Goodman Dissertation 2015
Goodman Dissertation 2015
Goodman Dissertation 2015
A Thesis
Presented to
The Academic Faculty
by
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy in the
School of Architecture
Approved by:
iii
ACKNOWLEDGEMENTS
This thesis proposal and thesis work completed to date has been supported by the
U.S. Department of Energy project D6596, SIMPLE BoS. Georgia Tech students,
and faculty have enabled the accomplishments to date. Florida International Uni-
versity collaborated to implement the wind tunnel test and process measurements
into pressure coefficients. Generous advise and patience have been provided by thesis
committee members. Dr. Bruce Ellingwood provided insight into the structural code
development process, intent and opportunities for improvement. Finally, the oppor-
tunity to work with Professor Augenbroe for the past five years has been valuable
beyond a degree or title. My gratitude for his time and patience know no limit.
iv
TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 PV Balance of System Importance and characteristics . . . . 1
1.1.2 Balance of system drivers and stakeholders . . . . . . . . . . 4
1.1.3 Structural reliability . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.4 Fragility curves and probability of failure: risk management
methods in practice . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Objective scope and limitations . . . . . . . . . . . . . . . . . . . . 14
1.3 Methodological road map . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Development of a general probabilistic model for assessing
structural response . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.2 Wind tunnel testing to address existing model error in pressure
coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.3 Development of a general reliability model . . . . . . . . . . 22
1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
v
III FRAGILITY ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.1 Uncertainties in Structural Systems . . . . . . . . . . . . . . 47
3.1.2 Exact Methods for Limit State Design . . . . . . . . . . . . . 49
3.1.3 Performance Based Design . . . . . . . . . . . . . . . . . . . 51
3.2 Structural Fragility . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Fragility model derivation methodology . . . . . . . . . . . . 54
3.2.2 Model estimation error . . . . . . . . . . . . . . . . . . . . . 55
3.2.3 Applications to wind hazards . . . . . . . . . . . . . . . . . . 56
3.2.4 Sources of Uncertainty and Error . . . . . . . . . . . . . . . . 57
3.3 Extension to residential PV systems . . . . . . . . . . . . . . . . . . 58
3.3.1 PV system wind load limit states . . . . . . . . . . . . . . . 60
3.3.2 Applicability of wind load Statistics . . . . . . . . . . . . . . 61
3.3.3 PV Structural Model . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Case Study: Fragility Analysis . . . . . . . . . . . . . . . . . . . . . 62
3.4.1 Stochastic wind loads . . . . . . . . . . . . . . . . . . . . . . 62
3.4.2 Stochastic resistance statistics . . . . . . . . . . . . . . . . . 63
3.4.3 Case Study Fragility Curves . . . . . . . . . . . . . . . . . . 72
3.4.4 Fragility Curve Validation . . . . . . . . . . . . . . . . . . . 73
3.4.5 Fragility Analysis Conclusions . . . . . . . . . . . . . . . . . 78
vi
4.2.3 Wind tunnel test design . . . . . . . . . . . . . . . . . . . . . 104
4.2.4 Experimental Test Plan . . . . . . . . . . . . . . . . . . . . . 109
4.3 Experimental Data Analysis . . . . . . . . . . . . . . . . . . . . . . 112
4.3.1 Case study fragility curves . . . . . . . . . . . . . . . . . . . 120
vii
LIST OF TABLES
viii
LIST OF FIGURES
ix
23 LS#1 fragility curves for code compliant system, 30 deg roof, 90mph
wind zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
24 LS#1 fragility curves for code compliant system, 45 deg roof, 90mph
wind zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
25 Validation curve for 15 deg roof . . . . . . . . . . . . . . . . . . . . . 77
26 Bronkhorst wind tunnel test reference [19] . . . . . . . . . . . . . . . 83
27 Meroney and Ne↵ wind tunnel test reference [57] . . . . . . . . . . . . 84
28 Kopp wind tunnel test reference [53] . . . . . . . . . . . . . . . . . . 85
29 Banks wind tunnel test reference [16] . . . . . . . . . . . . . . . . . . 86
30 Banks array layout [16] . . . . . . . . . . . . . . . . . . . . . . . . . . 87
31 Stathopoulos wind tunnel test reference [70] . . . . . . . . . . . . . . 88
32 Sample wind pressure time series [20] . . . . . . . . . . . . . . . . . . 89
33 Geurts field test reference [40] . . . . . . . . . . . . . . . . . . . . . . 90
34 Geurts residential PV module wind tunnel test reference [38] . . . . . 91
35 Erwin test article reference [34] . . . . . . . . . . . . . . . . . . . . . 93
36 Cp values (y-axis) mean(Left) and peak(right) from partial turbulence
simulation and full atmostpheric boundary layer simulation for 16 taps
(x-axis) on a gable roof at 45 degree angle of attack [36] . . . . . . . 96
37 Illustration of subinterval, mean flow velocity, low frequency fluctua-
tions and high frequency fluctuations [58] . . . . . . . . . . . . . . . . 97
38 Proposal values for a as a function of tributary area [71] . . . . . . . 99
39 FIU 12 Fan Wall of Wind Tunnel with sample test article mounted on
turn table [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
40 Test Section Wind Speed Profile [10] . . . . . . . . . . . . . . . . . . 102
41 Test Section Turbulence Intensity Profile [10] . . . . . . . . . . . . . . 102
42 Wall of Wind Power Spectra [10] . . . . . . . . . . . . . . . . . . . . 103
43 PV module pressure tap layout . . . . . . . . . . . . . . . . . . . . . 105
44 Test Article Array Layout . . . . . . . . . . . . . . . . . . . . . . . . 106
45 Test article drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
46 3D CAD model of the wind tunnel test article with transparent roof
surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
x
47 Physical test article installed in the wind tunnel . . . . . . . . . . . . 110
48 Structural zoning for tributary area, TA = 275 sq.ft., Points 0-180. . 112
49 Structural zoning for tributary area, TA = 275 sq.ft., Points 180-350. 113
50 Maximum module GCp by wind angle . . . . . . . . . . . . . . . . . . 114
51 Minimum module GCp by wind angle . . . . . . . . . . . . . . . . . . 115
52 Pressure coefficient sign notation . . . . . . . . . . . . . . . . . . . . 115
53 Asymmetric Envelope Approach . . . . . . . . . . . . . . . . . . . . . 116
54 Mean and standard deviation GCp + statistics by design case and roof
angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
55 Mean and standard deviation GCp statistics by design case and roof
angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
56 Fragility curves for 15 deg roof . . . . . . . . . . . . . . . . . . . . . . 122
57 Fragility curves for 30 deg roof . . . . . . . . . . . . . . . . . . . . . . 123
58 Fragility curves for 45 deg roof . . . . . . . . . . . . . . . . . . . . . . 124
59 Sampling error standard deviation vs sample size adapted from CPP
[63] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
60 Gumbel theoretical vs empirical quartiles plot of Atlanta extreme 3-
second gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
61 Frequency plot of Synthetic set of extreme wind speeds for Atlanta
(n=1000) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
62 Frequency Distribution of Atlanta annual maximum 3-second gust . . 137
63 Reliability index, 1 for limit state #1 for each design case in the
Atlanta case study. Benchmark target reliability values are illustrated
for reliability class, RC, 2 with upper spec RC3 and lower spec RC1. . 138
xi
SUMMARY
This thesis applies structural reliability measures for the performance based
codified practices prescribe global factors (allowable stress design) and partial factors
(load and resistance factor design) intended to provide an acceptable level of relia-
prescriptive approach has two flaws, (1) calibration e↵orts needed to ensure consis-
tency across structural system types have not kept up with the commercially available
system types and (2) the actual expected reliability is not quantified and available
ure conditioned to wind speed in a fragility curve and the reliability index , both
of which are commonly used in performance based design. The approach is demon-
strated through the application of the reliability measures to code compliant designs.
Diverse system types are utilized to demonstrate how the existing code prescribed
approach may lead to non-uniform structural performance. For each of the system
types on which the reliability measures are demonstrated, a code compliant design is
developed for three roof slopes, wind tunnel testing is conducted to provide an ex-
perimental measure of wind pressure coefficients, system specific fragility curves are
and then, a site specific wind model is applied to produce a probability of failure and
reliability index . Through the performance based approach proposed in this thesis,
systems designed according to the prescriptive code method. The two key outputs
xii
which illustrate this finding are fragility curves which illustrate the probability of
failure over a range of wind speeds and reliability index, values which couple the
xiii
CHAPTER I
INTRODUCTION
1.1 Motivation
1.1.1 PV Balance of System Importance and characteristics
A power generation device such as a fuel cell or photovoltaic (PV) cell may require
software and other technology required to fulfill these functions is generally character-
plied1 residential systems requires balance of system hardware including racking and
ment hardware (mounting clamps), structural members (rails) and roof attachment
hardware (L-feet) lag screws and flashing. Electrical protection is composed of wire
management hardware and ground protection. Typically2 all of this hardware is as-
technological factors including system layout and racking and mounting hardware
selection. For analysis purposes, labor and other soft cost, are often included in the
general balance of system cost category. [59].3 Until the 2010 Department of Energy
hosted BoS workshop focused on cost reduction, the majority of federal and privately
funded research was targeted at advancing PV cells and PV modules [28]. Due to
1
building applied refers to systems installed in addition to a complete building system in lieu of
building integrated in which the PV systems fulfills a part of the building envelope functionality.
2
some approaches under are exploring factory assembly of subsystems
3
PV inverters are often considered an independent power conversion component and not included
in the balance of systems definition
1
great cell and module research and development success, along with external factors,
PV markets have grown and system cost have declined. In an e↵ort to achieve aggres-
sive cost reduction goals , BoS cost and reliability are receiving increased industrial
2
3
Figure 1: Typical residential PV system with 1-D rail
Figure 2: U.S. PV Installation actuals and forecast for 2010-2016 by sector.
[49]
Among the areas of PV BoS investigation, reliability and risk management are
critical for supporting continued adoption. The importance of safety risk management
for comment on solar energy by Duke Energy with the following quote:
power grid reliability should never come second to advancing any energy
resource [30].
Hardware cost and risk management are not the only drivers of PV balance of system
design. Installation time, logistics, and layout flexibility have motivated development
of divergent BoS technologies that all meet a common set of functional requirements.
At a 2010 industry PV BoS workshop, the racking and electrical components of BoS
were functionally described as the electrical system for aggregating and conveying
power and the structural system for resisting natural forces. Forces applied to a PV
system include self weight, wind, snow, and seismic. Analysis of the structural system
4
revealed that in most high growth PV markets, wind forces have the greatest influence
of structural member sizing. Workshop attendees postulated that without wind loads
on a PV system, structural cost could be reduced by 75% [18]. While reducing wind
loads through improved aerodynamic design and understanding has resulted in cost
reductions for commercial flat roof PV systems, wind loads have had little influence
Requirements associated with installation time, logistics cost and layout flexibility
are reflected by system types introduced to the market over the past 5 years. The
installation time required for positioning and attaching the roof attachment hard-
ware is significantly greater in the U.S. than in Germany for reasons including the
installation tolerance to as built U.S. roof conditions such as imprecise rafter spacing
[22]. Logistical cost are influenced by the weight and length of racking structural
members, rails, that must be transported to site and conveyed to the roof. Logistical
cost have motivated elimination of continuous rails in favor of light weight discretized
racking configurations. The rapid change in structural systems and continued drive
for cost reduction further motivates an improved understanding of failure risk across
The ”reliability” concept is loosely applied throughout the literature with varying
implied and explicit definitions. Often the definition is excessively broad, for example
if a reliable system is thought of as one that does not fail, then the definition neglects
specificity of operating life and operating scenarios. Further, such a broad definition
dress these shortcomings, ISO 2394 defines reliability as the ”ability of a structure
or structural element to fulfill the specified requirements, including the working life,
5
for which it has been designed [2]. According to this definition ”requirements” shall
be defined with three components: a structural failure definition, a service life, and
a scenario of use. Structural failure definitions are specified by the structural states
beyond which the performance requirements are no longer satisfied; these states are
commonly referred to as limit states [6]. In other words, a limit state is the threshold
condition for failure. For example, a PV system limit requirement may specify that
the structural rails shall remain in an elastic state below the yield stress. The struc-
tures service life, defined in the time domain commonly with units of years, is either
established by the governing structural code or by the structures owner. For exam-
ple, EN1990 establishes a design working life of 10-25 years for replaceable structural
parts [3]. For many building applied PV system components, this definition may be
account for compatible load e↵ects that may occur simultaneously [6]. For example,
a load combination may account for the probable simultaneous occurrence of gravity
and wind load e↵ects [11]. Load combinations are necessary because in actual scenar-
ios of use single load rarely occurs in isolation. In combination limit states, service
life, and load combinations provide a basis for defining reliability requirements and
at a given hazard level as found in fragility models or over a specified time period as
found in Pf . When expressing Pf over a time period, the reliability index is often
Equation 1
1
= u (Pf ) (1)
1
where u () denotes the inverse standard normal distribution function.
6
Figure 3: Theoretical deterministic structural design process
Both fragility curves and single reliability measures Pf or are often useful to-
convolute structural attributes with scenarios of use, fragility curves provide a prim-
the hazard, wind speed. In order to further clarify the meaning of each reliability
mining load e↵ects, for use in load combinations, along with designing structural
design and the scenario of use, load e↵ects e may take a value from the real set, R (S
is also commonly used to represent load e↵ects due to the prevalence of stress design).
