Tali wood and Fraké wood were selected in this work. Table 1 presents some physical and mechanical properties of these species, referenced in the Tropix 7.0 database. Tali wood is a dense and emblematic hardwood from the Congo Basin. The colour of the wood varies from orange yellow brown to reddish brown. The colour of Frake wood is light yellow and does not have demarcated sapwood. These species are abundant and can be glued for a structural use. They present a good interest for a glulam use [1] . The differences of densities and elasticity moduli (MOE) should make Tali and Frake, respectively, suitable for external and internal lamellae of the glulam. For this work, we have selected Epoxy as our glue. Its characteristics were set up as presented on Table 2.

In this study we have considered the following dimensions of the beam, length: 30 m, width: 100 mm, height: 78 mm. Since our software could not consider more than 3 lamellae, we have set the number of lamellae to 3. The thickness of each lamella has been set to 25 mm, and the thickness of each glue layer was set to 1.5 mm. The characteristics of the bolts will be determined in the method.

The simulation of the load-displacement (P-Δ) behaviour was made according to the following assumptions:

· Wood is an elastoplastic orthotropic material [7] ;

· The transverse isotropy is considered, which assumes identical properties in the radial and tangential directions [4] [5] ;

· Glue is isotropic linearly elastic;

· Bolts are modelled as elastic-perfectly plastic material;

· The hole clearance and the splitting in timber due to tension stresses perpendicular to the grain don’t have a significant influence in the load-deformation analysis [20] [21] ;

· Out-of-plane deformations are not considered;

· The beam is subjected to uniformly applied long-term loads;

· Moisture content is constant;

· Bending strength is dominant in wood connections [22] .

In order to determine the characteristics of bolts, we have used the yield theory through Equations (1) to (3) [23] :

M e l = f y π d 3 32 (1)

M p l = f y d 3 6 (2)

M y , R k = 0.3 f u , k d 2.6 (3)

where M e l is the elastic moment of bolt, M p l is the plastic moment, f y is the yield strength, d is the diameter of bolt which is set to 12 mm in this study, M y , R k is the yield bending moment of bolt and f u , k its tensile strength.

We can express the shear strength F V , R k of bolt using Equation (4) from Eurocode 5 [23] :

F V , R k = min { f h ⋅ t ⋅ d f h ⋅ t ⋅ d ⋅ [ 2 + 4 ⋅ M y , R k f h ⋅ d ⋅ t 2 − 1 ] 4 ⋅ M y , R k ⋅ f h ⋅ d (4)

where f h is the embedding strength of wood, t is the thickness of wood and d the diameter of bolt. The embedding strength of wood can be expressed from Equation (5) [23] :

f h = 0.082 × ( 1 − 0.01 × d ) × ρ k (5)

where ρ k is the relative density of wood.

There are two ways to express the ultimate shear strength. According to Eurocode 5 we have Equation (6) [23] :

F V , R d = F V , R k ⋅ k m o d γ M (6)

where k m o d is the modification factor of wood and γ M its partial coefficient. The second way to express the ultimate shear strength is Equation (7) from Eurocode 3 [24] :

F V , R d = α V ⋅ f u b ⋅ A γ M 2 (7)

where α V is a coefficient depending on the class resistance of bolt, f u b is the notation of f u , k in Eurocode 3, A is the cross section of bolt and γ M 2 = 1.25 .

If F V , E d is the applied load, the number of bolts can be determined from Equation (8) [23] :

n = F V , E d F V , R d (8)

For the self-weight CC1, Figure 9 displays displacements over increments and iterations without bolted wood connection, and with a bolted wood connection. The results obtained in this Figure are found respectively on Table 4 and Table 5.

The curves on Figure 9 increase to reach maximum values. These values are maximum displacements. It stands at u max = 384.1   mm for the self-weight without bolted wood connection (see on Table 4), and u max = 333.6   mm with bolted wood connection (see on Table 5). The presence of the residual loads can increase these values to u max = 398.8   mm without bolted wood connection (see on Table 4, iteration n˚3), and u max = 334.9   mm with bolted wood connection (see on Table 5, iteration n˚3). Displacements are greater for the self-weight without bolted wood connection. This difference stands at Δ u max = 50.5   mm . It means that the presence of steel has reduced the maximum displacement by 50.5 mm (13.15% of the initial displacement). According to Eurocode 5, especially the standard NF EN 1995-1-1/NA, the maximum deflection W f i n caused by the set of loads and creep is W f i n = L / 125 = 240   mm (L = 30 m). Compared to the previous values, the gap between this maximum deflection and maximum displacements is 37.5% without bolted wood connection, and 28.06% with bolted wood connection. From iteration n˚1 to iteration n˚2, load factor has been increased by 0.436 (1.02 KN) and displacement has increased by 112.1 mm with bolted wood connection (see on Table 5). The strength-to-displacement ratio stands at 9 N for 1 mm. While, load factor has been increased by 0.279 (0.65 KN) and displacement by 115.9 mm without bolted wood connection (see on Table 4). The strength-to-displacement ratio is rather 5.61 N for 1 mm Thus, by reducing displacements and increasing load factors, steel brought more stability to the structure.

