Construction and Building Materials 47 (2013) 1225–1234

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Construction and Building Materials

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Numerical modelling of the hygro-thermal response of timber bridgesduring their service life: A monitoring case-study

0950-0618/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.conbuildmat.2013.06.009

⇑ Corresponding author. Tel.: +358 40 579 3891; fax: +358 20 722 7007.E-mail addresses: stefania.fortino@vtt.fi (S. Fortino), alessandrag_83@hotmail.it, andreagenoese_83@hotmail.it (A. Genoese), linanunes@lnec.pt (L. Nunes), p

lnec.pt (P. Palma).

Stefania Fortino a,⇑, Alessandra Genoese b, Andrea Genoese b, Lina Nunes c,d, Pedro Palma c

a VTT Technical Research Centre of Finland, P.O. Box 1000, FI-02044 VTT, Finlandb University of Calabria, Department of Engineering Modelling, Via P. Bucci, Cubo 39/b – 87036 Arcavacata di Rende (CS), Italyc National Laboratory for Civil Engineering (LNEC), 1700-066 Lisbon, Portugald Azorean Biodiversity Group (CITA-A) and Platform for Enhancing Ecological Research & Sustainability (PEERS), Universidade dos Açores, Dep. Ciências Agrárias,9700-042 Angra do Heroísmo, Terceira, Açores, Portugal

h i g h l i g h t s

� A 3D multi-Fickian numerical methodis proposed to integrate the sensor-based hygro-thermal monitoring oftimber bridges.� The method is implemented in

Abaqus FEM code by defining aspecial finite element in the Uel usersubroutine.� The hygro-thermal behaviour of a

timber pedestrian bridge in Lisbon issimulated during a period of itsservice life.� The method is found to be able to

simulate the moisture states inmembers of timber bridges undernatural environments.� Conclusions are given on the variable

moisture gradients which cangenerate moisture induced stressesand surface cracking.

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

Article history:Received 30 October 2012Received in revised form 6 June 2013Accepted 10 June 2013

Keywords:Timber bridgesVariable environmentMonitoring techniquesMoisture induced stresses (MIS)Multi-Fickian modelsFinite element methodAbaqus code

a b s t r a c t

The monitoring of timber bridges during their service life is important for the maintenance plans of thesestructures. Numerical modelling can integrate the monitoring techniques by reducing the needed inspec-tions and the maintenance costs. In this paper a 3D computational model based on a well assessed multi-Fickian theory is implemented in Abaqus FEM code. The hygro-thermal response of a timber pedestrianbridge is simulated during a period of its service life. The numerical results are in agreement with mea-surements taken by a sensor-based technique. Conclusions are given on the moisture gradients whichcould generate the so-called moisture induced stresses (MIS).

� 2013 Elsevier Ltd. All rights reserved.

palma@

1226 S. Fortino et al. / Construction and Building Materials 47 (2013) 1225–1234

1. Introduction

During the last two decades the development of timber bridgesin European Nordic countries has been promoted by the joint effortof timber industries, road authorities and research organizations[1–3]. The amount of timber bridges in Sweden is grown from adozen old structures in 1990 to about 600–800 in 2010 [3]. Timberbridges are important elements within environmental friendly de-sign plans in cities and in countrysides. On the other hand, the useof wood as a renewable resource in construction is promising forthe beneficial impact against CO2 emissions. However, since woodis a hygroscopic and biodegradable material [4], timber bridgesrequire systematic on-site inspections, sometimes under difficultaccess conditions [3,5,6]. Continuous and reliable measurementscan improve the maintenance plans providing economic benefitsfor owners and fewer disturbances for the users.

Both experimental and numerical issues in regards to monitor-ing timber bridges under natural environments are of greatinterest. The experimental research aims to build more efficientsystems for measurements of all the involved variables (hygro-thermal, mechanical, biological, etc.) while the scope of the com-putational research is to assess theories and numerical techniquesable to simulate the real behaviour of the wood material under var-iable environmental conditions [6]. From the experimental side,advanced technologies for sensors and the design of automaticmonitoring systems for timber bridges are attractive research top-ics that can provide useful and reliable tools to the owners [6].Commercially available wood moisture sensors and loggers cost30 to 60 EUR per channel (sensor) and a wireless data transmissionsystem can cost from 500 to 1000 EUR, as of 2010, not consideringthe installation and set-up of the monitoring system. If many mea-surement points are required (at different locations and/or depths),the cost of deploying such a system can escalate quickly.

The present research proposes a numerical approach to simu-late the hygro-thermal response of timber bridges subjected tovarying environmental conditions during their service life. In thefollowing, both humidity and temperature are considered to be ac-tions on the structure as strongly recommended in [8]. The recentliterature has pointed out the need of a deeper understanding onthe effect of moisture gradients in timber structures exposed tocontinuously variable climate [6–9]. Moisture gradients constrainswelling and shrinkage of the material and this results in the so-called moisture induced stresses (MIS). These stresses can lead tocracks that could significantly affect both the serviceability andthe safety of timber structures [10–12].

