Introduction and Literature ReviewTragic events resulting in loss of life, such as targeted attacks on buildings and accidental explosions, have highlighted the need for developing reliable and robust design provisions to address the behavior of structural elements subjected to blast loading.Globally, an increasing number of timber structures are being built, with primary emphasis on sustainability, cost efficiency, and safety. Common structural forms include low- to mid-rise light-frame structures, typical of residential buildings, as well as mid- to high-rise heavy- and mass-timber structures, commonly composed of engineered wood products (EWPs), such as glued laminated timber (glulam) and cross-laminated timber (CLT). Examples of such structures include the Richmond Olympic Oval in Richmond, British Columbia, Canada, the Tianning Temple in Changzhou City, China, and the UMass Design Building in Amherst, Massachusetts. Although these structures are recognized for their importance and novelty, they tend to carry a higher risk of exposure to the potential effects of explosions.Although building design codes do not explicitly require blast loads to be considered in structural design, the need for structural detailing that would ensure adequate performance of elements and assemblies when subjected to blast loading has led to the development of various analysis and design guidelines (e.g., ASCE 2011; CSA 2012; Office of the Deputy Prime Minister 2004; PDC 2008; UFC 2008, 2009). In Canada, a detailed and relatively recent design standard has been developed with the main objective to prevent human casualties and progressive collapse, limit damage level, and minimize flying debris (CSA 2012).The design provisions included in this standard for timber structures are based on limited testing conducted on small clear-wood specimens (e.g., Jansson 1992; Spencer 1978; Sukontasukkul et al. 2000) and qualitative experimental testing, whereby small light-frame wood shelters were subjected to live explosions (Marchand 2002). Extending the work performed on light-frame wood stud walls, quantitative studies have been performed using actual explosions (Parlin et al. 2014; Syron et al. 2010), impact loading (Starr and Krauthammer 2005, 2006, 2007), and simulated farfield explosions (Jacques et al. 2014; Lacroix and Doudak 2015b; Lacroix et al. 2014; Viau and Doudak 2016a; Viau et al. 2016). The blast behavior of glulam and CLT members has also been investigated and quantified using a shock tube apparatus (Lacroix and Doudak 2018a, b; Poulin et al. 2018) as well as live explosives (Oswald 2018; Sanborn et al. 2019; Weaver et al. 2017, 2018).Studies investigating the effect of realistic boundary connections in light-frame (Viau and Doudak 2016b), glulam (McGrath and Doudak 2021; Viau and Doudak 2021a, c), and CLT assemblies (Côté and Doudak 2019; Viau and Doudak 2019, 2021c) have reported that such boundary connections represent a source of inelastic energy dissipation that may provide improved blast resiliency. Use of fiber-reinforced polymers (FRPs) on wood assemblies—with the overarching goal to engage wood in compression, bridge natural defects in wood, and provide additional ductility in cases where brittle failure dominates the behavior—has also been investigated (Lacroix and Doudak 2020; Lacroix et al. 2021; Lacroix and Doudak 2018c; Lopez-Molina and Doudak 2019).There is limited guidance in the literature regarding appropriate design and modeling methods for timber members (Lacroix and Doudak 2015a; Lacroix et al. 2018; Viau et al. 2018, 2016); however, no complete and systematic design approach has been proposed. This frames the overarching objective of this paper, to develop and validate design methods for blast-loaded timber elements and assemblies. Additionally, full-scale static and dynamic experimental testing was conducted as part of this study to augment published data, thereby helping provide specific design parameters and modeling methodologies. The developed and validated design method was also used to evaluate the adequacy of current blast design provisions through comparison with the proposed method and experimental test results. It should be noted that the scope of the current paper—including recommendations and observations on failure modes, high strain-rate effects, etc.—are limited to farfield blast explosions, whereby the load acting on structural elements can be assumed uniform. Design of timber members subjected to close-in and contact explosions, which tend to produce nonuniform and localized damage, are outside the scope of this study.Presentation of Proposed Design MethodsGeneralThe following sections present the proposed design methods for light-frame wood stud walls, glulam members, and CLT panels, subjected to out-of-plane pressure generated by blast loading. Common design considerations for all components and assemblies are presented next.The load duration factor (KD), found in the current Canadian engineering wood design standard, CSA O86 (CSA 2019), is based on work by Foschi et al. (1989), which investigated the effect of long-term and transient loads on wood members. When the member strength is normalized relative to that obtained during “short-term” static testing, the value of the load duration factor for “standard” and “permanent” loading is taken less than unity. For example, the load duration factor for standard load is taken equal to 0.8. For simplicity, the O86 standard assigns a value of 1.0 to the standard term load duration, likely because it represents the most common design situation, resulting in a short-term load duration value of 1.25 (or 1/0.8). However, this value is conservatively taken as 1.15 in the Canadian standard, given that this category of load duration also covers construction loads lasting up to 7 days (e.g., form work). The Canadian blast design standard, CSA S850 (CSA 2012), requires that a load duration factor of unity be assumed when used concurrently with the