Dynamic characteristics and response to earthquakes in the single-eaves mountain type ancient building model
Dynamic characteristics and response to earthquakes in the single-eaves mountain type ancient building model
Zhou Qian1, Ji Jinbao2
(1.The Palace Museum, Beijing, 100009; 2.Beijing Key Laboratory of Engineering Seismic Resistance and Structural Diagnosis and Treatment, Beijing University of Technology, Beijing 100124, China)
Abstract: In order to better protect the ancient buildings, the dynamic characteristics and response of 8 degree earthquakes were further studied by using the finite element analysis method based on the existing experimental results. The spring element was used to simulate the mortise and tenon joints and the structure of the bucket arch, and the finite element model of the ancient building model was established by considering that the column base was a flat swing floating rest. Through modal analysis, the fundamental frequency and principal mode shape of the model are obtained, and the seismic performance of the model is evaluated by the horizontal bidirectional seismic wave (x:y=0.85:1) of the input PGA=0.1g(x-direction) into the model, and the displacement and acceleration response curves of typical nodes and the internal force response curves of typical elements are obtained. The results show that the fundamental frequency of the single-eaves mountain model is 1.34Hz, and the main mode shape is mainly translational, and the model can maintain a stable vibration state under the action of 8 degree earthquakes, and the internal force and deformation of the structure are within the allowable range. In addition, the results of finite element analysis are in good agreement with the test results.
Keywords: ancient architecture, timber structure, dynamic characteristics, seismic response, single eaves gable roof
CLC:TU366.2 Document Identification Code:A
The ancient buildings in the mainland are mainly wooden structures, which have important historical and cultural value and are of great significance for protection. In recent years, earthquakes have occurred frequently, posing a certain threat to the safety of ancient buildings [1-3]. Correspondingly, some scholars have carried out performance evaluation of ancient buildings and put forward scientific protection suggestions [4-7]. However, for the single-eaves gabled roof type ancient buildings, the relevant results are few. The author has made a 1:2 scale model of a single-eaves gable roof in the Forbidden City (Fig. 1) with Korean pine as the main material, and carried out research focusing on shaking table tests, and obtained the test results of such ancient buildings under different seismic intensities [8-9]. In this study, the modal analysis and time history analysis of the ancient building model were carried out by finite element analysis, and the dynamic characteristics and time response characteristics of the model under the action of 8 degree earthquakes were further discussed, so as to enrich the experimental results and better protect the ancient buildings in mainland China.
Fig.1. Model dimensions (unit: mm)
1 Finite element model
The establishment of the finite element model in this study mainly considers the assumptions of the following aspects:
(1) The finite element analysis program ANSYS was used to establish the finite element model of the ancient construction model of the single-eaves rest mountain.
(2) Column base: The constraint condition is considered to be semi-rigid connection. The main reason is that the bottom of the column floats flat on the capstone, and under the action of medium and low intensity earthquakes, the seismic force exceeds the static friction between the top of the column and the capstone, and the bottom of the column can produce relative slip and rotation around the capstone. The COMBIN40 element is used to simulate the semi-rigid connection characteristics of the column top-capstone, and the main input parameters include the sliding force F at the bottom of the column and the initial lateral stiffness k of the column. Considering that the coefficient of static friction between the bottom of the column and the capstone is 0.5 [10], after calculating the load on the upper part of the single column, F=2568N can be obtained. In addition, referring to the research results of Ref. [11], the value k=4.5×105N/m.
(3) Mortise and tenon joints and bucket arches: nonlinear spring units are used to COMBIN39 simulate mortise and tenon joints and bucket arches. In this analysis, the seismic action of medium and low intensity is mainly considered, so the relative rotation of the tenon and the joint in the elastic motion stage and the relative slip motion of the bucket arch member in the elastic motion state are mainly considered, and correspondingly, the COMBIN39 element only considers the elastic stiffness value. Referring to the test results of Ref. [12], the rotational stiffness of the mortise and tenon joint is Krotx= Kroty= Krotz=5.755kN∙m/rad, where Krotx, Kroty, and Krotz3 represent the rotational stiffness of the joints around the x, y, and z axes, respectively. Referring to the results of [13]-[14], the translational stiffness of the bucket arch Kx=Ky=0.3×106N/m and Kz=3.21×106N/m are taken in this study, wherein Kx and Ky represent the horizontal bidirectional stiffness of the bucket arch respectively, and Kz represents the vertical stiffness of the bucket arch.
