## 2 Time-Lag Effect
### Time-Lag Effect  
Concrete is a non-homogeneous material that consists of multiple different phases with complex thermal properties; heat transfer through complex geometries common in structural applications often  
Figure 1: Photo of Lieshihe River Bridge and deployment locations of sensors.  
results in nonuniform temperature distributions [22]. Temporal, spatial, and structural characteristics exist in temperature distributions on the bridge structure. Apart from the difference in temperature distributions (between different seasons and different days), a notable feature of the time delay between temperature and temperature-induced response indeed exists. The phenomenon that the temperature-induced structural response lags following the temperature is referred to as the time-lag phenomenon of the temperature-induced response. A considerable time-lag effect between temperature and temperature-induced strain can be found in concrete box-girder bridges.  
The measured data of temperature and strain history at location S10 on 4 April 2017 is shown in Figure 2a,b, respectively. Furthermore, the correspondence of temperature and strain is plotted in Figure 2c.  
In Figure 2a, from the 24 h time history data, starting from 00:00 AM to 23:59 PM, the temperature data shows a trend of decreasing between 00:00 AM and 04:15 AM, then an increasing between 04:15 AM and 15:00 PM, and a downward trend between 15:00 PM and 23:59 PM at last. By comparing Figure 2a,b, the strain is found to follow the same overall trend as the temperature. The curve of temperature vs. corresponding strain at the same time are plotted in Figure 2c. The relationship between the two represents a fusiform annular shape, showing a significant nonlinear correlation. As a structural response, strain changes cyclically due to the process of heating and cooling from sunshine, which lags the temperature change, as a result showing the annular feature. The strain data is inevitably contaminated by some live load, such as traffic load from moving vehicles, which determine a high number of local small-period fluctuations, as is shown in Figure 2b. The strain obtained after the removal of the live load is the temperature-induced strain. The separation methods will be introduced in Section 4.1.  
Figure 2: Time history plots of (**a**) temperature, (**b**) strain, and (**c**) temperature-strain correlation curve on 4 April 2017.  
The primary cause of the temperature time-lag effect may be the hysteresis of temperature transfer, as transfer of heat throughout the concrete cross section takes time. This hysteresis manifests as uneven temperature distributions, called thermal gradients or thermal inertia effects [19]. As strain represents an overall response of the structure measured at a single point, considering only one temperature measurement point may cause some inconsistencies. This paper investigates the temperature time-lag effect in concrete box-girder bridges, but further research is required to reveal the ultimate cause of this phenomenon. It's worth noting that the time-lag effects mentioned in this paper all refer to the time scale of a single day. Considering that the daily trend of temperature is most distinct and has a direct influence on the structure, this paper mainly deals with the daily time-lag effect.