Application of finite element analysis in the formal analysis of ancient architectural plaque section

Wen Su; Yixiong Hua

doi:10.1515/nleng-2024-0073

Article Open Access

Application of finite element analysis in the formal analysis of ancient architectural plaque section

Wen Su and Yixiong Hua

Published/Copyright: April 4, 2025

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From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

Ancient architecture in Tibet has attracted considerable research attention because of its historical, religious, and artistic significance. Because the design and construction of ancient buildings often do not have the same detailed documentation as modern buildings, it is difficult to accurately simulate their structural properties. To study the structural performance of ancient buildings, the Tibetan ancient architecture structural performance analysis method based on finite element analysis is proposed. To better reflect the actual situation of engineering, the genetic algorithm is introduced to fine-tune the finite element calculation. The results of the experiments showed that the column compressive strain was maximum at 700 $με$ . Moreover, the plastic strain occurred only when the compressive strain reached more than 1,900, so the column was in the elastic strain stage during the process. When subjected to a vertical load, the upper part of the beam was compressed and the lower part was stretched. Furthermore, the lower part of the bow wood was stressed, and the upper part was stretched. The proposed method effectively improves the accuracy of the structural performance analysis of ancient buildings, and the results can be used for reference in ancient buildings.

Keywords: finite element analysis; ancient Tibetan architecture; genetic algorithm; nodal force transfer mechanism

1 Introduction

Ancient Tibetan architecture (ATA) has received significant attention owing to its historical, religious, and artistic value. The internal structures of many of these buildings have deteriorated significantly due to their construction centuries ago [1]. It is essential to objectively assess the structural integrity of historic edifices to implement appropriate protective measures. Furthermore, comprehending nodal performance is crucial to comprehending the mechanics of structures. Finite element method (FEM) is a numerical analysis technique that is effective for assessing the force distribution and deformation computation of intricate structures. In comparing the homotopy perturbation method and the Laplace transform decomposition method as numerical methods for solving the epidemic SIR model through Morgan-Voysey series, the FEM is distinguished by its capacity to address complex geometric shapes and boundary conditions, as well as its ability to deliver high-precision solutions. The FEM breaks down complex problems into smaller, more manageable pieces, allowing detailed analysis of each small piece. According to Hu et al., finite element analysis (FEA) was carried out on the bracket set (upper arch of wooden architecture) form of ancient constructions to transform the building’s geometrical model into a finite element mesh [2,3]. The mesh was then input with material properties and constraints to enable force analysis and deformation calculation. Nevertheless, bracket set constructions in ancient buildings typically possessed distinctive characteristics and complexities. When employing FEM for analysis, it was crucial to accurately model the material properties, structural characteristics, and actual loading of the building. According to Maynard et al., this ensured the reliability and validity of the results obtained [4]. Therefore, it was of utmost importance to ensure a precise and detailed representation of the building in the analysis process. FEM presented challenges in determining with accuracy the intrinsic model and related parameters, resulting in calculated outcomes that frequently diverge largely from practical realities [5]. The genetic algorithm (GA) is a stochastic optimization algorithm that manipulates code and conducts parallel searches across multiple initial points. Its evolutionary search, which employs probabilistic heuristics, is suitable for explicit or implicit functions expressed through mathematical equations, as it does not mandate microscopist and continuity of the optimization function. Compared to other optimization algorithms, GA has demonstrated superior robustness. The aim of this study is to introduce GA to mitigate the drawbacks of FEM and bring finite element computation closer to engineering reality. This will ultimately enhance the effectiveness of structural analysis and optimization.

The four major parts of this study are as follows: Current research findings are reviewed in Section 2, and the research methods and design aspects used in this study are thoroughly explained in Section 3. Section 4 presents experimental results and analyses based on the research methodology within the second stage of the study. Finally, Section 5 summarizes the outcomes of this research.

