Antoine Al Hannoun
Ronnie Chtcot Brito
Anissa Sinojmeri

1. Introduction

Seismic analysis stands as a crucial part within structural engineering, mainly in the design, renovation, and maintenance of various structural entities. This report goes into analysing the behavior of a five-story building under three different seismic events. With Matlab, the analysis is conducted using the accelerograms corresponding to these earthquakes. Moreover, to identify the impact of material selection on structural response, the study incorporates two distinct stiffness models, one using reinforced concrete and the other using steel. Through this investigation, we aim to gain insights into structural dynamics and material behavior under seismic loading conditions.

2. Structure

The figure below illustrates a five-story building, with each story measuring a height of = 12 ft. For the columns, two structural materials are employed: firstly, steel with a stiffness of K $_{}$ = 31.54 kips/inch, as specified in Chopra's book. Additionally, reinforced concrete is utilized, characterized by a strength class of = 45 MPa. The stiffness of concrete K is determined by multiplying the elasticity modulus ratio of concrete to steel which is $_{0.171}$ .Thus K $_{}$ is computed as kips/inch. Furthermore, we simplify the columns using springs and represent the floors as rigid masses, as depicted in the figure below.

3. Methodology

To find the building's response, our primary focus lies on understanding its deformation. Therefore, we undertake a response spectrum analysis, aiming to derive the equation of motion for the structure:

$\begin{array}{l}\displaystyle \mathbf{M}\ddot{\mathbf{w_{r}}} + \mathbf{C}\dot{\mathbf{w_{r}}} + \mathbf{K}\mathbf{w_{r}} = -\mathbf{M}\ddot{\mathbf{w_{g}}(t)}\end{array}$

Here, M denotes the mass matrix, C represents the damping matrix, K signifies the stiffness matrix, w_r denotes the relative displacement and w_g the ground displacement.

We proceed by employing modal analysis to solve this equation. First, we compute and illustrate the natural frequencies of the system. Considering a multi-degree-of-freedom (MDOF) system, we model each storey as a degree of freedom. As the seismic excitation primarily affects the structure in the lateral direction due to horizontal ground motion.

In the case of a five-storey building, we encounter five degrees of freedom. This enables us to represent the system's differential coupled equations in matrix form. We begin by calculating the eigenvectors and eigenvalues of the MDOF system, which helps us finding the system's characteristics.

This approach leads us to solve an eigenvalue problem of fifth order, which we tackle efficiently using MATLAB. By obtaining the eigenvalues and corresponding eigenvectors, we gain valuable insights into the dynamic behavior of the structure and its response to seismic forces.

Modal analysis entails a crucial step involving the transformation of motion values into the normalized coordinates of the system, denoted as . The eigenmode shapes serve as a new basis, facilitating a deeper understanding of the structural deformation behavior. These mode shapes, being orthogonal to each other with respect to the stiffness and mass matrices, exhibit what is known as eigenform orthogonality. This inherent property results in the decoupling of the equations of motion, simplifying the analysis process.

Now the system of equations become:

$\begin{array}{l}\displaystyle [\phi]^T[M][\phi]\ddot{y}(t) + [\phi]^T[C][\phi]\dot{y}(t) + [\phi]^T[K][\phi]y(t) = -[\phi]^T[M]\mathbf{r}\ddot{w_g}(t)\end{array}$

Where each equation can be solved independently for each y_i(t). Due to orthogonality we have:

$\begin{array}{l}[\phi_i]^T[M][\phi_i] = 1 \\ [\phi_i]^T[C][\phi_i] = 2D_{i}w_{i} \\ [\phi_i]^T[K][\phi_i] = w^2_{i} \\ -[\phi_i]^T[M]\mathbf{r_i} = \beta_i\end{array}$

Where β_i represents the mass participation factor and r represents the activation vector.

This is rewritten in scalar form:

$\begin{array}{l}\displaystyle \ddot{y_i}(t) + 2D_{i}w_i\dot{y_i}(t) + w^2_iy_i(t) = -\beta_{i}\ddot{w_g}(t)\end{array}$

Through this decoupling process, we derive the eigenvalues and eigenvectors by solving for the undamped free vibration problem, where damping and external loading are both zero, as the following:

$\begin{array}{l}\displaystyle \det(-w^2_i\mathbf{M} + \mathbf{K}) = 0\end{array}$

The eigenvalues denote the natural frequencies of the structure. It is crucial to maintain a significant separation between these eigenfrequencies and any harmonic loading frequencies to prevent amplification of the building's response. By carefully managing the relative positions of these eigenfrequencies and the excitation frequencies, we can mitigate the risk of resonance and optimize the structural response to seismic forces.