dynamic increase factor (DIF). Although this approach is meant to be conservative for the design of the load-bearing member itself, it may not be adequate when the intent is to capacity protect a component or when the accurate prediction of the actual load-bearing capacity is required. In such situations, using a KD factor of 1.25 would be recommended.Another deviation in the design provisions between the Canadian wood design standard CSA O86 and the blast design standard CSA S850 is related to the material resistance factor, ϕ. This factor, for example, for bending members, is taken as 0.9 for static design, whereas a value of unity is used for the purpose of blast design.The value for DIF is currently set in the Canadian blast design standard, CSA S850, at a constant value of 1.4 for all wood products. Recent studies have shown that this value is appropriate for solid sawn lumber elements (e.g., light-frame studs), while engineered wood products tend to experience less dynamic increase, in part due to the presence of defects, finger joints, and transverse laminates. Due to this discrepancy, the DIFs used in the proposed methodology is based on published research results specific to the members for which the provisions are being presented.Values for the strength increase factors (SIFs) provided in the blast design standard (e.g., CSA S850) typically underestimate the actual average strength to ensure conservative design. However, more accurate and representative predictions can be obtained if information from tests on related products are available. This is, again, particularly useful when predictions are evaluated against test results or when capacity-based design considerations are utilized in design of, for example, connections.Light-Frame Wood Stud WallsBackgroundAs mentioned earlier, a DIF of 1.4 has been developed based on comparison of dynamic strength at strain rates ranging from 1.0×10−2 to 5.5×10−1 s−1, relative to those obtained during static testing (ε˙ < 10−3 s−1) (Jacques et al. 2014; Lacroix and Doudak 2015b). The current Canadian blast design standard specifies that a factor of unity should be assumed for the load-sharing factor, KH, when determining the wall strength (CSA 2012). This approach has been found to be too conservative (Lacroix and Doudak 2013, 2015a). As an alternative to using the load-sharing factor, a designer can use partial composite action (PCA) provided by the semirigid fastener between the sheathing and the framing. The resistance curves used as input into the equation of motion can be modeled based on a representative T-beam, composed of the stud (i.e., the web) and a sheathing panel (i.e., the flange) with a width equal to the stud spacing and with partial connection rigidity. McCutcheon (1977, 1984) developed a method that transforms the semirigid section into an equivalent rigid section. Information on the interlayer slip of the connection is required and can be determined experimentally or obtained from the connection manufacturer. This model was originally developed for light-frame wood floors with only out-of-plane loading, but was expanded to consider the combined state of stress generated from axial and transverse loading (Bulleit et al. 2005; Rosowsky et al. 2005).Although the methodology outlined in McCutcheon (1977, 1984) provides a simple and reasonable approach for implementing the effects of PCA, it should be noted that the method was originally derived for static loading, where it is generally assumed that the sheathing and studs act in unison. The model does not account for the localized bending of the sheathing panel occurring between the studs caused by a dynamic response to a shock wave, as reported in Lacroix and Doudak (2015b) and Viau and Doudak (2016a). It also does not account for the influence of the sheathing deformation on the overall dynamic response of the wall, nor does it account for the time-dependent difference in response period between the sheathing and the studs. To circumvent these limitations, it is essential that the sheathing panels selected for a design be of adequate out-of-plane strength and stiffness, to ensure that the assumptions of the T-beam modeling method are valid (Lacroix et al. 2014; Viau and Doudak 2016a). More robust models, including finite-element or other high-fidelity modeling analysis, may be required to adequately capture the complex interaction among the sheathing panels, studs, and the fasteners connecting them. Examples of such models have been developed and validated primarily for assemblies subjected to static loading (e.g., Foschi 1982; Polensek 1976; Thompson et al. 1977; Wheat et al. 1983). For the majority of design situations, it is submitted that the proposed T-beam model provides a reasonable balance between simplicity and adequacy in predicting the dynamic response of stud walls subjected to blast loads (Lacroix and Doudak 2015b; Viau and Doudak 2016b; Viau et al. 2016).A detailed assessment of the behavior of 33 full-scale light-frame wood stud walls subjected to a total of 48 shots of simulated blast loading was presented by Viau et al. (2016). Based on the results, a maximum ductility ratio of 2.0 was determined to be more representative of the ultimate state of failure (i.e., blowout) for light-frame wood stud walls than the maximum ductility ratio of 4.0 currently provided in the Canadian blast design standard (CSA 2012). The study also emphasized that the failure of a single stud is not expected to cause hazardous risk to the occupants, and therefore should not be considered to be representative of the ultimate resistance of the wall assembly.Proposed Design Method for Light-Frame Wood Stud WallsA bilinear resistance curve was proposed to capture the behavior of stud walls, particularly the interaction between the sheathing and the studs. The elastic region is characterized by the initial stiffness (K), until the peak resistance (Rpeak), which occurs at the elastic limit (xe). This is followed