International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Optimal design of deteriorating timber components under climate variations Bastidas-Arteaga, Emilio; Aoues, Younes; Chateauneuf, Alaa Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Optimal Design Of Deteriorating Timber Components Under Climate Variations Emilio Bastidas-Arteaga Associate Professor, University of Nantes, Nantes, France Younes Aoues Associate Professor, INSA de Rouen, Rouen, France Alaa Chateauneuf Professor, Blaise Pascal University, Aubière, France ABSTRACT: The mechanical and physical properties of timber structures could be affected by a combination of loading, moisture content, temperature, biological activity, etc. This paper focuses on the optimal design of new timber structures subjected to fungal decay. Among the optimization methods available in the literature, this study considers a Time-Dependent Reliability Based-Design Optimization approach. This method aims at ensuring a target reliability level during the operational life by considering deterioration and the uncertainties related inherent to materials properties, models and climate. This approach is applied to design optimization of a timber truss subjected to an aggressive (very humid) French climate. The performance of the optimized solution is compared, in terms of safety, with solutions estimated from Deterministic Design Optimization and the classical Reliability-Based Design Optimization approaches. The overall results indicate that the optimized solution obtained by the Time-Dependent Reliability Based-Design Optimization approach ensures the target reliability level during the whole structural lifetime.   1. INTRODUCTION The mechanical and physical properties of timber structures could be affected by a combination of loading, moisture content, temperature, biological activity, etc. This paper focuses on the optimal design of timber structures subjected to fungal decay. Structural optimization is widely used for searching the optimal design cost of civil engineering structures. The Deterministic Design Optimization (DDO) procedure is successfully applied for designing concrete and steel structures (Kravanja et al. 2013; Tomás and Martí 2010). Other works have used the DDO approach to optimize the design of timber trusses (Šilih et al. 2005) or finger-joints under bending solicitations (Tran et al. 2014).  The DDO procedure is based on minimizing objective functions defined in terms of the structural volume or costs for structures and subjected to geometric, stress and deflection constraints. These design conditions are considered in accordance with Eurocode 5 (NF-EN 1995 2005) in order to satisfy the requirements of both the ultimate and the serviceability limit states. In Fact, the real benefit of the DDO approach is cost reduction and effective use of structural capacity.  The partial safety factors introduced in the deterministic design are assumed to take account for uncertainties related to timber material, structural dimension and loading. These safety factors are applied in the design constraints to ensure the safety margin. These factors are calibrated for a large class of structures. Moreover, the safety margin produced with these partial safety factors is not directly linked to uncertainties. Thus, the use of these partial safety 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 factors in the deterministic design optimization can lead to over or under designing structures. The Reliability-Based Design Optimization (RBDO) offers a suitable framework for the consideration of the uncertainties in the design optimization and to find the best compromise between cost reduction and safety assurance. A practical formulation of the RBDO consists of minimizing the expected cost under probabilistic constraints. Furthermore, the long-term durability of timber structures depends on the effect of moisture that in combination with propitious temperature conditions and exposure time may deteriorate the material resistance capacity. The exposure of unprotected timber structures to humid environments increases the moisture content inside wood leading to fungal decay. This deterioration mechanism reduces the strength capacity of timber structures affecting its serviceability and safety.  