The 5% Eccentricity Rule
The 5% eccentricity rule according to Eurocode 8 (EN 1998-1): what it means, how it is calculated, where it comes from, and why it is crucial for the seismic resistance of buildings. A complete guide with practical examples.
The 5% eccentricity rule (in technical terminology “accidental eccentricity”) is a seismic design provision defined in Eurocode 8 (EN 1998-1, Article 4.3.2), according to which the calculated center of mass of each story is shifted in both directions by ±5% of the floor plan dimension perpendicular to the direction of the seismic action. The aim is to cover torsional effects caused by uncertainties – from spatial variability of seismic motion to differences between designed and actual stiffness of structural elements.
This rule, although at first glance it appears to be a technical detail, has a fundamental impact on the safety of every building in a seismic zone. It generates 32 load combinations, each of which represents a different stress scenario for the structure. In this article, we will analyze in detail its origin, mathematical basis, practical application, and the criticism it faces.
What is the 5% eccentricity rule?
Definition of accidental eccentricity
Accidental eccentricity is a design provision that requires, in the seismic analysis of a building, the center of mass of each story to be deliberately shifted from its nominal (calculated) position. This shift is performed in both horizontal directions and in both orientations (+ and −), by a value equal to 5% of the floor plan dimension of the story perpendicular to the direction of the analyzed seismic action.
In other words: if we analyze a building for seismic action in the X direction, we shift the center of mass in the Y direction by ±5% of the building’s width in the Y direction. If we analyze the action in the Y direction, we shift the center of mass in the X direction by ±5% of the building’s length in the X direction.
Why exactly 5%?
The value of 5% is not arbitrary. It is the result of decades of research, which began in the 1970s with the work of Nathan M. Newmark and Emilio Rosenblueth. Extensive studies by De la Llera and Chopra from 1994 (University of California, Berkeley) later confirmed that for most structures, uncertainties in stiffness and mass location lead to an increase in deformations averaging less than 5% for steel buildings and less than 10% for reinforced concrete buildings. The value of 5% was therefore adopted as a conservative but realistic estimate for standard buildings.
Where does the 5% eccentricity rule come from?
Pioneers of torsional analysis
The first systematic considerations of torsional effects during earthquakes were published by Nathan M. Newmark, who introduced the concept of accidental eccentricity as a way to account for torsional response caused by factors that are not explicitly modeled in static analysis. Newmark argued that even a perfectly symmetric building on paper will not actually be perfectly symmetric – there are differences between the designed and actual stiffness of elements, uneven distribution of live loads, and last but not least, the rotational components of seismic motion at the foundations.
In 1960, Emilio Rosenblueth published a groundbreaking study on the effects of the 1957 Mexico City earthquake, in which he showed that torsional effects introduce a combination of shear, bending, and normal forces into the load-bearing elements, dramatically complicating the stress state of columns. Rosenblueth was the first to quantify that it was precisely torsion – not direct horizontal forces – that was responsible for the collapse of many buildings.
The groundbreaking study by De la Llera and Chopra (1994)
The extensive study by Juan C. De la Llera and Anil K. Chopra from the Earthquake Engineering Research Center at UC Berkeley is still considered the definitive analysis of accidental torsion. The researchers:
- Analyzed the increase in building response caused by stiffness uncertainty (differences between calculated and actual element stiffness) – they found that the increase in deformations averages a maximum of 10% for reinforced concrete and 5% for steel structures.
- Analyzed the increase in response caused by rotational excitation of foundations – they found that the increase is less than 5% for torsionally stiff systems with a period above 0.5 s.
- Criticized the use of a constant eccentricity of 0.05b (or 0.10b) for all stories regardless of building height.
Monte Carlo simulation and validation
Later studies (Ramadan, Mehanny, and Mostafa, 2008) used the Monte Carlo technique to simulate accidental eccentricities in different height levels of multi-story buildings. The main finding: calculated accidental eccentricities are often lower than the typical value of 5%, especially for stories with a larger number of stories above (≥ 5). In other words – for taller buildings, 5% may be overestimated, while for low buildings it is adequate.
Why is exactly 5% eccentricity used in seismic design?
Three main sources of accidental torsion
The 5% eccentricity rule covers three fundamental sources of uncertainties that are not explicitly modeled in standard seismic analysis:
1. Uncertainty in mass localization — Live load (people, equipment, storage supplies) is never distributed perfectly evenly. The center of mass of the actual load can differ from the geometric center of the story by more than 5%, especially in buildings with an open floor plan (open-space offices, halls).
