Research on the Vibration Fatigue Characteristics of Ancient Building Wood Materials

Abstract

In this study, we selected ancient building timber as the research object. A series of static load tests were conducted to analyze the different performances of timber under tensile and compressive loads. After that, vibration fatigue tests on ancient timber samples were carried out under different upper limit stress ratios. Finally, a dynamic constitutive model of ancient timber was established based on the Ramberg–Osgood model. The static load test results show that the tensile strength was approximately 80% of the compressive strength. Meanwhile, the samples that failed under compressive pressure had obvious residual strength, and their failure strains were also much larger than those under tensile stress. In the vibration fatigue tests, the stress–strain curves were analyzed and the results showed that the curves displayed a trend moving to sparse from dense during the loading process. Meanwhile, the curves moved right with the increase in the upper limit stress ratios. The relationship between axial strain and the number of cycles appeared to be characterized by a three-stage form, i.e., damage occurrence, damage expansion, and damage penetration, and this relationship was formulated by a nonlinear function model. Finally, a dynamic constitutive model with high accuracy in describing the vibration fatigue characteristics of ancient timber was established by converting constant parameters to the variable parameters of the Ramberg–Osgood model.

1. Introduction

Timber was widely used as a building material in ancient times because of its abundance, excellent mechanical properties, and anti-seismic properties [1,2]. In China, the majority of palaces, temples, and other ancient buildings were mainly made of timber. Wooden dougong, beams, columns, and mortise and tenon joints were the main load-bearing components in these ancient buildings [3,4,5]. These unique timber buildings not only reflected superb ancient building skills but were also valuable pieces of cultural heritage. It should be noted that timber, as the main load-bearing material of ancient buildings, experienced obvious performance deterioration over hundreds of years due to biological, physical, and chemical erosion. Much research has demonstrated that for timber that has deteriorated due to physical erosion, the essence of its performance deterioration is the oxidative degradation of polysaccharides such as cellulose, hemicellulose, and lignin under external physical factors such as light energy and heat energy [6,7,8,9,10]. The deterioration of timber properties caused by years of erosion has had a very adverse effect on the wooden load-bearing components of ancient buildings. Yermán et al. [11] studied nailed slash pine sapwood timber assemblies exposed to foitopsis ostreiformis and pycnoporus coccineus for 34 weeks and found that the joint strength loss was 40~60%. Dong et al. [12] exposed thermally modified beech wood and linden wood to outdoor conditions for 12 months, and the mechanical properties of the two types of timber were found to have decreased by 41% and 42%, respectively. Many researchers have come to similar conclusions [13,14,15,16]. In addition, some research has also demonstrated that the modulus of elasticity (MOE) of timber increased under temperatures of less than 40 °C. Furthermore, the MOE of the timber decreased obviously when the thermal treatment temperature exceeded 55 °C, and this trend showed a significant dependence on time [17,18,19,20,21,22]. Clearly, the deterioration of the performance of ancient timber caused by long erosion requires great attention from cultural relic protection personnel.

In recent years, the scale of Chinese cities has been expanding rapidly. An increasing number of citizens are choosing the metro as their main mode of transport since it has faster run speeds and is a more convenient way to travel. There are usually many ancient buildings with a long history in these larger cities. As urban landmarks and tourism resources, these ancient buildings are usually surrounded by metro stations and main roads. However, due to the irregularity of metro rails and road surfaces, trains and cars constantly generate vibration when they are running. These vibrations are transmitted to the ancient buildings through the strata and cause adverse effects on these buildings. Research has shown that the vibration generated by metro operation had a serious impact on the surrounding ancient buildings [23,24,25,26,27,28]. There are several methods for monitoring damage to ancient timber, such as a pressure/particle-velocity impedance gun, which supports the detection of lesions on material surfaces according to variations in the sound absorption coefficient; electromechanical impedance, which uses the damage index-root mean square deviation to evaluate the severity of damage to timber; infrared thermography, which monitors the timber damage through the variance of surface temperature; the acoustic emission (AE) system, which analyzes the damage through cumulative AE energy; and so on [29,30,31,32,33]. There is one more problem that should be noted, namely, that test data may be lost due to the cursoriness of researchers and the malfunction of the equipment. In such cases, there are four commonly used methods for the recovery of missing vibration test data [34,35,36,37,38,39,40,41,42,43]: the finite element-based method, which aims to transform the recovery problem into an estimation problem of unknown inputs (the data that need to be estimated are input to a structural finite element model, and the model finishes the process of recovery); the sparse representation method, which is a type of optimization model that can recover the missing vibration test data; the statistical inference method, which recovers the data through probability estimation models; and the machine learning-based data reconstruction method, which recovers the missing data through end-to-end regression models, such as the convolutional neural network (CNN), conditional variational auto-encoder (CAVE), and so on.

