4.4 Nonlinear dynamic analysis
3 min read•july 25, 2024
is crucial for understanding how structures behave during earthquakes. It goes beyond simple linear models, capturing complex responses like material and large deformations that can make or break a building's survival.
This analysis uses advanced modeling techniques to simulate structural damage, , and material behavior under extreme loads. and specialized solution techniques help engineers predict a building's performance and identify potential weak points before disaster strikes.
Fundamentals of Nonlinear Dynamic Analysis
Limitations of linear analysis
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- Linear analysis assumes small displacements oversimplifies complex structural behavior
- Constant stiffness throughout loading neglects material property changes (steel yielding)
- Unable to capture material nonlinearity misses critical structural responses (concrete cracking)
- Fails to account for geometric nonlinearities (large deformations in tall buildings)
- Cannot accurately predict P-Delta effects destabilizing structures under lateral loads
Modeling nonlinear structural behavior
- simulates localized damage in frame structures
- approach idealizes nonlinear behavior at element ends
- approach allows for spread of inelasticity along element length
- define hinge behavior (elastic-perfectly plastic)
- represents energy dissipation through cyclic loading
- Characterizes material behavior under repeated stress reversals
- indicates energy absorption capacity (pinched vs. full loops)
- capture stress-strain relationships
- simplifies behavior with distinct yield point
- approximates post-yield stiffness (strain hardening)
- accurately represents smooth transition to plastic behavior
- Component-specific nonlinear models tailored for structural elements
- with fiber sections capture axial-flexural interaction
- modeled with multi-layer shell elements for in-plane and out-of-plane behavior
- Connections simulated with (semi-rigid behavior)
Implementation of nonlinear dynamic analysis
- Time-stepping methods solve equations of motion incrementally
- (central difference) computationally efficient but conditionally stable
- (Newmark-beta) unconditionally stable but require iteration
- widely used for structural dynamics problems
- Governing equations relate displacement, velocity, and acceleration
- Parameters and control stability and accuracy (, for average acceleration method)
- Time step size selection balances accuracy and computational cost (Δt ≤ T/10, T = fundamental period)
- Solution techniques for nonlinear equations handle material and geometric nonlinearities
- converges quadratically but requires frequent stiffness updates
- uses initial stiffness, reducing computational cost
- (BFGS) approximate stiffness updates for improved efficiency
- ensure solution accuracy
- check equilibrium (unbalanced forces < tolerance)
- monitor incremental displacements
- assess work done by unbalanced forces
Interpretation of nonlinear analysis results
- Key response parameters assess structural performance
- indicates overall deformation (roof drift ratio)
- highlight weak stories (typically limited to 2-2.5%)
- represents total lateral force demand on structure
- affect non-structural components and occupants
- criteria define acceptable behavior levels
- minimal structural damage, building remains functional
- significant damage, but low collapse risk
- severe damage, but structure remains standing
- Damage assessment evaluates structural integrity
- reveals load redistribution
- account for repeated cyclic loading effects
- Global stability indicators warn of potential collapse
- identifies structures prone to overturning
- observed through increasing displacements under constant load
- Energy dissipation analysis quantifies structural response mechanisms
- through various mechanisms (hysteretic, viscous)
- Comparison with design spectra and code requirements ensures compliance
- graphically compares demand and capacity
- correlation validates nonlinear dynamic results
Key Terms to Review (59)
ASCE 7: ASCE 7 is the standard for minimum design loads for buildings and other structures, developed by the American Society of Civil Engineers. It provides essential guidelines for assessing the impacts of various loads, including seismic forces, which are crucial for ensuring safety and performance in the design of structures in earthquake-prone areas.
Base Shear: Base shear is the total horizontal force that a building or structure experiences during seismic events, primarily caused by ground motion. It is crucial for understanding how a structure will respond to earthquakes, as it influences the design and analysis of structures, ensuring they can withstand seismic forces while maintaining stability.
