The mortise and tenon joints are the main connection forms used in ancient timber buildings, and damaged joints have a critical effect on the safety of a timber structure. There are three main damaged cases of dovetail joints which are pulling, contraction, and mixing damages. In this study, using a theoretical analysis of the stress distribution in a mortise and tenon joint resulted from the pullout damage, a theoretical equation for the resisting moment of the joint was proposed. A finite element model was used to simulate the cyclic displacement loading of a frame with intact joints and with different levels of pulling and contraction damaged joints. The results show that the moment capacities both for the test and the simulation were in good agreement with each other. The simulation results also indicated that there are no changes in the capacity and energy dissipation of the pulling damaged joint compared to that of the intact joint, and good seismic performance still was provided when the pulling damage was less than 2/5 of the joint length. However, the capacity of the contraction damaged joint was significantly reduced, and its seismic performance was tolerably lost. The seismic performance of a mixing damaged tenon with the same degree of pulling damage was between that of the pulling damaged tenon and the contraction damaged tenon, and generally, it was controlled by the contraction damage. The friction between the tenon and the mortise is the main source of resisting moment and energy dissipation ability.
Timber structures have been used widely around the world throughout the history, including China, Japan, Korea, Canada, the USA, and northern Europe [
Joints in Chinese ancient timber building. (a) Straight tenon joint and (b) dovetail joint.
Research on mortise and tenon connections has typically focused on intact joints. Zhao et al. [
In this field, there has been little research on the performance of damaged dovetail joints, and most damaged characteristics have been mainly determined by the numerical method. Pang et al. [
In recent years, some achievements about traditional timber frames and mortise and tenon joints have been obtained. Xie et al. [
The behavior of damaged dovetail joints conductive to improve the conservation and restoration of ancient timber structures require a theoretical method to determine the likely performance of the joint-containing structure under seismic loading. In order to theoretically derive a method for calculating the resisting moment of a joint, in this paper, the effects of damage caused by an external force (pulling damage), the shrinkage of the tenon (contraction damage), or a combination of the two are determined. The effects of these types of damages are simulated by performing a pseudostatic numerical simulation of a dovetail truss subjected to different deflection angles and loads. By comparing the results of the theoretical and numerical analyses, the changes in the seismic performance of the joints after damage can be illustrated, assisting conservation efforts by improving the ability of restoration workers to evaluate the condition of ancient timber structures.
An earthquake subjects a timber frame to a horizontal cyclic action, causing the joints to rotate, analogous to the reciprocal loading of a frame during a lab simulation, during which the overall deflected shape of the frame can be divided into five positions: normal (unloaded), upward loading, upward unloading, downward loading, and downward unloading. Typically, the frame shape under loading and unloading can be considered the same when the column is at the same displacement. Figure
Horizontal displacement loading directions. (a) Upward loading and unloading and (b) downward loading and unloading.
After rotating to a certain limit, the joint will resist the applied moment, restricting the rotation of the joint. For an intact joint, the contact between the tenon and the mortise becomes local contact at this time, consisting of an extraction force created by the moment and a corresponding friction force between the mortise and tenon in the resulting compression zone at the points of contact. These extraction, compression, and friction forces are shown in Figure
Forces on mortise and tenon joints under horizontal displacement of frame: (a) tenon compression zones, (b) upward loading, and (c) downward loading.
The following assumptions have been made in the derivation of the bending moment calculation formula: (1) when determining the displacement at each point on the tenon, the tenon is assumed to move as a rigid body within the mortise; (2) the effect of the upper and lower faces of the mortise are the same, and the compression embedment region is considered to deform along the width of the tenon; (3) the stress-strain relationship of the timber strip compression and the straight-grain compression is in line with the simplified bifurcation model; (4) in the elastic stage, the compressive stress in the contact area is proportional to the deformation due to deflection. When subjected to upward loading, the deformation area of the tenon head can be described as shown in Figure
Tenon deflection under upward loading.
The geometric parameters of the compression zone deformation at
The geometric parameters of the compression zone deformation at
The geometric parameters of the compression zone deformation at
Tenon deflection under downward loading.
The method for solving the extracted displacement of the tenon
Regardless of the direction of loading or unloading, the extraction deformation state is the same, so whether evaluating the upward or downward direction, the geometric extraction deformation equations change only by exchanging the
Deformation and stress distribution in local compression zones of the tenon. (a) Local compression deformation distribution and (b) local compression stress distribution.
