 Help Page for the Java-Powered Simulation for Nonlinear Two-Story Buildings

# Page created by Yong Gao, 01/08/2003

# Introduction

Welcome to the help page of the Java-Powered Simulation for Nonlinear Two-Story Buildings.

It is common to design structures to behave nonlinearly under extreme load conditions, e.g. earthquakes and hurricanes. To instruct students or practitioners to better understand the effect of nonlinear behavior of buildings, our research effort has focused on the development of the nonlinear dynamic analysis virtual laboratories.

In this simulation, the structure is modeled as a two-story building. Same analytical model are used to described the behavior of each story of the structure, while different parameters are allowed for different stories. Based on different models employed, this simulator consists of four cases: (i) linear structure with each story designed as linear model; (ii) nonlinear structure with each story designed as nonlinear damping model; (iii) nonlinear structure with each story designed as hysteretic bilinear model; (iv) nonlinear structure with each story designed as hysteretic Bouc Wen model.

The user is allowed to select different model to design the structure, to change the parameters of the structure and choose different earthquake ground motions to do analysis. This simulation is intended to be used to increase understanding and provide a conceptual "feel" for various parameter changes on the performance of nonlinear structures under different excitations.

This document offers a description of how to operate and use the Java-Powered Simulation for Nonlinear Structure, a picture of which is shown below, and also the technical background of this simulation. A number of "homework" problems (or exercises) are suggested and references are provided. Fig. 1 Java-Powered Simulation Applet

How to Use the Simulation

There are four components in this simulator that can be modified by the user to design the structure and to evaluate the responses: (1) Control Panel; (2) Animation Frame; (3) Excitation Frame; (4) Response Frame. These are each identified on the above picture of the simulator.

Control Panel

Located on the far right of the simulator, this panel is used to enter specific data for the structure and excitation. This panel also contains buttons to do calculation and animation and the button which links to this help page.

### Structure Parameters

• Floor Mass: The mass of the structure. The default values are 100 tons for the first story and 90 tons for the second story.
• Stiffness: The stiffness of structure. The default values are 5000 KN/m and 4000 KN/m for the first and second story.
• Natural Frequency: The natural frequency of the structure. These parameters are not editable and they are determined corresponding to other structure parameters.
• Damping: The damping coefficient of the structure. The default values are 30 KN*s/m for the first story and 10 KN*s/m for the second story.
• Damping Ratio: The damping ratio of the structure. These parameters are not editable and they are determined corresponding to other structure parameters.

### Structure Models

• Check Boxes: These check boxes allow user to select different structure designs of which the response will be displayed. The default is to display the response of the structure with linear model for each story.
• Involution Coefficients: The nonlinear damping multiplier of nonlinear damping model. The default value is 0.5 for each story.
• Yield Displacement: The yield displacement in each story for hysteretic models (Bouc Wen and Bilinear Model). The default value is 0.02 m for each story.
• Post Yield Stiffness: The post yield stiffness in each story for hysteretic models (Bouc Wen and Bilinear Model). The default value is 1000 KN/m for each story.

### Excitation Parameters

• Maximum Amplitude: The maximum amplitude of the earthquake acceleration. The default value is the same as the original maximum acceleration of the earthquake excitation. By changing the maximum acceleration of the earthquake excitation, one can compare the responses of the earthquake records with different amplitude.
• Sinusoid Frequency: The frequency of the sinusoid excitation. The default value is 1.0 Hz.

### Response Parameters

• Response Window: Width of the excitation/response windows (in seconds) used during the animation. If you computer can't display the animation smoothly, make this number smaller.

Animation Frame

The animation frame, located in the upper left portion of the simulation window, shows the behavior of this two story structure under the specified excitation.

Structures, designed as Linear Model, Nonlinear Damping Model, Bouc Wen Model and Bilinear Model, can be animated by the appropriate selection in the left choice of this frame.

The user can also choose to animate absolute or relative motion of the structure:
• Absolute Motion: Display the absolute motion of the structure. Thus, the ground is seen moving.
• Relative Motion: Display the motion of the structure relative to the ground. Thus, the ground is seen not moving.

Excitation Frame

Located in the top center of the simulator, the excitation frame shows current excitation signal. Five historical excitation records are available for simulation:
• Sinusoidal Input: The default value of the frequency is 1.0 Hz. The default value of the maximum acceleration is 0.3g.
• El Centro Earthquake: North-south component recorded at the Imperial Valley Irrigation District substation in El Centro, California, during the Imperial Valley, California earthquake of May 18, 1940. The magnitude is 7.1 and the maximum ground acceleration is 0.3495g.
• Tokachi-oki (Hachinohe) Earthquake: North-south component recorded at Hachinohe City during the Tokachi-oki earthquake of May 16, 1968. The magnitude is 7.9 and the maximum ground acceleration is 0.2294g.
• Northridge Earthquake: North-south component recorded at Sylmar County Hospital parking lot in Sylmar, California, during the Northridge, California earthquake of Jan. 17, 1994. The magnitude is 6.8 and the maximum ground acceleration is 0.8428 g.
• Hyogo-ken Nanbu (Kobe) Earthquake: North-south component recorded at Kobe Japanese Meteorological Agency (JMA) station during the Hyogo-ken Nanbu (Kobe) earthquake of Jan. 17, 1995. The magnitude is 7.2 and the maximum ground acceleration is 0.8337g.
The user can choose to display the displacement or the acceleration signal of the excitation by proper selection of the menu.

