Analysis II — Dynamic Analysis & Reliability

Dynamic response analysis and probabilistic design methodologies for modern offshore structures, including limit state design and risk-based approaches.

1. When Dynamic Analysis Is Required

Natural Period and Wave Excitation

A structure responds dynamically when its natural frequency is close to the frequency of applied loads. Wave frequency is related to period: f_wave = 1/T, typically 0.05–0.25 Hz (period 4–20 seconds). If the structure's fundamental period T_struct > 3 seconds (frequency < 0.33 Hz), the structure is flexible and dynamic amplification can occur.

• Short-Period Structures: T_struct < 2 s (jackets, GBS in shallow water). Static analysis often sufficient; dynamic effects are modest (DAF ≈ 1.1–1.3).
• Long-Period Structures: T_struct > 3 s (deepwater spars, TLPs, semi-subs). Dynamic analysis mandatory; can capture resonance amplification or rocking modes.

Design Criteria Triggering Dynamic Analysis

API RP 2A Section 5.2 and ISO 19902 Section 8 prescribe dynamic analysis when:

• Fundamental period T > 3 seconds
• Structure has significant non-linear behavior (soil flexibility, cable sag)
• Site-specific seismic design spectrum is available (required in seismic zones)
• Risk assessment or regulatory authority mandate full dynamic time-history analysis

2. Modal Analysis and Eigenvalue Problems

Free Vibration Analysis

Modal analysis solves the free-vibration eigenvalue problem to find natural frequencies and mode shapes:

The solver finds the smallest eigenvalues and associated mode shapes. Typically, the first 5–20 modes are extracted, capturing frequencies up to 1.0 Hz. Each mode represents a decoupled oscillation pattern; the mode shape describes the relative motion of all nodes.

Mode Shape Interpretation

Common modes for jacket platforms:

• Mode 1 (First Lateral Mode): Platform sways in one horizontal direction. Frequency typically 0.15–0.35 Hz depending on stiffness and mass. Most important for wave loading.
• Mode 2: Sway in perpendicular horizontal direction (similar frequency to Mode 1 for symmetric jackets).
• Mode 3: Rocking (tipping motion about base). Frequency typically 0.25–0.50 Hz; higher if piles are very stiff or very soft.
• Higher Modes: Local oscillations of legs, braces, decks; less coupled to global response.

3. Response Spectrum and Time-History Analysis

Response Spectrum Method (RSM)

For seismic design, the Response Spectrum Method is efficient and widely used. A design spectrum (plot of pseudo-acceleration vs. period) characterizes the hazard. The platform's response at each natural period is read from the spectrum, then combined (SRSS or CQC) to estimate total response:

The method is fast and conservative; suitable for preliminary design. Disadvantage: it does not capture nonlinear effects (soil yielding, material damping) or sequential failure modes.

Time-History Analysis

For complex structures or high-consequence projects, time-history analysis applies a recorded or synthetic earthquake ground motion (acceleration vs. time) and integrates the equations of motion using time-stepping algorithms (Newmark, Runge-Kutta). Output is full time-series of displacement, velocity, acceleration, and force at every node—far more detail than spectrum method. Disadvantages: computationally expensive, sensitive to input motion selection, requires expertise in nonlinear dynamics.

4. Dynamic Amplification Factor (DAF)

DAF Definition and Resonance

When wave loading frequency is near the structural frequency, resonance amplifies response:

For typical offshore steel structures with ζ ≈ 0.03, DAF ≈ 16 at exact resonance. In practice, wave spectra are broad, so exact resonance rarely occurs; typical DAF values are 1.5–2.5 for first-mode resonance.

Damping Sources

Structural damping arises from multiple sources:

• Material Damping: Internal friction in steel; typically 0.5–1% critical damping.
• Radiation Damping: Energy radiated as waves when the structure vibrates; 0.5–2% depending on frequency and water depth.
• Fluid Viscous Damping: Drag on oscillating members; 0.5–2% for slender cylinders, higher for bluff bodies.
• Connection Damping: Slip and friction at welds, bolted joints; difficult to quantify, often 2–5% when present.

API RP 2A Table 5-6 provides default damping values: 2% for jacket platforms in normal conditions, 1.5% for GBS. Designers may justify different values with engineering analysis or testing.

