2 DIEH-DOU STRUCTURE 2.1 Seismic vulnerability
Taiwan is an island located south of mainland China and between the Eurasian and Philippine Sea plates, an highly seismic area. Investigation carried out within the present project identified at least 52 historic architectural compounds built as Dieh-Dou structures over the Chi-Chi earthquake, 7.3 Richter Scale magnitude, stricken region. While for 29 of them was not possible to gather data about their condition after the event, for the remaining 23 data were collected on level and mode of damage as shown in table 1. Level 1 means no structural damage. Buildings with elements pulling out from joints (Figure 3) and damage of the roof decorated ridge are categorised as damage level 2.
Figure 2. Dou-Gon joint.
As the bearing capacity of the structure as a whole is not impaired, damage level 3 includes eaves rupture (Figure 4), masonry wall cracks and columns sliding from the original position: these are not considered an immediate threat to collapse but reduce the structures bearing capacity. Finally, Dieh-Dou frames with in plane or out of plane leaning or collapse are classified as damage level 4 or 5, respectively.
In Dieh-Dou buildings the purlins linking two main frames out of plane sit on the top of the Dou, connected with a simple scarfed joint. Under lateral excitation, the purlins easily slide out of their seating once the vertical loading is overcome. The field survey showed that under earthquake, in several cases, once the purlins failed the two main frames became virtually independent and started to take load separately. Hence in the following numerical analysis, conservatively, a single Dieh-Dou frame is considered neglecting the effect of the 3D assembly.
For a single frame, the damages from table 1 can be classified into two types, joint failure and material failure. Joint failures include element pull out and loss ofrotational stiffness or columns sliding from the original position. Material failures, such as timber element rupture, are caused by excessive stresses in the timber pieces.
3 FINITE ELEMENT MODEL EVALUATION 3.1 Description of FEmodel
Although precedent work on Dieh-Dou structures is relatively limited, research on other historic timber structures have proven that numerical analysis can be successfully applied to obtain a fair description of Far East Asian ancient timber structures seismic behaviour. Also, the complicated geometry configuration of Dieh-Dou structures is not well understood and finite element analysis methods can enable to identify the whole structural behaviour. To approach the final goal of this project, the finite element commercial package Algor © was chosen to analyse the structure.
Three different building typologies were defined. Results are discussed here for typology C, characterised and represented by the main hall of Guan-Shi family Temple, these buildings started in 1849 (Hsu, 2002). The difference between this frame and other main hall of temple compounds is the supporting element between column and main beam (see part A in Figure 1). The lack of part A may render the main hall of Guan-Shi weaker than other frames.
The frame was created by beam elements in the FE environment. Ever timber piece of the Dou-Gon joint
set (see Figure 2) was considered. Figure 5 shows one layer of the joint in perspective view and the corresponding FE modelling (in plane). The centre of joint is represented as Node 1 and all the parts of the joint set were modelled as beam element located at the neutral axis of each position. Part 1 is the out of plane bracket which is continuous through the joint; two rigid elements, B1 and B2, were created to connect it with part 2 and 3, which are the 'Dou' and are simulated by parts B3 and B4. Part 4 and 5 were defined as beam elements B5 and B6, respectively, and linked with the joint centre Node 1 with the appropriate stiffness by using boundary elastic elements E1 and E2. Here, the rotational stiffness was taken from test results (Tsai & D'Ayala & Lewis 2006). B5 and B6 represent the ends of the in plane beams connecting to the out of plane brackets by means of dove tail joints.
Apart from the complicated Dou-Gon joint set, there are other more common beam to column joints in the Dieh-Dou buildings. The stiffness of these joints of these mortise tenon type was assumed in relation to the connection geometry as proposed by Chang (2006) in which equation (1) is for beams continuous through column and equation (2) is for flat cut beam face to face inside the column.
Cw: column width Bw: beam width Bd: beam depth
E^: modulus of elasticity of timber perpendicular to grain.
Two type of analysis were performed with this FE model, elastic and nonlinear step by step. In the elastic analysis a total lateral load equal to 0.33 mg, Fl, is applied in correspondence of each purlin. The advantages of linear elastic analysis is the possibility of getting a rapid result and make a quick judgement of the capacity of the structure under a load magnitude of the order of the maximum which can be applied to the structure by the strongest earthquakes. However, elements are likely to fail during the earthquake, induced loads causing loss of structural stiffness and leading the frame to a diverse distribution of load. A nonlinear step by step analysis therefore was performed, where the model was modified at each load step according to the criteria explained below. Figure 6 shows the finite element model of the frame.
3.2 Failure criteria
The two categories of failure leading to the failure modes previously described, namely joint failure and material failure, are detailed below.
Three types of joint failure modes can be further grouped as pull out and bending failure (applying to horizontal members) and column failure.
A series of pull out tests were first performed in laboratory with full scale samples of one layer of Dou-Gon joint. The results indicated that the maximum pull out forces at failure were similar and did not depend on the magnitude of vertical load applied (with a value varying between 8000 N and 9000 N) if the dove tail connection was good, but reducing to 3000 N and 1500 N when applied loads were 6500 N and 3250 N respectively, in the case of poor fit of the
dove tail joint (then relying on the timber to timber friction, as expected).
Assuming proper dove tail connections which are normally guaranteed by accurate craftsmanship, a value of 8000 N was set to define the failure due to element pull out in the FE model. Once the forces are over this value in the model, the element capacity of transmitting axial force is considered lost and the corresponding degree of freedom is released on the beam element in the FE model.
