摘要: 目的建立随时间变化的古建筑残损木构件多因子强度退化模型，验证基于概率密度演化方法对残损木构件进行可靠度分析的适用性，提高古建筑残损木构件可靠度分析的计算精度和效率，为古建筑保护中构件力学性能的量化和评估提供科学依据、�/sec> 方法综合考虑长期荷载、腐朽、虫蛀和干缩开裂对木材损伤的影响，借助已有模型和理论，建立木构件多因子损伤时变模型并推导出强度退化模型。以某古建筑木结构的典型柱为例，根据强度退化模型确定影响参数，对残损影响参数进行灵敏度计算及排序，通过设定阈值确定影响构件残损的关键参数，将非关键参数常数化以实现参数的初步降维。通过拉丁超立方法在关键参数的参数域内选点得到1 000组代表点，基于强度退化模型计算残余强度变化率，根据概率密度守恒构建概率密度演化方程，利用差分法求解得到残余强度比与随机参数的联合概率密度函数，在随机参数域内积分求得残余强度比随时间变化的概率密度函数，最后将概率密度函数在安全域内积分得到构件可靠度；同时基于蒙特卡罗法随机抽样产生10 000组参数数据对残损木柱进行可靠度分析，对比分析两种方法计算出木柱服役时间为1 ~ 1 000 a时的失效概率、�/sec> 结果随服役时间增加，木柱损伤变量逐渐增大，服役时间为1 000 a�? 000组不同参数值的构件几乎都达到失效界限；概率密度演化曲面中概率最高的残余强度比逐渐减小；构件失效概率随时间增大，即可靠度降低；服役时间�?00�?00�?00�?00�?00 a�?个节点，蒙特卡罗法和概率密度演化方法计算所得的木柱失效概率差值分别为9.48%�?.92%�?.10%�?.40%�?.40%，在蒙特卡罗法参数取样量更多、计算用时更长的前提下，两者计算所得失效概率差值不超过10%、�/sec> 结论建立的多因子强度退化模型可用于木柱残余强度的评估和预测，但仍需要进一步的修正。基于概率密度演化方法对残损木构件进行可靠度分析是可行的，与蒙特卡罗法相比具有更高的效率、�/sec>

关键诌�
• 残损木构仵�/a> /
• 强度退化模垊�/a> /
• 概率密度演化方法/
• 可靠度评�?/a>

Abstract: ObjectiveThis paper establishes a multi-factor strength degradation model of damaged wood members of ancient buildings that changes with time, verifies the applicability of reliability analysis of damaged wood members based on probability density evolution method, so as to improve the calculation accuracy and efficiency of reliability analysis of damaged wood members of ancient buildings, and provide scientific basis for the quantification and evaluation of mechanical properties of members in the protection of ancient buildings. MethodConsidering the effects of long-term loading, decay, insect and shrinkage cracking on wood damage, a multi-factor damage time-varying model of wood components was constructed and a strength degradation model was derived with the help of existing models and theories. Typical columns of an ancient timber structure building were used as an example, the influence parameters were determined according to the strength degradation model, the sensitivity of the influence parameters of the damage was calculated and ranked, the key parameters affecting the wood damage were determined by setting the threshold, and the non-key parameters were stabilized to achieve the preliminary dimension reduction of the parameters. 1000 groups of representative points were selected in the parameter domain of key parameters by Latin overshot method. The changing rate of residual strength was calculated based on the intensity degradation model. The probability density evolution equation was constructed according to the probability density conservation, and the joint probability density function of residual strength ratio and random parameters was obtained by the difference method. The probability density function of the residual strength ratio over time was obtained by integrating in the random parameter domain. Finally, the reliability of the component was obtained by integrating the probability density function in the safety domain. At the same time, 10 000 sets of parameter data were randomly sampled by Monte Carlo method to analyze the reliability of the damaged wooden pillar, and the failure probability of the wooden pillar with service time of 1�?000 years was calculated by comparing the two methods ResultWith the increase of service time, the damage variable of wooden pillar gradually increased, and the 1 000 groups of members with different parameter values whose service time was 1 000 years almost reached the failure limit. On the probability density evolution surface, the residual intensity ratio with the highest probability decreased gradually. Component failure probability increased with time, which means the reliability decreased. For nodes with service time of 100, 300, 500, 700 and 900 years, the difference of failure probability calculated by Monte Carlo method and probability density evolution method was 9.48%, 3.92%, 6.10%, 8.40% and 4.40%, respectively. Under the premise of more parameter sampling and longer calculation time of Monte Carlo method, the difference of failure probability calculated by the two methods was less than 10%. ConclusionThe multi-factor strength degradation model can be used to evaluate and predict the residual strength of wooden columns, but it still needs further modification. It is feasible to analyze the reliability of damaged wood structures based on probability density evolution method, and compared with Monte Carlo method, probability density evolution method has higher computational efficiency.