2 Related works

The FEM is a widely used numerical analysis technique in various disciplines and engineering sectors. These include acoustics, vibration, optimization, earthquake engineering, groundwater flow, and mechanics of materials. A finite element model was created in the field of architecture by Upasiri et al. [6]. To simulate the fire behavior of lightweight steel framed walls exposed to real design fires, a verified FEM was used to conduct a parametric study. “Hunter” was a term that was oftentimes used. They described how stochastic simulation methods were applied in a non-linear FEA context, according to Hunter et al. [7]. Elettaria et al. employed parametric FEA to examine the impact of specific design factors on the global and local reaction of the nodes in steel non-destructive self-centered column foundations. Using data from experiments, an advanced finite element model was created in ABAQUS. Then, finite element models for three distinct case studies were created, considering 16 configurations for every instance with various design parameters and structural traits, according to Elettaria et al. [3]. Sosso et al. adopted a method using finite elements of rotating laminated beams [8]. The calculations’ findings for big reinforced concrete frames and reinforced concrete beams compared favorably to experimental data in the literature. They were successful in establishing a connection between structural degradation and information derived from cross-section properties using the laminated beam equation, which offered local insights into cross-section and material properties, according to Sosso et al. [8].

GA as a heuristic optimization algorithm in conjunction with FEM can be used for optimization problems. Spicer proposed a continuous curvilinear variable stiffness method using GA in conjunction with a finite element model in NASTRAN software to improve the load carrying capacity of notched fuselage panels. The load carrying capacity of the optimized laminate was increased by 57% compared to the quasi-isotropic design, according to Spicer [9]. An effective nodal kinematic optimization method was employed to develop a refined one-dimensional finite element model. The improved model belongs to the optimal theoretical map which is built using Carrera’s unified equation, axiomatic/asymptotic approach, and GA. The impact of diverse genetic trait types on the genetically enhanced model and genetic performance was examined. The results showed that it was possible to build refined composite load models from optimized models for individual loads, according to Mentari et al. and Liu [10,11].

In summary, experts have made many studies on the application of FEM in various fields and the combination of finite element and machine learning algorithms. However, experts are missing the importance of FEM in assessing the structural performance of ancient buildings, and the FEM combined with machine learning algorithms is less useful. Therefore, it is worthwhile to study how to use FEM to derive accurate analyses of the structural performance of ancient buildings.

3 FEA study on mechanics of beam-column connection in ancient buildings

In this study, FEM is used to analyze the mechanics of ATA beam-column connection. Considering that FEM is not easy to accurately determine the intrinsic model and related parameters, its calculation results are often quite different from the actual ones, according to Pan et al. [12]. In this study, GA is introduced so that the finite element calculation can be closer to the engineering reality. Moreover, based on the finite element intelligent calculation method integrating GA, the rotational capacity of beam-column connection and the force transfer mechanism of nodes under vertical loads are analyzed and modeled.

3.1 Intelligent computational optimization of finite elements by fusion GA

The fundamentals of GA are essentially an optimization problem to find the maximum value of a function, whose mathematical programming model is described by Eq. (1), according to Jain et al. [13].

(1)

$max , f ( X ) st . X ∈ R R ⊆ U ,$

where $f ( X )$ is the objective function, $X$ is the decision variable, i.e., $X = [ x 1 , x 2 , … , x n ] T$ , $st$ is the constraints, $U$ is the basic space set up, and $R$ is its subset. $X$ that satisfies the conditions is then a feasible solution. In GA, the n-dimensional decision variables are denoted as symbol strings by Eq. (2).

(2)

$X = [ x 1 , x 2 , … , x n ] T ⇒ X = X 1 , X 2 , … , X n ,$

where $X n$ is the notation and is regarded as a gene in the algorithm, which takes values called alleles, and $X$ can be regarded as a chromosome consisting of n genes. The simplest allele is made up of the numbers 0 and 1, and its corresponding chromosome can be visualized as a series of binary symbols. The corresponding individual fitness needs to be calculated based on the performance of everyone after randomly given individuals are generated, and it is required that the fitness of all the respective individuals must be positive or zero, according to Mulita et al. [14]. Eq. (3) describes the optimal transformation method for the maximum value of the objective function.

