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

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].

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).

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.

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.

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].

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.

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.

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

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].

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].

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).

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).

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).

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.

4.1 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).

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.

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.

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.

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.

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.