By Maha Kenawy, PhD, University of Nevada, Reno
The early 2000s witnessed the rise of performance-based earthquake engineering, and with it came the emphasis on understanding ‘extreme’ structural performance; i.e., how structures behave at very large deformation demands, and how rapidly they lose their ability to carry loads when subjected to extreme events. This fundamental change to the structural design philosophy was followed by continuing refinements and guidelines on how to assess expected structural performance under extreme loads (For example, FEMA P-58 (Applied Technology Council, 2018)). Today, estimating the expected collapse capacity of structural systems and components is a common practice for engineers and researchers. At the base of determining the collapse capacity of a structure is the fundamental assumption that numerical structural models are capable of simulating the deterioration of structural components and materials. Many challenges stand in the way of having such models.
Structural engineers often seek to utilize computationally inexpensive models. Naturally, computational efficiency is the result of creating a set of idealizations to simplify the way that the model works, without significantly compromising its performance. When the issue at hand is predicting the behavior of structural components subjected to extreme loading conditions (such as those of earthquakes), the ability to simulate anticipated structural damage is a critical component of our modeling tools. In this context, the structural engineering modeling philosophy lends itself to two major types of modeling tools; each comes with their own compromises. The first type is concentrated or lumped plasticity models, which idealize structural components as elastic elements with nonlinear hinges at their ends. This type of model typically possesses superior numerical performance; however, it requires extensive calibrations based on experimental tests of structural components (which makes it difficult to generalize its applicability), and suffers from other performance limitations that come from the very nature of its idealizations. Despite its limitations, this type of model remains the most common numerical tool today for assessing the collapse capacity of reinforced concrete (RC) structures.
The other common modeling approach offers more granularity - and therefore accuracy - to predicting the performance of structural components. The so-called fiber-discretized (FD) component model is a type of distributed-plasticity frame finite element (FE) approach that relies upon convenient and limited calibrations based on the fundamental behavior of materials (in this case, concrete and steel), which make the model easy to generalize to various applications. Because this type of model is more sophisticated than the former in terms of its predictive capability and versatility, it quickly gained much popularity as a superior tool for assessing structural performance.
Figure 1 - Schematic description of mesh bias in fiber-discretized frame finite elements. On the left, damage to a RC column in the 2016 Kaikoura, New Zealand earthquake is pictured (photo from EERI photo gallery by Dmytro Dizhur and Marta Giaretton). Simulation of damage as constitutive softening illustrates the dependence of the member deterioration on the size of the numerical model elements. The mesh bias appears at the global level (load-displacement behavior) of a column loaded with axial and lateral loads, as well as the local level (strain or curvature distribution along the member)
In the past few decades, however, the rising demand for predicting the deterioration of structural components has exposed the extent of the shortcomings associated with the FD element modeling approach. To break down the fundamental issue with FD models, one has to go back to the underlying modeling of progressive damage (or cracking) in quasi-brittle materials as constitutive “softening” or negative stiffness, which became popular in the 1970s (a large number of publications by Zdeněk Bažant discusses the subject in due depth - in addition to his unique anecdotal account in Bažant, 2002). The advent of computerized FE analysis exposed the major limitation of constitutive softening: mesh bias. In other words, the numerical solution to a physical problem describing material damage depends on the characteristic size of the FE model, leading to non-objective, and therefore unphysical, model predictions. This issue is schematically described in figure 1.
The 1980s were ripe with several approaches proposed by researchers to overcome mesh bias in constitutive softening problems. Such approaches were not widely adopted in structural frame models until the 2000s. Some structural engineering researchers proposed simple approaches to amend the current FD modeling tools (which suffer from the same mesh bias issues); others created fundamentally new FE formulations that suspend the traditional idea that individual finite elements are independent, and introduce some continuity between neighboring elements in a numerical model. This family of approaches is known as the nonlocal models (I review many of these approaches in Kenawy 2018). My former research group at the University of California, Davis, led by Amit Kanvinde and Sashi Kunnath, proposed new structural component and constitutive formulations that advance the latter approach for both reinforced concrete and steel structures.
Figure 2 - Results of the verification study of the proposed nonlocal model by Kenawy et al. (2018). In a pushover analysis of a RC column, the conventional fiber-discretized frame model (left) fails to converge to a unique solution. The novel nonlocal model (right) overcomes mesh bias and rapidly converges to a unique solution.
The frame FE formulation and concrete constitutive model I developed as a doctoral student overcome the mesh bias of FD models (as shown in figure 2), without imposing too many restrictions that would limit the applicability of the model (Kenawy et al. 2018; Kenawy et al. 2020). The major drawback of the proposed framework is an inevitable added computational expense to our numerical structural models. The model also requires additional assumptions regarding the extent of damage in a structural member (referred to as a characteristic length in a continuum, and more commonly approximated as a plastic hinge length by structural engineers). My doctoral work also proposes a rather simple approach to characterize this characteristic length for a wide range of structural problems. Figure 3 shows representative blind predictions of the behavior of RC columns along with their observed behavior in laboratory experiments. These plots are part of a large validation study which suggests that the proposed framework and underlying assumptions are capable of simulating the observed degradation of RC components due to crushing of the concrete material. Much work remains to extend this modeling approach to simulate different types of structural components (structural walls, for example) and failure modes, and to assess its ability to simulate the performance of entire structures (for which experimental test data are rather scarce). Nonetheless, the findings of my doctoral work, along with those of a few recent studies by structural engineering researchers, bring the promise of more sophisticated and reliable modeling approaches for assessing the performance of RC structures subjected to extreme loads, and utilizing these novel methods to improve structural design provisions.
Figure 3 - Representative results of the validation study of the nonlocal model. The simulated cyclic lateral load vs. drift response of several RC columns by the nonlocal model is compared against the observed component behavior in laboratory experiments. The experimental datasets were obtained from the PEER column database
Selected References
Applied technology Council (2018). FEMA P-58: Seismic Performance Assessment of Buildings. Federal Emergency Management Agency.
Bažant, Z. P. (2002). Reminiscences on four decades of struggle and progress in softening damage and size effect. Concr. J.(Japan Concr. Inst.), 40, 16-28.
Kenawy, M. (2018). Nonlocal Computational Framework for Simulating Extreme Limit States in Reinforced Concrete Structures. Ph.D. Dissertation, University of California, Davis.
Kenawy, M., Kunnath, S., Kolwankar, S., & Kanvinde, A. (2018). Fiber-based nonlocal formulation for simulating softening in reinforced concrete beam-columns. Journal of Structural Engineering, 144(12), 04018217.
Kenawy, M., Kunnath, S., Kolwankar, S., & Kanvinde, A. (2020). Concrete Uniaxial Nonlocal Damage-Plasticity Model for Simulating Post-Peak Response of Reinforced Concrete Beam-Columns under Cyclic Loading. Journal of Structural Engineering, 146(5), 04020052.
