Improved four-level criteria for reliability assessment of timber components subjected to decay

Abstract

Safety evaluation of existing timber structures is of paramount importance for making reliable and timely decisions on repair actions. Among the inspection items, biological decay is a key factor that seriously affects the structural performance of timber components. Given the unclear limits of safety grades in the existing assessment criteria and the increased reliability requirements by the latest Chinese code GB50068-2018, improved four-level criteria are proposed to update the grading rules for timber components subjected to decay. The criteria are developed by probabilistic models and limit state functions with the depth of decay as a key variable. The revised target reliability index and load partial coefficients are taken into account. The JC method and importance sampling method are applied to obtain the relationship between the reliability index and the relative depth of decay under various load conditions. The proposed four-level criteria give explicit grading rules for the safety evaluation of decayed timber components and satisfy the revised reliability requirements. It is suggested that the ratio of the decayed area to the whole cross-section ρ_cr is set to be 0, 5.58%, and 8.36% for main flexural components; 0, 7.40%, and 11.92% for main tension components; 0, 5.91%, and 8.61% for main compression components as limits between a_u and b_u, b_u and c_u, c_u and d_u.

Keywords

reliability assessment timber components biological decay safety appraisal probabilistic models

Introduction

Timber structures are commonly used around the world, with the easy availability of wood and the rapid development of manufacturing and construction techniques. Despite the advantages of sustainability and excellent seismic performance, timber structures are vulnerable to biological attack by decay fungi. Decay can reduce the mechanical properties of wood and the effective cross-section of components, thus decreasing the load-carrying capacity (Van De Kuilen, 2003; Wilcox, 1978). Therefore, the safety appraisal of the existing timber structures is a key task to decide whether, where, and when conservation and strengthening measures need to be taken.

Decay risk has been included in the critical inspection items of timber components in Chinese standard for appraisal of reliability of civil buildings GB50292-2015 (2015), in which a four-level criterion is adopted to evaluate the safety of decayed components. However, the existing standard only gives the grading rules of b_u and c_u, but omits the clear boundaries between a_u and b_u, c_u and d_u. In this case, the nominal four-level criterion is actually downgraded into a two-level one, which increases the workload of appraisers and makes the safety evaluation less objective and reliable. Furthermore, it should be noted that the standard GB50292-2015 (2015) becomes inconsistent with the latest Chinese unified standard for reliability design of building structures GB50068-2018 (2018) on reliability levels. The concept of the quality lower limit and the load partial coefficients adopted in GB50292-2015 (2015) has been discarded and raised respectively in GB50068-2018 (2018), which implies that the requirements of the target reliability index have been improved compared to the old version GB50068-2001 (2001). For these reasons, there is a strong need for research on improving the existing four-level assessment criteria for decayed timber components based on reliability methods.

In recent decades, the development of probabilistic methods, especially the promulgation of relevant structural codes (Probabilistic Model Code, 2001), has promoted the application of reliability-based methods in the safety evaluation of existing structures or components. Köhler et al. (2007) proposed comprehensive probabilistic models of timber material properties including spatial variability and duration of load effects. The proposal offers a guideline for probability-based code calibration of timber design codes. Based on reliability and mechanical methods, Tran et al. (2018) proposed a methodology for reliability assessment and updating of notched timber components subjected to environmental and mechanical loading. Yang et al. (2019) adopted damage accumulation approaches and proposed a Bayesian framework for long-term reliability analysis. Tian et al. (2020) updated the deflection criteria for existing flexural timber components based on the long-term deflection probabilistic model.

As a critical factor that causes the deterioration of material performance, decay has also been taken into account for the structural safety assessment. Brites et al. (2013) included the effects of decay in the safety analysis and applied probabilistic methods based on Monte Carlo simulations to assess the safety of a timber truss system subjected to decay. Huan et al. (2019) proposed an evaluation method for the safety states of ancient timber buildings based on attribute recognition theory. Besides, a dynamic Bayesian network approach was proposed by Tran et al. (2020) for updating the reliability of decayed timber structures using inspection data. Via this method, uncertainties in the decay process and the effect of spatial variability can be accounted for. Some measuring methods, such as non-destructive testing and drilling resistance tests, can also be used for reliability analysis when evaluating the structural safety of existing timber structures subjected to decay (Cabaleiro et al., 2018; Lourenço et al., 2013; Sousa et al., 2013). Although significant works have been done by the above researchers, the study on the reliability-based evaluation of timber structures is still insufficient. For the inspection item of decay risk, reliability-based assessment criteria are rarely addressed and discussed.

The objectives of this paper are to improve the existing four-level assessment criteria for timber components subjected to decay. Based on the reliability analysis, it is aimed to propose practical grading standards that give explicit limits for the safety evaluation and satisfy the revised reliability requirements. Initially, a simplified model for decay and probabilistic models for resistance and loads are presented. Based on these models, limit state functions of decayed timber components are established and reference values for relevant statistical parameters are summarized. The JC method and importance sampling method are applied to calculate the reliability index for three mechanical types of timber components under 21 different load conditions. Finally, the improved four-level assessment criteria for the inspection item of decay risk are proposed based on a comprehensive understanding of the reliability requirements in the latest Chinese codes.

