Vertical Distribution of
Seismic Base Shear
Learn exactly how earthquake forces are distributed floor-by-floor using the Qi formula — with interactive calculator, worked examples, and visual aids.
📋 Contents
The Big Picture — What is Base Shear?
🌍 Seismic Ground Motion
During an earthquake, the ground moves. Because of inertia, a structure tends to stay put while its base is dragged. This creates a horizontal inertia force at every mass point of the building.
🏗️ Total Base Shear (VB)
IS 1893 allows us to lump all seismic forces into a single horizontal resultant at the base called Base Shear (VB). This is computed using the Equivalent Static Analysis method.
📤 The Distribution Problem
Once VB is known, it must be shared among all floors. Each floor gets a portion Qi. The sum of all Qi must equal VB.
Why Distribute Forces Vertically?
In a real earthquake, every floor with mass experiences an inertia force. Lower floors carry more total weight above them, but upper floors move MORE (higher acceleration in the first mode shape). IS 1893 reflects this with a parabolic distribution:
- Floors with higher height AND higher weight attract more lateral force.
- The distribution is not uniform — it follows a triangular-to-parabolic pattern.
- This represents the first mode response of the building — the dominant mode in most regular buildings.
Physical Intuition
Think of a flagpole in the wind. The base resists ALL the wind force (= VB), but the force is felt most at the top. A building behaves similarly under ground shaking — upper floors whip back and forth, experiencing higher acceleration than lower floors.
The Qi Formula — IS 1893 Clause 7.7.7
IS 1893 (Part 1): 2016, Clause 7.7.7 gives the lateral force at floor i as:
The h² term comes from the assumption that the building vibrates in its first (fundamental) mode, and the first mode shape approximation is a straight-line (linear) displaced shape.
The inertia force at floor i is proportional to: mass × acceleration. The modal acceleration in the first mode is proportional to the mode shape ordinate, which is proportional to height (φᵢ ∝ hᵢ).
Modal participation × mode shape ordinate × mass gives: Wᵢ × hᵢ in numerator. When normalized by total Σ Wⱼhⱼ and the mode shape normalisation introduces another hᵢ, the combined effect becomes Wᵢhᵢ².
Wᵢ is the seismic weight of floor i. By Newton’s Second Law, Force = Mass × Acceleration. A heavier floor (higher Wᵢ) attracts more seismic force. The formula correctly weights each floor by both its mass and its height, so a heavy floor at a great height gets the maximum share of base shear.
Seismic weight = Dead Load of floor + Appropriate fraction of Live Load (as per Clause 7.3.1 of IS 1893:2016).
Variable Dictionary
Step-by-Step Calculation Procedure
Determine Seismic Zone & Parameters
Identify Zone (II–V), Importance Factor I, Response Reduction Factor R, and Soil Type. These set up Ah.
Compute Seismic Weight of Each Floor (Wᵢ)
Wi = Dead Load of slab + beams + columns (tributary to floor i) + λ × Live Load. Include floor finishes and partitions in DL.
Measure Heights (hᵢ) from Base
hi is the height of the floor level above the base (foundation level or plinth level). Ground floor slab may be at h = 0 m or the first level above ground.
Calculate Wi·hi² for Each Floor
Create a table: for each floor, multiply its seismic weight by the square of its height. Sum all these values to get the denominator Σ(Wj·hj²).
Compute Total Base Shear (Vᴮ)
VB = Ah × W where W = ΣWi. Use the fundamental natural period T to get Sa/g from Fig. 2 of IS 1893.
Apply the Qi Formula
For each floor: Qi = VB × [Wi·hi²] / [Σ(Wj·hj²)]. Verify that ΣQi = VB.
Apply Forces & Design
Qi is applied as a horizontal point load at each floor level. Use these forces to design structural members (beams, columns, shear walls) for the combined seismic load case.
Worked Example — 5-Storey Building
Problem: A 5-storey RC framed building has equal storey heights of 3.2 m. Seismic weight of each floor is as below. Total Base Shear VB = 650 kN. Find Qi at each floor.
| Floor (i) | Wᵢ (kN) | hᵢ (m) | hᵢ² (m²) | Wᵢ·hᵢ² (kN·m²) | Ratio (Wᵢhᵢ²/Σ) | Qᵢ (kN) |
|---|
Building Force Visualization
Schematic showing how lateral forces increase with height. Arrow length is proportional to Qi.
Special Cases & Codal Notes
⚠️ Ground Floor (h = 0)
If height of ground slab is taken as zero, then Q₁ = 0. This is physically correct — there is no relative inertia at ground level since the base moves with the ground. All seismic force is distributed to floors ABOVE ground.
🔄 Basement Floors
For buildings with basement, the height hi for basement levels can be taken as negative or zero depending on the assumption about the fixity point. IS 1893 typically considers the base at grade level.
🏠 Irregular Buildings
For plan/vertical irregularities, Dynamic Analysis (Response Spectrum Method) is mandatory (Clause 7.8). The Qi distribution is still used but derived from modal forces rather than the static formula.
🔝 Roof vs Top Floor
The topmost mass (roof) usually attracts the maximum Qi due to maximum hn and significant Wn. Always include water tanks, equipment, and parapet loads in Wn.
Seismic weight of each floor = Full Dead Load + Appropriate fraction of Live Load
| Imposed Load (Live Load) | Fraction to Include (λ) |
|---|---|
| LL ≤ 3.0 kN/m² | 25% (λ = 0.25) |
| LL > 3.0 kN/m² | 50% (λ = 0.50) |
| Roof (accessible) | No LL included |
| Storage loads | 100% |
The Equivalent Static Analysis (and hence the Qᵢ formula) is applicable only when:
- Building height ≤ 15 m in Zones IV and V
- Building height ≤ 15 m in Zones II and III with plan irregularities
- Regular buildings up to 40 m height in Zones II and III
The storey shear at any level i (the total shear force in the storey below floor i) is:
This is used to design columns, shear walls, and connections in each storey.
🧮 Interactive Base Shear Distribution Calculator
IS 1893 (Part 1): 2016 — Clause 7.7.7 | Enter floor data and compute lateral force distribution
Key Takeaways & Common Mistakes
✅ Remember
- ΣQᵢ MUST equal Vᴮ
- Height is measured from the BASE, not floor-to-floor
- Ground floor at h = 0 gets zero lateral force
- Higher floors attract MORE lateral force (h² effect)
- Seismic weight ≠ Gravity load
❌ Common Mistakes
- Using storey height instead of cumulative height
- Including full live load (use λ-fraction only)
- Including roof live load in seismic weight
- Forgetting to include self-weight of columns & beams
- Not verifying ΣQᵢ = Vᴮ at the end
Core Formula to Remember
Qi = VB × (Wi·hi²) / Σ(Wj·hj²) — IS 1893:2016, Clause 7.7.7
Reference: IS 1893 (Part 1): 2016 — Criteria for Earthquake Resistant Design of Structures. Bureau of Indian Standards, New Delhi. | Educational resource for academic use.
