Soil Structure Interaction during Earthquake using Boundary Nonlinear Analysis

What is Nonlinear Analysis?

Before delving into nonlinearity, let's first see what a simple linear analysis entails. A linear analysis is an analysis where the stiffness of the structure remains constant throughout the process; the stiffness matrix remains the same. Consequently, a linear correlation between applied forces and deformations is observed.

On the other hand, a nonlinear analysis involves applied forces and displacements showing a nonlinear relationship; the stiffness of the system is not constant throughout the analysis.

Nonlinear analysis can be categorized mainly into three: geometric, material, and boundary nonlinear analysis. In this blog, my primary focus will be on boundary nonlinear analysis, where the stiffness of the boundary changes concerning time or loading.

Fig 1. Nonlinearity types

Most of the formulas in soil mechanics are derived assuming soil to be an elastic material. However, in reality, soil P-Y curves are not linear, and their slope changes with loading. Thus, we can conclude that soil is nonlinear. 

Current Scenario:

Engineers use different methods for seismic analysis of bridges. Linear analysis includes response spectrum and linear time history analysis, while Nonlinear Seismic Analysis includes Pushover Analysis and Nonlinear Time History Analysis.

Many engineers prefer to simplify their approach and rely on linear analysis, such as response spectrum. However, in any linear analysis, the nonlinear behavior of the soil is not accounted for. Typically, only the initial stiffness of the soil is considered for analysis, treating it as a linear spring, which tends to yield slightly conservative results in most cases.

Let's explore this further, Imagine a man pushing a wall. As he applies more force, the wall either yields or topples, reducing the force he experiences due to decreased reaction from the wall.  Now, think of the scenario where the wall is not moving; as force increases, more reactions are exerted onto the man.

The same is true for soil. During response spectrum analysis, the forces in the structure will increase since the soil is considered linear and does not yield. However, incorporating nonlinear soil behavior allows for more realistic and often more economical seismic designs.

How to Consider Nonlinear Soil Behavior during Seismic Analysis?

Now, let's address the core question: How to simulate it? Using boundary nonlinear time history analysis.

During a Boundary Nonlinear Time History Analysis, the equations of motion for the structure are solved numerically considering the change in stiffness of the structure (Nonlinear Soil behavior) concerning loading for each time step. 

In other words, a certain portion of the structure is considered nonlinear during time history analysis, in this case, the entire structure as linear except for the soil whose stiffness changes with applied load.

One of the primary challenges engineers encounter during nonlinear time history analysis is the scarcity of earthquake data, often necessitating the scaling of earthquakes.

Simulation using Midas 

Currently, in Midas Civil, the multi-linear springs are converted to linear springs during any seismic analysis like Eigen Value, Response Spectrum and Time History (both linear and nonlinear), leading to slightly conservative results in most cases. 
For example, when a bilinear curve, as shown in the figure, is used to simulate soil, the software considers only the initial curve and treats the spring as linear with a stiffness of 2000 kN/m.

We can use the General Link feature in Midas to simulate the exact soil stiffness as a workaround. 

A general link is considered a dynamic boundary nonlinear element; a nonlinear seismic time history analysis can generate results considering the soil behavior. For any linear analysis, these links exhibit a linear behavior.

Consider a straightforward three-span bridge to compare the outcomes of linear and nonlinear time history analyses, aiming to grasp the impact of soil-structure interaction during seismic assessments. Nonlinear soil behavior is replicated by establishing general links connecting the substructure node to a fixed support, representing the stiffness of the soil continuum. Time history analysis is then conducted to replicate seismic loading.

Figure 2: Sample Bridge Model with General Links simulating Nonlinear Soil Behavior

Figure 3: Sample model - Nonlinear time history bending moment results.

Figure 4: Sample model - Linear Time History

Upon reviewing the results, linear time history analysis yields 1743.33 kNm, while nonlinear time history analysis produces 1253.9 kNm — 28% lower than the former. This discrepancy arises primarily because linear analysis assumes the soil to be elastic, thus offering greater stiffness to the substructure, leading to increased forces. Conversely, in boundary nonlinear time history analysis, the soil yields, resulting in reduced substructure stiffness and consequently lower seismic forces.

To sum up, bridges are longer and more complicated than ever before, and incorporating the dynamic nature of the soil is imperative to ensure resilient, safe, and economically viable seismic designs. By leveraging simulation techniques such as boundary nonlinear time history analysis, engineers can accurately capture the nonlinear behavior of soil and its impact on structural integrity.

If you want to explore further, Click on the downloadable materials and follow the step-by-step process to consider nonlinear soil behavior during Boundary Nonlinear Analysis. You can also access the model sample file and spring stiffness verification for a better understanding.

About the Author: Rohit Joseph is a structural engineer with over two years of experience at Midas IT, where he works as a Technical Support Engineer. He holds a Master’s degree in  Structural Engineering from NIT Surat. Rohit collaborates with bridge engineers worldwide, offering expert assistance with technical queries related to bridge engineering and Midas software.