For ancient buildings with natural weathering, peeling, and cracks, even if the vibration speed is very small, it will cause fatigue damage due to the continuity of the vibrations. Therefore, it is of great significance for the protection of ancient buildings to study the vibration fatigue characteristics and failure characteristics of deteriorated timber. At present, research on the impacts of vibration on ancient buildings is mainly focused on ancient city walls with brick structures, and few ancient buildings involving wooden structures. Moreover, this research is mainly focused on the response state of the whole ancient building under vibration and does not investigate the vibration fatigue characteristics of the ancient building materials after performance deterioration. Based on the above background, this research selected deteriorated ancient building timber as a study object, and its static and dynamic mechanical characteristics were explored through a series of static and vibration tests. The achievements of the article mainly include a three-stage model for the vibration failure process and a dynamic constitutive model of ancient timber, which can provide a significant reference for the formulation of industrial vibration resistance standards for ancient timber.

2. The Physical and Static Mechanical Parameters of Ancient Building Timber

2.1. Design and Physical Parameters of Ancient Building Timber Samples

The timber demounted in the maintenance of a historic building underwent testing. Specifically, the structural component in question was a jack rafter, serving as a bending element. During sample fabrication, efforts were made to circumvent any areas exhibiting defects. Subsequent to conducting the elastic wave examination, the velocity of these waves within the samples closely mirrored that of the building’s structure. Consequently, in accordance with the Technical Specifications for the Protection of Historic Buildings Against Man-Made Vibration, the samples may be considered to possess an equivalent structural status as the original building [44].

Based on the outcomes of the tests, it was observed that the samples featuring chamfering exhibited a superior bearing capacity. Additionally, in compliance with the specific regulations and the thickness criteria (7~8 mm) for sample dimensions set by the MTS fatigue testing machine manufactured by MTS System Corporation in United States; the developed sample is depicted in Figure 1 [45]. The actual thickness of the sample was maintained at 6 mm. Part I was constructed as rectangles of a larger dimension to ensure optimal engagement with the testing apparatus. To induce damage within the core stress region of part III, the design of part II incorporated an angle of 63.5°. An image of the sample is presented in Figure 2.

The age of the timber samples was determined using accelerator mass spectrometry. Prior to this determination, all extraneous background particles were thoroughly eliminated. The carbon dating method was used to determine the age of the timber samples, which was found to be 210 years old [46]. The basic physical parameters of the ancient timbers are detailed in

Table 1. The physical parameters of the samples.

Water Content	Density (kg/m3)	Dry Density (kg/m3)
2.71%	432.93	421.12

.

2.2. Static Tests of Ancient Building Timber Samples

Static and fatigue tests on ancient timbers were conducted using the MTS858 material fatigue testing machine (manufactured by MTS System Corporation in Eden Prairie, MN, USA). The fatigue testing machine comprises four main parts, as detailed in Figure 3. It is capable of applying a maximum static axial load of 30 kN and a dynamic axial load of up to 25 kN. The displacement range of the test spans from −75 mm to +75 mm, and the applicable frequency range extends from 0.001 Hz to 100 Hz.

Static tensile stress and static compressive stress tests were conducted on ancient timber samples following the protocols outlined in the Test methods for physical and mechanical properties of small clear wood specimens—Part 14: Determination of tensile strength parallel to grain (GB/T 1927.14-2022) [45]. The resultant stress–strain curves are presented in Figure 4. Within this figure, the sample labels No.mul 1~3 denote tensile stress application on the timber, whereas No.muy 1~3 indicate compressive stress application. Initially, during the static tensile test, elastic deformation was prominently observed. Axial stress at approximately 40 MPa marks the onset of plastic deformation, followed by a rapid increase in strain and abrupt unloading upon sample failure, leaving negligible residual strength. The maximum strain records in the tensile test exceeded 30%, approximately 4–5 times higher than that in the compressive stress test. In static compressive tests, the initial phase also displays significant elastic deformation. Plastic deformation begins at around 45 MPa axial stress, succeeded by a gradual stress increase until achieving peak strength at approximately 2% axial strain. Even after reaching peak strength, the timber samples retain some bearing capacity, exhibiting strain softening along with notable residual strength and strain. Generally, the maximum load sustained in the tensile stress test is about 80% of that in the compressive stress test, indicating that the tensile strength of the timber samples is lower than their compressive strength. The peak load value observed in both static tensile and static compressive stress tests for the timber samples was determined to be 5 kN.

