
### Numerical solution to El Centro ground motion

Now that we've _proven_ our code, we can easily apply it to simulate the response of the SDoF system to the El Centro ground acceleration [downloaded previously](/course/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/overview-of-earthquake-ground-motion). We'll start by importing the data. Make sure to place the csv file in the same folder as your Jupyter Notebook for the import code below to work. Otherwise, just provide the path to wherever you store the csv. file.

```python
#Import data
from numpy import genfromtxt #For importing data from csv
data = genfromtxt('El-Centro_NS.csv', delimiter=',')
time = data[:,0] #(s) Time
time[0] = 0 #Remove nan from initial time value
delta_t = time[1] - time[0] #(sec) The timestep
acc_g = data[:,1] #(g) Acceleration normalised by acceleration due to gravity, g
acc = acc_g*9.81 #(m/s^2)
```

In this case we obtain the effective inertia force simply by scaling the negative of the ground acceleration by the mass. This really emphasises the efficiency of the effective force formulation of the problem. We'll use the same approach for MDoF systems later in the course.

```python
Peff_EC = -m*acc #(N)
pos_EC, vel_EC = calculateResponse(time, Peff_EC) #Call the function
```

Next, we can plot the ground acceleration and the resulting structural displacement.

```python
#Plotting
fig, axes = plt.subplots(figsize=(12,10),nrows=2,ncols=1)
axes[0].plot(time,acc,'-',label='Ground acceleration')
axes[0].set_xlim([0,30])
axes[0].set_ylabel('Acceleration ($m/s^2$)')
axes[0].set_title('Ground acceleration')
axes[0].grid()

axes[1].plot(time,pos_EC,'-',label='Relative structural displacement')
axes[1].set_xlim([0,30])
axes[1].set_xlabel('Time (sec)')
axes[1].set_ylabel('Displacement (m)')
axes[1].set_title('Relative structural response')
axes[1].grid()
plt.show()
```

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/realistic-ground-motion-response/img1.jpg"
	alt="Ground acceleration record (top), structural response relative to the moving support (bottom). | EngineeringSkills.com"
	width="1000"
	height="816"
	caption="Fig 1. Ground acceleration record (top), structural response relative to the moving support (bottom)."
/>

From an engineering design point of view, the relative motion of the structure is more relavent than the absolute motion. Remember that it's the additional stresses associated with this motion that need to be considered within the design. Global motion of the structure, while alarming, will not in itself generate any additional stress within the structure.

Next we'll spend some time explaining how to obtain ground velocity and displacement from an acceleration record. This just requires a simple numerical integration, but it will be a helpful tool to have at our disposal.


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