Design Eccentricity
1.5eₛ ± 0.05b Explained
A complete student guide to torsional seismic design — from definitions to interactive calculations. Covers Clause 7.8 of IS 1893 (Part 1): 2016 (Sixth Revision).
Why Does Torsion Happen in Buildings?
Before jumping to formulas, let’s understand the physical problem that IS 1893 is solving.
The Centre of Mass is the point through which the resultant inertia (earthquake) force acts during shaking. It depends on the distribution of mass — heavier parts of the floor pull the CM towards them.
Clause 4.4 IS 1893 (Part 1): 2016
The Centre of Resistance is the point through which the resultant lateral stiffness of the floor acts. It depends on the distribution of lateral stiffness — stiffer columns/walls pull the CR towards them.
Clause 4.5 IS 1893 (Part 1): 2016
Visual Concept: CM vs CR on a Typical Floor
When CM ≠ CR, the earthquake force creates a twisting moment (torque) about the vertical axis — this is the torsion IS 1893 accounts for.
Earthquake force acts at the Centre of Mass (CM)
During shaking, inertia forces are distributed over the floor and their resultant acts at the CM.
CM and CR rarely coincide
Due to irregular mass distribution or irregular stiffness layout, CM and CR are offset. This offset = Static Eccentricity (eₛ).
The force at CM creates a torsional moment
Force × offset distance = Torque. This torque twists the building about the vertical axis, creating additional forces in the lateral load resisting elements.
Real-world eccentricity is amplified dynamically
Static eccentricity underestimates the actual torsion. IS 1893 applies a 1.5× dynamic amplification factor to account for dynamic coupling effects.
Accidental eccentricity is added/subtracted
Even if CM and CR are at the same point, uncertainties in mass distribution, unexpected loads, ground motion rotational components etc. cause accidental eccentricity = ±0.05b.
Design uses the WORST CASE of the two combinations
Both 1.5eₛ + 0.05b and eₛ − 0.05b are checked. The one that causes larger forces in the element governs.
Defined in Clause 4.6.2:
This is a purely geometric/structural parameter determined from the building layout. It is different for each floor.
Case 1: edᵢ = 1.5eₛ + 0.05b
The dynamic amplification (1.5×) and accidental eccentricity both act in the SAME direction, pushing the effective centre of mass further away from CR. This is the additive case.
Case 2: edᵢ = eₛ − 0.05b
The accidental eccentricity opposes the static eccentricity. This may seem to reduce the torsion — but it can reverse the direction or make another element critical.
IS 1893 (Part 1): 2016, Clause 7.8.2 states explicitly:
🔬 Why 1.5× for dynamic amplification?
During dynamic (earthquake) loading, the actual oscillation of the building amplifies the torsional response beyond what a static analysis predicts. This amplification arises due to coupling between translational and torsional modes of vibration. The factor 1.5 is a code-prescribed amplification to convert the static eccentricity to an equivalent dynamic design value.
📏 Why 0.05b for accidental eccentricity?
Even in a perfectly symmetric building, uncertainties exist: actual mass positions differ from computed ones, live load is not uniformly placed, stiffness varies with temperature, construction tolerances exist, and ground motion has rotational components. The 5% of plan dimension accounts for all these real-world uncertainties.
Clause 7.8 — Torsion Provisions
Provision shall be made in all buildings for increase in shear forces on the lateral force resisting elements resulting from twisting about the vertical axis of the building, arising due to eccentricity between the centre of mass and centre of resistance at floor levels.
The design forces calculated (as per Clauses 7.6 and 7.7.5) shall be applied at the displaced centre of mass so as to cause design eccentricity (as given by Clause 7.8.2) between the displaced CM and CR.