3. The Characteristics of Vibration Fatigue

3.1. The Test Process and Results Analysis

To analyze the vibration fatigue characteristics of the samples, fatigue vibration tests were conducted using the MTS858 fatigue testing machine. Based on the peak load determined from the static test results, alternating sine waves with constant amplitude were applied during loading. Given that the peak value of the static tensile stress strength was about 80% of the peak value of the static compressive stress strength, the selected upper limit stress ratio (ULSR) (the ratio of maximum applied force to peak load value) was in the range of 0.40~0.78. The frequency for the fatigue vibration test was set to 2 Hz. The test results depicted in Figure 5 indicate that as the ULSR increases, the number of cycles decreases, while the failure strain generally shows an increasing trend. However, due to the inherent nonuniformity of the timber (such as knots, splits, micro-cavities, etc.), some samples exhibit irregular variations.

When the ULSR is below 0.52, the timber tested exhibits substantial fatigue resistance, enduring on average of more than 250,000 cycles (with the exception of LYPL-2.6-2). Conversely, when the ULSR exceeds 0.70, the timber’s fatigue resistance significantly diminishes, leading to rapid damage after an average of fewer than 1500 cycles. Compared with the static test results, the maximum strain of timber under cyclic load is only 0.8%, markedly lower than the maximum strain observed under static load. This discrepancy suggests that timber may fail under low strains when subjected to repetitive loading.

3.2. Analysis of Stress–Strain Behavior

A sinusoidal load $F=A\sin t$ was applied during the test, where $A$ was equal to the ULSR of each sample multiplied by 5 kN. The stress and strain values at the peak and trough positions of applied force were selected as the data points, as illustrated by the dashed line in Figure 6.

The stress–strain scatter points obtained from various samples were organized into curves. For a detailed analysis, three samples with ULSRs of 0.40, 0.60, and 0.70 were successively selected, as depicted in Figure 7.

The curves vary from dense to sparse during loading due to the different stages of crack development in samples. In the early stage of loading, no cracks appear inside the samples, and the strain develops slowly. As microcracks gradually expand in the middle and later stages of loading, strain development accelerates, though the samples do not entirely lose their load-bearing capacity. Consequently, the cyclic dynamic stress and axial strain curves shift from dense to sparse. With the increase in ULSR, the curves tend to shift to the right. This is because that the strain generated by the tensile stress is greater than that generated by the compressive stress. In other words, when both the tensile and compressive stresses increase simultaneously, the strain generated by the tensile stress becomes more pronounced. As a result, the stress–strain curves of the timber samples shift towards the tensile stress, moving closer to the coordinate axis.

3.3. The Relationship of Axial Strain and the Number of Cycles

Figure 8 illustrates the relationship between axial strain and the number of cycles under different ULSR values. The slope of the curves reflects the rate change in fatigue strain in the samples. These curves can be divided into three distinct stages. The first stage corresponds to damage occurrence, representing the early stage of the fatigue life. This stage occupies a relatively small proportion of the fatigue life, with a noticeable increase in strain. The second stage corresponds to damage expansion, occurring during the middle stage of fatigue life and comprising the largest proportion of the fatigue life. In this stage, the strain remains relatively constant. The third stage represents damage penetration (i.e., the stage of material failure), occurring in the late stage of the fatigue life and occupying a small proportion of the overall fatigue life. During this stage, the strain increases rapidly.