Beam-column elements: Beam-column elements are structural components in finite element analysis that combine the characteristics of both beams and columns, allowing for the modeling of structures that experience both bending and axial loads. These elements are crucial in simulating the behavior of structures under various loads, particularly during nonlinear dynamic analysis where the interaction between axial and lateral forces becomes significant. Understanding how these elements behave under different loading conditions is essential for accurately predicting the response of a structure to seismic activity.
Bilinear Model: The bilinear model is a simplified mathematical representation used to describe the nonlinear behavior of structures subjected to dynamic loading, particularly during seismic events. This model accounts for the initial linear elastic response of the structure and then captures the yielding behavior as the load increases, effectively allowing for both stiffness degradation and energy dissipation. By using this approach, engineers can better analyze and predict how structures will respond under earthquake conditions, making it a critical component in the study of nonlinear dynamic analysis.
Calibration: Calibration is the process of adjusting and fine-tuning measurement instruments to ensure their accuracy and reliability. It involves comparing the output of a device to a known standard and making necessary adjustments so that the readings are precise, especially when analyzing complex systems like structures during seismic events.
Capacity Design: Capacity design is an engineering approach that ensures structures are designed to withstand seismic forces without experiencing significant damage. This concept involves intentionally designing certain components of a structure to yield or deform under extreme conditions, while other elements remain elastic and stable. By using this method, engineers can control how and where a structure will fail, enhancing safety and performance during earthquakes.
Capacity Spectrum Method: The capacity spectrum method is a seismic analysis technique that combines the capacity curve of a structure with the demand spectrum derived from seismic hazard assessments. This method allows engineers to evaluate how a structure will respond to seismic forces by comparing its inherent strength and ductility against the expected ground motion demands. By using this approach, engineers can identify potential vulnerabilities and make informed decisions on how to enhance the building’s performance during earthquakes.
Collapse prevention (cp): Collapse prevention (cp) is a design philosophy aimed at ensuring that structures remain stable and do not experience complete failure during extreme loading conditions, such as earthquakes. This approach focuses on enhancing the building's resilience by allowing it to withstand significant seismic forces while maintaining its overall integrity, preventing life-threatening situations for occupants. By prioritizing collapse prevention, engineers can design buildings that protect human lives even in severe seismic events.
Concentrated plasticity: Concentrated plasticity refers to a modeling approach used in structural engineering where nonlinear behavior, especially plastic deformations, is assumed to occur at specific locations within a structure rather than being distributed throughout. This concept simplifies the analysis of structures under dynamic loads, particularly in seismic design, by focusing on critical sections that are more likely to undergo yielding, thereby allowing for a more efficient assessment of a structure's response.
Convergence Criteria: Convergence criteria refer to the set of conditions used to determine when a numerical analysis, especially in nonlinear dynamic analysis, has sufficiently approximated a solution. These criteria are essential for ensuring that the iterative procedures used in these analyses reach a stable and reliable endpoint, indicating that further iterations will yield negligible changes in results.
Cumulative Damage Indices: Cumulative damage indices are quantitative measures used to assess the extent of damage that structures accumulate over time, especially during repeated seismic events. These indices help engineers evaluate the performance and resilience of structures under cyclic loading, enabling better predictions of structural failure and guiding repair strategies after earthquakes.
Damping energy dissipated: Damping energy dissipated refers to the energy that is lost from a vibrating system due to damping mechanisms, typically in the form of heat or sound. This concept is crucial in understanding how structures respond to dynamic loads, such as earthquakes, as it helps to reduce the amplitude of vibrations over time, enhancing the overall stability and resilience of structures.
Displacement demand: Displacement demand refers to the amount of relative movement between different parts of a structure during seismic events, highlighting how much a structural element is required to move to maintain safety and performance under dynamic loads. This concept is crucial in understanding how structures respond to earthquakes, as it helps engineers evaluate potential damage and design appropriate reinforcement strategies. Accurate assessment of displacement demand is essential for ensuring that buildings can withstand seismic forces without experiencing catastrophic failure.