Probabilistic models

Decay simplified model

Assessing the decay extent of timber components is of paramount importance for the safety appraisal of the existing timber structures. Visual grading is a basic approach for decay classification, which has been applied by British standard BS EN 252:2014 (2014) and Chinese forestry standard LY/T 2146–2013 (2013). Despite being simple and straightforward, this method requires observers equipped with considerable experience. Wilcox (1978) indicated that weight loss is a useful index to assess decay based on an extensive literature review on the effects of early decay on wood strength. The weight loss has also been regarded as assessment criteria for decay resistance of woods by American standard ASTM D2017-05 (2017). However, the weight loss index is mainly targeted at laboratory tests instead of in-situ tests.

For assessing existing structures, both Chinese standard GB50292-2015 (2015) and GB/T 50165–2020 (2020) take the area ratio of the rotten zone to the original cross-section as the main assessment index, which models decay as a reduction of the cross-section. The same concept is also applied to the structural fire design of timber construction since the effect of incipient decay can be compared with that of initial fire degradation (Brites et al., 2013). Similar to the thickness of the carbonization layer in fire-resistant design (GB50005-2017, 2017), the depth of decay is also an important concept for estimating the residual capacity of rotten members. For engineering practice, Leicester (2001) illustrated the process of fungus decay (Figure 1) and assumed that the transition zone is a narrow band and the undecayed wood suffers no strength loss. This assumption is widely accepted by the following researchers (Chen, 2019; Nguyen et al., 2008; Qiao et al., 2021; Wang et al., 2008). On this basis, the assumption made in this paper is that the superficial decayed layer of timber components is removed and the remaining cross-section is considered to retain its full strength capacity. Moreover, it is assumed that the pattern of the biological decay is a circumferential one with depth of decay t, as shown in Figure 2.

When subjected to decay, the geometric characteristics and stability of timber components are supposed to change correspondingly. This paper covers flexural, tension, and compression members and aims at sawn and log timber. The geometric parameter refers to section modulus for flexural members but section area for tension members. As for compression members, their load-carrying capacity is generally controlled by stability since defects are inevitable in practical engineering (Zhang, 2016). Decay can reduce the gyration radius and then increase the slenderness ratio, and finally cripple the structural stability of members.

According to Section 5.1.2 of GB50292-2015 (2015), the influence of decay on the geometric characteristics of members should be taken into account. The geometric parameters of timber members under undecayed and decayed conditions are represented by A and A_t respectively. By defining the relative depth of decay ε = t/r for log timber or ε = 2 t/b for sawn timber and aspect ratio n = h/b, the ratio of geometric parameters of undecayed members to decayed members (a = A/A_t) can be expressed in Table 1.

Table 1.

Ratio of geometric parameters of flexural and tension members.

Timber components		Undecayed	Decayed	Ratio
Timber components		A	A _t	A = A/A_t
Flexural members	Log	$π r^{3} / 4$	$π {(r - t)}^{3} / 4$	${(1 - ε)}^{- 3}$
Flexural members	Sawn	$b h^{2} / 6$	$(b - 2 t) {(h - 2 t)}^{2} / 6$	${(1 - ε)}^{- 1} {(1 - ε / n)}^{- 2}$
Tension members	Log	$π r^{2}$	$π {(r - t)}^{2}$	${(1 - ε)}^{- 2}$
Tension members	Sawn	$b h$	$(b - 2 t) (h - 2 t)$	${(1 - ε)}^{- 1} {(1 - ε / n)}^{- 1}$

The ratio of geometric parameters for log timber is determined by the relative depth of decay while that for sawn timber also depends on the aspect ratio. For compression and tension members, the sawn timber is usually processed into square sections, and the aspect ratio can be taken as one. For flexural members, the aspect ratio of sawn timber should not be over four; Otherwise, necessary measures must be taken to ensure lateral stability (GB50005-2017, 2017). In this paper, flexural members with square sections are taken as examples for the calculation of the reliability index.

For compression members, decay changes the slenderness ratio and thus affects the stable bearing capacity. According to Chinese standard for design of timber structures GB50005-2017 (2017), the stability coefficients of the sawn and log timber are related to the strength grade of tree species. Similar to geometric parameters, the stability coefficients of compression members under undecayed and decayed conditions are represented by φ and φ_t respectively. The ratio of stability coefficients of undecayed members to decayed members (η = φ/φ_t) can then be obtained, as shown in Table 2.

Table 2.

Ratio of stability coefficients of compression members.

Strength grades	Undecayed	Decayed	Ratio
Strength grades	φ	φ _t	η= φ/φ_t
TC15, TC17, TB20	${\begin{matrix} {[1 + {(\frac{λ}{80})}^{2}]}^{- 1} & , λ \leq 75 \\ \frac{3000}{λ^{2}} & , λ > 75 \end{matrix}$	${\begin{matrix} {[1 + {(\frac{λ_{t}}{80})}^{2}]}^{- 1} & , λ_{t} \leq 75 \\ \frac{3000}{λ_{t}^{2}} & , λ_{t} > 75 \end{matrix}$	${\begin{matrix} \frac{6400 {(1 - ε)}^{2} + λ^{2}}{{(1 - ε)}^{2} (6400 + λ^{2})} & , λ \leq 75 \\ {(1 - ε)}^{- 2} & , λ > 75 \end{matrix}$
TC11, TC13, TB11, TB13, TB15, TB17	${\begin{matrix} {[1 + {(\frac{λ}{65})}^{2}]}^{- 1} & , λ \leq 91 \\ \frac{2800}{λ^{2}} & , λ > 91 \end{matrix}$	${\begin{matrix} {[1 + {(\frac{λ_{t}}{65})}^{2}]}^{- 1} & , λ_{t} \leq 91 \\ \frac{2800}{λ_{t}^{2}} & , λ_{t} > 91 \end{matrix}$	${\begin{matrix} \frac{4225 {(1 - ε)}^{2} + λ^{2}}{{(1 - ε)}^{2} (4225 + λ^{2})} & , λ \leq 91 \\ {(1 - ε)}^{- 2} & , λ > 91 \end{matrix}$