Response Frame

There are two response frames, which are located at the bottom left corner and bottom center of the simulator. These two identical response frames facilitate the user to investigate the responses of different stories or the same story at the same time. It can show the displacement, velocity or acceleration of each story. It also can plot the displacement (or velocity) vs. shear force (or damping force, or spring force). Peak response information such as the maximum value of each response, peak reduction (= maximum peak of each case / maximum peak of Linear Model case * 100 (%)) is displayed in the bottom of this frame.

# Technical Background

Definition of the Primary Parameters

• Mass of the first floor (kg)
• Mass of the second floor (kg)
• Stiffness of the first floor (KN/m)
• Stiffness of the second floor (KN/m)
• Damping coefficient of the first floor (KN*s/m)
• Damping coefficient of the second floor (KN*s/m)
• The natural frequency of the structure can be obtained by solving the eigenvalue problem of the stiffness and mass matrix. The damping ratio of the structure can then be obtained from the eigenvalue problem nd the damping matrix of the structure.
• For the cases of hysteretic model, the elastic stiffness , post yielding stiffness and yielding force are defined as,

#### where is the yield displacement (m). Fig. 2 Restoring Force vs. Displacement

Mathematical Model

• Case I: Linear Model
The equations of motion for the structure with linear model are where, is the first floor structural displacement relative to the ground (m). is the first floor structural velocity relative to the ground (m/s). is the first floor structural acceleration relative to the ground (m/s2). is the second floor structural displacement relative to the ground (m). is the second floor structural velocity relative to the ground (m/s). is the second floor structural acceleration relative to the ground (m/s2). is the ground acceleration (m/s2).
• Case II: Nonlinear Damping Model
The equations of motion for the structure with nonlinear damping model are where, and are the nonlinear damping involution coefficients of the nonlinear damping model for the first and second story, respectively.
• Case III: Hysteretic Bouc Wen Model
The equations of motion for the structure with hysteretic Bouc Wen model are where the restoring forces. and are defined as In the above equations, and are the solutions of the following equations,  , , , and are the shape parameters for the hysteresis loops. In this case, these parameters are defined here as • Case IV: Hysteretic Bilinear Model
The equations of motion for the structure with hysteretic bilinear model are where the restoring forces. and are defined as and where and ; and are displacements relative to the center of the hysteresis loop. Fig. 3 Illustration of Displacement and Other Definitions

Spring Force: Force related to stiffness only.

Damping Force: Force related to damping only.

Shear Force: Summation of the shear and damping force.

# Homework

• Change the damping coefficients of the first and second story; and then calculate the response of the structure. Note that the damping ratio is automatically calculated when the damping coefficient is changed.
• Change the mass of the first and second floor; and then calculate the response of the structure. Note that both natural frequency and damping ratio are changed in this case.
• Change the stiffness of the first and second floor; and then calculate the response of the structure. Note that bother natural frequency and damping ratio are changed in this case.
• For the default parameters, change the involution coefficient to achieve a peak reduction of displacements greater than 35% in both floors under El Centro earthquake.
• For the default parameters, change the post yielding stiffness to achieve a peak reduction of displacements greater than 20% for Bouc Wen model and great than 35% for bilinear model for both floors under El Centro earthquak
• Change the maximum amplitude of the acceleration to 1.0 (g) and then design the structure with first floor 100 tons and second floor 90 tons as nonlinear structure. Try to achieve a peak reduction of displacements in both floors at least greater than 20% in all the cases under El Centro earthquake.
• Calculate the response of the structure and then evaluate the mitigation effect of the different models under different earthquake excitations.

# References

• Teb Belytschko and Thomas J.R. Hughes (1983). "Computational Methods for Transient Analysis", North-Holland.
• Glen V. Berg (1989). "Elements of Structural Dynamics", Prentice Hall.
• Joseph W. Tedesco, William G. McDougal and C. Allen Ross (1998). "Structural Dynamics: theory and applications", Addison-Wesley.

# Acknowledgements

The support of the National Science Foundation under Grant No. CMS 95-28083 (Dr. S.C. Liu, program director), and the support of the Multidisciplinary Center for Earthquake Engineering Research (MCEER) are gratefully acknowledged. In addition, we would like to thank Prof. Yozo Fujino of the University of Tokyo for his help in securing the Kobe and Hachinohe earthquake records.

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