For the long-term load CC2, Figure 10 displays displacements over increments

and iterations without bolted wood connection, and with a bolted wood connection. The results obtained from this Figure are found on Table 6 and Table 7.

On Figure 10, the maximum values are respectively u max = 1324.1   mm for CC2 without bolted wood connection, and u max = 1042.5   mm with bolted wood connection (see on Table 6 and Table 7). The difference stands at Δ u max = 281.6   mm (21.27% of the initial displacement). This difference is greater than the one obtained for CC1. It means that the more the loads are high, the more steel reduces their displacements. We can also notice that these displacements are greater than the limit W f i n provided by Eurocode 5. The gap stands at 81.87% without bolted wood connection, and 76.98% with bolted wood connection. Load factors

are greater and increase faster with bolted wood connection. From iteration n˚1 to iteration n˚8 it has increased by 0.836 (125.4 KN) without bolted wood connection, and by 0.998 (149.7 KN) with bolted wood connection (see on Table 6 and Table 7). In the meantime, displacements have increased by 1081.1 mm without bolted wood connection (115.99 N for 1 mm), and by 851 mm with bolted wood connection (175.9 N for 1 mm). At iteration n˚1, displacements remain the same ( u = 195.3   mm ). It means that for very low load factors ( λ ≤ 0.008 ), there is no need to use steel reinforcement. In other words, for loads lesser than P = 1.2   KN , there is no need to use steel reinforcement.

Figure 11 displays displacements over increments and iterations for the combinatoric CO1 with and without bolted wood connection. The results corresponding to these curves are respectively found on Table 8 and Table 9.

On Figure 11, the curves reach maximum values before stabilizing. These values are respectively u max = 466.4   mm without bolted wood connection and u max = 406.3   mm with bolted wood connection (see on Table 8 and Table 9). Here steel has reduced maximum displacements by Δ u max = 60.1   mm . This difference is greater than the one obtained from self-weight CC1. Indeed, this combinatoric is greater than self-weight (CO1 = 1.35 × CC1). The more displacements are high, the more impact of steel is visible. In other words, the more loads are high, the more bolted wood connection absorbs them. It is the elastic-perfectly plastic character of steel that is highlighted.

For the ULS (CO2 = 1.35 × CC1 + 1.5 × CC2), Figure 12 shows the plot of displacements over increments and iterations with and without bolted wood connection. The results of these curves are found on Table 10 and Table 11, respectively for CO2 without bolted wood connection, and with bolted wood connection.

On Figure 12, maximum displacements are respectively u max = 1670.4   mm without bolted wood connection, and u max = 1320.2   mm with bolted wood connection (see on Table 10 and Table 11). We notice a maximum reduction Δ u max = 350.2   mm (20.96% of the initial displacement). Indeed, in the ULS loads are greater than any other combinatoric. It is the combinatoric which has the most penalizing effect on the structure. Compared to the limit W f i n provided by Eurocode 5, the gap stands at 85.63% without bolted wood connection, and 81.82% with bolted wood connection. While there was no need for using steel in the applied load for λ ≤ 0.008 (1.2 KN), it is advised to use steel when one designs at ULS for λ ≥ 0.004 (912.6 N) (see on Table 10). For the structural analysis and design, we can define the plastic resistance value of steel as the intersection between the initial tangent stiffness and the tangent stiffness at the final part of the load-displacement curve ( F y , U y ) [6] . This value is determined at the ULS from the load-displacement curve CO2 with bolted wood connection. In order to make it easier we are going to use the original curve from the software (see on Figure 13).

From this curve, the plastic resistance F y corresponds to the 6.9 iteration. We obtain F y = 84.34   KN and U y = 1323.8   mm . To assess the reinforcement of walls with densified wood under shear strength, the stiffness was defined in [10] as the ratio between the maximum load and the maximum slip. Compared to this present study where bending strength is dominant, we can define the bending stiffness by the ratio between the maximum load and the maximum deflection. From Table 10 and Table 11, the bending stiffness has increased from 0.13 KN∙mm⁻¹ without bolted wood connection to 0.17 KN∙mm⁻¹ with bolted wood connection. The bending stiffness has been increased by 32.94%. Since the bending stiffness is proportional to the critical load in [1] , the increase of the bending stiffness results to the increase of the critical load by 32.94%. Thus, inducing the increase of the bending resistance by 32.94%.

For the SLS (CO5), Figure 14 displays the curves of displacements over increments and iterations with and without bolted wood connection. Table 12 and Table 13 provide their corresponding results.

The maximum displacements are respectively u max = 1441.9   mm mm and u max = 1141.4   mm . Here the difference is Δ u max = 300.5   mm (20.84% of the initial displacement). Displacements are lower than the ones obtained from ULS. In order to ensure the safety of the structure, it is advised to use steel reinforcement for load factors λ ≥ 0.006 (914 N) (see on Table 12 and Table 13). The SLS are used to limit deformations in the structure. During the design process, one should make sure that the value of the mid-span deflection caused by the set of loads applied to the structure remains lower than the limit value.