Displacement-based criteria: Displacement-based criteria are design and assessment methods that focus on the expected displacements of structures during seismic events, rather than solely on forces or stresses. This approach allows engineers to evaluate a structure's performance based on its ability to undergo specific displacements, ensuring that safety and functionality are maintained during and after an earthquake. By emphasizing the deformation response of structures, this method aligns with the nonlinear dynamic analysis needed for realistic earthquake scenarios.
Distributed plasticity: Distributed plasticity refers to a modeling approach in structural engineering where the inelastic behavior of materials and components is spread over a significant volume or length, rather than being concentrated in a small area. This concept is particularly important in understanding how structures respond to large loads or extreme events, as it allows for a more realistic representation of material degradation and energy dissipation throughout the structure during nonlinear dynamic analysis.
Ductility: Ductility is the ability of a material to deform plastically before fracture, allowing structures to absorb and dissipate energy during seismic events. This property is crucial for maintaining structural integrity and safety, as it enables buildings to withstand the forces generated by earthquakes without collapsing instantly.
Dynamic instability: Dynamic instability refers to the tendency of a system to experience uncontrolled oscillations or movements in response to external forces, often leading to catastrophic failure. This phenomenon is particularly important in the analysis of structures subjected to dynamic loading, such as earthquakes, where nonlinear behaviors can significantly amplify instability effects.
Elastoplastic model: The elastoplastic model is a mathematical representation used to describe the behavior of materials that exhibit both elastic and plastic deformation under load. In this model, the material initially behaves elastically, meaning it will return to its original shape when the load is removed. However, once a certain yield point is reached, it begins to deform plastically, permanently altering its shape. This dual behavior is crucial for accurately predicting how structures respond to various loads, particularly in nonlinear dynamic analysis.
Energy dissipation: Energy dissipation refers to the process by which energy is transformed into a less usable form, often through mechanisms such as damping and friction, during dynamic loading events like earthquakes. This process plays a crucial role in structural engineering by helping to absorb and mitigate the forces exerted on structures, ensuring their stability and integrity during seismic events.
Energy-based criteria: Energy-based criteria refer to assessment methods used to evaluate the performance of structures under seismic loading by analyzing the energy absorption, dissipation, and demand in relation to the capacity of the structure. These criteria focus on how much energy a structure can withstand during an earthquake, thereby providing a more accurate understanding of its resilience compared to traditional force-based methods. Energy-based approaches emphasize the importance of considering nonlinear behaviors in structures, which is critical for effective nonlinear dynamic analysis.
Explicit methods: Explicit methods are numerical techniques used to solve differential equations, where the solution at the next time step is directly calculated from known values at the current time step. These methods are often simpler to implement and can provide a straightforward approach to time-dependent problems, particularly in the realm of dynamic analysis and advanced modeling techniques. The key feature is that they allow for the calculation of future states without needing to solve a system of equations simultaneously.
Fema 356: FEMA 356 is a guideline published by the Federal Emergency Management Agency that provides procedures for the seismic rehabilitation of buildings. It outlines the necessary methodologies for assessing existing structures and determining appropriate retrofitting techniques to improve their earthquake resilience. The document focuses on both nonlinear dynamic analysis and nonlinear static procedures, offering engineers a comprehensive framework to ensure buildings can withstand seismic forces effectively.
Finite Element Method: The finite element method (FEM) is a numerical technique used for solving complex engineering problems by breaking down a large system into smaller, simpler parts called elements. This method allows for the analysis of structures and materials under various conditions, making it essential in predicting how structures will respond to loads, including dynamic forces such as those experienced during earthquakes.
Floor accelerations: Floor accelerations refer to the rate of change of velocity experienced by the floors of a structure during dynamic loading events, such as earthquakes. This measure is crucial as it influences how forces are transmitted through the structure and affects the overall response during seismic events. Understanding floor accelerations helps engineers design buildings that can withstand these forces effectively, leading to safer structures in earthquake-prone areas.
Force Reduction Factor: The force reduction factor is a numerical value used in seismic design to account for the nonlinear behavior of structures during an earthquake. It helps to reduce the design forces applied to a structure by acknowledging that structures can dissipate energy through inelastic deformation, allowing for a more economical and efficient design while still ensuring safety during seismic events.
