Chapter 11 of 11

Seismic analysis of high-rise buildings

In the TNPSC AE Civil syllabus under Structural Analysis · 2 parts

Seismic Zoning & Design Seismic Loads (IS 1893) · Structural Dynamics & Response Spectrum

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📑 Contents (22 sections)

Part 1 of 2

Seismic Zoning & Design Seismic Loads (IS 1893)

Last reviewed 16 Sept 2026 · Facts as of 16 Sept 2026 · 10 min read

Seismic zoning of India

India is divided into seismic zones based on expected intensity of shaking. Under IS 1893 (Part 1): 2016, there are four zones:

Zone	Seismic risk	Zone factor Z	Approx. MSK intensity basis
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

Zone V includes the north-east, parts of Jammu and Kashmir, Himachal Pradesh, Uttarakhand, the Rann of Kachchh, north Bihar and the Andaman and Nicobar Islands (2016 map).
The zone factor Z corresponds to the maximum considered earthquake (MCE) in the zone; Z/2 represents the design basis earthquake (DBE).

NoteCode revision

The Bureau of Indian Standards has been revising IS 1893 (Part 1), including the seismic zonation map (a revised map with a new highest-hazard zone has been reported). Most examinations continue to use the 2016 provisions summarised here; check the current edition and zone map before use in design.

Design philosophy (IS 1893)

Minor earthquakes (more frequent, less than DBE) — structures resist without damage.
Moderate earthquakes (DBE) — no significant structural damage, though some non-structural damage may occur.
Major earthquakes (MCE) — no collapse (life safety).

Structures are therefore designed for forces much lower than elastic forces of the MCE, relying on ductility, redundancy and overstrength — accounted for by the response reduction factor R.

Design parameters

Soil types (for spectra)

Type	Soil	Typical SPT N
I	Rock or hard soil — well-graded gravels, dense sands	N > 30
II	Medium or stiff soil	N 10–30
III	Soft soil	N < 10

Importance factor I

Structure	I
Important and community buildings — hospitals, schools, emergency buildings (fire stations, police), telephone exchanges, power stations, large assembly buildings	1.5
Residential/commercial buildings with occupancy more than 200 persons	1.2
All other buildings	1.0

Response reduction factor R (examples)

Lateral load resisting system	R
RC ordinary moment resisting frame (OMRF)	3.0
RC special moment resisting frame (SMRF)	5.0
Steel OMRF / SMRF	3.0 / 5.0
Ordinary RC structural walls	3.0
Ductile RC structural walls	4.0
Dual system — ductile structural walls with SMRF	5.0

(Unreinforced masonry and other systems have lower values; OMRFs are not permitted in higher zones for buildings as per code restrictions.)

Design horizontal seismic coefficient

FormulaDesign acceleration coefficient

A_{h} = \frac{Z}{2} \cdot \frac{I}{R} \cdot \frac{S _{a}}{g}

For structures with $T \leq 0.1$ s, $A_{h}$ is not taken less than $Z /2$ , whatever the value of $R$ (IS 1893: 2016).
$S_{a} / g$ = design spectral acceleration coefficient for 5% damping, multiplied by a damping factor for other damping ratios.

Design acceleration spectrum (5% damping)

For the equivalent static method (IS 1893: 2016):

Soil type	$S_{a} / g$
I (rock/hard)	$2.5$ for $T < 0.40$ s; $\frac{1.00}{T}$ for $0.40 \leq T < 4.00$ s; $0.25$ for $T \geq 4.00$ s
II (medium)	$2.5$ for $T < 0.55$ s; $\frac{1.36}{T}$ for $0.55 \leq T < 4.00$ s; $0.34$ for $T \geq 4.00$ s
III (soft)	$2.5$ for $T < 0.67$ s; $\frac{1.67}{T}$ for $0.67 \leq T < 4.00$ s; $0.42$ for $T \geq 4.00$ s

For the response spectrum method, the initial branch rises linearly as $1 + 15 T$ for $T < 0.10$ s, then follows the same values.

Softer soils give larger spectral values at longer periods.
Damping multiplying factors (to the 5% spectrum), e.g. 2% → 1.4; 5% → 1.0; 7% → 0.9; 10% → 0.8.
Commonly adopted damping: 5% for RC and masonry, 2% for steel structures.

Fundamental natural period

FormulaApproximate fundamental period (IS 1893: 2016)

Bare RC moment resisting frame buildings (without infill): $T_{a} = 0.075 h^{0.75}$
Bare RC–steel composite MRF buildings: $T_{a} = 0.080 h^{0.75}$
Bare steel MRF buildings: $T_{a} = 0.085 h^{0.75}$
All other buildings (including MRF with masonry infill, structural walls): $T_{a} = \frac{0.09 h}{d}$

$h$ = height of building (m); $d$ = base dimension (m) along the direction of shaking.

Seismic weight

Seismic weight of each floor = full dead load + appropriate percentage of imposed load:
- 25% of imposed load for loads up to 3 kN/m²;
- 50% for loads above 3 kN/m².
Imposed load on roofs need not be considered.
Weights of walls and columns are divided between the floors above and below.
Total seismic weight $W$ = sum of floor seismic weights.