In the elastic stage, the compressive forces are given by
For the elastoplastic stage, the compression force is given by
Thus, the
The friction force corresponding to the compression area is as follows:
To calculate the resisting moment due to friction on the sides of the tenon, first, the deformation of the different sides must be determined, and then the corresponding stress values can be obtained. From the stress values and geometric conditions, the friction moment between the sides of the tenon and the mortise can be obtained with reference to Figure
Side view of the tenon under (a) slippage and (b) contraction.
The horizontal deformation at the middle of the tenon head can be obtained through its relationship with the amount of tenon slippage (Figure
Further, stress on the sides of the tenon can be determined as follows:
The cross-section compression deformation is assumed to be evenly distributed along the length of the tenon. Under upward loading and unloading (Figure
Tenon stress and deformation under extraction. (a) Upward loading (or unloading), (b) downward loading (or unloading), and (c) side extraction stress.
Taking the position “o” at the origin of the coordinates shown in Figure
The unit friction moment ΔM in any microcell on the side of the tenon at coordinates
The friction moment
Combining the geometric conditions, physical conditions, and balance and coordination conditions, the bending moment
The dimensions of the timber frame in the finite element simulation were determined according to [
Frame dimensions.
Column | Strut | |||
---|---|---|---|---|
Diameter (mm) | Height (mm) | Length (mm) | Width (mm) | Height (mm) |
210 | 1650 | 1400 | 120 | 180 |
Through investigation and research, three main reasons were identified for the slippage of mortise and tenon joints. First, a large horizontal load (such as that caused by large earthquakes or strong winds) applied at the joints can result in an excessive lateral displacement causing tenon pulling damage. Second, due to the influence of the natural environment, a change in the material properties of the tenon can cause contraction damage allowing the tenon to more readily rotate or slide out under the action of wind loads or small earthquakes. The third reason for slippage in mortise and tenon joints is the combination of the first two situations, referred to as mixed damage. The interaction between the mortise and tenon is different for different causes of tenon slippage. Therefore, in this study, each of the three cases, pulling, contraction, and mixing, damage was evaluated individually. The slippage level was determined using the requirements for the assessment of “damage points” and the seismic structure requirements in the “
Simulation parameters for different tenon sizes and pulling levels.
Damage type | Specimen number | Slippage proportion | Applied tenon pulling damage (mm) | Applied tenon contraction damage (total, two sides) (mm) | Width of tenon head after pullout (mm) | Width of tenon neck after pullout (mm) | Length of tenon (mm) |
---|---|---|---|---|---|---|---|
Intact | W1 | 0 | 0.00 | 0.00 | 60.0 | 50.0 | 50 |
Contraction | Q1 | 1/5 | 10.00 | 1.00 | 58.0 | 48.0 | 50 |
Q2 | 1/4 | 12.50 | 1.25 | 57.5 | 47.5 | 50 | |
Q3 | 3/10 | 15.00 | 1.50 | 57.0 | 47.0 | 50 | |
Q4 | 7/20 | 17.50 | 1.75 | 56.5 | 46.5 | 50 | |
Q5 | 2/5 | 20.00 | 2.00 | 56.0 | 46.0 | 50 | |
Pulling | L1 | 1/5 | 10.00 | 0.00 |
|
50.0 | 50 |
L2 | 1/4 | 12.50 | 0.00 |
|
50.0 | 50 | |
L3 | 3/10 | 15.00 | 0.00 |
|
50.0 | 50 | |
L4 | 7/20 | 17.50 | 0.00 |
|
50.0 | 50 | |
L5 | 2/5 | 20.00 | 0.00 |
|
50.0 | 50 | |
Mixed | LQ1 | 1/5 | 10.00 | 0.50 |
|
49.0 | 50 |
LQ2 | 2/5 | 20.00 | 1.00 |
|
48.0 | 50 | |
LQ3 | 3/10 | 15.00 | 0.50 |
|
49.0 | 50 | |
LQ4 | 3/10 | 15.00 | 1.00 |
|
48.0 | 50 |
In the finite element analysis, the column and tenon were modeled as solids and the bottom of the column was allowed to rotate only in the direction of loading to simulate the articulation of the column foot. The column was extended 150 mm higher than the strut to enable easy loading. The empty space in the joint was filled with a cork material in the referenced experiments. The interaction between the cork, the column, and the tenon is surface-surface contact: in the normal direction, it is hard contact, and in the tangential direction, it is Coulomb friction with a coefficient is 0.3. In order to optimize both the computational efficiency and accuracy, an eight-node linear hexahedron element (C3D8R) was used to model the solids. In the mortise and tenon joint areas, the mesh was refined to improve the accuracy in this area of interest. The complete finite element model is shown in Figure
Meshes of (a) column frame and (b) mortise and tenon joint.