Similarly, a structural system may be designed with resistance, r, from the real set
design load e↵ect, ed and design resistance rd noted as (rd , ed ). As shown in figure
3 (center) (rd , ed ) may exist anywhere in the real domain which is divided into two
sets. Cases where rd ed are said to belong to the failure set according to the limit
state definition, while cases where rd > ed belong to the safe set. Graphically the
limit state is shown to be the condition that separates the safe set from the failure
set with the limit state included in the failure set. Mathematically the limit state,
7
Figure 4: General probabilistic structural design process
G(X) = O (2)
value for ed , the designer may engineer the structure with strength rd (or sti↵ness)
engineering community has recognized neither ed nor rd are deterministic values be-
must be taken into account when designing any construction work” [5]. Uncertainties
are recognized to occur throughout the basic set of variables X = [x1 , ...xn ] character-
izing material properties, geometries, load e↵ects and model uncertainties from which
r and e are determined [5]. Consequently, r and e must be treated as random vari-
ables, with example probability distributions for r and e illustrated in Figure 4 (left).
Random variables r and e are not compatible with deterministic design methods and
(center) represents a general approach for structural reliability where the stochastic
8
of e r. Transformation of the limit state from G(x) to G(u) by normalizing the
to Equation 4 [35].
Xi µxi
Ui = (3)
xi
µM
= (4)
M
Historically, limit state design has been incorporated into multiple prescriptive
methods. Each method has been structured with the intent to ensure that a design
falls within the safe set through the use of coefficients or factors that create a safety
margin between the design and limit state. Each of these methods rely on a linear limit
state. For a non-linear limit state a linear approximation is made using the tangent
line through the most central failure point (on the limit state)[29]. Implementation
of linear approximation methods is beyond the scope of this thesis, interested readers
In practice, three prescriptive methods have been utilized in U.S. codes to ensure
structural reliability, these include permissible stress, global safety factor, and partial
The permissible stress method reduces the critical stress to a permissible level
Where
9
crit
per = (6)
k
Coefficient k is greater than 1 and is intended to account for all sources of uncer-
which sources of uncertainty are accounted for and also precludes an engineer from
calibrating the coefficient for di↵erent levels of uncertainty. Further, permissible stress
does not allow for a more efficient design through global analysis methods including
A second prescriptive method, the global safety factor method, only partly ad-
dresses the short comings of permissible stress. A broad approach is taken by pre-
scribing a minimum safety factor, So that describes the ratio of resistance, R to load
variables.
R
s= > So (7)
E
In determination of the resistance, global behavior may be taken into account.
However, the single safety factor still lacks transparency required for calibration.
More recently introduced, the partial factor method addresses the remaining short-
coming by replacing the global factor with partial factors. This method accounts for
specific sources of uncertainty and ensures through Equation 8 the design value of
10
Xk . Depending on the design value, factors may include partial factors and reduc-
most common implementations of the partial factor method including the ASCE ver-
sion Load and Resistance Factor Design (LRFD) do not incorporate calibration for
factor method commonly fails to provide quantified reliability. Further, in the ab-
sence of quantified reliability it may not be reasonable to assume the method delivers
As described, common to each method is the use of a linear limit state or linear
[5]. Alternatively, when a non-linear limit is not approximated as linear, the design
may exist any where along the acceptable probability contour to provide a uniform
measure of safety according to G(X) > 0. Typically, this method results in a concave
limit state which ensures that any design point falling between two limit state points
falls within the safe set. This approach is the basis of ”exact” methods including
Monte Carlo analysis which have been more recently adopted in performance based
reliability codes. The EU code EN1990 provides a performance path based on meeting
target reliability levels with quantified design values for probability of failure Pd and
provide a reliability not less than that expected for similar components
when subject to the influence of dead, live, environmental, and other loads.
11
Consideration shall be given to uncertainties in loading and resistance.
Currently, structural life safety risk is managed locally by a city or county building
code official who may require stamped engineering drawings and Underwriter Labo-
the system’s structural design complies with the structural building code. This pro-
cess is intended to ensure that the structure has an acceptable likelihood of failure
during a design wind event. The design wind event is based on the prescribed recur-
rence interval for the occupancy category. In the U.S. building code, the threshold for
methods are more mature and more commonly used in practice. The result is that
with a calculated value while this is a nascent practice in the U.S. especially on
State of art structural risk management has moved beyond factored design meth-
ods to better quantify the risk to building systems from earthquake and hurricane
hazards have identified the need to provide stakeholders with quantified performance
measures. Stakeholders are recognized to be diverse in origin and may include build-
ing owners, financiers, and the public. Stakeholder concerns may be equally diverse
including not only life safety risk but also operational risk, financial risk, repair cost
12
Performance based engineering commonly utilizes performance measures to syn-
thesize risk analysis for communication with decision makers. These measures include
fragility curves and the reliability metrics, probability of failure and reliability index,
. An important distinction between fragility curves and reliability metrics lies in the
quantifies the conditional probability of failure along the y-axis given a specific hazard
level or structural demand on the x-axis, such as wind speed [75]. Multiple curves are
commonly plotted together for two purposes (i) illustration of the fragility of a single
technology with multiple limit states and (ii) illustration of fragility for multiple tech-
nology or component options with a common limit state. Regardless of the specific
purpose, a few noteworthy characteristics are common to all fragility curves, a low
demand region with 0 probability of failure and a high demand region with certain
failure. In between, the fragility curve illustrates the probability of failure over a range
of demand values. Fragility curves are commonly preferred by engineers because they
to derive a single scalar value that quantifies the expected probability of failure over
failure is most useful for understanding structural risk in a specific location or with
warranty reserves. Multiple methods exist in practice for the derivation of demand
models, they commonly depend on historical data and mathematical models which
both have limits of accuracy. Further, historical weather models have limited ability
to predict future weather because weather patterns and events are becoming more
extreme due to global warming. Nevertheless, fragility analysis and probability of fail-
ure analysis are the two available measures for synthesizing and reporting structural
13
Figure 5: illustration of multiple fragility curves [75]
risk management analysis. Each of these measures play a critical and complementary
The research objective of this thesis is to support the design of distributed PV sys-
tems by providing decision makers with explicit performance measures for PV system
structural reliability.
formed on three system types applied to three roof angles in multiple configurations
for a total of 30 design cases. The design cases have been selected to support the
the argument that reliability measures should and can be applied to residential PV
14
systems. The methodological sections are not a replacement for the foundational
literature found in the bibliography. Systems types used in this thesis may appear
similar to referenced commercially available system types but have been intentionally
altered. This thesis in no way attempts to make claims regarding actual commer-
cial systems and the specific results presented in this thesis should not be used for
fragility and probability of failure risk measures is predicated on the following four
assumptions:
ability through the application of safety factors that shift the design load from an
3) Both the design load and resistance are unknown and best modeled as random
variables because the variables that a↵ect each are uncertain and best treated as
random variables. In the absence of uncertainty, a safe system would constitute one
in which the load is just less than resistance for the most demanding scenario of use;
4) The building code accomplishes multiple objectives which include but are not
limited to imposing a level of safety accepted by society and simplifying the structural
Table 1 [15] contextualizes ASD and LRFD code methods in a hierarchy of struc-
tural reliability methods. ASCE code methods are considered a simplified ”level 1”
15
reliability method because they eliminate probabilistic methods from practice by ap-
4
plying safety factors to nominal or characteristic values as described in assumption
statistical model for each random variable based on their collective understanding
and believes [33]. The foundation for our current statistical modeling of the random
variables that influence wind loads was made by at least 20 experts from industry,
academia, and practice who individually demonstrated expertise in the 1980’s and
1990’s, then came together with facilitation by Bruce Ellingwood to codify their col-
lective understanding through the statistical process of a Delphi. Through two rounds
of questions and controlled feedback the group reached consensus on each variables
uncertainty [33]. Knowledge of the uncertainty in each random variable allows for
a factor to be derived which in combination with the mean value produces a design
value with acceptably low probability of exceedance. The threshold for acceptably
practice. This practice is premised on the assumption that if the risk was not ac-
cepted by society then change would have been instigated by stakeholders. Further,
the simplified level 1 approach implies a belief that stakeholders do not require an
For the numerous scenarios in which a quantified reliability measure enables im-
proved decision making a series of options exist with increasing accuracy. Common to
each of the quantified methods is the direct use of random variable distributions. One
option, second order moment methods utilizes the mean and coefficient of variation
of each random variable in a limit state function to estimate , the reliability index
4
A nominal value is di↵erentiated by a lack of statistical implication such as mean value and does
not imply a probabilistic distribution
16
Table 1: Hierarchy of structural reliability methods [15]
Level Calculation
Probability Limit Results
Method Distribu- State
tions Functions
1 Code Safety factor Not used Linear usu- Partial fac-
methods with nominal ally tors
values
2 Second Second mo- Normal dis- Linear factor
Moment ment algebra tributions and nomi-
methods implied nal failure
probability
3 ”Exact” Random normal Approximated Failure
methods Variable distributions as linear probability
Transforma-
tion5
Monte Carlo Any Any form Failure
probability
µM
= (9)
M
Where, µM is the mean value and standard deviation of the safety margin defined
µM = R M SM (10)
q
2 2
M = R + S (11)
In both second moment methods and exact methods, the direct use of random
factors from falling out of calibration. Further, the calculated reliability measures sup-
port explicit quantification for improved decision making. Second order methods are
codified in Eurocode EN 1990 through the provision of target values. In this thesis,
second order methods will not be used directly but rather used for benchmarking via
17
Another more ”exact” method is the use of Monte Carlo simulation which incor-
porates probability distributions for the set of random variables, X that influence the
limit state equation under consideration. When defining the probability distributions
the same uncertainties found by expert Delphi [33] are commonly used in conjunction
with any project specific random variables defined through testing. Each distribu-
as xˆj , j = 1, 2, ...N . Taken together, the vector x ”are realizations of the so-called
basic random variables X representing all the relevant uncertainties influencing the
probability of failure” [35]. For each xj a virtual experiment is conducted and ana-
lyzed through the limit state equation with a failure event F given by a functional
F = {g(x) 0} (12)
Using the failure definition, the probability of failure may be calculated exactly
through equation 13
Z
Pf = fx (x)dx (13)
g(x)0
Where fx (x) is the joint probability distribution of X, the set of random variables
[35].
Due to the challenges of solving Equation 13 in closed form, Monte Carlo simula-
tion methods are commonly used to provide an estimate of Pf . The method commonly
indicator function I[g(x) 0] which is set to equal 1 if g(x) 0 and 0 if g(x) > 0.
Z Z
Pf = fx (x)dx = I[g(x) 0]fX (x)dx (14)
g(x)0
18
Further, simulation of N realizations of the random variables composing X pro-
1 X
Pf = I[g(x) 0] (15)
N
Combination of Equations 12 and 15 shows the Pf estimated through Monte Carlo
simulation is literally the percent of failures from a large number of experiments which
nf
Pf = (16)
N
Monte Carlo is considered an ”exact” method because neither parameter distri-
butions nor limit state functions are simplified. They are however not exact because
of uncertainty in the parameter distributions, limit state functions, and model form.
These sources of uncertainty will be identified throughout the thesis and are gener-
to code for a 90 mph wind zone, will be evaluated through Monte Carlo simulation to
ing the scope of the Monte Carlo simulation, two sequential performance measures
will be produced to characterize PV system reliability. First a wind speed array will
be combined with the set of random variables that influence wind loads along with
the random variables that influence strength. Together, this provides a probabilistic
model for assessing the structural state of failure or non failure (survival) in response
to a given wind speed. The results of this analysis will be synthesized as a set of
fragility curves. Second, the Monte Carlo scope will be expanded by modeling the
extreme annual wind speed as a random variable to assess reliability over time with
19
Pf and based on historical wind speeds distributions. The fragility model and
the reliability model both provide measures useful for quantifying structural perfor-
mance. However a critical di↵erence exists that will be explored in greater depth
through the thesis, the fragility model provides a probability of failure conditioned
on a wind speed while reliability model treats wind speed as a random variable and
is dependent on time.
At the core of the Monte Carlo method is the use of statistical distributions for
random variables. Evidence reported in the literature review suggests that the statis-
tical distribution used to develop pressure coefficients for building roofs is erroneously
utilized in the design of building applied PV systems. Consequently, wind tunnel test-
ing for the development of refined pressure coefficient statistics has been included in
this thesis to reduce model error and to illustrate how the simplified code method
fails to provide uniform performance across actual systems types when engineered
according to current code application guidance. In contrast, steel design and struc-
tural component design are well established fields with proven methods for translating
member loads into member sections and properties [7]. Furthermore, once the loads
tural systems with similar scale members as seen in light weight roof and wall frames
and warehouse storage racking. Because the wind load determination is the primary
source of risk, this thesis is focused primarily on these loads and will use a simple
example failure mode for resistance so that a complete limit state may be analyzed.
Based on the thesis goals and assumptions, the following tasks are structured
fied code method through a baseline Monte Carlo simulations that is first refined to
20
1.3.1 Development of a general probabilistic model for assessing struc-
tural response
The proposed risk measure will be based on fragility analysis to establish a method-
ology for performance based engineering risk assessment of code designed residential
PV systems. The code designed systems for evaluation will follow the American
Society of Structural Engineering (ASCE) method for wind load determination and
the American Iron and Steel Institute (AISI) method for structural member sizing.
tural response to a set of wind speeds that result in wind loads. The outcome of
tial PV system engineered according to the structural building code and prescribed
wind speed map. The probability of failure quantifies the likelihood of failure, when
exposed to a specific wind speed from the set of wind speeds. The wind speed set
includes the design wind speed (e.g. 90 MPH) along with lower and higher wind
found in the wind load structural demand and the PV system’s structural supply.
