Performance-Based Design of Timber Structures for Earthquake ...

Performance-Based Design of Timber Structures for Earthquake Demand
Ricardo O. FOSCHI
Ph.D, Applied Mechanics
Professor, Civil Engineering
Univ. of British Columbia
Vancouver, B.C.
Canada V6T 1Z4
rowfa1@civil.ubc.ca
Dr. Foschi is a Civil Engineer
(Rosario, Argentina, 1962), and
holds Master and Ph.D. degrees in
Applied Mechanics (1964-1966)
from Stanford University. He is a
Professor of Civil Engineering at
UBC since 1983. He received the
Marcus Wallenberg International
Prize (1982) for wood engineering
research.
Summary
The objective in performance-based design is to obtain optimum values for specific design
parameters, so that the different performance requirements are satisfied with associated target
reliability levels.This paper presents a general approach to performance-based design of timber
structures under earthquake excitation. The discussion also focuses on the sources of uncertainty
and the implementation, via neural networks, of either reliability assessment or performance-based
design optimisation. The discussion is illustrated with the reliability evaluation and design of a
shear wall.
Keywords: earthquake response, dynamics, reliability, performance, design
1. Introduction
The problem of performance-based design under earthquake demands is conceptually simple but
complex to solve. Consider, for this discussion, the response of a single degree of freedom
oscillator of mass M subjected to a ground acceleration history a(t), Figure 1. The dynamic
equilibrium of the mass requires that
M( x
a)
M
Fig. 1 SDOF Oscillator
F(x)
a(t)
x
x F(
x)
/ M a(
t)
(1)
in which x(t) is the mass displacement and F(x) is the
structure’s restoring force. The restoring force is a
nonlinear function of the displacements x, and this
relationship constitutes the hysteretic response of the
structure. A nonlinear time-step analysis is required to
find the function x(t) and, ideally, this would permit the
calculation of an associated damage function d(x).
Performance requirements can then be written in terms
of the associated damage:
G d LIM d(
x)
(2)
in which dLIM is a performance limit. At the same time that the damage d(x) accumulates due to
lateral deformation, the capacity of the structure to sustain vertical loads, C(x), also diminishes.

Thus, the probability of collapse must be studied with another performance function
G C(
x)
W
(3)
in which W is the weight supported by the structure. The objective is to determine structural
parameters (sizes, mechanical properties) so that the performance requirements are met with target
reliability levels.
Equation (1) shows clearly the major sources of uncertainty in the problem: 1) the ground motion
a(t) and 2) the characteristics of the restoring force F(x). There are also additional uncertainties
related to the relationship between the displacements x and the damage function d(x) or the axial
capacity C(x). Let us consider these uncertainty sources one at a time.
1.1 The Ground Motion
The dynamic equation requires the ground acceleration function in its right-hand side. This
function will have a peak value, a corresponding frequency content, and a duration for the
segment of strong motion. Essentially, these are all basic random variables in the ground motion
characterization. A sample of earthquake records, likely to affect the site, need to be used to
characterize the randomness in the input motion. These records could be artificially generated or
could be historical. It does not make too much sense to study the problem mixing in records from
far away places (e.g., for a structure in Canada to use records from Japan or Chile). It is a major
problem in seismology to pinpoint site-specific ground motion statistics, and this shortcoming
must be recognized and be the subject of continuing studies.
1.2 The Restoring Force
The characterization of the restoring force, or hysteretic behaviour, as a function of displacements
is also quite difficult. It must be recognized that F(x) is a function of x(t) or of the input history.
Therefore, it is not a material property and it cannot be measured ahead of time using any other
history. Frequently, however, F(x) is characterized by a standard cyclic test following an agreedupon
protocol. The results are then fitted with a specific form for F(x), usually one appearing in
standard structural analysis programs, and this form for F(x) is then used to analyze the response
for any other history or earthquake. This, basically, is not right. Protocols, in this sense, are of no
use unless just utilized to compare one structural shape against another, both under the same
demand. Although this approach is not right, the question remains as to whether it is sufficiently
accurate. Of course, this question cannot be answered without a trusted benchmark but, again,
Equation 1 gives us a hint of when the errors in F(x) might be of minor importance. Obviously, for
structures with a heavy mass M, (long period), the importance of errors in F(x) will be small.