NOT required for: Time History Analysis (1.5 amplification is not applied; only ±0.05b is added).
| Parameter | Case 1: 1.5eₛ + 0.05b | Case 2: eₛ − 0.05b |
|---|---|---|
| Dynamic Amplification (1.5×) | ✓ Applied | ✗ Not applied |
| Accidental Eccentricity | +0.05b (adds to eₛ side) | −0.05b (opposes eₛ) |
| Resultant ed vs eₛ | Always ≥ 1.5eₛ | May be less than eₛ; can be negative |
| Critical for element at… | Far flexible edge (away from CR) | Near stiff edge; or when torsion reverses |
| Governs when… | Element is on the torsionally flexible side | Accidental reversal makes an element worse |
| Used in | SCM & RSM | SCM & RSM |
| Clause | Topic | Key Content |
|---|---|---|
4.4 | Centre of Mass (CM) | Point through which resultant inertia force acts. Depends on mass distribution. |
4.5 | Centre of Resistance (CR) | Point through which resultant internal resistance acts with no floor rotation. |
4.6.1 | Design Eccentricity (edᵢ) | Value of eccentricity used in torsion design calculations at floor i. |
4.6.2 | Static Eccentricity (eₛ) | Distance between CM and CR at floor i. |
7.8.1 | Torsion Provision | All buildings must account for torsion due to CM-CR offset. |
7.8.2 | Design Eccentricity Formula | edᵢ = max{1.5eₛ + 0.05b, eₛ − 0.05b} |
Table 5(i) | Torsional Irregularity | Max displacement > 1.2× average displacement; torsional mode period check. |
6.4.2 | Design Seismic Coefficient Ah | Ah = (Z/2)(I/R)(Sa/g); the lateral force basis. |
Once the design eccentricity is known, the design torsional moment at each floor is computed as:
This torsional moment is then distributed to each lateral load resisting element (column, shear wall, frame) in proportion to their stiffness and their distance from the CR.
A building is torsionally irregular when the maximum horizontal displacement at one edge exceeds 1.2× the average of the displacements at both edges (as per Amendment No. 2). Additional checks:
- If Δₘ is in range 1.2Δₐ to 1.4Δₐ → revise configuration AND use 3D dynamic analysis
- If Δₘ > 1.4Δₐ → structural configuration must be revised
Design Eccentricity Calculator
Compute design eccentricity per IS 1893 (Part 1): 2016, Clause 7.8.2 for one or multiple floors. The calculator shows step-by-step working and highlights the governing case.
Worked Numerical Examples
Problem: A G+4 storey RC building in Seismic Zone IV has the following data for 2nd floor:
- Static eccentricity, eₛ = 0.9 m
- Floor plan dimension perpendicular to seismic force, b = 15 m
- Analysis method: Seismic Coefficient Method
Find: Design eccentricity edᵢ.
Step-by-Step Solution
Problem: For a floor in an office building:
- eₛ = 1.2 m, b = 20 m
- Lateral seismic force at this floor, Qᵢ = 350 kN
- Method: Response Spectrum Method
Find: ed and torsional moments for both cases.
Step-by-Step Solution
Case 2 (70 kN·m) may govern for elements on the opposite side if it reverses the sign of shear.
Problem: Same building data as Example 2, but using Time History Analysis. eₛ = 1.2 m, b = 20 m.
Step-by-Step Solution (THM)
A 5-storey building (SCM). Each floor has different eccentricity and plan dimensions:
| Floor | eₛ (m) | b (m) | 0.05b (m) | 1.5eₛ+0.05b (m) | eₛ−0.05b (m) | Remarks |
|---|---|---|---|---|---|---|
| 5th | 0.5 | 10 | 0.50 | 1.25 | 0.00 | ed = 1.25 m; Case 2 = 0 (boundary) |
| 4th | 0.8 | 12 | 0.60 | 1.80 | 0.20 | ed = 1.80 m governs flexible side |
| 3rd | 1.0 | 15 | 0.75 | 2.25 | 0.25 | ed = 2.25 m governs |
| 2nd | 1.2 | 15 | 0.75 | 2.55 | 0.45 | ed = 2.55 m governs |
| 1st | 0.6 | 18 | 0.90 | 1.80 | −0.30 | ed = 1.80 m; Note negative Case 2! |
Frequently Asked Questions
No! Even if eₛ = 0 (CM coincides with CR), there is still accidental eccentricity. The formulas become:
Case 2: 0 − 0.05b = −0.05b
So a “symmetric” building still has an effective eccentricity of ±5% of its plan dimension for torsion design. This is why IS 1893 requires torsion provisions for ALL buildings.
Yes. When eₛ < 0.05b, Case 2 (eₛ − 0.05b) gives a negative value. A negative ed means the effective torsion is in the opposite direction to what the static eccentricity suggests. This can load elements on the opposite side more severely. Always check all lateral load resisting elements under both cases.
