Joint analysis in wood trusses Traditional carpentry joints Samuel Soares Instituto Superior Técnico, Universidade de Lisboa 1. Introduction This study consists in analyzing two traditional joints of roof timber structures, the front notched joint with a single tooth and the front notched joint with two teeth. The first one is the most common joint in existing timber roof structures. The analysis covers these two types of connections for different skew angles between the elements forming the roof. The analysis were performed using the automated calculation program SAP2000. 2. Calculation models in joints between the bottom chord and the top chord of a wooden truss The traditional carpentry joints are connections that are only able to transmit compressive stresses, without any type of device to make the connection between the elements other than their surfaces. These joints rely on the compression of internal forces to keep facing surfaces in close contact and seldom in metal fasteners. This practice is more common in Western countries than in the Eastern. (Palma, P. & Cruz, H., 2007). 2.1. Front notched joint with a simple tooth According to the model calculation based on the Switzerland regulation (SIA), the contact surface on the edge of the cutout transmits the axial loads. The wood has more resistance in the parallel direction of the grain than the perpendicular one. So with the top chord of the truss cut at 90, it will provide maximum strength for this element, but the bottom chord resistance will be reduced, because the force will be acting perpendicularly to the grain. With a cutting angle, ε, equal for the two members the optimal resistance is attained.
In cases where the notch is located near the end of the bottom chord, it's necessary to ensure a minimum length, ν, to resist the shear stress that appears on that location. The lower limit of ν length is 15 cm to prevent the weakening of the natural wood nib (Natterer, J., et al 2004). Figure 1 Geometry and forces in the front notched joint with a single tooth 2.2. Front notched joint with double teeth The birdsmouth joint with two teeth is the result of the combination of the characteristics from the birdsmouth joint with a single tooth and the birdsmouth joint with a rear tooth. Although the analysis of the birdsmouth joint with a rear tooth is not considered on this paper, it is important to present it, in order to understand where the front notched joint with double teeth came from. In this type of joint, the cutting angle is generally positioned perpendicularly to the axis of the top chord (Figure 2). This avoids the transmission of loads from the front cutout that would create tensile stresses perpendicular to the fibers. Figure 2 Geometry and forces in the joint with a rear tooth. The force is conducted by contact on the cut surface that is perpendicular to the longitudinal axis of the top chord. Only the back of the connection resists to the applied force, so it requires a verification of the oblique resistance of the bottom chord and the compression strength parallel to the grain of the top chord. If the unit resistance value of the bottom chord and top chord are the same, the oblique resistance of the grain will be the most conditioned for the calculation, because the parallel resistance is higher than the oblique resistance.
Therefore, as stated before, the joint with double teeth combines the advantages of the two previous types of joints. The double tooth joint has two different depths (t 1 and t 2). The first one is configurated as a joint with a single tooth and the second one has a joint with a rear tooth. The most of the applied force is transmitted by two surfaces with different inclinations. The angle of the design values of resistance is different for the two surfaces, with α = β/2, characteristic of the front notch with a single tooth with a depth t1, and α = β, characteristic of the birdsmouth joint with a rear tooth with a depth t2. To simplify the calculation, it is assumed that the angle of the design value of resistance is the mean between the two, ie, α = 3/4β. Therefore, the condition is given by the thickness (Natterer, J., et al 2004): t N dcosβ bf c,α,d with t = t 1 + t 2 e α = 3 4 β (2-1) Figure 3 Geometry and forces in the joint with two teeth 3. Modeling The modeled joints were constructed by taking in account the structure of a wooden truss. The main objective of this paper is to study the transmission of the forces on the joints, so the model was done locally using only half of the truss. The characteristics used for all the carpentry joints are the same and the class C18 of the wood was used (Table 1). Wood C18 Modulus of elasticity (kn/mm 2 ) EO,mean 9 Modulus of distortion (kn/mm 2 ) G,mean 0.56 Characteristic density (kg/m 3 ) ρk 320 Mean density (kg/m 3 ) ρmean 380 Table 1 Physical wood characteristics used on the model The geometric characteristics used are shown on the Table 2
