Structural Dynamics — Earthquake, Wind & Base Isolation — Engineering Reference

1. At a glance

Structural dynamics is the response of structures to time-varying loads — i.e. everything that is not the static gravity case of structural-analysis. Three application clusters dominate the practising engineer’s day:

1. Earthquake engineering — design and evaluation against seismic ground shake. The regulatory stack in the US is IBC 2024 → ASCE 7-22 ch.11–22 (loads, design spectrum, system R-factors) → AISC 341-22 / 358-22 (steel) + ACI 318-25 ch.18 (concrete). In Europe Eurocode 8 (EN 1998) plays both roles. Existing buildings use ASCE 41-23 (evaluation + retrofit).
2. Wind engineering — vortex shedding, gust response, flutter, galloping, buffeting. ASCE 7-22 ch. 26–31 plus boundary-layer wind-tunnel testing for tall and special buildings.
3. Human-induced + machinery-induced vibration — pedestrian-bridge synchronization, floor liveliness, rotating-machine foundations. AISC Design Guide 11; ISO 10137; CEB-FIP Bulletin 209.

This note presupposes the Tier 1 vibration-dynamics foundation (SDOF/MDOF math, modal decomposition, damping models, Miles equation) and the Tier 1 structural-analysis foundation (matrix stiffness, load combinations, drift limits). Here we apply both together to seismic + wind + dynamic-loading problems at building, bridge, and tower scale.

Where it sits: statics → mechanics-of-materials → structural analysis (static) → structural dynamics (this note) → performance-based seismic design + advanced FEM. The mass matrix [M], stiffness matrix [K], and modal decomposition introduced in vibration-dynamics become the engine of seismic response-spectrum, time-history, and base-isolation analysis.

2. Why it matters

Every major structural dynamics event has reshaped the codes:

• Tacoma Narrows Bridge (Washington, 1940) — torsional-bending flutter at 0.2 Hz in 19 m/s wind; bridge destroyed itself within hours. Catalyzed modern aeroelastic analysis and the wind-tunnel testing industry.
• San Fernando earthquake (CA, 1971, M6.6) — Olive View Hospital and Veterans Hospital wing collapses drove the NEHRP program and the modern site-coefficient + R-factor framework.
• Mexico City earthquake (1985, M8.0) — distant epicentre, but soft lake-bed sediments amplified 1-second motion ~5×, resonating with 6-to-15-story buildings (T ≈ 1–2 s). ~400 buildings collapsed. Drove site-class amplification factors (F_a, F_v).
• Loma Prieta (CA, 1989, M6.9) — Cypress Viaduct (I-880) double-deck collapse + Bay Bridge upper-deck failure. Drove post-tensioned bridge column retrofits and ATC-32.
• Northridge (CA, 1994, M6.7) — welded steel moment-frame connections fractured brittly at the column flange weld; ~150 buildings damaged. Drove SAC Joint Venture → AISC 358-22 prequalified connections (RBS, WUF-W, BFP, etc.).
• Kobe (Hyogoken-Nanbu) (Japan, 1995, M6.9) — 6 434 dead; expressway pier failures; first major test of base-isolation (West Japan Postal Center survived with isolators while neighbours collapsed). Drove JSCE/AIJ revised provisions and explosive growth of isolation worldwide.
• Christchurch (NZ, 2011, M6.3) — close-source pulse-type ground motion + widespread liquefaction; CTV Building collapse killed 115. Drove NZS 1170.5 revisions on near-source factors.
• Tohoku (Japan, 2011, M9.0 + tsunami) — long-duration shaking with strong long-period components; tested Tokyo’s many isolated buildings (most performed well, some bearing tension uplift identified as issue).
• Mexico (Puebla) (2017, M7.1) — directly under Mexico City; ~370 dead, ~40 building collapses, again concentrated in mid-rise soft-soil zones.
• Turkey–Syria (Kahramanmaraş, 2023, M7.8 + M7.5 doublet) — > 50 000 dead; widespread soft-story and lap-splice failures in RC buildings; reignited code-enforcement debate.
• Tacoma Narrows is the canonical wind example, but Tower of Pisa pedestrian retrofits, Citicorp Center crisis 1978 (NYC; LeMessurier discovered quartering-wind under-design after construction → secret retrofit with TMD + welded plates), Millennium Bridge (London 2000; synchronous lateral excitation closed 3 days after opening), and FIU pedestrian bridge (Miami 2018; failed during PT-tendon work) are all structural-dynamics events.

Modern practice: explicit dynamic analysis is now required for (a) Risk Category III + IV buildings, (b) base-isolated structures, (c) buildings with significant vertical/torsional irregularity, (d) tall buildings (typically > 73 m / 240 ft in high seismic zones), and (e) any structure with non-standard lateral systems.

3. First principles

3.1 Equation of motion (MDOF)

The same canonical equation that appears in vibration-dynamics is the backbone of every dynamic structural analysis:

$[M]\{\ddot{x}\}+[C]\{\dot{x}\}+[K]\{x\}=\{F(t)\}$

For base-excitation (earthquake, the typical case), let x_g(t) be the ground displacement and {x} be the relative displacement of the structure with respect to the ground. The effective force vector becomes:

$[M]\{\ddot{x}\}+[C]\{\dot{x}\}+[K]\{x\}=-[M]\{r\}{\ddot{x}}_g(t)$

where {r} is the influence vector (= {1} for uniform horizontal base motion in a shear-frame; more complex for multi-support excitation, e.g. bridges with spatially-varying ground motion).

3.2 Free vibration → mode shapes

Setting {F} = 0 and {C} = 0, assume harmonic {x} = {φ}·e^(iωt). The generalized eigenvalue problem:

$([K]-{\omega }^2[M])\{\varphi \}=0$

yields N eigenpairs (ω_i, {φ_i}). Mass-normalize: {φ_i}^T [M] {φ_i} = 1. Modes are orthogonal: {φ_i}^T [M] {φ_j} = δ_ij; {φ_i}^T [K] {φ_j} = ω_i²·δ_ij. The modal matrix [Φ] = [{φ_1} … {φ_N}] transforms physical {x} into modal {q} via {x} = [Φ]{q}, decoupling the N-DOF system into N independent SDOF problems — the workhorse of structural dynamics.