In the last decade several studies have shown that the multi-Fickian approach is particularly able to accurately model the mois-ture transport in wood by taking into account the real phenomenaoccurring at the cellular level: diffusion of water vapour in the celllumens, sorption of bound water, and diffusion of bound water inthe cell walls (see [13] and the related literature review). Forvalues of relative humidity RH higher than 80%, the bound waterdiffusion is a relevant phenomenon while the sorption is low. Forlower values of RH a different behaviour occurs since the watervapour diffusion is dominant and the sorption is very fast. Thesedifferences are well modelled by the multi-Fickian theory.

Recently, the moisture induced stresses have been a key topicwithin Cost Action E55 [14] to assess the modelling of moisture-re-lated degradation of timber components and connections duringtheir service life. The main outcomes of this Cost Action in regardsto multi-Fickian models and MIS are discussed in [8]. Furthermore,a first attempt to identify MIS in different European climatic zoneswas done in [9].

In [8], the fully coupled multi-Fickian model with sorption hys-teresis of wood presented in [15,16] was implemented into MatLab

code and applied for the first time to case-study timber elementsunder constantly variable humidity actions. The research was fo-cused on the effects of a pure harmonic excitation on the moisturestates in wood for isothermal conditions. Interesting relationshipswere found between the parameters of the sinusoidal load (ampli-tude, average and period) and three identified parameters thatcharacterise the response to this load (the moisture content ampli-tude on the surface of the timber element, the distance from thissurface to the point with variations of moisture content (MC) lessthan 5% and the largest positive moisture gradient). An importantresult of the study is that daily humidity variations (correspondingto short periods of RH loads) and relevant changes of humidity(large loads amplitudes) at high levels of relative humidity RH(high mean values of loads) provide higher moisture gradients(and higher MIS) and, consequently, a major risk of surface crack-ing in timber elements. In particular it is observed that dailyhumidity variations have more influence on moisture inducedstresses compared to yearly humidity variations with the sameamplitudes.

The above outcomes can be exploited by referring to the realdaily and yearly variations of RH in different climatic zones (seefor example the values of RH in [9]). In paper [9] a moisture-stressanalysis based on a single-Fickian moisture transfer approach andon the viscoelastic-mechanosorptive model presented in [21] wasimplemented into Abaqus. No sorption hysteresis of wood wassimulated. The moisture gradients in different types of timber ele-ments were calculated by referring to the real humidity historiesprovided by the Finnish Meteorological Institute (FMI) during a10 years period. MIS were evaluated for 1 year period. The influ-ence of different types of coatings was also investigated. The re-sults showed that the timber members located in regions withcolder climates (Northern Europe), characterised by larger yearlyvariations of RH (characteristic values over 90% and average valuesover 80%), provide higher moisture gradients (and, consequently,higher MIS) with respect to Southern European climates. The pres-ence of a protective coating was found to be an effective solution toreduce the moisture gradients (as well as the penetration depth ofmoisture into the cross section due to daily variations of MC).

A successful implementation of the full coupled multi-Fickiantheory into Abaqus FEM code was recently presented in [17] wherethe macroscopic material properties used in the transport modelwere estimated by using an advanced multiscale approach andcontinuum micromechanics. In this case the sorption was de-scribed separately using a sub-model at cellular scale and the hys-teresis was not implemented. The differential problem as well asthe FEM discretization were written by taking into account alsothe equation of internal energy including temperature as a thirdvariable of the problem and implemented in a user subroutine ofAbaqus [18,19]. The method was used to simulate laboratoryexperiments of small cylindrical wood samples under varyinghumidity loads and constant temperature.

An alternative approach to implement the multi-Fickian meth-od in Abaqus was proposed in [20] where the two equations ofbound water and water vapour are handled separately and a se-quence of moisture-type analyses at constant temperature is usedto reach the final solution. Neither sorption hysteresis nor theequation of internal energy were implemented. This approachwas used to simulate the same isothermal laboratory experimentstested in [17].

The approach proposed in the present is to use a multi-Fickiannumerical method, alongside with easily deployable temperatureand relative humidity sensors, for monitoring timber structuresunder variable hygro-thermal conditions.

In this work the multi-Fickian model with hysteresis presentedin [15] is used. The coupled equations of the model are

Fig. 1. Longitudinal view of the bridge.

S. Fortino et al. / Construction and Building Materials 47 (2013) 1225–1234 1227

implemented in Abaqus FEM code by defining a 3D user element[19] characterised by 3 variables: bound water concentration,water vapour concentration and temperature. The multi-Fickianmethod is shortly described in Section 2 and the material parame-ters of the model are summarised in Appendix A. The implementa-tion in Abaqus can be found in Appendix B.