The development of axial deformation in materials can be regarded as the gradual progression of microscopic damage [47,48]. Research indicates that the fundamental mechanism of fatigue failure in materials under cyclic loads is the process of microcrack formation, expansion, and penetration within the materials. In the fatigue life diagram, the development can be simplified into a three-stage model due to the distinct differences in strain development rates. The three-stage model is applicable to certain materials. Given the nonlinearity of damage progression across the three stages, a three-stage nonlinear model is employed. In this model, a logarithmic function is used in the first stage, a linear function is applied in the second stage, and an exponential function is adopted in the third stage. The specifics of the model are provided in Formula (1).

$$S=\left\{\begin{array}{l}{a}_1\ln \left(N\right)+b_1; \mathrm{for} \mathrm{Stage} 1\\ a_{2}N+b_2;\hspace{1em}\hspace{1em}\mathrm{for} \mathrm{Stage} 2\\ a_3{e}^{b_{3}N};\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{for} \mathrm{Stage} 3\end{array}\right.$$

(1)

where both $a$ and $b$ are constants related to the material.

3.4. Discussion of Vibration Fatigue Characteristics of the Samples

The deformation development pattern of Sample LYPL-3.5-1 accurately mirrors the general damage progression trend, making its axial strain and number of cycles ideal for analysis using the aforementioned model. As shown in Figure 9, the curve distinctly encompassed three stages. During the initial stage, with a cycle ratio (the ratio of elapsed cycles to total cycles) below 0.12, there is a marked development of axial strain, reaching approximately 50% of failure strain at a relatively low number of cycles. In the intermediate stage, spanning a cycle ratio from 0.12 to 0.88, the axial strain displays a stable linear relationship with the number of cycles. This stage constitutes the majority of the fatigue life, approximately 76%, yet it features a minimal axial strain increment, around 0.1%. The final stage, characterized by a cyclic ratio exceeding 0.88, demonstrates a significant surge in axial strain in the approach to failure—accounting for about 38% of the total strain—as microscopic cracks coalesce and expand, ultimately culminating in macroscopic fracture and sample failure. This stage represents about 12% of the total fatigue life.

4. Dynamic Constitutive Relation Based on the Ramberg–Osgood Model

4.1. Ramberg–Osgood Constitutive Model with Variable Parameters

To derive a dynamic constitutive equation that aligns with test findings, this study modified the Ramberg–Osgood model by transforming its constant parameters into variable ones, taking into account the stages of fatigue life [49]. The expression of the constitutive model is developed within the stress space framework. The Ramberg–Osgood model proposed the following formula to describe the stress–strain skeleton curve of the material under cyclic load.

$$\epsilon ={\displaystyle \frac{\sigma }{E_0}}\left(1+\alpha {\left|{\displaystyle \frac{\sigma }{C{E}_0}}\right|}^{R-1}\right)$$

(2)

where $\alpha$, $C$, and $R$ are the constant parameters that determine the shape and position of the curve, $E_0$ is the maximum elastic modulus, and ${\sigma }_0$ is the maximum normal stress. The relationship between $E_0$ and ${\sigma }_0$ can be described in Formula (3).

$${\sigma }_0=E_0{\epsilon }_0$$

(3)

where ${\epsilon }_0$ is the maximum strain, which can be determined from Formula (4).

$${\epsilon }_0={\displaystyle \frac{\left|{\epsilon }_{\max }^{+}\right|+\left|{\epsilon }_{\max }^{-}\right|}{2}}$$

(4)

where $\left|{\epsilon }^{+}{ }_{\max }\right|$ and $\left|{\epsilon }^{-}{ }_{\max }\right|$ are the absolute values of the maximum tensile strain and maximum compressive strain, respectively.

According to Masing’s criterion, the mathematical expression of the hysteretic circle can be written as Formula (5) [50].

$$\epsilon -{\epsilon }_c={\displaystyle \frac{\sigma -{\sigma }_c}{E_0}}\left(1+\alpha {\left|{\displaystyle \frac{\sigma -{\sigma }_c}{2C{E}_0}}\right|}^{R-1}\right)$$

(5)

where $\left({\epsilon }_c,{\sigma }_c\right)$ and $\left(-{\epsilon }_c,-{\sigma }_c\right)$ are, respectively, the loading and unloading turning points in the current circle (i.e., the two acmes of the hysteretic circle). The skeleton curve and hysteretic circle based on the Ramberg–Osgood model are demonstrated in Figure 10.