3.3 Damping ratios — typical values

Material / structural system	Damping ratio ζ
Welded steel structure (small amplitude)	0.005 – 0.010
Bolted steel structure	0.010 – 0.020
Steel moment frame (yielding)	0.030 – 0.050
Steel braced frame (no slip)	0.005 – 0.015
Reinforced concrete (uncracked, small amp.)	0.020 – 0.030
Reinforced concrete (cracked, design level)	0.040 – 0.070
Prestressed concrete	0.015 – 0.030
Masonry (URM)	0.050 – 0.080
Wood-framed building	0.050 – 0.100
Cable-stayed bridge cable	0.001 – 0.005
Soil (SSI, foundation radiation)	0.05 – 0.30
Base-isolated structure (LRB lead-core)	0.15 – 0.30 (effective)
Viscous-damped retrofit	0.10 – 0.30 (added)

Source: Chopra Dynamics of Structures 5th ed. Table 11.2.1; ASCE 41-23 Table 7-3; PEER NGA documentation.

Code-default ζ = 0.05 (5 % of critical) for life-safety earthquake design unless project-specific values are justified. This is the value baked into ASCE 7-22 design-spectrum ordinates.

3.4 Rayleigh damping

For nonlinear time-history analysis we still need a [C] matrix. Rayleigh proportional damping is the universal compromise:

$[C]=\alpha [M]+\beta [K]$

with α, β chosen so that ζ_i = α/(2ω_i) + β·ω_i/2 hits desired values at two anchor frequencies ω_a, ω_b. Pick ω_a near the dominant elastic mode and ω_b near a representative inelastic / higher mode; modes between them get less damping (the bathtub of the Rayleigh curve), modes outside get more. Caprini-Chopra and modified Rayleigh variants address the high-frequency over-damping issue.

3.5 Analysis-procedure ladder (ASCE 7-22 §12.6)

Procedure	When permitted	Effort	Output
Equivalent Lateral Force (ELF)	Regular structures with T ≤ 3.5 T_S (most low-rise)	Hand or simple software	Static-equivalent base shear V, vertical distribution
Modal Response Spectrum Analysis (RSA / MRSA)	Almost all buildings; required for irregularity	ETABS/SAP2000 modal solve	Modal SRSS or CQC response
Linear Response History Analysis (LRHA)	Permitted for all; required for some	7–11 record suites	Element-level demand histories
Nonlinear Response History Analysis (NRHA / NLTHA)	Required for Risk Cat IV; base-isolated; > 73 m in SDC D/E/F	Days to weeks per model	Inelastic hinge histories, ductility demands

Section 16 (formerly Chapter 16) covers NLTHA; Chapter 17 covers seismic isolation; Chapter 18 covers structures with damping systems.

4. Seismic engineering basics

4.1 Ground motion characterization

Parameter	Symbol	Typical range	Notes
Peak ground acceleration	PGA	0.05–1.5 g	High-frequency content metric
Peak ground velocity	PGV	0.05–2.0 m/s	Mid-frequency; correlates with damage to flexible structures
Peak ground displacement	PGD	0.01–1.0 m	Low-frequency; relevant for tall buildings, base isolation
Spectral acceleration	S_a(T, ζ)	function	Response of 5%-damped SDOF; the design quantity
Significant duration	D_5-95	5–30 s	5 %–95 % Arias intensity
Pulse period	T_p	1–10 s	Near-fault directivity

4.2 Design spectrum (ASCE 7-22)

USGS produces risk-targeted maximum considered earthquake (MCE_R) maps with the design ground-motion parameters:

• S_S — 5 %-damped MCE_R spectral response acceleration at short period (~0.2 s)
• S_1 — 5 %-damped MCE_R spectral response acceleration at 1.0 s

Site-modified values: S_MS = F_a · S_S, S_M1 = F_v · S_1, where F_a, F_v are site coefficients (ASCE 7-22 Tables 11.4-1 and 11.4-2) depending on Site Class A–F from geotechnical V_S30 measurement.

The design spectrum uses 2/3 of MCE_R: S_DS = (2/3) S_MS; S_D1 = (2/3) S_M1. The full multi-period design spectrum has four branches:

Period range	Spectral acceleration S_a
T < T_0 = 0.2·S_D1/S_DS	S_DS · (0.4 + 0.6·T/T_0)
T_0 ≤ T ≤ T_S = S_D1/S_DS	S_DS
T_S < T ≤ T_L (long-period transition)	S_D1 / T
T > T_L	S_D1 · T_L / T²

T_L from USGS maps; ranges 4 s (CA) to 16 s (Pacific Northwest, Cascadia).

4.3 Design coefficients (ASCE 7-22 Table 12.2-1, excerpt)

Seismic-force-resisting system	R	Ω₀	C_d	Height limit (SDC D)
Special moment frame (SMF), steel	8	3	5.5	NL
Intermediate moment frame (IMF), steel	4.5	3	4	10 m
Ordinary moment frame (OMF), steel	3.5	3	3	Prohibited
Special concentrically braced frame (SCBF)	6	2	5	49 m
Buckling-restrained braced frame (BRBF)	8	2.5	5	NL
Eccentrically braced frame (EBF)	8	2	4	NL
Special reinforced concrete moment frame (SMRF)	8	3	5.5	NL
Special reinforced concrete shear wall	6	2.5	5	NL
Ordinary RC shear wall	4	2.5	4	Prohibited
Light-frame wood shear wall	6.5	3	4	20 m
Cantilevered column system (e.g. lighting pole)	1.25–2.5	1.25	1.25–2.5	11 m

• R — response modification factor; divides the elastic demand to give design demand (accounts for ductility + overstrength)
• Ω₀ — overstrength factor; capacity check for non-ductile elements (collectors, foundation columns)
• C_d — deflection amplification factor; multiplies elastic-analysis drift to give expected inelastic drift
• I_e — importance factor (1.0, 1.25, 1.5 for Risk Cat I/II, III, IV)

4.4 ELF base shear

$V=C_s\cdot W,C_s=\min \left(\frac{S_{DS}}{R/I_e},\frac{S_{D1}}{T\cdot R/I_e},\frac{S_{D1}{T}_L}{T^2\cdot R/I_e}\right)$

with a lower bound C_s ≥ 0.044·S_DS·I_e ≥ 0.01 (and additionally 0.5·S_1/(R/I_e) in SDC E/F).