As an application of the computational approach, the hygro-thermal response of a glulam beam of a pedestrian bridge locatedin the city of Lisbon is simulated for a period of its service lifeduring 2007. The beam was monitored using a sensor-based tech-nique and, since it is not directly exposed to sun radiation or pre-cipitation, only the relative humidity (RH) and the temperature (T)are considered as loads in the analysis. The analysed case-study ispresented in Section 3. The description of the monitored bridge(including protection against sun and rain) and details on theadopted sensor-based technique are given in Section 3.1. Thenumerical simulations are described in Section 3.2 where somecomments on the possible extension of the computational methodwithin sequential hygro-thermo-mechanical simulations [21–23]for the evaluation of MIS are given. The obtained results and theiruse within monitoring of timber structures are discussed in Sec-tion 3.3. The conclusions are drawn in Section 4.

2. A multi-Fickian model for moisture transfer in wood

The multi-Fickian theory [13] is a multiphase approach basedon the identification of the following three phenomena occurringin wood during moisture transfer: (1) diffusion of water vapourin the cell lumens, (2) sorption of bound water, and (3) diffusionof bound water in the cell walls. The phenomenon of sorption isresponsible for the coupling between the two phases of water(vapour and bound water). For a more complete and realisticdescription of the moisture transfer in wood, a sorption hysteresischaracterised by two isotherms of adsorption and desorption hasto be introduced in the multi-Fickian model as described in [15].As pointed out in [8], the hysteresis of wood has important effectson the moisture gradients that play a fundamental role in thedevelopment of moisture induced stresses (MIS).

Following the notation used in [17], from a macroscopic point ofview the hygro-thermal differential equations governing the multi-Fickian model are expressed as

@cb

@t¼ r � ðDbrcbÞ þ _c ð1Þ

@cv

@t¼ r � ðDvrcvÞ � _c ð2Þ

cw.@T@t¼ r � ðKrTÞ þ r � ðDbrcbÞhb þr � ðDvrcvÞhv þ _chbv ð3Þ

where (1) and (2) are the transport equations of the model and (3) isthe equation describing the energy conservation. In (1)–(3) the con-centration of bound water cb, the concentration of water vapour cv

and the temperature T are the unknowns of the problem while thecoupling term c represents the sorption rate, accounting for theconcentration increase in the bound water phase. The concentra-tions are defined by referring to the dry volume of wood. Db andDv represent the macroscopic diffusion tensors for bound waterand vapour water, respectively, K is the thermal conductivity ten-sor, hb, hv are the specific enthalpies of bound and vapour water,hbv = hb � hv the specific enthalpy of the water phase transitionfrom the bound state to the vapour one, . is the dry density of woodwhose specific heat is cw.

The following homogeneous Neumann boundary conditions forbound water concentration, water vapour concentration and tem-perature are used on the insulated surfaces [16,17]:

rcb ¼ 0; rcv ¼ 0; rT ¼ 0 ð4Þ

On the surfaces in contact with air, the boundary condition onthe bound water concentration is assumed to be (see also [8]):

rcb ¼ 0; ð5Þ

and the fluxes of water vapour and heat fluxes are expressed as

�n � Dvrcv ¼ kvðc0v � cavÞ; �n � KrT ¼ kTðT � TaÞ ð6Þ

In (6) cva and Ta are the water vapour concentration and the

temperature in the surrounding ambient, kv and kT represent thesurface emission coefficients for water vapour and temperature,respectively, and c0v ¼ cv=u is the concentration of water vapourbased on the volume of the cell lumens, u being the porosity, i.e.the volume of lumens with respect to that of the dry wood. Asummary of the adopted material properties for the governingequations and details on the used emission coefficients also inthe presence of coatings (see [24]) can be found in Appendix A.The equations for the sorption rate and the sorption hysteresisare also summarised in Appendix A. In the model used in thepresent paper, the diffusion tensors, the specific heat and theenthalpies are functions of the current temperature. However,the model does not account for temperature dependence of theequilibrium states for moisture in wood and surrounding humiditydescribed by the sorption isotherm. This phenomenon, already ex-plained in [25], has been recently revisited in [26] within a criticalliterature review of old and new research works related to theinteractions between wood and water.

In order to implement the model in the commercial code Aba-qus 6.9 by defining a user-defined element [19], a weak form ofEqs. (1)–(3) is required (see also [17]). The implementation detailscan be found in Appendix B.

3. Case-study: a pedestrian timber bridge in Lisbon

3.1. Measurements

The monitored glued laminated timber pedestrian bridge(Fig. 1), erected in mid/late 2006, is located in the city of Lisbon,Portugal, not far from the Tagus river estuary and the airport. Fol-lowing the Köppen-Geiger climate type classification of Europe, thecity of Lisbon is subjected to ‘‘Warm Mediterranean Climate’’ [9].

