Approximate Natural Period Tₐ
A complete student guide to computing the fundamental translational natural period of oscillation for MRF buildings, structural wall buildings, and all other buildings — as per IS 1893 (Part 1) : 2016.
What is the Natural Period of a Building?
Every building has a natural tendency to vibrate at a specific rate when disturbed — like a pendulum. This rate, called the natural (or fundamental) period of oscillation T, is the time (in seconds) the building takes to complete one full back-and-forth sway cycle. It is the single most important parameter in earthquake engineering because it determines how strongly the ground shaking excites the building.
The design acceleration Sa/g from the response spectrum depends directly on T. A shorter period means more acceleration; a longer period generally means less — but the relationship is non-linear. Getting T right is essential to computing the design base shear accurately.
Exact vs. Approximate Period
The true natural period can be computed by solving an eigenvalue problem from the full structural stiffness and mass matrices. However, for design using the Equivalent Static Method (ESM), IS 1893 provides simple empirical formulae to compute an Approximate Natural Period Tₐ — derived from statistical regression of measured building periods during earthquakes worldwide.
When to Use Tₐ
IS 1893 Clause 7.6.1 mandates the use of Tₐ (from 7.6.2) for computing the design horizontal coefficient Aₕ in the Equivalent Static Method. Even when dynamic analysis is performed (Clause 7.7.3), the base shear from dynamic analysis must be compared to that from Tₐ and scaled up if it falls short.
The Equivalent Static Method (ESM) with Tₐ is applicable only for regular buildings with height less than 15 m in Seismic Zone II. For all other buildings, dynamic analysis (Clause 7.7) is mandatory — but Tₐ is still needed to set the floor on the dynamic base shear.
The Three Formula Categories
IS 1893 divides all buildings into three categories based on their lateral load-resisting system, each with its own Tₐ formula:
- k Frame material coefficient (s·m⁻⁰·⁷⁵)
- h Height of building above base, in m
- h Height of building, in m
- Aᵤ Total effective wall area (m²) in first storey
- d Base dimension in direction of shaking, in m
- h Height of building, in m
- d Base dimension in direction of shaking, in m
The value of Tᵣ (period from any formula or refined analysis) shall neither be taken as more than that given in Clause 7.6.2(a) nor less than that given in Clause 7.6.2(c). This sets upper and lower bounds on the design period.
The original Aᵤ formula in the 2016 edition has been corrected. The term (Lwᵢ/h) is now squared: Aᵤ = Σ Aᵥᵢ [0.2 + (Lwᵢ/h)²]. Always use the amended formula.
Bare MRF Buildings
A Moment Resisting Frame (MRF) building where no masonry infill panels are present or relied upon for lateral resistance. If infills exist but are not modelled for stiffness, the building uses the Category C formula instead (Clause 7.6.2(c)).
The Formula
| Building Type | Material | Coefficient k | Example: h = 15 m → Tₐ (s) |
|---|---|---|---|
| RC Moment Resisting Frame | RC | 0.075 | 0.075 × 15⁰·⁷⁵ = 0.67 s |
| RC-Steel Composite MRF | Composite | 0.080 | 0.080 × 15⁰·⁷⁵ = 0.71 s |
| Steel Moment Resisting Frame | Steel | 0.085 | 0.085 × 15⁰·⁷⁵ = 0.76 s |
How to Measure Height h
Height h is the total height of the building measured from the base to the topmost floor level. Key rules for basements:
- Basement walls connected to ground floor deck / tied between columns: Basement storeys are excluded from h (stiff basement acts like a fixed base).
- Basement walls NOT so connected: Basement storeys are included in h (the building sways from the foundation up).
The exponent 0.75 reflects empirical data: actual building periods scale with height to the power ≈ 0.75–0.80, not linearly. Taller buildings are more flexible (longer T), but the increase is sublinear because inter-storey height often decreases in upper floors and overall stiffness distribution changes.
Buildings with RC Structural Walls
When the building’s lateral resistance is provided by RC structural walls (shear walls), the formula accounts for the wall area in the first storey facing the direction of shaking. More walls → greater stiffness → shorter period.
Step 1 — Compute Effective Wall Area Aᵤ
| Symbol | Definition | Notes |
|---|---|---|
| Aᵤ | Total effective wall area in first storey (m²) in direction of shaking | Sum over all Nᵥ walls |
| Aᵥᵢ | Effective cross-sectional area of wall i in first storey (m²) | = thickness × length of wall i |
| Lwᵢ | Length of wall i in first storey in direction of shaking (m) | Lwᵢ/h ≤ 0.9 (cap) |
| h | Total building height (m), same as in 7.6.2(a) | Basement rules apply |
| Nᵥ | Number of walls in direction of shaking | Count only walls facing shaking direction |
The ratio Lwᵢ/h used in the formula shall not exceed 0.9. If a wall is very long relative to the building height, use 0.9 in place of the actual Lwᵢ/h ratio. This prevents unrealistically stiff results from very squat walls.
Step 2 — Compute Tₐ
Here, d is the base dimension (in metres) of the building at plinth level in the direction of earthquake shaking being considered. Note that d is measured in the same direction as the earthquake — so separate values of Tₐ exist for X-direction and Y-direction shaking.
Physical Meaning of the Formula
The formula says: the period decreases as Aᵤ increases (more/bigger walls → stiffer → shorter period). The minimum value (floor) is the “all other buildings” formula, which assumes no walls contribute. This is logical — walls can only reduce T, never increase it beyond the bare-frame value.