by a perfectly plastic plateau, which is limited by the maximum midspan deflection equal to twice the elastic limit (2xe). A normalized resistance curve is presented in Fig. 1.Assuming simply supported boundary conditions and uniformly distributed load, the peak resistance, Rpeak, can be calculated using the following equation: (1) where L = clear span; and Mr−D = dynamic bending resistance, defined as the static bending resistance, Mr, modified for high strain-rate effects (DIF) and reflective of in-situ values (SIF), as outlined in (2) The static bending resistance, Mr, as defined in the timber design standard CSA O86 (CSA 2019), is provided as follows: (3) where ϕ = material resistance factor; fb = specified strength in bending; KDKHKSbKT = load duration, load sharing, serviceability, and treatment factors, respectively; S = section modulus of the load-bearing member; KZb = size factor in bending; and KL = lateral stability factor.The PCA can be accounted for through the modification of the flexural rigidity considering the interlayer slip between the sheathing and the stud. The composite flexural rigidity, EIPCA, and the modified axial stiffness of the flange, EA¯f, can be determined according to Eqs. (4) and (5), respectively (McCutcheon 1986) (4) EIPCA=EIu+(EA¯f)(EAw)EA¯f+EAwh2(5) where EIu = bending stiffness of the web (i.e., stud) only; EAw = axial stiffness of the web; h = distance between the centroid of the web and flange; EAf = axial stiffness of the flange (i.e., sheathing); Sk = interlayer stiffness; and Lf = distance between the sheathing gaps along the stud length.The section modulus accounting for partial composite action, SPCA, can be obtained using the transformed area method for the composite moment of inertia; then, SPCA can be substituted into Eq. (3) to obtain the modified static bending resistance. Similarly, EIPCA can be utilized to obtain the bending stiffness of the T-section. The stiffness for a stud with simply supported end conditions subjected to uniformly distributed load can be calculated as (6) Alternatively, if experimental test data are not available, the appropriate load-sharing factor, KH, of 1.2 and 1.4 to the bending resistance of machine stress rated (MSR) and visually graded lumber, respectively, may be applied. It should be noted that this approach does not involve any modification to the flexural rigidity, and thus, the bending stiffness can be obtained using the following equation: (7) where EI = bending stiffness of the stud without PCA. This will result in a more flexible system than that obtained using the PCA method.Glued-Laminated TimberBackgroundAn experimental program investigating the behavior of 38 glulam beams of three different cross sections was undertaken and the results reported in Lacroix and Doudak (2018a). Static and dynamic reactions were measured to obtain both the modulus of elasticity (MOE) and the modulus of rupture (MOR), and the failure modes were documented. An analysis of variance (ANOVA) test was conducted and the results showed that, under static loading, there was no significant difference between the resistances of the specimens that failed at a finger joint (FJ) and those that failed in a region with no defects. When considering the dynamic test results, it was clear that specimens with continuous defects or FJ had a significantly different and lower dynamic resistance than the group with staggered or no defects. This observation is consistent with findings by Nadeau et al. (1982), who reported that clear wood specimens, which were intentionally notched on the tension side, lacked the increase observed in specimens without a notch. A DIF of 1.14 on the flexural strength of glulam beams was determined to be statistically significant only in the absence closely aligned defects and FJs in the outer tension layer. Similar analyses were conducted for the stiffness and failure strain, and the results showed no evidence of a dynamic increase (Lacroix 2017). These findings were corroborated in another experimental program of 22 glulam beams with staggered FJs (Viau and Doudak 2021a, c), and the DIF was found to be 1.10. Lacroix and Doudak (2018a) also reported that glulam beams under both static and dynamic loading showed no significant postpeak resistance, and therefore suggested that the beams be designed as linear elastic with no postpeak resistance (i.e., μ=1).The effect of varying axial load levels on the response of six columns subjected to combined axial and simulated out-of-plane blast loads was investigated by Lacroix and Doudak (2018b). The axial load was observed to decrease as the columns displaced laterally (i.e., out-of-plane). The contribution of axial loads in actual structures would highly depend on the connection detailing between the columns and the elements they support. A moment-curvature analysis procedure was developed (Lacroix and Doudak 2018b) to implement variable axial load into the resistance curve of glulam columns. The moment curvature analyses were shown to be sensitive to the ratio of tensile-to-compressive strengths of the lumber used in the glulam element.A linear resistance curve with no postpeak behavior was proposed to capture the behavior of glulam beams. The elastic region is characterized by the initial stiffness (K) up to peak resistance (Rpeak), which occurs at the elastic limit (xe). Flexural failure is assumed to govern the design, as shear does not tend to govern for typical length-to-depth ratios of glulam elements used as beams and columns. Assuming simply supported boundary conditions and uniformly distributed loading, Rpeak can be calculated using Eqs. (1) and (2). The stiffness for a glulam beam with simply supported end conditions subjected to uniformly distributed loading can be calculated using Eq. (7). When axial loading is applied to the glulam member, the resistance curve would require the use of moment-curvature analysis to capture secondary moments.Proposed Design Method for Glulam