Using numerical procedures to quantify seismic reliability of wood shear walls and braced frames

Abstract

This article discusses the use of numerical procedures to quantify the probabilistic seismic behaviour of panel-sheathed wood shear walls and diagonally braced frames made with three species including Japanese sugi, Canadian hemlock and European whitewood. Three probabilistic methods were implemented in the analysis, based on the assumption that ground motion records follow a uniform distribution of representing earthquake characteristics. The first method is the traditional one in seismic reliability analysis. The second method calculates the exceeding probability from conditional distributions at given ground motions. The third method is the Monte Carlo simulation method. Confidence curves from two different methods were also used to visualize the comparison among structural configurations and species. The results showed that panel-sheathed walls exhibit better performance than braced frames. The results also show that systems built with hemlock are better than other systems with whitewood or sugi.

Keywords

earthquake engineering probability analysis reliability shear walls timber wood structures

Introduction

Wood construction includes both light-frame construction and post-and-beam construction. Light-frame construction is dominating low-rise residential buildings in North America. Wood shear walls are commonly used as a seismic force resisting system (SFRS) to develop strength, stiffness and energy dissipation under seismic loading. In some oriental countries including China, Japan and Korea, post-and-beam construction is a traditional style in building construction. In these countries, semi-rigid moment frames are traditionally used as the SFRS to build post-and-beam construction. Because of the increasing concern of seismic performance of traditional post-and-beam construction, as was observed in the 1995 Kobe earthquake, diagonally braced frames interested some builders, engineers and researchers.

With various configurations of SFRSs, wood construction has two paths to compliance with local building codes and standards. One path is to build non-engineered buildings following prescriptive requirements specified by codes and standards. The prescriptive requirements are typically based on statistics of historic performance of buildings combined with engineering judgements of representative studies. The other path is to construct engineered buildings with design methods and procedures specified by the codes and standards. In the traditional engineering design method, base shear is used as the design parameter with consideration of force modification factors to account for ductility and energy dissipation. Both the prescriptive requirements and the force modification factors in the codes and standards are essentially based on judgement from limited laboratory tests. As the public has an increasing concern on performance of wood buildings at different seismic hazards, rational methodologies are needed to accurately quantify seismic behaviour of these buildings. Probabilistic-based analysis can be implemented to achieve a uniform performance level of building.

In the past two decades, many researchers performed reliability and probabilistic analysis for wood structures subjected to earthquake loads. Rosowsky (2002) developed a risk-based methodology for seismic design of shear walls. This study conducted a sequence of sensitivity studies to evaluate the contributions of various sources of uncertainty using the CASHEW programme (Folz and Filiatrault, 2001). This study also evaluated the variability in the peak response from the selected ground motions. Zhang and Foschi (2004) reported their work in performance-based seismic design and reliability analysis using optimization methods to deal with the databases of structural response. The designed experiments and neural networks were implemented to improve computational efficiency. van de Lindt (2005) presented the development of a damage-based seismic reliability model for light-frame wood structures. The damage model used in this work expressed damage as a linear combination of the maximum displacement and the hysteretic energy dissipated by shear walls. Li and Ellingwood (2007) used stochastic nonlinear dynamic analysis to simulate the behaviour of shear wall systems. The probability of failure under a spectrum of possible earthquakes was determined by convolving the structural fragility derived from dynamic analysis. Pang et al. (2009) examined the seismic performance of typical one- and two-storey wood-frame structures in the central and eastern United States. Conditional limit state probabilities for six structures with two foundation types were evaluated considering three possible failure mechanisms. Their results showed that seismic damage to wood-frame structures may result in significant financial losses. Li et al. (2010) investigated the implications of designing for uniform hazard versus uniform risk for light-frame wood residential construction. They found that the collapse capacity of wood-frame construction was sensitive to the ground motions. They also found that the collapse probabilities of light-frame wood residential construction in different regions of the United States remain non-uniform. Pei and van de Lindt (2010) presented their discussion about the financial loss for residential light-frame structures.

The purpose of this article is to accurately quantify seismic reliability analysis of eight types of Japanese style panel-sheathed shear walls and braced frames. Three reliability methods were used in the analysis: a traditional method based on conditional distributions at given intensity levels, a method based on conditional distributions at given ground motion records and the Monte Carlo simulation (MCS). Numerical procedures of the first and second methods were implemented to quantify the analysis. These numerical procedures do not involve any data fitting technology to obtain parameters and thus retain the accuracy of results from nonlinear time history analysis (NTHA). The results of probability failure and confidence curves were compared for the performance of the eight types of walls and frames.

