth versus non-smooth material models, (2) static
versus dynamic analysis, and (3) insufficiently small time steps to integrate the equations of
motion, on discontinuities in FE response sensitivities and convergence to the design point(s)
when performing reliability analyses. As an illustration, Fig. 3 shows the FE response sensitivity
surface for the roof relative displacement of a 3-story shear frame modeled using the Menegotto-
Pinto constitutive law (smooth model) and subjected to the El Centro 1940 earthquake scaled by
a factor 3. The discontinuities along the parameter axis (interstory initial yield strength F_{y0}) are
due to the fact that the integration time step (Δt = 0.02 sec) is too large.

We developed and implemented in OpenSees analytical tools for performing probabilistic quasi- static response analysis (e.g., push-over analysis) of structural and/or geotechnical systems. These tools are based on the First-Order Second-Moment (FOSM) method of probabilistic analysis and allow to compute first-order approximations of the first moment (mean) and second moment (standard deviation and correlation coefficient) of any FE response quantities based on the first and second moments of basic random/uncertain loading/material/geometric parameters. The approximate analytical FOSM results were validated with Monte Carlo Simulation and were found to be accurate up to the range of moderate nonlinear behavior. We also investigated the use of FOSM in the case of dynamic earthquake response analysis and found that it does not provide satisfactory results in terms of accuracy.

Problems of time-invariant and time-variant reliability analysis of structural or SFSI systems reduce to solving a large-scale nonlinear constrained optimization problem in the search for the design point(s), which represents the heart of the finite element reliability analysis methodology. This is in general a very challenging computational task, which often suffers of non-convergence problems. We have already augmented the computational optimization capabilities previously implemented in OpenSees by Haukaas and Der Kiureghian by linking SNOPT with OpenSees. SNOPT is a state-of-the-art software package for nonlinear constrained optimization based on sequential quadratic programming (developed by Prof. Philip Gill at UCSD). The resulting combined software OpenSees-SNOPT can also be used for general purpose optimization (e.g., structural optimization, finite element model updating). The topology of the limit-state surface (in both the physical and standard normal spaces) for time-invariant and time-variant reliability problems will be investigated. Based on the insight gained from these studies, more efficient and more robust approaches/schemes for the design point(s) search and for evaluating the probability content of the failure domain will be developed and implemented in OpenSees. Some of these approaches will consist of customizing state-of-the-art optimization algorithms so as to exploit the physics and geometry of the specific problems at hand. Alternative approaches will consist of reformulating the problem in a form that can be solved efficiently by using the state-of-the-art algorithm in computational optimization. For accurate evaluation of the probability content of the failure domain, we are also investigating the use of hybrid methods combining the use of the design point(s) with variance-reduction simulation techniques such as importance sampling, directional simulation and subset simulation.

Using the finite element response sensitivity and reliability analysis framework in OpenSees, we are in the process of performing time-variant reliability analysis (i.e., computation of the mean rate of a critical FE response quantity exceeding a specified threshold) of deterministic SDOF and simple MDOF benchmark problems subjected to stochastic broad-band earthquake excitation. Initial results on the search for the design point(s) are promising. However, it appears that in the case of large nonlinearities, approximation of the probability content of the failure domain (for mean up-crossing rate calculation) using the First-Order Reliability Method (FORM) is not satisfactory (i.e., not accurate enough). We are investigating alternative procedures/methods to obtain more accurate results, keeping the computational cost within acceptable limits.

## Other similar work being conducted within and outside PEER and how this project differs

FE reliability codes have been developed and used by NASA, Boeing, SouthWest Research Institute, Det Norske Veritas, and a number of other large engineering enterprises as well as by the University of Munich (reliability software STRUREL), the Technical University of Denmark (PROBAN software), and the University of Innsbruck (COSSAN software). To our knowledge, none of this software is under an objected-oriented platform, or aimed specifically at soil- structure-foundation systems and seismic reliability problems. In this sense, the framework in OpenSees for sensitivity and reliability analysis of SSF systems will be unique.

## Expected milestones & deliverables

- - Developments in OpenSees for sensitivity, probabilistic response, and reliability analyses of SSF systems.
- - Documentation of demonstration applications.
- - PEER technical report, papers, and User's Guide

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