Table 2 Geometric characteristics of the elements. Geometric Characteristics Top Chord Bottom Chord Thickness (m) 0.2 0.2 Width (m) 0.1 0.15 The depths, the skew angles and the ν length used for the joints are presented on the Table 3. Front Notched Joint t 1 (cm) t 2 (cm) v (cm) β ( ) Single Tooth 3-10-30 25-45 Two Teeth 2 3 10-30 25-45 Table 3 Depth, v length and skew angles on the different joints Those values were defined using the equations that exist on the Switzerland regulation (SIA). One of those equations was presented on the previous chapter. Examples from the used models are presented on the Figure 4 and Figure 5. Figure 4 Model of the front joint with a single tooth. Figure 5 - Model of the front joint with two teeth
Links elements were used to make the connection between the elements. While separating the top and bottom chords, different pairs of nodes that use the same coordinates were created, with one member of the pairs being assigned to the top chord and the other member to the bottom chord. The connection between the node pairs was made through links with an equivalent rigidity of the axial stiffness of the bottom chord. This is a hypothesis used to ensure that the transmission of the loads occurs without large relative deformations. The program allows one to choose the directions in which the links work. The choice of those directions is related to the way that the force is transmitted between the elements. As already mentioned above, the load is transmitted perpendicularly to the notched surfaces, and the horizontal component is primarily conducted through the front surface. Therefore, to approximate the model in SAP2000 to the theoretical model shown on the previous chapter, three links were created with different axes, for the two types of joints studied in this paper. The joints characteristics are shown on the Table 4. Table 4 Link characteristics. Name Rigid Direction Stiffness (N/mm) LIGACAO 1 U2, U3 270000 LIGACAO 2 U2 270000 LIGACAO 3 U3 270000 Figure 6 Links characteristics on the front notched joint with a single tooth. 3.1. Non linear analysis By abruptly applying a force, a variation in stresses is created, due to the sudden load application. To prevent these fluctuations, a non-linear analysis was performed with a gradual application of the load. The reason why the nonlinear analysis was chosen is related to the conditions that are considered in the support, that allows only the transmission of compressive forces. The loading took place in a gradual way in order to avoid the dynamic response that stems from the instant application of the force. A time interval was defined in which the action is increased to it's maximum value and remaining constant until the end of the analysis. We defined a total analysis time of 10 seconds and the load reached it's maximum value in the 4 th second.
Force transmitted (kn) 4. Analysis and Results For the analysis to take place, it was necessary to put a compression load on the top chord. The value of the force is not relevant, because, in this paper, there isn t any verification for the security of the structure, however, as an indicative figure, and to be able to compare the different analysis, the force placed had the value of 30 kn. After the end of the analysis, it was possible to receive the value of the forces transmitted by each link between the top and the bottom chord, and compare the results for each skew angle. Through the stress results obtained, it was also possible to compare the stress along the joints with different v lengths. 4.1. Front notched joint with a single tooth The force F1 and F2 are the forces transmitted by the front surface and the rear surface, respectively. 35 30 25 20 15 10 F1 F2 5 20 30 40 50 60 70 Angle ( ) Figure 7 Force transmitted in function of the skew angle - Front notched joint with one tooth. From the last figure, it is possible to examine the evolution of the importance of each surface by the increase of the skew angle. For small angles, the rear surface almost does not take part of the transmission of the applied force and it starts to become more relevant with a higher angle. In the next graphics, the results for different v lengths are shown, for the trusses with an angle of 40 and 30 degrees, in the direction 1 (direction perpendicular to the first surface). As the length increases, the position of the supports gets further from the joint, but keeping in mind not to have a big eccentricity between the vertical supports and the line of action of the applied force.
Stress S11 (kn/m2) Figure 8 Vertical Support location for the joint with a skew angle of 40 a) with v = 10 cm b) with v = 30 cm. The point number 1 in the axis of the graphics represents the first node in the right of the first surface and the point 23 the last node on the left of the second surface. 0.00-1000.00 1 3 5 7 9 11 13 15 17 19 21 23-2000.00-3000.00-4000.00-5000.00-6000.00 v=10 v=15 v=20 v=25 v=30-7000.00-8000.00 Node Figure 9 Stress in the direction 1 along the joint with a single tooth β = 40. In the Figure 9, despite the difference of the stress not being high, it is possible to see that in the nib of the joint (position 8 and 9), the lower stress happens for the bigger v length. This way, when the v length is smaller the concentration of stress in the nib of the joint is higher.