The data, provided by LNEC, were gathered in the scope of theproject EU-INTERREG ‘‘MEDACHS – Marine environment damage

Fig. 2. View of monitored glulam beam.

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to Atlantic coast structures and buildings: methods of assessmentand repair’’.

The measurements of moisture content in a glulam beam of thebridge were obtained on the basis of the electrical resistance andtemperature of wood. Measuring the electrical resistance of timberis a widely used method to locally evaluate its moisture content[27] and fit to long-term monitoring ([28,6]). In [29], 17 differenthand held resistance-type wood moisture content meters weretested. The study reports an average absolute accuracy (95%

Fig. 3. Illustration of the beam wi

confidence interval) of ±2.5% under industrial conditions and mois-ture content in the range 7% 6MC 6 25%. In the present case study,the electrical resistance was measured with a Scanntronik Materi-alfox Mini sensor/logger (with three reading channels) andpin-type penetrating electrodes (stainless steel screws). Two pairsof electrodes were placed near to the ends of the beam, next to theconnection’s steel plates, and the third pair in the centre of the ele-ment (see Figs. 2 and 3). Tests performed in laboratory showed thatthere was no significant influence of the steel plates in the mea-surements. Electrodes were inserted through larger pre-drilled1 cm deep holes to a depth of 2 cm. The gap between the electrodeand the hole surface was sealed. This set-up assures that any liquidwater on the surface does not lead to high (and wrong) values ofmoisture content and that, since non-insulated stainless steel elec-trodes were used, the measurement of electrical resistance is doneapproximately in the wood ‘‘layer’’ between the depths of 1 and 2cm (along the wettest path between the electrodes).

The glulam structural members are made of Norway spruce(Picea abies (L.) Karst.), the longitudinal members are protectedfrom direct exposure to sun and rain by wooden cladding and zincsheet capping, while the transversal members are protected by thedeck boards. The monitored transversal glulam beam has a rectan-gular cross-section of 115 � 360 mm2 and a length of 1700 mmwith large steel-to-bolted connections at both ends (Fig. 3). Allthe structural members were coated with a thick layer of paint thatcan be considered as a weak paint particularly when directlyexposed to weathering.

The in situ measured data provide values of electrical resistanceR, air temperature Tair and relative humidity RH in correspondenceof the three sensors. These data are continuously available from

th the position of the sensors.

Fig. 4. Abaqus model: mesh and moisture (or temperature) fluxes on wood.

Fig. 5. Curves of relative humidity (RH), top, and temperature (T), bottom, FinnishMeteorological Institute (FMI). Period from 01.02.2007 (time 0 days) to 11.09.2007(time 222 days).

S. Fortino et al. / Construction and Building Materials 47 (2013) 1225–1234 1229

17.01.2007 to 11.9.2007. The used equations which relate woodtemperature, electrical resistance and moisture content are theones proposed in [29]. A first equation is the following:

logðlogðRÞ þ 1Þ ¼ a �MC þ b ð7Þ

where R represents the electrical resistance in MOhm, MC is themoisture content (%), a and b are calibration parameters. Then,the following expression for the temperature-corrected moisturecontent MCcorr (%) is:

MCcorr ¼0:0014T lnð10Þþ lnðexpðaMCmeas lnð10Þþb lnð10ÞÞþ1Þ�1:075lnð10Þ

lnð10Þ � ð0:000158Tþ0:0262Þ

where MCmeas is the measured moisture content (in %) and T is thewood temperature (in �C).

In [29] a calibration in the temperature range�10 �C 6 T 6 70 �C was performed. The obtained calibrationparameters for Norway spruce are a = �0.037 and b = 1.054.

3.2. Numerical simulations

As pointed out in the previous section, the data of moisture con-tent are available from 17.01.2007 to 11.09.2007 and consist of anaverage of 4 daily measurements (one data value per day). Withinthe numerical simulations, these data are not sufficient to take intoaccount the effect of daily variations of RH which is important forestimating the variations of moisture gradients close to the surfaceof the beam. In addition, there is a lack of information on the initialconditions of the glulam beam at the beginning of its service lifebefore the starting of measurements.

Then, to carry out the numerical simulations, the followingassumptions have been made:

– Due to the fact that the glulam beams of the bridge are notdirectly exposed to precipitations and sun radiation (seeFig. 2), only the effects of RH and T are considered.

– The climate acting on the bridge is assumed to be the same asthe one monitored at the meteorological station located in theLisbon airport zone (Fig. 5). The data, provided every 6 h, areavailable from the Finnish Meteorological Institute (FMI). Thisassumption is considered to be reasonable because the studiedbridge is located not so far from the airport area. For cases oftimber structures located considerably far from meteorologicalstations, the general computational approach suggested in [8],which assumes sinusoidal humidity loads, should be preferablyused.