(3)

$F ( X ) = f ( X ) + C min , if f ( X ) + C min > 0 0 , if f ( X ) + C min < 0 ,$

where $C min$ is the number of relatively small values and $C min$ is the individual fitness. Eq. (4) describes the optimal transformation method for the minimum value of the objective function.

(4)

$F ( X ) = C max − f ( X ) , if C max > f ( X ) 0 , if C max ≤ f ( X ) ,$

where $C max$ is the number with relatively large value. Considering the slow convergence of GA, this study adopts the linear scale transformation method to do adaptation scale transformation for individual adaptation, which is described by Eq. (5), according to Yuan et al. [15].

(5)

$F ′ = a F + b ,$

where $F ′$ is the new individual fitness after scale transformation, $F$ is the original fitness, and a, b are the transformation coefficients. If the fitness is low, the probability of being inherited into the next generation is relatively small. Eq. (6) describes the probability of an individual being selected.

(6)

$p i s = F i / ∑ i = 1 M F i ,$

where $p i s$ represents the probability of the $i$ th individual being selected, $M$ is the population size, and $F i$ is the fitness of individual $i$ . Next a selection operation is performed on the individuals, and all the individuals are arranged in descending order according to the size of the fitness. The operation steps of GA are displayed in Figure 1. The operation steps of the GA are as follows. First, randomly generate $M$ individuals as the initial population $P ( 0 )$ . Then, calculate the fitness of each individual in $P ( t )$ . A selection operation is performed, a crossover operation is performed, and a mutation operation is performed to obtain the next generation population $P ( t + 1 )$ after the selection, crossover, and mutation operations, as well as to terminate the conditional judgment. Determine whether population $P ( t + 1 )$ satisfies the termination condition, if not, repeat the calculations in steps 2 through 5. If the termination condition is satisfied, output the fittest person as the ideal solution to the problem.

Figure 1

Operational steps of GA.

In this study, the joint inversion method of GA and finite element is used to solve the mechanical properties of wooden beams and columns of ancient buildings. First, the GA is used to solve and calculate the simulated stress value at the corresponding measured point of everyone in the initial population. Then, based on the measured stress value, the corresponding value is calculated, and this value is taken as the individual’s adaptation degree. Eq. (7) describes the finite element stiffness matrix for elastic deformation.

(7)

$K = K ( E , μ ) ,$

where $K$ represents the overall stiffness matrix, $E$ and $μ$ are constants. Eq. (8) describes the finite element stiffness matrix for elastic–plastic deformation.

(8)

$K = K [ E ( ε ) , μ ( ε ) ] ,$

where $E ( ε )$ and $μ ( ε )$ are strain and stress functions. Elastic–plastic deformation can be regarded as elastic deformation with the changes in $E$ and $μ$ . Therefore, as long as the changes in $E$ and $μ$ can be determined, any deformation can be calculated in terms of its displacement, strain, and stress according to the elastic deformation. This means that it is no longer necessary to consider what their principal relations are or what the parameters corresponding to these principal relations are equal to when performing finite element calculations, according to Papez and Varalika [16]. The values of $E$ and $μ$ are converged to the actual values in engineering by finding the very small value of the error function. Eq. (9) describes the error function, according to Karamanlis and Vo [17].

(9)

$W = 1 n ∑ i = 1 n [ u i ( E , μ ) − u i 0 ( E 0 , μ 0 ) ] 2 ,$

where $W$ represents the error function, $n$ represents the total number of measurement points, $u i ( E , μ )$ is the measured point displacement, and $u i 0 ( E 0 , μ 0 )$ is the calculated displacement. Usually in GA structures, the Relu function is used in the output data activation operation step, described by Eq. (10), according to Hwang et al. [18].

(10)

$Relu ( x ) = max ( 0 , x ) ,$

where $x$ is the input value. When $x ≥ 0$ , the Relu function has a direct output for it, conversely the output value is 0. However, if none of the values input into the Relu function are greater than or equal to 0, i.e., the output values are all 0, the learning performance of the GA will be reduced. Therefore, this study uses Celu activation function described by Eq. (11).