All Other Buildings
The Formula
What Falls Under “All Other Buildings”?
| Building System | Why Category C? |
|---|---|
| RC MRF with masonry infill walls | Infills stiffen the frame. Their contribution is implicitly captured by the base dimension d in the formula. Separate MRF formula (a) not applicable. |
| RC frames with masonry walls (non-structural) | Even non-structural infills add stiffness empirically; Category C captures this conservatively. |
| Masonry buildings (load-bearing walls) | Load-bearing masonry is stiff; d reflects plan extent of stiffness distribution. |
| Precast / other systems | No specific formula; general formula applies. |
The vast majority of Indian residential and commercial buildings are RC frames with unreinforced masonry (URM) infills. These use the Category C formula. URM infills significantly reduce the natural period compared to a bare frame, making these buildings stiffer and attracting more earthquake force per unit weight.
Effect of Plan Dimension d
A wider building (larger d in the direction of shaking) is stiffer in that direction → shorter period. The formula correctly captures this: T decreases as d increases (√d in the denominator).
Step-by-Step Procedure
Identify Building Category
Determine whether the lateral system is: (a) Bare MRF — no infill walls modelled; (b) RC structural wall building; or (c) All others (infilled frames, masonry, etc.). This sets which formula applies.
Measure Building Height h
Measure h from the base of the building to the topmost occupied floor level. Apply the basement inclusion/exclusion rules per Clause 7.6.2(a): if basement walls are properly connected to the ground floor deck or inter-column bracing, exclude the basement height.
Choose Direction of Analysis
Tₐ must be computed separately for X-direction and Y-direction shaking. The base dimension d (for Categories B and C) and the effective wall areas (for Category B) change with direction. Perform full calculation for both directions.
Apply the Formula
Substitute h (and d or Aᵤ as relevant) into the correct formula. For Category B, compute Aᵤ first using the amended formula, cap Lwᵢ/h at 0.9, then compute Tₐ and verify the minimum bound.
Check Bounds (Amendment 2 — Clause 7.6.2.1)
As per the 2022 amendment: Tₐ from any refined analysis or formula must be ≤ Tₐ from Category A formula [7.6.2(a)] and ≥ Tₐ from Category C formula [7.6.2(c)]. This creates a mandatory band for the design period.
Use Tₐ to Find Sa/g
Enter the computed Tₐ into the design acceleration spectrum (Clause 6.4.2) for the appropriate soil type (Hard/Medium/Soft). Read off Sa/g at 5% damping. Then compute: Aₕ = Z·I·(Sa/g) / (2R).
Compute Design Base Shear
VB = Aₕ × W, where W is the seismic weight from Clause 7.4. This completes the equivalent static analysis pipeline.
Tₐ Interactive Calculator
Select building category, enter parameters, and get instant results with step-by-step workings
📋 Step-by-Step Workings
Structural Wall Data — First Storey
| # | Wall Length Lwᵢ (m) | Wall Area Aᵥᵢ (m²) |
|---|---|---|
| 1 | ||
| 2 |
📋 Step-by-Step Workings
📋 Step-by-Step Workings
Comparative Analysis
For the same building height and plan dimension, the three categories give very different periods:
| h (m) | d (m) | MRF RC [Cat A] | MRF Steel [Cat A] | All Other [Cat C] | Ratio A/C |
|---|
Bare MRF buildings always have a longer natural period than infilled-frame buildings of the same height. A longer period generally maps to a lower spectral acceleration (for T > 0.67s on most soils), meaning bare frames attract less base shear per unit weight. This is why infilled buildings, despite being “stiffer”, need careful detailing — the stiffness increases force demand significantly.
Key Takeaways for Students
Height is the primary variable
In all three categories, building height h is the dominant parameter. Taller buildings always have a longer (larger) natural period.
Walls shorten the period
Adding RC structural walls increases lateral stiffness, which reduces Tₐ. The reduction is captured by the Aᵤ term in Category B.
Plan size also matters
For Categories B and C, a wider base (larger d) means a stiffer building → shorter period → more seismic force per unit weight.
Infills increase earthquake force
RC frames with URM infill have shorter periods (Category C) than bare frames (Category A), meaning they attract more seismic force — an important practical point for Indian buildings.
Always apply both amendments
Use the squared (Lwᵢ/h)² in the Aᵤ formula (Amd. 1), and check the period bounds from Clause 7.6.2.1 (Amd. 2).
Do it for both directions
Compute Tₐ separately for X and Y directions. Use the corresponding d in the relevant direction for each calculation.
Tₐ feeds into Aₕ → VB
The entire ESM pipeline hinges on Tₐ: Tₐ → Sa/g (from spectrum) → Aₕ → VB. A small error in Tₐ can cascade into design base shear errors.
Exponent 0.75 is empirical
The h⁰·⁷⁵ exponent comes from regression of measured building periods. It reflects that real buildings become relatively stiffer (per unit height) as they get taller.
IS 1893 (Part 1) : 2016 — Criteria for Earthquake Resistant Design of Structures, Part 1: General Provisions and Buildings (Sixth Revision). Bureau of Indian Standards, New Delhi. As amended by Amendment No. 1 (September 2017) and Amendment No. 2 (May 2022). Clause 7.6.2, pp. 21–22.