– The initial moisture content is assumed to be MC0 = 17%, that isthe mean value of the measurements in correspondence of thethree sensors at the date of 17.01.2007. The initial moisture

1230 S. Fortino et al. / Construction and Building Materials 47 (2013) 1225–1234

state is then identified by MC0 and by the corresponding initialvalue of relative humidity RH0 = 0.828 taken on the absorptioncurve of the hysteresis model illustrated in Appendix A.3. In thisway, the analysis starts from an equilibrium point (see also [8]).The used dry wood density is q = 450 kg/m3.

– To avoid a strong influence of the initial conditions on thenumerical results, the comparisons with the measurementsstart from 01.02.2007. So, the period of analysis includes238 days (from 17.01.2007 to 11.09.2007), while the compari-sons between numerical and measured values refer to a periodof 223 days (from 01.02.2007 to 11.09.2007).

Fig. 6. Numerical results and measurements from 01.02.2007 (time = 0 days) to11.09.2007 (time = 222 days). Top: Sensor 1. Centre: Sensor 2. Bottom: Sensor 3. MCvalues on the surface and at a depth of 1.5 cm.

The Abaqus model for the beam is shown in Fig. 4. During theanalysis, both fluxes of moisture and temperature are assumed toact only on the wood surface and not on the steel plates.

4. Results and discussion

The numerical results in terms of MC during time are taken overthe surface as well as at a depth of 1.5 cm from the beam surface. Thislast location is chosen because the electrodes measure the electricalresistance between 1 and 2 cm deep as explained in Section 3.1. Thenumerical results show a good agreement with the measurements(see Fig. 6). It has to be noted that the measurement curves representaveraged daily values. Bigger differences between the numerical re-sults and the measurements for Sensor 1 (Fig. 6, top) could be due tothe location of this sensor, closer to the external surfaces with re-spect to Sensor 2 and Sensor 3 and therefore prone to be influencedby some free water running from the walkway above.

A case-study analysis with a permeance coefficient correspond-ing to an Alkyd oil paint coating was also performed in order tostudy the influence of more protective coatings. The results areshown in Fig. 7 for the case of Sensor 2 at 1.5 cm depth. The ob-tained values of moisture content during time in the presence ofAlkyd oil paint appear much smoother than the ones for the realsituation with weak paint on the surface. In particular, the higherpeaks of MC are avoided. This confirms the results obtained forexample in [9] about the capability of coatings to noticeably reducethe moisture gradients in the vicinity of the surface of timbermember exposed to external variable climates. Coating can thenavoid possible surface damages caused by common weak paintsthat are very thin and usually flaking.

To evaluate the effect of the daily variations in the vicinity ofthe external surface, moisture gradients were calculated as(uDL � usurf)/DL where uDL is the moisture content at a distanceDL = 10 mm from the external surface and usurf is the moisture

Fig. 7. Numerical results and measurements from 01.02.2007 (time = 0 days) to11.09.2007 (time = 222 days). Sensor 2: results of the analysis with the used weakpaint and of a case-study analysis with Alkyd oil paint.

Table 1Maximum values of DRHd (over 48 h) in the studied periods and corresponding peaksof RH.

10.3.2007–10.4.2007 June 2007 August 2007

DRHd,max 0.521 (reached 10–12.3.2007)

0.442 (reached 5–7.6.2007)

0.581 (reached 24–26.8.2007)

RHpeak 0.861 (during 10–12.3.2007)

0.788 (during 5–7.6.2007)

0.902 (during 24–26.8.2007)

Table 2Maximum positive gradients calculated at 10 cm from the external surface of the glulam beam (weak paint) in correspondence of the 3 sensors in the studied periods.

10.3.2007–10.4.2007 max grad (m�1) (reached11.3.2007)

June 2007 max grad (m�1) (reached6.6.2007)

August 2007 max grad (m�1) (reached25.8.2007)

Sensor 1 1.952 1.642 2.246Sensor 2 2.082 1.643 2.195Sensor 3 1.878 1.662 2.169

S. Fortino et al. / Construction and Building Materials 47 (2013) 1225–1234 1231

content on the surface. In this work the daily variation DRHd is de-fined as the difference between the maximum and the minimumvalues of RH monitored over 48 h of reference period. The gradi-ents calculations were done in different seasonal periods of year2007: (a) starting of spring (10.3.2007–10.4.2007), (b) starting ofsummer (June 2007), and (c) end of summer (August 2007).

Fig. 8. Moisture content on a path along the thickness of the beam (in correspondenceweak paint and the numerical case-study Alkyd oil paint.

