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MAT063-1
CSCE Annual Conference
Growing with youth – Croître avec les jeunes
Laval (Greater Montreal)
June 12 - 15, 2019
ANALYTICAL MODELLING OF HEAVY TIMBER ASSEMBLIES WITH
REALISTIC BOUNDARY CONDITIONS SUBJECTED TO BLAST LOADING
Viau, Christian1,2 and Doudak, Ghasan1
1 University of Ottawa, Canada
2 cviau037@uottawa.ca
Abstract: Several existing studies, investigating the performance of heavy timber assemblies with realistic
boundary conditions, have concluded that using simplified modelling tools such as single-degree-of-
freedom (SDOF) modelling may not be sufficient to adequately describe their behaviour and predict the
level of damage observed during blast events. A two-degree-of-freedom (TDOF) model, dubbed BlasTDOF,
that captures the effects of boundary conditions in the overall system response and includes considerations
for high strain-rate effects and semi-rigid boundary conditions is presented and discussed in this paper.
Sensitivity analyses were conducted for various cases using single- and two-degrees-of-freedom modelling
in order to make recommendations on the needs and appropriateness of using more advanced modelling.
It was determined that the use of SDOF modelling is adequate when the connection resistance and stiffness
exceed those of the timber element by ratios of one and ten, respectively. For the cases where these
conditions could not be met, the use of TDOF modelling was determined to be required in order to
accurately model the timber assembly.
1 INTRODUCTION
Threats of blast explosions on buildings and infrastructure are typically addressed through blast design
provisions (e.g. Unified Facilities Criteria Program 2008, ASCE 2011, CSA 2012). These standards provide
designers with guidance to perform blast analysis and design, and include items such as high strain-rate
effects, response limits, pressure-impulse diagrams, etc. The majority of these provisions deal with the
response of the load-bearing elements (e.g. columns, walls) under idealized boundary conditions and
requires that connections be overdesigned relative to the loaded structural elements. This approach may
not adequately reflect the performance of structural systems with relatively flexible end conditions. This is
generally the case for structural steel and timber assemblies, where the joints are generally assumed to be
pinned or (for properly detailed steel connections) fully fix. Having some deformation in the connections of
timber assemblies may even be desired since the wood structural elements are likely to experience brittle
failure (mainly in flexure and/or shear) when subjected to a blast load. The connections could help in
absorbing some of the imparted energy on the structure, however, it is imperative that the connections
themselves do not fail prior to achieving full capacity in the main structural element. This balancing act
between providing energy dissipation in the connections while still maintaining the deformation capacity
such that premature failure in the system is not experienced requires careful investigation of the behaviour
of the connections in isolation as well as systems containing such connections.
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It is common for designers to use single-degree-of-freedom (SDOF) analysis, based on idealized boundary
conditions, which does not explicitly consider the behaviour of the connections. As stated in CSA (2012),
more refined methods should be used if the dynamic response of the structural system cannot be
represented by SDOF methods. While this statement provides some guidance for designers, it does not
explicitly specify when it is appropriate or even necessary to resort to more robust modelling techniques.
The majority of studies investigating the applicability of simplified modelling methodologies (such as SDOF)
in blast research have dealt primarily with idealized boundary conditions (Jacques et al. 2012, Lacroix and
Doudak 2015, Viau and Doudak 2016a, Poulin et al. 2017, Lacroix and Doudak 2018). Studies investigating
structural elements with realistic boundary conditions subjected to blast loads have generally concluded
that limiting the modelling to SDOF will often lead to inaccurate predictions (Viau and Doudak 2016b, El-
Hashimy et al. 2017, Côté and Doudak 2019). Without resorting to a more resource-intensive finite element
analysis (FEA), other methods have been used effectively to model these assemblies, including energy
methods (Lavarnway and Pollino 2015), SDOF analysis with modified resistance curve (Whitney 1996,
Gagnet et al. 2017), and two-degree-of-freedom (TDOF) modelling (Park and Krauthammer 2009, Jacques
and Saatcioglu 2018).