Test configurations and results

Test configurations

Eight types of full-size shear walls and braced frames were tested by the Center for Better Living of Japan (CBL, 2001). The structural configuration of the diagonally braced frames is shown in Figure 1(a), while the structural configuration of the plywood and oriented strand board (OSB) single-side panel-sheathed shear walls are shown in Figure 1(b) and (c), respectively. The overall dimensions of all walls and braced frames were 1.82 m × 2.73 m, with posts spaced at 455 mm on centre. The top girders were 105 mm × 180 mm members. Bottom sills, end posts and central posts were 105 mm × 105 mm members, while other internal posts were 30 mm × 105 mm. The diagonal braces of the braced frames were 45 mm × 90 mm.

Figure 1.

Configurations of shear walls and braced frames: (a) braced frame, (b) plywood-sheathed wall and (c) OSB-sheathed wall.

All of the perimeter and central members were connected together with mortise-and-tenon connections, as commonly seen in Japanese post-and-beam construction. The small internal posts were toe-nailed to the girders and sills. S-HD 20 hold-downs were used at four corners. In Figure 1(a), the diagonal braces were connected to the frames with ‘BP2’ metal connection plates. Each of the metal plates was connected to the frame with sixteen 50-mm-long galvanized spiral nails and one M12 bolt. In Figure 1(b), the plywood panels were 9.5-mm thick and in compliance with Japanese Agricultural Standard (JAS) grade two. The dimension of the two panels on the bottom was 910 mm × 1820 mm, while the two panels on the top had the dimension of 910 mm × 910 mm. In Figure 1(c), the walls were sheathed with 9-mm-thick full-height JAS grade OSB panels. All sheathing panels were connected with 50-mm-long common wire nails spaced at 150 mm along the panel edges and 300 mm in the interior. All other connection hardwares were approved for residential construction in Japan (CBL, 2001).

The top girders of all specimens were built with Canadian Douglas Fir. Three wood species were used to build other framing members: Japanese sugi, Canadian hemlock and European whitewood glulam. Sugi and hemlock were used to build all three structural configurations as shown in Figure 1. Whitewood was only used to build the braced frames and plywood-sheathed walls. Therefore, there were eight types of walls and braced frames in total, each of which had three replications.

Test procedure and results

All walls and braced frames were tested according to Japanese Industrial Standard JIS A1414 without a tie-rod. The reversed cyclic loading had 12 amplitudes with maximum value of 1/15 (6.67%), which is defined as a ratio between the lateral displacement and the wall height. Each of the amplitude was repeated for three cycles. No vertical load was applied during the tests. The relative density of all framing members was measured and adjusted to a moisture content of 15%. The relative density of all sugi braces was 0.36 on average, while other sugi framing members had an average density of 0.43. The relative density of hemlock framing members was 0.49 on average, except that the braces had an average density of 0.45. The whitewood braces had a density of 0.43 and other members had a density of 0.48 (CBL, 2001).

The results of reversed cyclic tests of three replications of each type appear to be very consistent. Some key parameters, such as the stiffness and ultimate strength, were within a 10% variation among three replications (CBL, 2001). Therefore, the specimen with the moderate response may represent structural behaviour of each type of walls and braced frames and thus can be used to analyse their seismic reliability.

Dynamic model and parameters

A single-degree-of-freedom (SDOF) system with a nonlinear spring representing the hysteretic behaviour was used to simulate the dynamic behaviour. The nonlinear spring is an analogue model based on a detailed finite element model considering the behaviour of the nail shank, the contacting behaviour between the nail and its surrounding medium, and the formation of the gap (Figure 2) (Foschi, 2001). The contacting behaviour of the surrounding medium is represented by compression-only springs with the embedment function shown as

p (w) = {\begin{matrix} (Q_{0} + Q_{1} w) [1 - \exp (- Kw / Q_{0})] (if w ⩽ D_{\max}) \\ p_{\max} \exp [Q_{3} {(w - D_{\max})}^{2}] (if w > D_{\max}) \end{matrix}