The model developed in this study could find applications in frameworks, bridges and any structure subjected to high bending strength, where it is important to undermine deformations which can lead to instabilities. For wood beams subjected to bending strength, checking deformations at the SLS is the most important criterion for structural analysis and design [23] . This study enables to predict load-displacement (P-Δ) behaviour in structures subjected to uniformly be applied long-term loads, which is difficult to carry out through testing. The elastoplastic steel properties provide good estimates of load-displacement [28] . Mechanical properties of wood and glue used for the simulation in the software are realistic and obtained from various tests (see on Table 1 and Table 2). Realistic characteristics of steel were used as inputs of the model, since they were drawn from Eurocode 3. The bi-linear trends of load-displacement curves obtained are confirmed by authors such as in [5] [7] and [28] . Knowing the load-displacement behaviour of the structure helps designers reduce costs of construction. Indeed, they are able to know when to use steel reinforcements that are very costly.

Moreover, in the literature it is difficult to find investigations about load-displacement behaviour on large span structures made of tropical hardwood species. Most of studies are dealing with small size structures made of softwoods. For instance, load-displacement behaviour was studied in [12] on a 1200 mm length beam with and without bolted wood connection. They obtained from bending tests approximately 6000 N for 40 mm displacement without steel and 6000 N for 20 mm with steel. In this present study, after performing a linear interpolation from Figure 11, Table 6 and Table 7, 6000 N corresponds to 505.9 mm displacement without steel and 399.45 mm with steel. This significant gap could be explained by the differences in the sizes of beams, wood nature and the characteristics of bolts. The plastic resistance obtained in [6] was F y = 86.1   KN . In this present study, the plastic resistance is F y = 84.34   KN . This slight difference could be explained by the fact that in their work, they have used dowels with class resistance 4.8 which means that the nominal yield and ultimate tensile strength are respectively equal to 320 MPa and 400 MPa [24] . While in this study, class resistance of bolts was set to 4.6 which means that the nominal yield and ultimate tensile strength are respectively equal to 240 MPa and 400 MPa [24] .

Conducting experimental investigations on large span structural product would be difficult. Indeed, the process is a priori time-consuming and expensive. For these reasons, the present paper is devoted to modelling by using a 3D FE nonlinear analysis. FE analysis is suited for modelling and evaluating parallel and perpendicular to the grain loading of wood with bolted wood connections [28] . That is the reason why many authors used 3D FE models to predict load-displacement behaviour in timber with bolted wood connections [5] [6] [9] [28] .

The iterative Newton-Raphson method was used in [16] to study the nonlinear behaviour of a large span steel frame structure (10 m). A load of 45 tons was uniformly step by step applied to the structure. Their simulation was performed through a CEPAO program. They carried out loads versus displacements curves considering steel as elastic-perfectly plastic material as we did in this work. The trends of the curves were the same as we obtained in this work. The elasto-plastic analysis done in their work revealed that the curves stabilize after a load factor λ = 0.754 as on Table 5, Table 7, Table 9, Table 11 and Table 13 with bolted wood connection, and displacements can go beyond 30 cm. Reference [29] simulated the nonlinear behaviour of a large span glulam beam using the Newton-Raphson method. The beam length was set to 8000 mm and the dimensions of the cross section 200 × 1000 mm². Here the tangent stiffness matrix was analytically calculated before being implemented in a MATLAB code. Since the length was smaller than the one used in this present work (30 m), there were less nodes, thus, less nonlinear equations to resolve. This made it possible to calculate analytically the tangent stiffness matrix. In this present work, the tangent stiffness matrix was numerically computed by the software during the simulation. Besides, their nonlinear study was also numerically performed through the step-by-step iterative Newton-Raphson method. Instead of carrying out loads versus displacements curves as we did in this work, they were interested in carrying out loads versus warping angles curves. Results matched well with the ones obtained from the analytical Euler method. Several other authors as in [30] [31] and [32] successfully performed a numerical nonlinear analysis on large span glulam beam columns using the iterative Newton-Raphson method. In most of these cases, a uniform load was step by step applied to the beam. Their works proved the efficiency of the Newton-Raphson method compared to the classical finite element method.

However, this study focused on the nonlinear analysis of the load-displacement behaviour without predicting failure. The stability check for bolts was performed in a different section of the RFEM 5 software called RF-JOINTS (see on Figure 7). In order to enable the software to run the simulation, we had to select a load that meets stability conditions for both wood and steel. This made it difficult to predict failure for both materials despite the lack of accuracy in this investigation [28] . Geometrical imperfections due to the curvature of the beam-column (initial out of straightness) were not considered in this work. Such imperfections may create an additional bending moment along the axis of the beam, thus perturbing the loading of the glulam beam [1] . Moreover, the study was carried out with the assumption that the constant climate in the wood moisture content was 12%. The natural climate condition, corresponding to the exposure environment of these beams, generally varies, thus inducing some mechano-sorptive effects in wood which may be characterized by more important deformations.