Stress S11 (kn/m2) 0.00-1000.00 1 3 5 7 9 11 13 15 17 19 21-2000.00-3000.00-4000.00-5000.00-6000.00 v=10 v=15 v=20 v=25 v=30-7000.00-8000.00 Node Figure 10 - Stress in the direction 1 along the joint with a single tooth β = 30. The same happen for the angle of 30 degrees as shown in Figure 10. 4.2. Front notched joint with two teeth For the double tooth joint, four different areas transmit the force, as illustrated in Figure 11. Figure 11 Forces transmitted in each surface. As was done for the birdsmouth joint with a single tooth, the forces were calculated in all surfaces for various angles, with the results given by the links between the elements. Those results are shown in the Figure 12.
Stress S11 (kn/m2) Total Force Transmitted (kn) 24.50 19.50 14.50 9.50 4.50 F1 F2 F3 F4-0.50 20 30 40 50 60 70 Ângle ( ) Figure 12 - Force transmitted in function of the skew angle Front notched joint with two teeth. Analyzing the Figure 12 it is possible to see how each surface evolves with the increasing of the skew angle. On the 4th surface, the force transmitted between the element is null or close to null for the trusses with a lower skew angle. For example, in the cases of the trusses with 25⁰ and 30⁰, the 4th surface is parallel to the direction of the applied force, so there isn t any percentage of the applied force transmitted by this surface to the bottom chord. For the angles of 40⁰ and 45⁰ a percentage of transmission already exists, this happens because the surface and the applied force are no longer parallel, and has the skew angle increases, the percentage will be higher. As expected, the forces F1 and F3 are those with the highest load transmission, and when the skew angle increases, it appears that the force F2 is gaining prevalence. The analysis of these curves cannot be treated as exact results because the joint is optimized for an angle of 30 degrees, and not to one of 45 degrees. Therefore, depending on the arrangement of the several geometry variables of a wood truss, the results can be different. 1000.00 500.00 0.00-500.00-1000.00-1500.00-2000.00-2500.00-3000.00-3500.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Node Série1 Série2 Série3 Série4 Série5 Figure 13 - Stress in the direction 1 along the joint with two teeth β = 45.
Stress S11 (kn/m2) 1000 500 0-500 -1000-1500 -2000-2500 -3000-3500 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Node v=10 v=15 v=20 v=25 v=30 Figure 14 - Stress in the direction 1 along the joint with two teeth β = 30. In Figure 13 it is observed a clear tendency toward one surface (1-8) and the third surface (12-19) of the joint with two teeth. For a smaller length ν, the first surface will have a lower stress concentration and, in turn, the surface 3 will have a greater stress concentration. To a greater length ν the reverse happens. This happens by the change that happens to the direction of the transmission of the charges from one surface to another depending on the support location. For bigger ν lengths, the vertical supports were placed further apart from the join, taking care not to create a large eccentricity between the position of the supports and the line of action of the applied force. Having the support further, the surface number 2 forwards more load to the supports. In Figure 14 the tendency is identical, as expected. 5. Conclusions After performing the analysis, it can be concluded that the geometry of the truss has a great influence in terms of the transmission of efforts. This is so because, depending on the geometry, the most conditioning areas are different. Due to the various geometrical solutions, the wooden trusses are complex elements. So it is important to collect the maximum data about those type of elements to be able to give the best solutions for their rehabilitation. 6. References Natterer, J., Sandoz, J-L., Rey, M. (2004). Construction en bois: matériau, technologie et dimensionnement. Presse Polytechniques et Universitaires Romandes. Palma, P. & Cruz, H. (2007). Mechanical Behaviour of traditional timber carpentry joints in service conditions Result of monotonic tests. ICOMOS IWC XVI Internation Symposium: Florence, Venice and Vicenza.