Within this framework, this paper applies a Time-dependent Reliability-Based Design Optimization approach (TD-RBDO) (Aoues et al. 2009) to find the best design configuration of a roof structure under fungal decay. The TD-RBDO approach aims at searching the optimal design that minimizes the structural cost and to ensure a target reliability level during the operational life. This approach is applied to design optimization of a timber truss subjected to a very humid French climate (Nantes City). The performance of the optimized solution is compared with the optimal design estimated by the DDO and the classical RBDO approaches. The overall results show that an optimized solution obtained by the TD-RBDO method ensure the target reliability level during the whole structural lifetime. 2. DETERMINISTIC DESIGN OPTIMIZATION The Deterministic Design Optimization aims at minimizing an objective function as the structural volume or cost subjected to geometric, stress and deflection constraints, as defined in the design codes, for instance by the Eurocode 5:   mind= h,b{ }i=1nb∑ ρLihibisubject to GULS ,id,xk,γ( )≤0GSLS ,id,xk,γ( )≤0ψid( )≤0⎧⎨⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪          (1) where, d is the vector of design variables, for rectangular cross-section d is composed by the depth b and the breadth h of the members, Li is the length of the ith member and ρ is the timber density. GULS,i and GSLS,i are respectively the ith Ultimate Limit State (ULS) and Service Limit State (SLS). The limit state functions are defined in terms of the design variables d, the characteristic values of load actions and material properties collected in the vector xk and the partial safety factors γ and ψ are the feasibility constraints (e.g. upper and lower bounds of design variables).  3. TIME-DEPENDENT RELIABILITY BASED-DESIGN OPTIMIZATION For structural systems, The Reliability Based Design Optimization is formulated as the minimization of the cost function under reliability constraints.  mind= h,b{ }i=1nb∑ ρLihibisubject to βid,X( )≥βitψid( )≤0⎧⎨⎪⎪⎪⎩⎪⎪⎪          (2) where βi(d, X) are the reliability indexes for the ith ultimate or serviceability limit state Gi and  βit  is the target reliability index for the ith limit state. In the above formulation, the reliability constraints define the feasible domain, such as the reliability indexes β are kept upper than the target reliability indexes βt. Several approaches have been recently developed to solve the above formulation in Eq. 2. Aoues and Chateauneuf (2010) have compared and discussed different 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 RBDO approaches regarding robustness and numerical performance. The SORA method (Du and Chen 2004) appears more robust and more accurate than the other methods. This approach is used in the study to search the best reliable design of the timber structures. However, when degradation is considered, the time-dependent Reliability-Based Design optimization is proposed to find the optimal design by satisfying appropriate safety levels during the whole structure lifetime. The TD-RBDO is formulated as:  mind= h,b{ }i=1nb∑ ρLihibisubject to βid,X,t( )≥βit ,TLψid( )≤0⎧⎨⎪⎪⎪⎩⎪⎪⎪⎪∀t ∈ 0,TL⎡⎣⎤⎦          (3) where, βi(d, X, t) is the reliability index at the time t taken the lifetime interval [0, TL],  βit ,TL  is the target reliability index at the allowable life time TL, depending on the target reliability related to one year reference period by the following relation (NF-EN 1990 2003):   βTLc= Φ−1Φ β1t( )TL⎛⎝⎞⎠          (4) where  β1t  is the target reliability index for one year. The classical TD-RBDO formulation given in Eq. (3) is based on the time-dependent reliability, that aims at computing the probability of failure during the whole structure lifetime. The time dependency lies mainly in the degradation phenomena. This formulation is not suitable for real engineering structures because considerable computational effort is required, and convergence can hardly be achieved. The major drawback lies in the time-dependent reliability analysis, which requires considerable computational efforts. In this work, the Sequential Optimization and Time-Variant Reliability Analysis (SOTVRA) approach developed and implemented by Aoues et al. (2009) is used to perform the TD-RBDO methodology. The SOTVRA approach is based on transforming the TD-RBDO problem into a sequence of equivalent deterministic design optimization sub-problems. This transformation is defined by the mean of optimal safety factors, linking the reliability requirement to the equivalent deterministic optimization. At the end of each optimization sub-problem, the reliability constraint is verified by performing the time-dependent reliability. The safety factors corresponding to the target reliability level at the initial time are calibrated by a probabilistic approach. Finally, these safety factors are provided to the following sub-problem of the equivalent deterministic optimization and so on, until convergence. 