Based on the data provided by FMI, the maximum values of dai-ly variations of relative humidity (DRHd,max) for the studied periodsare reported in Table 1 together with the dates in which these val-ues are reached and the peaks of relative humidity (RHpeak) at thesedates. The above values are related to the seasonal periods as wellas to the specific climatic zone (Lisbon). In the same periods, the

of Sensor 2) every 3 days during August 2007. Results of simulations with the used

1232 S. Fortino et al. / Construction and Building Materials 47 (2013) 1225–1234

maximum numerical values of the positive gradients (calculated at10 mm from the external surface in correspondence of the threesensors) are reported in Table 2 for the case of weak paint. Alsothe dates in which these values are attained are reported.

From Tables 1 and 2 it can be observed that the maximum gra-dients in all the considered periods are reached in correspondenceof the maximum daily variations of relative humidity DRHd,max andfor higher peaks of relative humidity RHpeak. Furthermore, Table 2shows that the gradients reached in August (end of Summer) areonly slightly higher than the ones at the beginning of Spring whilesmaller values are attained in June (beginning of Summer) with re-spect to the other studied periods. This is related to more uniformchanges of RH peaks in June 2007.

In Fig. 8 the distributions of moisture content during August2007 are reported (every 3 days) on a path along the cross sectionof the beam in correspondence of Sensor 2. In the same figure, thestrong smoothing of both moisture gradients and peaks of moisturecontents due to the possible use of an Alkyd oil paint is also shown.

Although the numerical solution depends on the used modelparameters, these have been previously used within successfulmulti-Fickian simulations where numerical results showed goodagreement with experimental laboratory data (see [17,20,30]).

In case of timber members directly exposed to sun and rain, lar-ger moisture content gradients could occur between the outer layerand the inner wood layers, therefore increasing the chance ofweathering cracks, but no evaluation was made or any data col-lected to assess the effects of direct exposure compared to protectedexposure. As already observed in [9], the inclusion of solar radiationand rain in the numerical modelling would make the model verycomplex. For example, in the presence of rain also the diffusion ofliquid water and capillary transfer in wood should be taken into ac-count. Further computational work is needed to address this topic.

The presented approach can be easily extended to simulate thehygro-thermo-mechanical response of timber elements underthe service life of bridges. This can be obtained by transferringthe results of the hygro-thermal analysis described in this paperto a mechanical analysis where the mechanical actions on thebridge (also dynamics) are taken into account. Suitable modelsfor wood characterised by mechanical quantities depending onthe hygro-thermal variables where proposed for example in[21–23]. The results of a full hygro-thermo-mechanical analysiscan provide the evaluation of moisture induced stresses (MIS) inthe vicinity of the external surfaces.

The results of this research show that numerical modelling canbe used alongside with weather data to improve the assessmentand monitoring of timber bridges. The presented numerical modelcan also be used to assess the detailing of sensitive areas regardingmoisture during the design phase (such as recesses, connections,and other possible water traps) and avoid expensive repairs after-wards. As discussed in [11], half of failures in timber and otherstructures are related to the design phase. Improved designaccounting for moisture influence is important for the serviceabil-ity and safety of timber bridges especially to prevent risks of failurerelated to significant variations of relative humidity.

5. Conclusions

This paper proposed a computational approach to simulate thehygro-thermal response of timber bridges and other timber struc-tures under natural climates. Accurate information on the moisturestates in the timber members and, in particular, on the moisturegradients close to the external surfaces, is important to identifythe zones with possible development of MIS which could causecracking.

The computational approach is based on a well-assessedmulti-Fickian theory and the model has been implemented in a

user subroutine of Abaqus FEM code. The performed computa-tional tool is characterised by the following features:

– a coupled transport model with variable temperature;– use of sorption hysteresis;– capability to solve three dimensional problems at the structural

scale under real environmental conditions.

As a case-study, the hygro-thermal behaviour of a glulam mem-ber of a timber pedestrian bridge located in the city of Lisbon, be-tween the airport area and the river, was studied under thevariable relative humidities and temperatures registered by theFMI from 17.01.2007 to 11.09.2007. In that period in situ measure-ments were carried out by means of a sensor-based technique.

The main results of the study are the following:

– the method is able to simulate the moisture states in each pointof a timber member of the bridge (sheltered from sun and rain)by using as external loads the relative humidity and tempera-ture histories provided by a meteorological station close tothe bridge itself;

– high moisture gradients in the vicinity of the external surfacesare caused by daily variations of relative humidity in differentseasonal periods;

– the influence of a possible protective coating is confirmed to beimportant to strongly reduce the peaks of moisture content inthe timber member as well as the moisture gradients.

From a computational point of view, the presented numericaltool represents a first step for a full hygro-thermo-mechanicalanalysis aimed at calculating MIS in timber elements under realenvironmental conditions.

The proposed approach can be a useful support for the nowa-days techniques used to monitor the hygro-thermal response oftimber structures during their service life. This can be exploitedto reduce the overall costs for monitoring in the context of modernmaintenance plans.

