## 1 Introduction  
Pile foundation has the characteristics of high bearing capacity, small settlement, good stability and strong adaptability, which has been widely applied in engineering. The positive skin friction (PSF) of the pile is mobilized when the pile moves downward relative to its surrounding soils. When the pile foundation is installed in soft soil area, negative skin friction (NSF) will be observed due to the consolidation of the surrounding soils. NSF develops from downward shear stresses induced by relative movements along the pile-soil interface. Generally, NSF has two effect aspects on pile foundations: the development of additional compressive axial load and excessive settlement [1, 2]. These two aspects are important indicators to measure whether the pile foundation is safe or reliable. NSF is always an important direction of the research on pile foundation.  
A series of investigations of NSF in the past have shown that down-drag forces are dependent on interaction time, soil features, and relevant displacement of pile and soil [3, 4, 5]. The essence of soil consolidation is the excess pore water pressure dissipation. In this process, there is a corresponding increase in soil settlement and effective stress. Current analytical methods for the predictions of NSF on pile foundations fall into four broad categories: the empirical method [6], the differential quadrature method [7], the load transfer method [8, 9], and the continuum method [10].  
On the other side, the finite element method (FEM) is also the main research method of the mechanical response of piles under vertical loads. [PERSON] et al. studied the bearing capacity characteristics of rock socketed short piles in weathered rock site by the FEM [11]. [PERSON] and [PERSON] utilized the FEM to predict the termination criteria of vertically loaded rock socketed bored piles [12]. [PERSON] and [PERSON] carried out FEM-BEM coupling analysis on mechanical properties, deformation, and settlement of pile-soil interface under vertical load [13]. Furthermore, [PERSON] et al. investigated NSF of a single pile by FEM and found that the distribution and magnitude of NSF is influenced mainly by the pile/soil interface, soil compressibility, and the surcharge intensity [14]. Additionally, a number of field tests on instrumented piles have been carried out in the past to study NSF [15; 16; 17]. It can be concluded that drag loads are time dependent in that they are related to the pore water pressures.  
Several studies have been focused on the estimation of the development of drag load of conventional uniform cross-section piles [18; 19; 20; 21]. The different distribution cases of NSF based on the range of fully mobilized skin friction have been discussed [22]. The distribution and magnitude of NSF considering pile-soil interaction under surcharge has also been analyzed [23] and simple design method was proposed based on a numerical investigation. The evolution of local friction mobilized at the pile-soil interface has been investigated, focusing on small amplitude and large number cycles of strain-controlled tests, corresponding to fatigue behavior [24].  
However, field experiments on NSF are time-consuming and it is difficult to simulate large surcharge loads in experimental research. In this paper, a simple theoretical analysis method is proposed based on the effective stress method (ESM). The measured data proved the reliability of the present method. The closed-form analytical solutions for calculating the pile skin friction and axial load corresponding to the plastic and elastic state were derived, respectively. Furthermore, a numerical investigation was performed to study the effects of various influencing factors, including the consolidation time, friction coefficient, drainage condition, consolidation pressure, and pressure ratio.