Where λ and λ_t represent the slenderness ratio of undecayed and decayed members respectively, and their relationship can be expressed by λ_t=λ/(1-ε).

Resistance probabilistic model

Based on the above assumptions, decay can be modeled as a reduction of the cross-section by introducing the depth of decay as an indicator of deterioration. The resistance of timber components is equivalent to the residual capacity of the rotten members. For flexural and tension members whose capacity is controlled by material strength, the resistance can be expressed by equation (1). For compression members whose capacity is mainly determined by stability, the resistance should be calculated by equation (2).

R = K_{Q 3} K_{P} K_{A} f A_{t}

(1)

R = K_{Q 3} K_{P} K_{A} f φ_{t} A_{t}

(2)

The adopted probabilistic models for the mechanical resistant properties of timber components are based on the theory described in GB50005-2017 (2017), which includes several reference variables, namely the influence coefficient of long-term loads K_Q3, the uncertainty coefficient of resistance calculation modes K_P, the uncertainty coefficient of section size K_A and material strength of timber components f. Besides, A_t and φ_t represent the geometric parameters and the stability coefficients of decayed members respectively, which have been illustrated when modeling decay.

Load probabilistic model

Loads can be distinguished between dead loads and live loads according to the duration of the applied action. The dead loads are always present in structures and their values barely change over time, which can be assumed as normal variables. Unlike dead loads, live loads have changeable values over time, which mainly includes floor live loads, snow live loads, and wind live loads. Strictly, live loads are stochastic processes in time. The probabilistic modeling of live loads is complex and only simplified extreme value distributions are considered in this paper.

Supposing only a single live load Q is applied, the effects of applied actions can be expressed by equation (3). Here, C_G and C_Q are the load effect coefficients of dead load and live load respectively. K_B is the uncertainty factor that considers the error existing in the calculation of the load effect.

S = K_{B} (C_{G} G + C_{Q} Q)

(3)

Limit state functions and reference values

Limit state functions

Limit state functions (LSFs) are used to determine whether a given component or structure continues to perform its function as desired. For material deterioration resulting in failure due to loss of resistance, the ultimate limit state is defined when the resistance of the component or structure becomes equal to, or less than, the internal mechanical force (ISO 13823:2008(E), 2008). In this case, the simplest way of defining LSFs can be achieved via subtraction of the mechanical resistance with the effects of applied actions. If assuming negative values, LSFs indicate that the given elements or structures fail to fulfill their required functions in terms of probability. Based on the resistance and load probabilistic models above, the LSFs of decayed timber components can be expressed as

Z = R - S = {\begin{matrix} K_{Q 3} K_{P} K_{A} f A_{t} - K_{B} (C_{G} G + C_{Q} Q) \\ K_{Q 3} K_{P} K_{A} f φ_{t} A_{t} - K_{B} (C_{G} G + C_{Q} Q) \end{matrix}

(4)

When the residual capacity reduces to the effects of applied actions, the decayed members are considered on the verge of the ultimate limit states of load-carrying capacity. In this case, the basic combination with the load partial coefficient should be adopted to calculate the effects of applied actions. The limit state equations for both flexural and tension members, and axial compression members can be expressed as

γ_{0} (γ_{G} C_{G} G_{k} + γ_{L} γ_{Q} C_{Q} Q_{k}) = f_{k} K_{DOL} A / γ_{R}

(5)

γ_{0} (γ_{G} C_{G} G_{k} + γ_{L} γ_{Q} C_{Q} Q_{k}) = φ f_{k} K_{DOL} A / γ_{R}

(6)

where γ₀ is the importance coefficient of structures; γ_G and γ_Q are partial coefficients of dead load and live load, respectively; γ_L is the revised coefficient of load design reference period; γ_R is the partial coefficient of resistance; K_DOL is the design coefficient of long-term loads; f_k is the characteristic value of timber strength.

If the dead load and live load are assumed to have the same distribution type, their load effect coefficients can be extracted as a common factor. The relationships between the actual values (G, Q) and characteristic values (G_k, Q_k) are substituted by ρ = Q_k/G_k, g = G/G_k, and q = Q/Q_k. By eliminating the common factor via Equations (5) and (6) and introducing the ratios of geometric parameters and stability coefficients (a, η) to consider decay, LSFs are redefined as Equations (7) and (8).