This theoretical background has been implemented in Matlab in order to perform the analysis.

4. Results

This chapter contains the graphical results of the natural modes, displacements and equivalent static forces of a building with five stories for two different stiffness values (representing a steel and a concrete building). We compared the response of this setup for two different earthquakes: the El Centro earthquake and the San Fernando Castaic earthquake. For the latter we looked at two different incident directions: one coming from N69W and the other from N21E, in order to obtain the influence of a 90 degree tilt of the earthquake direction. The results are presented below.

4.1. El Centro

The graph below depicts the ground acceleration caused by the El Centro earthquake plotted for the time. The following graphs show the response of the building for this accelerogram.

As can be seen in the figure above, the natural mode shapes are not dependent on the stiffness of the building since they are the same for K1 (steel) and K2 (concrete). The figure below depicts the displacements of the building caused by the El Centro earthquake. As expected, the less stiff concrete building has a higher displacement than the stiffer steel building.

The graph below shows the equivalent static forces for K1 and K2, which are of course higher for the stiffer building.

Below is a summary table with assemble values of peak response to both structures, K1 and K2, that can illustrate what was said before:

With respect to the relative displacements, the structure calculated with K2 (more flexible) has presented greater displacements of the floors for all modes, especially for modes 1 and 2. On the other hand, the static forces have shown greater results for the structure with greater rigidity since they are more resistant for the first 3 modes, but for modes 4 and 5 there is no significant difference between them.

4.2. San Fernando Castaic N69W

The San Fernando Castaic N69W earthquake causes the ground acceleration that is given in the graph below.

Since we are dealing with the same building as above, the natural mode shapes are the same as before. The figure below depicts the displacements of the building caused by the San Fernando Castaic N69W earthquake. Again, the less stiff concrete building has higher displacement than the stiffer steel building.

The graph below shows the equivalent static forces for K1 and K2, which are of course higher for the stiffer building.

Below is a summary table with assemble values of peak response to both structures, k1 and k2, that can illustrate what was said before:

In the same way as the calculations carried out with the El centro earthquake, the results have the same trend. With respect to the relative displacements, the structure calculated with K2 (more flexible) has presented greater displacements of the floors for just the first mode, for the others, they have almost the same values. On the other hand, the static forces have shown greater results for the structure with greater rigidity since they are more resistant for all modes.

4.3. San Fernando Castaic N21E

Finally, the accelerogram corresponding to the San Fernando Castaic N21E earthquake is shown in the graph below.

The natural mode shapes can again be taken from the picture in the El Centro subchapter. The figure below depicts the displacements of the building caused by the San Fernando Castaic N21E earthquake. Again, the less stiff concrete building has higher displacement than the stiffer steel building.

The graph below shows the equivalent static forces for k1 and k2.

In the same way as the calculations carried out with the previous earthquakes, the results have the same trend.

With respect to the relative displacements, the structure calculated with K2 (more flexible) has presented greater displacements of the floors for just the first mode, for the others, they have almost the same values. On the other hand, the static forces have shown greater results for the structure with greater rigidity since they are more resistant for all modes.

Below is a summary table with assemble values of peak response to both structures, K2 and K2, that can illustrate what was said before:

Again the structure calculated with K2 (more flexible) has presented greater displacements of the floors for just the first mode, for the others, they have almost the same values. On the other hand, the static forces have shown greater results for the structure with greater rigidity since they are more resistant for all modes.

5. Conclusion

To summarize, in this work we took a closer look at the response of a five storey building with two different stiffnesses to the different earthquakes: El Centro, San Fernando Castaic N69W and San Fernando Castaic N21E. From this work we can conclude that:

The modal shapes, for all modes, have not presented differences with the two rigidity values adopted for this work.
The earthquakes San Fernando Castaic N21E and San Fernando Castaic N69W do not lead to any significant differences regarding the displacements and the equivalent static forces of the building.
The static forces have shown higher values for the structure with higher stiffness.
The displacements of the main modes are bigger for a more flexible stucture (lower stiffness).

6. References

Dynamics of structures/ Chopra/ Fourth edition

Seitenhierarchie

Response Spectrum Analysis of a shear building with recorded accelerograms