# Effect of local nonlinearity in cohesionless soil on optimal radius minimizing fixed-head pile bending by inertial and kinematic interactions

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## Abstract

This study presents the effects of a local nonlinearity in cohesionless soil upon the optimal radius minimizing the bending strains of a vertical, cylindrical fixed-head pile embedded in a layered soil stratum in a soil–pile–structure system where the kinematic interaction dominates. The seismic deformation method (SDM) with discretized numerical models is applied since the SDM is a static numerical method that can easily consider realistic conditions of layered soil strata and the nonlinearity of the soil. In the numerical models, the local nonlinearity of the soil in the vicinity of the pile is represented by subgrade springs having bi-linear skeleton curves with a simple hysteretic loop. Various amplitudes of the lateral displacements of the soil and the lateral forces at the head of the pile are considered as numerical parameters. The results of parametric analyses reveal the presence of an optimal pile radius that locally minimizes the bending strains of the piles under strong nonlinearity of the soil, and the optimal pile radius tends to increase as the degree of nonlinearity increases. Criteria are presented for predicting the increment in the optimal radius of soil–pile–structure systems under strong nonlinearity in the soil.

## Keywords

Dynamic analysis Nonlinearity Seismic design Seismic deformation method Soil–structure interaction Soil–pile interaction## List of symbols

*a*Radius of pile

- \( {a \mathord{\left/ {\vphantom {a {H_{{{\text{opt}} .}} }}} \right. \kern-\nulldelimiterspace} {H_{{{\text{opt}} .}} }} \)
Optimal slenderness ratio

*E*_{b}Young’s modulus of bedrock

*E*_{g}Young’s modulus of soil

*E*_{gi}Young’s modulus in the

*i*th layer of soil*E*_{p}Young’s modulus of pile

*H*Length of pile

*h*_{i}Length of beam element in the

*i*th layer*I*Geometrical moment of inertia of pile

*K*_{i}Horizontal deformation of subgrade of the

*i*th layer of soil*K*_{i}^{e}Initial stiffness of the horizontal subgrade spring of the soil at the

*i*th node of pile*K*_{r}Rotational stiffness at the toe of pile

*k*_{i}Horizontal subgrade spring of soil at the

*i*th layer of soil*N*Total number of layers

*P*_{i}^{e}Ultimate strength of the horizontal subgrade spring of the soil

*P*_{ i }^{e}at the*i*th node of pile*p*_{i}^{I}Ultimate soil reaction of the

*i*th layer of soil*ϕ*_{i}Angle of shearing resistance of the

*i*th soil layer*ϕ*_{r}Phase lag of lateral load with respect to mean shear strain

*u*_{p}Horizontal displacement of pile with respect to bedrock

*u*_{si}Relative displacement of the

*i*th soil layer with respect to bedrock*V*Lateral load acting at the top of the pile

*V*_{s}Shear velocity of soil

*V*_{sb}Shear velocity of bedrock

- \( \gamma_{i}^{\prime } \)
Effective unit weight of the

*i*th soil layer*γ*_{s}Mean shear strain of soil stratum

- ν
Poisson’s ratio of soil

*ν*_{b}Poisson’s ratio of bedrock

*ρ*_{b}Mass density of bedrock

*ρ*_{g}Mass density of soil

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