5.15. SDOF Systems in Time History Analysis

Numerical methods used for solving the SDOF response to ground motion:

• Newmark-beta method
• Wilson-θ method
• Runge-Kutta method

The response of a Single Degree of Freedom (SDOF) system to ground motion during seismic events can be effectively analyzed using various time-stepping numerical methods. These methods approximate the system's response by breaking it down into discrete time steps, allowing for easier calculations over time as the ground motion varies. Three commonly used methods are:

1. Newmark-Beta Method: This method is widely used due to its versatility and stability. It allows for both implicit and explicit integrations of motion, making it adaptable to a range of problems.

2. Wilson-θ Method: This method provides a more stable solution for dynamic problems, especially those involving oscillatory motions, and it can be fine-tuned for different scenarios by adjusting the θ parameter.

3. Runge-Kutta Method: Known for its accuracy, this method uses multiple evaluations of the system's dynamics within each time step to provide a more precise solution, especially useful for non-linear systems.

Imagine you are trying to predict the path of a bouncing ball. Just as you would step through time in small increments—watching where the ball goes after each bounce—time-stepping methods break the seismic response into tiny segments to understand how structures react throughout an earthquake. Each method is like choosing a different camera angle to capture the bouncing ball’s trajectory: some angles give you a clearer view, while others might miss the nuances.

For realistic seismic analysis, SDOF systems are modeled with bilinear or elasto-plastic behavior.

• Hysteresis loops (force–displacement) show energy dissipation, ductility, and residual deformation.

In order to achieve a more accurate depiction of a structure’s behavior during earthquakes, SDOF systems are often modeled to account for non-linear characteristics. This includes:

1. Bilinear Models: These models represent a change in the stiffness of the structure once it reaches a certain force or displacement, commonly used to simulate yielding behavior in materials.

2. Elasto-Plastic Models: These models allow for permanent deformation (plasticity) of the material after the yield point, reflecting more accurately how real structures behave under high stress.

The key aspect of these models is the representation of hysteresis loops, which depict the relationship between the applied force and the resulting displacement over time. These loops are essential as they show how energy is dissipated during seismic events, and they provide information on ductility (how much deformation a structure can take without failing) and any permanent deformations that might occur post-event.

Think of a rubber band. When you stretch it gently, it returns to its original shape—this is like elastic behavior. However, if you stretch it too far, it may not return completely and could be permanently deformed—similar to what happens in elasto-plastic behavior. Hysteresis loops, therefore, help visualize how structures 'remember' the forces they're subjected to during an earthquake, just like a rubber band remembers its stretched position after being overextended.

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