In this study, the timber was considered to be northeast Korean pine and finite element code ABAQUS was utilized to obtain numerical simulations of wooden frame with dovetail mortise and tenon joints. Timber is a typical orthotropic anisotropy material with the simplified constitutive model (Figure
Simplified constitutive model of timber. (a) Parallel to the grain; (b) compression in radial direction.
Wood can be divided in radial, chordwise, and longitudinal directions, each with different physical and mechanical characteristics. The material behavior in the elastic phase in ABAQUS was defined by the nine engineering constants shown in Table
Northeast red pine wood constants.
|
|
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|
650 | 275 | 10000 | 0.035 | 0.02 | 0.02 | 210 | 650 | 275 |
After entering the plastic stage, Hill’s potential function in the property module can be used to define the yield criterion of the timber. By defining the yield stress ratios in six directions to consider the difference in the yield strength of the material in three orthogonal directions, [
Yield stress ratios.
R11 | R22 | R33 | R12 | R13 | R23 |
---|---|---|---|---|---|
0.2 | 0.2 | 1 | 1 | 0.35 | 0.2 |
To simulate the presence of a roof, a vertical load of 10 kN was applied to the top of each column and a horizontal cyclic load was applied at the top of one of the columns using displacement control. After the good results in [
Horizontal displacement loading amplitude.
The frame was first loaded by the vertical load and then by the horizontal load as follows. For the pullout damage and mixed damage frame tests, the column was drawn to the one side until it reached the specified deflection. For the contraction damage test, the degree of contraction, defined as when the tenon and the mortise contacted each other without gaps, was used to determine the applied deflection. The same loading modes were applied to the intact joint.
The finite element simulation was designed based on a real experiment, so the simulation of the column frame can be verified by the results in [
Hysteresis curves of (a)–(c) reference tests and (d) simulation.
Comparison of bending moments from the proposed theoretical calculation and the simulation. (a) No damaged joint, (b) pulling damaged joint, (c) contraction damaged joint, and (d) mixed damage joint.
It can be seen that the simulation results are in good agreement with the experimental results in which loading deflection was 100 mm. Furthermore, the comparison between the theoretical calculation and the numerical simulation shown in Figure
The results of the proposed theoretical calculation method considering the change in the loading and in the size of the tenon head for frames with 1/5 pullout damaged joints and a similar degree of cutting damaged joints are compared with the simulation results at the 100 mm loading stage in Figures
The hysteresis curve provided by the parametric analysis depicts the relationship between the resisting moment and the rotation angle of the joints. In Figure
The (M-
It can be observed that the behavior of Frames L1–L4 is similar to that of W1 in general, with a maximum bending capacity of around 1.6 kN·m. The hysteresis curves of the frames with pulling damage have a spindle shape in the initial stage of loading. The larger the deflection, the greater the fullness of the curve. In the later stages, the overall stiffness of the frame is relatively large, and the energy dissipation effect is good. The main reason for this is that in the process of forming a pulling failure, the tenon is compressed against the mortise, increasing the initial compression force between the tenon and the mortise along with the friction force at the contact surface, enhancing the rigidity of the joint. The drop in the peaks of several subsequent cycles was due to the sliding of the tenon. The hysteresis curves under continued loading show that when the lateral displacement is sufficiently large, the difference in deflection has little effect on the seismic performance of the frame. The hysteresis curves of Frames Q1–Q4 with contraction damage are flat and the bearing capacities are significantly lower than those of W1. The larger the extent of contraction damage, the more the substantial this reduction is. After the column frame is stressed, the tenon rotation relative to the column is close to 0.06 rad, the bending stiffness slightly increases, and the maximum bending capacity is about 0.75 kN·m. The reason for this change is that the size of the tenon is decreased as a result of contraction damage, meaning that the tenon surfaces make almost no contact with the surfaces of the mortise, so there is only a small friction force between the upper and lower surfaces and the bottom of the mortise. When the axial force in the strut is greater than the friction between the mortise-tenon, the tenon will slide, and when the rotation angle reaches at 0.06 rad, the tenon slides into the deeper part of the mortise where the inside of the mortise and the front face of the tenon finally come into contact. During the rotation of the joint, the frictional force generated by this contact increases the resistance moment of the joint.
