## 1 Introduction
Therefore, in addition to the sorption rate between the two phases of water vapour and bound water, also the sorption rate between the free water and bound water phases, as well as the evaporation/condensation rate between the free water and the water vapour phases, need to be defined. These phenomena are schematized in Figure 2. For a complete description of the moisture transfer in wood, a sorption hysteresis characterized by two isotherms of adsorption and desorption was originally introduced in the multi-Fickian model by [PERSON] [17]. In the present work, only case-studies with moisture states below the FSP are studied. As discussed by [PERSON] et al. (2018) and [PERSON] et al. (2019), high values of moisture gradients due to high yearly and daily variations of relative humidities are the main causes of moisture induced stresses (_MIS_) perpendicular to the grains in wooden members. Larger yearly variations of \(RH\) (average values above 80%) and larger moisture gradients and _MIS_ in timber cross sections were found under Northern European climates when compared to Southern European climates (2019). In (2019, 2019) it was observed that, under Norther climates, high gradients in the vicinity of surfaces of bridge gulam beams during drying periods of the year are caused by high peaks of \(RH\) (above 85%) in conjunction with high daily variations of \(RH\) (above 50%). Knowledge on moisture gradients is therefore important to identify the zones prone to crack risk in wooden components, as shown in (2019) for the case of a bridge gulam beam where the _MIS_ were also calculated and discussed in relation to the moisture gradients. In (2019) it was found that the most critical _MIS_ are the tensile stresses perpendicular to the grain that can be also greater than the limits prescribed by the Eurocodes. Due to this, the uncoated bridge wooden beams may be exposed to the formation of moisture induced cracks and delamination. The structural significance of cracks in timber bridges under outdoor environments is discussed also in (2029) where the asymmetric damage (longitudinal splitting cracks) is especially investigated. Models for moisture transport, coupled with mechanical models, can be used to calculate the moisture induced cupping in stress laminated timber decks. The models can also allow the evaluation of the bar force losses during time, as shown in Section 4 of (2019), were a single-phase model for moisture transport was used. The novelty of the present paper is the use of a recent multi-phase model, proposed by some of the authors in (2019), to assist the monitoring of SLTDs of bridges under Nordic European climates carried out by integrated humidity-temperature sensors. In particular, the hygro-thermal monitored data are collected from a previous study of the untreated deck of Sorliveien Bridge in Norway (2019, 2020) and from the on-going monitoring of the painted and thick deck of Tapiola Bridge in the city of Espoo, Finland. Untreated and painted bridges are interesting cases to study in terms of their hygro-thermal performance. In this paper, the monitoring systems of the Sorliveien and Tapiola bridges are presented, and selected measurements are used for simulation by the finite element method (FEM). While the monitoring provides the \(RH\) and \(T\) in some locations of the analysed decks, the numerical model completes the health monitoring providing the overall hygro-thermal response of a representative volume of the deck in terms of distribution of \(MC\), vapour  
Figure 2: Scheme of the water phases and sorption phenomena in wood. **(a)** Below the fibre saturation point (FSP): bound water in the wood cell walls, water vapour in the lumens and sorption rate between bound water and water vapour phases (\(\hat{c}_{\text{lo}}\)).