(11)

$Celu ( x ) = max ( 0 , x ) + min ( 0 , α × ( exp ( x / α ) − 1 ) ) .$

In Eq. (11), the Celu function outputs normally when $x ≥ 0$ , and outputs $α × ( exp ( x / α ) − 1 )$ , $α = 1$ when $x ≥ 0$ is less than 0. The GA optimization counting technique is used in this study, and the genetic-finite element intelligent computation process is given in Figure 2, where it can be seen that, first, the individual $( E i , μ i )$ is randomly generated as the initial population $P ( 0 )$ , where $i = 1 , 2 , … , M$ . Substituting the initial population $P ( 0 )$ into the elastic finite element to perform the operation, the node displacements are derived and compared with the measured displacements comparison. The error function is then acquired, and this error function is changed into a fitness function to determine each person’s fitness within population $P ( K )$ . It is stopped if the condition is met, or it is coded. Next selection, crossover, and mutation operations are carried out, and the population is then subjected to the resulting selection, crossover, and mutation operators to produce the next-generation population $P ( K + 1 )$ . Finally, decoding is performed and then substituted into the elasticity finite element computation to obtain the computed displacements of the nodes and the fitness of the new population. Then, the termination condition judgement is performed to determine whether the population $P ( K + 1 )$ satisfies the termination condition, if not, the calculation is repeated. if it has been satisfied, the individual with the maximum fitness is output as the optimal solution of the problem $P ( K + 1 )$ .

Figure 2

Genetic finite element intelligent calculation process.

3.2 Finite element modeling of mechanical properties of wooden beam-column connections in ancient buildings

Each single building in Tibetan architecture is an independent structural unit, and the plan of these structural units is mostly rectangular or square. The internal plan is arranged in several up-and-down walls according to the tic-tac-toe or nearly tic-tac-toe. Inside the surrounding walls, the space is a timber frame system. Usually, a longitudinal row of beams and columns is used, and the beams are covered with dense rafters. If there is a need to increase the internal space, it consists of several rows of longitudinal rows of frames. One of the characteristics of the shape of the ATA is the flat roof, which is unique in the construction of the floor and roof, and the floor system is closely connected to the wooden beams, which play a role in the process of force transmission in the timber frame, according to Bryden et al. [19]. Figure 3 gives the common structural form of column network, i.e., one-column type, which can be seen that the column network structural form can make more reasonable and full use of beams and columns of smaller length, thus improving the stability and seismic capacity of the building. One-column type is the most basic structural form and structural unit, and two-column type, three-column type, and complex column network structural form are developed and evolved from one-column type.

Figure 3

Column grid structure form.

To analyze the mechanical properties of beam–column connections in typical ancient Tibetan buildings, it is necessary to establish a structural analysis model capable of investigating the force transfer mechanism of its nodes under vertical loading. According to Figure 3, the following nodes are selected for load and constraint analysis. Figure 4 shows the structure of the analytical model for the force transmission mechanism of the beam–column connection under vertical loading. The beams are subjected to uniform pressure from above, while being supported by the lower arch member. The force state of the two beams is such that the upper portion experiences compression while the lower portion experiences tension. The bow timber is subjected to vertical loads transmitted from the beams and is supported by the lower mat. Compliant bending may damage the bow timber, with the upper part in tension and the lower part in compression. If the length of the mat is insufficient, the contact area may experience cross-grain compression damage first. The mat carries the vertical loads transmitted from the bow timber and is simultaneously supported by the lower bucket. Since the bucket size is generally much smaller than the mat size, the mat is most susceptible to transverse compression damage at the contact area rather than bending damage.

Figure 4

Analysis model structure of the force transmission mechanism of beam column connection.

The stiffness will gradually decrease when subjected to repeated cycles of external forces, i.e., stiffness degradation. The cut line stiffness is described by Eq. (12).

(12)

$K i = ∣ F i ∣ ∣ Δ i ∣ ,$

where $K i$ is the cutline stiffness of the $i$ cycle, $F i$ is the peak load of the wood frame of the $i$ cycle, and $Δ i$ is the displacement corresponding to the cycle. Eq. (13) describes the specific expanded form of Eq. (12).