Z = K_{Q 3} K_{P} K_{A} f - \frac{a f_{k} K_{DOL} K_{B} (g + q ρ)}{γ_{0} γ_{R} (γ_{G} + γ_{L} γ_{Q} ρ)}

(7)

Z = K_{Q 3} K_{P} K_{A} f - \frac{a η f_{k} K_{DOL} K_{B} (g + q ρ)}{γ_{0} γ_{R} (γ_{G} + γ_{L} γ_{Q} ρ)}

(8)

Reference values

Load parameters

Assuming that the safety level of timber structures is second and the design reference period is 50 years, both the importance coefficient of structures γ₀ and the revised coefficient of load design reference period γ_L can be taken as 1.0 (GB50068-2018, 2018). With regard to the uncertainty coefficient of load effect calculation mode K_B, it is considered to follow a normal distribution with an average value of 1.0 and a variation coefficient of 0.05 (Zhu and Pan, 2017).

In this paper, three load combinations are considered: (1) dead load and residential live load, (2) dead load and office live load, (3) dead load and snow load. According to GB 50005–2017 (2017), the statistical parameters of loads are shown in Table 3, which will be used for the subsequent calculation of the reliability index.

Table 3.

The statistical parameters of loads.

Type of loads	Ratio of average values to characteristic values (g, q)	Variation of coefficient	Distribution
Dead load G	1.060	0.070	Normal
Office floor live load Q_o	0.524	0.288	Extreme value Ⅰ
Residential floor live load Q_r	0.644	0.233	Extreme value Ⅰ
Snow live load Q_s	1.139	0.225	Extreme value Ⅰ

The ratio of the characteristic value of live load to dead load ρ reflects the relative size of the structural self-weight. Due to the relatively small ratio of self-weight to strength, the maximum value of the parameter ρ of timber structures is generally larger than that of concrete structures, steel structures, and masonry structures (Tian et al., 2021). Considering the wide application range of timber structures, the parameter ρ is taken as (0.2, 0.3, 0.5, 1.0, 2.0, 3.0, 4.0). In passing, the load partial coefficients are determined by the latest Chinese code GB50068-2018 (2018), taking γ_G = 1.3, γ_Q = 1.5, respectively.

Resistance parameters

In accordance with GB50005-2017 (2017), the statistics of random variables that affect the resistance of timber components are shown in Table 4. The design coefficient of long-term loads K_DOL is a fixed value of 0.72, which is different from the random variable K_Q3.

Table 4.

Statistical parameters relating to resistance of timber components.

Types of components				Flexural	Tension	Compression
K _Q3	Long-term load effect	Mean value	μ_Q3	0.72	0.72	0.72
K _Q3	Long-term load effect	Coefficient of variation	δ_Q3	0.12	0.12	0.12
K _P	Uncertainty of resistance calculation modes	Mean value	μ_P	1.00	1.00	1.00
K _P	Uncertainty of resistance calculation modes	Coefficient of variation	δ_P	0.05	0.05	0.05
K _A	Uncertainty of geometric parameters	Mean value	μ_A	0.94	0.96	0.96
K _A	Uncertainty of geometric parameters	Coefficient of variation	δ_A	0.08	0.06	0.06

The resistance of components is also associated with timber strength. f is the measured actual strength of existing timber components while f_k is the characteristic value of timber strength with 95% guarantee rate in the design stage. Assuming timber strength follows a log-normal distribution, the relationship between f_k and f is given by equation (9). Based on the formula, the ratio value f_k/f can be calculated according to corresponding coefficients of variation (Qiao, 2015). The results are shown in Table 5. Then, LSFs can be converted to Equations (10) and (11) in the limit state, taking the ratio f_k/f as a key variable.

f_{k} / f = {[(1 + V_{f}^{2}) e^{1.645 \sqrt{\ln (1 + V_{f}^{2})}}]}^{- 1}

(9)

Z = K_{Q 3} K_{P} K_{A} - \frac{f_{k}}{f} \frac{a K_{DOL} K_{B} (g + q ρ)}{γ_{0} γ_{R} (γ_{G} + γ_{L} γ_{Q} ρ)}

(10)

Z = K_{Q 3} K_{P} K_{A} - \frac{f_{k}}{f} \frac{a η K_{DOL} K_{B} (g + q ρ)}{γ_{0} γ_{R} (γ_{G} + γ_{L} γ_{Q} ρ)}

(11)

Table 5.

The statistics information of timber strength.

Types of components	Flexural	Tension	Compression
The ratio value f_k/f	0.71	0.54	0.71
CoV of timber strength	0.20	0.35	0.20
CoV of measured timber strength	0.12	0.27	0.10

Actually, the dispersion of timber strength of existing structures is smaller than code values since the assessed timber components are usually made from the same raw material and processing. Zhang and Chen (2010) manufactured over 200 standard test specimens and measured the material strength, which did show smaller dispersion compared with the characteristic values. According to GB50292-2015 (2015), the material properties should be determined by measured values. The results measured by Zhang and Chen (2010) are adopted as variation coefficients of timber strength in this paper and are also summarized in Table 5.

Given the specified reliability index (GB50068-2018, 2018), the resistance partial coefficient γ_R is not only associated with the variation coefficient of timber strength, but also with load combinations and the ratio of dead load to live load. GB50005-2017 (2017) has provided the δ_f-γ_R benchmark curves, where δ_f is the variation coefficients of material strength and can be determined by the CoV of measured timber strength shown in Table 5. On this basis, the resistance partial coefficient γ_R can be obtained and summarized in Supplementary Table S1.