Equivalent static (seismic coefficient) method

FormulaBase shear and distribution

Design base shear

V_{B} = A_{h} W

Minimum design base shear (IS 1893: 2016): $V_{B} \geq ρ W$ , with $ρ$ = 0.7% (Zone II), 1.1% (III), 1.6% (IV), 2.4% (V).

Vertical distribution (lateral force at floor $i$ ):

Q_{i} = V_{B} \frac{W _{i} h _{i}^{2}}{\sum _{j = 1}^{n} W _{j} h _{j}^{2}}

$W_{i}$ = seismic weight of floor $i$ ; $h_{i}$ = height of floor $i$ from the base.

The parabolic distribution approximates the first mode with some higher-mode effect.
Storey shear at any level = sum of lateral forces above that level.

Part 2 of 2

Structural Dynamics & Response Spectrum

Last reviewed 16 Sept 2026 · 7 min read

Dynamic loads

A load is dynamic when its magnitude, direction or position varies with time fast enough to produce significant inertia forces. Examples: earthquakes, wind gusts, machine vibrations, moving vehicles, blasts, waves.

In dynamic analysis, inertia ( $m \overset{u}{¨}$ ) and damping ( $c \overset{u}{˙}$ ) forces are considered with elastic forces.
Degrees of freedom (DOF) — number of independent displacements needed to define the displaced position of all masses.
Lumped mass idealisation — mass concentrated at floor levels (e.g. shear building with rigid floors and axially inextensible columns → one lateral DOF per floor).

SDOF system

FormulaEquation of motion

m \overset{u}{¨} + c \overset{u}{˙} + k u = F (t)

$m$ = mass; $c$ = viscous damping coefficient; $k$ = stiffness; $u$ = displacement.

Undamped natural circular frequency $ω_{n} = \frac{k}{m}$ (rad/s) Natural period $T_{n} = \frac{2 π}{ω _{n}} = 2 π \frac{m}{k}$ Natural frequency $f_{n} = \frac{1}{T _{n}}$ (Hz)

Stiffer structures → shorter periods; heavier structures → longer periods.
For a cantilever column with a lumped top mass: $k = 3 E I / L^{3}$ ; for a column fixed at both ends (sway): $k = 12 E I / L^{3}$ ; storey stiffness = sum of column stiffnesses.

Damping

FormulaDamping relations

Critical damping $c_{cr} = 2 m ω_{n} = 2 k m$ Damping ratio $ζ = \frac{c}{c _{cr}}$ Damped natural frequency $ω_{D} = ω_{n} 1 - ζ^{2}$ (≈ $ω_{n}$ for small ζ)

Free vibration response (underdamped): $u (t) = e^{- ζ ω_{n} t} [u_{0} cos ω_{D} t + \frac{u ˙ _{0} + ζ ω _{n} u _{0}}{ω _{D}} sin ω_{D} t]$

Logarithmic decrement:

δ = ln \frac{u _{i}}{u _{i + 1}} = \frac{2 π ζ}{1 - ζ ^{2}} \approx 2 π ζ, ζ \approx \frac{1}{2 π j} ln \frac{u _{i}}{u _{i + j}}

Case	Behaviour
$ζ < 1$ — underdamped	Oscillates with decaying amplitude (all practical structures, ζ ≈ 2–7%)
$ζ = 1$ — critically damped	Returns to rest fastest without oscillation
$ζ > 1$ — overdamped	Returns slowly without oscillation

Forced harmonic vibration

For $F (t) = F_{0} sin ω t$ , the steady-state amplitude is $u_{0} = (u_{s t})_{0} \times R_{d}$ where $(u_{s t})_{0} = F_{0} / k$ .

FormulaDynamic amplification factor

R_{d} = \frac{1}{( 1 - r ^{2} ) ^{2} + ( 2 ζ r ) ^{2}}, r = \frac{ω}{ω _{n}}

At resonance ( $r = 1$ ): $R_{d} = \frac{1}{2 ζ}$
$r ≪ 1$ : $R_{d} \to 1$ (quasi-static); $r ≫ 1$ : $R_{d} \to 0$ (mass hardly moves)

Transmissibility (force transmitted to support / applied force):

T R = \frac{1 + ( 2 ζ r ) ^{2}}{( 1 - r ^{2} ) ^{2} + ( 2 ζ r ) ^{2}}

Vibration isolation ( $T R < 1$ ) requires $r > 2$ .

Earthquake excitation

With ground acceleration $\overset{u}{¨}_{g} (t)$ and relative displacement $u$ :

m \overset{u}{¨} + c \overset{u}{˙} + k u = - m \overset{u}{¨}_{g} (t) \Rightarrow \overset{u}{¨} + 2 ζ ω_{n} \overset{u}{˙} + ω_{n}^{2} u = - \overset{u}{¨}_{g} (t)

The response depends only on $ω_{n}$ (or $T_{n}$ ) and $ζ$ for a given ground motion.
Base shear $V_{b} = k u_{ma x}$ .

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