(4) Roof: The SHELL181 unit is used to simulate the gray back and tile of the roof panel and the upper part. According to the roof layering method and material composition, the main input parameters of the SHELL181 unit are as follows: density 1.8×103kg/m3, equivalent thickness 0.134m. In addition, for the masses of normal ridges, vertical ridges and ridges, they were simplified into uniformly distributed particle units, which were attached to the position of each ridge, and the MASS21 unit was used to simulate.
(5) Beams and columns: linear beam elements BEAM189 elements are simulated, and the main input parameters include elastic modulus E=9x109N/m2, Poisson's ratio γ=0.3, and density ρ=500kg/m3.
Based on the above assumptions, the finite element model for this analysis is established, as shown in Figure 1. It should be noted that this model is displayed in the ANSYS program after turning on the shape switch (/esha,1) in solid mode. In the subsequent analysis, in order to reduce the computing memory occupied by the model, the display mode of turning off the shape switch (/esha, 0) was adopted, and the model was displayed in line and surface mode.
Fig.2. Finite element model
2 Power characteristics
The modal analysis of the finite element model of the single-eaves mountain is carried out to obtain the first 10th order natural frequency of the model, as shown in Table 1, and the first 1st and 2nd order mode shapes of the model are obtained in Figure 3. It is easy to know: (1) The principal mode shapes of the model are concentrated in the first and second orders. The first-order mode shape is dominated by the longitudinal (length direction) vibration of the model, and the second-order mode shape is the longitudinal (width) vibration of the model, and the vibration correlation between the two directions is very small. (2) Due to the structural characteristics of the single-eaves gable, its mass is distributed differently in the longitudinal and horizontal directions, resulting in a slight plane shaking during vibration. (3) Because the bottom of the column floats flat on the capstone, there is a relatively obvious relative slip movement between the bottom of the column and the capstone when the model vibrates, but this movement form is only seen in the middle column, but for the side columns (the columns on both sides of the model), due to the restraining effect of the embedded wall, the vibration form is manifested as the rotation of the bottom of the column around the capstone. (4) The bottom of the wall is fixed to the ground, and its vibration form is manifested as longitudinal and horizontal swing around the ground. Because the seismic performance of the wall is worse than that of the timber frame [15], it is destroyed before the timber frame through this mode of movement under the action of earthquake.
In addition, the fundamental frequency of the model obtained in this study is 1.34Hz, which deviates less than 10% from the field measurement results [8], which can approximate the validity of the finite element model.
Table 1 Natural frequency of the model
(a) The first mode shape façade
(b) Side façade of the second mode shape
Fig.3. Diagram of the principal mode shape of the structure
3 Earthquake response
3.1 Structural damping determination
In the time history analysis, Rayleigh damping is used as the structural damping of the finite element model in this study, and the calculation formula is shown in Eq. (1):
In Eq. (1), α and β are proportional constants, which are used as the damping parameters of the program input and can be determined according to Eq. (2)-(3), and [C], [M], and [K] are the damping matrix, mass matrix, and stiffness matrix of the model, respectively.
In Eq. (2)-(3), ξi and ξj are the damping ratios of the i and j mode shapes, respectively, and ωi and ωj are the circular frequencies of the i and j mode shapes, respectively. Transform Eq. (2)-(3) to obtain the solution formula for α and β, see Eq. (4)-(5):
again
Generally, i=1, j=2, ξi=ξj=0.05 is taken, and the first two frequencies of the model, f1=1.34Hz and f2=1.64Hz, are substituted into equation (6), and the formula (4)-(5) is paralleled to obtain α=0.152 and β=0.016.