Numerical solutions available through the use of FEA are generally not justified when considering the
computational efforts involved in the development and validation of these models. This is particularly the
case when dealing with non-homogenous materials (e.g. cross-laminated timber) and nonlinear
connections. A good balance between simplicity and accuracy can be obtained with TDOF modelling, which
consists of lumping the behaviour of each subcomponent (i.e. connections and load-bearing elements) into
equivalent subsystems. This inherently allows for two failure modes, as well as the effects of realistic
boundary conditions (i.e. translational and rotational flexibility), to be captured by the model. This paper
summarizes the findings of an investigation on the applicability of SDOF and TDOF modelling for timber
assemblies with realistic boundary connections subjected to blast loads. This investigation was conducted
through sensitivity analyses of various parameters such as capacities and stiffness of both the connections
and the load-bearing timber element.
2 TWO-DEGREE-OF-FREEDOM (TDOF) ANALYSIS
The following section describes the development and process of the proposed TDOF model. While the
model is developed for cross-laminated timber (CLT) and glued-laminated timber (glulam) assemblies with
various end connections, the methodologies can be extended beyond this application provided that proper
material and connection characteristics are obtained.
2.1 Model Definition
The wood assembly can be represented as a continuous frame element connected at its ends with
translational and rotational springs. In order to discretize the continuous wood beam element, the deflected
shape function must be determined in order to obtain the appropriate load-mass factor (kLM). The factor is
obtained by equating the kinetic energy and strain energy of the real structural system, based on the
assumed static deflected shape, to that of the equivalent system. The end springs account for the
translational and rotational stiffness of the connections by associating their behaviour to a load-
displacement relationship. As the end translational connections are acting in parallel, they can be lumped
together into a single equivalent translational spring. The equivalent mass of the wood member and
connections are represented as mwood and mconn., respectively, while their respective stiffnesses are
represented by kwood and kconn. In the full-scale test, the force enacted onto the system consists of a pressure
collected by a load-transfer-device (LTD) and applied to the specimens via two concentrated point loads
(see Figure 1). The stiffness of the assembly can be modelled as two springs in series, each represented
with a SDOF, as shown in Figure 1c.
MAT063-3
(a) Actual CLT Test Assembly
(b) Actual Glulam Test Assembly
(c) Idealized TDOF System
Figure 1: Two-Degree-of-Freedom Idealization
For the undamped TDOF system shown in Figure 1c, the following two equations of motions must be solved
simultaneously:
[1]
[2]
where is the load-mass transformation factor, used to transform the continuous wood member into an
equivalent SDOF, m and R are the component masses and resistances, respectively, is the applied
concentrated blast force, and are the component accelerations.
The system described through Equations 1 and 2 can be solved numerically using the constant average
acceleration method (Newmark 1959). The absence of the stiffness terms can be observed in Equations 1
and 2, as they have been replaced with the respective resistance terms. The nonlinear response expected
though yielding in the connections as well as the post-peak response of the CLT panels make it desirable
to introduce the resistance term since this makes for a more stable numerical solution. Numerical
instabilities may be encountered in cases where the stiffness term approaches zero or becomes negative.
MAT063-4
2.2 Model Inputs
The linear-elastic portion of the resistance curve can be defined by the maximum resistance () which
occurs at the elastic limit (). For a beam with two equal point loads at third spans, these parameters can
be obtained from Equations 3 and 4:
[3]
[4]
where is the maximum dynamic moment, which can be obtained experimentally or from published
static data, modified for high strain-rate effects (CSA 2012), and is the clear-span of the flexural wood
member.
The initial stiffness of the wood member for two concentrated point loads () can be modified to consider
the rotational stiffness of the connections at the beam ends. This is done through the derivation of an
analytical solution of an Euler-Bernoulli beam with semi-rigid springs at its ends. The solution considers a
nondimensional constant () defined as the ratio of the rotational stiffness () to that of the beam stiffness,
() (Equation 5). The solution for the modified stiffness is presented in Equation 6.
[5]
[6]
By setting , the solution in Equation 6 corresponds to the case of simply-supported beam, and by
setting , the solution corresponds to a beam with fully-fixed ends. The input values of the rotational
stiffness can also be obtained via experimental testing of the joints in question. It should be noted that in the
case of timber joints, the effects of rotational stiffness are generally low and tend not to affect the response
significantly.
Research done on glulam beams subjected to blast loads has shown that the dynamic behaviour can be
modelled using a linear-elastic resistance curve since little-to-no post-peak behaviour was observed (Lacroix
and Doudak 2018). For CLT, the cross-laminations allow for some post-peak resistance, which can be
described as ratios of the maximum resistance. Research on CLT under blast loads shows that the post-
peak behaviour tends to be consistent, in that failure of the outer tension laminates causes a drop-in load to
an intermediate region based on the remaining transverse and longitudinal layer (Poulin et al. 2017).
