40.5. Distribution of Base Shear along Height

The lateral force at any floor level is given by:

$F_i = \frac{W_i \cdot h_i^2}{\sum W_j \cdot h_j^2} \cdot V$

Where:

• F_i : Lateral force at level i
• W_i : Seismic weight at level i
• h_i : Height of level i from base

This formula expresses how the lateral force (the force acting horizontally due to seismic activity) is distributed across different floors of a building. $F_i$ represents the lateral force at a specific floor (level i). The seismic weight ($W_i$) of that floor and its height ($h_i$) from the base contribute to the calculation. The sum in the denominator accounts for all the floors below it, ensuring that the force distribution reflects the building's entire height and weight. The total base shear, $V$, is applied to determine the forces acting on each individual floor based on these heights and weights.

Imagine you're on a multi-story parking garage during a shake. The floors at the top might sway more than the ground floor. This is similar to how the forces distribute; lighter (or less stable) sections of a building might experience greater shaking, just like lighter cars at the top of a tall garage moving more than those on the ground.

Where:

• W_i : Seismic weight at level i
• h_i : Height of level i from base
• j : Index running over all floors in the building

In the equation, $W_i$ is the seismic weight specific to each floor, which includes the weight of structural elements and a portion of the live load (like furniture or occupants). The height, $h_i$, measures how far that floor is from the base, which influences its contribution to the overall force experienced during an earthquake. Essentially, higher floors have a greater effect on the distribution of seismic force due to their height.

Think of how a swing moves. The higher you pull the swing, the more it will sway. Similarly, in buildings, the height of the floor affects how much force it feels during an earthquake. The floors higher up experience more movement, just like a swing at its peak.

Where:

• j : Index running over all floors in the building

The sum in the denominator totals the seismic contributions of all the floors in the building. Each floor has a certain weight and height combination, and adding these together helps calibrate how strong the forces are at different levels. This ensures that the most accurate calculations for the lateral forces at each floor are derived, reflecting the overall structure's response during seismic activities.

Imagine a packing scale, where each item (floor) has a different weight. When you put everything on, the scale shows the total weight. In our case, the 'scale' balances the effects of all the different floors to find out how forces change across the structure when an earthquake occurs.

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