## 2 Materials and Methods
### 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:  
* The model is a 3D slice of the deck far from the ends. The bottom face is exposed to the humidity and temperature of the air. The top surface is exposed only to temperature, because the top of the deck is protected from moisture by the asphalt layer. * The asphalt layer is not modelled. * The lateral, back and front faces are internal surfaces and therefore are not exposed to the air temperature or moisture fluxes. * The bottom surface is sheltered from rain and without water traps. * The model does not include the effect of solar radiation. * The height of the model represents the thickness of the timber deck and its width varies depending on the width of the lamella, with the mesh size of the FEM typically between 5 and 10 mm. * The effect of glue between lamellas is not considered. The initial values for the variables of the differential problem are the following:  
* The temperature \(T_{0}\) is chosen equal to the air temperature at the beginning of the analysis. * The concentration of bound water is calculated as \(c_{b0}=\varrho_{0}MC_{0}\), where \(MC_{0}\) is the moisture content in equilibrium with the initial air relative humidity \(RH_{0}\) at the beginning of the monitoring. This is obtained from the temperature dependent sorption isotherm listed in Table A2 of Appendix A. * The concentration of water vapour \(c_{v0}\) is calculated by using Equations (7)-(9) and the \(RH_{0}\) and \(T_{0}\). The fluxes acting on the 3D slice of the deck are as follows:
* The first boundary condition of Equation (6) applies on the top and bottom surfaces. * The heat flux and thermal flux act on the bottom surface exposed to the air temperature and relative humidity. * Only the heat flux acts on the top protected by the asphalt. * There are no fluxes on the lateral (internal) surfaces. The input material data used for both case-studies of the paper are the dry wood density \(\varrho_{0}\)= 450 kg/m\({}^{3}\), the porosity \(\varphi\) = 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 (\(k_{w}\)) and for the weak paint used in the second case-study (\(k_{p}\)) 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 \(p_{v}\) (obtained from the water vapour concentration \(c_{v}\)), and the temperature \(T\) in each element of the 3D model. instrumented in August 2005 and monitored since then. The instrumentation is situated at the northern end and is logged every fourth hours.