(1)

where

p_{\max} = (Q_{0} + Q_{1} D_{\max}) [1.0 - \exp (- K D_{\max} / Q_{0})]

(2)

Q_{3} = \log 0.8 / [D_{\max} (Q_{2} - 1)]^{2}

(3)

and w = lateral displacement of the springs, K = initial tangential stiffness, p_max = peak load, D_max = displace-ment at the peak load, Q₀ = intercept of the asymptote, Q₁ = slope of the asymptote and Q₂ = ratio of the displacement at $0.8 p_{\max}$ on the descending branch to $D_{\max}$ . Figure 3 illustrates the geometric meanings of these parameters. The behaviour of unloading and reloading is defined by the following function

p = {\begin{matrix} 0 (if w ⩽ D_{0}) \\ \min [p_{1} = K (w - D_{0}), p_{2} = p (w)] (if w > D_{0}) \end{matrix}

(4)

with

D_{0} = {\begin{matrix} w - p / K (if p ⩽ p_{2}) \\ unchanged (if p = 0 or p = p_{1}) \end{matrix}

(5)

where D₀ = maximum zero-force displacement in the unloading history from point ‘b’, as indicated by point ‘a’ in Figure 3 and p₀ = load in the unloading history at point ‘b’, as indicated in Figure 3. The function in equation (4) may be linear or nonlinear depending on the previous loading history and the current displacement.

Figure 2.

Springs between nail shank and wood medium.

Figure 3.

Embedment function of p(w) and unloading rules.

Two additional parameters, the diameter of the pseudo nail, $D_{i}$ , and the length of the pseudo nail, L, were needed to represent physical shear walls and frames. Hence, there are seven parameters, $Q_{0}$ , $Q_{1}$ , $Q_{2}$ , $K$ , $D_{\max}$ , $D_{i}$ and L, to be validated by the test results of the shear walls and frames in order to represent them. The input of parameter identification was the backbone curve of the reversed cyclic test data. The parameter identification was obtained from optimization using five non-gradient search methods (Gu, 2006). The resulted parameters of the discussed walls and braced frames are listed in Table 1. With these parameters, the examples of comparisons between the predicted behaviour and test results are shown in Figure 4. The details of the wall analysis, including key parameters of test results, the nail model, optimization methods and the input of parameter identification, were reported in other documents (CBL, 2001; Gu, 2006).

Table 1.

Model parameters for each type of walls and braced frames.

	Q₀ (kN/mm)	Q₁ (kN/mm²)	Q ₂	K (kN/mm²)	D_max (mm)	D_i (mm)	L (mm)
SG-BR	4.093	0.1065	1.958	0.5187	30.28	3.626	72.49
SG-PL	0.3376	0.02759	1.809	0.08346	45.26	9.279	426.7
SG-OS	3.672	0.0001	1.196	0.1596	65.08	7.167	193.9
HF-BR	0.01215	0.05834	1.254	0.05874	54.67	9.416	461.2
HF-PL	0.8021	0.02910	1.115	0.1901	75.72	10.77	41.55
HF-OS	0.2211	0.02880	1.295	0.06188	54.52	10.83	207.9
WW-BR	0.1523	0.06751	1.081	0.3841	68.81	7.676	47.66
WW-PL	0.8765	0.04502	2.929	0.09967	22.13	8.822	410.2

SG: sugi; HF: hemlock; WW: whitewood; BR: braced frames; PL: plywood-sheathed walls; OS: oriented strand board–sheathed walls.

The first two letters indicate wood species, and the last two letters indicate structural forms.

Figure 4.

Comparison between test results and model prediction for hemlock: (a) SG-BR, (b) SG-OS and (c) SG-PL.

It is worth noting that this wall model is an SDOF system based on an analogue to a pseudo giant nail. The hysteresis energy dissipation of this model is realized through the bearing resistance of the imagined contacting surface surrounding the nail shank determined by the finite element programme HYST (Foschi, 2001). Similar to other simulation-based models (Pang et al., 2009; Pei and van de Lindt, 2010), this model relies on parameter calibration for further application, while it is relatively accurate in describing the details of the pinching behaviour due to its complex modelling techniques resulted from HYST. Compared with those purely analytical models, such as He et al. (2001), this model is computationally efficient, reasonably accurate and suitable for all discussed structural configurations. The accuracy of this model may be partially shown by the comparison in Figure 4 and an earlier work (Gu, 2006), which is essential to produce reliable results in the analysis. This model was successfully implemented in ABAQUS to model timber–steel hybrid structures by other researchers (Li et al., 2014).