Calculation

Method

The JC method is adopted to calculate the reliability index and the importance sampling method is employed to estimate the structural failure probability, verifying the results the JC method obtained. As a representative approach of the second-order reliability methods (SORM), the JC method is recommended by the Joint Committee on Structural Safety (JCSS) for approximate calculation of the reliability index. Compared to the center point method, the JC method is improved by taking the practical probability distributions of basic random variables into consideration. In such cases, non-normal random variables are transformed into equivalent normal variables according to the Rackwitz-Fiessler algorithm (Rackwitz and Flessler, 1978). To apply this method, an iterative calculation is usually needed and the detailed steps are illustrated in Figure 3. The formulas involved in the calculation are expressed by Equations 12–16.

σ_{i} = \frac{φ {ϕ^{- 1} [F_{x i} (x_{i}^{*})]}}{f_{x i} (x_{i}^{*})}

(12)

μ_{x i} = x_{i}^{*} - ϕ^{- 1} [F_{x i} (x_{i}^{*})] σ_{x i}

(13)

\cos θ_{x i} = \begin{matrix} \frac{{- \frac{\partial g}{\partial X_{i}} |}_{P^{*}} σ_{x i}}{{[\sum_{i = 1}^{n} {({\frac{\partial g}{\partial X_{i}} |}_{P^{*}} \cdot σ_{x i})}^{2}]}^{1 / 2}} & (i = 1,2, \cdot \cdot \cdot, n) \end{matrix}

(14)

X_{i}^{*} = μ_{x i} + β σ_{x i} \cos θ_{x i}

(15)

Z = R - S = g (X_{1}^{*}, X_{2}^{*}, \cdot \cdot \cdot, X_{n}^{*}) = 0

(16)

The Monte Carlo method, used hereafter, is a simple probabilistic method that employs random sampling to simulate a large number of experiments and evaluates LSFs to ascertain whether the structural failure has occurred. The probability of failure, p_f, is estimated from the fraction of trials leading to failure divided by the total number of trials. To enhance the efficiency of the Monte Carlo simulation, importance sampling, a more advanced simulation method, is adopted to reduce the variance of the estimate of p_f. Basically, this is achieved by concentrating the distribution of the sample points in the vicinity of likely failure points, such as the design point obtained from SORM analysis. Figure 4 presents the calculation flow of the importance sampling method.

Results and discussion

The Matlab software is applied to design programs for calculating the reliability index based on the above LSFs, reference values and the calculation flows. For three types of components made from sawn and log timber (flexural, tension and compression components), 21 load conditions including 3 load combinations and 7 load ratios are considered. In order to ensure the error within the tolerance range, the reliability index is calculated through the JC method by iterating around 10 times, and 10⁶ sampling simulations are implemented by the importance sampling method.

For flexural and tension components, the decay ratio of geometric parameters a associates the reliability index β with the relative depth of decay ε. Forty-one cases of a varying from 1.00 to 1.40 with the gradient of 0.01 were calculated. The results under 21 load conditions are shown in Figures 5 and 6. Supplementary Table S2 takes the case of a = 1.00 as an example to present the results of the reliability index and failure probability obtained.

As shown in Figures 5 and 6, the reliability index of flexural and tension components decreases as the ratio of geometric parameters increases, showing an approximately linear relationship. It can be considered that the reliability index reduces with the increase of the relative depth of decay, given the positive relationship between a and ε implied by Table 1. Furthermore, the reliability index β increases when the load ratio ρ becomes larger. Two reasons may account for this issue. One is that the load partial coefficients have been raised in GB50068-2018 (2018), indicating overall higher setting levels of reliability. Another is that a larger load ratio ρ means heavier live loads the structures or components tend to endure, which deserves a higher reliability index for safety. Basically, the growth of the reliability index is not obvious with the increase of the load ratio, especially at higher ratios (such as ρ≥2.0). Besides decay depth and load ratios, load combinations also have an effect on the reliability index levels of timber components. It can be found that the reliability indexes under the combination of dead load and snow load are relatively higher than those under the other two load combinations. This is understandable because the consequences of structural failure caused by snow load can be serious, especially for timber buildings with light self-weight. From the perspective of reference values for resistance parameters, the resistance partial coefficient γ_R for the combination of dead load and snow load is indeed relatively larger than those for the other two load combinations.

For compression components subjected to decay, the reliability index is closely related to the ratio of stability coefficients η and that of geometric parameters a, which come down to the relative depth of decay ε and the slenderness ratio λ. According to the formulas in Table 2, the ratio of stability coefficients η is independent of the slenderness ratio when λ exceeds the critical value [λ]. Linked to strength grades of tree species, the value of [λ] is equal to 75 for TC15, TC17, TB20, and 91 for TC11, TC13, TB11, TB13, TB15, TB17 (GB50005-2017, 2017). Fifty-one cases of the reliability index under common slenderness ratios were calculated, where ε varying from 0.00 to 0.10 with the gradient of 0.002. Figure 7 presents the results of the reliability index of compression components with ε of 0.05.

