🤙🏼Earthquake Engineering Unit 4 – Structural Dynamics in Earthquakes

Key Concepts and Terminology

• Structural dynamics studies the behavior of structures subjected to dynamic loading (earthquakes, wind, impact)
• Degrees of freedom (DOF) refer to the number of independent parameters required to define the motion of a system
  • Single degree of freedom (SDOF) systems have one independent parameter (displacement)
  • Multi degree of freedom (MDOF) systems have multiple independent parameters (displacements at each floor of a building)
• Natural frequency is the frequency at which a system tends to oscillate in the absence of any external force
• Damping is the dissipation of energy in a vibrating system, which reduces the amplitude of oscillations over time
  • Critical damping is the minimum amount of damping required to prevent oscillation
  • Damping ratio ($\zeta$) is the ratio of actual damping to critical damping
• Response spectrum is a plot of the maximum response (displacement, velocity, or acceleration) of an SDOF system as a function of its natural frequency or period
• Time history analysis involves subjecting a structure to a recorded or simulated earthquake ground motion and analyzing its response over time

Fundamentals of Structural Dynamics

• Equation of motion for an SDOF system: $m\ddot{u} + c\dot{u} + ku = p(t)$
  • $m$ is mass, $c$ is damping coefficient, $k$ is stiffness, $p(t)$ is external force, and $u$, $\dot{u}$, $\ddot{u}$ are displacement, velocity, and acceleration, respectively
• Natural frequency ($\omega_n$) of an undamped SDOF system: $\omega_n = \sqrt{\frac{k}{m}}$
• Damped natural frequency ($\omega_d$) of an SDOF system: $\omega_d = \omega_n\sqrt{1-\zeta^2}$
• Logarithmic decrement ($\delta$) is used to determine the damping ratio from the decay of free vibrations: $\delta = \frac{1}{n}\ln\frac{u_0}{u_n}$
  • $u_0$ and $u_n$ are the amplitudes of the first and $n$-th cycles, respectively
• Duhamel integral is used to calculate the response of an SDOF system to an arbitrary forcing function
• Rayleigh damping is a common method for constructing the damping matrix in MDOF systems, which assumes the damping matrix is a linear combination of the mass and stiffness matrices

Earthquake Characteristics and Ground Motion

• Earthquakes are caused by the sudden release of energy due to the rupture of rock along a fault
• Seismic waves propagate from the source of an earthquake and cause ground motion at the Earth's surface
  • Body waves (P-waves and S-waves) travel through the interior of the Earth
  • Surface waves (Rayleigh waves and Love waves) travel along the Earth's surface
• Ground motion is characterized by amplitude, frequency content, and duration
  • Peak ground acceleration (PGA), peak ground velocity (PGV), and peak ground displacement (PGD) are common measures of ground motion amplitude
• Seismometers record ground motion as a function of time, producing seismograms
• Accelerograms are time histories of ground acceleration recorded during earthquakes, which are used as input for structural analysis
• Intensity measures (Mercalli scale) and magnitude scales (Richter scale, moment magnitude) are used to quantify the strength of an earthquake

Single Degree of Freedom Systems

• SDOF systems are the simplest models for studying structural dynamics, consisting of a mass, spring, and damper
• The response of an SDOF system to ground motion is governed by its natural frequency, damping ratio, and the characteristics of the ground motion
• The equation of motion for an SDOF system subjected to ground acceleration $\ddot{u}_g(t)$ is: $m\ddot{u} + c\dot{u} + ku = -m\ddot{u}_g(t)$
• The transfer function $H(\omega)$ relates the Fourier transforms of the input (ground motion) and output (structural response) of an SDOF system: $H(\omega) = \frac{1}{-\omega^2m + i\omega c + k}$
• The impulse response function $h(t)$ is the inverse Fourier transform of the transfer function and represents the response of an SDOF system to a unit impulse
• Convolution integral relates the input ground motion to the output structural response using the impulse response function: $u(t) = \int_{0}^{t} h(t-\tau)\ddot{u}_g(\tau)d\tau$
• Elastic and inelastic response spectra are used to characterize the maximum response of SDOF systems with different natural frequencies and damping ratios subjected to a given ground motion

Multi Degree of Freedom Systems

• MDOF systems are used to model more complex structures, such as multi-story buildings or bridges
• The motion of an MDOF system is described by a set of coupled differential equations: $\mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{K}\mathbf{u} = \mathbf{p}(t)$
  • $\mathbf{M}$, $\mathbf{C}$, and $\mathbf{K}$ are the mass, damping, and stiffness matrices, respectively
  • $\mathbf{u}$, $\dot{\mathbf{u}}$, and $\ddot{\mathbf{u}}$ are the displacement, velocity, and acceleration vectors
• Modal analysis is used to decouple the equations of motion by transforming them into a set of independent SDOF systems (modal equations)
  • Mode shapes ($\boldsymbol{\phi}_i$) represent the relative displacement of the DOFs in each mode of vibration
  • Modal frequencies ($\omega_i$) and modal damping ratios ($\zeta_i$) characterize the dynamic properties of each mode
• Modal superposition method combines the responses of the individual modes to obtain the total response of the MDOF system
• Lumped mass and consistent mass matrices are used to represent the distribution of mass in an MDOF system
• Stiffness matrix is obtained from the force-displacement relationships of the structural elements (beams, columns, etc.)