MembersThe static design parameters can be determined using the provisions in the wood design standard (CSA 2019). The bending resistance, Mr, can be expressed as follows: (8) Mr=ϕ[fb(KDKHKSbKT)]SKXmin(KL,KZbg)where KL = lateral stability factor; and KZbg = size effect factor for glulam in bending. The dynamic strength [Eq. (2)] can be determined by modifying the value obtained in Eq. (8) by an SIF of 1.2, while a DIF of 1.1 on the resistance may be applied where no continuous defects or FJs are found in the outer tension laminations (Lacroix and Doudak 2018a). However, if designers do not know or cannot ensure that this is not the case, a DIF of unity can be conservatively assumed.Cross-Laminated TimberBackgroundAn experimental program investigating the out-of-plane behavior of CLT under static and simulated blast loading of 18 panels with different panel thicknesses was undertaken by Poulin et al. (2018). A DIF ranging from 1.2 to 1.3 for the flexural strength of CLT was reported for strain rates ranging 0.02 to 0.21 s−1 (Poulin et al. 2018). In another study, an average DIF of 1.31 for the flexural strength was observed for strain rates ranging from 0.15 to 0.32 s−1 (Viau and Doudak 2019). A DIF equal to unity was recommended for stiffness when deriving the dynamic resistance curves.The CLT panels were observed to have some level of postpeak resistance, defined as a resistance following peak of at least 50% (Lacroix and Doudak 2020), under both static and dynamic loading. Based on the experimental static and dynamic test results, a maximum ductility ratio of 2.5 was deemed appropriate (Poulin et al. 2018). The initial drop in resistance observed in the CLT specimens represents the loss of the tension longitudinal and transversal laminates. For the 5-ply specimens, which consist of three longitudinal and two transverse layers, this failure mechanism implies that the specimen would now behave as a 3-ply specimen, consisting of two longitudinal and one transverse layers with little to no damage. The level of postpeak resistance for the panels, once all primary longitudinal layers had failed, corresponded to approximately 20% of peak resistance (Poulin et al. 2018). The authors noted that the 5-ply specimens tended to experience significant rolling shear failure prior to flexural failure and, in some instances, would ultimately fail in rolling shear.In general, the proposed resistance curve for CLT panel will differ as function of the number of plies of which it is composed. Based on experimentally obtained resistance curves, Viau et al. (2018) proposed a generalized approach for constructing the resistance curve for flexure failures. The flexural behavior can be described as initially linear elastic, after which the loss of the longitudinal tension laminates would cause a sudden drop in resistance. For an n-ply CLT panel whose laminas follow an alternating pattern, the loss of the outermost ply would cause the panel to behave as an n-2 ply panel. Further loading will result in progressive loss of longitudinal layers, until only the innermost longitudinal laminates exist, which can be modeled as a sudden drop in resistance to a value equal to 20% of the ultimate resistance, based on Poulin et al. (2018).Quantification of Rolling Shear Behavior at High Strain RatesWhile the flexural behavior of CLT had been documented in recent studies, the effect of high strain rates on rolling shear failure has only been qualitatively assessed. Design parameters, such as SIF and DIF for rolling shear failure, have yet to be established. The primary implication of this relates to the governing failure mode of CLT panels, where an assumed failure mode (i.e., flexural) may not actually govern the panel capacity in certain design scenarios.To establish the needed values for determination of dynamic rolling shear strength, four 7-ply CLT panels from the same manufacturer as those tested by the two aforementioned studies (Poulin et al. 2018; Viau and Doudak 2019) were tested statically and dynamically under four-point bending and simply supported boundary conditions. The panels were of E1 grade, measuring 300 mm in width, 245 mm in depth, and with a clear span of 2,235 mm, correlating to a span-to-depth ratio of 9.1, to promote rolling shear failure. Fig. 2 shows representative specimens in both the static and dynamic test setups. For the static testing, loading was applied at the third points using a hydraulic jack, which was connected to a load cell. Midspan deflections were measured using a wire gauge attached to the tension face of the specimens, while the midspan strains on the tension and compression face were measured using strain gauges. The dynamic tests were conducted using the University of Ottawa shock tube, a test facility allowing the simulation of farfield blast explosions without the need for live explosives. The shock tube is capable of generating up to 100 kPa of peak reflected pressure, lasting over a positive phase duration of up to 70 ms. A load-transfer device (LTD) was used to convert the pressures generated from the shock tube to two point loads, to replicate the loading used during static testing. Dynamic piezoelectric pressure sensors, load cells, strain gauges, and linear-variable differential transformers (LVDTs) were used to measure the reflected pressures, reaction forces, strain deformations, and deformations at a sampling rate of 2,000 samples/s. The average static and dynamic strain rates were measured to be 9.56 E−6 s−1 and 3.22 E−1 s−1, respectively.In both static and dynamic specimens, rolling shear failure was initiated in the transverse layers, followed ultimately by flexural failure of the outer laminates (Fig. 3). An example of the response of a 7-ply CLT panel to a blast load is shown in Fig. 4. As the shock wave arrives and strikes the LTD [Fig. 4(a)], the specimen will deflect out-of-plane as it is loaded by the reflected pressure. As shown in Fig. 4(b) for specimen CLT7.3, pronounced rolling