It can be found that the reliability index of compression members decreases slightly as the slenderness ratio increase. For both strength grades of log and sawn timber, the steep decline of the reliability index occurs around the critical value [λ] under various load conditions. The reliability index is independent of the slenderness ratio after the decline according to the above analysis. The phenomenon of the steep decline in reliability index can also be observed and even more apparent in modern timber structures or members (Zhang, 2016). The main reason behind the sheer drop can be that the function for stability coefficient applied in existing codes is unsmooth around the critical value [λ]. Conservatively, the minimal results of the reliability index were taken as the basis for subsequent classification.

The results of the relationship between the reliability index and the relative depth of decay under various load conditions are shown in Figure 8. Similar to flexural and tension components, the reliability index β of compression components decreases in approximate linearity with the increase of the relative depth of decay ε. It can be found that the reliability index of undecayed components (ε = 0) meets the requirement of ultimate limit states in the persistent design situation. The reliability index is also enhanced as the load ratio increases, although the growth declines sharply at higher load ratios. Basically, the load ratio does not significantly affect the reliability index.

Improved four-level assessment criteria

Safety rating basis

To update the assessment criteria for safety grades of decayed components, it is necessary to have a comprehensive understanding of the existing reliability index adopted in Chinese codes. Boundary values of safe grades in GB50292-2015 (2015) are determined based on the reliability index specified by GB50068-2001 (2001). Table 6 illustrates the grading rules for the inspection item of load-carrying capacity in GB50292-2015 (2015). The above classification involves two quality limits, specifically the reliability index required by design codes β₀ and the corresponding lower limit. As defined by GB50068-2001 (2001), the former is obtained based on the calibration of the average values of statistical parameters of the material and component performance, representing the average quality levels in China. The lower limit is determined by subtracting 0.25 from the target reliability index, which is equivalent to an increase of failure probability by half an order of magnitude.

Table 6.

The existing grading rules adopted by GB50292-2015.

Grade		a _u	b _u	c _u	d _u
Main components	k = R/γ₀S	≥1.0	≥0.95	≥0.90	<0.90
Main components	Reliability index β	≥β₀	≥β₀-0.25	≥β₀-0.5	< β₀-0.5
General components	k = R/γ₀S	≥1.0	≥0.90	≥0.85	<0.85
General components	Reliability index β	≥β₀	≥β₀−0.5	≥β₀−0.75	< β₀−0.75

Due to relatively lower safety levels in China and the increasing structural risk in the future, the concept of the lower limit has been abolished in GB50068-2018 (2018). This standard prescribes that the target reliability index of structural components in ultimate limit state design should not be less than the specified values [β]. Specifically, [β] for ductile and brittle failure modes are 3.2 and 3.7 respectively under the second safety level. This rule virtually improves the requirements of reliability for structural design, taking the initially average reliability index as the new lower limit without subtracting 0.25. Given this change, there is a strong need for the existing assessment standard GB50292-2015 (2015) to update the relevant grading rules.

Based on the above analysis, the target reliability index [β] defined by GB50068-2018 (2018) is the lower limit that must be satisfied by all structural design codes. So the design reliability index β₀ implied by GB50292-2015 (2015) is commonly larger than [β]. In accordance with the principle of minimizing reinforcement recommended by the international standard ISO13822:2010 (2010), it is considered that there is no need to take reinforcement measures for the components whose reliability index is lower than β₀ but higher than [β]. Therefore, the target reliability index [β] is suggested to be the boundary value between b_u and c_u, and that between c_u and d_u can be decreased by 0.25 on this basis. The above rules apply to the main components and relatively lower reliability indexes are formulated for general components with reference to the grading rules shown in Table 6.

Besides, the failure modes and mechanical types of components should also be taken into account when updating the assessment criteria. According to GB50005-2017 (2017), the failure mode of flexural and compression components is ductile while that of tension components is brittle. For each mechanical type of component, the representative value of β₀ is conservatively determined by the minimum value of reliability indexes under typical working conditions. In this way, the subsequent grading standards can apply to various common working conditions. Finally, the updated grading standard is obtained, as shown in Table 7.

Table 7.

Grading rules of reliability index for the load-carrying capacity of timber components.

Grade	a _u	b _u	c _u	d _u
Main components	≥β₀	≥[β]	≥[β]−0.25	< [β]−0.25
General components	≥β₀	≥[β]−0.25	≥[β]−0.50	< [β]−0.50

Where β₀ for flexural, compression and tension components are 3.64, 3.52 and 4.10 respectively while [β] for flexural, compression and tension components are 3.20, 3.20 and 3.70 respectively.

Practical rating criteria

To obtain the practical grading standards for decayed members, the relative depth of decay ε should be determined according to the reliability index provided by the updated grading rules. For flexural and tension components, the decay ratio of geometric parameters a under various load combinations and load ratios can be first calculated. Similarly, the minimum value of a is considered as the representative value. Subsequently, the relative depth of decay can be obtained in accordance with the formulas in Table 1. It is noteworthy that the relationship between a and ε is related to the aspect ratio n for flexural members with rectangular sections. For a given ratio of geometric parameters, the relative depth of decay increases with the increase of the aspect ratio. In order to make grading rules apply to most members, the minimum value of relative depth of decay is adopted conservatively, taking the aspect ratio as one. In this case, the relationship between a and ε for circular-section components can be consistent with that for rectangular-section components, so it is not necessary to distinguish between the two types of sections. For compression members, the reliability index is related to the slenderness ratio and the relative depth of decay. According to the analysis in the previous section, the reliability index reduces to the minimum value and remains constant when the slenderness ratio exceeds a certain value [λ]. The corresponding relative depth of decay is considered as the boundary value of grading rules. Finally, boundary values of the relative depth of decay for timber components are presented in Table 8.