$\alpha$ and $R$ in Formula (3) are independent parameters. In order to express the stress–strain relationship at different cycle stages, these two parameters are taken as functions of the cycle ratio as Formula (6) shows.

$$\left\{\begin{array}{l}\alpha =\alpha \left(n/N\right)\\ R=R\left(n/N\right)\end{array}\right.$$

(6)

The parameter $\alpha$ is a positive number, indicating the elongation trend of the hysteretic circle. A larger parameter $\alpha$ value causes the peak of the hysteretic loop to stretch toward both ends, reduces the slope of the hysteretic curve, and tilts the hysteretic loop toward the axis, assuming other parameters remain constant. The parameter $R$ is a positive number greater than 1, representing the degree of nonlinearity of the hysteretic circle. As the parameter $R$ increases, the nonlinearity of the hysteretic circle intensifies, resulting in a larger loop area, a greater ratio of the long axis to the short axis, and a fuller loop shape, provided the other parameters are unchanged. The shape characteristics of the hysteretic curve significantly differ before and after the cycle ratio of 0.1, allowing $\alpha$ and $R$ to be regarded as piecewise functions related to the cycle ratio. The following mathematical expressions for parameters $\alpha$ and $R$ represent the constitutive models of the hysteretic circle at different cycle stages.

$$\alpha =\left\{\begin{array}{l}-2n^{*}+0.5,\hspace{1em} n^{*}\le 0.1\\ 0.05n^{*}+0.295, n^{*}>0.1\end{array}\right.$$

(7)

$$R=\left\{\begin{array}{l}-2n^{*}+2,\hspace{1em}\hspace{1em}\hspace{1em} n^{*}\le 0.1\\ -0.222n^{*}+1.822, n^{*}>0.1\end{array}\right.$$

(8)

where $n^{*}$ is the cycle ratio (i.e., $n/N$). By substituting Formulas (7) and (8) into Formula (2), a dynamic constitutive equation of timber based on the Ramberg–Osgood model with variable parameters can be obtained.

$$\epsilon =\left\{\begin{array}{l}{\displaystyle \frac{\sigma }{E_0}}\left[1+\left(-2n^{*}+0.5\right){\left|{\displaystyle \frac{\sigma }{C{E}_0}}\right|}^{-2n^{*}+1}\right],\hspace{1em}\hspace{1em}\hspace{1em} n^{*}\le 0.1\\ {\displaystyle \frac{\sigma }{E_0}}\left[1+\left(0.05n^{*}+0.295\right){\left|{\displaystyle \frac{\sigma }{C{E}_0}}\right|}^{-0.222n^{*}+0.822}\right], n^{*}>0.1\end{array}\right.$$

(9)

4.2. Verification and Discussion of the Dynamic Constitutive Equation

To ensure the applicability of the dynamic constitutive equation, the measured hysteretic curves of NPL2.6-1, NPL2.6-3, and NPL2.6-8 were compared with those generated by the constitutive model, as shown in Figure 11, Figure 12 and Figure 13. The hysteretic curves at different cycle ratios were compared and analyzed, revealing that the hysteretic curves derived from the dynamic constitutive equation with variable parameters based on the Ramberg–Osgood model closely match the measured data. This demonstrates the high reliability of the model.

The previous research (reviewed in Section 1), on the one hand mainly focused on the static mechanical properties, and on the other hand only the entire vibration responses of ancient buildings were studied. The dynamic properties of ancient building materials were not researched deeply. This research not only analyzed the vibration fatigue characteristics of ancient building timbers but also established a constitutive model with high accuracy in describing their vibration fatigue characteristics. This research will provide a theoretical basis for regulation of the anti-vibration standards of wooden ancient buildings.

5. Conclusions

The vibration fatigue characteristics and dynamic constitutive model of ancient timbers were researched through theoretic analysis and calculations based on static load and vibration tests. The main conclusions are shown below.

• The tensile strength is approximately 80% of the compressive strength. The samples which fail under compressive pressure have obvious residual strength and their failure strains are also much larger than those under tensile stress.
• The stress–strain curves of the ancient timber in vibration fatigue tests change to sparse from dense when the number of cycles increases. And the curves’ position moves right when the ULSR enhances.
• The vibration failure process of the samples has three stages: the damage occurrence stage, the damage expansion stage, and the damage penetration stage. A three-stage model is established through nonlinear functions.
• A dynamic constitutive model with high accuracy is established based on the Ramberg–Osgood model.

The future works can devote attention to the damage identification and damage monitoring of ancient building materials. These materials may not be confined to timbers and can also include bricks, terracotta, mortar, and so on.