It is worth noting that the model input parameters in Table 1 do not necessarily relate to the mechanical properties of wood species used to construct the frames of the walls. As a simulation-based model, these parameters are obtained from optimization algorithms and do not have particular physical meanings. There may be multiple sets of parameters (or multiple pseudo nail models) corresponding to the same wall test results. Table 1 only lists the best results that are not very sensitive to the hysteresis behaviour. It is reasonable to assume that a stable solution of these parameters do not significantly change the outcome of analysis, as long as the experimental results can be reasonably fitted. It should be mentioned that different experimental results may significantly affect these parameters, which may significantly change the results of performance analysis.

Reliability methods

Demand and limit state functions

The seismic loads are believed to be among the main sources of uncertainty in structural reliability analysis under seismic loadings. It is well recognized that the earthquake hazard is typically represented by a suite of ground motion records and intensity measure. The uncertainty from the ground motion records and intensity measure is typically assumed to be independent. Since the ground motion records are assumed to be equally important, they can be viewed as discrete uniform samples of their representing probability domain. If R denotes the random variable of ground motion records, r, the probability density function of records, $f_{R} (r_{j})$ , equals to $1 / N$ . Here, N = total number of selected ground motion records and j = jth record.

Since the demand, $d$ , is a function, $g (\cdot)$ , of the ground motion records, $r$ , and intensity measure, $im$ , the random variable for drift demand, D, follows a bivariate distribution about ground motion records, R, and intensity measure, IM. The distribution function of drift demand can be expanded to incorporate other random variables, such as seismic weight and material properties. Figure 5 illustrates the concept of the density function of drift demand. Depending on the nature of the density function for intensity, the shape of all curves in this figure may vary. All curves are supposed to have the same shape, since all ground motion records are assumed to contribute equally and thus scaled to have the same density function, $f_{IM} (\cdot)$ .

Figure 5.

Probability density function of drift demand.

The probability of structural failure, $P_{f}$ , can be expressed as

P_{f} = \int_{0}^{+ \infty} f_{C} (y) [1 - \int \int_{g (r, x) ⩽ y} f_{R} (r) f_{IM} (x) drdx] dy

(6)

where $f_{C} (\cdot) = density function of drift capacity, C$ ; x = dummy variable for intensity measure with its domain subjected to the function, $g (\cdot)$ ; and y = dummy variable for demand with its domain from zero to infinite.

Theoretically, the double integral of drift demand as specified in equation (6) may be further expressed as two simple integrals under some special circumstances. Since the results from NTHA are discrete and typically nonlinear, it is difficult, if not impossible, to simplify equation (6) with multiple simple integrals. If other uncertainty sources are considered, it is almost impossible to use the multivariate distribution of drift demand to find a closed-form solution for nonlinear structures. Following methods may be implemented to solve the probability of failure shown in equation (6).

Method 1: the traditional seismic reliability method

This method was used by the SAC Steel Moment Frame project to study probabilistic behaviour (Cornell et al., 2002). With this method, the exceeding probability of drift demand is calculated from the conditional distributions for given intensity levels, shown as

F_{D} (d) = P (D ⩽ d) \approx \int_{0}^{+ \infty} P (D ⩽ d | IM = x) f_{IM} (x) dx

(7)

where P(D ≤ d|IM=x) = conditional cumulative distribution function (CDF) of drift demand, D, not exceeding the value d, given the intensity level of $IM = x$ . Therefore, the probability of failure can be expressed as

P_{f} \approx \int_{0}^{+ \infty} f_{C} (y) [1 - \int_{0}^{+ \infty} P (D ⩽ y | IM = x) f_{IM} (x) dx] dy

(8)

Due to the nature of drift demand as discussed above, a closed-form solution to equation (7) or (8) is generally difficult. From numerical analysis, a numerical format of equation (8) can be expressed as

P_{f} \approx \int_{0}^{+ \infty} f_{C} (y) {1 - \sum_{i = 1}^{M} P (D ⩽ y | I M = x_{i}) [F_{I M} (x_{i + 1}) - F_{I M} (x_{i - 1})] / 2} d y