Table 8.

Boundary values of the relative depth of decay.

Types	The relative depth of decay under different reliability indices (%)
Types	[β]	[β]−0.25	[β]−0.5
Flexural components	2.83	4.27	5.63
Tension components	3.77	6.15	8.94
Compression components	3.00	4.40	5.80

Following the existing rating index for the decay item in GB50292-2015 (2015), the ratio of decayed area to the whole cross-section is defined as ρ_cr. The index is only associated with the relative depth of decay, as expressed by equation (17). Based on this formula and Tables 7 and 8, the improved four-level criteria for the decay item are finally obtained, as shown in Table 9.

ρ_{cr} = 1 - {(1 - ε)}^{2}

(17)

Table 9.

Improved four-level rating criteria for the decay item.

Safety grade		a _u	b _u	c _u	d _u
Flexural components	Main	ρ_cr = 0	0<ρ_cr ≤ 5.58%	5.58%<ρ_cr ≤ 8.36%	ρ_cr > 8.36%
Flexural components	General	ρ_cr = 0	0<ρ_cr ≤ 8.36%	8.36%<ρ_cr ≤ 10.90%	ρ_cr > 10.90%
Tension components	Main	ρ_cr = 0	0<ρ_cr ≤ 7.40%	7.40%<ρ_cr ≤ 11.92%	ρ_cr > 11.92%
Tension components	General	ρ_cr = 0	0<ρ_cr ≤ 11.92%	11.92%<ρ_cr ≤ 17.08%	ρ_cr > 17.08%
Compression components	Main	ρ_cr = 0	0<ρ_cr ≤ 5.91%	5.91%<ρ_cr ≤ 8.61%	ρ_cr > 8.61%
Compression components	General	ρ_cr = 0	0<ρ_cr ≤ 8.61%	8.61%<ρ_cr ≤ 11.26%	ρ_cr > 11.26%

Conclusions

Given the imperfections of existing assessment criteria and the raised reliability requirements, improved four-level assessment criteria are proposed for timber components subjected to decay. In addition to the comprehensive understanding of the existing relevant Chinese codes, probabilistic models and limit state functions involving the depth of decay are developed for use in structural reliability assessment. The relationship between the relative depth of decay and the reliability index is obtained by the JC method and the importance sampling method. The ratio of decayed area to the whole cross-section of timber components is chosen as the practical rating index of safety grades. The main conclusions include:

(1) The biological decay can be modeled as a reduction of the effective cross sections by introducing the relative depth of decay as an indicator of deterioration. The decay ratios of geometric parameters and stability coefficients, i.e. variables a and η, can serve as key links between the reliability index and the relative depth of decay during the reliability analysis of decayed timber components.

(2) The reliability index of timber components increases with the decrease of the relative depth of decay, showing an approximately linear relation. Also, the reliability index is enhanced as the load ratio increases, although the growth is not obvious at high load ratios (especially when ρ ≥ 2.0). Besides decay depth and load ratios, load combinations also affect the reliability index of timber components. The reliability index under the combination of dead load and snow load is higher than that under the other two load combinations, which can be attributed to the relatively larger resistance partial coefficient.

(3) In the latest code GB50068-2018 (2018), the concept of the quality lower limit is abandoned and the load partial coefficients are raised, which implies that the requirements of the target reliability index have been improved compared to the old version GB50068-2001 (2001). On this basis, the new load partial coefficients have been adopted for reference values in the LSFs, and the revised reliability requirements have also been considered in the improved four-level criteria. In accordance with the principle of minimizing reinforcement, the target reliability index [β] is suggested to be the limit between b_u and c_u for main components.

(4) By reliability analysis, the four-level assessment criteria are improved to update the rating rules for the inspection item of decay. The proposed four-level criteria not only satisfy the target reliability index required by the latest Chinese codes but also give reliability-based and explicit limits of existing safety grades for decayed timber components. New insight into the reliability assessment of decayed timber components has been provided by the improved four-level criteria, which can be seen as a guideline for the performance evaluation and safety appraisal of existing timber components.

Supplemental Material

Supplemental Material - Improved four-level criteria for reliability assessment of timber components subjected to decay

Supplemental Material for Improved four-level criteria for reliability assessment of timber components subjected to decay by Weijie Lu, Hongxing Qiu, Panpan Tian and Junyang Gu in Advances in Structural Engineering

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Thirteenth Five-Year National Key Research and Development Program of China (Grant No. 2017YFC0703503).

ORCID iDs

Weijie Lu

Panpan Tian

Supplemental material

Supplemental material for this article is available online.

References

ASTM D2017-05 (2017) Standard Test Method of Accelerated Laboratory Test of Natural Decay Resistance of Woods. Philadelphia: ASTM International.

Brites

Neves

Saporiti Machado

, et al. (2013) Reliability analysis of a timber truss system subjected to decay. Engineering Structures 46: 184–192.

BS EN 252:2014 (2014) Field Test Methods for Determining the Relative Protective Effectiveness in Ground Contact. British: BSI Standards Publication.

