**2. Materials and Methods**

#### *2.1. Description of the Multi-Phase Model*

Summarizing the multi-Fickian model presented in [11] for wood below the *FSP*, the variables for transient moisture transport are the concentration of bound water in the cell walls *cb*, the concentration of water vapour in the cell lumens *cv*, and the temperature *T*. Denoting by **D***b* and **D***v* the diffusion tensors for bound water and water vapour phases and by **K** the thermal conductivity tensor, the governing equations of the problem are:

$$\frac{\partial \mathbf{c}\_b}{\partial \mathbf{f}} = -\nabla \cdot \mathbf{J}\_b + \dot{\mathbf{c}}\_{bv} \tag{1}$$

$$\frac{\partial \mathbf{c}\_{\upsilon}}{\partial t} = -\nabla \cdot \mathbf{J}\_{\upsilon} - \dot{\mathbf{c}}\_{\upsilon} \tag{2}$$

$$\mathbf{c}\_w \boldsymbol{\varrho} \frac{\partial T}{\partial t} = \begin{bmatrix} -\nabla \cdot \mathbf{J}\_H - \nabla \cdot \mathbf{J}\_b h\_b - \nabla \cdot \mathbf{J}\_v h\_v + \dot{c}\_{bv} h\_{bv} \end{bmatrix} \tag{3}$$

where ∇ is the nabla operator, **J***b* and **J***v* are the fluxes of bound water and water vapour, and **J***H* represents the thermal flux:

$$\mathbf{J}\_b = -\mathbf{D}\_b \nabla \mathbf{c}\_b \mathbf{J}\_v = -\mathbf{D}\_{\overline{v}} \nabla \mathbf{c}\_{\overline{v}}, \mathbf{J}\_H = -\mathbf{K} \nabla T \tag{4}$$

In Equation (3), *cw* represents the specific heat and the wood density, the coupling term . *cbv* is the sorption rate between the two water phases (see Figure 1), *hb* and *hv* are the specific enthalpies and *hbv* = *hb* − *hv* is the specific enthalpy of the transition from the bound water to the water vapour. The moisture content *MC* is defined as *cb*/0 where 0 is the dry wood density. The sorption rate in Equations (1)–(3) is defined as:

$$\dot{c}\_{bv} = H\_c(\varrho\_0 M \mathcal{C}\_{bl} - \mathfrak{c}\_b) \tag{5}$$

where *Hc* represents the moisture dependent reaction rate and *MCbl* is the moisture content in equilibrium with the relative humidity. In Equation (5), the *MCbl* has the meaning of temperature-dependent sorption isotherms. These are defined by using the Anderson– McCarthy model (see Appendix A). In the present work, according to [11], an average between the temperature dependent adsorption and desorption isotherms is used, while a model for sorption hysteresis is not included.

Since the bound water cannot pass the external surfaces and it is restricted in the cell walls, the model includes only exchanges of vapour and heat with the ambient air. Therefore, the first boundary condition of Equation (6) holds on all the external surfaces in relation to variable *cb*. For the other variables, the second and third boundary conditions in Equation (6) apply for the external surfaces exposed to the variable *RH* and *T*:

$$\mathbf{n} \cdot \mathbf{J}\_{\flat} = \mathbf{0}, \; \mathbf{n} \cdot \mathbf{J}\_{\upsilon} = \; k\_{\upsilon}^{w} \; c\_{\upsilon}' - \; k\_{\upsilon}^{a} c\_{\upsilon \prime}^{a} \; \mathbf{n} \cdot \mathbf{J}\_{\upsilon} = \; k\_{T} (T - T^{a}) \tag{6}$$

where **n** represents the outward normal direction to the surface, *cav* and *Ta* are the water vapour concentration and temperature of the air, *kwv* and *kav* the surface permeances corresponding to wood temperature and air temperature, and *kT* is the thermal emission coefficient. The expressions of the permeances are reported in Appendix A. In Equation (6), *cv* = *cv*/*ϕ* represents the concentration of water vapour divided by the wood porosity *ϕ*. The concentration *cv* is related to the partial vapour pressure *pv* through the ideal gas law:

$$\mathbf{c}\_{\upsilon} = \varrho \ p\_{\upsilon} M\_{H2O} / RT \tag{7}$$

where *R* is the gas constant and *M H*2*O* the molecular mass of water. The vapour pressure can be expressed as a function of the relative humidity *RH*:

$$p\_{\upsilon} = RH \cdot p\_{\upsilon s} \tag{8}$$

where *pvs* is the saturated vapour pressure given by the semi-empirical Kirchhoff expression for the thermal ranges above the freezing point and by Teten's fitting for ice in the subfreezing temperature range [26]:

$$p\_{\rm res} = \begin{cases} \exp\left(53.421 - \frac{6516.3}{T} - 4.125 \text{ ln}(T)\right) & \text{for } T \ge 0^{\circ}\text{C} \\\ 100 \times 10^{\frac{9.5(T-273.15)}{1-L\beta\Theta} + 0.7858} & \text{for } T < 0^{\circ}\text{C} \end{cases} \tag{9}$$

All material parameters of the model are summarized in Table A1 of Appendix A. The model is suitable for wooden members sheltered from rain and without water traps or other contacts with water. It does not allow the modelling of liquid water in pores and can simulate only moisture states below the FSP.