Force-based criteria: Force-based criteria refer to the design and evaluation methods used in structural engineering that focus on the forces acting on a structure during events such as earthquakes. This approach emphasizes ensuring that structures can withstand predefined force levels to maintain their integrity and functionality under seismic loading. By establishing acceptable limits for these forces, engineers can design buildings that are capable of enduring the dynamic impacts associated with earthquakes, thereby enhancing safety and resilience.
Hysteresis loop shape: The hysteresis loop shape refers to the graphical representation of the relationship between stress and strain in materials exhibiting nonlinear behavior under cyclic loading. This shape illustrates how a material's response differs during loading and unloading, highlighting energy dissipation, plastic deformation, and the history-dependent nature of its mechanical properties.
Hysteretic Damping: Hysteretic damping is a phenomenon in structural dynamics where energy is dissipated through the inelastic behavior of materials during cyclic loading. This type of damping plays a crucial role in the nonlinear dynamic response of structures, particularly during seismic events, as it accounts for the way materials yield and deform while still providing resistance to motion. Understanding hysteretic damping is essential for accurately predicting how structures will behave under dynamic loads and for improving their resilience against earthquakes.
Immediate occupancy (io): Immediate occupancy (IO) refers to a performance objective in structural engineering indicating that a building can remain functional and safe for occupants immediately after an earthquake event. This standard is crucial for structures that house critical operations, as it ensures that they can continue to be used without significant repairs or evacuation after seismic activities.
Implicit methods: Implicit methods are numerical techniques used to solve differential equations, particularly in dynamic analysis, where future states depend on both current and future values. These methods are characterized by their formulation where the unknown variables are located on both sides of the equation, allowing for greater stability and convergence in complex, nonlinear systems. In contexts like dynamic analysis and advanced numerical modeling, implicit methods are crucial for accurately predicting system behavior under various loading conditions.
Inelastic behavior: Inelastic behavior refers to the response of materials or structures that do not return to their original shape after being subjected to stress, meaning they undergo permanent deformation. This behavior is crucial in understanding how structures react during dynamic loading, such as earthquakes, where materials may experience significant strains that exceed their elastic limits.
Input energy from ground motion: Input energy from ground motion refers to the energy transmitted from seismic waves generated by earthquakes to structures and the surrounding soil. This energy plays a crucial role in determining how a structure responds to seismic activity, especially in the context of nonlinear dynamic analysis, which considers the material and geometric nonlinearities that can arise during significant ground shaking.
Inter-story drift ratios: Inter-story drift ratios measure the relative displacement between two adjacent floors in a building during seismic events. This term is crucial in assessing the performance of structures under earthquake loading, as it helps engineers understand how much one floor moves compared to another, which is vital for ensuring structural integrity and occupant safety.
Kinetic energy of structural masses: The kinetic energy of structural masses refers to the energy possessed by a structure due to its motion, particularly during dynamic events like earthquakes. When a building or bridge moves as a result of ground shaking, its mass is set into motion, which generates kinetic energy that can influence the overall performance and stability of the structure. Understanding this energy is crucial for designing structures that can withstand seismic forces effectively.
Life safety (ls): Life safety (ls) refers to the measures and designs implemented in structures to ensure the protection of human life during emergencies, particularly in situations like earthquakes. This concept emphasizes that buildings should be designed to minimize risks to occupants, allowing them to evacuate safely and efficiently. Ensuring life safety often involves both structural integrity and the provision of safe egress routes.
Material nonlinearity models: Material nonlinearity models describe how materials behave under loading conditions that cause them to exhibit nonlinear stress-strain relationships. These models are essential in predicting the response of structures during events such as earthquakes, where materials can undergo significant deformation and damage beyond their elastic limits.
Maximum displacement: Maximum displacement refers to the greatest distance a point on a structure moves from its original position during seismic activity. This movement is crucial in understanding how structures respond to earthquakes and is essential in designing buildings that can withstand these forces. It helps engineers evaluate the potential damage to structures and ensures that buildings can return safely to their initial position after seismic events.
