Dynamic interaction of various beams with the underlying soil – finite and infinite, half-space and Winkler models

Various beams lying on the elastic half-space and subjected to a harmonic load are analyzed by a double numerical integration in wavenumber domain. The compliances of the beam–soil systems are presented for a wide frequency range and for a number of realistic parameter sets. Generally, the soil stiffness G has a strong influence on the low-frequency beam compliance whereas the beam parameters EI and m ′ are more important for the high-frequency compliance. An important parameter is the elastic length l = ( EI / G ) 1 / 4 of the beam–soil system. Around the corresponding frequency ω l = v S / l , the wave velocity of the combined beam–soil system changes from the Rayleigh wave v R ≈ v S to the bending wave velocity v B and the combined beam–soil wave has typically a strong damping. The interaction frequency ω l is found not far from the characteristic frequency ω 0 = ( G / m ′ ) 1 / 2 where an amplification compared to the static compliance is observed for special parameter constellations. In contrast, real foundation beams show no resonance effects as they are highly damped by the radiation into the soil. At medium and high frequencies, asymptotes for the compliance of the beam–soil system are found, u / P ∼ ( ρ v P a i ω ) − 3 / 4 in case of the dominating damping and u / P ∼ ( − m ′ ω 2 ) − 3 / 4 for high frequencies. The low-frequency compliance of the coupled beam–soil system can be approximated by u / P ∼ 1 / G l , but it also depends weakly on the width a of the foundation. All numerical results of different beam–soil systems are evaluated to yield a unique relation u / P 0 = f ( a / l ) . The integral transform method is also applied to ballasted and slab tracks of railway lines, showing the influence of train speed on the deformation of the track beam. The presented results of infinite beams on half-space are compared with results of finite beams and with infinite beams on a Winkler support. Approximating Winkler parameters are given for realistic foundation-soil systems which are useful when vehicle-track interaction is analyzed for the prediction of railway induced vibration.