
In this lecture we'll perform some relatively simple static calculations to get a sense of the overturning stability of the structure and the actions developed at the base. We start by calculating the base shear. This is simply equal to the sum of the effective inertia forces.

```python
#Calculate base shear
baseShear = np.sum(Peff,0)
```

Next we calculate the overturning moment. The correcting moment is also calculated assuming the lateral oscillation of the structure has negligible impact on its magnitude.

```python
#Calculate base moment
moment = np.zeros([N,len(acc)])
for i in np.arange(N):
    moment[i,:] = Peff[i,:]*floorHeights[i+1]

baseMomentOverturning = np.sum(moment,0)
selfWeight = sum(mArray)*9.81
baseMomentRighting = selfWeight*buildingWidth/2
```

The axial push/pull forces developed in the columns is then calculated.

```python
#Calculate base push/pull
basePushPull = -baseMomentOverturning/buildingWidth
```

And finally, some output statements.

```python
Fs = max(abs(np.round(baseShear/1000,2)))
print(f"The maximum base shear is {Fs} kN")

Mo = max(abs(np.round(baseMomentOverturning/1000,2)))
Mr = round(baseMomentRighting/1000,2)
print(f"The maximum overturning moment is {Mo} kNm compared with a righting momnet of {Mr} kNm")

F_down = round(max((0.5*selfWeight + basePushPull)/1000),2)
F_up = round(min((0.5*selfWeight + basePushPull)/1000),2)
print(f"The maximum axial foundation force is {F_down} kN")
print(f"The maximum axial foundation uplift force is {F_up} kN")
```

- The maximum base shear is $531.7 \:kN$
- The maximum overturning moment is $10790.3 \:kNm$ compared with a righting momnet of $6670.8\: kNm$
- The maximum axial foundation force is $2096.21\:kN$
- The maximum axial foundation uplift force is $-514.94 \:kN$

We can plot the base shear, overturning moment and axial forces at the base of the columns. In each case these are simply going to be scaled versions of the ground acceleration record. Plotting the correcting moment magnitude as a horizontal line allows us to easily see when the overturning moment exceeds the correcting moment. If the structure behaves as a rigid body, this suggests overturning.

```python
fig, axes = plt.subplots(figsize=(12,14),nrows=3,ncols=1)
axes[0].plot(time, baseShear/1000)
axes[0].set_xlim([0,max(time)])
axes[0].set_ylabel('Force (kN)')
axes[0].set_title('Base Shear')
axes[0].grid()

axes[1].plot(time, baseMomentOverturning/1000,
             label="Overturning moment")
axes[1].plot([0, max(time)], [baseMomentRighting/1000, baseMomentRighting/1000],'r',
             label="Righting moment")
axes[1].set_xlim([0,max(time)])
axes[1].set_ylabel('Moment (kNm)')
axes[1].set_title('Base Overturning Moment')
axes[1].legend(loc='lower right')
axes[1].grid()

axes[2].plot(time, (0.5*selfWeight + basePushPull)/1000)
axes[2].set_xlim([0,max(time)])
axes[2].set_ylabel('Force (kN)')
axes[2].set_title('Foundation axial forces (Self-weight + Base Push/Pull)')
axes[2].grid()
```

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/base-shear-and-overturning-moments/img1.jpg"
	alt="Base shear force, overturning moment and axial column force (top to bottom). | EngineeringSkills.com"
	width="1000"
	height="1086"
	caption="Fig 1. Base shear force, overturning moment and axial column force (top to bottom)."
/>


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