While the focus of this thesis is on the structural demand incorporation of a simple
demand on reliability. Therefore, each Monte Carlo sample will combine the supply
and demand associated with a simple limit state example to determine a sample
outcome of either survival or failure. A large set of samples are then aggregated for
each wind speed to develop a probability of failure or its inverse, the probability of
survival.
The conditional probability of failure developed in this task is most useful in cases
when a decision maker benefits from maintaining separate structural and climate
models. This is common, when a project developer has conducted technical due
21
diligence on a site and developed a more accurate wind model than available in the
code. Alternatively custom wind models may be developed to account for uncertain
As discussed the Monte Carlo method requires statistical models for uncertain pa-
rameters. Most of the nominal values and the expert derived uncertainty statistics
accepted for use in reliability analysis of ASCE and AISI are also accepted for analysis
of residential roof mounted PV systems. However, the ASCE wind pressure coeffi-
cients are reported to be erroneous when modeling wind loads on roof mounted PV
systems [17]. Wind tunnel testing has been conducted to reduce error by updating
the wind pressure coefficients for residential rooftop mounted PV systems. The wind
tunnel experiment was designed to address the relevant set of parameters identified
through a literature review of prior PV system wind tunnel studies and by following
useful for decision makers who maintain independent structural and climate models.
For some decision makers, combination of these models into a general reliability risk
of failure. This is accomplished by coupling the fragility model with the wind hazard
wind load model will be quantified and reported. The reliability risk measure will be
used to compare the reliability of alternative system types engineered to code, and
also used to benchmark against system types and also against codified target values.
22
1.4 Organization
Chapter 2 will then review the prevalent aerodynamic e↵ects that govern wind loads
from PV system failure under wind loads will be presented. Finally, engineering design
methods used for residential PV system structures are presented and demonstrated
from the literature. A general fragility model for PV system structural failure will be
developed. Finally Chapter 3 will demonstrate the fragility model on the three system
types under consideration using a single limit state example, that is representative of
Chapter 4 will present the experimental e↵ort to reduce model error in the wind
sive literature review will be presented on the current state-of-the-art approaches and
Next, the wind tunnel experimental approach will be described and related to the
literature review and ASCE code requirements. Finally, the measured wind load
wind models. The reliability model will be demonstrated via case study on each sys-
tem type with the consistent limit state example. The results will be used to assess
whether code engineered systems have a consistent reliability across system types and
Chapter 6 will summarizes the main points, reviews key findings and conclusions.
23
Chapter 6 also identifies areas of investigation for future works.
24
CHAPTER II
Despite the significant material and labor devoted to a residential PV systems struc-
tural response to wind loads, relatively little scientific attention has been given to the
determination of wind loads on residential PV systems [17]. This Chapter will present
fects on residential PV systems and wind related losses. Finally, a case study will
review and demonstrate current design practices on ten design cases applied to three
roof configurations.
Among the current commercially available PV systems, silicon solar modules applied
over the residential roof surface dominate the market with over 90% market share
[9]. Thin film technologies, including CIGS, CdTe and amorphous Silicon make up
the rest of the residential market, most of which are also applied in modules over
the roof top with a small percentage applied as building integrated Photovoltaic
(BIPV) roof shingles or integrated panels replacing roof components. Within the
silicon solar module market, equipment providers have responded to unique costumer
requirements by developing divergent racking system types and have gained significant
market share. In comparison to the residential market at the time of the Solar ABC’s
in structural racking systems. Currently, there is risk that our ad-hoc approaches do
Racking system types are categorized in this thesis according to the number of
25
Figure 6: The 0-D, 1-D, and 2-D Racking System types
26
(0-D), 1-Dimensional (1-D), and 2-Dimensional (2-D) configurations. Figure 6 shows
examples of a 0-D system with isolated connection, a 1-D system with linear rail mem-
ber spanning multiple modules, and a 2-D PV racking system with members spanning
tension members in the Z plane but this has not yet been introduced to the market.
the structural spanning member is the primary mode used for aggregating modules
in a shared structural system. The length and area of module surface supported by
a single structural member impacts the area averaged peak wind load experienced by
Common to each racking systems type are modules mounted parallel to the roof
top with a gap of 2-6” between the roof surface and module back. Modules are
configured on the roof with unique layouts defined by solar access, fire code set backs,
wind zones, and power capacity requirements. All configurations utilize a gap between
typical gaps range from 1/8 to 1/2”. Configuration parameters are believed to impact
the aerodynamic e↵ects that create wind loads on the array [17] [60] [71].
More holistically, the structural system impacts a broad set of cost including
material costs, roof penetration hardware and flashing, labor requirements and even
module reliability. Due to these broad implications, it is not suggested that one
system type should be preferred over the other based on the wind loads, but rather
systems and devices for PV modules has reinforced the importance of wind loads
and, designated a static load test designed to simulate wind loads. Additionally, the
standard specifies a snow design load and test procedure as well as electrical bonding
27
Figure 7: Schematic of daytime atmospheric boundary layer[48]
terms and phenomena will be provided. Early work to characterize wind and wind
e↵ects on structures was conducted by the American Society of Civil Engineers. Wind
can be simply described as air with a non-zero velocity relative to a frame of reference.
The velocity field may be along a single axis or circulate with a spectrum of eddy
building, the velocity field of wind may include components along the building length,
transverse to the building, and orthogonal to the building plan [43] [11].
Uniform wind, also referred to as free stream flow, typically only occurs in the ab-
28
such as the ground plane a boundary layer forms with zero velocity at the bound-
ary. Farther away from a boundary is a transition zone, known as the atmospheric
boundary layer (ABL) which exists up until wind flow is no longer e↵ected by the
boundary and free stream flow conditions exist. Within the ABL large scale eddies
form that break up into small eddies and eventually convert to heat or aggregate back
Obstructions such as trees, buildings and solar PV systems alter flow conditions
by forcing a change in the velocity field’s speed and direction. Bernoulli’s theorem
establishes a relationship between free stream flow and flow around an obstruction
by relating the dynamic pressure, qo (Equation 17) and static pressure, po with the
Equation 18.
q = 1/2⇢(V )2 (17)
and
A relevant phenomena occurs when the local velocity around an obstruction be-
comes greater than the free stream and a negative pressure is developed contributing
to a normal force vector away from the surface. Alternatively, the local velocity may
decrease and pressure increase resulting in an increased local pressure and normal
force vector towards the surface. When an obstruction such as a solar module has
both top surface and bottom surface, both surfaces see a local pressure and the corre-
sponding net force vector is the resultant of top and bottom vectors. For the purpose
of predicting structural responses, it is convenient to resolve the net force into a com-
ponent parallel to wind at the roof surface, drag, and normal to wind at the roof
29
Drag( f orceparalleltowind) = CD qA (19)
Where A is the surface area and CD and CL are experimentally determined wind
pressure coefficients for drag and lift with sign convention shown in Figure 8.
Figure 8: Sign convention for aerodynamic lift applied to residential solar PV systems
Early wind tunnel work to determine wind pressure coefficients was conducted
on flat plates. Figure 9 illustrates that drag was observed to monotonically increase
lift coefficient of 0.7 to 0.9 occurring between 35 and 50 . At 0 , parallel to the free
Generally, the trends from flat plate research apply to buildings, a low pitch roof
top experiences less drag and more lift than a steep pitch roof top [11]. However,
and Bernoulli e↵ect lift and drag. Furthermore, wind that approaches a building
corner is subject to being turned along the building face then flows over the roof
edge in a vortex [16]. State of the art emperical models lack the capability to predict
behavior associated with the complex flow behaviors around a building with adequate
accuracy for engineering design [39]. Current practices rely on experimentally derived
30
Figure 9: Experimentally derived pressure coefficients for flat plates with aspect ratio
= 1 adapted from Hoerner [43]
systems.
Despite the challenges in characterizing aerodynamic phenomena, the e↵ects are often
but not always readily apparent. Wind is known to be a contributing cause to module
failure modes including broken interconnects, solder bond failure, broken glass, me-
chanical connection failures, and module frame structural failure [54] [74]. As shown
multiple failure modes share common e↵ects making failure forensics a challenging
and growing field. For example causes of solder bond failure include wind deflection,
thermal cycling, vibration during shipping, poor manufacturing quality, and poor
module design. The e↵ects of module failure can include, power loss, ground fault,
31
Table 2: Failure Mode E↵ects Analysis for Structural failure
been reported in the literature or through the National Renewable Energy Labora-
from residential PV system installations in New York, New Jersey and Louisiana
after Hurricane Katrina and Hurricane Sandy. It is possible that the critical wind
speed did not occur at any specific residential PV systems or that critical wind speeds
approached the system from a direction that did not cause peak structural e↵ects.
Over time, as the total installed fleet of residential PV systems grows and accumu-
lates more wind exposure, failures should be expected to occur at a rate consistent
with predictive reliability models if the sources of uncertainty are characterized and
accounted for.
In the event of a wind caused structural failure the nature and severity of the
e↵ects will be influenced by the configuration and size of system components. Figure
1 (See Chapter 1) illustrates a typical residential PV system with 1-D structural rails
attached to a residential roof composed of rafters, decking and water proofing layer. A
L-foot is attached to the rafters with lag bolt and provides a pin connection to support
a rail, which is the primary structural member. Flashing around the L-foot prevents
water intrusion at the penetration. The rail is visible on the left side of the roof
with no modules attached and in section. Modules mounted parallel to the roof are
attached to the rail with a module clamp that transfers wind pressure to the rail as a
point load. Modules are certified with a test load of 45 pounds per square foot (PSF)
[4]. Under wind pressure, multiple failure e↵ects are possible including plastic rail
32
hinge will either deflect away from the roof or into the roof. In either case a change
in the gap between modules will occur and change the aerodynamic performance of
the system in an unpredictable manner due to the limitations in the current body
of knowledge. Furthermore, a plastic hinge may also change the modules angle of
attack with respect to the wind and also result in an unpredictable change in wind
loads. With the propensity for a sudden change in wind loads it is possible that
loads will exceed the design loads for other system components, such as lag bolts,
rail with plastic hinge may damage the attached module and be electrified. If a rail
and result in a system shut down. Alternatively, if the grounding is not operational
due to improper installation or other failure, the rail may remain electrified at high
mounting rail experiences a plastic hinge, multiple scenarios are possible ranging from
Other wind related structural failures may occur in the module clamps, lag screws,
rail attachment bolt (T-bolt) pull out or even failure of the module frame or laminate.
Non structural failure modes may also occur due to wind, these include electrical wire
movement and chafing that results in an electrical fault, deflection of roof penetrations
that results in water intrusion. Due to the focus on uncertainty in wind loads, this
thesis will use rail plastic hinge formation as an example failure mode to demonstrate
The PV system structural design process consists of wind load determination, stress
analysis, then member and connection strength check. The methodology for each
33
of each system types under consideration.
In the design of PV racking systems, load combinations with wind as the principal
action govern member sizing unless designing for high earthquake prone or high snow
regions. A consequence of wind loads governing is that the wind loads used in design
significantly a↵ect hardware and labor cost [18]. Currently, an engineer is required
by code to use American Society of Civil Engineers (ASCE) Chapter 7 2005 or 2010
dealing with residential roof wind load calculations to determine the structural loads
for the design of racking systems that maintain a gap between modules and the
roof plane. Wind loads are determined through calculation of a dynamic pressure q,
[11] [18].
Dynamic pressure The first step in determining wind loads is estimation of the
where
kz = Exposure dynamic pressure accounts for acceleration due to height and terrain
roughness
kd = Directionality factor accounts for the likelihood of peak structural action occur-
V = Basic wind speed representing the 50 year mean recurrence interval (MRI) 3-
34
This approach for evaluating dynamic pressure combines risk management factor,
I, with physics based factors kz , kzt , kd and V . The importance factor, is established
based on human life and property hazard commensurate with a the ASCE building
for a 25 year mean recurrence interval (MRI) for structures with low hazard while,
I = 1.15 calibrates velocity to a 100 yr MRI for high hazard structures [11]. A
consequence of this risk management structure is that when the physics based factors
in the dynamic pressure calculation are inconsistent with the actual scenario of use,
the structural risk of the as-built system is also inconsistent with code intentions for
risk management [49]. In recognition of this opportunity for error, ASCE provides
may be updated through analysis of local wind patterns and the gust e↵ect pressure
Design Pressure
calculated using equation 22 and one of the alternative ASCE methods to evaluate
GCp and GCpi where, GCp is the pressure coefficient applied to the external surface
oriented away from the building in the negative lift direction and GCpi is the pres-
sure coefficient applied to the internal surface oriented towards the building in the
positive lift direction. For a PV module applied above a residential roof, the concept
of external and internal are loosely applied since both surfaces are external of the
The main wind force resisting system (MWFRS) method is intended for the design
of structural systems experiencing wind pressure loading from a large surface area
35
and the components and cladding (C&C) method is used for the design of structural
members which are in direct contact with the applied wind load and have a small
surface area in contact with the wind. Components or cladding structural elements
”transfer” the applied loading to the the MWFRS [11]. ASCE application guidance,
published by the Solar American Board for Codes and Standards advises use of the
MWFRS method for Low Rise Buildings for the design of PV racking systems and
advises the use of C&C for Low Rise Buildings for module mounting hardware [17].