4. MODELING TIMBER DECAY 4.1. Background Decay models are based on in-field or in-lab measures and report results for specific wood species and locations (Brischke and Thelandersson 2014). Among the models based on in-field observations, the Timber Life empirical model was derived based on 20-year data obtained by exposing 4000 small clear timber specimens in 11 sites around Australia, augmented by more than 1500 bits of data obtained from examining existing timber constructions (Leicester et al. 2009). This study focused on Australian species. This model has been adapted and used for many researchers worldwide (de Freitas et al. 2010; Lourenço et al. 2012; Ryan et al. 2014; Sousa et al. 2014). However, it does not provide direct relationships between decay and specific climate conditions for a given zone that we would like to consider in an optimal design.  Isaksson et al. (2012) also proposed a decay model based on in field measurements on Scots pine sapwood (Pinus sylvestris L.) and Douglas fir heartwood (Pseudotsuga menziesii Franco) specimens. The specimens were subjected to real exposure conditions in 24 European test sites with different climate regimes between 2000 and 2008. The model developed by Isaksson et al. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 (2012) aims at estimating decay in terms of ‘dose-response functions’. The ‘dose’ is expressed as a function of daily wood moisture content and wood temperature and the level of decay is defined according to (EN 252 1990). EN 252 (1990) proposes 5 levels of decay condition from no decay (sound) until very severe decay. These levels are useful when condition assessment is based on serviceability limit states. However, they cannot provide quantitative information about the loss of effective section that can be used for estimate the strength loss. Consequently, this model is not considered in this work. 4.2. Adopted decay model On the basis of previous in-lab experimental studies (Viitanen 1996), Viitanen et al. (2010) developed a model for the decay growth of brown rot in pine sapwood under variant climate conditions. Such a model is divided into two processes: (i) activation process and (ii) mass loss process. 4.2.1. Activation process A parameter α is used as a relative measure of fungi deterioration activity. α is set initially to 0. Once it reaches the limit value α=1, the mass loss initiates. The parameter α varies with time according to:  α(t) = Δαi=0t∑(i)  with  α(t) ∈ 0,1⎡⎣⎤⎦ (5) where  Δα(i) =Δttcrit(RH (i),T (i))if T (i) > 0ºC and RH (i) > 95%−Δt17520otherwise⎧⎨⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪ (6) where RH(i) and T(i) are the ith air relative humidity (in %) and temperature (ºC), respectively, Δt is the time step between two consecutive climatic records (hours), and tcrit (in hours) is estimated as follows:   tcrit(i) =2.3T (i)+ 0.035RH (i)−0.024RH (i)T (i)−42.9+ 0.14T (i)+ 0.45RH (i)⎡⎣⎢⎢⎤⎦⎥⎥              ×30×24 (7) Eq. (6) shows that Δα(i) increases when T>0ºC and RH>95%. Under dry and cold conditions α(t) decreases linearly from 1 to 0 in two years (17520 hours). 4.2.2. Mass loss process Mass loss (in % of initial weight) occurs once the fungi activation process is reached, (α(t) = 1) and it is estimated as:  ML(t) =ML(RH (i),T (i))dt×Δt×1α(i)⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟⎟i=0t∑ (8) where  1α(i) =0 if   0≤α(i) <11 α(i) =1⎧⎨⎪⎪⎪⎩⎪⎪⎪ (9) and   ML(RH (i),T (i))dt=−5.96 ⋅10−2+1.96 ⋅10−4T (i)                         + 6.25⋅10−4RH (i) [% loss/hour] (10) According to eq. (6), mass loss only takes place when the temperature is above 0°C and the relative humidity is above 95%. Otherwise mass loss process is stopped.   5. NUMERICAL EXAMPLE 5.1. Problem description Building roofs are usually made of timber trusses. The design of timber structures requires the verification of a certain number of rules resulting from the codes of practice, such as Eurocode 5 (NF-EN 1995 2005). Where, these rules should satisfy given requirements related to their ultimate of capacity in ultimate limit state (ULS) and their deformation in service limit state (SLS). In practice, roof trusses are made and composed of wood members connected by steel plates. Generally, the timber joints are considered completely flexible (free rotations 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 hinges in the connections of the timber members). However, this situation is rarely found in real structures, due to eccentricity and geometrical imperfection of the timber members. A rigid joints hypothesis can be considered as an intermediate solution with respect to the flexible model. The best model of timber connections is to consider the semi-rigid law behavior by using springs or contact elements (Riahi et al. 2011).  