The mass of the wood assembly is assumed to be that of the wood member as well as the weight of the load
transfer device (Figures 1a and 1b). While the mass of the connections is negligible, a non-zero mass must
be entered in the TDOF model, otherwise the dynamic analysis will not converge to a solution.
For the purpose of TDOF modelling, the translational (i.e. out-of-plane) stiffness of the end connections can
be idealized as a separate axial spring with associated load-displacement relationship. Considerations of
high strain-rate effects in timber connections are not well developed yet, however, ongoing research is
underway at the University of Ottawa to address this issue (McGrath et al. 2019, Viau and Doudak 2019).
MAT063-5
2.3 BlasTDOF Algorithm
In order to conduct TDOF analysis for a wide range of structural components, a numerical algorithm
(BlasTDOF) was developed. BlasTDOF is capable of analyzing two-component systems subjected to blast
loads, and permits the user to input custom resistance curves and masses, as well as the pressure-time
histories. BlasTDOF is comprised of three modules; an input module, a dynamic analysis module, and an
export module. A flowchart of the program’s algorithm is presented in Figure 2.
Figure 2: BlasTDOF Algorithm
MAT063-6
3 SENSITIVITY ANALYSES
The main objective of the sensitivity analyses was to establish cases where the use of TDOF modelling
would be required and where SDOF modelling could be justified and not lead to significant erroneous
results. Two parameters, namely the ratio of the connection stiffness and maximum resistance to the
corresponding values for the wood member, were evaluated. For all analyses, a bi-linear resistance curve,
with a ductility limit of 2.0, was used to represent the behaviour of the connections. This was consistent
with observed behaviour from experimental studies. The reference resistance curves of the CLT and glulam
members are shown in Figures 3a and 3b, respectively. These are based on proposed models from recent
studies on CLT panels (Poulin et al. 2017) and glulam members (Lacroix and Doudak 2018) subjected to
blast loads. A mass of 385 kg for the CLT panel and load-transfer-device was used for the CLT groups,
while a mass of 321 kg was used for the glulam groups. Forcing functions described by idealized triangular
pressure-time histories were used and are presented in Figures 3c and 3d for the CLT and glulam cases,
respectively. The forcing functions represent a reflected pressure and impulse (i.e. area under the pressure-
time curve) combination which will allow the CLT and glulam member to reach their respective ultimate
failure displacement. The sensitivity analyses are summarized in Table 1.
(a) CLT Resistance Curve
(b) Glulam Resistance Curve
(c) Pressure-Time History for CLT Groups
(d) Pressure-Time History for Glulam Groups
Figure 3: Reference Resistance Curves and Pressure-Time Histories for Sensitivity Analyses
0
20
40
60
80
100
120
140
160
180
020 40 60 80
RCLT (kN)
Displacement (mm)
0
20
40
60
80
100
120
140
160
0 5 10 15 20 25 30
Rglulam (kN)
Displacement (mm)
0
10
20
30
40
50
60
010 20 30
Reflected Pressure (kPa)
Time (ms)
0
10
20
30
40
50
60
010 20 30
Reflected Pressure (kPa)
Time (ms)
MAT063-7
Table 1: Sensitivity Analyses
Groups
Variable
Variable Ratios
Constants
CLT-K
Kconn / KCLT
0.1, 0.5, 1.5, 10, 50
Case 1: Rconn / RCLT = 0.5
Case 2: Rconn / RCLT = 1.5
CLT-R
Rconn / RCLT
0.5, 0.8, 1, 1.05, 1.125, 1.25
Case 1: Kconn / KCLT = 0.5
Case 2: Kconn / KCLT = 1.5
Case 3: Kconn / KCLT = 10
LAM-K
Kconn / Kglulam
0.1, 0.5, 1.5, 10, 50
Case 1: Rconn / Rglulam = 0.5
Case 2: Rconn / Rglulam = 1.5
LAM-R
Rconn / Rglulam
0.5, 0.8, 1, 1.05, 1.125, 1.25
Case 1: Kconn / Kglulam = 0.5
Case 2: Kconn / Kglulam = 1.5
Case 3: Kconn / Kglulam = 10
Groups CLT-K and LAM-K consisted of varying the stiffness ratios for two cases of maximum connection
resistance, one being smaller and one larger than the maximum resistance of the wood member. This is
meant to represent two different design philosophies, where the connection is overdesigned to reach the
ultimate capacity of the structural member, or where the connection is intentionally under-designed in order
to dissipate energy in the ductile connections rather than the brittle wood member. Figures 4a and 4b show
that for the case where the connection is stronger than the wood member, SDOF analysis can accurately
predict the response only if the stiffness of the connection is at least ten times that of the wood member.