Lighter structures might not be so fortunate, and wood structures might fall into this category.
In general, F(x) shows characteristics which will influence the structural response: 1) pinching of
the hysteretic loop; and 2) degradation in stiffness and strength. These, in turn, will be influenced
by basic properties like yielding strengths, initial moduli of elasticity, different behaviours in
tension and compression; appearance of cracks, etc. What is needed is a mechanical model for
F(x), with proper constitutive equations for the intervening elements. Such a model would permit
the estimation of F(x) no matter what the demand history, and only then the response x(t) can be
estimated with some confidence. A mechanical model for hysteretic response of timber fasteners
has been developed [1] and implemented in the dynamic analysis of timber frames and shear
walls [2].
For reliability analysis, the use of ad-hoc hysteresis models would require the introduction of a
model error with a larger uncertainty than what would be needed in the case of a more detailed
and robust approach to calculating F(x).
A reliability-based comparison between using a calculated hysteretic loop and a fitted model
(Bouc-Wen-Baber-Noori) [3,4] for a pile foundation is shown by Foschi in [5]. Although the
BWBN model can fit the test hysteresis quite well for the chosen cyclic protocol, reliability under
earthquake demands can show substantial differences. Similarly, [5] shows substantial differences
in performance-based design, when the design parameter is the mass carried by the pile. The

ehaviour of a pile is conceptually identical to that of a pin fastener joining wood layers, and the
conclusions in [5] can be equally applied to timber fastening systems.
1.3 Damage and Axial Capacity
Performance requirements must be ultimately written in terms of damage, because only then can
one assess the probabilities of different consequences and make decisions about cost of repairs,
replacements or retrofits. Of course, collapse is an extreme form of damage due to axial or vertical
capacity degrading below the weight of the structure. The relationship between damage states and
displacements, d(x), and the axial capacity function C(x), are quite difficult to obtain and form
part of the general current research in earthquake engineering. As an intermediate step, however,
damage itself could be assimilated with displacements, requiring, for example, that the largest
sway of a shear wall during an earthquake should be allowed to exceed a tolerable displacement
only with a small target probability. Similarly, “collapse” could be assimilated with a rather large
sway, on the assumption that such a situation would lead to vertical collapse. However, for the
moment, these would be assumptions to be improved by extensive testing, and this should be a
priority in earthquake research.
2. Dynamic Analysis
Equation (1) needs to be solved to obtain the output time history x(t). Time-stepping integration of
this equation should be the default approach. Several other strategies have been proposed, based
on an elastic structure, mainly to save time or to make use of much simpler analyses like modal
superposition. This approach, while holding the promise of a fast integration, leaves open the
question of what to do about a correction to the elastic analysis in order to obtain the desired
nonlinear response. These corrections rely on assumptions about the equality of the displacements
in the two cases, an assumption which may be sufficiently approximate in some cases, completely
false in others. In any case, there is no need to resort to such assumptions when direct integration
of the equations of motion is facilitated by good algorithms and fast computers.
3. Reliability and Performance-Based Design
The study of structural reliability requires the definition of a performance function G, in terms of
the variables entering into the models for capacity and demand. Assuming that damage d(x) can
be represented by the maximum displacement Xmax produced by the earthquake, Equation (2) can
be rewritten
G = X Lim - Xmax ( x1 , x2 , … , xN ) (4)
in which XLim is the limiting displacement and the set of ( x1 , x2 , … , xN ) includes all the
intervening variables in the problem. Note that some of these may not be random but actually a
design parameter like nail spacing in a shear wall. The probability of failure associated with
Equation (4) corresponds to the probability of the event G

FORM and an approximating quadratic response surface. The simulations around that point are
then implemented using a neural network representation of Xmax. This approach has proven to be
very effective and the corresponding algorithms have been implemented in the software RELAN
developed at the University of British Columbia [8].
