Report

Report#

A building, considered as a rigid mass, is subject to an earthquake. The foundation of the building, interacting with the ground, is represented by a spring and a dashpot (damper).

The dynamic motion of the building is defined by the following differential equation:

\[ k \left(u_b(t)-u_g(t) \right) + c \left(\frac{d u_b}{d t} - v_g(t) \right) + m \left(\frac{d^2 u_b}{d t^2} \right) = 0 \]

where:

  • \(u_b\) = horizontal displacement of the building

  • \(u_g\) = horizontal displacement of the ground (= earthquake outcrop motion)

  • \(v_g\) = \(du_g / dt\) = horizontal velocity of the ground

  • \(t\) = time

  • \(k\) = (shear) stiffness between the building foundation and the ground

  • \(c\) = damping between the building foundation and the ground

  • \(m\) = mass of the building

This is how the original earthquake data looks like:

After having performed the tasks and computer exercises for this group assignment, you are requested to write a report by answering the following questions:

Question 1

Formulate the numerical expression for the acceleration \(a_g(t)\) = \(\frac{d v_g(t)}{d t}\) according to the Forward Difference scheme.

Question 2

Formulate the numerical expression for the displacement \(u_g(t)\) = \(\int{v_g(t) d t}\) according to the Trapezoidal Rule.

Question 3

Conclude on the accuracy of the calculated results from the comparison of your Python implementation with the original data (acceleration \(a_g\) and displacement \(u_g\)). Explain possible differences.

Were the results (\(u_g\), \(v_g\), \(a_g\)) according to your expectation? For example, comment on the values at the end of the recorded time series.

Question 4

Formulate the differential equation in a numerical scheme using the Central Difference scheme for the first and second derivative and by considering \(u_b(t)\) = \(u_b^i\) as the central point.

Elaborate the numerical scheme such that the solution for the building displacement at the next time step \(u_b(t + \Delta t)\) = \(u_b^{i+1}\) is expressed explicitly in the other (known) >components.

Question 5

How many initial conditions are required to solve the differential equation?

Formulate these initial conditions for the numerical scheme.

Question 6

Based on the variations of parameters k, c and m, what is the influence of individual parameters on the building response?

  • What is the influence of the stiffness k on the solution?

  • What is the influence of the damping c on the solution?

  • What is the influence of the mass m on the solution?

  • If ‘oscillations’ occur, try explain their origin.

By Ronald Brinkgreve and Anna Störiko, Delft University of Technology. CC BY 4.0, more info on the Credits page of Workbook.