Nonlinear structural analysis is the main motivation for most users to turn to OpenSees. Therefore, having a sufficient understanding of the why and how of nonlinear analysis is crucial for successfully starting work with OpenSees. To begin introducing nonlinear structural analysis, we need to answer a few important questions:
- Why is nonlinear structural analysis necessary?
- What is the difference between linear and nonlinear structural analysis?
- What is the general process of performing a nonlinear analysis?
In this article from the OpenSees home, we will first discuss the important applications of nonlinear structural analysis. A review of the basics of linear structural analysis will be presented in the article titled “Linear Analysis Process of Structures.” Based on the knowledge gained, we will then explain the process of nonlinear analysis in another article, so that the differences and complexities of nonlinear analyses, compared to the linear method, can be clearly revealed.
Necessity of Nonlinear Structural Analysis
Nonlinear or non-elastic analysis of structures is essential for several important reasons:
- All code provisions used in the design of structures under severe loads, such as earthquakes, could not exist without the use of nonlinear analysis. These provisions are derived from the results of precise nonlinear analyses that have been confirmed by experimental observations.
- The adequacy and performance of any new structural system must be evaluated by considering its ultimate behavior (behavior at the threshold of failure), and the failure of structural members typically occurs after experiencing plastic and nonlinear behavior.
- When we want to design a structure using a performance-based approach, if the expected performance involves nonlinear deformations, nonlinear analysis will be required. For more information on performance-based assessment and design, we have dedicated a separate article on the OpenSees Home website.
When does a structure enter the nonlinear inelastic behavior region?
The occurrence of nonlinear behavior is a natural mechanism that the material forming the structure adopts to dissipate the energy applied to it from external sources. In other words, without utilizing the capacity for plastic behavior in structures, energy dissipation caused by intense excitations like earthquakes would be difficult or even impossible. If you’re interested in understanding the role of the structure’s plastic response in controlling its vibrations from a mathematical perspective, we invite you to read the article published by Dr. Seyed Alireza Jalali and his group of authors on this topic. In part of this article, the calculation of various energy modes at different time intervals of the dynamic response of bridges under seismic loading is performed using the customized version of OpenSees software, developed by OpenSees House. Reading the theoretical and mathematical foundations related to various energy modes and the process of converting them into each other in the structure’s vibration process, as well as the results obtained for the studied bridge, will certainly be interesting for those who are interested in publishing credible research articles.
What we present in this writing aims to introduce the structure as a system to which input energy is applied, and the structure is responsible for dissipating this energy, or in other words, converting it into thermal modes that can radiate to the surrounding environment. The energy applied to the structure, if not dissipated by a proper mechanism, will be converted into kinetic energy, resulting in uncontrolled vibrations of the structure, progressive displacements, and potentially failure. This energy-based perspective is an effective and valuable approach, through which many of the structural behavior complexities can be analyzed and interpreted.
Nonlinear Structural Analysis Based on Energy
In order to utilize the energy-based approach in structural analysis, the energy absorption modes in structures must first be identified. The mechanisms that contribute to energy dissipation or absorption can be classified into several groups.
- Hysteretic energy absorption of structural members
- Hysteretic energy absorption of damping devices
- Viscous energy absorption of intrinsic members
- Viscous energy absorption in damper devices
- Energy absorption through out-of-phase oscillation of Tuned Mass Dampers (TMD)
Hysteretic energy absorption is one of the fundamental modes of energy dissipation, relying on loading-unloading cycles within the nonlinear range. As shown in the figure below, when a member undergoes a nonlinear cycle, the “work done on the member” during the loading path (the energy entering the member) is unequal to the “work done by the member” during the unloading path. As a result, a portion of the energy, which is equal to the area under the force-displacement curve and is referred to as hysteretic energy, is dissipated in the system in the form of thermal energy. It is natural that if the member’s behavior remains in the elastic range, no area will be formed under the curve, and no energy will be dissipated. Hysteretic energy is one of the most important sources of dissipating input energy in structures.
The role of damping devices in the structural response
In addition to the main structural elements (beams, columns, walls, braces), sometimes equipment is attached to the structure that is designed for nonlinear behavior and hysteretic energy dissipation. These elements are called hysteretic dampers, and are typically made from metal plates in various shapes, as well as friction-based slip plates. These dampers increase the energy dissipation capacity of the structure, reducing the need for energy absorption by the main elements, thus protecting them from failure during vibrations. Furthermore, added dampers are often easily replaceable after vibrations cease, with a reasonable cost.
In addition to the hysteretic mechanism, or cyclic nonlinear behavior, the viscous mechanism can also lead to the dissipation of input energy to the structure. This mechanism can be understood by considering door dampers in buildings. It is also applied in damper devices that can be attached to structures, where the viscosity of fluids and the resistance they offer to moving bodies results in negative work and energy absorption by the moving member. Besides auxiliary equipment, the main structural elements also act as high-viscosity fluids that absorb energy during deformation in the elastic range. Additionally, some other parts of the structure, such as bolted connections that experience friction during loading and unloading, also exhibit energy dissipation mechanisms, though they are not directly accounted for in damping calculations. All these factors contribute to the overall inherent viscous damping of the structure.
The explanation regarding energy absorption through TMD (Tuned Mass Dampers) oscillation will be addressed in other materials presented in the OpenSees repository. The calculation and prediction of energy absorbed through viscous mechanisms and by TMDs are not directly related to the nonlinear behavior of the structure. However, their accurate prediction, when the role of a serious partner, hysteretic energy, is ignored in the analysis process, is essentially meaningless. In other words, in real conditions, the energy distribution between different modes results from the competition between the mechanisms supplying them, and if one of the competitors is mistakenly overlooked, the outcome of the competition will not be predicted correctly.
Complexities of Nonlinear Behavior of Structures
The competition between different structural elements (beams, columns, braces, dampers, or shear walls) located in various floors to absorb the input energy will ultimately determine the structural response. The overall result of this competition is the sequence of yielding in different structural members. You have likely heard the term “soft story” and its role in structural collapse, or perhaps studied it in other sections of the OpenSees repository. This phenomenon is the result of yielding occurring first in a specific floor and concentrating deformations there due to a greater capacity to absorb hysteretic energy. The strong-column, weak-beam principle is another important example that prevents a destructive yielding sequence where the column yields before the beam and absorbs the hysteretic energy. Accurate prediction of the softening distribution mechanism as plastic zones form and the subsequent chain of events can all be addressed in nonlinear structural analysis.
Based on the above content, nonlinear analysis examines the simultaneous effects of complex phenomena, each of which has a profound impact on the behavior of structures under severe loads. Some of these phenomena include the sequence (order of occurrence) of member yielding, failure and removal of members, and the absorption and dissipation of input energy by them. Accurate prediction and modeling of the behavior of each member, as well as careful consideration of its effect on the overall structural behavior, is the responsibility of a nonlinear structural behavior evaluator. Dealing with the weaknesses and strengths of member models, analysis algorithms, and confronting analysis divergence are other challenges of nonlinear structural analysis.