Modal analysis: Modal analysis is a mathematical technique used to study the dynamic behavior of structures by identifying their natural frequencies and mode shapes. This analysis helps in understanding how structures respond to dynamic loads, like those from earthquakes, by breaking down complex motion into simpler components. By applying modal analysis, engineers can assess multi-degree-of-freedom systems, implement response spectrum methods, and conduct nonlinear dynamic analysis, ensuring that structures are designed to withstand seismic events.
Modified newton-raphson method: The modified Newton-Raphson method is an iterative numerical technique used to find approximate solutions to nonlinear equations. This method enhances the traditional Newton-Raphson approach by incorporating strategies to improve convergence, especially in scenarios where the original method may struggle, such as with highly nonlinear systems or when starting points are far from the solution. It's particularly useful in applications like nonlinear dynamic analysis, where understanding complex structural responses during seismic events is crucial.
Moment-rotation relationships: Moment-rotation relationships describe how a structural element behaves under bending moments, specifically illustrating the connection between the applied moment and the resulting angle of rotation. These relationships are crucial in understanding the nonlinear behavior of structures, as they provide insights into how materials yield and deform beyond their elastic limits during dynamic loading events.
Newmark-Beta Method: The Newmark-Beta Method is a numerical integration technique used to solve differential equations in structural dynamics, especially for analyzing the response of structures subjected to dynamic loading. This method is particularly useful in nonlinear dynamic analysis as it provides a reliable way to evaluate how structures behave under seismic events and other dynamic forces.
Newton-Raphson Method: The Newton-Raphson method is an iterative numerical technique used to find approximate solutions to real-valued functions, particularly useful for solving nonlinear equations. This method relies on the derivative of the function to predict the roots more accurately with each iteration, making it highly efficient for problems often encountered in nonlinear dynamic analysis.
Nonlinear dynamic analysis: Nonlinear dynamic analysis refers to the evaluation of structural response under time-varying loads, particularly seismic forces, considering the inelastic behavior of materials and structural components. This approach captures the complex interactions between a structure and seismic waves, allowing engineers to assess performance beyond elastic limits. It connects to various essential concepts, such as the incremental dynamic analysis method, which systematically evaluates a structure's response over increasing ground motion intensities, and detailing requirements that ensure ductile behavior during extreme loading events.
Nonlinear springs: Nonlinear springs are mechanical components whose force-displacement relationship does not follow a straight line, meaning the stiffness of the spring changes with deformation. This nonlinearity can arise from material properties, geometric factors, or boundary conditions, making them essential in modeling real-world systems where simple linear models fall short. They play a significant role in dynamic analysis, especially in understanding how structures behave under varying loads, such as during earthquakes.
P-delta sensitivity ratio: The p-delta sensitivity ratio is a measure that quantifies the influence of lateral displacements on the internal forces of a structural system, particularly under lateral loads such as earthquakes. This ratio helps in understanding how much additional moments are generated due to these lateral displacements, which is crucial in nonlinear dynamic analysis where structures may behave unpredictably under extreme conditions.
Performance-Based Design: Performance-based design is an approach to earthquake engineering that focuses on meeting specific performance objectives during seismic events, rather than solely adhering to prescriptive building codes. This method emphasizes the functionality and safety of structures, allowing engineers to evaluate how buildings will perform under different seismic scenarios and ensuring that they meet the desired levels of resilience and safety.
Plastic hinge formation sequence: Plastic hinge formation sequence refers to the progression of localized plastic deformations in structural elements when subjected to seismic or cyclic loading. This sequence is critical for understanding how structures respond beyond their elastic limits, allowing for energy dissipation during events like earthquakes, which is a key aspect of nonlinear dynamic analysis.
Plastic hinge modeling: Plastic hinge modeling is a method used in structural analysis that represents the behavior of beams and frames under inelastic conditions, particularly during significant loading events such as earthquakes. This approach simplifies the complex nonlinear behavior of materials by assuming that plastic hinges form at specific points, allowing for rotation and energy dissipation while maintaining equilibrium. This concept is crucial in understanding how structures behave during nonlinear dynamic analysis, particularly in predicting failure mechanisms and ensuring safety.