The MWFRS method for low rise buildings provides a zoning figure and corresponding
table of pressure coefficients shown in Figure 11 that provides load cases to account
for orientation relative to the wind, leading edge or trailing edge, and proximity to
the eaves. The MWFRS method simplifies the physics and assumes a discrete change
in GCp occurs across an infinitely thin boundary between the interior and edge. The
components and cladding method takes a similar approach but zoning accounts for
proximity to the roof eaves, edge, and ridge. Also, each zone’s pressure coefficient
accounts for the area ”supported by a single structural member [55]” this area is
referred to as the tributary area [11]. In the case of a PV module with a structural
rail along (or o↵set from) each edge (Figure 10), half the area between each rail
constitutes the tributary area. In contrast, the full area between rails may a↵ect the
rail’s structural load, it is refereed to as the influence area and is used later in this
chapter. Figure 10 illustrates the di↵erence in tributary area and influence area for
a 1-D structural system with interior rails BH and CD adjacent to an empty column
defined by BCIH.
ASCE prescribes GCpi values for use with both MWFRS and C&C methods.
Table 3 shows the prescribed value is dependent on the degree to which the structure
roof flashing, wires, wire management devices and in some locations pest management
36
Figure 10: Diagram of tributary area and influence area for interior edge rails
devices. Classification of this cavity as open, partially open, or closed has received
”Based on discussions with experts in the field of wind tunnel testing and
the ASCE Standard, [the authors of the Solar ABC’s guidance publication]
believe that a value of +/- 0.1 to +/-0.3 is a reasonable choice for systems
the wind load determination approach and to generate results for discussion. For
the purpose of the case study, a two story residential home in Atlanta Georgia will
37
Figure 11: ASCE 7-05 Fig 6-10 MWFRS External Pressure Coefficients
be assumed with limited topographic variation in the vicinity. Each system type is
system results in 20 discrete structural influence areas, the 1-D system results in 4
discrete structural influence areas, and the 2-D system results in 5 possible structural
influence areas. From all possible 2-D influence areas, only the 3 influence areas
composed of full columns are selected for the case study. As will be shown, a design
case is not only defined by the influence area but also by the placement on the roof,
relative to the zoning. Each influence area is assigned two locations, one in which the
roof edge is along the right side of the array and the other in which the roof edge is
along the left side of the array. In Figure 14 the possible influence area design cases
are represented by 18-1 through 18-20 when the roof edge is along the right side and
38
18-21 through 18-40 when the roof edge is along the left side. From the 68 possible
design cases, 10 design cases shown in Table 4 are selected for analysis by eliminating
Figure 12: Influence Areas for OD, 1D, and 2D PV systems configurations
To proceed with the case study, the dynamic pressure is calculated through deter-
mination of the following parameters for use in equation 21, resulting in q = 10.9psf
h = 20 ft
39
Table 4: Design case parameters
kd = 0.85 is a nominal value accounting for the likelihood of peak wind pressure
Zoning according to ASCE Figure 6-10 (Figure 11) is the next step in defining the
wind pressures for each design case [11]. Sample zoning diagrams are illustrated in
Figure 13. Four load combinations are considered to determine the maximum positive
pressure and minimum negative pressure. For structural systems that span multiple
zones, an area averaging approach is taken to determine the pressure coefficient. Zone
percentages and corresponding GCp and GCp + valued are reported for roof angles
mph dynamic pressure from Equation 21 and a nominal internal pressure coefficient
of 0.18 into Equation 22 yields design pressures for each design case shown in Table
6.
40
Figure 13: Sample zoning of 1D, and 2D systems
41
Figure 14: Influence Areas for OD, 1D,m and 2D PV systems configurations
42
Table 5: Calculated design case GCp values
43
Table 6: Calculated design case design pressures
Design P+ 15 P- 15 P+ 30 P- 30 P+ 45 P- 45
Case
18-101 -10.36 -15.67 5.80 -9.01 5.80 -9.01
18-102 -10.36 -15.67 5.80 -9.01 5.80 -9.01
18-103 -7.90 -10.98 5.06 -7.77 5.06 -7.77
44
18-104 -7.90 -10.98 5.06 -7.77 5.06 -7.77
90-101 -10.36 -15.67 5.80 -9.01 5.80 -9.01
90-102 -7.90 -10.98 5.06 -7.77 5.06 -7.77
180-101 -9.13 -13.32 5.43 -8.39 5.43 -8.39
180-102 -7.90 -10.98 5.06 -7.77 5.06 -7.77
270-101 -8.76 -12.58 5.31 -8.14 5.31 -8.14
270-102 -7.90 -10.98 5.06 -7.77 5.06 -7.77
Wind pressure determination conclusions This section discusses relevant
observations from the calculated pressure coefficients and corresponding design pres-
sures. For a roof angle of 15 the positive design pressure is negative. Flow separation
is principally caused by the building wall area. A positive pressure occurs when flow
streamlines impinge on the array surface, yet because of the low 15 roof slope, the
streamlines and array do not come into close proximity resulting in a negative pres-
sure for all approach angles. Higher pitch roofs do have a positive pressure because
the roof slopes up into the separated flow resulting in an impinged flow and positive
pressure.
The tributary area of a design case does not a↵ect the pressure coefficients within
a zone. This is an e↵ect of using the MWFRS method and is inconsistent with the
C&C method which would apply a tributary area factor to scale the design pressure
Finally, design pressures for the 30 and 45 roofs are the same for each design case.
ASCE attempts to balance accuracy with usability resulting in the need to envelope
cases that are believed to have similar load e↵ects. However, if a PV system configured
in one portion of the roof such as the top half, it may experience significantly di↵erent
45
CHAPTER III
FRAGILITY ANALYSIS
State of the art practice for residential PV system design utilizes prescriptive methods
oped for nuclear power structures, and applied to light frame wood structures, damns
and other structures have not been applied to solar structures. Barriers to the ap-
plication of PBE tools include a large number of complex failure modes, proprietary
components with limited published actual performance statistics, and hard to de-
termine wind load statistics. Nevertheless, the shift to PBE is motivated by the
perspective that adoption of residential PV systems does not present a new isolated
risk but is actually a revenue generating asset that displaces existing risk from cur-
of providing a net benefit to society measured in lives lost per trillion kWh. Relia-
bility quantification provides a foundation for decision makers who are designing PV
A comprehensive reliability assessment for wind loads may be viewed in two parts,
fragility analysis and coupling with extreme wind load models. Fragility analysis
results of a fragility analysis provide a probability of failure for each wind speed in the
set of wind speeds. Typically, the set of wind speeds is selected to span from near zero
probability to near certain probability of failure. Extreme wind load modeling entails
46
statistical analysis of a site or regional historic wind speeds. The result is an extreme
annual wind speed model and error quantification. Coupling a site specific wind speed
This chapter will present background on reliability and structural engineering top-
ics common to both steps in reliability assessment. Subsequently, the fragility analysis
case study. The methodology for fragility analysis incorporates Monte Carlo analy-
sis to evaluate the probability of failure at a wind speed given uncertainties in the
structural resistance, R and load e↵ects, E for a given wind speed. The Monte Carlo
method is considered a high fidelity method compared to low fidelity code method
because parameter distributions are utilized for uncertain variables in lieu of nomi-
nal values and global or partial safety factors. This chapter will also present a brief
distributions applicable to both resistance and load e↵ects required for Monte Carlo
analysis.
3.1 Background
To support the development of fragility based risk performance measure for residential
certainty. This section will provide a foundation for specific sources of uncertainty
47
ASCE code and Monte Carlo methods recognize uncertainties do exist throughout
ties of the resistance and load e↵ect models due to simplifications of actual
ments; gross errors in design, during execution and use; lack of knowledge
Sources of randomness include variation in the topography, extreme wind speed, and
extreme wind direction [33]. Randomness in material properties and geometries oc-
uncertainties occur due to limited records of historical extreme load including ex-
treme wind speed and direction [8]. Additional statistical uncertainty stems from
limited records of load combinations. Uncertainty of the resistance and load e↵ect
models occur due to simplified loading patterns, treatment of dynamic wind loads
with quasi static state assumptions and idealized boundary conditions. Vagueness
performance such as low deflection and low vibration are treated with simplified pre-
tunnel studies of residential PV systems suggest that gross errors in design may occur
[40]. Errors in execution are also believed to occur due to multiple causes, but most
48
visible indication of rafter location. Finally, lack of knowledge concerning material
degradation has been reported due to combined operating and contextual variables.
For example the combination of low temperature and structural load e↵ects have
While each of the sources of uncertainty can not be explicitly accounted for within
the scope of this thesis, the general objective relates to them in two ways. First, the
lished. while this may be generally true across structural and building engineering,
rapidly evolving materials and methods and also dependent on long term perfor-
PV systems will hopefully enable experts throughout the PV supply chain to under-
limit states as the basis for prescriptive safety factor based code methods. This section
will review key points and present an expanded view of limit sates for use with Monte
Carlo analysis.
boundary condition and load variables. Each variable is free to have a value from
the subset of all possible values existing in the n-dimensional space Rn [29]. Each
characterized by loads e↵ects e which exceed the resistance, r, these states compose a
49
failure set. In the context of a structural requirement, the failure set indicates an un-
satisfied requirement. In the simplified linear approximation method the tangent line
through the closest point of intersection between limit state surface and probability
density function is used to represent the limit state. Alternatively the Rn dimensional
boundary of the limit state characterizes a failure surface where each point on the
surface is a structural state at the limit of failure, for this reason the surface is also
known as the limit state surface. Further, the limit state surface separates a safe set
from the failure set. Mathematically, the limit state surface is defined by ”the set of
zero points for a piecewise di↵erentiable function g(x1, ..., xn) (Equation 23) which
is di↵erentiable everywhere in the domain and takes positive values in the internal of
the safe set and negative values in the internal of the failure set” [29].
G(X) = 0 (23)
Exact Solutions
Given the random variables E and R are function of X, an exact solution for the
Similarly an event B must be defined as the event that occurs when R < X,
50
Given P(A) and P(B) and using Equation 13 a di↵erential probability of failure
Z1
Pf = R (x)'E (x)dx (27)
1
Use of Monte Carlo simulation as an exact method for limit state design is one of
the established options for meeting explicit design requirements established through
performance based design. A benefit of this method is the ability to examine the
modify nominal values as needed to meet specified performance targets. The process
Traditional engineering practice has focused on ensuring life safety through the appli-
cation of safety factors, first through ASD and more recently through LRFD. These
approaches have been criticized for three limitations (1) failure to provide explicit
safety and (2) exclusion of other ”desirable attributes” including serviceability and
(3) lack of documented intent which encumbers development of novel designs [65].
51
Figure 15: Probability density function of load e↵ect, E and Resistance R illustrating
failure region [5]
mance under a specific hazard severity such as continued use during an annual extreme
ysis requires definition of a limit state that relates a specific structural action such
used to relate occurrence of the limit state to a hazard’s demand. Given the uncer-
tainty in both demand and structural response, performance based design commonly
expresses the probability of a limit state through the conditional likelihood of a struc-
tural action given a hazards demand, D reaching state x joint with the probability of
D = x as shown in Equation 28
X
P (LS) = P (LS|D = x)P (D = x) (28)
52
The conditional limit state probability represented by P (LS|D = x) is the struc-
focused on nuclear facilities due to the high hazard levels [76]. In the early use of
fragility models, structural outcomes have been expressed in terms of both limit states
[33] and damage states [26]. While a limit state expresses the notion of a structural
demand exceeding the structural capacity for a specific failure mode; a damage state
extends this concept to include the physical and/or monetary impact of the failure
mode.
Fragility models may be represented with fragility curves or with the lognormal
ln D
✓i
Fi (D) = ( ) (29)
i
the median value of the probability distribution and i is the logarithmic standard
deviation .
Graphically, the fragility function is represented with the fragility curve in Figure
16 with structural demand on the X axis and failure probability on the Y axis. The
fragility curve may be interpreted by the statement ”at demand ✓ there is a 50% prob-
ability of failure”. Multiple fragility curves may be plotted together to illustrate how
53
Figure 16: Illustration of lognormal fragility function [26]
Alternative methods for the development of fragility models are recognized by the Fed-
eral Emergency Management Agency (FEMA) with the preferred methods including
test data or field data documenting system reliability under a range of structural
demands. When no field data is available, fragility curves may be derived through
[26].
In order to account for the random variables that a↵ect structural response,
fragility model derivation utilizes Monte Carlo simulation ”to explore the e↵ect of
the resulting capacity” [26]. After sufficient iterations are completed to sample from
the full distributions of each random variable, a median value and calculated stan-
mends the model standard deviation be corrected for epistemic model uncertainty
54
based on the standard error required for the intended analysis. A discussion of error
assumed [26]
✓ = 0.92Q (30)
The single calculation method is based on expert judgment for typical model
dispersion and uncertainty. While this is not a preferred method it does o↵er a
viable validation approach when actual failure data or a reliable model with known
The Monte Carlo method is useful for estimating the distribution for a random vari-
ables which itself is a function of multiple random variable. The distribution estimate
cording to the central limit theorem, the sum of n random variables will be normally
mean of the distribution converges faster than the tails. This is particularly relevant
for reliability analysis which is predicated on the occurrence of low frequency events.
The error with with which a sample distribution estimates a parent distribution is
given by Equation 31 where n the standard deviation of the sample quantifies the
must be increased by k 2 .