3.2 Selection of seismic waves
In this paper, the response characteristics of single-eaves gables under the action of moderate and low intensity earthquakes are studied. The same seismic wave action model as in Ref. [8] is adopted: the El-Centro wave of 1940 is suitable for the Forbidden City site type (ІІ type), the x and y directions, where the acceleration peak in the x direction is PGA=0.1g, the horizontal bidirectional acceleration peak ratio is x:y=0.85:1, the time interval is 0.01s, and the continuous action time is 15s. Beijing is an 8-degree seismic fortification area, and the peak value of seismic wave acceleration is slightly larger than that in Table 5.1.2-2 of the 2016 edition of the Code for Seismic Design of Building Structures (GB50011-2010), the peak of the acceleration time history of 8-degree earthquakes, so the calculation results are conservative. The seismic wave waveforms used in this analysis are shown in Figure 4.
(a) X-direction
(b) Y direction
Fig.4 El-centro waveform
3.3 Seismic response curves
In order to fully understand the internal force and displacement response of the structure under the action of earthquake, based on the results of shaking table test, representative nodes or elements were selected for analysis: (1) the bottom of the golden pillar (node No. 125) and the middle of the roof ridge of the bright house (No. 730) to study the displacement and acceleration response characteristics of the structure, (2) the middle span of the peach tip beam (element number 157) to study the bending moment response characteristics of the structure, and (3) the end of the peach tip beam (element number 153) to study the shear response characteristics of the structure. The locations of the above nodes and elements are shown in Figure 2.
The displacement response curves of nodes 125 and 730 are shown in Figure 5, and the main features are as follows:
(1) Due to the low seismic wave intensity and the restraint effect of the embedded walls on both sides of the model, the friction-slip motion between the floating column base and the column capstone is not obvious, and the motion form is mainly rotational, which is more consistent with the experimental phenomenon in Ref. [8]. The peak displacement of node 125 in the X direction is 0.92 mm (t=3.06s) and 0.91 mm (t=1.76s) in the y direction, which is similar to the experimental results in Ref. [8]. Correspondingly, the peak displacement of node 730 at the top of the model is 7.08 mm (t=3.06s) in the x-direction and 10.55 mm (t=1.76s) in the y-direction, which can reflect that the displacement of the roof is significantly amplified relative to the bottom of the column.
(2) The displacement response curve of the model shows a uniform vibration centered on the equilibrium position and approximately reciprocating, which reflects the stable motion state of the structure under the action of earthquake, and is consistent with the experimental phenomenon in Ref. [8].
(a) X-direction
(b) Y direction
Fig.5 Displacement response curves of nodes 125 and 730
The acceleration response curves of nodes 125 and 730 are shown in Figure 6, and the main features are as follows:
(1) From the perspective of the peak acceleration response, the peaks of nodes 125 and 730 are 0.091g (t=1.24s) and 0.046g (t=1.07s) in the X direction, respectively, and the peaks of nodes 125 and 730 are 0.101g (t=0.59s) and 0.062g (t=1.67s), respectively, in the y direction.
(2) From the perspective of the variation characteristics of the acceleration peak, for the same node, the acceleration peak in the y direction is greater than that in the x direction, and the main reason is that the input seismic wave peak in the y direction is larger. In addition, the acceleration peak of node 125 is slightly smaller than the acceleration peak of the input seismic wave, which can reflect that under the action of medium and low intensity seismic waves, the bottom of the column floated on a flat swing can produce a certain friction and shock absorption. The peak value of the acceleration response of node 730 is less than that of node 125, which can reflect the energy dissipation effect of mortise and tenon joints, bucket arches and other structures.
(a) X-direction
(b) Y direction
Fig.6 Acceleration response curves of nodes 125 and 730
In addition, because node 730 is located at the ridge of the roof, its acceleration peak is small, which can reflect that the seismic force on the roof tile is small. Combined with the test phenomenon, the roof tiles did not loosen or fall under the action of seismic waves with PGA=0.1g intensity (Fig. 7).