The hysteresis curve of the mixed damage frame combines the characteristics of frames with pulling and contraction damage. The deflection of Frame LQ1 is the same as that of Frames L1 and Q1. The stiffness at the initial stage of loading is large, and then sliding of the tenon occurs, causing the bearing capacity to decrease. At this stage, the shape of both Frames LO1 and L1 changes similarly, but the stiffness of LO1 is significantly smaller and is affected by the contraction-reduced size of the tenon, as the performance of a joint depends on the compression forces within the joint and the associated friction to limit its rotation and sliding. The entire hysteresis of the LO1 curve is located within those of L1 and Q1. The behavior of Frame LQ2 is consistent with that of Frames L5 and Q5 at similar deflections. In this case, although in the proportion of pulling to contraction damage, the pulling damage is equal, the load capacity of the mixed damage joint is clearly smaller. The development of the hysteresis curve of Frame LO2 is also within that of L5 and Q5 but closer to Q5. The joint in Frames LQ3 and LQ4 both have the same degree of pulling damage of 3/10, similar to Frames L3 and O3, and their hysteresis curves are accordingly between those of L3 and Q3. The comparison of the performance of Frames LQ1 and LQ3 and Frames LQ2 and LQ4 indicates that although the total applied deflection is different, the hysteresis characteristics are similar when the combined contribution of pulling and contraction damage is the same.
Figure
Bending moment-angle (M-
The bending moment in the joint can be observed to increase with the increase in the rotation angle during the loading of the frame W1 and does not decrease until the loading is complete, exhibiting excellent capacity. From the change in the slope of the skeleton curve, it can be inferred that the joint exhibits different stiffnesses during rotation: when the rotation angle is small, the squeezing between the mortise and tenon is tight and the bending stiffness is large, but as the rotation angle increases, the sliding of the tenon increases as well, some plastic deformation occurs, and the stiffness of the joint decreases. The stability of the load capacity in the later stages indicates that an undamaged mortise and tenon joint generally provide good ductility.
The skeleton curves of frames L1–L5 with pulling damaged joints are initially “S” shaped. The linear change from the start of loading to the first turning point is due to the fact that the increasing internal compression forces cause the friction force to increase, delaying the sliding of the tenon head, keeping the joint in the elastic deformation stage. The “S” shaped top descending segment is caused by the sliding of the tenon when the friction force between the mortise and tenon is smaller than the axial force in the strut. After this segment, the behavior of these frames is close to that of the intact column frame, and the difference in the degree of tenon pullout is very small. Compared to the frames with intact and pulling damaged joints, the capacities of the frames with contraction damaged joints were significantly smaller. The bending moments of Frames Q1 and Q2 at the joints increase after a rotation of 0.06, exhibiting a maximum bending moment only 40% that of the intact column frame. The variation in capacity of frames Q3–Q5 is very small, though generally the capacity of the frame gradually decreases with the increase in applied deflection.
In Figure
The relationship between the equivalent viscous damping coefficient and the rotation angle of the frames with intact and pulling damaged joints is shown in Figure
Equivalent viscous damping coefficient of frames with pulling damaged joints.
Note that the area within the hysteresis curve of all frames with contraction damaged joints is much smaller than within those of frames with intact and pulling damaged joints. When the degree of joint pulling allowed by the contraction of the tenon material exceeds 1/5, the joints basically lose their energy dissipation ability. Frames LQ1 and LQ3, which have a small amount of pulling damage relative to contraction damage, exhibit an energy dissipation capacity between the pulling and contraction damaged joints. Frames LQ2 and LQ4 are also similar. This is due to the fact that cutting damage to the tenon weakens the contact friction between the mortise and tenon, reducing the friction energy consumption, which is an important component of the energy consumption of the joint.
The relationships between the bending moments in the joints and deflections applied to the frames are shown in Figure
Relationship between bending moment and degree of slippage of the (a) pulling damaged frame, (b) contraction damaged frame, and (c) mixed damage frame.
There was no significant yield point after the frames were sufficiently stressed. The increase in bending moment gradually decreased after the frames with intact and pulling damaged joints were deformed about 0.05 rad until reaching 0.12 rad, during which time the capacity did not decrease and the column frame ductility was better. As the bending moments applied to frames with contraction damaged and mixed damaged joints began to increase, the deformation of the frames was comparatively larger than that of the frames with intact and pulling damaged joints. From the overall view of the frame, a joint rotation angle of 0.12 rad (0.2 m) at the end of loading was greater than 1/30 (0.033 rad), the limiting rotation angle of a timber member provided in the specification, which is also greater than the overall “damage point” of timber components when evaluated with a limit of 0.016 m for a building structure with a seismic tilt limit of 0.014 m and 0.035 m (for a fortification intensity of 8 and 9 degrees, respectively).