(9)

where M = number of intensity levels. In equation (9), the exceeding probability, $F_{IM} (x_{i})$ , is used to determine the uncertainty of intensity measure. The traditional method shown in equation (9) is independent of any sampling strategy of intensity measure, which is convenient for calculation. However, in order to determine the conditional distribution, $P (D ⩽ d | IM = x_{i})$ , all ground motion records are required to be scaled to the same intensity level, $x_{i}$ . Moreover, the number of selected ground motion records has to be sufficient to generate the conditional distributions from ranking. If the intensity measure is sampled uniformly in its distribution domain, the incremental cumulative probability, $[F_{IM} (x_{i + 1}) - F_{IM} (x_{i - 1})] / 2$ , will be close to 1/M. Thus, the calculation of $F_{IM} (x_{i})$ in equation (9) can be avoided.

Method 2: a method based on conditional distributions at given ground motion records

The joint probability distribution of demand deals with the uncertainty from ground motion records and intensity measure. As mentioned above, the traditional reliability method uses conditional distributions at given intensity measure. By switching the random variable, the distribution of drift demand may be determined from the conditional probability density function of intensity measure given at ground motion records, shown as

F_{D} (d) = P (D ⩽ d) \approx \int_{0}^{+ \infty} P (D ⩽ d | R = r) f_{R} (r) dr

(10)

where P(D ≤ d|R = r) = conditional CDF of drift demand not exceeding the value d, given the ground motion record of $R = r$ . The integral sign in equation (10) is based on a continuous random variable for ground motion records. For engineering analysis, the ground motion records cannot be continuous. Therefore, a practical form of equation (10) is a numerical format, shown as

F_{D} (d) = P (D ⩽ d) \approx \sum_{j = 1}^{N} P (D ⩽ d | R = r_{j}) f_{R} (r_{j}) Δ r_{j}

(11)

where j = jth ground motion record.

Since the selected ground motion records are uniformly distributed in the representing probability distribution domain, each record contributes equally to the conditional distribution of drift demand. With reference to Figure 5, the values of the probability density function, $f_{R} (r_{j})$ , for all records are the same. The intervals between adjacent ground motion records, $Δ r_{j}$ , are also the same. Therefore, the drift demand in equation (11) can be rewritten as

F_{D} (d) \approx \frac{1}{N} \sum_{j = 1}^{N} P (D ⩽ d | R = r_{j})

(12)

With equation (12), the probability of failure of structures can be determined as

P_{f} = P (C < D) = \int_{0}^{+ \infty} f_{C} (y) [1 - \frac{1}{N} \sum_{j = 1}^{N} P (D ⩽ y | R = r_{j})] dy

(13)

The probability of failure shown in equation (13) has several features compared with the traditional method (equation (9)). Equation (13) uses the mean of the conditional exceeding probabilities of demand over the selected ground motion records, which is convenient and accurate when only limited records are available. The calculation of the conditional distribution for each record is independent of other records, so that some records can be added or deleted from analysis without affecting the results for other records. The sampling strategy for different records is flexible, which is convenient to construct different incremental dynamic analysis (IDA) curves. Considering different structural nonlinearities for different records, flexible sampling strategies are useful for some large-scale problems.

The conditional probability function at given ground motion records, $P (D ⩽ y | R = r_{j})$ , is calculated from a single IDA curve for the jth record. The uncertainty of the exceeding probability comes from intensity measure. It should be mentioned that one does not have to rank the results of drift demand to produce the conditional exceeding probability. Instead, a mapping procedure can be implemented to determine the conditional exceeding probability, as shown in Figure 6. In this figure, the relationship between drift demand and intensity measure with two IDA curves is shown on the right, while the relationship between intensity measure and the conditional probability distribution is shown on the left. The calculation of the probability of drift demand not exceeding value y given the jth ground motion record, $P (D ⩽ y | R = r_{j})$ , starts from the drift on the right side of the abscissa. With the jth IDA curve for the record $r_{j}$ , the corresponding intensity can be determined from the ordinate. Using the cumulative distribution curve of the intensity on the left side, the corresponding cumulative probability at this intensity level can be determined from the left side of the abscissa. This cumulative probability is the conditional exceeding probability $P (D ⩽ y | R = r_{j})$ in equation (13). This mapping procedure can be programmed using appropriate numerical methods, such as the linear interpolation.

Figure 6.

Mapping relationship between drift demand, intensity and conditional distribution.

