Closure to “Kinematic Bending of Fixed-Head Piles in Nonhomogeneous Soil” by Raffaele Di Laora and Emmanouil Rovithis

Abstract

The writers would like to thank the Discussers for their comments and solutions, which contribute to the state of the art on the topic. The writers believe that the closed-form solution for active pile length derived by the Discussers in Eq. (3) is helpful in computations, especially when non-typical values of problem parameters are examined (e.g., very high pile–soil stiffness ratios). In the latter conditions, if no iterations are performed to assess the active pile length via the approximate procedure suggested in the original paper, the errors in the computed bendingmoment could reach 5–10%. The writers also agree with the Discussers that the absolute value of pile-head bending moment fluctuates with frequency, yet this is true if soil amplification is taken into account (e.g., when acceleration is specified at rock level). In this case, the absolute value of bending moment fluctuates with frequency due to the fluctuation in ground surface acceleration. In the original paper, the dynamic modification factor employs a pre-specified soil surface acceleration to allow (1) a straightforward physical interpretation, and (2) a direct implementation in design codes, which often prescribe a value for surface acceleration that encompasses (frequency-dependent) site effects. With reference to the decrease of the dynamic reduction factor with frequency, while the writers agree as to the validity of the formula derived by the Discussers [Eq. (13)], they believe that it should not be correlated with an “increased dynamic resistance of the pile, mainly due to inertia,” as mentioned by the Discussers. This is evident in Eq. (13) in the discussion, which does not involve pile mass in the right-hand side. The decreasing trend is rather explainable by Fig. 11 in the original paper: the excitation frequency modifies the distribution of soil shear strain with depth, indicating that soil deformations loading the pile are the only source of frequency effects. Pile inertia is proved to be negligible in the kinematic interaction problem, as also recognized by Anoyatis et al. (2013). In other words, although the development of pile bending moments is a frequency-related phenomenon, the dynamic effect is limited to the dynamic response of soil, while pile–soil interaction is essentially a static phenomenon. In a Winkler framework, applying statically, at any time step, the free-field soil displacements (derived in the dynamic regime) to the pile-supporting springs will produce accurate predictions of kinematic pile bending. Finally, the writers correctly highlight that, due to a clerical error in Eq. (22), the value of surface acceleration is missing. The correct solution for shear strain reads