Differently from pull out tests, the rotational tests showed that the results, in terms of stiffness, depend on the magnitude of the vertical load. However, although the rotational stiffness were different, the maximum moment resisted by the joint was found to be always around 1000Nm (Tsai & D'Ayala & Lewis 2006), after which the capacity dropped. For the beam-column joints an elastic limit value of 1500 Nm was taken from Chang (2006). If the bending moment in the model was found to exceed that of the criteria mentioned above at some point in the analysis, then the joint will be considered failed in bending, which implies its capacity to transmit moment to the neighbour elements is lost. In this case, the beam element is released in rotation. Finally, the foot of the column has a shallow pin into a stone base and relies on friction to resist lateral action.
The coefficient of static friction were calculated from laboratory tests of the timber pieces, obtaining values of 0.53 and 0.58, with an average of 0.55, inline with codes indications (which range from 0.5 to 0.6 for friction of timber to timber in dry specimens). Dividing the resultant base shear applied at the bottom of the column by the vertical load applied, if the value is over 0.55, then the columns are assumed to be sliding from their original position and so no more horizontal force could be transmitted to the ground anymore. Due to the particular connection with the ground described above, it is further assumed that the columns are able to rotate and do not transmit moment.
For material failure, ombined bending and tension, combined bending and compression, and shear stress, are examined. The material capacities are taken from the material tests (Tsai & D'Ayala & Lewis 2006) and from the Wood Handbook (1999). The failure criterion for bending stress was set at 4.6 E + 7 (N/m2), for shear at 5.5 E + 6 (N/m2) and for tension and compression at 4.5 E + 7 (N/m2) and 2 E + 7 (N/m2) respectively.
For the elements subject to axial compression and bending, the interaction formula of British Standard (BS 5268-2; 2002), shown as equation (3), is used as failure criterion:
where am,a, j is the applied bending stress; am,adm, || is the permissible bending stress; ac,a, j is the applied compression stress; ac,adm, || is the permissible compression stress (including buckling); ae is the Euler critical stress.
Equation (3) is for permissible stress design. However, because the present defined analysis ofDieh-Dou building focuses on failure, the values above are inserted instead of permissible values. The value K12 can be found in Table 22 of the British Standard (BS 5268-2; 2002).
If the resulting stresses at one step are over the values set above, then the corresponding element is regarded as failure and completely deactivated in the corresponding part of the FE model.
and the failure criterion (Miyamoto al, 2004), and is matched by the observation of the real situation after the earthquake (Tsai & D'Ayala & Lewis 2006). For the step by step analysis, a total of 30 steps were performed until the collapse criterion was reached. The lateral load at the thirtieth step was 80% of Fl. In both cases (elastic and step by step analysis), the columns did not reach the failure criterion. At the seventeenth step (50% of FL), the roof apex has already met the damage criterion. The structure drifts results are shown in Figure 8.
When compared with the results of the elastic analysis where the full lateral load was applied, the step by step analysis shows a considerable decrease of overall stiffness. This is because during these 30 steps several joints lost their rotational stiffness and several beams pulled out and lost bending transmitting capacity, rendering the structure more deformable. The result of elastic analysis compared with the evidence after the earthquake (Tsai & D'Ayala & Lewis 2006), confirmed that finite element model can catch the structural behaviour under horizontal forces.
However, the physical tests showed that the maximum pull out load in Dieh-Dou joint is 8000 to 9000 N, and in the linear analysis results of FE model, there are several elements already over 8000 N, implying that several joints could have suffered element pull out leading the frame to a diverse distribution of load which may also cause additional decrease of capacity. The step by step analysis proved to be more conservative than the elastic analysis. Furthermore, in the FE model analysis, timber material were assumed in good condition, which may not always be the case.
In the elastic analysis the maximum load FL is applied all at once, while the step by step analysis consists of a series of models in which 10% of the FL is initially applied and further increases of 10% of FL are added if the results of the previous step shows that no damage occurred. Once elements are found to reach the failure criteria they are treated accordingly and the same load level reapplied until no further element failure is attained before increasing the load again. The step by step analysis stops when the roof apex lateral displacement are over the collapse criteria proposed by Miyamoto et al 2004 for Asian timber structures. The procedure of analysis implemented in the program is shown in the flow chart of Figure 7. The results of both analyses are discussed in the following section.
3.3.1 Structure drift
In the static elastic analysis, the displacement of the roof apex is 55.2 mm, which lies between the damage
In order to find out the critical elements that may govern the structural behaviour, the maximum horizontal displacements at each load level are reported in Figure 9 for the step by step analysis.
It can be noted that until the horizontal load is below 20% of FL, the structure is stiffer; after this level the displacement rate increases and is maintained constant until failure. This implies that the elements and joints whose capacity is lost before 20% of Fl are critical to the stiffness of the Dieh-Dou frame.
In figure 10 the elements that failed up to 20% of Fl are marked. It is seen that they represent 8 main beams and corridor beams ends, together with 2 binding element ends, and they failed mostly for loss of rotational stiffness. Three main beams span the main ceremonial space of the Dieh-Dou temples (see Figure 1). The results show that the second and third main beams, failed at both ends, can be the first critical elements under earthquake.
Figure 10. Element failed up to 20% of FL .
Figure 8. Structure drift of static and step by step analysis.
Figure 8. Structure drift of static and step by step analysis.
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