2. Spatial variability of seismic motion — An earthquake does not produce identical motion at all points of the foundation soil. Waves pass through the foundations gradually, which generates rotational components around the vertical axis. This effect is more pronounced in buildings with a large floor plan (> 50 m).
3. Differences between designed and actual stiffness — Concrete in one corner of a building can have a modulus of elasticity 10–15% different from concrete in the opposite corner. Steel profiles have manufacturing tolerances. Masonry infills (which are often not modeled) can significantly increase the stiffness of some frames.
How does Eurocode 8 define accidental eccentricity?
Exact wording of EN 1998-1
Article 4.3.2(1)P of Eurocode 8 (EN 1998-1:2004) states:
(1)P To account for uncertainties in mass localization and in the spatial variability of seismic motion, the calculated center of mass of each story i shall be considered as displaced from its nominal position in each direction by accidental eccentricity:
e_ai = ±0,05 · L_i
where:
- e_ai is the accidental eccentricity of the mass of story i from its nominal position, applied in the same direction on all stories,
- L_i is the floor plan dimension of the story perpendicular to the direction of seismic action.
The designation (P) means that it is a principle – a binding provision from which one cannot deviate. The national annex (STN EN 1998-1/NA) may specify supplementary conditions, but leaves the value of ±5% unchanged.
How is accidental eccentricity calculated according to EN 1998-1?
Practical calculation procedure
The procedure for calculating accidental eccentricity can be summarized in these steps:
- Determination of floor plan dimensions — For each story, we determine the dimensions L_x (in the X direction) and L_y (in the Y direction).
- Calculation of accidental eccentricity — e_ai,x = ±0.05 × L_x, e_ai,y = ±0.05 × L_y.
- Determination of the four center of mass positions — (+e_x, +e_y), (+e_x, −e_y), (−e_x, +e_y), (−e_x, −e_y).
- Calculation of the accidental torsional moment — M_ai = e_ai × F_i, where F_i is the horizontal seismic force acting on story i.
Calculation example for various floor plans
| Floor plan dimension (L_x × L_y) | e_ai,x = ±0.05·L_x | e_ai,y = ±0.05·L_y | Max. center of mass shift (diagonal) |
|---|---|---|---|
| 10 m × 15 m | ±0.50 m | ±0.75 m | 0.90 m |
| 15 m × 20 m | ±0.75 m | ±1.00 m | 1.25 m |
| 20 m × 30 m | ±1.00 m | ±1.50 m | 1.80 m |
| 30 m × 40 m | ±1.50 m | ±2.00 m | 2.50 m |
| 50 m × 60 m | ±2.50 m | ±3.00 m | 3.91 m |
The larger the building floor plan, the greater the absolute shift of the center of mass – and the greater the accidental torsional moment. For large-area halls (e.g., 50 × 60 m), the center of mass shift represents almost 4 meters, which can dramatically change the distribution of internal forces on perimeter columns and walls.
What is the difference between natural and accidental torsion?
Two sources of torsional effects – comparison
| Criterion | Natural torsion | Accidental torsion |
|---|---|---|
| Cause | Mismatch between center of mass (CM) and center of stiffness (CR) – static eccentricity e_s | Uncertainties in mass position, stiffness variability, rotational motion of foundations |
| Origin | Architectural design, layout of load-bearing elements | Random, unpredictable factors |
| Calculation | e_s = distance CM – CR (calculated from geometry and stiffness) | e_ai = ±0.05 × L_i (normative value) |
| Sign | Unambiguous (determined by position of CM relative to CR) | Bidirectional (±), creates 4 combinations |
| Design eccentricity | e_d = α·e_s + β·b (combined) | Included in the second term ±β·b |
| Height dependence | Yes (stiffness changes with story) | Standard prescribes a constant value |
| Elimination | Possible by symmetric design (CM = CR) | Cannot be eliminated – always considered |
Key insight: Natural torsion can be minimized with good architectural-structural design (symmetric floor plan, even distribution of load-bearing walls). Accidental torsion, however, acts always – even on a perfectly symmetric building – and therefore the standards require it to be considered without exception.
How is the 5% eccentricity rule applied in practice?
Four center of mass positions and their consequences
When calculating according to EN 1998-1, the center of mass of each story is shifted to four different positions:
- (+e_x, +e_y) — shift in direction +X and +Y
- (+e_x, −e_y) — shift in direction +X and −Y
- (−e_x, +e_y) — shift in direction −X and +Y
- (−e_x, −e_y) — shift in direction −X and −Y
Each of these four positions generates a different distribution of internal forces in the load-bearing elements. Although the shift may seem small (0.5–3 m depending on building size), its effect on perimeter columns can be dramatic – a perimeter column on the “unloaded” side can have significantly lower stress than the one on the “loaded” side.