‘b’ is specifically the floor plan dimension perpendicular to the direction of seismic force. For example:
- If seismic force is applied along the X-direction → b = Y-dimension of the floor
- If seismic force is applied along the Y-direction → b = X-dimension of the floor
Since seismic analysis is done for both X and Y directions separately, ‘b’ takes a different value for each direction of analysis.
The 1.5 dynamic amplification factor is needed when using the Seismic Coefficient Method or Response Spectrum Method because these are approximate methods that use simplified (smoothed) spectral values and don’t fully capture dynamic coupling between translational and torsional modes.
In the Time History Method, the building is analysed with actual ground motion records step-by-step. The dynamic response (including mode coupling and torsional amplification) is inherently computed in the analysis itself. So the 1.5 factor would be double-counting the dynamic effect.
No. Design eccentricity is calculated independently for each floor. The CM can shift from floor to floor (if mass distribution varies), the CR can shift (if column/wall stiffness changes with height), and the plan dimension b may vary in setback buildings. Therefore eₛ, b, and hence edᵢ must be computed separately for each floor level.
For each lateral force resisting element (frame, shear wall, column), you calculate the shear force under:
- Direct shear (from translational response) + torsional shear using edᵢ from Case 1
- Direct shear + torsional shear using edᵢ from Case 2
The maximum (absolute value) of these two combinations governs the design of that element. Elements on different sides of the building will be governed by different cases.
| Property | Static Eccentricity (eₛ) | Design Eccentricity (edᵢ) |
|---|---|---|
| Definition | Actual distance CM to CR | Code-prescribed value for design |
| Basis | Structural geometry and mass | eₛ × amplification ± accidental |
| Value | Single value ≥ 0 | Two values: Case 1 and Case 2 |
| Purpose | Input to design eccentricity formula | Used to calculate torsional moment |
| Always > eₛ? | — | Case 1 always ≥ 1.5eₛ; Case 2 may be < eₛ |
When torsional irregularity is significant, IS 1893 Table 5(i) requires:
- Check if building falls under torsional irregularity (Δₘ vs Δₐ check)
- If Δₘ > 1.2Δₐ to 1.4Δₐ: revise configuration AND perform 3D dynamic analysis
- If Δₘ > 1.4Δₐ: building configuration must be revised entirely
- Use 3D modelling to properly capture torsional modes and response
The simple design eccentricity approach (Cl. 7.8.2) is a simplified method suitable for regular structures. Highly irregular structures need more rigorous treatment.
IS 1893 Reference Tables & Data
| Seismic Zone | Z | Intensity (MSK) |
|---|---|---|
| II (Low) | 0.10 | VI or less |
| III (Moderate) | 0.16 | VII |
| IV (Severe) | 0.24 | VIII |
| V (Very Severe) | 0.36 | IX and above |
| Structure Type | I |
|---|---|
| Critical / Lifeline structures | 1.5 |
| Business continuity structures | 1.2 |
| All others (residential, offices) | 1.0 |
| Scenario | Formula | Note |
|---|---|---|
| SCM or RSM — Case 1 | 1.5eₛ + 0.05b | Dynamic amplification + accidental |
| SCM or RSM — Case 2 | eₛ − 0.05b | Reversed accidental eccentricity |
| Time History — Case 1 | eₛ + 0.05b | No 1.5× (THM captures dynamics) |
| Time History — Case 2 | eₛ − 0.05b | Same as SCM/RSM Case 2 |
| Symmetric building (eₛ=0) | ±0.05b | Minimum accidental eccentricity |
| Torsional moment | Mᵢ = Qᵢ × edᵢ | Qᵢ = lateral force at floor i |
- Using b as the dimension parallel to force — b must be perpendicular to the seismic force direction
- Forgetting both cases — both ed values must be checked; using only Case 1 is incomplete
- Applying 1.5× in Time History analysis — the amplification is not required for THM
- Same ed for all floors — edᵢ must be computed separately for each floor
- Ignoring negative ed (Case 2) — negative values indicate reversal of torsion and must be checked
- Using eₛ from one direction for both X and Y — eₛ may differ for X and Y shaking directions
