S - Kam Ng PhD Dissertation Final.pdf - Digital Repository of CCEE ...

242.3.3. Wave mechanicsThe principle of wave mechanics is the basis for the Case method derivation thatdetermines the static pile resistances. As described by Rausche et al. (1985), when a uniformelastic rod of cross sectional area (A), elastic modulus (E), and wave speed (C), is axiallyloaded by an impact force, the relationship between the force (F(t)) in the rod and thevelocity of particle motion (v(t)) can be expressed using Eq. (2.3) as long as there are noresistance effects on the rod or no reflections arrive at the point under consideration. Thetermis also known as rod impedance (Z).( ) ( ) ( ) ( ) (2.3)where,F(t) = force in a uniform rod, kN or kip,E = elastic modulus of a uniform rod, kN/mm 2 or ksi,A = cross sectional area of a uniform rod, mm 2 or in 2 ,v(t) = particle velocity in the a uniform rod, m/s or ft/s,C = wave speed of a uniform rod, m/s or ft/s, andZ = rod impedance, kN-s/m or kip-s/ft.The wave speed can be expressed in terms of mass density (ρ) and elastic modulus(E) of a uniform rod material using Eq. (2.4). The detailed derivative of this equation isgiven by Timoshenko and Goodier (1951).√ (2.4)where,C = wave speed of a uniform rod, m/s or ft/s,E = elastic modulus of a uniform rod, kN/m 2 or ksf, andρ = mass density of a uniform rod; kN-s 2 /m 4 or kip-s 2 /ft 4 .