The numerical application is carried out for the roof truss described in Figure 1. The depth b and breadths {h1, h2, h3} of the cross-section of the truss members are optimized by accounting for material and loading uncertainties. Table 1 gives the statistical parameters for the truss parameters, loading and material properties. It is assumed that all random variables follow lognormal distributions (Köhler et al. 2007).   Figure 1:roof truss.  For plane timber truss, the ultimate and serviceability limit state functions for each ith member are defined as:  σm,dfm,d+σt ,0,dft ,0,d≤1      in tension ,  σm,dkcritfm,d⎛⎝⎜⎞⎠⎟2+σc,0,dkc,zfc,0,d≤1   in compression ,  τdkν fν ,d( ) ≤1 ,  δinstδinst ,lim≤1 SLS( ),  δfinδfin,lim( )≤1 SLS( ) (11) where, σm,d, σt,0,d, and σc,0,d and  are respectively  the design values of bending stress, tensile stress along the grain and compressive stress along the grain. fm,d, ft,0,d, and fc,0,d are respectively the design values of bending strength, tensile strength and the compressive strength along the grain. kcrit and kc,z are respectively factors which take into account the reduced bending strength due to lateral buckling and compressive strength due to buckling about the y and z axes in accordance with Eurocode 5. τd is the design value of the shear stress and fν,d is the design value of shear strength and kv is the reduction factor of the shear strength at the notched member. The strength design values are defined by :  fd= kmod× fk/ γm (12) where fd and fk are respectively the design and the characteristic values of the strength, kmod is the modification factor, which takes into account the effect of the duration of the load and the moisture content, γm is the partial safety factor for a material property. In the deterministic design the values for kmod and γm are taken from Eurocode 5. Eq. (11) considers the following stress criteria: i) tension and bending: where the tension is parallel to the grain; ii) compression: where members are checked for compressive strength as well as for buckling; iii) shear: for all the truss members; and iv) deformation: corresponding to the serviceability state functions, where δinst and δfin are respectively the instantaneous deflection and the final deflection composed with the instantaneous and creep deflections. δinst,lim and δfin,lim are respectively the limit values for instantaneous and final deflections, taken respectively to Li/300 and Li /200 in mm for the ith component. To find the optimal design that minimizes the structural volume of the roof truss, three optimization methods are applied: • The DDO method on the basis of the safety factors prescribed by the Eurocode 5. • The RBDO method using the SORA approach without considering degradation model.  • The TD-RBDO method using the SOTVRA approach considering degradation model.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 Table 1: Statistical parameters for materials and loads. Name parameter xk xm cov fm (MPa) 24 33.9 0.2 fc (MPa) 21 29.66 0.2 ft  (MPa) 14 19.77 0.2 fc,90 (MPa) 2.5 3.53 0.2 fν (MPa) 4 5.65 0.2 E (GPa)  10.8 11 0.2 Permanent load (kN/m²) 620 466.52 0.2 Snow (kN/m²) 1193 798.81 0.3 Wind (kN/m²) 1320 883.53 0.36  The target reliability index for one year for the ultimate limit state is set to 3.8 and for serviceability limit state is set to 2.9 (NF-EN 1990 2003). In the TD-RBDO method, the target reliability for allowable lifetime TL fixed to 30 years is estimated with Eq. (4). For all the optimal solutions found by these methods, a time-dependent reliability analysis considering the decay model is performed. The time-dependent reliability analysis is based on the design equation established by Eq. 11. However, the partial safety factors are neutralized (Brites et al. 2013) Timber decay is assessed by the deterioration model presented in section 4 that depends mainly on specific environmental conditions of a given place. It allows for estimating the mass loss during the time. For illustrative purposes, the roof was exposed to a very humid environment corresponding to the city of Nantes (France). Nantes is close to the Atlantic Ocean and has a temperate oceanic climate with annual mean temperature and relative humidity of 12.7 ºC and 81%, respectively. Hourly variations of temperature and relative humidity for 30 years [1980-2010] were used in the example.  