Pile raft foundation (PRF) is a common foundation type for buildings and bridge piers which has been commonly subjected to eccentric vertical load in engineering applications. Dissimilar PRF is often adopted to reduce the excessive settlement and differential settlement of superstructures. The behavior of dissimilar PRF under eccentric vertical load is a significant issue and investigated with the boundary element method in this paper. In this method, the dissimilar pile–soil system is decomposed into extended soil elements and fictitious pile elements. The second kind of Fredholm integral governing equation of the axial force of fictitious piles is established based on the compatibility condition of axial strain between the extended soil and fictitious piles. An iterative procedure is adopted to analyze the average settlement w and rotation slope θ of raft stemming from the settlement compatibility condition of the top of each element and the equilibrium condition of the raft. Furthermore, the axial force and settlement of each element along its depth can be predicted. The corresponding results agree well with a reported case and the finite element method. The characteristics of 3 × 1 and 3 × 3 dissimilar PRFs under eccentric vertical load, including non-dimensional vertical stiffness N₀/wE_sd, differential settlement w_d and the load sharing ratio of typical elements N_i/N₀, are systematically investigated by considering different eccentricity e, length/diameter ratios of pile l/d and pile–soil stiffness ratio E_p/E_s conditions. The N₀/wE_sd increases with l/d, while the load sharing ratios of the raft N_raft/N₀ and w_d decreases with l/d. The eccentricity e has a significant effect on w_d and N_i/N₀ and a neglect effect on N₀/wE_sd and N_raft/N₀. The N₀/wE_sd, w_d and N_i/N₀ are significantly increased with E_p/E_s. This research is expected to provide insights to the practitioners into the dissimilar PRF design under eccentric vertical load.

Pile composite foundations (PCFs) have been commonly applied in reinforcement engineering to adjust the vertical stiffness of foundations, due to the displacement control design criteria for foundations. PCFs that have dissimilar pile lengths, located over inclined bedrock, have shown significantly different vertical behaviors from PCFs with equal pile lengths, placed over a semi-infinite medium. However, the vertical behaviors of dissimilar PCFs over inclined bedrock cannot be predicted with the current theoretical methods, although they have been widely adopted in engineering. An analytical method is proposed in this investigation to analyze the vertical bearing characteristics of dissimilar PCFs over inclined bedrock. A pile–soil system is decomposed into fictitious piles and extended soil, and then a control equation to determine the axial force along the fictitious piles is established, stemming from the compatibility conditions between them. The vertical behaviors of dissimilar PCFs can be obtained by solving the control equation with iterative procedures, and the equation is verified by two field load tests of single piles from the Honghe bridge and a numerical case. Then, the settlement and load transfer behaviors of 3 × 1 dissimilar PCFs and their influence factors are analyzed, and the results are as follows. (1) Obvious differences can be observed concerning the axial force distribution, settlement w, and load-sharing ratio (LSR) of each pile element for different pile–soil stiffness ratios (E_p/E_s). (2) The LSR of pile 1 increases from 0.074 to 0.253 for the rigid pile and from 0.062 to 0.161 for the flexible pile condition when the cushion stiffness K_c changes from 1 × 10⁴ kN/m to 3 × 10⁸ kN/m. The non-dimensional vertical stiffness of the foundation, N₀/wdE_s, increases from 10.21 to 28.95 for the rigid pile condition and increases from 8.69 to 14.44 for the flexible pile condition, when K_c increases from 1 × 10⁴ kN/m to 4 × 10⁵ kN/m. (3) The neutral layer depth of the pile z_n, the average settlement w, and the differential settlement w_d of each element head decrease with K_c, and no negative friction zone exists (z_n = 0 m) for all the pile elements when K_c> 2 × 10⁵ kN/m. (4) The N₀/wdE_s decreases with the distance between the pile bottom and the inclined bedrock Δ. For the rigid and flexible pile conditions, the N₀/wdE_s is 22.16 and 13.48 for Δ = 1 m, and 13.13 and 10.10 for Δ = 8 m. The w_d reaches 16.7 mm and 4.0 mm for Δ = 1 m and Δ = 8 m, respectively. (5) The N₀/wdE_s increases almost linearly with an increase in l/d for the rigid pile condition, and it gradually decreases for the flexible pile condition. The developed model can improve the design and analysis of PCFs located over inclined bedrock under vertical loading.