In earthquake engineering, one of the random variables is the earthquake itself. If the peak ground
acceleration is taken on its own, the remaining randomness of the earthquakes can be represented
by a set of records normalized to a unit peak acceleration. Thus, it is convenient to develop two
neural networks for a desired response R ( Xmax in this case): one for the average value of R over
the set of normalized records, and the other for the standard deviation or the coefficient of
variation of the response R over the same set of records. Both of these networks use as input the
set of remaining variables. Thus, the response R can then be written, for example, as having a
lognormal distribution :
R
2
R exp[ RN ln( 1
VR
)
(5)
2
( 1
V )
R
in which R is the average obtained from its neural network, and VR is the coefficient of variation
calculated from its corresponding network.
RN is a Standard Normal variable which thus represents the uncertainty due to different
earthquake records.
For performance-based design, the problem must be formulated as follows: obtain a set of design
parameters D so that the cost of the structure C(D) (including the cost of failure) is minimized,
subject to target reliability levels being met or exceeded for a set of performance requirements.
Thus,
(D) = C(D) Minimum (6)
Subject to i (D) Ti ( i = 1, M)
and Lj < Dj < Uj (j = 1, ND)
in which i(D) are the reliability levels achieved with the set D, for each of M performance
requirements, and Ti are the corresponding target levels. In addition, for practical reasons, each of
the ND components of the design parameters D have to be between a corresponding lower bound L
and an upper bound U. If cost is not addressed, and the objective is to calculate D so that the target
reliabilities are achieved as closely as possible, Equation (6) can be modified as follows,
(D) = 2
M
T i i ( D)
i1
Minimum (7)
The algorithms for performance-based design, according to either Equations (6) or (7), have been
implemented in the software IRELAN developed at the University of British Columbia [8].
4. Example: A Shear Wall
A shear wall, shown in Figure 2, supports a mass M and is exposed to earthquakes with
accelerations a(t). The shear wall is of standard construction, using Oriented Strand Board
sheathing (one side only) and vertical framing at 400mm spacing, with 63.5mm Bright Common
nails. The wall height and length are both 2.44m. The earthquake accelerograms, normalized to a
peak of 1 m/sec 2 , were obtained from the spectral properties and duration of the 1992 event at
Landers, CA, Joshua Tree Station.

The random variables in this problem are the mass M, the peak ground acceleration aG , and the two
nail spacings e1 and e2 . Other wood properties for the frame and the sheathing were considered
deterministic, as the behaviour of the fastenings is the most important factor for the wall response.
The response of interest is the maximum wall lateral displacement . Table 1 shows the ranges for
the variable combinations adopted for creation of the response database, used for training of neural
networks for the mean response ( e1,
e 2 , M,
a G ) and coefficient of variation V( e1
, e 2 , M,
a G ) . The
structural dynamic analysis implemented hysteresis loops for each fastener, calculated using the
mechanics-based model HYST [1].
The combinations were selected with an optimal grid-based experimental design, achieving
coverage of the variable space that maximized the minimum distance between combinations.
Framing
Member
e2
Fastener
Mass
a(t)
Sheathing
Panel
e1
Table 1. Variable range for the response databases.
Table 2. Variable statistics
Fig. 2 Shear Wall and Earthquake Demand
With the statistics of Table 2, the reliability of the
wall was evaluated for two performance criteria, as shown in Table 3, with achieved reliability
indeces also shown in the same table.
Table 3. Reliability indices , variable statistics from Table 2.