(13)

$K i = ∣ + F i ∣ + ∣ − F i ∣ ∣ + X i ∣ + ∣ − X i ∣ ,$

where the peak load under positive loading is denoted by + $F i$ , and the peak load under negative loading is $− F i$ . The displacement corresponding to the peak load under positive loading is $+ X i$ , and the displacement corresponding to the peak load under negative loading is $− X i$ . Due to the small modulus of elasticity of the transverse grain bearing of the wood, the transverse grain bearing deformation of the matting is large, which is the main source of vertical deformation of the node. If there is unevenness in this deformation, it will lead to skew deformation of the superstructure. The bucket bears the vertical load transferred by the matting, and at the same time bears the support force of the lower column. The columns bear the vertical loads transmitted from the bucket and are axial compression members. The pin between the column and the bucket and the pin between the bucket and the bedding are mainly subjected to vertical compliant pressure, while the pin between the bedding and the bow and the pin between the bow and the beam are mainly subjected to transverse shear force.

The load carrying and deformation capacities of beam–column nodes are related to the rotational stiffness of the nodes, according to Mahmood and Ali [20]. To analyze the rotational capacity of beam–column connections of wooden frames of typical Tibetan ancient buildings, a corresponding analytical model needs to be established. According to Figure 3, the lower node is selected to analyze its loading and constraints, assuming that no out-of-plane lateral deformation of the structure occurs. The columns will not produce torsion and bending deformation, no columns are set, and the bottom of the bucket is used as a fixed constraint. The relative rotations of the beams, the bow timbers, and the pads are consistent during the loading process. Moreover, the rotations of the unloaded beams are not taken into account in the loading process. In the actual structure, the vertical displacement at the connection of the two beams is very small. Therefore, the test can be set up with jacks pressed on the plates, which can simulate the concentrated force transmitted between the columns and ensure the vertical displacement at the beam connection. In the numerical simulation, the upper part of the plate can be fixed to ensure the convergence of the calculation. The structural analysis model of the rotational performance of the beam–column connection is given in Figure 5. $A$ represents the relative angle or position between beams and columns. $B$ represents the connection point between columns. $F$ represents the force acting on the beam. The angle $θ$ of the beam’s end section on the right side may be used to indicate the relative angle between the beam and the column, and that $M = P L$ stands for the bending moment at the beam’s end. Using these parameters, it is possible to determine the node’s rotational stiffness.

Figure 5

Structural analysis model for rotational performance of beam–column connections.

4 Experimental results and analysis of nodal force transfer mechanism (NFTM) and rotational performance of wooden beams and columns in ancient buildings

ABAQUS/Standard module in the FEA software ABAQUS is used to experimentally analyze the force transfer mechanism of ATA timber frame beams and columns as well as the rotational performance of the connection. Structured meshing for simple shaped structures is utilized, with a mesh size of 100 × 100 × 100, taking the accuracy of the calculation results and the length of convergence time into account.

4.1 Experimental results and analyses of NFTM for wooden beams and columns in ancient buildings

To cooperate with the static test, the timber of the processed specimen is tested for its timber properties. The timber used in this test is camphor pine, originating from the north-eastern region of China. According to the Design Manual for Wood Construction (Third Edition), there are several major factors that should be considered in accurately determining the discount of the strength of the test material relative to the strength of the specimen, described by Eq. (14).

(14)

$K Q = K Q 1 K Q 2 K Q 3 K Q 4 .$

where $K Q 1$ is the natural defects influence factor, $K Q 2$ is the drying defects influence factor, $K Q 3$ is the strength influence factor of the wood frame under long-term loading, and $K Q 4$ is the dimensional influence factor. Eq. (15) assumes mutual independence for each of these four random variables.

(15)

$K Q → = K Q 1 → K Q 2 → K Q 3 → K Q 4 → .$

where $K Q →$ represents the mean value of its corresponding influence coefficient. This test is discounted according to the existing ATA materiality test results, taking the discount factor of 0.8, Table 1 gives the adjusted experimental results.

