## 3 Verification by Numerical Simulation
### Simulation Result Analysis  
Using the software _ANSYS_ (ANSYS, Inc., Canonsburg, PA, USA) [25], a simply supported beam, a clamped-clamped beam and three-span continuous beams are simulated. The applicability of the deflection calculation method for three loading cases was analyzed in these simulations. The average strain distribution along the beams (40 cm of sensor gauge length) was extracted using Equation (9).  
Under the concentrated loads, the simply supported and clamped-clamped beams were under two point loads P\({}_{1}\) and P\({}_{2}\) at 1/4 and mid-span of the beams, respectively. The deflection distribution along the beams is shown in Figure 4(a),b when P\({}_{1}\) and P\({}_{2}\) are equal to 2 kN and 4 kN, respectively. For Figure 4(c), a point load of P = 10 kN was exerted at the middle of the second span of the continuous beam. \"Actual deflection\" means directly extracted deflection from the nodes of finite element models. The relative errors between the predicted and actual deflection at the middle of the simply supported, clamped-clamped and three-span beams were 1.05%, 4.40% and 5.73%, respectively.  
Under the even load, a uniform load q of 300 N/m is exerted on these beams. The vertical deflection distribution along the beams is demonstrated in Figure 5(c). From the results of the analysis, the relative errors of deflection at mid-span of the simply supported and clamped-clamped beams are  
Figure 4: Strain and rotation angle.  
Figure 5: Deflection distribution along the different beams under point loads.  
0.9% and 4.4%, and the relative errors at the middle of each span of the continuous beam are 4.21%, 10.53% and 4.20%, respectively. The reason for the relative error of the second span is larger than those of other spans is that deflection of the second span under the uniform load is far less than that of other spans (Figure 6c).  
According to the simulation, the presented method is applicable to diverse forms of beams and different types of static loads. Generally, the relative error between the actual and predicted deflection for simply supported beams is less than those of clamped-clamped and continuous beams. This could be a result of the actual strain distribution having been substituted by average strain distribution in the method, and the relative error between the actual and average strain near the support is relatively larger for clamped-clamped and continuous beams subjected to the hogging moment. However, the relative error can be reduced by decreasing the gauge length of sensors near supports according to actual precision requirements.  
Under the dynamic moving load, a double-axle vehicle load with the speed of 18 km/h and wheelbase of 0.4 m are exerted on these beams, and the weight of front and rear axles is 5 kN and 10 kN, respectively. Typical strain-time histories are demonstrated in Figure 7, and time-varying deflection at 1/4 span and mid-span is illustrated in Figure 8.  
Figure 6: Deflection distribution along the different beams under uniform loads.  
Figure 8: Time-varying deflection of structures under dynamic moving loads.  
Figure 7: Typical strain-time histories of structures under dynamic moving loads.  
The applicability of the method under dynamic loads is verified based on transient analysis. During the process of a vehicle passing through the structure, the maximum relative error of deflection estimation of the simply supported beam is less than 1% and 2% at mid-span and 1/4 span, respectively. At time \(t\) = 0.65 s, the relative errors of deflection at mid-span and 1/4 span are 5.03% and 5.40% for the clamped-clamped beam, respectively. For the continuous beam, the relative error of deflection at the middle of the first span is 4.80% at time \(t\) = 0.47 s, while the error is 5.21% at the middle of the second span at time \(t\) = 1.28 s.