# Research on Lateral Bearing Behavior of Spliced Helical Piles with the SPH Method
## 2 Numerical Description  
Finite Element Analysis is a mathematical method of simulating and approximating actual physical systems for various complex geometric and loading conditions. It is a widely used and effective tool for transforming complex practical problems into simpler ones. This method involves discretizing a complete solution range into a number of elementary units, which are connected at the nodes, and the approximate solution of the problem is subsequently obtained by solving each unit. The Navier-Stokes (NS) equation governs the hydrodynamic problem, which can be written in the following formulation:  
\[\frac{d_{\rho}}{d_{t}}\ =\ -\rho\
abla\cdot\
u \tag{1}\]  
\[\frac{d\
u}{dt}\ =\ -\frac{1}{\rho}\
abla p\ +\ \frac{\mu}{\rho}\
abla^{2}\
u\ +\ g \tag{2}\]  
In the above equation, \(\rho\), \(\
u\), \(p\), \(g\) denote the density, velocity vector, pressure, gravity, and dynamic viscosity, respectively. In the particle approximations, the computational domain is discretized with a group of particles. The field functions and their derivatives can be written as:  
\[\langle f(x_{i})\rangle\ =\ \sum_{j=1}^{N}\frac{m_{j}}{p_{j}}f(x_{i})W\big{(}x_{i} \ -\ x_{j},h\big{)} \tag{3}\]  
\[\langle\
abla f(x_{i})\rangle\ =\ \sum_{j=1}^{N}\frac{m_{j}}{p_{j}}f\big{(}x_{j} \big{)}\
abla_{i}W_{ij} \tag{4}\]  
where \(\langle f(x_{i})\rangle\) is the approximate value of particle \(i\); \(f(x_{j})\) is the value of \(f(x)\) associated with particle \(j\); \(x_{i}\) and \(x_{j}\) are the positions of corresponding particles; \(N\) is the number of the particles in the support domain; h is the smooth length; \(W\) is the smoothing function and represents a weighted contribution of particle \(j\) to particle \(i\)[23]. By substituting the SPH approximation of the function and its derivative into the N-S equation, the SPH motion equation of the N-S equation can be obtained, as follows:  
\[\frac{d\rho_{i}}{dt}\ =\ \sum_{j=1}^{N}m_{j}v_{ij}\cdot\
abla_{i}W_{ij} \tag{5}\]  
\[\frac{dv_{i}}{dt}\ =\ -\sum_{j=1}^{N}m_{j}\bigg{(}\frac{p_{i}}{p_{i}{}^{2}}\ +\ \frac{p_{j}}{p^{2}}\bigg{)}\
abla_{i}W_{ij}\ +\ \sum_{j=1}^{N}\frac{4m_{j}\big{(}u_{i}\ +\ u_{j}\big{)}x_{ij}\cdot\
abla_{i}W_{ij}}{ \big{(}p_{i}\ +\ p_{j}\big{)}^{2}\big{(}x_{ij}{}^{2}\ +\ 0.01h^{2}\big{)}v_{ij}}\ +\ g \tag{6}\]  
In SPH, an artificial compressibility technique is often used to model an incompressible flow as a slightly compressible one. The artificial compressibility considers that every theoretically incompressible fluid is actually compressible. Therefore, it is feasible to use a quasi-incompressible equation of state to model the incompressible flow. In this work, the artificial equation of state is:  
\[p\ =\ c^{2}\rho \tag{7}\]  
where \(c\) is the sound speed of the concerned artificially compressible fluid (e.g., water). [PERSON] [24] argued that the relative density variation, \(\delta\), is related to the fluid bulk velocity and sound speed in the following way:  
\[\delta\ =\ \frac{\Delta\rho}{\rho_{0}}\ =\ \frac{|\rho\ -\ \rho_{0}|}{\rho_{0}} \ =\ \frac{v_{b}{}^{2}}{c^{2}}\ =\ M^{2} \tag{8}\]  
where \(\rho_{0}\), \(\Delta\rho\), \(v_{b}\) and \(M\) are the initial density, absolute density variation, fluid bulk velocity, and Mach number, respectively. \[\frac{dv_{I}}{dt}\ =\ -\sum_{j=1}m_{I}(\frac{\sigma_{I}}{\rho_{I}^{2}}\ +\ \frac{\sigma_{I}}{\rho_{I}^{2}}\ +\ \Pi_{IJ})\cdot\
abla W_{IJ} \tag{9}\]  
I corresponds to a particle, \(\sigma_{I}\) is the stress tensor in the Kelvin model representation, \(v_{I}\) is the velocity tensor, and \(\
abla W_{IJ}\) is the gradient of the kernel function associated with particle \(j\) evaluated at particle \(i\). \(\Pi_{IJ}\) is an artificial viscosity, usually set by the keyword Control-Bulk-Viscosity or the hourglass card in LS-DYNA. The additional viscoplastic strain moves the yield surface outwards with time and the yield surface gradually widens with time, which means that the mechanical properties of the soil have gradually changed.