Using SDOF analysis to analyze a case containing connections with lower stiffness may lead to significant
error in results, and in those cases, the use of TDOF may be required. For the case of overdesigned wood
member, the results clearly show that using SDOF analysis can no longer adequately predict the correct
displacement or failure mode, regardless of the stiffness ratio. Additionally, it can be observed that
convergence is faster for the CLT panel, which may be attributed to its significant post-peak region.
Groups CLT-R and LAM-R consisted of varying the maximum resistance ratio for three values of connection
stiffness. As shown in Figures 4c and 4d, for both groups, a plateau seems to form when the ratio of
connection to wood member resistance becomes greater than one. This can be explained by the fact that
beyond a ratio of one, the behaviour of the wood member will tend to govern the overall displacement, and
additional connection resistance will no longer play a role. It is also observed that the more flexible the
connections are, the higher the TDOF/SDOF ratio is for the plateau values. As seen in the previous analysis,
once the resistance of the connection becomes less than that of the wood member, the SDOF predictions
will diverge from the actual displacement and failure mode. It is interesting to note that when CLT is used,
the use of SDOF analysis seems adequate for all stiffness ratios as long as the connection capacity is
greater than the panel capacity. A better fit is obtained when the stiffness of the connection is relatively
higher than the panel stiffness, however in general all cases produce a reasonable agreement between the
two analysis methods. Interestingly, the outcome looks significantly different for glulam, where only the case
with very high relative connection stiffness ratio (i.e. > 10) yields adequate use of the SDOF modelling
methodology. As expected, the scenario where the connection stiffness is ten times that of the wood
member and with a resistance that is half that of the CLT panel yields the least accurate prediction when
using SDOF modelling. This is attributed to the fact that the connection will fail at a significantly lower
displacement than that of a more flexible connection, and the wood member will play a significantly lesser
role in the response of the system as a whole.
MAT063-8
(a) CLT-K
(b) LAM-K
(c) CLT-R
0.0
0.5
1.0
1.5
2.0
2.5
0 5 10 15 20 25 30 35 40 45 50
TDOF/SDOF
Kconn/KCLT
Rconn / RCLT = 0.5
Rconn / RCLT = 1.5
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0 5 10 15 20 25 30 35 40 45 50
TDOF/SDOF
Kconn/Kglulam
Rconn / Rglulam = 0.5
Rconn / Rglulam = 1.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
TDOF/SDOF
Rconn/RCLT
KConn / KCLT = 0.5
KConn / KCLT = 1.5
KConn / KCLT = 10
MAT063-9
(d) LAM-R
Figure 4: Results of Sensitivity Analyses for Predictions of Maximum Deflection
4 CONCLUSIONS AND RECOMMENDATIONS
This paper discusses the applicability of SDOF and TDOF modelling for timber assemblies with realistic
boundary connections subjected to blast loads. A TDOF analysis program, BlasTDOF, was presented, and
the developed numerical algorithm was described. The sensitivity of varying stiffness and capacity of
connections relative to the timber structural elements was investigated. The results from the sensitivity
analyses show that:
- In the case where the connection is stronger than the wood member, SDOF analysis can accurately
predict the response only if the stiffness of the connection is at least 10 times that of the wood
member. For the case of overdesigned wood members, the results clearly show that using SDOF
analysis can no longer adequately predict the correct displacement or failure mode, regardless of
the stiffness ratio.
- In general, a consistent ratio of SDOF to TDOF results is obtained when the ratio of connection to
wood member resistance becomes greater than one. Once the resistance of the connection
becomes less than that of the CLT panel, the SDOF predictions will diverge from the actual
displacement and failure mode.
- When CLT is considered, the use of SDOF analysis seems adequate for all stiffness ratios as long
as the connection capacity is greater than the panel capacity. For glulam, only the case of
overdesigned connections with a very high relative stiffness ratio yields adequate use of the SDOF
modelling technique.
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0.0
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2.0
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KConn / Kglulam = 1.5
KConn / Kglulam = 10
MAT063-10
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