5. Reliability and Limit State Design

Limit State Framework

Modern codes emphasize performance at distinct limit states rather than fixed safety factors:

• Ultimate Limit State (ULS): Structure approaches or reaches collapse. Design for rare 100-year (or more rare) event. Target annual probability of failure: 10^-4 to 10^-5.
• Serviceability Limit State (SLS): Structure remains functional but may experience deflection, acceleration, or fatigue. Design for frequent operational conditions (annual probability ~10^-1 to 10^-2).
• Accidental Limit State (ALS): Structure survives low-probability accidents (dropped objects, vessel collision) without loss of life or environmental catastrophe. Target conditional failure given accident: ~10^-2 to 10^-3.
• Fatigue Limit State (FLS): Cumulative fatigue damage controlled over design life. Target cumulative damage ratio D ≤ 0.3–0.5.

ISO 19902 Consequence Categories

ISO 19902 classifies structures by consequence of failure, which determines design factors:

Category	Consequence Level	Examples	Design Factor γ_f (example)
L1	Low (minimal)	Small unmanned platform, wellhead	1.2–1.3
L2	Moderate	Manned platform, small production field	1.3–1.4
L3	High (loss of life likely)	Large manned platform, major export pipeline	1.4–1.6

Higher consequence → higher safety factors → more conservative design. Conversely, a low-consequence structure in deep water might justify reduced factors, permitting smaller and cheaper design.

6. Probabilistic Design and FORM Analysis

First-Order Reliability Method (FORM)

FORM is a probabilistic approach to design that accounts for uncertainties in loads, resistances, and geometry:

FORM is used in advanced projects to optimize designs by directly controlling failure probability rather than applying fixed factors. It requires characterization of uncertainty distributions for loads, material properties, and geometry—substantial effort but yields risk-informed designs.

Monte Carlo Simulation

Monte Carlo directly samples the random variables (load, resistance) thousands or millions of times and counts failure occurrences, giving direct estimate of P_failure. More accurate than FORM but computationally expensive; often used for validation of FORM estimates or for non-linear problems where FORM assumptions break down.

7. Structural Redundancy and Ductility

Redundancy Ratios

A structure with redundancy can tolerate failure of individual members without overall collapse. ISO 19902 Section 5 requires assessment of redundancy via:

A jacket with 4 legs has inherent redundancy; loss of one leg does not cause immediate collapse. A single-buoy mooring system has zero redundancy. Modern codes require η ≥ 0.2 for L2 structures, encouraging designs with backup load paths.

Ductility and Plastic Hinges

Ductile failure (plastic deformation before fracture) is preferable to brittle fracture, as it provides warning and energy dissipation. Moment-resisting connections in jacket decks allow plastic hinging at gusset plates before brace fracture. Limit state design accounts for ductility by:

• Permitting member stresses to reach yield in isolated regions
• Allowing inelastic redistribution of forces to secondary load paths
• Designing connections to have adequate deformation capacity

Brittle failure (circumferential cracks in welds, lamellar tearing in thick plates) is actively avoided through material selection (high-toughness steel), weld quality (radiography, PWHT), and stress relief.

8. Progressive Collapse Assessment

Single-Member Failure Scenarios

To assess robustness, designers evaluate the structure's response if a single critical member fails (e.g., a large-diameter leg is damaged by vessel collision). Methodology:

1. Identify critical members (largest diameter, highest utilization, unique load paths).
2. Remove the member from the FEA model (set stiffness to near-zero).
3. Re-run analysis under operational loads (reduced environmental case, not design extreme).
4. Check if remaining structure can redistribute loads without exceeding 1.5× allowable stress or 30% deflection limits.

A robust structure continues to function safely after single-member loss. A fragile structure may collapse or require emergency evacuation. This is increasingly mandated by regulators in high-consequence areas.

9. Risk-Based Assessment Framework

Quantitative Risk Assessment (QRA)

QRA combines frequency of occurrence of hazardous events with severity of consequence:

QRA is used to allocate design effort and resources. High-frequency low-consequence events (e.g., dropped objects, vessel near-miss) may justify protective systems (nets, barriers). Low-frequency high-consequence events (e.g., systemic structural failure) require robust design margins.

Modern Risk Management Philosophy

Traditional prescriptive codes (API RP 2A ASD) ensure a baseline safety level but may not address project-specific risks efficiently. Risk-based approaches allow operators to:

• Identify and focus on the highest-consequence failure modes
• Justify design reductions where redundancy or monitoring mitigates risk
• Optimize lifecycle costs by balancing design conservatism vs. operational risk