Approximate fundamental period T_a = C_t · h_n^x where (C_t, x) = (0.028, 0.8) for steel moment frames; (0.016, 0.9) for RC moment frames; (0.020, 0.75) for all other systems (ASCE 7-22 §12.8.2.1).

Vertical distribution: F_x = (w_x h_x^k / Σ w_i h_i^k) · V, with k = 1 for T ≤ 0.5 s, k = 2 for T ≥ 2.5 s, linear interpolation in between.

4.5 Drift limits

ASCE 7-22 Table 12.12-1: allowable story drift Δ_a in terms of h_sx (story height):

Structure type	Risk Cat I, II	Risk Cat III	Risk Cat IV
Masonry cantilever shear wall	0.010 h_sx	0.010 h_sx	0.010 h_sx
Other masonry	0.007 h_sx	0.007 h_sx	0.007 h_sx
All other (typical building)	0.025 h_sx	0.020 h_sx	0.015 h_sx
4 stories or fewer (interior partitions etc. designed for drift)	0.025 h_sx	0.020 h_sx	0.015 h_sx

Computed inelastic drift δ_x = C_d · δ_xe / I_e (elastic analysis drift amplified by C_d).

5. Response spectrum analysis (RSA / MRSA)

5.1 Procedure

1. Build a linear-elastic 3-D model (ETABS / SAP2000 / RAM SS / STAAD.Pro / RISA-3D).
2. Modal extraction until ≥ 90 % of total effective mass is captured in each principal horizontal direction (ASCE 7-22 §12.9.1.1).
3. For each mode i, compute:
   • Modal participation factor: Γ_i = {φ_i}^T [M] {r} / ({φ_i}^T [M] {φ_i})
   • Effective modal mass: M*_i = Γ_i² · ({φ_i}^T [M] {φ_i})
   • Spectral acceleration: S_a(T_i, ζ_i) from the design spectrum
4. Modal response: r_i = Γ_i · {φ_i} · S_a(T_i)/ω_i² (displacement) or analogous for forces.
5. Combine modes:
   • SRSS (square-root-of-sum-of-squares): r = √(Σ r_i²) — acceptable when modes are well-separated (T_j / T_i > 1.5).
   • CQC (complete quadratic combination, Wilson-Der Kiureghian 1981): r² = Σ_i Σ_j ρ_ij r_i r_j with correlation coefficient ρ_ij accounting for closely-spaced modes. Default in modern software.
6. Scale to ELF base shear: ASCE 7-22 §12.9.1.4.1 requires V_RSA ≥ 0.85 V_ELF (computed with same T but capped per §12.8.2). Scale all responses proportionally if the modal base shear is below this floor.
7. Combine 100 %/30 % directionally per §12.5 (or 30 %/100 %), and add accidental torsion (5 % eccentricity).

5.2 Mode-truncation tips

• 90 % effective mass in each principal direction independently — soft-story buildings often need 15+ modes to reach 90 % in the direction with high torsion coupling.
• Ritz vectors (load-dependent) capture base-excitation response more efficiently than eigenvectors for the same target accuracy — supported by SAP2000, ETABS, OpenSees.
• For tall buildings (> 30 stories) modes 5–10 carry the higher-mode shears that govern collector and outrigger forces.

6. Time-history analysis (THA / NLTHA)

6.1 Ground motion selection

ASCE 7-22 §16.2 (NLTHA) and §17 (isolated):

• 11 ground motions for two-component analysis (per current edition; older 7 records still seen).
• Records selected from PEER NGA-West2 (shallow crustal), NGA-Sub (subduction), or KiK-net / K-NET databases matching:
  • Magnitude bin (e.g. M ≥ 6.5 within 25 km of fault for site-specific hazard)
  • Fault mechanism (strike-slip, normal, reverse)
  • Site class consistent with project V_S30
  • Pulse content (near-fault) if applicable
• Spectral matching (SeismoMatch, RspMatch09) or amplitude scaling to the target spectrum. Matching is more conservative for individual records; scaling preserves natural variability.

6.2 Time integration

• Newmark-β (1959) — implicit, unconditionally stable for γ = 0.5, β = 0.25; the de-facto standard. Variants:
  • Hilber-Hughes-Taylor α (HHT-α, 1977) — adds numerical damping of higher modes (α ∈ [−1/3, 0]); preferred for nonlinear seismic and contact problems
  • Generalized-α (Chung-Hulbert 1993) — optimal high-frequency dissipation with second-order accuracy
• Wilson-θ (θ ≥ 1.37) — implicit, popular before HHT-α
• Central difference — explicit, conditionally stable (Δt < T_min/π); used for explicit codes (LS-DYNA, Abaqus/Explicit) on short-duration high-strain-rate problems (blast, impact)

6.3 Element-level inelastic models

• Concentrated plasticity — lumped plastic hinges at member ends with M-θ or P-M-θ rules (modified Ibarra-Medina-Krawinkler, FEMA P-440A backbone). PERFORM-3D, OpenSees beamWithHinges.
• Distributed plasticity — fiber sections (steel + concrete fibers with uniaxial stress-strain models). OpenSees forceBeamColumn, ETABS fiber hinge, ABAQUS *REBAR in beam elements.
• Continuum FEM — full 3-D meshing of beam-column joints and shear walls (Abaqus, ANSYS, LS-DYNA). Used for blast, progressive-collapse, performance validation studies.

7. Three worked examples

7.1 Example A — Single-story ELF base shear

Given: Single-story commercial building, dead load W = 5 000 kN, S_DS = 1.0 g, S_D1 = 0.6 g, R = 8 (steel SMF), I_e = 1.0, Risk Cat II, approximate T from §12.8.2.1: h_n = 4 m steel SMF → T_a = 0.028 · 4^0.8 = 0.085 s. Use T_a (capped) = 0.5 s for example clarity.