Combining the relevant geometry and physical parameters of the frame, the loading amplitude was set to 100 mm per cycle. For the intact column frame, the upward loading mode is taken as an example. The different contact sites and bending moment contribution trends with respect to rotation angle are shown in Figures
Bending moment distribution between joint areas. (a) Rotated joint and (b) moment contribution of each area.
In the upward loading stage, the moment due to contact and the moment due to friction at the contact point rapidly increase, causing the bending moment to increase rapidly with the increase in the rotation angle. As the rotation angle continues to increase, the tenon begins to slip in the mortise, and the amount of tenon pulling is increased. With this increase, the moment continues to increase but at a significantly reduced rate. The bending moment then drops rapidly when upward unloaded, and the compression and friction forces change little as the frame continues to unload. The main direction of load change is in the direction of the friction force, indicating that the lateral friction bending moment accounts for the bending moment in the entire joint. The additional bending moment created by the vertical load applied to the top of the column must be balanced by horizontal tension during the unloading of the column. During the unloading process, the deformation of the compression zone gradually decreases and slip occurs between the mortise and tenon.
Table
Percentage of bending moment contribution from each area.
Angle ( |
|
|
|
Side friction (%) |
---|---|---|---|---|
0.0036 | 4.53E-05 | 8.00 | 0 | 91.99 |
0.0144 | 0.00311 | 35.10 | 0 | 64.89 |
0.0184 | 0.00627 | 42.88 | 0 | 57.10 |
0.0243 | 0.01161 | 43.41 | 0.10 | 56.46 |
0.0332 | 0.02297 | 42.38 | 0.40 | 57.19 |
0.0467 | 0.04900 | 39.96 | 0.20 | 59.79 |
Through an analysis of the forces in a timber mortise-tenon joint, a theoretical equation for calculating the resisting moment of the joint was established that can account for joint damage. The theoretical calculations and numerical simulations indicate that when the degree of pulling damage to a joint is within 2/5 of the length of the joint, the same applied deflection results in only a small difference in frame behavior compared to that of a frame with intact joints, while a similar degree of contraction damage causes the stiffness of the frame and joint to decrease significantly compared to that of a frame with intact joints, and the load-carrying capacity and resistance of a frame with mixed damage joints are between those of a frame with intact joints and a frame with contraction damaged joints. The evaluated dovetail mortise and tenon joints exhibit a good semirigid character. The seismic performance of a frame with pulling damaged joints is not significantly different from that of a frame with intact joints; the seismic performance of a frame with contraction damaged joints is significantly weakened, with the joint losing half of its rigidity, while the seismic performance of a frame with mixed damaged joints is controlled by the proportion of pullout damage to contraction damage. The higher the proportion of contraction damage, the weaker the seismic performance. The friction force between the tenon and mortise is the main source of bending moment resistance and energy consumption. The moment generated by the lateral friction contributes more than 50% of the overall bending moment resistance of the joint. This paper sets the wood properties and the finite element model according to the ideal state. Some simplified calculation methods are used in the calculation of bending moments, and the accuracy of the results can still be optimized. This paper carried out numerical simulations without experiments and only studied a single dovetail joint. Using this theory in the overall structural simulation analysis will provide greater help in assessing structural damage.
Contact areas of the top, bottom, and front surfaces of the tenon head
Compression forces on the top, front, and bottom faces of the tenon
Friction force on the top, front, and bottom compression areas of the tenon
Maximum deformation in each compression zone
Deformation corresponding to the ultimate compressive strengths of the crushed zones
Length of the elastic zones in
Elastic and plastic zones in
Distances between the top, front, and bottom compression forces of the tenon head and the center of the neck cross section under upward loading and unloading
Distances between the top, front, and bottom compression forces of the tenon head and the center of the neck cross section under downward loading and unloading
Stress transformation coefficients of each zone
Length, height, width of the tenon
Rotation angle of the joint
Extracted displacement of the tenon from the mortise
Extracted displacements at the upper and lower faces of the tenon
Initial displacement
Extrusion compression conversion coefficient
Extrusion deformation at the corresponding position along the height of the tenon
Stress on the sides of the tenon
Compression stress of the compression zone
Compressive strength of the timber strip
Elastic modulus of the timber
Length of the tenon head
Unit friction moment
Friction moment
Yield strain under compression
Ultimate compressive strain
Ultimate tensile strain
Compressive strength
Tensile strength.
The program data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
The authors would like to acknowledge the financial support from the National Key Research and Development Program of China (project no. 2016YFC0701500).