Project Title/ID Number Computational Reliability Tools for Design—4142002
Start/End Dates 10/1/02—9/30/03
Project Leader Armen Der Kiureghian (UCB/Faculty)
Team Members Terje Haukaas (UCB/Grad Student)
Project goals and objectives

Develop, implement, and document reliability and response sensitivity methods in the OpenSees computational simulation framework. Apply to test-bed examples to assess reliability.

Role of this project in supporting PEER’s vision

This project supports the PEER strategic plan by providing an advanced tool for uncertainty analysis in OpenSees. These tools are essential for performance-based earthquake engineering analysis and design. Analysis options include reliability assessment, fragility analysis, and sensitivities measures.

Methodology employed

Use is made of the following methodologies:

  1. first-order second-moment (FOSM) uncertainty analysis,
  2. first-order reliability method (FORM),
  3. importance sampling and Monte Carlo simulation,
  4. response sensitivity analysis by the direct differentiation method (DDM).

Furthermore, FORM techniques are used to obtain reliability sensitivities with respect to model parameters, as well as mean out-crossing rates for response to dynamic loading. One specific stochastic ground motion model in the form of filtered random pulse train has been implemented.

Brief description of past year’s accomplishments and more detail on expected Year 6 accomplishments

During the previous phases of this project, we developed a software framework for FORM and Importance Sampling analyses for inelastic static finite element reliability problems in OpenSees.

Implementation of the direct differentiation method is now extended to include response sensitivities for inelastic dynamic problems. Figure 1 shows a sample plot of a displacement time history and the corresponding displacement sensitivity with respect to yield strength. Response sensitivities may exhibit discontinuities. This could cause convergence problems in reliability analysis applications by FORM. We have implemented “smooth” material models to remedy this problem. Currently, these include Bouc-Wen, Generalized Plasticity and smoothed “Steel01” and “Concrete01” models in OpenSees. Figure 2 shows the desired effect of smoothing of “Steel01” on typical displacement sensitivities with respect to yield strength for a static problem.


Figure 1. Dynamic response sensitivities
Larger View


Figure 2. Gradient discontinuities remedied by smooth material models
Larger View

A particular ground motion model has been implemented in OpenSees to facilitate estimation of mean out-crossing rates of nonlinear dynamic response to stochastic ground motion. Derived quantities, such as an upper bound to probability of excursion during a specified time interval, are available. It is emphasized that the ground motion model has a physical interpretation in seismic applications. The train of random pulses may be interpreted as the impulses originating at the fault rupture. Filters and modulating functions provide means to take into account the propagation of these pulses through soil layers.

A user’s guide and an array of example files are available from

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

Finite element reliability codes have been developed and used by NASA, Boeing, Southwest Research Institute, Los Alamos and a few other entities. General-purpose reliability codes have been developed at the Technical University of Munich (STRUREL), Det Norske Veritas (PROBAN), the University of California, Berkeley (CalREL) among others. The on-going development in OpenSees is unique in several aspects. This is the first finite element reliability code developed in the object-oriented programming paradigm and it is the first code to handle nonlinear structures. Other researchers within PEER (G. Fenves at UCB, J. Conte at UCSD) are also working on DDM sensitivities in OpenSees. The individual groups are using the framework developed by our prior research. Their work is complementary, as each group addresses a particular element or material within OpenSees. We are also aware of theoretical work in this area by researchers in MCEER and are following it with interest.

Plans for Year 7 if this project is expected to be continued

The software is currently being used for reliability evaluations of the I-880 Testbed bridge model provided by researchers at UC Davis. This is a process involving continuous refinement of our implementations. We also expect that this work will highlight needs for further developments. Several areas may already be pointed out for further research:

  • Development and implementation of random field schemes to characterize structural properties. Part of this work will focus on software design issues related to parameterization of the finite element software.
  • Develop sensitivity equations for a range of material models suitable for advanced reliability analysis.
  • Implementation of additional algorithms to assure convergence of high-dimensional non-linear reliability analysis problems.
  • Promote use of advanced reliability methods in practical performance-based engineering by demonstrating the analysis process/options on comprehensive real-world examples.
  • Implementation of alternative stochastic ground motion models and corresponding simulation algorithms.
Describe any instances where you are aware that your results have been used in industry
We are communicating with a diverse group of approximately 25 researchers worldwide who have reported interest in performing reliability and response sensitivity analysis with OpenSees. In particular, our implementations is being used by other researchers at UCB to develop fragility curves for a bridge bent. We hope that our application to the I880 testbed will facilitate the use of reliability methods by practitioners.
Expected milestones

PEER report on the implemented software and the numerical studies of the I-880 Testbed bridge model (June 2003).


A comprehensive and modern computer software framework for finite element reliability and response sensitivity analysis. Complete on-line documentation including user’s guide and examples.