Table 1

Material test results

Index	Density		Tensile strength parallel to the grain	Compressive strength parallel to grain of wood	Bending strength	Bending modulus of elasticity
Index	Basic density	Air dry density	MPa	MPa	MPa	MPa
Mean value	0.477	0.457	101.23	31.04	66.12	9,891
Standard deviation	0.034	0.049	1.31	7.16	12.07	3.26
Coefficient of variation (%)	11.30	11.30	14.10	11.50	12.40	11.10

To analyze the NFTM under vertical loading, three manual hydraulic jacks fixed on counter stands are used as the loading devices in the test, and the maximum pressure load that can be applied by each jack is 300 kg. The beam ends are designed as articulated and the column legs are also designed as articulated. According to the strain gauge results collected during the test, the force state of each component during the loading process can be judged. The beam exhibits a bending pattern, with the bottom half primarily susceptible to tensile stresses and the higher section primarily subject to pressure, according to the findings of the beam testing, which are shown in Figure 6. The commencement of the seventh loading phase is when the tensile stress grows significantly, as can be seen from the strain gauge. This behavior precisely matches the test loading regime, where the load is mostly delivered to the top of the columns during the first six steps and to the beams on both sides at the start of the seventh step.

Figure 6

Beam test results. (a) Tensile strain at the lower part of the beam. (b) Pressure strain on the upper part of the beam.

Figure 7 gives the data of column strain gauge strain variation under vertical loading and there is an eccentricity phenomenon, which is since the left and right beams cannot be loaded synchronously and there are serious initial defects in the guard bucket. The column compressive strain is maximum at 700 $με$ , and the plastic strain occurs only when the compressive strain reaches more than 1,900, so the column is in the elastic strain stage during the process.

Figure 7

Strain change data of column strain gauge under vertical load. (a) Column strain gauge data. (b) Column strain gauge arrangement.

Table 2 gives the sequence of damage for each member of the node, and damage occurs when the beam yields under lateral twist, which means that the beam is subjected to bending forces. At this point, the node damage has occurred. However, the bow timbers and columns are maintained in the elastic phase at the time the damage to the node occurred.

Table 2

Destruction sequence of components of nodes

Destruction order	Member	Failure mode
1	Skid	Central transverse grain pressure failure
2	Bucket	Transverse stress failure
3	Beam	Bending failure
4	Arrowwood	Bending failure
5	Column	Pressure failure along grain

As a result of the experiments and analyses, the beam, as a bending element, is subjected to pressure in its upper part and tension in its lower part. Moreover, the bow timber is also a bending element, although it is subjected to tension in its upper part and pressure in its lower part. Turning to the matting, it is subjected to transverse compressive stresses and yields, while the buckets are transverse compressive elements. The post, on the other hand, is the element that is subjected to compression along the grain, and the concealed pin has the duty of resisting shear or bearing pressure. These components transfer loads and work in concert through friction and pressure on the contact surfaces and shear on the concealed pins. Since such nodes are weakly stiff out-of-plane and the loads on the left and right sides are almost asymmetric when the actual force is applied, the loads are subject to eccentricity effects both in-plane and out-of-plane. The eccentricity effect rapidly increases in the compressive strain as the load is conveyed from top to bottom, according to the strain data gathered throughout the studies. The difference in the data collected from symmetric strain gauges for the remaining components, except for the beam, indicates that the eccentricity effect produces a lesser effect on the tensile strain.

4.2 Experimental results and analysis of rotational performance of beam–column connections in ancient wooden frames

The stiffness of a rotating node may be affected by many factors, which include the cross-sectional dimensions of the elements, the modulus of elasticity of the timber, and the lengths of the pads and bows. In Tibetan architecture, the building modulus is not set as a fixed standard, so the cross-section and length of each element of the node usually change according to the room changes, and the dimensions of the matting wood change relatively little. Therefore, in the current test, the focus is on the effect of the length of the bow timber and the position of the concealed pins on the stiffness of the node after fixing the cross-section and span of the beam and designing four specimens for static tests. The variation in specimen parameters is given in Table 3.