Acknowledgments

The Doctorate School of Science and Technique BernardinoTelesio (Doctorate Degree Programme in Computational Mechan-ics) and the International Mobility funding of University of Cala-bria, Italy, that allowed the authors Alessandra Genoese andAndrea Genoese to undertake a research stage at VTT, are grate-fully acknowledged.

Lina Nunes is grateful to Fundação para a Ciência e Tecnologia(FCT) for project MONITOR (ref. PTDC/ECM/099121/2008).

The authors would like to particularly acknowledge Cost ActionE 55 ‘‘Modelling the performance of Timber Structures’’ (http://www.coste55.ethz.ch/) that made possible the cooperation workneeded for this research.

Appendix A

A.1. Material properties of the governing equations

In the following the matrix form of the model tensors are used.The material properties for the equations of bound water andwater vapour are taken from ([13] and related references).

� Bound water diffusion matrix:

Db ¼ D0 exp � Eb

RT

� �ðm2 s�1Þ

S. Fortino et al. / Construction and Building Materials 47 (2013) 1225–1234 1233

in which Eb = (38.5 � 29m) � 103 (J mol�1) is the activation energyfor bound water diffusion, T the temperature in K and R = 8.314(J mol�1 K�1) the universal gas constant, where m = cb/.0 is themoisture content and .0 the dry density of wood (kg m�3). Diagonalterms of D0 are D0T = 7 � 10�6 m2 s�1, D0L = 17.5 � 10�6 m2 s�1 forthe radial and tangential direction and for the longitudinal one,respectively.� Vapour water diffusion matrix:

Dv ¼ n 2:31� 10�5 Patm

Patm þ Pv

� �T

273

� �1:81 !

ðm2 s�1Þ

where Patm = 101,325 Pa is the atmospheric pressure and Pv the par-tial vapour pressure in Pa. The ideal gas law furnishes Pv ¼ c0v

RTMH2O

where MH2O ¼ 18:02� 10�3 kg mol�1 is the molecular mass ofwater. The components of matrix n represent reduction factors inthe different directions of wood. Their values are assumed to benL = 0.9 and nT = 0.03 (see [20]).

The material properties for the internal energy equation are ta-ken from [31] and [30] (and related references):

� Thermal conductivity in the longitudinal direction:

kqL ¼ 2:5kqT ðW m�1K�1Þ

with kqT = G(0.2 + 0.38 m) + 0.024 (W m�1 K�1), being G ¼ 0:693G00:653þm the

specific gravity of wood in the presence of moisture and G0 = .0/.w

the one of dry wood, .w = 103 kg m�3 being the density of water.� Specific heat and the dry density of wood:

cpW ¼0:0011T þm� 0:0323

1þmðJ kg�1 K�1Þ;

. ¼ Gð1þmÞ.w ðkg m�3Þ

� Enthalpies:Expression for enthalpies of liquid and vapour water are the

same used in [30]:

hb ¼ �7:8955� 105 � 4:476206� 102T þ 2:274399� 10T2

� 4:9553577� 10�2T3 þ 4:041035� 10�5 � T4 ðJ kg�1Þ

hv ¼ 1:891879� 106 þ 2:56352� 103T � 1:2360577T2 ðJ kg�1Þ

A.2. Surface emission coefficients and influence of coatings

The used surface emission coefficient for temperature used inboundary conditions (6) is:

kT ¼ 20 W m�2 K�1

.Following [24], in the presence of a coating the resistance to the

humidity exchanges between wood and the surrounding environ-ment is calculated as

1kv¼ 1

kvsþ 1

kp) kv ¼

11

kvsþ 1

kp

where kvs is the surface emission coefficient for the uncoated woodand kp the coefficient for coating. The paint permeance is expressedin terms of partial pressure. By indicating this value with �kp, thecoefficient in the presence of paint is kp ¼ �kp

RTMH2O

where the consid-ered value of T changes when referring to the external air or to thewood and MH2O is the molecular mass of water. The following flux ofmoisture is then considered:

�n � Dvrcv ¼ kwv c0v � ka

vcav

where kwv and ka

v are calculated exploiting the previous definitionand by using the surface and the air temperatures respectively.

The expressions of emission coefficients used in the numerical anal-ysis are:

kwv ¼

11

�kvsþ 1�kp

RTw

MH2O; ka

v ¼1

1�kvsþ 1

�kp

RTa

MH2O

with the following coefficients:� permeance coefficient for the uncoated wood surface and the

weak paint:�kvs ¼ 5� 10�9 Kg m�2 s�1 Pa�1;

� permeance coefficient for Alkyd oil (from [24]):�kp ¼ 3:1� 10�10 Kg m�2 s�1 Pa�1:

A.3. Sorption rate, sorption hysteresis and used material parameters

The sorption rate is defined as in [13]:

_c ¼ Hcðcbl � cbÞ ðkg s�1 m�3Þ

in which cbl = q0mbl is the bound concentration in equilibrium witha given relative humidity, according to the Hailwood – Harrobinisotherm

mbl ¼h

f1 þ f2hþ f3h2

where the humidity h represents the ratio of vapour pressure pv andsaturated vapour pressure [8]. The shape parameters fi for thecurves of adsorption and desorption used within the hysteresismodel are the ones proposed by Ahlgren in [32] for Norway Spruce:– parameters for the adsorption curve: f1 = 1.804, f2 = 13.63,

f3 = -12.12;– parameters for the desorption curve: f1 = 1.886, f2 = 7.884,

f3 = -6.526.