The mapping procedure shown in Figure 6 can be expressed mathematically. If the function for drift demand, $g (\cdot)$ , has its inverse form, $im = g^{- 1} (d, r)$ , the conditional distribution in Figure 6 can be expressed as

\begin{array}{l} P (D ⩽ y | R = r_{j}) = \int_{0}^{y} f_{D | r_{j}} (z) d z \\ = \int_{0}^{y} f_{I M} [d^{- 1} (z, r_{j})] | \frac{\partial g^{- 1} (z, r_{j})}{\partial z} | d z \end{array}

(14)

where $f_{D | r_{j}} (z) = density function of P (D ⩽ z | R = r_{j})$ .

Since the function of drift demand, $g (\cdot)$ , is typically discrete and may not be able to be expressed in a closed form, the partial derivative of the inverse function for drift demand in equation (14) only helps to understand Figure 6. In engineering, one still has to use the mapping relationship shown in Figure 6.

Method 3: the MCS method

The seismic demand of structures follows a bivariate distribution with respect to ground motion records and intensity measure. Both the traditional reliability method and the second method are approximate solutions based on the conditional probability distributions. The accuracy of either method depends on the nature of structural nonlinearity and is to be verified by further applications in reliability analysis.

If computational efficiency is not a problem, the MCS and its variations may be implemented to generate a relatively accurate result. This is based on the fact that the selected ground motion records are natural samples from their representing probability distributions. The samples of all other random variables have to follow their own probability distributions. Using the combination of the samples for all random variables, repeatable NTHA can be performed to obtain seismic demand. Then all demand values are ranked together to generate the cumulative probability distribution, with which the probability of structural failure can be calculated.

The MCS procedure can solve equation (6) directly without any approximation or data fitting to other distribution types. In order to reduce the computational time of NTHA, some strategies including the sequence samples (Mckay et al., 1979) were employed in this study.

Confidence curves from IDA

Riddell and Newmark (1979) used 84% cumulative probability (i.e. one sigma) curves over an ensemble of earthquake records to construct the linear design spectra. For nonlinear problems, similar confidence curves can be constructed to simplify the randomness of the selected earthquake records so that comparisons between different configurations can be visually shown. Confidence curves can be viewed as a simplified relationship among records, intensity and demand from IDA. The confidence curves can be constructed either at given levels of intensity (Figure 7(a)) or at given levels of drift demand (Figure 7(b)). In Figure 7(a), the probability distribution of drift demand given an intensity level can determine a point representing a certain magnitude of probability of exceedance, for instance, 84%. At another intensity level, the probability distribution of drift demand will be different, which therefore defines another point at the same probability of exceedance. Linking the points from different intensity levels establishes a curve representing the exceeding probability of 84% at any intensity level over all selected records. This curve is noted as an intensity-based confidence curve. Figure 7(a) also shows another 50% confidence curve.

Figure 7.

Confidence curves constructed with different methods: (a) from given intensity levels and (b) from given drift demand.

Alternatively, at a given level of drift demand, the intensity levels corresponding to selected records can be ranked to establish conditional distributions. With this distribution, a point can be determined at certain magnitude of exceeding probability. The points at different levels of drift demand can be linked together to construct a confidence curve. This confidence curve is noted as a demand-based confidence curve (Figure 7(b)). Another 50% confidence curve is also shown in this figure.

Confidence curves present a simplified relationship between drift demand and intensity levels. If other random variables are considered, the concept of confidence curves can be further employed to simplify their relationships. These confidence curves can be used to visually compare the behaviour of different systems or structures under seismic loading.

Results of reliability analysis and discussion

A suite of 10 ground motion records were used in the analysis (Table 2). These records were recommended either by Building Center of Japan or other researchers in this field. They were scaled to multiple levels of intensity measure as determined by its probability distribution discussed below, while these records are assigned with an equal weight. The viscous damping ratio is assumed to be 1% of the critical. Peak ground acceleration (PGA) was used as the intensity measure and assumed to follow a lognormal distribution with a mean of 0.25 g and a coefficient of variation of 0.55. According to Zhang and Foschi (2004), this statistics is consistent with a site design acceleration of 0.717 g (corresponding to a return period of 475 years) and a mean arrival rate of earthquakes of 0.2 (average of one every 5 years). This type of PGA was used to study shear wall performance on the west coast of Canada by Zhang and Foschi. In this study, only the uncertainty from ground motion records and intensity were considered. They were assumed to be uncorrelated. Other sources of uncertainty were not considered here. NTHA were carried out at three levels of effective seismic weight to study the potential impact: 20, 30 and 40 kN. The drift limit was chosen to be 2.5% of the wall height. With the three methods discussed above, the probability of failure $(P_{f})$ was calculated and is shown in Tables 3 to 5. The numbers in the brackets show the corresponding reliability indices, defined as $Φ^{- 1} (P_{f})$ , where $Φ (\cdot)$ is the standard normal distribution.