Compute C_s:

C_s,1 = S_DS / (R/I_e)          = 1.0  / (8/1.0)         = 0.125
C_s,2 = S_D1 / (T · R/I_e)      = 0.6  / (0.5 · 8/1.0)   = 0.150
C_s   = min(C_s,1, C_s,2)        = 0.125
C_s,min = max(0.044 S_DS I_e, 0.01) = max(0.044, 0.01)   = 0.044

C_s = 0.125 governs (between min and the cap).

Base shear:

V = C_s · W = 0.125 · 5 000 = 625 kN (~ 141 kips) per principal direction.

Vertical distribution (single story → 100 % at the diaphragm level): F_roof = 625 kN.

Drift check (Risk Cat II, “all other”): Δ_a = 0.025 · h_sx = 0.025 · 4 000 = 100 mm. Run elastic model → δ_xe; compute δ_x = C_d · δ_xe / I_e = 5.5 · δ_xe; require δ_x ≤ 100 mm. For a typical SMF, δ_xe ≈ 10–15 mm → δ_x ≈ 55–82 mm < 100 ✓.

Overstrength check (collectors, foundation): demand = Ω_0 · E = 3.0 · E_seismic.

7.2 Example B — Wind vortex shedding on a slender steel stack

Given: Cylindrical steel stack, H = 30 m, D = 1.0 m (constant), wall t = 10 mm. Steel A572-50 (ρ = 7 850 kg/m³, E = 200 GPa). Design wind V_design = 40 m/s. Cantilever from foundation.

Mass per unit length: m’ = π · D · t · ρ = π · 1.0 · 0.010 · 7 850 = 246.6 kg/m.

Section: I = π/64 · (D⁴ - (D-2t)⁴) = π/64 · (1.0⁴ - 0.98⁴) = π/64 · (1 - 0.9224) = π/64 · 0.0776 = 3.81 × 10⁻³ m⁴. EI = 200 × 10⁹ · 3.81 × 10⁻³ = 7.62 × 10⁸ N·m².

First natural frequency (cantilever, β_1 L = 1.8751):

ω_1 = (1.8751)² · √(EI / (m’ L⁴)) = 3.516 · √(7.62 × 10⁸ / (246.6 · 30⁴)) = 3.516 · √(7.62 × 10⁸ / 1.997 × 10⁸) = 3.516 · √3.815 = 3.516 · 1.953 = 6.87 rad/s

f_1 = 6.87 / (2π) = 1.09 Hz, T_1 = 0.92 s.

Strouhal vortex-shedding frequency at design wind (St ≈ 0.20 for Re in the post-critical regime, Re ~ 2.7 × 10⁶ for V·D/ν = 40·1/1.5e-5):

f_vortex = St · V / D = 0.20 · 40 / 1.0 = 8.0 Hz

8.0 Hz vs 1.09 Hz — vortex shedding is ~7× the first natural frequency at design wind. Resonance is not at design wind, but critical wind speed for lock-in to mode 1:

V_cr = f_1 · D / St = 1.09 · 1.0 / 0.20 = 5.45 m/s

5.45 m/s is well within the operational wind range and will excite the first mode every windy day. Mitigation (covered in detail in vibration-dynamics §11p):

• Helical strakes (Scruton 1955) — three helical strips, 0.1 D high, 5 D pitch — disrupt coherent shedding; standard for chimneys and offshore risers
• Tuned mass damper at the tip — mass 1–2 % of generalized modal mass, tuned to f_1, ζ ≈ 0.05
• Perforated shrouds — alternative to strakes for tall stacks

ASCE 7-22 §26.11 + ASME STS-1-2016 Steel Stacks code formalize this analysis with the Scruton number Sc = 2m·ζ/(ρ_air · D²); Sc > 8 generally avoids destructive lock-in.

7.3 Example C — Base isolation period shift

Given: 8-story RC office building, plan 30 m × 40 m, story heights 4 m → h_n = 32 m. Total mass M = 5 000 t (5 × 10⁶ kg). Fixed-base fundamental period T_1,FB = 0.8 s. Design spectrum: S_DS = 1.0 g, S_D1 = 0.6 g, T_L = 8 s, T_S = 0.6 s, ζ = 0.05.

Fixed-base fundamental ω:

ω_FB = 2π / 0.8 = 7.85 rad/s

Spectral acceleration at T_1,FB = 0.8 s: since 0.6 < T_1 < T_L, S_a = S_D1 / T = 0.6 / 0.8 = 0.75 g.

Base shear (elastic, ζ = 0.05): V_e = S_a · W = 0.75 · 5 000 · 9.81 = 36 800 kN. Divide by R (= 5, special RC shear wall, I_e = 1.0): V_design = 7 360 kN. C_s = 0.15.

Add 24 lead-rubber bearings (LRB) with effective stiffness K_eff = 25 kN/mm each at design displacement. Total isolation stiffness:

K_iso = 24 · 25 × 10³ N/mm · 1 000 mm/m = 24 · 25 × 10⁶ = 6.0 × 10⁸ N/m

Isolated frequency:

ω_iso = √(K_iso / M) = √(6.0 × 10⁸ / 5.0 × 10⁶) = √120 = 10.95 rad/s … hmm, that’s higher than fixed-base, can’t be right. Re-check: K_iso for a low-period target is low, not high. For target T_iso = 2.5 s:

K_iso,target = M · ω² = 5.0 × 10⁶ · (2π / 2.5)² = 5.0 × 10⁶ · 6.32 = 3.16 × 10⁷ N/m

That’s the total needed: 3.16 × 10⁷ / 24 = 1.32 × 10⁶ N/m = 1 320 N/mm per bearing. A typical LRB is 1–5 kN/mm effective at design displacement — well within standard product range (DIS, Bridgestone, Maurer, FIP Industriale).

So with appropriate K_eff,bearing:

T_iso = 2.5 s → S_a(2.5 s, ζ = 0.20 effective from LRB lead-core) ≈ 0.6 / 2.5 · B_D = 0.24 · 0.80 = 0.19 g

where B_D is the damping coefficient from ASCE 7-22 Table 17.5-1 (0.80 for β_eff = 20 %, 0.67 for 30 %). Compare to fixed-base 0.75 g → ~75 % reduction in spectral demand.