5.2. Results Tables 2 and 3 present the optimal solutions corresponding to different optimization methods considering flexible and rigid joints, respectively. It is observed that the optimized volume obtained by TD-RBDO doubles the values calculated by the other approaches. This is expected because DDO and RBDO did not include directly the deterioration process. Small volumes are found when the design considers rigid joints.  Table 2: Design optimization results for flexible joints.  DDO RBDO TD-RBDO b (mm) 166.89 175.22 240.30 h1 (mm) 235.81 248.80 375.32 h2  (mm) 333.78 350.44 480.60 h3 (mm) 222.52 233.62 320.40 Volume (cm3) 1182.65 1305.4 2524.11  Table 3: Design optimization results for rigid joints.  DDO RBDO TD-RBDO b (mm) 162.24 143.80 236.27 h1 (mm) 216.24 191.74 315.03 h2  (mm) 324.50 287.61 472.55 h3 (mm) 216.33 191.74 315.03 Volume (cm3) 1100.12 864.26 2333.04  Figure 2 depicts the time-dependent reliability indexes of the optimal design of the roof truss given by the DDO approach. The initial reliability index at t = 0 of the ultimate limit state satisfies the target reliability index of one year. However, the initial reliability index at t = 0 of the serviceability limit state is not checked. In other words, the use of the partial safety factors in the design optimization cannot guarantee the target reliability. When timber degradation is considered, the reliability indexes of the serviceability and ultimate limit state fell suddenly, where the serviceability reliability index reaches the target after 8 years when the flexible joints are considered. This is explained by the fact that the structure is less rigid with flexible joints increasing the critical deflections.  Figure 3 shows the time-dependent reliability profiles of the RBDO solution, where this optimal solution is estimated without the decay model and by considering only the target reliability of one year. Figure 3 indicates that the reliability indexes for ultimate and serviceability 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 limit states at the initial time are checked regarding the target values for one year (3.8 for ULS and 2.9 for SLS). However, considering the timber decay, the reliability indexes decrease quickly and the serviceability and ultimate target reliability indexes are reached after 13 and 9 years respectively for the rigid joints.   Figure 2: Time-dependent reliability index of DDO solution for flexible (F) and rigid (R) joints.  Figure 3: Time-dependent reliability index of RBDO solution for flexible (F) and rigid (R) joints.  Figure 4 indicates that the TD-RBDO method combined with the SOTVRA approach give the optimal design that satisfies the serviceability and reliability requirements during the 30 years lifetime. However, the proposed design solution is more expensive. The optimal volume is about 2.7 times larger than the RBDO volume (Tables 2 and 3). This large volume ensures the reliability and serviceability requirements under a very humid exposure that accelerates timber decay. Different results could be obtained for other climates. Besides, if the failure cost is considered in the RBDO problem, the total cost of the TD-RBDO may be lower. For all cases, the target reliability is reached early when rigid joints are considered because these joints generate additional bending moments in the truss.   Figure 4: Time-dependent reliability index of TD-RBDO solution for flexible (F) and rigid (R) joints. 6. CONCLUSIONS This study focused on the design optimization of timber truss structures by accounting for uncertainties, climate variations, and serviceability and safety constraints. The preliminary results indicate that the TD-RBDO solution ensures the serviceability and reliability requirements during the whole lifecycle. However, the optimized solution needs a large material volume (larger construction costs) in comparison with DDO and RBDO for the environmental exposure. Further work in this area will focus on: (1) the consideration of climate uncertainties, (2) the design optimization for other locations, and (3) the consideration of real costs (including the failure cost). 7. ACKNOWLEDGMENTS: The authors would like to acknowledge the National Agency of Research (ANR) for its financial support of this work through the project CLIMBOIS ANR-13-JS09-0003-01 as well as the labeling of the ViaMéca French cluster. 