55
p
n = / n (31)
Fragility practice was extended to treat wind hazards in the 1990’s in response to the
residential construction [65]. Observed failure rates due to wind hazards were not
consistent with the reliability predicted from LRFD. This inconsistency challenged
insurance premium underwriters to set premium rates and policy makers to estab-
lish hazard management policy. This motivation led to the application of fragility
analysis to wind hazards [65]. In the application of fragility analysis to wind hazards
(Equation 29), the demand has been treated as both the 3-second wind speed and the
wind pressure. Use of the 3-second gust necessitates that the random variables and
must be embedded into the fragility curve. This incorporates the physics required to
estimate an applied pressure based on a 3-second gust wind speed. Alternatively, use
an applied pressure.
model for the load and resistance. Wind load models combine a basic wind speed,
along with empirically derived site factors to determine the velocity pressure on a
structure per Equation 21. For example, an ASCE 7 compliant calculation calls
for the selection of an exposure coefficient kz based on surface roughness and mean
actual exposure coefficient that varies from the calculated model value. Similarly, the
actual structural response may vary due to uncertainties in the structural analysis.
56
Uncertainty, in both the load and resistance, must be accounted for in the application
The accuracy with which a wind load and structural response may be predicted is
structure used to represent measured data and predict future occurrences is charac-
for hazards of all types including wind must account for both aleatory and epistemic
The mathematical model (Equation 17) for velocity pressure attempts to account
for physical parameters which a↵ect the velocity pressure. The approach embeds
eling. As a result of the model form, model form uncertainty is introduced which has
been shown to contribute significantly to the total uncertainty [72]. Limited e↵orts
have been reported on the quantification of velocity pressure model form uncertainty.
Nevertheless, validation e↵orts have been completed which suggest that model input
uncertainty is the dominant cause of total uncertainty [50], [33]. Model validation
tion measures the capability to predict samples that have not yet been measured [72].
The scope of this thesis is limited to the treatment of model input uncertainty found
Wind Load statistics The ASCE approach to wind load determination incor-
porates both aleatory and epistemic uncertainty. Aleatory uncertainty stems from
57
natural variation in the basic wind speed, surface characteristics and wind direction.
wind speed maps, model error in the velocity exposure coefficient, and uncertainty in
and Gcp are random variables, and necessitate statistical modeling [31]. Starting in
the 1970’s e↵ort was made and reported on characterizing statistical distributions for
beliefs of 20 experts from industry and academia to characterize the wind load statis-
tics for use in PBD as shown in Table 7 [33]. The characterized wind load statistics
are for certain scenarios such as Exposure B with mean height z = 20f t. There is
limited evidence in the literature to suggest how these statistics vary across scenarios
but wind tunnel testing provides a practical approach for quantifying statistics for
specific scenarios of interest. The case studies developed in this thesis are developed
to match the cases for which statistical models have been published.
Pm is the ratio of mean to nominal value and COV is the coefficient of variation.
Pm = 1 indicates the mean strength and nominal value are equal, while Pm < 1
While the broad notion of wind hazards including air-born debris is generally appli-
58
structural loads that may contribute to excessive stress or deflection. Currently, in
most U.S. jurisdictions, the structural wind loads used in the design of PV systems
are determined through ASCE 7-2005 or 7-2010 ASD or LRFD. In both cases, a stake-
holder does not have publicly available information to discern the statistical likelihood
of a wind load induced structural failure during the 25 year expected performance
period. The information regarding structural performance and location specific wind
speeds are not coupled. In fact, it is not uncommon for a project developer to mea-
sure site specific wind speeds or select a sub-set of super-station historical data sets
to develop a more accurate assessment of local wind speeds than is available through
ASCE national wind speed maps. In this case, a structural analysis should not embed
wind speed such that local wind speed knowledge may be coupled on demand with
wind speed.
statistics. Each of these steps will be presented in the following sections and demon-
59
Table 8: Sample limit states for PV systems
of limit states. At low wind speeds one may judge the probability of occurrence for any
wind related limit state to be near zero. While at extreme high wind speeds, one may
PV system at a specific wind speed. Further, structural limit states in the current
code only incorporate failure modes that may result in significant damage or collapse
rather than serviceability related failures. So, while a code engineered PV system
may ensure reasonable yet un-quantified catastrophic failure risk, no such assump-
tion may be made regarding production risk or other serviceability risk. Given the
method for serviceability failure modes is needed for PV systems. The established
approach to quantify both serviceability and life safety risk is through limit state
based performance measure [29], [5], [51]. Over time limit states and associated per-
building protection, and life safety. Table 8 provides a sample set of limit states for
The concept of damage states also has a potential role in improved PV system
60
decision making by engineers, policy makers, financiers and insurance agents. How-
ever, the current body of knowledge may not yet support development of damage
states because of the complex interaction and uncertain relationship between struc-
tural limit states and system damage [74], [54]. Alternatively, probabilistic treatment
future work.
The wind load statistics generally accepted for Main Wind Force Resisting System
and Component and Cladding systems are considered applicable to PV systems [17].
However, PV systems attached above a residential roof surface have been shown to
experience unique fluid dynamics phenomena that substantially a↵ect the net pressure
coefficient [39]. These recent findings suggest that application of MWFRS pressure
Translation from the proposed limit states to fragility models requires identification
of the structural and mechanical behaviors for analysis or testing. For example, LS
#1 related to local bending stress may utilize beam theory to relate the wind load
into bending stress. Typical structural models for PV systems will include stress and
strain and may require multi-physics coupling with thermal and electrical models.
Dynamic models are required when the structures natural frequencies is less than
dynamic response below the 1Hz threshold [11]. Most frequently, wind loads applied
models developed for static wind load bending stress analysis will be demonstrated
61
3.4 Case Study: Fragility Analysis
an approach for fragility analysis on residential PV systems using sample LS#1, rail
with a stochastic analysis that accounts for the random variables in Equations 21 and
22. Subsequently, structural analysis for each design case supports determination of
the American Iron and Steel Institute (AISI) compliant member strengths. Finally,
the random variables that influence resistance will be introduced into the structural
analysis through Monte Carlo simulations to generate a CDF for G(X), the fragility
model.
Stochastic treatment of wind loads first requires a statistical model of the wind veloc-
ity pressure based on treating kz , kd and GCp as random variables defined by Table
7. Monte Carlo Simulation is used to sample values for each of the random variables
Application of the wind load statistics in Table 7 to the case study building and
site requires two assumptions to be made 1) Pm and COV values for zone 2 and zone
2E can be combined through a weighted average to generate Pm and COV values for
systems that span multiple zones and 2) Pm and COV for a 0 roof are also applicable
to 15 and 45 roofs.
Using the weighted averaged wind load statistics, a probability density function for
q can be developed through Monte Carlo Simulation with 1000 runs. The frequency
11.1psf and COV = 0.22. An additional 1000 runs results in less than 1% di↵erence
The nominal q value, (Calculated in Chapter 2 as 10.9 psf for the case study)
62
used for ASCE design has a 53% chance of exceedance. Introduction of the load
and resistance factored design (LRFD) load factor of 1.6 for principal action results
in a factored q = 17.44 which is 2.5 standard deviations from the mean and has a
result is made by replacing the 50 year return period wind speed V(50) with the
700 year return period wind speed V(700) calculated with Equation 32 [63]. Use of
V(700) in lieu of V(50) results is q = 17.5. This in only a partial validation because
the validity of wind load statistic used for sampling. For the purpose of this thesis,
the statistics derived from 20 expert opinions are considered to be a valid starting
project specific decision of great consequence, and the decision has high sensitivity to
the statistics used, a site specific wind tunnel study is recommended to update the
statistics.
p
V (700)/ 1.6 = V (50) (32)
Analysis of the required moment resistance is conducted for each O-D, 1-D, and
2-D design case. The wind pressures for each design case shown in Table 5 and
the configuration influence area diagrams shown in Figure 14 are treated as inputs.
structural analysis for each systems is then conducted through definition and analysis
of the following steps 1) tributary area, 2) boundary conditions, 3) free body diagram,
shear diagram and bending moment diagram 4) maximum and minimum bending
moment as a function of wind pressure for each design case and 5) factored load and
resistance.
63
Figure 17: Likelihood of exceedance nominal pressure in 90mph wind pressure distri-
bution
64
Figure 18: Likelihood of exceedance factored pressure in 90mph wind pressure distri-
bution
65
Tributary Area Determination of the tributary area for each case requires iden-
tification of the structural load path and geometry. As discussed in Chapter 2, the
tributary area di↵ers from the influence area identified in Chapter 2 because the
tributary area is the actual area over which the average pressure from the influence
area is applied to calculate a load [23]. O-D systems transfer loads from a module
attachment hole to a roof attachment bracket located at an adjacent roof rafter. De-
pending on the location in the array, a 0-D system may attach a single corner module,
up to two edge modules and up to 4 interior modules with a bracket. Given a roof
attachment bracket is located in the corner of each module, the tributary area for
the aforementioned cases are 1/4, 1/2 and 1 module area as illustrated in Figure 19.
For the purpose of this design case, the worst case tributary area of 1 module will
for analysis [24]. Currently, there are indications that the market could shift to ei-
ther larger modules for component count reduction or smaller modules for improving
layout flexibility and handling. Similar analysis is conducted for the 1-D and 2-D
system types. Figures 20 shows inset parallel structural rails under a column of mod-
ules with shaded tributary area extending from the module edge (aligned with guide
D) halfway across the module. The resulting tributary area is 1/2 module wide by
5 modules long for a total area of 2.5A. Similarly, Figure 21 shows a structural rail
located below the union of two module columns with half the area from each column
contributing to the 5A tributary area. While the assumed layouts for tributary area
calculations are representative of actual system layouts alternative layouts are also
common. For the purpose of this case study tributary area is considered critical and
Boundary Conditions and Free Body Diagram The tributary area along
with assumed module dimensions and typical residential rafter spacing allows for a free
body diagram to be produced. The free body diagram concentrates wind load applied
66
Figure 19: Example corner, edge, and interior tributary areas
67
Figure 21: Example 2-D interior rail tributary area
to one quarter of module surface as a point load at each of four module attachment
hole locations. The structural member transfers load from the module attachment
location to the roof rafter, the length of the load path depends on the permitted PV
system layout. For a 0-D system, a rafter may be centered between two modules or
o↵set to one side, alternative 0-D strategies can be found in industry [79], [78], [44]
however this case study is not representative of any specific commercially available
systems. For the purpose of this case study the rafter attachment is constrained to the
inner third of the member length, L as shown in the free body diagram and associated
shear and moment diagrams as shown in ??. Because the 0-D system type has a single
towards factory assembled 1-D systems to achieve greater overall labor productivity.
This approach utilizes two parallel structural for a column of modules, both located
along the module attachment holes. Roof attachment strategies for 1-D members vary,
68
for this case study the members will be treated as simply supported, a conservative
midspan supports[25]. 2-D systems, utilize a traditional 1-way structural system [23]
with a primary member located under two adjacent module edges and extending to the
primary members and transfers loads into the roof structure. For this analysis both
the primary and secondary members are treated as simply supported. Finally, for
each design case, free body diagrams are used to derive shear and moment diagrams.
Maximum and Minimum Bending Moment Using the location of the max-
imum moment found in the moment diagrams, first order statics are applied to de-
X
M =0 (33)
For example, the maximum moment from wind on a 0-D system is given by equa-
tion 34.
Mw = Pw A(2L/3) (34)
ASCE load and resistance factor design (LRFD) approach provides a framework for
combining probabilistic loads through load factors , along with a resistance factor
69
LRFD recognizes that transient loads due to unique hazards such as wind and
snow are unlikely to have peak values simultaneously and therefore assigns a load
factor max to the principle action Qmax,i and separate companion load factor i to
each of the companion loads Qni . Permanent loads Dn are allocated load factor
D which may be greater or less than one based on uncertainty and whether the
Using this framework, ASCE provides a series of load combinations that are used
to envelope the design. Each member or even member feature such as connection
details and cross section are typically governed by specific load combinations. For
the PV rail members under consideration, Equation 36 governs the required cross
section positive bending moment by accounting for the combination of actual dead
load D exceeding the nominal value by 20% , wind load W exceeding nominal by
2.5 standard deviations, and a companion snow load S of 50% nominal occurring
concurrently. Based on solar panel installation and operating manuals, no live load L
37 is used for uplift based on the likelihood of the actual dead load being 10% less
than nominal and no snow load counteracting the wind uplift [11]. Earthquake loads
also require consideration in some jurisdictions but are not considered further in this
case study.
Maximum moments for snow load, Ms and dead load, MD on the 0-D system for
Ms = Ps A(2L/3) (38)
70
MD = 2.5(4/18)L2 + 2 ⇤ Wmod L/3 (39)
by the American Iron and Steel Institute (AISI). Resistance factors for cold formed
steel are based on the failure mode. For lateral torsional bucking resistance factor
form as shown in Equation 40. Treatment of the principal action wind load, as a
moment capacity.
1
Rd = ( D Dn + max Qmax,i + ⌃ i Qni ) (40)
used to predict member strength. For common steel, the material yield strength
steel sections similar to those commonly used in PV system rails, moment strength
statistics (Table 9) have been experimentally measured and account for both the
greater than nominal strength and 21% greater than the factored strength. Further,
with a COV = 0.12 the factored strength has a 93% chance of exceedance.
71
Table 9: Cold formed steel section strength statistical model adapted from Schafer
2008 [67]
Parameter Pm COV
Mc r 1.09 0.12
Mcr
Ma = Md (41)
Mnom
Such that by combination of Equation 41 and 40 the actual moment strength is
Mcr 1
Ma = ( D Dn + max Qmax,i + ⌃ i Qni ) (42)
Mnom
.