Pushover analysis: Pushover analysis is a nonlinear static analysis method used to evaluate the seismic performance of structures by applying a gradual lateral load until failure occurs. This technique helps engineers understand how a structure will respond to seismic forces, identifying potential weaknesses and assessing ductility, which is essential for effective seismic design.
Quasi-newton methods: Quasi-Newton methods are a category of iterative optimization algorithms that aim to find local maxima or minima of a function without needing to compute the Hessian matrix directly. Instead, these methods build up an approximation to the inverse Hessian matrix, allowing for more efficient calculations while still maintaining convergence properties similar to those found in Newton's method. This approach is particularly valuable in nonlinear dynamic analysis as it facilitates solving complex optimization problems arising from the behavior of structures during seismic events.
Ramberg-Osgood Model: The Ramberg-Osgood model is a mathematical representation used to describe the nonlinear behavior of materials under cyclic loading, particularly focusing on their stress-strain relationship. This model is crucial in understanding how materials, especially soils and structures, respond to dynamic loads during events like earthquakes, capturing both elastic and plastic deformations effectively.
Shear Walls: Shear walls are structural elements designed to resist lateral forces, primarily from wind or earthquakes, by providing stiffness and strength to a building. These walls are typically vertical and made from reinforced concrete or masonry, allowing them to effectively transfer horizontal loads down to the foundation. Their role is crucial in multi-degree-of-freedom systems, where they enhance stability and reduce sway during seismic events. Additionally, shear walls can be analyzed under nonlinear dynamic analysis to assess their performance during extreme loading scenarios, making them essential for retrofit strategies in existing structures.
Strain energy stored in elastic deformation: Strain energy stored in elastic deformation refers to the energy accumulated within a material when it is deformed elastically under stress. This energy is recoverable and is proportional to the amount of deformation, making it a key concept in understanding how materials behave when subjected to forces. In the context of nonlinear dynamic analysis, this term highlights how structures can absorb and release energy during seismic events, affecting their performance and stability.
T. Paulay: T. Paulay is a prominent figure in the field of earthquake engineering, best known for his contributions to the understanding of nonlinear dynamic analysis of structures under seismic loading. His work emphasizes the importance of accurately modeling structural behavior during earthquakes, particularly focusing on how materials and systems respond when they undergo large deformations, which is critical for designing safer buildings and infrastructure.
Time-History Analysis: Time-history analysis is a method used in earthquake engineering to assess the dynamic response of structures to seismic events by applying recorded ground motion data over time. This approach allows engineers to capture the effects of varying intensity and frequency of seismic forces on structures, providing insights into their behavior during actual earthquake scenarios.
Time-stepping methods: Time-stepping methods are numerical techniques used to solve differential equations by breaking the time domain into discrete steps. This approach allows engineers to analyze dynamic systems over time, making it particularly useful in nonlinear dynamic analysis where systems may exhibit complex behaviors under loading conditions. These methods enable the evaluation of response histories for structures subjected to dynamic forces, such as earthquakes.
Verification: Verification is the process of ensuring that a model, analysis, or simulation accurately represents the intended real-world system and its behavior under specified conditions. This step is crucial as it confirms that the assumptions, calculations, and algorithms employed are valid, ultimately leading to trustworthy results in assessing structural performance during seismic events.
Walter C. Z. Wong: Walter C. Z. Wong is a prominent figure in the field of earthquake engineering, recognized for his contributions to nonlinear dynamic analysis. His work focuses on developing advanced methodologies to assess the seismic performance of structures under varying conditions, emphasizing the importance of accurately modeling material behavior and structural response during earthquakes.
Yielding: Yielding refers to the process where a material or structure undergoes deformation when subjected to stress, particularly beyond its elastic limit. This behavior is crucial in engineering, especially when assessing how structures respond to dynamic loads such as earthquakes. Understanding yielding helps engineers design systems that can absorb energy and prevent catastrophic failures, ensuring safety during seismic events.