#### *2.2. Implementation of the Hygro-Thermal Model for Stress-Laminated Timber Deck in Abaqus Code*

The selected commercial finite element software Abaqus provides a comfortable environment for the 3D model construction and the evaluation of results. The finite element to be used for the hygro-thermal analysis was defined in the user subroutine UEL to accommodate the three differential equations that describe the material model. The subroutine is reading the weather data from the database of measured temperatures and air relative humidities at every time increment and applies them as external loads on the exposed model surfaces. The shape functions for 8-nodes isoparametric brick elements are used and a weak form of the governing equations and their boundary conditions with three variables per node (bound water concentration, water vapour concentration and temperature) is implemented in the UEL.

The time integration is carried out using the fully implicit Euler scheme and the nonlinear system is solved using the Newton method at each time step. The subroutine allows to implement the FEM contributions to the residual vector and to the Jacobian iteration matrix.

The general scheme for the hygro-thermal modelling of the timber deck is shown in Figure 3 and the simplifications used are the following:


**Figure 3.** Scheme of a 3D vertical slice of the timber deck for the hygro-thermal analysis. The asphalt layer is not modelled.

The initial values for the variables of the differential problem are the following:


The fluxes acting on the 3D slice of the deck are as follows:


The input material data used for both case-studies of the paper are the dry wood density 0= 450 kg/m3, the porosity *ϕ* = 0.65 and the coefficients of the diffusion tensors that are listed in Appendix A. The permeances for the uncoated wood used in the first case-study (*kw*) and for the weak paint used in the second case-study (*kp*) are listed in Table A3, and the thermal emission coefficient are listed in Table A1 of Appendix A.

The outputs are the moisture content *MC*, the vapour pressure *pv* (obtained from the water vapour concentration *cv*), and the temperature *T* in each element of the 3D model.

## *2.3. Case-Study: Sørliveien Bridge*

Sørliveien Bridge (Figures 4 and 5) is a pedestrian bridge built in summer 2005 in Akershus County, Norway, crossing a local road [24,25]. The owner was the Norwegian Public Road Administration. It is a slab bridge with eight spans and a total length of 87 m. The longest span is 17 m. The stress-laminated timber deck (48 × 333 mm) is composed of spruce glulam planks, which are untreated except for the edge planks of creosote-impregnated pine wood. The top layer consists of 60 mm asphalt with a moisture membrane of polymer modified bitumen (Topeka 4S) underneath. The bridge has been

instrumented in August 2005 and monitored since then. The instrumentation is situated at the northern end and is logged every fourth hours.

**Figure 4.** Sørliveien Bridge in Norway. (**a**) Side view with location of sensors (red circle). (**b**) Cross section showing the SLT deck with 333 mm thickness.

**Figure 5.** Sørliveien Bridge. Detail of the bottom deck with the monitoring equipment.

The monitoring equipment (Figure 5) collects data about the loading of the highstrength steel bars, temperature and humidity of the wood at different depths from the surface, and temperature and humidity of the air from the weather station positioned on the bridge. The collected data is processed directly on the embedded computing unit and regularly transmitted to the central monitoring server over the internet.

Three load cells were installed to the prestressing bars loaded to 227 kN, on the northeast side of the bridge. The measurements from load cells are not discussed in this paper, because they are not directly needed for the hygro-thermal simulations. Temperature and relative humidity were measured by ten integrated humidity-temperature sensors Vaisala Humitter 50Y [27]. This type of sensor has an operating range from −40 ◦C to +60 ◦C and from 0 to 100% of the *RH*. Its length is 70 mm and the diameter 12 mm. Nine sensors were installed in three different depths from the bottom surface (20 mm, 166 mm and 308 mm) and three different planks, and one additional sensor was measuring the temperature and relative humidity of the ambient air.

The FEM model for the Sørliveien deck is a 3D slice of the lamella, with the width of 48 mm, height 333 mm and thickness 5 mm. The weather data in Figure 6 was the primary information needed for the hygro-thermal simulation, because the boundary conditions of the model are based on the external *RH* and *T*. The numerical analysis is carried out from August 2005 until the end of January 2010, since in this period the measurements are continuous. The initial relative humidity and temperatures are equal to those of the air (*RH*0 = 65%, *T*0 = 26 ◦C) and the initial moisture content in equilibrium with *RH*0 is *MC*0 = 13.3%.

**Figure 6.** Sørliveien Bridge. Weather data measured between 2005 and 2010.