Performance Requirement
< H/300 (Serviceability, “no damage”) 1.870
< H/200 (Minor damage) 2.693
Finally, assume that the target reliability levels required for the two performance criteria are as
follows:
= 1.65 (or 5% exceedence probability) for < H/300
= 2.56 (or 0.5% exceedence probability) for < H/200
Variable Lower Limit Upper Limit
e1 (m) 0.010 0.075
e2 (m)
aG (m/sec
0.025 0.150
2 )
M
(kN.sec
0.500 3.500
2 Variable Mean COV Distributio
n
e1 (m)
e2 (m)
aG (m/sec
/m)
0.050
0.125
2.000
0.10
0.10
Normal
Normal
8.000
2 )
M
(kN.sec
0.981(0.1g) 0.55 Lognormal
2 /m)
6.0 0.10 Normal
and that the design parameters are the mean values for the nail spacing e1 and e2 , allowing in each
case a coefficient of variation of 0.10 for nailing innacuracies. Table 4 shows the result of the
performance-based design optimization. Alternatively, given the nail spacing, the design parameter
could have been the mean mass M, or the mean peak ground acceleration aG which would be
tolerated by the wall with the required target reliabilities.

Table 4. Performance-based design, target reliabilities and
optimum mean nail spacings
Performance
Requirement
< H/300
(Serviceability,
“no damage”)
< H/200
(Minor damage)
5. Conclusions
This paper has discussed the problem of reliability evaluation and performance-based design under
earthquake conditions. Sources of uncertainty have been identified, and it has been concluded that it
is necessary to have site-specific ground motion characteristics, and that the response of the
structure and its connections to cyclic loading must be properly modelled. The heavy task of
dynamic analysis coupled to a reliability simulation can be drastically reduced by first training
neural networks to structural responses of interest obtained a-priori for a range of the intervening
variables. With these neural representations of the response, reliability estimates and optimised
solutions for performance-based design can be readily obtained.
Current codified guidelines for earthquake design leave much to be desired, and the level of
achieved reliability is open to question. The use of elastic solutions to estimate the nonlinear
response or “ductility demand” of the earthquake depends on assumptions of equality of
displacements between the linear and the nonlinear system, and this assumption is generally not
true. The use of a single force reduction factor R cannot provide for uniform reliability for a range
of conditions and earthquake records. The introduction of proper, direct reliability evaluations,
based on consistent definitions of performance, is then an imperative and also required for proper
performance-based design.
6. References
e1 (m) e2 (m)
Target
Achieved
1.65 1.75
0.065 0.135
2.56
2.56
[1] Foschi, R.O. “Modeling the hysteretic response of mechanical connections for wood structures”, Proc.
World Timber Engineering Conference, Whistler, B.C. Canada, July 2000.
[2] Foschi, R.O., Ventura, C., Lam, F. and Prion, H. “Reliability and Design of Innovative Wood Structures
under Earthquake and Extreme Wind Conditions”, Department of Civil Engineering, University of British
Columbia, Vancouver, B.C. Canada, www.civil.ubc.ca/FRBC, 2002.
[3] Baber, T. and Noori, M.N. “Random Vibration of Degrading, Pinching Systems”. Journal
of Engineering Mechanics, ASCE, 111(8): 1010-1026, 1985.
[4] Foliente, G. “Hysteresis Modeling of Wood Joints and Structural Systems”, Journal of
Structural Engineering, ASCE, 121(6): 1013-1022, 1995.
[5] Foschi, R.O. and Zhang, J. “Performance-based design of a pile foundation under earthquake excitation”, Proc.
IFIP WG7.5 Reliability and Optimization of Structural Systems, Banff, Alberta, November 2003.
[6] Bucher, C.G and Bourgund, U. A fast and efficient response surface approach for structural reliability problems.
Structural Safety, Vol. 7, pp.57-66. 1990.
[7] Foschi, R.O. and Zhang, J. “Neural networks application in seismic reliability and performance-based design”,
Proc. International Conference on Applications of Statistics and Probability to Engineering, ICASP9, San
Francisco, CA, July 2003.
[8] Foschi, R.O., Li, H., Zhang, J. and Yao, F. “RELAN and IRELAN: Software Packages for Reliability Evaluation
and Performance-Based Design, Department of Civil Engineering, University of British Columbia, Vancouver,
B.C. Canada.