Key words:
• damaged wood components/
• strength degradation model/
• probability density evolution method/
• reliability assessment

�?nbsp; 1技术路线图

Figure 1.Technology roadmap

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�?nbsp; 2干缩裂缝类型

TR指裂纹面的方向为弦向，且裂纹沿径向扩展。图片引自参考文献[17]。TR means the direction of the crack surface is tangential and the crack extends along the radial direction. The picture is cited from reference [17].

Figure 2.Shrinkage crack type

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�?nbsp; 3木柱轴心受压示意国�/p>

Figure 3.Diagram of a wooden column under axial compression

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�?nbsp; 4参数灵敏度排庎�/p>

Figure 4.Parameter sensitivity ranking

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�?nbsp; 5损伤变量随时间变化规徊�/p>

损伤变量 � 1时，构件失效。The component failures when the damage variable is greater than or equal to 1.

Figure 5.Variation pattern of damage variable with time

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�?nbsp; 6残余强度比随时间变化规律

Figure 6.Variation law of residual strength ratio with time

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�?nbsp; 7残余强度比的概率密度演化曲面

Figure 7.Probability density evolution curved surface of residual strength ratio

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�?nbsp; 8失效概率随时间变化规徊�/p>

Figure 8.Changing law of failure probability with time

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�?nbsp; 1参数统计特征

Table 1.Parameter statistical characteristics

参数 Parameter	均� Mean value	变异系数 Variation coefficient
初始应力� Initial stress ratio $ ({\tau }_{0}) $	0.3	0.1
腐朽层厚� Thickness of decay layer $({\delta }_{0}) $/mm	4	0.1
截面边长 Section side length (d)/mm	400	0.05
虫蛀初始期阶�?/a Initial stage of insect infestation 1 ($ {t}_{1} $)/year	16.7	0.646
虫蛀初始期阶�?/a Initial stage of insect infestation 2 ($ {t}_{2} $)/year	4.9	0.813
虫蛀初始期阶�?/a Initial stage of insect infestation 3 ($ {t}_{3} $)/year	37.9	0.675
裂缝宽度和深度的比� Ratio of crack width to depth ($ \lambda $)	0.2	0.08

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�?nbsp; 2参数灵敏度因子（ $ {S}^{*} $ ）及其对应比重（ $ {\gamma }^{*} $ （�/p>

Table 2.Parameter sensitivity factor ( $ {S}^{*} $ ) and specific gravity ( $ {\gamma }^{*} $ )

参数 Parameter	$ {\tau }_{0} $	$ {\delta }_{0} $	d	$ {t}_{1} $	$ {t}_{2} $	$ {t}_{3} $	$ \lambda $
$ {S}^{\mathrm{*}} $	4.339 2 × 10�?	0.505 4 × 10�?	0.505 4 × 10�?	0.043 1 × 10�?	0.012 6 × 10�?	0.097 7 × 10�?	0.017 2 × 10�?
$ {\gamma }^{\mathrm{*}} $	0.786 0	0.091 5	0.091 5	0.007 8	0.002 3	0.017 7	0.003 1
注：$ {\tau }_{0} $为木柱初始应力比�? {\delta }_{0} $为腐朽层厚度（mm），d为截面边长（mm），$ {t}_{1} $为虫蛀初始期第1阶段的时间（a），$ {t}_{2} $为虫蛀初始期第2阶段的时间（a），$ {t}_{3} $为虫蛀初始期第3阶段的时间（a），$ \lambda $为干缩裂缝宽度和深度的比值，下同。Notes: $ {\tau }_{0} $ is the initial stress ratio of the wooden columns, $ {\delta }_{0} $ is the thickness of decay layer (mm),dis the section side length (mm), $ {t}_{1} $ is the time of the first stage of the initial period of insect infestation (year), $ {t}_{2} $ is the time of the second stage of the initial period of insect infestation (year), $ {t}_{3} $ is the third stage of the initial period of insect infestation (year), $ \lambda $ is the ratio of crack width to depth. The same below.

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�?nbsp; 3概率密度演化方法（PDEM）与蒙特卡罗法（MC）失效概率计算结果对毓�/p>

Table 3.Comparison of failure probability calculation results between probability density evolution method (PDEM) and Monte Carlo method (MC)

时间/a Time/year	失效概率 Failure probability/%		差� Difference value/%
	PDEM	MC
100	2.90	12.38	9.48
300	38.50	34.58	3.92
500	67.20	61.10	6.10
700	91.60	100.00	8.40
900	95.60	100.00	4.40

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�?nbsp; 4PDEM与MC计算效率对比

Table 4.Comparison of PDEM and MC calculation efficiency

研究方法 Research method	抽样次数 Number of sampling	运算时间 Operation time/s
PDEM	1 000	1.28
MC	10 000	25.83

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