The moisture-dependent reaction rate function Hc (s�1) is:

Hc ¼C1 exp �C2ðcb

cblÞC3

� �þ C4 cb < cbl

C1 exp �C2ð2� cbcblÞC3

� �þ C4 cb > cbl

8><>:

where Ci are the H-function shape parameters. In particular C2 var-ies with the relative humidity according with the equation C2 = c21

exp (c22h) + c23 exp (c24h). The values for Ci are the ones used in[20]. As explained in [15], the sorption hysteresis allows taking intoaccount the dependency of the equilibrium bound water concentra-tion cbl on the history of variations in relative humidity. It is as-sumed that a feasible state (mbl, h) lies in the domain bounded bythe adsorption and desorption boundary curves ma(h) and md(h).Following [30], the equilibrium moisture content cbl is defined ascbl = cbds + cba(1 � s), where cba and cbd are obtained through theadsorption and desorption isotherms, while s represents the degreeof exploited sorption (scanning curve) defined by the equation

s ¼

�1þ 21�h

1�hd;0

� � d1lnðd2 ð1�hd;0 ÞÞ

cb < cbl ^ s0 > 00 cb < cbl ^ s0 ¼ 0

2� 2ð h

ha;0Þ

d1lnðd2ha;0 Þ

cb > cbl ^ s0 < 11 cb > cbl ^ s0 ¼ 1

8>>>>>>>><>>>>>>>>:

ha;0 ¼ h0ðd2h0Þq1 ;hd;0 ¼ 1� ð1� h0Þðd2ð1� h0ÞÞq2

q1 ¼ �lnðlnð2ÞÞ � lnðlnð2� s0ÞÞ

lnðlnð2ÞÞ � lnðlnð2� s0ÞÞ � d1;

q2 ¼ �lnðlnð2ÞÞ � lnðlnð1þ s0ÞÞ

lnðlnð2ÞÞ � lnðlnð2þ s0ÞÞ � d1

1234 S. Fortino et al. / Construction and Building Materials 47 (2013) 1225–1234

where [h0, s0] are referred to the initial state (being ha,0 and hd,0 theorigins of scanning curves on the adsorption and desorption curve,respectively) while d1 and d2 are the shape parameters taken from[30].

Appendix B

B.1. Implementation details

To implement the model in Abaqus code, Eqs. (1)–(3) are mul-tiplied by three weight functions (wb, wv and wT) and the resultingequations are integrated over the volume V. The backward Eulermethod for time integration is used for the variables cb, cv and T.It exploits the linearization u;t ffi u;tþDt � @u

@t

��tþDt

Dt for a generic func-tion u at the time t starting from the corresponding values at thetime t + Dt that implies @u

@t

��tþDtffi 1

Dt ðu;tþDt � u;tÞ.After the usual FEM manipulations adopted within the FEM

technique the following equations are obtained:

1Dt

ZVðcb;tþDt � cb;tÞwb þ

ZVrwb � Dbrcb �

ZV

_cwb �Z

Swbn � Dbrcb

¼ 0;8wb ð8Þ

1Dt

ZVðcv;tþDt � cv;tÞwv þ

ZVrwv � Dvrcv þ

ZV

_cwv �Z

Swvn � Dvrcv

¼ 0;8wv ð9Þ

1Dt

ZV

cw.ðT ;tþDt � T ;tÞwT þZ

VrwT � KrT þ

ZVrwT � Dbrcbhb

þZ

VrwT � Dvrcvhv �

ZV

_chbvwT �

ZS

wT n � KrT

�Z

SwT hbn � Dbrcb �

ZS

wT hvn � Dvrcv

¼ 0;8wT ð10Þ

Since the sorption rate _c and the material parameters depend onthe variables cb, cv and T at the time t + Dt, the equations are non-linear and coupled. The discrete solution at the time t + Dt isreached as a series of improved approximations by means of aNewton–Raphson type iterative procedure [18]. To obtain the dis-crete formulation of the problem in Uel subroutine of Abaqus [19],the primary variables cb, cv, T and the weight functions are interpo-lated in the same form over each element. The user subroutineneeds the definition of the element residual vector at each iterationand the element contribution to the overall Jacobian matrix that isobtained by deriving the expressions of the residual vector with re-spect to the element nodal variables. The shape functions for aneight-node brick element transformed by the isoparametric changeof coordinates are chosen.

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