shear damage could be observed to occur near the load application points, where shear stresses are highest. Localized failure that initiated in the tension laminates [Fig. 4(c)] would ultimately lead to a flexural failure of the panel, accompanied with significant rolling shear damage [Fig. 4(d)].Table 1 summarizes the results from the experimental tests, which includes the peak resistance (Rpeak), ultimate displacement when complete failure of the panel is achieved (Δmax), initial stiffness based on experimental load resistance curve from 10% to 40% (Ki), and average strain rate (ϵ˙).Table 1. Summary of static and dynamic test results for 7-ply CLT panelsTable 1. Summary of static and dynamic test results for 7-ply CLT panelsLoading regimeSpecimenRpeak (kN)Δmax (mm)Ki (kN/mm)ϵ˙ (s−1)StaticCLT7.1118.958.77.21.02×10−5CLT7.2126.96.36.199.92×10−6Average121.5188.8.131.52×10−6DynamicCLT7.3139.086.47.13.07×10−1CLT7.4147.7184.108.40.206×10−1Average143.4220.127.116.11×10−1Comparing the static test results to the design-level strength, an experimental SIF of 2.0 was observed for rolling shear strength. Similarly, by comparing the average static and dynamic resistances of the CLT panels tested as part of this study, an apparent experimental DIF of 1.2 was observed. As such, an SIF of 2.0, as well as a DIF of 1.2, is deemed appropriate when the dynamic rolling shear strength of a CLT panel is determined.Proposed Design Method for CLT PanelsDue to the different orientation and grade of the laminates of the CLT panels, the dynamic resistance will depend on the shear and flexural strengths. For a simply supported CLT panel under uniformly distributed loading, the dynamic resistance of a CLT panel, equivalent to the maximum total load applied prior to failure, can be calculated as (9) Rpeak=min(2Vr−D,8Mr−DL)where Vr−D = dynamic shear resistance; Mr−D = dynamic bending resistance; and L = clear span of the CLT panel. The static shear capacity, Vr, can be calculated based on the wood design standard for the major strength axis (CSA 2019), as in (10) where fs = specified rolling shear strength; and Ag = gross cross-sectional area of the panel for the major strength axis. Based on the experimental results reported in Table 1, the dynamic shear resistance, Vr−D, is obtained by modifying the value calculated in Eq. (10) by an SIF of 2.0 and a DIF of 1.2.The static bending strength can be expressed according to the wood design standard for the major strength axis (CSA O86) as follows: (11) Mr=ϕ[fb(KDKHKSbKT)]SeffKrbwhere Seff = effective section modulus of the CLT member; and Krb = calibration factor (0.85 in the major strength axis). To obtain the dynamic flexural strength of a CLT member, the static strength is modified by an SIF of 1.2 and a DIF of 1.2.The flexural stiffness of the CLT panels under uniformly distributed load cannot be directly determined using Eq. (7), since the stiffness needs to be modified to account for shear deflections. The CSA O86 standard provides an equation to determine the out-of-plane deflection of CLT panels subjected to a uniformly distributed load, which accounts for the effective bending stiffness and in-plane shear rigidity, as in (12) K=15384*L3(EI)eff+18*L(GA)effwhere L = clear span; (EI)eff = effective bending stiffness; and (GA)eff = effective in-plane (planar) shear rigidity.A maximum ductility ratio of 2.5 is proposed based on the observations made in the aforementioned studies on CLT (Poulin et al. 2018; Viau and Doudak 2019). For a simply supported panel under uniformly distributed load, the maximum displacement can be expressed as (13) Fig. 5 shows representative resistance curves for 3-ply, 5-ply, and 7-ply CLT panels.DiscussionValidation of Single-Degree-of-Freedom Modeling MethodPrior to the verification of the proposed design methods, a validation of the modeling approach was undertaken. Blast events involve loading durations significantly smaller than those observed during earthquakes and wind loading. The interaction between the shockwave and the structure lasts only a fraction of a second, and generates inertial forces and kinetic energies that must be accounted for in analyses. Typically, minimizing computational cost while obtaining accurate results presents one of the main challenges in blast research. Due to the uncertainties associated with blast design, designers typically resort to the equivalent single-degree-of-freedom (SDOF) method. Various official publications outlining the methodology are available in the literature (e.g., Biggs 1964; Crawford et al. 1974; PDC 2008; UFC 2008; USACE 1957). The method consists of transforming an actual structure with distributed properties into an equivalent system, which can be represented by an SDOF, consisting of a spring to describe the stiffness and a lumped mass at the DOF. This is done through the use of transformation factors, which take into account the actual load distribution, boundary conditions, and deflected shape (Biggs 1964). This methodology has been successfully used in prior studies investigating the behavior of timber (e.g., Jacques et al. 2014; Lacroix and Doudak 2013; Lacroix and Doudak 2020; Lacroix et al. 2014; Viau and Doudak 2016a), reinforced concrete (Hammoud et al. 2021; Jacques et al. 2012, 2015; Krauthammer 1984; Krauthammer et al. 1986, 1990; Morrill et al. 2004), structural steel (e.g., Lee and Shin 2016; Nassr et al. 2012, 2013), and masonry (e.g., El-Hashimy et al. 2017; Urgessa and Maji 2010) members with idealized boundary conditions subjected to blast loading. The general equation for an equivalent SDOF system is as follows: (14) KLMmx¨(t)+cx˙(t)+R(x,t)=P(t)AThe load-mass factor, KLM, is used to transform the total mass of the real system, m, into an equivalent lumped mass, which is derived based on boundary conditions and equivalency of energies. Depending on the experimental test setup, the total mass may include an LTD, which may be utilized to convert a pressure into concentrated point loads