# A New Method for Evaluating the Bearing Capacity of the Bridge Pile Socketed in the Soft Rock
## 2 Research Methodology  
### The Calculation Method of the Vertical Fundamental Frequency of Bridge Piers  
The multi-component complex system consists of the mass element and the deformation element. The mass element is the mass state of each part of the system, and the deformation element is the deformation state of each part of the system. Each mass element and deformation element can be combined as a single subsystem, and its fundamental frequency can be calculated by the Rayleigh energy method. Then, the fundamental frequency of the whole multi-component complex system is synthesized by the frequency synthesis method. As the natural frequency calculated by the energy method is the upper-limit value of the fundamental frequency, the frequency calculated by the combination of the Rayleigh energy method and frequency synthesis method may not necessarily be the lower-limit solution. However, according to practical applications, this method is considered accurate in calculating the spatial fundamental frequency of bridge piers. Because of the pile side resistance, the pile end resistance, and the compressive capability of concrete, the pile foundation has vertical stiffness. The stiffness also acts on the pier bottom and gives the pier bottom a specific vertical restraint stiffness \(K_{v}\). In the condition of the bare pier, the constraint stiffness can be expressed by vertical translational spring, and the top of the pier is unconstrained. As shown in Figure 1, the pier displacement caused by vertical vibration consists of two parts: the pier's vertical translation \(x_{m}(z)\) and the pier's vertical deformation \(x_{n}(z)\). The vertical translation amplitude and the vertical deformation amplitude of the pier are expressed as \(c_{m}\) and \(c_{n}\), respectively. As the mass and stiffness of piers are continuously distributed, it will result in low accuracy when calculated by the discrete multi-degree-of-freedom method. The Hamilton principle is used to derive the vibration shape function of the pier with infinite degrees of freedom. As shown in Figure 1, \(H\) is the pier height, \(m\) is the continuous mass, \(EA\) is the vertical stiffness, \(M\) is the mass at the pier top, \(m_{c}\) is the mass of the platform, and the \(h\) is the height of the platform. With the vertical vibration caused by the distributed load on the pier top, the vertical displacement on the central axis of the pier is expressed as \(u(z,t)\). The initial position under the self-weight of the pier is the equilibrium position. The kinetic energy \(T\) and the potential energy \(V\) of vertical vibration of the pier are expressed as:  
\[T=\frac{1}{2}\int_{0}^{H}m\bigg{(}\frac{\partial u}{\partial t}\bigg{)}^{2}dz \tag{1}\]  
\[V=\frac{1}{2}\int_{0}^{H}EA\bigg{(}\frac{\partial u}{\partial z}\bigg{)}^{2}dz \tag{2}\]  
Without considering the work done by an external load, according to [PERSON]'s principle, there is an equation defined as:  
\[\int_{t1}^{t2}\delta(T-V)dt=0 \tag{3}\]  
where \(t1,\ t2\) are any two moments. Introducing Equations (1) and (2) into (3) gives the equation:  
\[\int_{t1}^{t2}\int_{0}^{h}\bigg{[}EA\frac{\partial^{2}u}{\partial t^{2}}-m \frac{\partial^{2}u}{\partial z^{2}}\bigg{]}\delta udzdt=0 \tag{4}\]  
Figure 1: The vertical translation and deformation of the pier. For any \(\delta u\), Equation (4) can be expressed as the bridge pier's partial differential equation of vertical vibration. \[m\frac{\partial^{2}u}{\partial t^{2}}-EA\frac{\partial^{2}u}{\partial z^{2}}=0 \tag{5}\]  
In \(u(z,t)=\varphi(z)q(t)\), \(\varphi(z)\) is the vertical vibration shape function, and \(q(t)\) is the primary coordinate function, which represents the vibration amplitude of the vertical pier vibration. Substituting \(u(z,t)\) into Equation (5) and using the separating variables method gives the following equation:  
\[\frac{EA\varphi(z)^{\prime\prime}}{m\varphi(z)}=\frac{\tilde{q}(t)}{q(t)} \tag{6}\]  
The left side of Equation (6) is a function of the coordinate \(z\), and the right side is a function of time \(t\).