Check our first post on Seismic Waves to get some background on the information discussed in this post. Be sure to comment and contact us!
Stiffness
The total stiffness of a single-degree of freedom (SDOF) system is calculated as:
Total K = F/Δ (kips/in, lbs/in)
Delta (sometimes represented as Gamma) is the deflection of the member, measured in inches.
The stiffness, K, and deflection, Δ, of a fixed base-free top or pinned base-fixed top system equate to:
K = 2EI/h^3 and Δ = Fh^3/3EI
and a fixed base-fixed top system:
K = 12EI/h^3 and Δ = Fh^3/12EI
where,
E = Modulus of Elasticity
I = Moment of Inertia
The stiffness of the system can be applied to the frequency and the period of a given system, where:
Angular Natural Frequency and Period,
ω=2πf=2π/T=√(k/m)=√(kg/W) (rad/sec) and
T=1/f=2π/ω (sec)
Linear Natural Frequency and Period,
f=1/T (Hz, cycles/sec) and
T=2π√(W/kg) (sec)
where,
W = Weight of the Structure
g = 32.2 ft/sec^2 = 386.4in/sec^2
Tedious as it may be, these are core concepts for designing structural systems with seismic load effects.
Damping
Damping is the dissipation of energy from an oscillating system. Critical damping, Bcrit, is the amount of damping that the system to equilibrium in the shortest amount of time. Underdamped and overdamped bring the system back to static position over longer periods of time.
The Critical Damping Ratio, β, is defined as:
β = B/Bcrit
Defined sources of damping include external and internal viscous damping, body friction damping, radiation damping, and hysteretic damping.
Considering the equations mentioned above, when damping slows the oscillation of the system, its period increases slightly. Typical damping ratios range from 2% for welded steel structures, 5% for concrete structures, 10% for masonry shear walls, up to 15% for wood structures.
Multiple Degrees of Freedom System
In design of multi-story buildings and systems with multiple degrees of freedom, the mode of vibration with the longest period is called this first fundamental mode (Mode 1). Mode 1 has the longest period, T1, which equates to the lowest frequency. Other modes with short-period of vibration are termed higher modes (e.g. Mode 2, 3, etc.).