Design displacement (ASCE 7-22 §17.5.3.1):

D_D = g · S_D1 · T_iso / (4π² · B_D) = 9 810 · 0.6 · 2.5 / (39.48 · 0.80) = 466 mm

Bearings must accommodate ~470 mm of horizontal travel + buckling stability margin + uplift restraint per §17.2.4. Maximum displacement D_M (under MCE): typically 1.5 × D_D ~ 700 mm.

System realisations in service (a non-exhaustive list):

• USC University Hospital, Los Angeles (1991, friction-pendulum + lead-rubber)
• San Francisco City Hall retrofit (1999, lead-rubber, 530 bearings)
• Salt Lake City & County Building retrofit (1989, first major US LRB retrofit)
• Long Beach V.A. Hospital retrofit (1989)
• Apple Park Cupertino (2017, double-pendulum, ~700 bearings on world’s largest isolated building)
• Tokyo Skytree base (2012, oil dampers + tuned mass)

8. Base isolation + supplemental damping

8.1 Isolation system types

Type	Mechanism	Period range	Damping	Re-centering	Notable example
Elastomeric (HDRB)	High-damping rubber compound	2.0–3.5 s	β = 10–15 %	Rubber elasticity	Foothill Communities Law Hall, San Bernardino (1985)
Lead-Rubber Bearing (LRB)	Rubber + central lead plug yields	2.0–3.5 s	β = 20–35 %	Rubber elasticity	William Clayton Building, Wellington NZ (1981, world first LRB)
Friction Pendulum (FP / SFP)	Sliding on concave PTFE surface	T = 2π√(R/g), fixed by radius	μ-controlled	Gravity restoring	San Francisco International Airport (2001, ~270 bearings)
Double / Triple Friction Pendulum (DFP / TFP)	Multiple sliding surfaces, staged μ	1.5–4 s (multi-stage)	Stage-dependent	Gravity	Apple Park (TFP); Hayward City Hall (DFP)
Sliding Isolator (Pure)	Flat PTFE-stainless, separate restoring	n/a alone	μ ~ 0.05–0.15	None (needs aux. spring)	Less common; bridges
Spherical Sliding (Eradiquake)	Stainless ball + concave surface	Similar to FP	Low	Gravity	Bridge bearings; lower seismic

Inventor lineage: Bill Robinson (DSIR / Robinson Seismic, NZ) — LRB 1975+; Victor Zayas (Earthquake Protection Systems, EPS) — Friction Pendulum 1985+; James Kelly (UC Berkeley) — HDRB theory.

8.2 Tuned Mass Dampers (TMD)

A secondary mass (m_TMD typ 0.5–2 % of generalized building modal mass) on a spring + damper tuned to the dominant mode (Den Hartog 1928 optimal tuning). At resonance, the TMD response is in anti-phase with the primary, absorbing energy.

Optimal Den Hartog parameters (for an undamped primary):

$f_{\text{opt}}=\frac{1}{1+\mu },{\zeta }_{\text{TMD,opt}}=\sqrt{\frac{3\mu }{8(1+\mu {)}^3}}$

where μ = m_TMD / m_primary and f = ω_TMD / ω_primary.

Building	TMD configuration	Year
Citicorp Center, NYC	410 t concrete block, oil bearings	1977 (secret retrofit 1978)
John Hancock Tower, Boston	Two 300 t blocks on oil films	1976
Taipei 101	660 t steel sphere, 5.5 m diameter, 4 fluid dampers	2004
Shanghai World Financial Center	150 t pendulum × 2	2008
One Wall Centre, Vancouver	TLD (tuned liquid damper), 7 m diameter tank	2001
Comcast Tech Center, Philadelphia	TLCD (column liquid damper), 1.3 ML water	2018
Burj Khalifa, Dubai	Aerodynamic shaping + multiple TMDs	2010
Millennium Bridge, London	17 viscous + 26 inertial TMDs (retrofit)	2002
Pedro e Inês footbridge, Coimbra	6 TMDs + 1 pendulum TMD	2007

8.3 Supplemental damping devices (ASCE 7-22 Ch. 18)

Type	Mechanism	Force-velocity / displacement	Hysteresis loop
Viscous fluid damper (VFD)	Silicone oil through orifices	F = C · sgn(v)·	v
Visco-elastic (VE) damper	Sheared polymer	F = K_d · u + C_d · v (linear)	Tilted ellipse
Friction damper (Pall, Sumitomo)	Brake-lining sliding	F = μN, rectangular	Rectangular
Hysteretic — ADAS / TADAS	X-plate yield	Bi-linear yield	Parallelogram (Bauschinger)
Buckling-Restrained Brace (BRB)	Yielding core in restrained sleeve	Stable hysteretic	Symmetric bi-linear (no buckling pinch)
Magnetorheological (MR)	MR fluid yield-stress modulated by current	Semi-active, tunable	Variable
Shape-memory alloy (SMA)	Superelastic NiTi	Flag-shaped	Re-centering

Manufacturers: Taylor Devices (VFD, the LA market leader), Maurer (Germany; VFD + LRB + ER fluid), Kawakin / Yokogawa Industries (Japan), FIP Industriale (Italy), EPS (Earthquake Protection Systems, FP bearings), Star Seismic (BRB), CoreBrace (BRB), Lord Corporation (MR fluid).

8.4 BRBF in modern practice

Buckling-restrained braced frames (BRBF) are the fastest-growing modern seismic system in the US. The brace consists of a yielding steel core (rectangular, cruciform, or T-section, A36 / A572) inside a mortar-filled steel HSS sleeve. The mortar + sleeve restrain global buckling; the core yields symmetrically in tension and compression. Hysteresis is nearly elasto-plastic with high energy dissipation per cycle — orders of magnitude better than ordinary concentrically braced frames (OCBF) where compression buckles and tension-yields asymmetrically.

ASCE 7-22 Table 12.2-1: R = 8, Ω₀ = 2.5, C_d = 5 — better than OCBF (R = 3.25) and SCBF (R = 6) and comparable to SMF (R = 8) but at lower cost. AISC 341-22 §F4 governs design with qualification testing (ATC-24 / AISC 341 Appendix K3).