Combination with bidirectional seismic action
EN 1998-1 further requires the simultaneous action of seismic action in both horizontal directions:
- Dominant direction: 100% of seismic force
- Orthogonal direction: 30% of seismic force
Thus, for the dominant direction X: combination E_x + 0.3·E_y
For the dominant direction Y: combination 0.3·E_x + E_y
How many load combinations arise when considering accidental eccentricity?
Mathematical derivation of 32 combinations
The total number of combinations is the product of three independent factors:
- 4 center of mass positions (±e_x, ±e_y in four combinations)
- 2 dominant directions (X or Y)
- 4 sign variants (± dominating direction × ± orthogonal direction)
4 × 2 × 4 = 32 seismic combinations.
Overview of all 32 combinations
| Center of mass position | Dominant direction | Combinations |
|---|---|---|
| 1: (+e_x, +e_y) | X | +E_x + 0.3·E_y, +E_x − 0.3·E_y, −E_x + 0.3·E_y, −E_x − 0.3·E_y |
| 1: (+e_x, +e_y) | Y | +0.3·E_x + E_y, +0.3·E_x − E_y, −0.3·E_x + E_y, −0.3·E_x − E_y |
| 2: (+e_x, −e_y) | X | +E_x + 0.3·E_y, +E_x − 0.3·E_y, −E_x + 0.3·E_y, −E_x − 0.3·E_y |
| 2: (+e_x, −e_y) | Y | +0.3·E_x + E_y, +0.3·E_x − E_y, −0.3·E_x + E_y, −0.3·E_x − E_y |
| 3: (−e_x, +e_y) | X | +E_x + 0.3·E_y, +E_x − 0.3·E_y, −E_x + 0.3·E_y, −E_x − 0.3·E_y |
| 3: (−e_x, +e_y) | Y | +0.3·E_x + E_y, +0.3·E_x − E_y, −0.3·E_x + E_y, −0.3·E_x − E_y |
| 4: (−e_x, −e_y) | X | +E_x + 0.3·E_y, +E_x − 0.3·E_y, −E_x + 0.3·E_y, −E_x − 0.3·E_y |
| 4: (−e_x, −e_y) | Y | +0.3·E_x + E_y, +0.3·E_x − E_y, −0.3·E_x + E_y, −0.3·E_x − E_y |
Practical note: Modern structural software (SCIA Engineer, Dlubal RFEM, Autodesk Robot, Graitec Advance Design) can generate these 32 combinations automatically. However, the designer must understand their origin – especially when manually checking results, to identify which combination is decisive for which load-bearing element.
Are there simplified methods for accounting for accidental eccentricity?
Simplified method according to EN 1998-1, Art. 4.3.3.2.4
Eurocode 8 offers a simplified method for buildings that meet the criteria of regularity in plan. Instead of explicitly shifting the center of mass and calculating 32 combinations, the forces in the load-bearing elements can be multiplied by a coefficient δ:
δ = 1 + 0.6 · (x / L_e)
where:
- x is the distance of the assessed element from the center of the floor plan in the direction perpendicular to the seismic action,
- L_e is the distance between the two outermost load-bearing elements in the same direction.
This method is conservative – it overestimates torsional effects for elements near the center and can lead to oversizing of internal load-bearing elements. However, it is significantly faster, especially for preliminary designs or for buildings with simple geometry.
Conditions of applicability
The simplified method can only be used if all of these conditions are met:
- The building is symmetric in plan (CM ≈ CR).
- The floor plan meets the criteria of regularity according to EN 1998-1, Art. 4.2.3.2.
- The analysis is performed spatially (3D model).
- The effects of accidental eccentricity are not determined by a more accurate method (e.g., by shifting the center of mass in the model).
What are the limitations and criticism of the 5% eccentricity rule?
What the research says
Despite wide acceptance, the 5% eccentricity rule faces professional criticism:
1. Constant value for all stories. Research by Ramadan, Mehanny, and Mostafa (2008) showed that accidental eccentricities calculated from Monte Carlo simulations are lower than 5% on stories with a larger number of floors above. For 15- and 20-story buildings, the effective accidental eccentricity can be less than half the normative value. The use of a constant 5% at all levels thus leads to oversizing of upper stories.
