## 3 Modelling of Wave Propagation in Wooden Poles
### Modelling of Soil-Pole Interaction  
Interaction behaviour simulation between soil and wood utility pole is another important issue in modelling since it will affect the results directly. In the FE analysis, the modelling of interaction behaviour can be realised via the contact analysis considering different contact conditions, in which the restraints and contact conditions should be carefully defined ahead of time. There are two commonly used contact analysis methods reported in FE modelling. One is perfectly bonded contact analysis, which is on basis of small deflection or rotation and supposes that two components are bonded together without friction [32]. The other one is frictional contact, which indicates that the contact surfaces could be divided and regarded as nonlinear behaviour. There are two main difficulties during the analysis: (1) the contact region is not predictable due to the many factors such as loading, material properties and boundary conditions; (2) the friction should be considered and the responses may be chaotic, which will make the solution not convergent. In this study, both bonded and frictional contacts are employed to demonstrate the contact state in ANSYS. The Lagrange Multiplier method is applied on the contact normal while the penalty method is applied on the frictional plane, which allows a very small amount of slip for a sticking contact condition. Moreover, the boundary condition of wood pole plays a significant role on the wave propagation behaviour. The hard boundary will change the phase direction of wave propagation compared with the soft boundary. To clearly illustrate this phenomenon, two numerical models with the length of 12 m and average diameter of 0.3 m are created. The model with a free end is achieved by modelling the pole hanging through two steel cables while the fixed boundary is achieved by coupling the contact node of the pole and bedrock. The hammer impact is applied on the top centre of the pole and the sensor is deployed at the top side of the pole, as shown in Figure 3. The mesh conditions are exactly same for the contact surface to make use of the coupling method, shown in Figure 4.  
The friction interface representing high nonlinear behaviour allows modelling of slipping and gapping between the soil and the pole, therefore taking more calculation time and computer resources. However, if the slipping and gapping is negligible or will not affect the analysis results, perfect bonding can be considered [23; 33]. In this study, negligible slipping and gapping between soil and pole is caused by low strain testing. In this case, perfect bonding is considered to simulate the interaction  
Figure 3: Finite Element (FE) model with the fixed boundary. behaviour, which is achieved by coupling the contact node of the pole and bedrock. Figures 4 and 5 show the mesh conditions of FE model for the contact surface. A measurement sensor is located on the top of the pole and a hammer excitation is applied on the top centres of two models. The material properties for different structural components are shown in Table 3. Figure 5 compares the results of wave behaviour from two different boundary conditions. It is noticeable that the incident waves, directly generated by the hammer impact, uniformly propagate until they arrive at the boundary of the pole. For the free end boundary condition, the wave reflects back from the boundary and keeps the same phase direction as shown by the red solid curve in the figure: all the peaks are in the same direction. However, for the bedrock boundary condition, when the wave reflects back, the phase direction is opposite to the incident wave and the phase changes again when it gets to the hard boundary as illustrated by the blue dashed curve: the peak directions are opposite to each other. This phenomenon can be intuitively explained in accordance with [PERSON]'s wave theory: when the wave pulse approaches the fixed end, the internal restoring forces which allow the wave to propagate exert an upward force on the end of the pole [34].