Table 3

Changes in specimen parameters

Test piece number	Bow length/beam length	Dowel pin position	Parameter changes
1	0.8	$L 1$ = 0, $L 2$ = 0	Change in length of bow wood
2	0.9	$L 1$ = 0, $L 2$ = 0	Change in length of bow wood
3	0.7	$L 1$ = 100 mm, $L 2$ = 0	Change in pin position
4	0.7	$L 1$ = 0, $L 2$ = 100 mm	Change in pin position

The nodal moment-rotation curves for the four specimens are given in Figure 8 and it can be seen that test piece number 1, 2 and 4 reach the yielding state at 15–25 kN m and the limit state in the range of 25–40 kN m, while specimen number 3 reaches the yielding state close to 6 and reaches the limit state at 6.85. The inward movement of the concealed pins influences the rotational capacity of the nodes. The rotational stiffness of the node shows a tendency to decrease when the dark pins between the bow timber to the beam are internally displaced. Similarly, the rotational stiffness of the node decreases when the concealed pin between the pad to the bow timber undergoes inward movement. However, under the same conditions, the effect on the rotational stiffness of the node due to the inward movement of the pin from the bow timber to the beam is much smaller than that due to the inward movement of the pin from the pad timber to the bow timber.

Figure 8

Node moment angle curve of four specimens.

The study of the experimental data leads to the conclusion that the bending moment-turning angle curves of all four specimens exhibit bilinear properties, allowing the bilinear model to be utilized to describe the node curves. The node rotation model curves are given in Figure 9, $K i$ represents the initial rotational stiffness of the node, $K p$ represents the plastic stiffness, $θ y$ represents the corner when the node reaches the yield state, and $θ μ$ represents the corner when the node reaches the limit state.

Figure 9

Node rotation model curve. (a) Node 1 rotation model curve. (b) Node 2 rotation model curve. (c) Node 3 rotation model curve. (d) Node 4 rotation model curve.

5 Conclusion

To improve the reliability of the structural performance analysis of ancient buildings and to solve the problem of analyzing bracket set-style historical buildings, a representative beam and column node from the ancient Tibetan wooden structure was selected as a structural prototype. This study combined GAs to improve the accuracy of finite element calculation and made it more compatible with practical engineering applications. The FEM combined with the GA is used to investigate and model the rotation ability and force transmission mechanism under vertical load. The results of the experiments demonstrated that the column exhibited maximum compressive strain at 700 $με$ , with the onset of plastic strain occurring when the compressive strain reached 1,900 or above. This indicated that the column was in the elastic strain stage throughout the process. Specimens 1, 2, and 4 attained the yielding state between 15 and 25 kN m, and the ultimate state fell within the range of 25–40 kN m. Specimen 3 reached the yielding state at approximately 6 kN m and the ultimate state at 6.85 kN m. The proposed method effectively analyzes the structural properties of ancient Tibetan buildings and improves the reliability of the analysis. However, FEA methods for simulating complex structures may require significant computational resources and time. It is anticipated that future developments will result in the enhancement of existing methods, thereby reducing the requisite computational resources and time.

Funding information: The research is supported by General Project of Humanities and Social Science Research of the Ministry of Education in 2020: Research on Palace Communication of Environmental Construction Skills in Ming and Qing Dynasties from the Perspective of Cultural Consumption (No.: 20YJCZH045); 2022 Social Science Foundation of Jiangsu Province: Research under the Oral History Method (No.: 22YSD008).
Author contributions: Wen Su and Yixiong Hua: conceptualization, investigation, formal analysis, methodology, and writing – original draft. Yixiong Hua: project management, writing – original design, and writing – review and editing. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2024-08-23

Revised: 2024-10-16

Accepted: 2024-10-31

Published Online: 2025-04-04

This work is licensed under the Creative Commons Attribution 4.0 International License.

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DOI: doi.org/10.1515/nleng-2024-0073

Keywords for this article

finite element analysis; ancient Tibetan architecture; genetic algorithm; nodal force transfer mechanism

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