Fig.7 Screenshot of the shaker test
In order to explore the characteristics of the internal force distribution of the model under the action of earthquakes at different times, t=1.67s was taken as an example for analysis. Based on the ANSYS program, the principal stress distribution of the model at this moment was obtained, as shown in Figure 8. It is easy to know that the peak value of the large tensile stress (0.325MPa) and the peak value of the main compressive stress (-0.325MPa) of the model are located at the tenon and tenon joint joint position where the side column and the forehead intersect, and the stress values along the column direction are larger around the periphery. The tenon and joint section here are weakened in size, and the relative motion of the node under the action of seismic force is obvious, and this position is the intersection where the vertical load of the roof is transmitted downward, coupled with the extrusion of the wall, so it is more likely to be tensile or compressive failure than other positions. Although the above-mentioned stress peak value is lower than the allowable stress tolerance value of Korean pine (tensile stress 8 MPa and compressive stress 10 MPa) stipulated in the Design Standard for Timber Structures (GB50005-2017) [16], the position of the stress peak is the location where the model is prone to failure, and it is easy to cause failure under the action of high-intensity earthquakes.
(a) Principal tensile stress
(b) Principal compressive stress
Fig.8. Principal stress diagram of the model (t=1.67s, unit: MPa)
In Ref. [9], shaking table tests were carried out on this model under extreme intensity earthquakes, and post-earthquake investigations were carried out. The survey photographs show the typical seismic damage locations and characteristics of the model (Fig. 9): at the intersection of the columns and the forehead on both sides of the model, a long coarse crack (indicated by a dashed straight line) and an extension along the direction of the column body, and a staggered gap between the tenon and the joint (see the dotted circle). The above shows that the above positions have been significantly damaged, and there are similarities with the characteristics of the principal stress diagram of the model shown in Figure 1.
(a) Left side
(b) Right side
Fig.9 Seismic damage photos of the model in Ref. [9].
The bending moment response curves of unit 157 in the x and z directions (considering the self-weight of the structure) are shown in Fig. 10, the peak bending moments are 91.8 N·m (x-direction) and 25.9 N·m (z-direction), and the maximum bending stress values of the corresponding cross-section (240×325 mm) are 0.03 MPa (x direction) and 0.006 MPa (z direction), respectively. The above peak value is much lower than the allowable bending stress tolerance of Korean pine (13 MPa) specified in the Design Standard for Timber Structures (GB50005-2017) [16]. Therefore, the peach tip beam has an ample reserve of flexural strength. On the other hand, the results [1,9] show that the cross-sectional dimensions of the beams of ancient buildings in mainland China are generally sufficient, so there are few bending failures.
Fig.10. Bending moment response curve of unit No. 157
The shear response curve of element 153 is shown in Figure 11. The square and square root method were used to synthesize the above curves, and the peak shear force of the unit was F=324N, and the maximum shear stress fs=0.017MPa of the corresponding cross-section (the cross-sectional size of the peach tip beam is 240×325 mm, considering that the upper part is made into a peach tip shape, the end is carved out, and the actual cross-sectional size is 120×160 mm), which is lower than the allowable value of shear stress (1.4 MPa) specified in the Design Standards for Timber Structures (GB50005-2017) [16]. That is, there will be no shear failure at this location.
Fig.11 Shear response curve of element 153
4 Conclusion
Based on the finite element analysis method, the dynamic characteristics and the response of 8-degree earthquakes were analyzed on the single-eaves mountain test ancient building model that had been tested on the shaking table, and the following conclusions were drawn:
(1) The calculated fundamental frequency of the model is 1.34Hz, the main mode shape is mainly translational, and the roof has a slight horizontal swing.
(2) Under the action of 8 degree earthquakes, the relative friction slip between the base of the model column and the capstone is not obvious, and the superstructure maintains a relatively stable vibration state.
(3) Under the action of 8 degree earthquakes, the peak values of the internal forces of the typical units of the model are within the allowable range.
(4) The numerical simulation results in this study are in good agreement with the shaking table test results.
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(Journal of Hydraulic and Architectural Engineering, Issue 5, 2023)