9. Wind engineering basics

9.1 Static (gust-equivalent) wind

ASCE 7-22 Ch. 26 main wind-force resisting system (MWFRS):

P = q_z · G · C_p − q_i · (GC_pi)

with q_z = 0.613 · K_z · K_zt · K_d · K_e · V² (Pa, V in m/s) = 0.00256 · K_z · K_zt · K_d · K_e · V² (psf, V in mph). Factors:

• K_z — velocity pressure exposure (Exposure B urban, C open, D shoreline)
• K_zt — topographic (hills, escarpments per Fig. 26.8-1)
• K_d — directionality (typically 0.85 for buildings)
• K_e — ground elevation
• V — basic wind speed (3-s gust, 700-year MRI for Risk Cat II in 2022 ASCE 7)
• G — gust-effect factor (0.85 for rigid; computed for flexible)
• C_p — external pressure coefficient (Fig. 27.3-1 etc.)
• GC_pi — internal pressure coefficient (function of enclosure classification)

9.2 Dynamic wind phenomena

Phenomenon	Trigger	Excitation type	Mitigation
Vortex-induced vibration (VIV)	Strouhal alternating wake	Cross-wind harmonic at f_s = St·V/D	Strakes, shrouds, TMD, increase Sc
Lock-in	f_s pulled to f_n over ~ ±25 % wind range	Self-synchronization	Add damping, perturb cross-section
Galloping (1-DOF)	Non-circular section, negative aerodynamic damping (Den Hartog criterion: dC_L/dα + C_D < 0)	Cross-wind divergent	Add damping; modify section
Flutter (2-DOF)	Coupled torsion-bending at V_flutter	Aeroelastic divergent	Mass distribution, torsional stiffness
Buffeting	Atmospheric turbulence	Random gust response	Quasi-static gust factor; full random-vib analysis
Wake-galloping	Downstream cable in wake of upstream	Quasi-steady wake instability	Cable spacing; cross-ties
Rain-wind vibration	Cable-stayed bridges; rain forms rivulets that modify C_L	Aerodynamic	Helical fillets, cable damping
Parametric excitation	Cable + pylon coupling	Time-varying stiffness at 2ω_n	Detune; add damping

9.3 Tall-building wind design

For buildings with H > 4·min(B, L) or T > 1 s, ASCE 7-22 §26.2 classifies them as flexible — requires computed gust-effect factor G_f rather than rigid 0.85. For supertall (typically > 250 m), boundary-layer wind-tunnel testing per ASCE 49-21 is virtually universal. Test types:

• Rigid pressure model (RPM) — measure cladding and structure pressures; static wind-tunnel approximation
• High-frequency force balance (HFFB / HFBB) — rigid model on stiff sensitive balance; base reactions converted to mode-generalized forces via mode shapes
• Aeroelastic model (full or 2D taut-strip) — flexible model that exhibits actual aerodynamic feedback; required for flutter-prone bridges (Akashi Kaikyo, Storebælt, Messina design studies)

Major wind-tunnel labs: RWDI (Guelph, ON — supertall specialist), CPP (Fort Collins, CO), BLWTL (Boundary-Layer Wind Tunnel Laboratory, Western University, Ontario), Force Technology (Denmark), CSTB (France), NRC-IAR (Canada).

10. Soil-structure interaction (SSI)

10.1 Why it matters

Real foundations are not rigid. Soil flexibility:

• Lengthens the building period (often 10–30 % for soft-soil sites)
• Adds damping through radiation (energy leaving through P/S waves into the half-space) and material hysteresis
• Redistributes demands — softer soil → larger displacements but generally smaller member forces (since you slide further down the design spectrum)

ASCE 7-22 §19 provides optional SSI procedures; ASCE 41-23 §8.4 mandates SSI for evaluation/retrofit of existing buildings on soft soil (Site Class D/E/F).

10.2 Modelling

• Sub-structure approach (Wolf 1985; Veletsos-Verbic 1973): compute foundation impedance K(ω) + C(ω) per DOF for each foundation footprint; attach to structural model as frequency-dependent spring-dashpots. Cone models give analytical closed forms for surface foundations.
• Direct approach: model soil as a meshed continuum (FE or finite-difference), with absorbing boundaries (Lysmer-Kuhlemeyer dashpots, PML perfectly-matched layers). Computationally expensive but captures spatial variation, layered media, and kinematic interaction.
• Inertial vs kinematic interaction: kinematic = soil filters input motion (averaging, base-slab embedment) before it reaches the structure; inertial = structure pushes back on the soil, modifying impedance. Both are present; both can be analyzed.

10.3 Site-effect catastrophes

• Mexico City 1985 — soft lacustrine clay basin amplified 1-second motion ~5×, T_site ≈ 2 s, devastated 6–15 story (T_struct ≈ 0.5–1.5 s, but inelastic period extension brought them to resonance)
• Loma Prieta 1989 Marina District — bay-fill on soft mud + reclaimed sand; double resonance + liquefaction
• Christchurch 2011 — extensive liquefaction of saturated sands; lateral spreading damaged ~50 % of foundations in CBD even for buildings whose superstructure performed adequately
• Niigata 1964 — first major documented liquefaction event; apartment buildings tipped over intact

10.4 Liquefaction screening

V_S30 (shear-wave velocity of upper 30 m, averaged) is the primary site-class metric (ASCE 7-22 Table 20.3-1):

Site Class	V_S30 (m/s)	Description
A	> 1 500	Hard rock
B	760 – 1 500	Rock
C	365 – 760	Very dense soil / soft rock
D	180 – 365	Stiff soil
E	< 180	Soft clay (PI > 20, w > 40 %, su < 25 kPa)
F	—	Special: liquefiable, quick clay, peat, > 3 m highly organic, > 7.6 m high-plasticity, etc.; site-specific study required

Simplified liquefaction triggering (Seed-Idriss 1971, Boulanger-Idriss 2014): compute cyclic stress ratio CSR vs cyclic resistance ratio CRR from SPT-N or CPT-q_c; FS = CRR / CSR < 1.3 → vulnerable.