Cabaleiro

Branco

Sousa

, et al. (2018) First results on the combination of laser scanner and drilling resistance tests for the assessment of the geometrical condition of irregular cross-sections of timber beams. Materials and Structures 51(4). DOI: 10.1617/s11527-018-1225-9.

Chen

(2019) The Study on Decay and Shrinkage Cracks of Timber componentsPhD Thesis. Nanjing, China: Southeast University.

GB/T 50165-2020 (2020) Technical Standard for Maintenance and Strengthening of Historic Timber Building. Beijing: China Architecture and Building Press, pp. 50165–52020.

GB50005-2017 (2017) Standard for Design of Timber Structures. Beijing: China Architecture and Building Press.

GB50292-2015 (2015) Standard for Appraisal of Reliability of Civil Buildings. Beijing: China Architecture and Building Press.

GB50068-2018 (2018) Unified Standard for Reliability Design of Building Structures. Beijing: China Architecture and Building Press.

10.

GB50068-2001 (2001) Unified Standard for Reliability Design of Building Structures. Beijing: China Architecture and Building Press.

11.

Huan

Wang

, et al. (2019) Safety state evaluation method based on attribute recognition model for ancient timber buildings. Advances in Civil Engineering 2019: 1–13. DOI: 10.1155/2019/3612535.

12.

ISO 13823:2008 (E) (2008) General Principles on the Design of Structures for Durability. Switzerland: ISO.

13.

ISO 13822: 2010 (2010) Bases for Design of Structures-Assessment of Existing Structures. Switzerland: ISO, p. 2010.

14.

Joint Committee of Structural Safety (2001) Probabilistic model code. Available at: www.jcss-lc.org/jcss-probabilistic-model-code/ (accessed 4 November 2021).

15.

Köhler

Sørensen

Faber

(2007) Probabilistic modeling of timber structures. Structural Safety 29(4): 255–267.

16.

Lourenço

Sousa

Brites

, et al. (2013) In situ measured cross section geometry of old timber structures and its influence on structural safety. Materials and Structures 46(7): 1193–1208.

17.

LY/T 2146-2013 (2013) Method for Non-destructive Testing and Decay Classification of Wood Element of Ancient Architecture. Beijing: Standards Press of China, pp. 2146.

18.

Leicester

(2001) Engineered durability for timber construction. Progress in Structural Engineering and Materials 3(3): 216–227.

19.

Nguyen

Leicester

Wang

C-H

, et al. (2008) Probabilistic procedure for design of untreated timber piles under marine borer attack. Reliability Engineering & System Safety 93(3): 482–488.

20.

Qiao

Z-h

Jiang

S-f

Tang

W-j

, et al. (2021) Dual-indicator prediction model for the safety of Chinese ancient wooden structures subjected to bioerosion. Journal of Building Engineering 43: 102868.

21.

Qiao

(2015) Reliability Analysis of Timber Structures and Method for Determining the Strength Design Value of Wood productsMaster Dissertation. Harbin, China: Harbin Institute of Technology.

22.

Rackwitz

Flessler

(1978) Structural reliability under combined random load sequences. Computers & Structures 9(5): 489–494.

23.

Sousa

Sørensen

Kirkegaard

, et al. (2013) On the use of NDT data for reliability-based assessment of existing timber structures. Engineering Structures 56: 298–311.

24.

Tian

P-P

Qiu

H-X

Liang

Z-Z

, et al. (2020) Reliability-based assessment of flexural timber components with ultimate deflection performance. Advances in Structural Engineering 23(16): 3469–3480.

25.

Tian

Qiu

Liang

, et al. (2021) Reliability analysis of the improved four-level assessment standardization for bearing capacity of timber components. Structures 33: 318–329.

26.

Tran

T-B

Bastidas-Arteaga

Aoues

, et al. (2018) Reliability assessment and updating of notched timber components subjected to environmental and mechanical loading. Engineering Structures 166: 107–116.

27.

Tran

Bastidas

Aoues

(2020) A dynamic Bayesian network framework for spatial deterioration modelling and reliability updating of timber structures subjected to decay. Engineering Structures 209: 1–9.

28.

Van De Kuilen

JWG

(2003) The influence of deterioration on the lifetime of timber structures. In 8th international conference on structural studies, repairs and maintenance of heritage architecture, Halkidiki, Greece, 07–09 May 2003, pp. 419–428.

29.

Wilcox

(1978) Review of literature on the effects of early stages of decay on wood strength. Wood and Fiber 9(4): 252–257.

30.

Wang

C-h

Leicester

Nguyen

(2008) Probabilistic procedure for design of untreated timber poles in-ground under attack of decay fungi. Reliability Engineering & System Safety 93(3): 476–481.

31.

Yang

C-H

Zidek

Wong

SWK

(2019) Bayesian analysis of accumulated damage models in lumber reliability. Technometrics 61(2): 233–245.

32.

Zhang

(2016) Stability of Wood Members in Axial Compression and the Reliability analysisMaster Dissertation. Harbin, China: Harbin Institute of Technology.

33.

Zhu

Pan

(2017) Discussions on Problems in the Design of Wood Structures. Beijing: China Architecture and Building Press.

34.

Zhang

Chen

(2010) Reliability assessment and strengthening of a school wooden roof structure in Harbin. Structural Engineers 26(04): 107–113.

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