Evaluation of the fragility curves is conducted using the criterion that a limit state
is exceeded when G(x) < 0. As discussed in Chapter 1 the exact method based
variables Ma and Md for wind speeds V ranging from 70 to 170 mph as defined by
Equation 43. Numerical integration of the area exceeding the limit state provides an
Moment demand can be calculated from the load combinations for each value in V
using Equation 42. According to the ASCE commentary, the value of the companion
load factor, i is set to represent an ”arbitrary point in time value” and the value of 0.5
the choice of a i value for use in the fragility curve is also arbitrary and up to
the decision maker using the curve. In practice, multiple fragility curves should be
72
composed with varying i values. Moreover, in the combined event of wind and snow
govern wind loads such that the uncertainty in pressure coefficient statistics should be
updated. For the purpose of this case study a value of 0psf will be used for i because
pressure coefficients measured without snow. Further investigation into the historical
frequency of combined wind and snow events is recommended along with evaluation
of snow a↵ects on wind load. Iteration of the G(X) function 1000 times for each
design case applied to 15 , 30 and 45 roofs yields the LS#1 fragility curves shown
in Figures 22 through 24 for a compliant system designed for a 90mph wind zone.
For the 15 and 30 roofs, the average ✓ is approximately 130 mph with less than 4
mph standard deviation. The 45 curve is shifted left with the average ✓ occurring at
The fragility curves shown assume that input uncertainty and model form uncertainty
are adequate to support a decision. The American Technology Council single calcu-
lation method method for fragility curves (Equation 30) provides a benchmark for
validation. The single calculation method is based on a nominal wind speed capacity
Q which is determined by setting the load factors to unity and solving for V given Md .
On average for the design cases, the nominal wind speed capacity Q = 110M P H.
Using Q and the ATC recommended value of = 0.4 the dashed curve shown in
Figure 25 is produced which has a theta less than 5% greater than the average ✓ and
While the single calculation method does provide limited validation that the Monte
Carlo simulation accounts for a typical amount of uncertainty in the structural calcu-
lation, it can not validate the underlying physical models. As discussed, the literature
73
Figure 22: LS#1 fragility curves for code compliant system, 15 deg roof, 90mph wind
zone
74
Figure 23: LS#1 fragility curves for code compliant system, 30 deg roof, 90mph wind
zone
75
Figure 24: LS#1 fragility curves for code compliant system, 45 deg roof, 90mph wind
zone
76
Figure 25: Validation curve for 15 deg roof
77
suggest the greatest risk to model validity is erroneous GCp values. Experimental
investigation of GCp values will be presented in the following Chapter along with
From the fragility curves presented, a clear performance statement for LS#1 can be
constructed to read, that design case 1 engineered to code for a 90mph wind zone and
installed on a 15 degree roof has a 50% probability of failure in a 130 mph 3-second
gust. Further, one can di↵erentiate between system types noting that 270 sq.ft. 2-D
systems types have a 44% average probability of failure at 130 mph wind speed on
a 30 roof compared to 58% for the 1-D system types. In conjunction with upfront
system cost, this is a valuable performance measure that can support the selection
Finally, system layouts can be evaluated based on the LS#1 performance measure
by comparing the design cases of similar types. For 1-D systems installed on a 15
roof, the edge design case has a 54% probability of failure compared to 64% for the
interior case when designed according to code. This illustrates that although the edge
may be subject to higher loads, the wind load statistics for the edge introduce more
conservatism than the interior because of the higher uncertainty, and corresponding
Clearly, the fragility curves are valuable for making a reliability comparison at a
given wind speed. However, some project specific decisions benefit from incorporating
a belief about predicted wind speeds. Most generally, one may believe that any
wind speed is possible and formulate a preference accordingly. Alternatively one may
believe that a wind speed greater than 90 mph has an acceptably low likelihood of
occurrence during the service life. In that case the fragility curves do not support a
preference between system types. Chapter 5 will explore and demonstrate methods
78
for incorporating wind speed statistics into the reliability assessment. First, Chapter
4 will examine the literature reporting on solar panel wind pressure coefficients and
79
CHAPTER IV
Current state of the art PV system design either utilizes pressure coefficients, GCP
derived from the building code or from system specific wind tunnel experiments.
Pressure coefficients prescribed by the building code are also derived from wind tunnel
residential buildings without PV systems [11], [12]. The current body of literature
[17].
In response to the call for residential PV system wind tunnel studies, a set of
experiments has been conducted as part of this thesis. To support the design of ex-
periment, a literature review has been conducted in three parts. This chapter will
present findings from the literature review. Also presented is this chapter are the
series of wind tunnel experiments conducted on the design cases under consideration.
Finally, the experimentally measured pressure coefficients are used to generate inde-
pendent wind load statistics for use in revised fragility curves and further reliability
analysis in Chapter 5.
Underwriters laboratory (UL) and the American Society of Civil Engineers (ASCE 7)
establishes structural loads for PV modules and racking system. UL loads are used
80
for general module and racking product design, while ASCE loads are used for specific
PV systems installations in the built environment. UL 1703, the safety standard for
PV modules and UL 2703 the standard for module racking systems both prescribe
a design load of 30 psf for wind forces and requires the product to survive a test
condition of 45 psf [4],[1]. A module or system is required to withstand the test load
the building code ASCE, Chapter 7-05 or 7-10 and must be designed according to
Recently guidance has been issued for how best to navigate the multiple methods
available in ASCE given that no one method accurately reflects the condition of
solar modules mounted over roof surfaces [17]. Wind tunnel testing has also been
from a cross section of the commercial and ground mount PV systems tested. Next,
the limited literature on residential PV system wind tunnel test will be reviewed
both for methodology and findings. Finally, guidance on the application limitation of
existing codes, along with potential changes in future code revisions will be reviewed.
The Solar Energy Research Institute analyzed wind loading on tracking and field
mounted solar collectors. Flat plate solar collectors are noted to be constructed with
over-built support structures which can lead to substantial costs. According to the
method for conducting scale wind tunnel experimentation using dynamic similarity
81
is identified such that force coefficients can be experimentally determined and then
applied to calculate lift forces for scenarios of constant Reynolds number Re. Tests
were conducted to identify e↵ects of varying conditions on lift coefficients GCp for
flat plate collectors. GCp for at solar panel mounted at 35 angle of attack was
found to be -0.9. Fences and wind shields with porosity of 0.3 to 0.4 were found to
Wind tunnel testing to determine wind pressure on solar energy systems was
conducted by Bronkhorst at the TNO atmospheric boundary layer (abl) wind tunnel
[19]. A boundary layer roughness length of 2.4mm was applied in 1:50 scale testing
for full scale roughness length Z of 0.12 m. The wind speed in the tunnel was
10.7 m/s with turbulence intensities of 15% longitudinal and 12% vertical. The
building full scale dimensions were 10 m heigh, 30m wide and 40m depth. Use of
The solar energy system was modeled with an inclination angle of 35 and a solid
base as shown in Figure 26. The configuration shown does not allow for any pressure
equalization to occur and would essentially result in a GCpi = 0. These results were
compared against CFD analysis that solved the Reynolds Averaged Navier-stokes
model or di↵erential reyonlds stress turbulence model (DSM). Both turbulence models
were found to produce well matching results with the greatest deviations when large
Wind tunnel tests for commercial flat roof PV systems was compared to CFD
analysis that employed alternative turbulent models. For the wind tunnel test, a
one-half scale module of a 10 sloped PV tile was constructed and connected to a five
component load balance. An additional 8 modules were placed around the connected
series of 92 tests. Pressure coefficients for all cases reported ranged between +/-0.2
82
Figure 26: Bronkhorst wind tunnel test reference [19]
Boundary layer wind tunnel testing was conducted by Kopp [53] to investigate
the e↵ect of building size, and e↵ective wind area on the pressure coefficients of low
tilt PV modules mounted on flat roofs. The experimental setup utilized 1:30 scale
and 30 were tested with inter-row spacing sized to prevent self-shading and 1.2 m
building edge set back (Figure 28. Modules were fabricated in 3 module panels with
12 top and 4 bottom pressure taps per module. The array was tested on three building
roof heights with the intent of varying the intensity of cornering vortices. Reynolds
number based on roof height was reported to be 1.9E5 about half the full scale value
but still greater than required by ASCE 67 [46]. Results showed that increasing
the building size and in particular the North South wall area, HL, increased the
wind loads. Further Kopp found that if one scales the e↵ective wind (or tributary)
area by the wall size, HL, then the area-averaged pressure coefficients, GCp , will
83
Figure 27: Meroney and Ne↵ wind tunnel test reference [57]
collapse onto a single curve for a particular array geometry [53]. This phenomena is
attributed to the conical vortices which form during cornering winds and strengthen
Banks reported on the e↵ect of cornering vortices on flat roof PV systems through
setups utilized exposure C with some exposure B, and typical wind tunnel blockage
buildings with 20Hx10H were also tested for large PV arrays. As shown in Figure 29 9
blocks compose a full 6H building. PV systems were tested on each block with 10 deg
rotation with six taps on top surface, and two on lower surfaces. Over 80% of test were
conducted between 1:40 and 1:50 scale. Pressure was sampled at 250 or 500 Hz for
30-120 sec per rotation angle. Geometric distortion from module thickness occurred
but it was considered less important than the gap between modules. The analysis
included use of transfer functions to correct for the frequency response of tubing. Area
averaging measurements from multiple taps was conducted to determine pressures for
each module. Due to model scaling, investigators utilized high frequency spectrum
84
Figure 28: Kopp wind tunnel test reference [53]
matching method as described by Banks (2011) and Dyrbye and Hansen (1997). This
method matches the energy in the spectrum at frequencies above the quasi-steady
Results indicated GCn varies with the gap above the roof h2, peak height above
the roof h1, row spacing and deflector design as illustrated in Figure 30. Modules
closest to the 0 and 90 edge with an o↵set of 0.03H and 0.2H were not found to have
separation bubble has little impact. Modules between 1H and 3H from the 180 edge
experienced uplift due to the reversed flow inside the separation zone. The author
recommends utilizing an edge zone of 3H wide along the edge to which modules are
oriented. The e↵ect of parapet height ph normalized to building width B (ph/B) was
found to increase uplift forces because the vortices are not disturbed up until a ph/B
ratio of 0.04 at which point the vortices are safely above the module height although
still stronger. Further, wind deflector design was found to have significant e↵ect on
wind pressures experienced by the module. The most dominant flow phenomena on
85
Figure 29: Banks wind tunnel test reference [16]
86
Figure 30: Banks array layout [16]
uplift forces was found to be corner vortices. In fact, cornering vortices were found to
significantly impact uplift of modules beyond the ASCE traditional edge zones. As
a result the authors recommended adoption of expanded edge and corner zones for
evaluate the e↵ect of tilt angle, building height and module location of solar modules
on flat roofs [70]. The experimental setup utilized a 1:200 module scale of a 20.6m
by 30.6m building and heights of 7m and 16m. Module tilt angles ranged from 20
to 45 (Figure 31). Positive pressure was found to be greatest for windward modules
and increased with module tilt angle. Wind uplift (suction) was found to be greatest
for the leeward location and decreased with inclination. The reference of the model
apparatus show in Figure 31 suggests the model building faade was not enclosed with
a wall which has been reported to influence the formation of cornering vortices a
Browne utilized wind tunnel experimental data to evaluate the e↵ect of load shar-
ing in ballasted roof-top solar arrays. The experimental setup utilized a 1:70 scale
building and array model with pressure taps distributed throughout the array to
record the variation in wind loads. The results indicated a high level of spatial and
temporal variation in loads due to the high frequency content of building generated
turbulence as illustrated in the sample time history of peak forces on a module (Figure
32. Due to the rapid rate of change, adjacent modules are not expected to see a peak
87
Figure 31: Stathopoulos wind tunnel test reference [70]
load simultaneously, thereby enabling the potential for load sharing among modules
that are structurally connected. Although the concept of load sharing is clear, de-
termination of how many modules actually participate in load sharing is less obvious
given the wide array of beam materials (E) and sti↵ness (I) used to connect modules.
Browne proposes a method for determining the number of load sharing modules that
and ballast load at each support provides a negative force that resists uplift. Lifting
a single module 50mm will engage any load sharing modules resulting in a higher
measured force used to perform the lift. Further, the participating modules will also
experience lift and have a decreased resultant force at the connection. Once the num-
ber of load sharing modules are determined a calculation may be made to determine
the ballast required at each module. Based on the results Browne concludes that a
significant reduction in ballast weight may be achieved but the actual amount should
vary based on the proximity to array edge. Further, Browne points out that this re-
duction in ballast may actually improve the safety of building occupants by reducing
the roof-top live-loads and should not put pedestrians in harms way because during
the design wind speed it is impossible for pedestrians to remain on their feet and
88
Figure 32: Sample wind pressure time series [20]
A full scale experiment was conducted on a residential 42 hipped roof in the Nether-
with dimensions 1.6m by 0.8m and 18mm thick were mounted on opposing roof sur-
faces (Figure 33). Each panel had 3 pressure taps on the top surface and 3 pressure
taps on the bottom surface mounted along the centerline. Pressure data was collected
for wind speeds greater than 7m/s and analyzed to provide top, bottom and di↵eren-
tial (net) pressure coefficients. Pressure coefficients recommended for use in building
code are -0.3 for uplift and 0.2 for downward acting loads. These recommendations
are only advised for use with a single row of modules in the center of the roof, all
other configurations are advised to follow conservative values in NVN 7250 or BRE
digest 489. The author recommends further work be conducted to explore actual
additional data was collected and compared against a wind tunnel study of the model
scale site. Geurts motivates the work by noting: ”The results from [the current body
of work] are appropriate for a limited range of roof forms”. The results are also
assumed to be very conservative, and hence uneconomical in many cases. Also, the
authors note the studies available do not cover common installation configurations
89
Figure 33: Geurts field test reference [40]
(Geurts and Blackmore 2013). The experimental setup utilized the same test roof and
modules as Geurts [40] and a 1:100 model scale with 1:200 atmospheric boundary layer
scale as shown in Figure 34. The model scale PV module utilized 24 pressure taps
disturbed over the top and bottom surfaces, which necessitated a 200mm full scale
panel thickness, approximately 3-6 times thicker than a typical panel. The apparatus
was rotated every 10 between 180-360 and 30 increments for the remainder. In the
analysis, data was screened to omit time series in which the mean pressure di↵erence
coefficient was not between -0.3 and 0.3 because this was believed to indicate a likely
value analysis was conducted on the pressure coefficients to derive values with a 0.02
distance from eaves due to the separated flow region along the windward edge. The
wind pressure was found to be independent of the gap between the module and roof
surface. The wind tunnel experimental data for Cp was found to be conservative
compared to full scale values for the 1 second averaging with 20% greater pressure
coefficients. Geurts concluded that for cases similar to those tested with a single
module away from the edges, design pressure coefficients of +0.4, -0.3 may be used.