11. Standards + software

11.1 Seismic standards by jurisdiction

Country / region	Primary standard	Notes
US (federal + most states)	ASCE 7-22 + AISC 341 + ACI 318 ch.18, adopted via IBC 2024	Performance-based via ASCE 7 §16, 17, 18
US (CA)	CBC 2022 (= IBC 2021 with amendments) + DSA / OSHPD / DSA-SS	Hospital, school overlay
US existing buildings	ASCE 41-23 (evaluation + retrofit), FEMA 547	Tier 1/2/3 procedures
Europe	Eurocode 8 (EN 1998-1 buildings, -2 bridges, -3 evaluation, -4 silos/tanks, -5 foundations, -6 towers)	National annexes
Japan	Building Standard Law (BSL), AIJ standards, Notifications 1457, 1791	Pre-1981 vs post-1981 (Shin-Taishin) distinction
Canada	NBCC 2020 + CSA S16, A23.3	UHS-based hazard
New Zealand	NZS 1170.5, NZS 3404 (steel), NZS 3101 (concrete)	Near-fault factor
Mexico	NTC-Sismo 2017 + Reglamento de Construcciones para el DF	Soft-soil zoning
China	GB 50011-2010 (revision in progress)	Three-level performance design
India	IS 1893 (Part 1):2016	Five zones
Chile	NCh 433.Of1996 mod.2009 + NCh 2369.Of2003 (industrial)	Post-Maule (2010) revisions
Turkey	TBDY 2018	Post-1999 Kocaeli; revised again post-2023

11.2 NEHRP + supplemental documents (US)

• FEMA P-2082 (NEHRP Recommended Provisions, 2020 edition) — source document for ASCE 7-22 seismic chapters
• FEMA P-58 + P-58-1/-2/-3 — performance-based seismic design + loss assessment (PACT software)
• FEMA P-695 / P-795 — quantification of building seismic performance factors (R, Ω₀, C_d)
• ATC-40 (1996) — RC seismic evaluation, predecessor to ASCE 41
• NIST GCR 12-917-21 — soil-structure interaction guidance
• NIST GCR 17-917-46 — guidelines for nonlinear structural analysis

11.3 Software

Software	Vendor	Niche
ETABS	CSI	Buildings; integrated AISC/ACI/EC design checks; the building default
SAP2000	CSI	General 3-D dynamic; bridges, towers, special structures
PERFORM-3D	CSI	Nonlinear seismic (concentrated hinges); ATC-58 / FEMA P-58 PBSE
CSI Bridge	CSI	Bridge-specific staged construction + seismic
RAM Structural System	Bentley	Integrated steel/RC building workflow
STAAD.Pro	Bentley	Industrial, multi-code
midas Gen / Civil	MIDAS IT	Strong nonlinear and seismic; bridges
OpenSees	UC Berkeley	Open-source research; fiber sections; the academic reference
OpenSeesPy / OpenSeesMP	UC Berkeley	Python wrapper + MPI parallel
ABAQUS	Dassault	General nonlinear FEM; implicit + explicit
LS-DYNA	ANSYS	Explicit dynamics; blast, impact, progressive collapse
DIANA	DIANA FEA	Concrete-cracking specialty (TNO heritage)
ADINA	ADINA R&D	Research-grade nonlinear
SeismoMatch / SeismoStruct	SeismoSoft	Ground-motion spectral matching + frame nonlinear
EDP-tools	various	PEER’s open-source PBSE post-processing
PACT	FEMA / ATC	FEMA P-58 loss assessment

11.4 Ground motion databases

• PEER NGA-West2 — shallow crustal active tectonic (most California / world); ~ 21 000 records
• PEER NGA-East — central + eastern North America; stable continental
• NGA-Sub — subduction (Tohoku, Maule, Cascadia design)
• KiK-net + K-NET — Japan dense network (~ 1 700 stations)
• CESMD — California Strong Motion Instrumentation Program + USGS
• ESM (Engineering Strong Motion) — European database
• NZ GeoNet — New Zealand records (Kaikoura, Christchurch)

12. Cross-references

• vibration-dynamics — Tier 1 foundation: SDOF/MDOF math, modal analysis, damping models, Miles equation, all of which this note presupposes.
• structural-analysis — Tier 1 companion: matrix stiffness, ASCE 7 load combinations, drift limits, FEM element types.
• steel-design (planned) — AISC 360-22 member checks; AISC 358-22 prequalified moment connections (RBS, WUF-W, BFP, ConXL, Kaiser bolted bracket); AISC 341-22 SMF / SCBF / EBF / BRBF detailing.
• reinforced-concrete (planned) — ACI 318-25 Ch. 18 special moment frames + special structural walls; confinement, lap splice, joint shear.
• fem-fea (companion in current batch) — discretization, element selection, nonlinear time-integration for NLTHA.
• fatigue-analysis — low-cycle fatigue under cyclic seismic demand; FEMA 350 / 351 connection fatigue criteria.
• mechanics-of-materials — stress-strain prerequisite; plasticity for hinge models.
• statics-fundamentals — equilibrium foundation that becomes [K].
• beam-theory — Euler-Bernoulli + Timoshenko PDE roots; mode shapes for chimneys.
• fasteners-bolts — moment-resisting connection design; AISC 358 bolt-and-weld details.
• soil-mechanics (planned) — SSI foundation impedances; liquefaction triggering; geotechnical input to seismic design.
• structural-dynamics (planned) — full wind-tunnel + atmospheric boundary-layer treatment.
• construction-bim (planned) — ETABS .e2k, SAP2000 .s2k, OpenSees .tcl/.py, PEER record format, NGA flatfile schema.
• pid-control (planned) — active and semi-active structural control overlaps with MR-damper algorithms (clipped-optimal, modulated homogeneous friction).