to be applied on beam and column elements. The resistance function, R, represents a mathematical formulation of behaviour, and x¨ x˙ x˙ are the acceleration, velocity, and displacement of the system, respectively, as a function of time, t. The damping coefficient, c, is typically set to zero for the purpose of blast analysis, because the effect of damping on the maximum response is insignificant (Biggs 1964). The right-hand side of Eq. (14) represents the forcing function, which can be calculated as the product of the reflected pressure, P, and the loaded area, A.A total of 89 experimental test results from the aforementioned studies on light-frame stud walls, glulam beams, and CLT panels were used for the validation of the modeling approach. For each specimen, SDOF modeling was conducted using the experimentally obtained pressure–time histories, average experimental resistance curves, and member masses (including LTD) as inputs. As shown in Fig. 6, an average predicted-to-experimental displacement ratio of 0.99 (COV=20%) was obtained, underlining the appropriateness of the proposed SDOF methodology in predicting the behavior of various timber members and assemblies.It can be observed that the predictions for the CLT panels tend to produce the most variability. This may be attributed to the idealized shape of the CLT resistance curves, which are sensitive to variation due to the presence of sudden drops in resistance attributed to ply failure. Such drop, in reality, may not be as abrupt as presented in the proposed approach. It should also be noted that average resistance curves for each specimen group were used in the analysis, which may also have contributed to the spread in the results found in Fig. 6. Using the actual parameters for each individual specimen—including, for example, using actual maximum displacement rather than a maximum ductility ratio of 2.5—is expected to further reduce the variability.Summary of Design-Level Parameters for Verification of Proposed Design MethodsThe validation of the SDOF modeling methodology presented in the previous section was conducted using experimental inputs from each of the aforementioned studies. For the purpose of verifying the applicability of the proposed design methods, material properties obtained from the wood design standard, in tandem with the strength modification factors and material predictive models presented in this paper, were used. Comparisons to the published experimental test results are then presented and discussed. Table 2 presents a summary of the design-level properties used in the comparison. The values were selected for each group of specimens obtained from the various test campaigns. For example, design parameters such as the specified strengths (fb,fs), moduli of elasticity and shear (E, G, GS), and load-sharing factor (KH), were obtained directly from the CSA O86 (CSA 2019). The load-duration factor, KD, was set to 1.25 for all the specimens, based on the prior discussion relating to the findings by Foschi et al. (1989). Modification factors, such as the DIF and SIF, and member-specific resistance curves, all consistent with the proposed design approach, were used. This verification method was chosen to represent the steps a designer would take to model the various timber members without any knowledge of in-situ material properties.Table 2. Summary of design parametersTable 2. Summary of design parametersPropertyCLTMember description1450Fb-1.3E MSR lumber, 11 mm OSB/18.5 mm Plywood sheathing, 400-mm stud spacingNo. 1/2 visually graded lumber, 11 mm OSB/18.5 mm Plywood sheathing, 400-mm stud spacing24f-E, multiple lams across width24f-E, multiple lams across widthE1 grade, 1950Fb-1.7E MSR (L), No. 3/Stud (T)20f-E, full width lamsSpeciesMember dimension86×318 mm137×267 mm445×105 mm (3-ply)137×222 mm445×175 mm (5-ply)80×228 mm300×244 mm (7-ply)E9,000 MPa9,500 MPa12,400 MPa12,400 MPa11,700 MPa (L), 9,000 MPa (T)10,300 MPaGN/AN/AN/AN/A731 MPa (L), 563 MPa (T)GSN/AN/AN/AN/A73.1 MPa (L), 56.3 MPa (T)Clear spanDIF1.2SIFbending18.104.22.168.21.2SIFrolling shearN/AN/AN/AN/A2.0fb21.0 MPa11.8 MPa25.6 MPa30.7 MPa28.2 MPa30.7 MPafsN/AN/AN/AN/A0.5 MPaKDKH1.21.4PCAK1.331.27PCAMr1.211.18Krb0.85Reference studyLacroix and Doudak (2015b)Viau and Doudak (2016a)Lacroix and Doudak (2018a)Viau and Doudak (2021a, b)Poulin et al. (2018); Viau and Doudak (2019, 2021c); Current studyVerification of Proposed Design MethodsThe peak resistance was used as a metric to quantify the accuracy of the proposed design method in cases where load-cells were utilized and thus experimental resistance curves were available. Where no load-cells were utilized (e.g., light-frame wood stud walls), the metric used to quantify the method accuracy was maximum midspan deflection. For each test specimen, the design equations and parameters outlined in the previous section and Table 2 were used to predict the member response to blast loading. Table 3 presents the ratios of the predicted peak resistances, based on the proposed design method, to the experimental peak resistances for the CLT and glulam specimens. In both Tables 3 and 4, the reference specimen ID from each respective study is provided, in addition to IDs assigned in this paper, to facilitate the discussion. Glulam, CLT, and light-frame wood stud wall specimens have been assigned the prefixes GL, CLT, and W, respectively, followed by an index according to their order within the tables. In addition, for cases where the same specimen was subjected to multiple shots, the nomenclature includes a suffix, which provides the sequence of shots to which the specimen was subjected. For example, W19-2 represents the second shot on the 19th wall specimen.Table 3. Relative accuracy of resistance predictions for glulam and CLTTable 3. Relative accuracy of resistance predictions for glulam and CLTReference studyReference study IDIDRDesignRExpReference study IDIDRDesignRExpMean (COV)Lacroix and Doudak (2018a) (static)B1-GL11.03B3-GL110.640.87 (18%)B2-GL21.10B4-GL120.77B3-GL31.12B5-GL130.85B4-GL41.08B6-GL140.80B1-GL51.06B7-GL150.79B2-GL60.86B8-GL160.74B3-GL71.05B9-GL170.77B4-GL80.98B10-GL180.70B1-GL90.75B11-GL190.80B2-GL100.63———Lacroix and Doudak (2018a) (dynamic)B12.2-GL200.72B5.3-GL271.030.87 (16%)B13.2-GL210.73B6.2-GL280.88B14.2-GL220.73B7.2-GL291.03B15.2-GL230.80B5.2-GL301.11B16.1-GL240.75B7.2-GL311.00B17.2-GL250.70B8.2-GL321.00B18.1-GL260.77B9.2-GL330.90Viau and Doudak (2021a) (dynamic)G1GL341.05G10GL401.111.00 (9%)G2GL351.03G11GL411.14G3GL360.95G12GL420.99G4GL370.93G13GL430.94G5GL381.09G14GL440.88G8GL391.06———Poulin et al. (2018) (static)CLT3-1CLT3.11.15CLT5-1CLT5.10.991.06 (8%)CLT3-2CLT3.21.00CLT5-2CLT5.20.98CLT3-3CLT3.31.19CLT5-3CLT5.31.06Poulin et al. (2018) (dynamic)CLT3-4CLT3.41.11CLT5-4CLT5.40.881.02 (9%)CLT3-5CLT3.51.13CLT5-5CLT5.50.90CLT3-6CLT3.61.03———Viau and Doudak (2019) (dynamic)FS1CLT5.60.98FS6CLT5.100.951.00 (9%)FS2CLT5.70.88FS8CLT5.110.99FS3CLT5.80.90FS10CLT5.121.09FS5CLT5.91.02FS11CLT5.131.16Viau and Doudak (2021c) (dynamic)CLT1-1CLT5.140.95CLT3-1CLT5.160.940.96 (7%)CLT2-1CLT5.150.99CLT4-1CLT5.170.94G1.2GL451.13G4.1GL480.90G2.7GL460.97G5.2GL490.91G3.1GL470.91———Current study (static)N/ACLT7.11.03N/ACLT7.20.981.01 (2%)Current study (dynamic)N/ACLT7.31.06N/ACLT7.40.991.02 (3%)Table 4. Relative accuracy of predictions for light-frame wood stud wallsTable 4. Relative accuracy of predictions for light-frame wood stud wallsReference studyReference study IDIDReference study IDIDMean (COV)PCA methodKH methodPCA methodKH methodLacroix and Doudak (2015b) (dynamic)W6-2W11.181.50W16-3W9-31.021.301.07 (22%)W7-1W20.921.16W17-1W10-11.241.60W8-1W30.850.96W17-2W10-21.071.35W8-2W40.891.05W18-1W11-11.381.58W9-1W51.241.61W18-2W11-20.911.07W9-2W60.821.06W19-1W12-11.361.75W9-3W70.841.07W19-2W12-21.101.40W10-1W8-11.682.18W20-1W13-11.121.46W10-2W8-20.760.97W20-2W13-20.841.08W16-1W9-11.401.80W20-3W13-30.901.13W16-2W9-21.061.36Viau and Doudak (2016a) (dynamic)W1-1W14-10.961.20W6-1W19-11.251.441.14 (10%)W1-2W14-21.231.45W7-1W20-11.251.38W2-1W15-11.101.16W8-1W21-11.271.52W3-1W16-11.061.21W10-1W22-11.271.37W4-1W17-11.251.67W11-1W23-11.001.50W5-1W18-11.201.32W12-1W24-11.011.54A clear trend around a value of unity can be observed in most specimens, indicating that the design method can predict the peak resistance with reasonable accuracy. On average, the peak resistance was underpredicted for glulam beams by a factor of 0.91 (COV=16%), which can be considered reasonably conservative for the purpose of blast design.The predicted peak resistance for the CLT panels tended to closely match the observed experimental peak resistance, as shown by the average predicted-to-experimental peak resistance ratio presented in Table 3. On average, the proposed design method for CLT panels overpredicted the peak resistance slightly by a factor of 1.01 (COV=8%). The highest deviation from the experimental peak resistances was observed in the panels tested by Poulin et al. (2018). Such deviations, as well as those observed in other specimens (e.g., CLT5.13 and GL41), can be attributed to the inherent variability of the mechanical properties found in wood, which are known to deviate in a similar magnitude to those reported in the current study. It can also be observed that the proposed method matched the results well for CLT where shear failure dominated the behavior (i.e., based on testing results reported in this paper). However, these results were solely based on testing of four 7-ply panels, and further testing is needed to independently confirm the validity of the proposed approach for CLT panels dominated by rolling shear failure.For light-frame wood stud walls, both the PCA design method and the simplified method using the KH factor were evaluated. Lack of load-cell use during experimental testing did not permit the peak resistance to be used as a metric for the validation of the proposed design method. As shown in the summary presented in Table 4, more variability was seen in the predictions for the stud walls than in the predictions for the glulam beams and CLT panels (Table 3). This can be attributed to the fact that the reported experimental maximum displacements were based on an average of the four middle studs of the wall specimens. On average, the proposed designed method utilizing PCA as input overpredicted midspan displacements on average by a factor of 1.10 (COV=18%), whereas the simplified method, as expected, yielded more conservative results and overpredicted the midspan displacements by a factor of 1.36 (COV=19%).Comparison between Proposed Design Methods and Current Code ProvisionsThis section presents a comparison between the proposed design approaches and the current design provisions found in the Canadian blast design standard, CSA S850 (CSA 2012). A summary of the resistance curve parameters as obtained from CSA S850 and the proposed design method are summarized in Table 5. Comparison between the design methods and test results are presented in Fig. 7.Table 5. Design parameters for proposed and current design provisionsTable 5. Design parameters for proposed and current design provisionsDesign parameterLight-frame wood studsGlulamCLTLight-frame wood studsGlulamCLTDIF22.214.171.124KDKHVariesN/AN/A1.0N/AN/AMaximum ductility ratio2.01.02.5Resistance curve typeBilinearLinearMultilinear staircase functionFrom Fig. 7(a), it can be observed that the data points associated with the static results tend to deviate by a significant margin, whereas little to no discrepancy can be observed with the rest of the dataset. The former can be attributed primarily to a load-duration factor of unity required by the blast design provisions. Regarding the dynamic data, it appears that the proximity of the results between the proposed design method and that of the current design standard is coincidental. It is noted that the product of the proposed DIF (1.1) and KD (1.25) is approximately equal to the DIF of 1.4 found in the current code provisions. It should be stated