Dynamics of Structures

Individual Project

1. Objective

The main objectives of this individual project are: 1) to further enhance the understanding of the numerical time integration method - Newmark's algorithm - introduced in the lectures by numerically investigating its accuracy and stability, and by implementing it using a programming language (e.g. MatLab, C, C++ etc), 2) to generate the earthquake response spectra for a particular earthquake; and 3) to undertake an earthquake/dynamic analysis of a simple structure.

2. Project description

One particular time stepping scheme ( and a value of of your choice) of the Newmark method is required to use and a proper programming language is employed to perform all the simulations. The project consists of three parts:

Part 1: Accuracy and stability analysis of Newmark's method

For a given undamped free vibration dynamic system with a zero initial displacement and a nonzero initial velocity (choose any values for and), assess the accuracy of the algorithm by plotting the error of the computed period for a wide range of different time steps. Also record the maximum kinetic energy of the system over a time integration interval of [0, 100T], where T is the natural period of the system. If a significant increase of the energy is observed, it would indicate that the algorithm is unstable for the corresponding and larger time steps.

Part 2: Response spectra of SDOF systems under a ground motion

Further consider a general damped dynamic system under ground motion:

With both zero initial displacement and zero initial velocity. The ground acceleration is taken from the El Centro earthquake shown inThe corresponding data file elcentro.dat can be downloaded from the BLACKBOARD. Note that the acceleration values in the file have been scaled down by the gravitational acceleration g. Choose one damping ratio from 2.0%, 3%, 4% or 5%. Taking values of the natural period T between 0.1 and 10 seconds and using the same time stepping scheme as you used in Part 1, determine the relative displacement and total acceleration as functions of time. Then find the peak values of displacement, velocity and total acceleration for different periods and plot the resulting displacement and acceleration response spectra.

Part 3: Earthquake analysis of a simple frame structure

Now consider a two-storey building that is supported by six square concrete columns of dimensions 0.2 x 0.2 m2, as shown in Figure 2. The total masses of the bottom and top floors are 400,000 kg and 200,000 kg respectively. Young's modulus of concrete is assumed to be E=12 GN/m2.

Use the response spectra obtained in Part 2 to compute the peak displacements and shear forces at each floor. The same solution procedure as introduced in the lectures needs to be used, i.e.: 1) determine the natural frequencies and the corresponding modes; 2) obtain the maximum displacements and forces for each mode according to the response spectra you obtained in Part 2; and 3) use the SRSS modal combination to find the peak displacements and shear forces at the two floors.

3. Project report

One report should be submitted, and should describe the numerical procedures used, the issues to be investigated, and results obtained. The results should be presented in the forms of tables and/or diagrams, and are expected to be companied by detailed explanations and discussions in the reports.