Response Spectrum Analysis

• Response spectrum analysis is a method for estimating the maximum response of a structure to earthquake ground motion without performing a time history analysis
• The response spectrum represents the maximum response (displacement, velocity, or acceleration) of a family of SDOF systems with different natural frequencies and damping ratios subjected to a given ground motion
• Acceleration response spectrum $S_a(T, \zeta)$ is the most commonly used in seismic design and is a function of the natural period $T$ and damping ratio $\zeta$
• Modal response spectrum analysis combines the response spectra of the individual modes using methods such as the square root of the sum of squares (SRSS) or the complete quadratic combination (CQC)
• Design response spectrum is a smoothed and idealized representation of the response spectrum used in seismic design codes
  • Often specified as a function of soil type, seismic zone, and importance factor of the structure
• Spectral acceleration $S_a$, spectral velocity $S_v$, and spectral displacement $S_d$ are related by the natural frequency $\omega$: $S_v = \omega S_d$ and $S_a = \omega^2 S_d$
• Seismic base shear $V$ is calculated using the spectral acceleration and the effective modal mass participation factors $\Gamma_i$: $V = \sum_{i=1}^{N} \Gamma_i S_a(T_i, \zeta_i) W$, where $W$ is the total weight of the structure

Time History Analysis

• Time history analysis involves subjecting a structure to a recorded or simulated earthquake ground motion and analyzing its response over time
• The equation of motion for an MDOF system subjected to ground acceleration $\ddot{u}_g(t)$ is: $\mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{K}\mathbf{u} = -\mathbf{M}\mathbf{I}\ddot{u}_g(t)$, where $\mathbf{I}$ is the influence vector
• Numerical integration methods, such as the Newmark-beta method or the Hilber-Hughes-Taylor (HHT) method, are used to solve the equations of motion step-by-step in the time domain
• The Newmark-beta method assumes a variation of acceleration over a time step and uses parameters $\beta$ and $\gamma$ to control the stability and accuracy of the solution
• Nonlinear time history analysis accounts for material and geometric nonlinearities, such as the formation of plastic hinges or P-delta effects
  • Requires the use of nonlinear constitutive models for materials (e.g., concrete, steel)
  • Incremental-iterative solution strategies, such as the Newton-Raphson method, are used to solve the nonlinear equations of motion
• Selection and scaling of ground motion records are critical for obtaining reliable results from time history analysis
  • Records should be selected based on the seismicity of the site and the characteristics of the structure
  • Scaling is performed to match the intensity and frequency content of the design response spectrum

Seismic Design Considerations

• Performance-based seismic design aims to ensure that structures meet specific performance objectives under different levels of seismic hazard
  • Performance objectives are defined in terms of acceptable levels of damage and functionality (e.g., immediate occupancy, life safety, collapse prevention)
  • Seismic hazard is characterized by the probability of exceeding a certain level of ground motion intensity over a given time period
• Capacity design is a design philosophy that aims to control the location and mechanism of structural failure by ensuring a hierarchical strength relationship between structural elements
  • Ductile elements (e.g., beams) are designed to yield before brittle elements (e.g., columns) to prevent sudden and catastrophic failures
• Ductility is the ability of a structure to undergo large deformations without significant loss of strength
  • Ductility capacity is quantified by the ductility factor $\mu$, which is the ratio of the maximum displacement to the yield displacement
  • Ductility demand is the actual ductility experienced by a structure during an earthquake
• Energy dissipation devices (e.g., viscous dampers, friction dampers) and seismic isolation systems (e.g., lead-rubber bearings) are used to reduce the seismic demand on structures
• Soil-structure interaction (SSI) refers to the mutual influence between the ground and the structure during an earthquake
  • SSI can affect the natural frequencies, damping, and seismic forces experienced by the structure
  • Foundation flexibility and kinematic interaction are important aspects of SSI

Practical Applications and Case Studies

• Seismic retrofit of existing buildings involves strengthening or modifying structures to improve their seismic performance
  • Techniques include the addition of shear walls, braces, or dampers, and the strengthening of columns and beam-column joints
• Seismic design of bridges considers the unique characteristics of these structures, such as their length, irregularity, and the potential for soil liquefaction
  • Isolation and energy dissipation devices are commonly used in bridge seismic design
• Seismic design of nuclear power plants requires a high level of safety and reliability due to the potential consequences of failure
  • Seismic probabilistic risk assessment (SPRA) is used to quantify the risk of seismic-induced accidents and to guide design decisions
• Seismic design of offshore structures (e.g., oil platforms) must account for the combined effects of earthquakes, waves, and currents
  • Pile-soil interaction and the potential for submarine landslides are important considerations
• Post-earthquake damage assessment and rapid screening techniques are used to quickly evaluate the safety and functionality of structures after an earthquake
  • Visual inspection, nondestructive testing, and structural health monitoring systems are used to assess damage
• Case studies of notable earthquakes (e.g., 1971 San Fernando, 1994 Northridge, 1995 Kobe, 2011 Tohoku) provide valuable lessons for seismic design and disaster preparedness
  • Analysis of the performance of structures during these events helps to identify weaknesses and improve design practices