2. Insensitivity to the ratio of torsional to lateral stiffness. De la Llera and Chopra (1994) showed that the increase in response due to accidental torsion strongly depends on the frequency ratio ω_t/ω_l (torsional vs. lateral). Torsionally stiff buildings with a high frequency ratio are much less sensitive to accidental eccentricity than torsionally flexible systems. The value of 5% ignores this dependence.
3. Underestimation for very flexible systems. For short-period, torsionally flexible systems (T < 0.5 s) with a large floor plan, 5% may be insufficient. Rotational excitation of foundations in these buildings causes an increase in response that may exceed the 5% threshold.
4. Difference between static and dynamic analysis. De la Llera and Chopra identified discrepancies between the increase in response predicted by code static analysis (center of mass shift + static forces) and dynamic analysis (center of mass shift in 3D model). The differences are greatest at a low frequency ratio and can be of the same order as the intended response increase itself – which in practice means that static and dynamic analyses are not mutually consistent.
How does the 5% eccentricity rule differ in various international standards?
International comparison
| Standard | Accidental eccentricity value | Amplification factor | Note |
|---|---|---|---|
| EN 1998-1 (Eurocode 8) | ±5% L_i | None | Applied equally to all stories |
| ASCE 7-22 (USA) | ±5% L_i | Yes, A_x | Amplified for torsionally irregular buildings: A_x = [δ_max / (1.2·δ_avg)]² |
| NTC 2018 (Italy) | ±5% L_i | None | Identical to Eurocode 8 |
| IS 1893:2016 (India) | ±5% L_i (default) | None | Allows specifying a different value |
| ECP-201 (Egypt) | ±5% L_i | None | Identical to international practice |
| Some national standards | ±10% L_i | None | More conservative approach, used in some Asian countries |
| New Zealand (NZS 1170.5) | ±5% L_i | Yes | Dynamic analysis preferred over static |
Key difference: ASCE 7 and the A_x factor
The American standard ASCE 7-22 uses the basic value of ±5%, but introduces the amplification factor A_x for torsionally irregular buildings. If the maximum edge displacement of the story δ_max exceeds 1.2 times the average displacement δ_avg, the accidental eccentricity is multiplied by the factor:
A_x = (δ_max / 1.2·δ_avg)², where A_x ≤ 3.0.
This approach reflects the above-mentioned criticism – that torsionally flexible systems are more sensitive to accidental eccentricity and need a higher value.
Frequently asked questions (FAQ)
Is the 5% eccentricity value mandatory for all buildings?
Yes. Article 4.3.2(1)P of Eurocode 8 is designated as a principle (P) – it is a binding provision that applies to all buildings in seismic zones. The only exception is buildings in areas with very low seismicity (reference acceleration a_gR < 0.04 g), where the effects of earthquakes can be neglected entirely.
How does the application differ in static and dynamic analysis?
In static analysis (equivalent static force method), accidental eccentricity is accounted for by applying an accidental torsional moment M_ai = e_ai × F_i at the center of mass of each story. In dynamic analysis (modal or time-history), the center of mass of each story is physically shifted in the 3D model by ±5%. Dynamic analysis is more accurate – especially for irregular and tall buildings.
Can accidental eccentricity be neglected for very low buildings?
No. The standard does not provide an exception for low buildings. On the contrary – research shows that for 1–4 story buildings, the effect of accidental eccentricity can be more significant than for tall buildings, because in them the first lateral mode dominates without a pronounced torsional component.
Does accidental eccentricity affect foundation design?
Yes, indirectly. The torsional moment generated by accidental eccentricity changes the distribution of vertical reactions into the foundation structures. Edge footings and strips on the “loaded” side are more stressed. When designing foundations, it is recommended to use the envelope of all 32 combinations.
Is there software that automatically generates 32 combinations?
Yes. Most modern FEM software for static and dynamic analyses – SCIA Engineer, Dlubal RFEM, Autodesk Robot Structural Analysis, Graitec Advance Design, ETABS, MIDAS Gen – includes a module for automatic generation of seismic combinations including all four center of mass positions. The designer enters only the basic parameters (spectrum, direction, eccentricity) and the software generates all 32 combinations. Despite automation, it is essential that the designer understands the logic behind the combinations – especially when checking for anomalous results.
This article is prepared according to EN 1998-1:2004 (Eurocode 8), STN EN 1998-1/NA, as well as according to the scientific works of De la Llera & Chopra (1994), Ramadan, Mehanny & Mostafa (2008) and other cited sources. It provides the current state of knowledge as of June 2026.
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