For system configurations with a larger array of modules, as typically found in U.S.
90
Figure 34: Geurts residential PV module wind tunnel test reference [38]
Erwin et. al. reported on a comparative study of wind loads on PV modules per-
formed both at the FIU wall of wind (WoW) and RWDI boundary layer wind tunnel
[34]. The experimental setup utilized a single module mounted on three buildings
with roof slopes of 0 22.6 and 30.2 . The scale at WoW was 1:1 while at RWDI a
1:10 scale was used (Figure 35. Five tilt tangles were tested with the flat roof and two
with each of the pitched roofs. Forces were measured with four multi-axis load cells in
the full scale test and pressures were measured with 28 pressure taps distributed top
and bottom module surfaces. Analysis of the test data translated measurements into
of the parallel mounted case indicated less than 5% change in lift pressures between
the two pitched roof angles with 0 angle of attack, but a nearly 100% increase in lift
was observed from the 30.2 to the 22.6 roof pitches with 180 angle of attack (i.e.
module on leeward roof pitch). The cases with the module mounted on an incline
relative to roof pitch are more relevant to solar thermal applications and not relevant
to the scope of this solar photovoltaic study. For the flat roof, significant pressure
changes were observed to occur with varying module inclination angles for all config-
urations. Further, increases in lift forces were observed between measurements from
91
WoW and RWDI for flush mounted configurations on all roof pitches. The authors
suggest the di↵erence may be due to the mean wind profiles but do not provide a
conclusive explanation.
92
93
Figure 35: Erwin test article reference [34]
Motivated by discrepancy in the limited prior residential PV system testing which
reported low pressures on PV modules relative to bare roof surfaces for single modules
[38] and approximately equivalent pressures for PV arrays compared to roof planes
when the PV arrays are modeled as a monolithic plane Stenabaugh [71] studies the
ratio of space between modules, G, and the gap between modules and the roof plane,
H. In order to maintain adequate Reynolds number for laminar flow between the
module and roof plane, a model scale of 1:20 was used for testing. Stenabaugh noted
that choice of model scale prevented simulation of the full atmospheric boundary
layer due to missing low frequency content. The proposed solution was to conduct
for the reduction in net pressure compared to the pressure experienced by a bare roof.
ered to represent the bare roof condition. Comparison of the bare roof measurements
with prior testing at much smaller model scales 1:400 provided ”reasonably similar”
results. The most significant reduction in Ceq was achieved when the module height
was between H = 2cm and H = 4cm and larger gaps between modules, G, were found
to further reduce Ceq up to G = 12cm the maximum value tested. These results al-
lowed the authors to conclude that ”gaps between modules are essential for e↵ective
pressure equalization” with typical values of Ceq = 0.6 while the lowest values for
a single module were Ceq = 0.2 [71]. Further when area averaging across multiple
modules was considered, an even greater reduction to Ceq = 0.1 was reported. These
results indicate that both G/H and load sharing have significant e↵ects on Cpnet .
1:20 and 1:40 using partial turbulence simulation [36]. The work was motivated by
a desire to test residential building features such as PV systems which are not easily
94
reproduced at typical wind tunnel scales of 1:100 and greater. Fu cites a body of
literature that establishes small scale turbulence as being approximately equal to the
separated shear layer, 1:10 of a low rise buildings height. The small scale turbu-
lence is thought to be primarily responsible for the flow phenomena flow separation
slow moving gusts which may be accounted for in a wind tunnel experiment by post
processing the data with an analytical correction of the mean wind speed U by U.
Using procedures reported on by Tu, mean and peak pressure coefficients produced
through the partial turbulence simulation method matched well with values produced
from full atmospheric boundary layer simulation as shown in Figure 36. The greatest
discrepancy occurred in the leeward corner taps with the partial turbulence simula-
tion producing peak pressures with 20% lower values. The author does not suggest
which method has greater accuracy but rather concludes that they are comparable.
For the purpose of future testing including wind tunnel test conducted as part of
this thesis, this article indicates that wind tunnel testing is subject to error at least
some of the time, and that the magnitude of error may be 20% or greater in isolated
locations. While the spatial resolution of the test was not adequate to extrapolate
the results to estimate error when pressures are averaged across a whole module or
array of modules, the results do suggest that the error may decrease with tributary
area.
Mooneghi et. al. [58] reported on an improved approach for matching the high
interest to the authors because of the ability to test small structures with minimal
Reynolds number e↵ects, and high measurement resolution. The continued e↵ort was
cur because of high frequency turbulence intensity mismatch. The authors reference
95
Figure 36: Cp values (y-axis) mean(Left) and peak(right) from partial turbulence
simulation and full atmostpheric boundary layer simulation for 16 taps (x-axis) on a
gable roof at 45 degree angle of attack [36]
past works which showed high frequency turbulence mismatch e↵ects peak suction
pressures, most acutely near the roof corners. As part of the approach, the authors
calculate the desired model scale turbulence intensity based on ratios of integral length
scale and building dimension. Next the authors calculate an appropriate cut o↵ fre-
from the low frequency turbulence that will be analytically incorporated. After the
wind tunnel experiment the authors employ a probabilistic approach for determining
the mean and peak pressures based on the measured distribution of subinterval pres-
sures along with an assumed Gaussian distribution of the low frequency component
defined in Figure 37. Experiments conducted to validate the proposed PTS approach
showed improved results especially when the transverse and vertical components of
the low frequency turbulence were incorporated. Further, the authors conclude that
the method is adequate to allow larger scale models to be tested with PTS and even
engineering community and has been criticized as not provid[ing] adequate guidance
96
Figure 37: Illustration of subinterval, mean flow velocity, low frequency fluctuations
and high frequency fluctuations [58]
to the design professionals and code officials tasked with assessing PV installations”
[17]. The authors recommend engineers follow the main wind force resisting system
for the most common rooftop PV installations to verify methods and calculations
and ultimately incorporation of results into codes and standards. In order to execute
selected based on the building enclosure , with an open building have a GCpi = 0 and
partially enclosed building GCpi = +/ 0.55. The authors recommend use of a GCpi
between +/-0.1 and +/-0.3. To support the recommendation for testing, they discuss
that the pressure equalization phenomenon recognized by ASCE to occur with air
permeable cladding likely also occurs with PV modules and may reduce wind loads
on modules by 50% to 80% or more but they ultimately defer to wind tunnel testing.
tems the structural engineers association of California, SEAOC, issued Wind Loads
97
on Low Profile Solar Photovoltaic Systems on Flat Roofs to bridge the gap between
ad-hoc ASCE interpretations and adoption of PV module specific language. Net pres-
sure coefficients are provided for shielded modules that are within a range of allowable
configurations and located in either a corner zone, 3, edge zone, 2, interior zone 1 and
deep interior, 0 where each zone has a progressively higher pressure coefficient. The
depth of the edge zone and corners are equal to twice the building height, based on
the size of cornering vortices which only develop to full strength for building widths
four times the height. This points out the importance of conducting wind tunnel test
with commercial building width at least four times greater than the height. A reduc-
tion factor to account for module size is applied to modules with inclination angles
greater than 5 to account for array induced turbulence but not less than 5 because
of the decreased aerodynamic e↵ect. An edge factor is incorporated to account for the
occurrence of flow reattachment when a gap greater than the module characteristic
height, hc , where
hc = 1 + lp ⇤ sin(!). (44)
The edge factor increases linearly between hc and 8hc after which it remains
constant. Guidance for wind tunnel tests per ASCE 7 specifies inclusion of building
features that influence flow environment such as varied parapet heights, and other
roof top obstructions. It requires an array to be located in each of the roof zones and
even greater reduction may be used if a third party review of the wind tunnel test
Most recently, proposed revisions to ASCE 7-16 have been released by Kopp et al
combined with an array edge factor E. For modules within 1.5 chord lengths from an
98
Figure 38: Proposal values for a as a function of tributary area [71]
exposed edge E = 1.5 and for all interior modules E = 1.0. As shown in Figure 38
the allowable reduction due to equalization is 0.8 for a single module and 0.4 for a 4x4
configuration that shares loads. These values are 2-4 times the values reported [37]
for G/H ratios of 1 because the proposed revision imposes minimal restrictions on G
and H and therefore envelopes all allowable cases. Further the proposal recommends
that the new equalization factors be used with components and cladding pressure
coefficients as shown in Equation 45. The proposal does allow for greater reductions
p = qh (GCp )( E )( a ) (45)
ponents and cladding values. Ci‘ting prior research Vickery showed components and
99
cladding values to be overly conservative for low to moderate roof slopes with low
tributary area (7 ✓ 27 ) and for hip roofs with slopes greater than 27 . For
example, zone 1 GCp- pressure coefficients are proposed to increase from -0.9 to -2
for tributary area less than 10f t2 and decrease from -0.8 to -0.5 for tributary area
greater than 100f t2 . In combination with [37] these two proposals propose a refined
approach based on the exclusive use of components and cladding and in combination
the proposed values for a and GCp- result in design pressures 8 times greater for
components with a 10f t2 tributary area than compared to a 100f t2 tributary area.
From the available literature, module location [77], load sharing area [20], module
tilt and layout [34], [70], [53], roof geometry [77] and wall geometry [16] and most
recently array configuration measured by G/H [71] are critical test parameters that
have been proven to e↵ect measured wind pressure. Moreover, experimental facili-
ties and analysis techniques have been shown to impact results [34], reinforcing the
Based on the available literature and ASCE guidelines for wind tunnel testing, [11],
[13] wind tunnel testing has been conducted to refine the wind pressure coefficient
statistics for the residential PV system design cases under consideration. This section
will provide a summary of the test with a focus on reporting experimental conditions
ature. Finally, wind tunnel results for each design case will be presented and used to
re-evaluate the fragility curves from Chapter 3 with independent pressure coefficient
The wind-tunnel experiment was conducted in the 12 Fan Wall of Wind (WoW) open
circuit wind tunnel depicted in Figure 39 and described in detail by Fu[36]. Generally,
100
Figure 39: FIU 12 Fan Wall of Wind Tunnel with sample test article mounted on
turn table [10].
air is sucked through the 12 fans then contracted to reach the desired wind speed and
entering turbulence. Once at speed, the air passes through triangular spires and floor
mounted roughness blocks to generate the desired boundary layer characteristics and
turbulence.
NEL PROCEDURE [11]. The experimental approach for complying with each of the
1. Wall of Wind fans, trips and spires were configured and operated to produce
a velocity and turbulence profile shown in Figure 40 and 41 consistent with the
2. The spires and trips were also configured to generate micro-length scale tur-
bulence matching the atmospheric boundary layer (Figure 42). Marco-length scale
101
Figure 40: Test Section Wind Speed Profile [10]
102
Figure 42: Wall of Wind Power Spectra [10]
[10].
3. The residential building and modules were 1:20 scale. The PV module thick-
ness was scaled to typical module frame thickness’s and was uniform throughout the
4. The maximum model projected area was less than 8% of the wind tunnel cross
5. The longitudinal pressure gradients in the WOW were small enough to con-
scale and high enough test speed for Reynolds number independence.
103
7. 512 simultaneous pressure measurements were collected at 120 Hz and any reso-
nance in the pressure tubing was digitally corrected with FIU’s proprietary algorithms
Pressure taps on the top and bottom surfaces of the scaled solar modules allowed
tubing to the module manifold. As shown in Figure 43 the 12 pressure taps were
evenly distributed across both surfaces for a total of 24 taps and allowed for uni-
form weighting of measurements. The taps were laser cut through acrylic sheets and
connected to the manifold with embossed flow channels sealed by adhering top and
bottom layers. Each module was individually checked for flow continuity and leakage.
The modules were then attached over the roof surface with a 6” (full-scale) gap. To
resist wind loads during testing the module attachment clamp material thickness was
1/8” which corresponded to a 2.5” full scale. This hardware resulted in less than
10% blockage under the modules in the eaves to ridge direction and created a 2.5”
full scale gap between modules. The PV module array was configured on a 5x5 grid
with an isolated 1x5 column of modules followed by an empty column then a 3X5
sub-array as shown in Figure 44. This layout provided an exposed edge column (left)
and partially exposed edge column (right), as well as a partially exposed (center-left)
and fully surrounded (center-right) interior columns. Each of these scenarios are com-
loads on modules within the same roof zone. As depicted the array was permanently
Additional blockage under the modules was created by the module pressure tab
104
Figure 43: PV module pressure tap layout
tubing bundles attached to the module tube manifold and passing through the roof.
