Authored by Kun Cai*
Abstract
Based on continuum topology optimization, a subdomain approach for achieving optimal pile layout in a composite pile foundation (CPF) is presented. In general, the computational time is excessively long when the span of a CPF is far greater than the depth of pile and the nonlinearity of soils and soil-pile contact are considered. To reduce computational time, we suggest dividing the original pile foundation into a number of subdomains. Only a few representative subdomains (RSs), such as the corner representative subdomain (CRS), side representative subdomain (SRS), and inner representative subdomain (IRS), are required to find the optimal pile layout through topology optimization. In optimization, the pile volumes, maximum settlement, and settlement uniformity of foundation surface are considered simultaneously. Numerical experiments are presented, and the validity of the algorithm is verified. In the present method, a CPF with an initial uniform layout of piles of the same length becomes the foundation for heterogeneous layouts and long/short piles. The final design also depends on the inter-pile distance.
Keywords: BComposite foundation; Topology optimization; Pile foundation; Subdomain method; Settlement
Introduction
Composite foundation is widely used in civil and hydraulic engineering [1,2]. Accurate estimations of pile-bearing capacity and settlement are required in the design of a pile foundation. Hence, in the design of a composite pile foundation (CPF), two major factors are usually considered. One is the materials properties of soil and piles comprising the CPF; the other is the layout of piles in the foundation. To improve the bearing capacity of a CPF more piles can be added or the friction between soil and pile can be increased. These two factors have received considerable attention in the study of CPFs through in-situ experiments. For instance, Wang et al [3] presented an analysis of the bearing capacity of piles and soil in a multiple composite foundation using an in-situ experiment. Tamura et al [4] applied cyclic lateral-loading centrifuge tests to investigate the effect of a new pile added at the center of existing piles on the new pile’s lateral resistance. Bao et al [5] studied the seismic enhancement effect of group-pile foundation on the partial ground improvement using both a shaking table test and numerical simulation. Subsequently, Suzuki et al [6] simulated the pile-soil effect under seismic load and revealed the effects of loading distribution on the mechanical behavior of piles. Gotman & Khurmatullin [7] experimentally investigated the performance of vertically loaded soil-cement piles installed by a jet-grouting procedure in clayey soils.
Besides in-situ experiments, numerical simulations have often been adopted in practical design to investigate the response of a CPF using mechanical models of soil and pile and the coupling effect between soil and pile built-in in-situ test. For example, Ge et al [8] studied the response of a long/short-pile composite foundation under certain loading using finite element method. Bao et al [5] studied the seismic enhancement effect of group-pile foundations using a partial ground improvement method. Shahin [9] investigated the load-settlement response of steel driven piles under axial load using CPT-based recurrent neural networks. Hariri-Ardebili [10] used a simulation method to predict the effects of a nonlinear foundation on the crack of either a gravity dam or an arch dam under seismic load. Zhou et al [11] analyzed the stress and displacement near a rectangular rigid plate supported by laterally loaded rectangular cross-sectional piles in an infinitely elastic soil using a conformal mapping method. Using finite element software ABAQUS, Rose et al [12] presented a numerical investigation of the dependency of pile-group behavior on such factors as number of piles, pile spacing, length/diameter ratio, and soil strength. To obtain a more effective and rational reinforcement foundation in water, Isobe et al [13] proposed a steel pipe sheet pile reinforcement method, and the static lateral bearing capacity and seismic performance of reinforced foundations were investigated by three-dimensional elastoplastic finite element method. Gotman [14] examined the stress-strain state of a horizontally loaded foundation with inclined precast piles sunk into a tamped-in pit and proposed a computational scheme.
In the traditional design of a CPF, bearing capacity and settlement are considered separately. In fact, soil resistance and settlement have a coupling effect and, therefore, simultaneous consideration of the two factors should produce a more reasonable CPF design [2]. In particular, the control of uniform settlement is more significant. Traditionally, the piles are laid out with the same spacing. Sometimes the lengths of piles are identical. It should be mentioned that the pile-group effect leads to non-uniform settlement of such pile foundations [15]. With this feature as inspiration, optimization of pile layout, both of positions and lengths of piles, is important for controlling uniformity of settlement. Continuum optimization theory [16-20] is a powerful computational tool for finding the optimal layout of a CPF. Following this line, Sheng and Qiu [21] developed a topology optimization method that could achieve the optimal layout of equal-length piles in a foundation. However, their method could only identify the positions of a few piles. Chan et al [22] carried out the pile group optimization using fully stressed design-controlled genetic algorithm. The major reason for this deficiency is that the computational cost is too high in the pile layout optimization of a CPF. The computational cost usually depends on three factors: (1) the foundation size may be far greater than the pile length; (2) material nonlinearity and state nonlinearity of soils and piles may exist, leading to a large amount of structural reanalysis when searching for an acceptable deformation configuration of the foundation; (3) the number of design variables in topology optimization may be too large for it to be solved directly by any traditional method. To bypass these difficulties, in the present study a subdomain method integrated with topology optimization models is proposed for achieving the optimal pile layout in a CPF with uniform settlement.
Methodology
Mesh scheme of subdomains in a CPF
In the pile layout design of a CPF, the whole structure is assumed to be composed of a slab foundation, piles and soils. When such a structure is used in an airport or a highway, its horizontal size is far greater than the depth of its piles. If we take the whole structure as a design domain, the computational cost would be too high to be acceptable in practical engineering. Furthermore, the final pile layout obtained by such approach is commonly not feasible for the final design, because the pile layout is significantly non-uniform, which can lead to severe differences in settlement of the foundation. Therefore, a critical need exists to reduce the computational cost before carrying out pile layout optimization. It is noted that, in continuum topology optimization, the computational cost depends on two major factors. The first is the analysis of structural deformation and the second is the updating of design variables. Based on this understanding, a subdomain approach is proposed in the present study to reduce computational time. The major principle of the subdomain method is to mesh the whole CPF into many square subdomains first (Figure 1). Considering computational cost and accuracy, the lengths of horizontal sides of each subdomain should be no more than 10 times of the minimal pile distance. On the other hand, to demonstrate variation of topology of pile layout in a subdomain, the horizontal side lengths should also be no less than 3 times of the minimal pile distance. Then, these subdomains are classified into 3 groups, corner subdomains, side subdomains and inner subdomains. In each group, at least one RS should be chosen. As an example, the foundation shown in Figure1A is meshed with 4 corner subdomains (dark blue), 18 side subdomains (yellow) and 18 inner subdomains (light blue). Each group has one RS, namely a corner RS (CRS), a side RS (SRS), and an inner RS (IRS). The boundaries of the three RSs are illustrated in Figure 2. Next, we need to determine the optimal layout of piles in each RS using topology optimization. Finally, we assemble the final piles layout in the whole foundation using the RSs.
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