0 5 10 15 20 25 300123456β(t)Time (years)  βF,ULSβF,SLSβR,ULSβR,SLSβULS,30t =2.85βSLS,30t =1.60 5 10 15 20 25 3001234567β(t)Time (years)  βF,ULSβF,SLSβR,ULSβR,SLSβULS,30t =2.85βSLS,30t =1.60 5 10 15 20 25 30024681012β(t)Time (years)  βF,ULSβF,SLSβR,ULSβR,SLSβULS,30t =2.85βSLS,30t =1.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 REFERENCES Aoues, Y., Bastidas-Arteaga, E., and Chateauneuf, A. (2009). “Optimal design of corroded reinforced concrete structures by using time-variant reliability analysis.” 10th International Conference on Structural Safety and Reliability ICOSSAR, CRC Press., Osaka, Japan, 1580–87. Aoues, Y., and Chateauneuf, A. (2010). “Benchmark study of numerical methods for reliability-based design optimization.” Structural and multidisciplinary optimization, 41(2), 277–294. Brischke, C., and Thelandersson, S. (2014). “Modelling the outdoor performance of wood products – A review on existing approaches.” Construction and Building Materials, 66, 384–397. Brites, R. D., Neves, L. C., Saporiti Machado, J., Lourenço, P. B., and Sousa, H. S. (2013). “Reliability analysis of a timber truss system subjected to decay.” Engineering Structures, 46, 184–192. Du, X., and Chen, W. (2004). “Sequential optimization and reliability assessment method for efficient probabilistic design.” Journal of Mechanical Design, ASME, 126(2), 225–233. EN 252. (1990). Wood preservatives. Field test methods for determining the relative protective effectiveness in ground contactitle. European Committee for Standardization. De Freitas, R. R., Molina, J. C., and Calil Junior, C. (2010). “Mathematical model for timber decay in contact with the ground adjusted for the state of São Paulo, Brazil.” Materials Research, scielo, 13(2), 151–158. Isaksson, T., Brischke, C., and Thelandersson, S. (2012). “Development of decay performance models for outdoor timber structures.” Materials and Structures, 46(7), 1209–1225. Köhler, J., Sørensen, J. D., and Faber, M. H. (2007). “Probabilistic modeling of timber structures.” Structural Safety, 29(4), 255–267. Kravanja, S., Turkalj, G., Šilih, S., and Žula, T. (2013). “Optimal design of single-story steel building structures based on parametric MINLP optimization.” Journal of Constructional Steel Research, 81, 86–103. Leicester, R. H., Wang, C., Nguyen, M. N., and MacKenzie, C. E. (2009). “Design of Exposed Timber Structures.” Engineers Australia, 9(3), 217. Lourenço, P. B., Sousa, H. S., Brites, R. D., and Neves, L. C. (2012). “In situ measured cross section geometry of old timber structures and its influence on structural safety.” Materials and Structures, 46(7), 1193–1208. NF-EN 1990. (2003). Eurocode 0: Basis of structural design Final draft prEN 1990-, Brussels, European Comitee for Standardization. NF-EN 1995. (2005). Eurocode 5: design of timber structure, Part 1-1 General rules and rules for buildings, Brussels, European Comitee for Standardization. Riahi, H., Moutou Pitti, R. Bressolette, P., Chateauneuf, A., and Fournely, E. (2011). “Relaibility based design of structures under seismic loading  : application to timber structures.” Conference Proceedings of the Society for Experimental Mechanics Series., 417–423. Ryan, P. C., Stewart, M. G., Spencer, N., and Li, Y. (2014). “Reliability assessment of power pole infrastructure incorporating deterioration and network maintenance.” Reliability Engineering & System Safety, 132, 261–273. Šilih, S., Premrov, M., and Kravanja, S. (2005). “Optimum design of plane timber trusses considering joint flexibility.” Engineering Structures, 27(1), 145–154. Sousa, H. S., Branco, J. M., and Lourenço, P. B. (2014). “Characterization of Cross-Sections from Old Chestnut Beams Weakened by Decay.” International Journal of Architectural Heritage, Taylor & Francis, 8(3), 436–451. Tomás, A., and Martí, P. (2010). “Shape and size optimisation of concrete shells.” Engineering Structures, 32(6), 1650–1658. Tran, V.-D., Oudjene, M., and Méausoone, P.-J. (2014). “FE analysis and geometrical optimization of timber beech finger-joint under bending test.” International Journal of Adhesion and Adhesives, 52, 40–47. Viitanen, H. (1996). “Factors affecting the development of mould and brown rot decay in wooden material and wooden structures. Effect of humidity, temperature and exposure time.” PhD Dissertation, University of Uppsala. Viitanen, H., Toratti, T., Makkonen, L., Peuhkuri, R., Ojanen, T., Ruokolainen, L., and Räisänen, J. (2010). “Towards modelling of decay risk of wooden materials.” European Journal of Wood and Wood Products, 68(3), 303–313. 


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