13. Citations

1. Chopra, A. K. Dynamics of Structures: Theory and Applications to Earthquake Engineering, 5th ed., Pearson, 2017. ISBN 978-0-13-455512-6. The canonical graduate text; comprehensive coverage from SDOF through nonlinear MDOF and base isolation.
2. Clough, R. W.; Penzien, J. Dynamics of Structures, 3rd ed., Computers & Structures Inc., 2003. ISBN 978-0-923-90750-1. The earlier canonical reference; especially strong on numerical methods (Newmark, Wilson-θ) and random vibration.
3. Naeim, F. (editor). The Seismic Design Handbook, 2nd ed., Springer, 2001. ISBN 978-0-7923-7301-5. Practitioner reference with chapters by leading experts.
4. Bozorgnia, Y.; Bertero, V. V. (editors). Earthquake Engineering: From Engineering Seismology to Performance-Based Engineering, CRC Press, 2004. ISBN 978-0-8493-1439-1. Multi-author treatise spanning seismology to design.
5. Skinner, R. I.; Robinson, W. H.; McVerry, G. H. An Introduction to Seismic Isolation, Wiley, 1993. ISBN 978-0-471-93433-1. The foundational base-isolation text from the inventors of LRB.
6. Constantinou, M. C.; Whittaker, A. S.; Kalpakidis, Y.; Fenz, D. M.; Warn, G. P. Performance of Seismic Isolation Hardware Under Service and Seismic Loading, MCEER-07-0012, University at Buffalo, 2007. The qualification-testing reference for LRB and FP bearings.
7. Den Hartog, J. P. Mechanical Vibrations, 4th ed., McGraw-Hill, 1956 (reprinted Dover 1985). ISBN 978-0-486-64785-2. Canonical TMD analysis (chapter 3); still the clearest derivation of optimal tuning.
8. Simiu, E.; Yeo, D. Wind Effects on Structures: Modern Structural Design for Wind, 4th ed., Wiley, 2019. ISBN 978-1-119-37588-9. The senior wind-engineering text from NIST.
9. Holmes, J. D. Wind Loading of Structures, 3rd ed., CRC Press, 2015. ISBN 978-1-4822-2920-5. Strong on the random-process and tall-building treatment.
10. Filiatrault, A.; Christopoulos, C.; Constantinou, M. C.; Pekcan, G.; Wanitkorkul, A. Elements of Earthquake Engineering and Structural Dynamics, 4th ed., Presses Polytechniques et Universitaires Romandes, 2024. ISBN 978-2-88915-518-0. Modern practitioner-oriented text.
11. Christopoulos, C.; Filiatrault, A. Principles of Passive Supplemental Damping and Seismic Isolation, IUSS Press, 2006. ISBN 978-88-7358-037-3. Definitive on damping devices.
12. Kelly, J. M. Earthquake-Resistant Design with Rubber, 2nd ed., Springer, 1997. ISBN 978-3-540-76131-6. The rubber-bearing reference.
13. Naeim, F.; Kelly, J. M. Design of Seismic Isolated Structures: From Theory to Practice, Wiley, 1999. ISBN 978-0-471-14921-7.
14. Newmark, N. M. “A Method of Computation for Structural Dynamics,” J. Engineering Mechanics Division, ASCE, vol. 85 (EM3), 1959, pp. 67–94. Original Newmark-β paper.
15. Hilber, H. M.; Hughes, T. J. R.; Taylor, R. L. “Improved numerical dissipation for time integration algorithms in structural dynamics,” Earthquake Engineering & Structural Dynamics, vol. 5, 1977, pp. 283–292. Original HHT-α paper.
16. Wilson, E. L.; Der Kiureghian, A.; Bayo, E. P. “A replacement for the SRSS method in seismic analysis,” EESD, vol. 9, 1981, pp. 187–192. Original CQC paper.
17. Vanmarcke, E. H. Random Fields: Analysis and Synthesis, MIT Press, 1983 (revised 2010). ISBN 978-981-4366-04-0. Random-vibration extension for spatial wind/ground-motion.
18. ASCE/SEI 7-22 — Minimum Design Loads and Associated Criteria for Buildings and Other Structures. American Society of Civil Engineers, 2022. The umbrella loads + seismic + wind standard.
19. ASCE/SEI 41-23 — Seismic Evaluation and Retrofit of Existing Buildings. ASCE, 2023.
20. AISC 341-22 — Seismic Provisions for Structural Steel Buildings. American Institute of Steel Construction, 2022.
21. AISC 358-22 — Prequalified Connections for Special and Intermediate Steel Moment Frames for Seismic Applications. AISC, 2022.
22. ACI 318-25 — Building Code Requirements for Structural Concrete, Ch. 18 (Earthquake-Resistant Structures). American Concrete Institute, 2025.
23. IBC 2024 — International Building Code. International Code Council, 2024.
24. FEMA P-58-1/-2/-3 — Seismic Performance Assessment of Buildings, Methodology + Implementation. ATC-58 / FEMA, 2018 update.
25. FEMA P-695 — Quantification of Building Seismic Performance Factors, FEMA, 2009.
26. FEMA P-2082 — 2020 NEHRP Recommended Seismic Provisions for New Buildings and Other Structures, FEMA, 2020.
27. ATC-40 — Seismic Evaluation and Retrofit of Concrete Buildings, Applied Technology Council, 1996.
28. EN 1998-1:2004 + A1:2013 — Eurocode 8: Design of structures for earthquake resistance — Part 1: General rules, seismic actions and rules for buildings. CEN.
29. PEER NGA-West2 Flatfile and Documentation. Pacific Earthquake Engineering Research Center, UC Berkeley, 2014. https://ngawest2.berkeley.edu/
30. OpenSees User’s Manual and Examples Repository. Pacific Earthquake Engineering Research Center, UC Berkeley. https://opensees.berkeley.edu/
31. ETABS Analysis Reference Manual + PERFORM-3D User Guide. Computers and Structures Inc., Berkeley CA.
32. Murray, T. M.; Allen, D. E.; Ungar, E. E.; Davis, D. B. AISC Design Guide 11: Vibrations of Steel-Framed Structural Systems Due to Human Activity, 2nd ed., AISC, 2016. Pedestrian-induced floor vibration.
33. ISO 10137:2007 — Bases for design of structures — Serviceability of buildings and walkways against vibrations. ISO.
34. ASME STS-1-2016 — Steel Stacks. American Society of Mechanical Engineers. Vortex-shedding design for industrial stacks.
35. Strouhal, V. “Über eine besondere Art der Tonerregung,” Annalen der Physik und Chemie, 1878. The original Strouhal-number paper.
36. Scruton, C. “On the Wind-Excited Oscillations of Stacks, Towers and Masts,” Proc. International Conference on Wind Effects on Buildings and Structures, NPL, 1963. Helical-strake mitigation.
37. Tamura, Y.; Kareem, A. Advanced Structural Wind Engineering, Springer, 2013. ISBN 978-4-431-54336-7. Modern wind engineering for tall structures.