Table 2.

Ground motion records.

Event	Year	Direction	PGA (g)	Station
Imperial Valley	1940	EW	0.214	El Centro
Kern County	1952	EW	0.180	Taft Lincoln School Tunnel
Tokyo	1956	NS	0.0755	Tokyo 101
Northern Miyagi Prefecture	1962	EW	0.0485	Sendai 501
Headland of Echizen	1963	EW	0.0255	Osaka 205
Tokachi	1968	EW	0.187	Hachinohe
Miyagi-ken oki	1978	NS	0.263	Tohoku
Lander	1992	EW	0.284	Joshua Tree
Northridge	1994	NS	0.416	Beverly Hills 14145
Kobe	1995	NS	0.243	Shin-Osaka

PGA: peak ground acceleration.

Table 3.

Probability of failure and reliability indices at 20 kN seismic weight.

	Method 1	Method 2	Method 3
SG-BR	7.97% (1.41)	9.24% (1.33)	5.57% (1.59)
SG-PL	3.23% (1.85)	3.53% (1.81)	3.45% (1.82)
SG-OS	5.57% (1.59)	5.35% (1.61)	2.69% (1.93)
HF-BR	1.32% (2.22)	1.39% (2.20)	1.20% (2.26)
HF-PL	1.25% (2.24)	1.40% (2.20)	0.66% (2.48)
HF-OS	2.46% (1.97)	3.15% (1.86)	2.12% (2.03)
WW-BR	5.22% (1.62)	5.54% (1.60)	3.18% (1.86)
WW-PL	1.92% (2.07)	2.43% (1.97)	1.06% (2.30)

SG: sugi; HF: hemlock; WW: whitewood; BR: braced frames; PL: plywood-sheathed walls; OS: oriented strand board–sheathed walls.

Table 4.

Probability of failure and reliability indices at 30 kN seismic weight.

	Method 1	Method 2	Method 3
SG-BR	22.9% (0.743)	25.8% (0.648)	24.5% (0.690)
SG-PL	13.8% (1.09)	14.9% (1.04)	12.9% (1.13)
SG-OS	14.8% (1.05)	15.5% (1.02)	13.1% (1.12)
HF-BR	7.16% (1.46)	7.93% (1.41)	6.12% (1.55)
HF-PL	5.19% (1.63)	5.87% (1.57)	4.54% (1.69)
HF-OS	10.3% (1.26)	11.1% (1.22)	5.17% (1.63)
WW-BR	12.6% (1.14)	14.3% (1.07)	11.3% (1.21)
WW-PL	9.24% (1.33)	9.59% (1.31)	7.02% (1.47)

SG: sugi; HF: hemlock; WW: whitewood; BR: braced frames; PL: plywood-sheathed walls; OS: oriented strand board–sheathed walls.

Table 5.

Probability of failure and reliability indices at 40 kN seismic weight.

	Method 1	Method 2	Method 3
SG-BR	36.7% (0.339)	37.1% (0.328)	29.4% (0.541)
SG-PL	22.4% (0.759)	22.9% (0.743)	18.3% (0.903)
SG-OS	25.1% (0.670)	26.2% (0.636)	21.2% (0.798)
HF-BR	16.1% (0.991)	17.8% (0.922)	13.5% (1.11)
HF-PL	11.6% (1.197)	12.9% (1.13)	10.8% (1.24)
HF-OS	19.5% (0.861)	20.4% (0.828)	17.8% (0.922)
WW-BR	21.3% (0.795)	22.6% (0.752)	18.0% (0.915)
WW-PL	18.6% (0.891)	19.0% (0.878)	14.9% (1.04)

SG: sugi; HF: hemlock; WW: whitewood; BR: braced frames; PL: plywood-sheathed walls; OS: oriented strand board–sheathed walls.