that this does not justify the use of 1.4 for DIF while requiring that no load duration factor increase is used concurrently, because, based on several studies, a DIF of a significantly lower value to that provided in the design standard is observed. The proposed approach provides modification values that are consistent with the physical behavior of the glulam members. Even if the blast design standard maintains the more conservative approach of recommending the use of KD of unity, it is recommended that the DIF for glulam be reduced to 1.1.The comparison between proposed and code design provisions for the CLT panels differs significantly, as shown in Fig. 7(b), where the code design provisions tend to overpredict the dynamic peak resistance of the CLT panels. The discrepancies are primarily caused by the values for DIF and SIF in the current blast design standard (CSA 2012) as well as its inability to account for rolling shear as a possible failure mode. The latter will likely cause incorrect failure mode predictions, particularly for the 5- and 7-ply panels. In a design situation, this could lead to unconservative design, thereby making the structure susceptible to higher-than-expected levels of damage, and possibly risk of progressive collapse.When comparing the maximum displacement for light-frame wood stud walls [Fig. 7(c)], it can be observed that the displacement predictions following the proposed design method tended to be more closely aligned with those observed from the experimental testing, with significantly less variability and lower magnitude than those obtained based on the current code provisions. Although overpredicting the maximum displacements may be interpreted as conservative, due to the level of expected damage correlating with the displacement, these overpredictions are the direct result of the maximum ductility ratio of 4.0 that is presently stipulated in the blast design standard. While the proposed design method for light-frame wood stud walls may predict lower displacements, the damage levels are stricter and based on lower ductility ratios and, therefore, are more accurate than the provisions currently found in the blast standard. It can be expected that the use of the simplified modeling methodology, whereby the load-sharing factor (KH) is used, would further increase variability in the results of the proposed design method, albeit leading to more conservative predictions.Connection DesignThe proposed design methods presented in this paper do not account for the design of the boundary end connections, which have been shown to be critical in blast-loaded timber assemblies. Studies investigating the performance and optimization of bearing connections for light-frame wood stud walls (Lacroix et al. 2021; Viau and Doudak 2016b), bolted connections for glulam beams and columns (McGrath and Doudak 2021; Viau and Doudak 2021a), self-tapping screws and bearing angle connections for CLT panels (Côté and Doudak 2019; Viau and Doudak 2019), as well as energy-absorbing connections (EACs) for the primary purpose of dissipating blast energies in heavy-timber assemblies (Viau and Doudak 2021c), have all demonstrated that significant energy dissipation may be achieved in the connections if adequate detailings are provided and brittle failure modes are suppressed. The omission in providing specific design guidance for connections in this paper is primarily related to the lack of extensive testing and robust modeling available in the literature. In general, it is noted that lack of consideration of connections may lead to premature connection failure, which ultimately will lead to the failure of the entire assembly prior to the load-bearing elements reaching their respective ultimate capacity (Syron 2010; Viau and Doudak 2016b). Neglecting connections may also lead to a design that is too conservative because the contribution of the connections to the energy dissipation in the structural assembly is neglected. Current provisions for timber address the design connections in a cursory manner, where an overstrength factor of 1.2 is applied to ensure that failure of the connection is prevented (CSA 2012). This factor, however, has been shown to be inadequate (Viau and Doudak 2021a). In terms of strength and dynamic increase factors, the CSA S850 specifies values of 1.0 to be used, unless otherwise justified through experimental test data, which are rarely available for designers.Recent studies focusing on timber connections and assemblies have shown that boundary connections tend to experience dynamic increases in strength that are related to the failure mode and type of connection (Côté and Doudak 2019; McGrath and Doudak 2021; Viau and Doudak 2016b, 2019, 2021a, c). Due to the inherent lack of ductility found in timber members, the connections have been shown to be a viable source of inelastic energy dissipation through controlled yielding and deformations, while ensuring that ultimate failure of the timber assembly occurs within the load-bearing member (Viau and Doudak 2021a, b, c). This has been demonstrated in connections for light-frame (Lacroix et al. 2021; Viau and Doudak 2016b), CLT (Côté and Doudak 2019; Viau and Doudak 2019, 2021c), and glulam (McGrath and Doudak 2021; Viau and Doudak 2021a, c). Shifting the design philosophy away from overdesigning the connections has been shown to optimize the energy dissipation in the assembly by allowing some inelastic dissipation in connections (Côté and Doudak 2019; Viau and Doudak 2019, 2021a, c). The successful use of this design approach, however, would require more stringent provisions for the design of brittle failure modes in connections and emphasizes that ultimate failure occurs in the load-bearing timber element.References ASCE. 2011. Blast protection of buildings. ASCE/SEI 59-11. Reston, VA: ASCE. Biggs, J. M. 1964. Introduction to structural dynamics. New York: McGraw-Hill. Côté, D., and G. 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