The onset of structural collapse is used as the acceptance criterion in performance-based seismic engineering, but there is no consensus on how to define collapse in structures. One difficulty of such definition is that investigating structural collapse in the laboratory is difficult and testing to total collapse is not typically done due to safety concerns. This project proposes to use energy as the indicator of structural collapse. The advantages of using the amount of energy dissipation as the collapse indicator include: (1) being the product of force and deformation, energy can be used to assess the conditions of force-controlled elements as well as deformation-controlled components; (2) being cumulative over time, energy can be used to measure the amount of work associated with monotonic loadings, cyclic loadings, or the combination thereof and the progression of damage with repeated loading; (3) structural components have already been tested to failure, and therefore the amount of energy dissipation associated with structural failure can be quantified at the component level without testing the entire structure to collapse; and (4) energy is positive definite making it a stable, useful parameter. Major efforts of this research will focus on the development of a methodology to analytically quantify the energy demand in structures during seismic events and correlation of these energy demands with energy capacities from existing component experimental data. The result will be a framework for analyzing and evaluating nonlinear structural behavior and performance at collapse with practical applications, including the possible use of energy as the damage index for the evaluation of building performance.

**Objective** - The objective of this project is to develop a framework for evaluating structural performance and collapse due to seismic excitations based on an energy approach. The structural response obtained from nonlinear dynamic analysis is evaluated based on the amount of energy stored in the system and the amount of energy dissipated via damage to the components in the structure. This dissipative energy in turn is compared with the energy capacity of the components based on existing experimental data. Through this framework, an analytical tool will be developed to assess the state of structure and identify when collapse can occur, which is crucial for the advancement of performance-based seismic engineering.

**What is the new technical idea?**

Nonlinear dynamic analysis of structures subjected to seismic excitations is now a standard procedure for performance-based seismic engineering, where designs are considered adequate when the maximum responses from the analysis are within the acceptance limits. ASCE/SEI 7 (ASCE 2016) requires that all structures be designed to have at most 1% probability of collapse in 50 years, where structural collapse is defined in terms of maximum global response of a deformation quantity (such as story drift) exceeding certain acceptance limits in the nonlinear dynamic analysis. Similarly, ASCE/SEI 41 (ASCE 2017) identifies structural collapse when maximum damage in deformation-controlled yielding components that are local to the structure (such as plastic hinge rotation) reaches the collapse-prevention acceptance limits. Regardless of whether structural collapse is identified globally or locally, the objective of looking at deformation quantity in nonlinear dynamic analysis is to prevent structural response going beyond the corresponding acceptance limits such that other devastating phenomena begin the occur, such as buckling or connection failure is steel structures or spalling and reinforcing bar buckling in concrete structures. Structural behavior evaluation based on using the maximum deformation from structural performance testing is the best available option that is typically used. However, maximum deformation by itself ignores the effects of damage accumulating from the effects of cycling of structural elements that is typically observed under seismic excitation. The end result is that maximum drift alone may overstate the deformation capacity, especially during long duration strong shaking where damage can accumulate. This issue could impact the acceptance limits for response when cyclic effects are not addressed.

Field reconnaissance after seismic events have indicated that damage in structures may not match damage observed in tests that have been conducted in laboratory due to differences in loading patterns. Earthquakes tend to cause one-sided cyclic yielding on structures, especially on those that are tall and flexible. This occurs because once yielding occurs in one direction, the action of geometric nonlinearity can force the structure to gain momentum for further damage in the same direction while reducing the chance of yielding in the reverse direction. This observation is contrary to either the monotonic tests or the fully reversed cyclic tests that forms the basis for the current ASCE/SEI 41 standard in arriving at the components’ acceptance limits (i.e., structural response due to earthquake is neither monotonic nor fully reversed cyclic). This suggests that there needs to be an investigation regarding improvements to the acceptance criteria where the limits should take into consideration both monotonic and partial and fully reversed cyclic loadings. This problem is of significance and is the subject of a parallel project under the Earthquake Risk Reduction Program titled “Improved Assessment Criteria for Performance-based Seismic Design”, led by Matthew Speicher, who will examine the acceptance criteria question concerning unbalanced cyclic response by conducting experiments to examine the needed changes to the ASCE 41 criteria to account for this phenomenon. These projects will be cooperative with each providing information to the other.

This research proposes to address the problem of identifying structural collapse by developing a framework based on an energy approach for evaluating structural performance. Energy is computed by multiplying force and deformation. At the same time, positive forces cause positive yielding and negative forces cause negative yielding, and therefore the energy dissipation due to the multiplication of force and deformation is always positive and is cumulative. In view of this, the energy-based approach to evaluate structural performance is proposed that is independent of the loading protocol. In doing so, it addresses the lack of knowledge whether the loading is monotonic, fully-reversed cyclic, or somewhere in between. More importantly, energy, being the product of force and deformation, can be used to assess the behavior and damage in force-controlled elements as well as deformation-controlled components. Quantification of structural damage and collapse can be performed by comparing energy demand from earthquake to the corresponding energy capacity of the structural element obtained in the laboratory.

The use of seismic energy in damage assessment is not new. Energy approaches such as those based on cyclic responses for assessing structural damage dates back to the 1980s (Park and Ang 1985, Tembulkar and Nau 1987, McCabe and Hall 1989). Currently, energy is calculated using empirical formula, where the stored hysteretic energy is calculated by the area underneath the force-displacement curve and the input energy is calculated by integrating the square of ground acceleration time history (Fajfar 1994). One study recently published on correlating energy demand with structural collapse has been proposed (Deniz et al. 2017), but it is also based on the empirical formula. The major problem of using empirical formula is that the equations used to quantify such energy are not based on engineering mechanics, and therefore the law of energy conservation may be violated. An analytical method for calculating the energy demand of linear structures was also developed (Uang and Bertero 1990), yet only limited analytical research has been conducted since then, especially on nonlinear structures. Most of the research on dissipation of earthquake energy today is based on empirical formula for quantifying response spectra of simple structures, which is impractical for evaluating damage in complex nonlinear structures.

In view of these gaps, it is therefore the goal of this project to extend the knowledge of quantifying energy dissipation of linear structure to the nonlinear domain, and at the same time addressing the collapse issues related to performance-based seismic engineering. This framework – to identify structural collapse and evaluate nonlinear structural performance based on an energy approach – will be extended to investigate the correlation between energy and structural collapse to link the research to previously completed work.

**What is the research plan? **

To limit the scope of this project, the focus is placed on investigating the structural performance of steel moment frames. Structural models will include the use of NIST-designed 4-story and 8-story frames (Harris and Speicher 2015a, 2015b). These will include special moment frames and eccentric braced frames for each height building, a total of four different structures. Both geometric nonlinearity and material nonlinearity will be considered in the structural models. By using these nonlinear structural models, the following tasks will be carried out for each of the four frames:

- Derive the energy equations and define each energy type in the dynamic analysis of framed structures considering both geometric and material nonlinearities.
- Perform literature search and data mining to quantify the energy capacity of yielding components for the evaluation of structural performances.
- Address the consistency in energy dissipation for various loading conditions, including monotonic, fully reversed cyclic, or partially reversed cyclic response.
- Evaluate the energy demand of individual components in framed structures and perform structural damage assessments using experimental data from Step 2 as well as available data from NIST testing by Speicher
- Correlate energy caused by damage to structural collapse for each frame
- Investigate the practical use of the energy approach, including the use of energy as a damage index for building evaluation.

Created October 15, 2019, Updated January 6, 2023