Tables 3 to 5 indicate that Methods 1 and 2 produce similar results compared with Method 3. The differences among three methods appear to become smaller when the seismic weight is increased from 20 to 40 kN. At 20 kN seismic weight, the difference of $P_{f}$ between Methods 1 and 2 is up to 11%, while the difference of $P_{f}$ between Methods 2 and 3 is up to 36%. At 40 kN seismic weight, the difference of $P_{f}$ between Methods 1 and 2 is up to 5%, while the difference of $P_{f}$ between Methods 2 and 3 is up to 19%. These differences may be explained by the accumulation of errors in determining the exceeding probability from the conditional distributions. These errors become more significant at a lower exceeding probability (at 20 kN seismic weight) when very few data points (or samples) exceed the limit state. With more samples exceeding the limit state at higher exceeding probability (at 40 kN seismic weight), these errors become smaller and results are closer to those from Method 3 (the MCS). The MCS only uses one distribution curve with many samples exceeding the limit state and thus may produce relatively accurate results.

It appears that plywood-sheathed walls and OSB-sheathed walls exhibit similar $P_{f}$ or reliability indices. The results also indicate that panel-sheathed walls made with hemlock and whitewood can achieve reliability indices of 2.0 (a $P_{f}$ of 2.1%) at 20 kN seismic weight. At 30 kN seismic weight, the reliability indices for the panel-sheathed walls with either hemlock or whitewood is about 1.5 (a $P_{f}$ of 7%). At 40 kN seismic weight, the reliability indices of these walls decrease to about 1.0 (a $P_{f}$ of 15%). Braced frames appear to be a relatively weak structural form, which may be explained by their brittle post-failure behaviour. The walls and braced frames made with sugi exhibit worse performance than whitewood or hemlock, which may be explained by the relatively weak nail holding strength of sugi, as observed from the results of the relative density. The $P_{f}$ of the braced frames with the same species increases dramatically with the increase in the seismic weight compared with sheathed walls, which indicates that braced frames may be not as reliable as sheathed walls. The probability of failure shown in these tables can be used to validate the ductility factor and shear design strength in codes and standards and mitigate potential losses in future events.

The difference of probabilistic behaviour among different wood species with the same structural form can be further illustrated by comparing 84% confidence curves, as shown in Figure 8. This figure only shows the results with the effective seismic weight of 40 kN. It was found that both the intensity-based confidence curves and the demand-based confidence curves are very similar (Gu, 2006). Only the results from the intensity-based confidence curves are shown in Figure 8. It is clear that, at the same intensity level, the drift demand of the systems made with hemlock is smaller than that with whitewood or sugi. The results at other levels of seismic weight produce similar conclusions.

Figure 8.

Comparison of 84% confidence curves for seismic weight of 40 kN: (a) braced frames at 40 kN, (b) plywood-sheathed walls at 40 kN and (c) OSB-sheathed walls at 40 kN.

Conclusion

Eight types of shear walls and braced frames were studied to quantify their probabilistic behaviour under seismic loading. Three species were examined: Japanese sugi, Canadian hemlock and European whitewood. Three methods were used in the analysis: a traditional method based on conditional distributions at given intensity, a new method based on conditional distributions at given records and the MCS. The results from three methods are generally consistent, although the probability of failure from the first and second methods appears to be slightly larger than the third method. The results of probability failure and confidence curves were compared for the performance of the eight types of walls and frames. In terms of structural form, the results of reliability analysis show that plywood- and OSB-sheathed walls exhibit similar probability of failure or reliability indices. The probability of failure of the braced frames at varying levels of seismic weight shows that the performance of such type of frames may not be very reliable compared with sheathed walls. This conclusion appears to be consistent with the observation from test results and engineering practice. But it is based on limited test data and needs to be further verified once further experimental data are available.

In terms of wood species, the panel-sheathed walls made with whitewood and hemlock exhibit similar probabilistic seismic behaviour, which have reliability indices of about 2.0 at 20 kN seismic weight and 1.5 at 30 kN seismic weight. The sheathed walls made with sugi have lower reliability indices compared with other two species at the same seismic weights. The results of this study can be further used to validate the ductility factor and shear design strength towards code improvement and mitigate potential losses in future events.

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) received no financial support for the research, authorship and/or publication of this article.

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