The pressure taps were considered to represent a bundle of electrical strings commonly
passing from a module back surface into a roof mounted combiner box. An air sealing
gasket was used to minimize artificial air flow between the building interior and PV
system. A gap between the ridge and top module edge was varied with set values
of 6”, 21”, and 36” (full-scale). To enable the variable ridge o↵set, the test article
roof was detachable from the building so that it can be moved up and down and
spacers were inserted to seal the gap. The test article allows for two unique ridge
conditions, exposed roof apex and contoured ridge cap. Alternative ridge conditions
were used in the experiment to further account for uncertainty in actual scenarios of
use due to variation in actual ridge conditions. The construction strategy described
necessitated the use of tape during the experiment to seal seams between roof surfaces
and wall surfaces. Tape introduced a surface condition that is di↵erent from the test
article construction but is considered to have small impact on wind induced pressure
Three residential building models were utilized for testing with three roof pitches
105
Figure 44: Test Article Array Layout
106
corresponding to ASCE published tables [11], these include 3 on 12 (15 ), 7 on 12 (30 )
and 12 on 12 (45 ) as illustrated in Figure 45. Due to the importance of wall area [16]
[11], the eaves height and E-W length was controlled for by having an instrumented
section with equal length that could be added to with modular extensions if needed.
Fixed pitch and height necessitated the N-S length to vary between buildings possibly
influencing aerodynamic similitude most notably the interaction with the turbulent
107
108
Figure 45: Test article drawings
??
The experimental test plan utilized a hierarchical structure composed of runs, points
Test article physical parameters varied between runs including ridge condition,
roof angle, and ridge o↵set. A full factorial experiment was utilized with roof angle
and ridge o↵set. Only the 6” ridge o↵set was tested with the ridge cap installed
and not installed as shown in Table 10. The ridge cap was considered most likely to
350 . Typically wind tunnel test employ symmetry to reduce the test duration by only
testing from 0 to 180 . However, the array layout asymmetry (Figure 44) enabled
a unique ASCE zoning from 0 to 180 compared to 180 to 360 when defining the
zone based on the leading edge, ridge and eaves and not including the trailing edge
foot influence area. Both ASCE Components and cladding zoning (red lines) and
1
Statistical analysis using paired T-test was conducted to verify this strategy by comparing wind
pressures measured in each zone.
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Figure 46: 3D CAD model of the wind tunnel test article with transparent roof
surface.
110
Table 10: Experimental Configurations
For each run and point the test article is in a fixed position for 30 seconds of
continuous testing and pressure data collection from each of the 240 pressure taps
(24 per module) to comply with ASCE 7-05 31.2 condition 7. Through the time scale
of 50 scaled 3-second wind gusts experiments were conducted at each point. In total
tm bm Um
= (46)
tp b p Up
where T =time scale, b = length scale and U = velocity scale.
111
Figure 48: Structural zoning for tributary area, TA = 275 sq.ft., Points 0-180.
The measured tap pressures were analyzed to provide area averaged pressure coef-
ficients for each design case influence area under consideration. This analysis was
tion Methodology For Small Structures” [58]. While the detailed methodology is
Local top surface and bottom surface pressure time series, Pi,net (t) were applied
to the local measurement area, Ai and summed for each influence area to generate a
net force time series F (t) for each influence area under consideration using Equation
47.
X
F (t) = Ai Pi,net (t) (47)
Force time series were then converted into net pressure coefficient time series,
Cpnet (t) using the reference height dynamic pressure, qref , and influence area, A,
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Figure 49: Structural zoning for tributary area, TA = 275 sq.ft., Points 180-350.
F (T )
Cpnet (t) = (48)
qref A
Subsequently, the pressure coefficient time series’ for each experiment, was cor-
rected for the macro-scale turbulence not experimentally modeled and analyzed in a
and minimum pressure coefficients, GCn for a 3 second gust [10] as originally de-
scribed and validated by Mooneghi and Irwin [58]. According to this method and
more general wind tunnel testing, GCn is reported with the sign notation illustrated
pressure towards the roof surface (downforce) while a minimum is typically negative
and denotes a pressure away from the roof (Uplift). For some roof angle, wind direc-
tion combinations, the maximum and negative are of the same sign indicating only
uplift or downforce can be expected for that specific configuration. Figures 50 and
51 graph the maximum and minimum GCp values by wind angle and illustrate two
critical conclusions.
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Figure 50: Maximum module GCp by wind angle
For a fixed wind angle there is significant variation of the maximum and minimum
on turbulent vortices that show the high pressure zones of the vortex to have a
narrow contact zone with the roof and that the location of the vortex contact zone
is sensitive to roof angle [16]. This finding suggests that as multiple modules are
aggregated to form a larger influence area, the area averaged pressure coefficients
should be significantly below the maximum and minimum local values. Second, the
extreme peak maximum and minimum are nearly 3 standard deviations above the
mean maximum and minimum indicating the low probability that the peak wind speed
direction will coincide with the direction of peak structural response. This e↵ect is
addressed by the directionality factor, kd which has published statistics [33] assumed
Once a set of maxima and minima for each 10 increment wind direction between
0 and 350 are determined, the sets are divided into two subsets corresponding to 0-
180 and 180-350 based on the unique zoning for each subset that supports multiple
design cases. Subsequently the subsets are enveloped by selecting the global maximum
114
Figure 51: Minimum module GCp by wind angle
Uplift
Pressure (-)
Downward
Pressure (+)
115
Figure 53: Asymmetric Envelope Approach
[33] and represented with mean and standard deviation as shown in Figure 54 for the
positive pressure coefficients and Figure 55 for the negative pressure coefficients. Both
data sets illustrate the single module (18 sq.ft. influence area) have large absolute
mean values and standard deviations compared to aggregations of modules with in-
fluence of 90, 180, 270 sq. ft for all roof angles. These results are consistent with the
variation observed in Figures 50 and 51 along with the rational for the components
116
Table 11: Wind Tunnel GCp+ statistics
117
118
Figure 54: Mean and standard deviation GCp + statistics by design case and roof angle
119
Figure 55: Mean and standard deviation GCp statistics by design case and roof angle
4.3.1 Case study fragility curves
In the last section the experimentally measured pressure coefficients were shown to be
statistically di↵erent than the code prescribed values. this section will examine the
the e↵ect on structural performance through revised fragility curves. The importance
Sub-Hypothesis Application of state of the art design guidance [17] to the di-
Revision of the case study fragility curves with the wind tunnel derived pressure
coefficient statistics results in the updated fragility curves in Figures 56, 57 and 58.
The only change in the revised curves is the replacement of the delphi’s pressure
coefficient statistics with the experimentally derived statistics. In each Figure, design
cases with increased risk outcomes relative to ASCE derived curves are illustrated
with dashed lines while those with decreased risk outcomes are illustrated with sold
lines. For the 15 roof, each of the 0-D design cases experienced an increase in risk
with the probability of failure going from 50% at 130mph to over 80%. Conversely,
the 1-D and 2-D design cases experienced a decrease in risk with 2-D cases dropping
from 50% probability of failure at 130mph to less than 10%. The Fragility curves for
the 30 and 45 illustrate the trend that as the roof slope increase an ASCE design
PV system becomes more likely to have a LS#1 failure. In the 45 case each system is
more likely to experience a LS#1 failure during a wind speed gust with V between 90
and 150 mph compared to what ASCE code values suggest. Also there is a significant
120
For both the hypothesis and sub-hypothesis, the results allow rejection of the null-
engineered system.
Further, the results provide motivation for an improved approach to risk manage-
ment in future code revisions. One approach could be to increase the uncertainty in
the wind load statistics so that the diverse system types fall within the distribution.
This approach is not recommended because some system types would be significantly
over-engineered. Another approach is to update the wind pressure statistics for res-
develop regression models similar to those published in the SEAOC guidance. When
planning this approach the literature suggest wind tunnel facilities should be selected
to include multiple testing and analysis methods including smaller scale facilities to
model the complete boundary layer and full scale facilities to quantify testing error.
Given the solar industries high sensitivity to cost, this path is preferable since it
Finally, the results for wind speeds V 70 mph and V 170 mph should be
with near 0 likelihood of failure for V 70 mph and near certain failure for V 170
when installed on a 45 roof. A logical question is which of the wind speed ranges
matters for a system with 25 year design service life. This question is addressed
121
h
122
Figure 57: Fragility curves for 30 deg roof
123
Figure 58: Fragility curves for 45 deg roof
124
CHAPTER V
Chapter 4 presented the process and e↵ects of reducing epistemic error in wind pres-
plication of the measured pressure coefficients to the case study fragility models. The
range of wind speeds. This chapter will address the yet unanswered question of how
wind speeds and required service life. This question is answered through application
the probabilistic structural analysis with a probabilistic extreme wind speed model.
Ultimately, this chapter in conjunction with Chapter 3 will show that the fragility
analysis and reliability analysis may be conducted in two distinct steps. First the
fragility analysis may be conducted for a specific technology and defined scenario of
use as demonstrated in Chap 3. Then the fragility analysis can be coupled to a site
specific wind speed model to provide a site specific reliability performance measure.
5.1 Background
annual wind speed for a location of interest based on limited historical data. In the
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analysis of historical data sets, daily maximum wind speeds compose an annual set
which is typically well characterized by the general Weibull or special case, Raleigh
distribution [66]. Weibull distributions have been used to conduct fatigue analysis
[56]. However, the short spans of residential PV system structural rails tend to
Equation 49 [11].
⇡ EI 1
! = ( )2 ( )2 (49)
L ⇢m A
When fatigue is not a design constraint and the yield stress governs, extreme loads
are of interest, which are not well characterized by Weibull distributions [66]. Methods
Extreme Value Theory [8]. Application of extreme value theory to wind loads has been
completed for tornado, hurricane, thunderstorms and mixed sources [63]. A focus of
current extreme value research is on the accurate shape of the distribution’s tale.
The Type 1 extreme value distribution, also known as the Gumbel distribution, has
a CDF given by Equation 50 and has been recognized as a best fit for extreme wind
speeds [66].
e (x µ)/
F (x) = e (50)
Alternative methods and software [45] are available for the estimation of maximum
annual wind speeds based on historical data. The method of moments shown in
Equation 51 has been shown to accurately predict a wind speed VN with recurrence
interval N measured in years. This method has been used with weather data from
126
single stations and more recently from superstation weather series1 data [62].
Where
The method of moments is valuable for setting a design wind speed based on
a specified recurrence interval, but does not support stochastic modeling of annual
maximum wind speeds. Alternatively, a graphical method has also been shown to
produce accurate parameter fits for the Gumbel distribution. The graphical method
for Gumbel parameter estimation utilizes the linear fit of a probability plot with wind
speed plotted against ordered sample count. In the graphical method, the Gumbel
shape parameter, is estimated with the Y intercept and the scale parameter, µ is
estimated with the slope [8]. The graphical fitting methodology will be demonstrated
records with limited length and accuracy. Up until the 1993 ASCE wind maps, wind
speeds utilized the fastest mile measurement collected from 129 airport stations with
10-50 years of operation. A switch to peak gust wind speed maps was motivated by
the obsolescence of fastest mile sensors and prevalence of peak gust sensors. BY 1988
487 National CLimatic Data Centers (NCDC) stations were operating with 5-45 years
of data. The limited operating history of both fastest mile and peak gust stations is
1
Superstations, are aggregations of weather-years from regional stations proven to have statisti-
cally independent but belonging to the same parent distribution[62]
127
Figure 59: Sampling error standard deviation vs sample size adapted from CPP [63]
1 1
SD(VN ) = 0.78(1.64 + 1.46(ln(N 0.577)) + 1.1(ln(N 0.5772 )) 2 S/n 2 (52)
As shown in Figure 59 the e↵ect of sample size on sampling error from typical
weather station records with less than 100 years of data is between 5% and 10% or
Investigation of extreme wind speed modeling methods for renewable energy applica-
tions has been principally focused on wind power applications. Investigations of wind
speed modeling for solar power have not been identified through literature review con-
ducted as part of this thesis. A primary reason for this gap in e↵ort is the extreme
wind speed is by definition a reference wind speed at 10 meters above ground. The
128
extreme wind speed is not a↵ected by the building structure nor the presence of a PV
system but rather the surrounding environment. The building structure and other
site specific contextual variables influence the dynamic pressure calculation, wind
speed is an input into the the dynamic pressure calculations along with variables that
account for these contextual factors. For this reason, the body of knowledge charac-
terizing extreme wind speed distributions is considered adequate for the evaluation
of residential PV systems.
Current codified design practice condenses the wind speed distribution into a
single design wind speed based on a recurrence interval [11]. In order to evaluate the
wind speed from the Gumbel probability distribution function for each sample, n, in
an experiment. This approach should lead to the benefits identified in the following
hypothesis.
bining structural risk with stochastic wind speeds for a single reliability
measure.
alternative system types given a stochastic model for extreme wind speeds.
reliability targets.
This Hypothesis is distinct from the fragility hypothesis because the latter hy-
pothesized capability to make comparison at a particular wind speed and the former
Evaluation of the hypothesis will be conducted through this chapter’s case study.
129
5.4 Case study
This section of the case study demonstrates coupling of an extreme wind speed model
For the case study location, Atlanta GA, 42 years of extreme 3-second wind speeds
are avail able from NIST [8]. The 42 year data set has a mean value of 59 mph, and
maximum value of 90 mph. Utilization of the linear fit approach [8] produces a
Gumbel distribution with location, µ = 54.1 and scale, = 8.6. The linear fit shown
Figure 60 has an R2 = 0.98. Using Equation 50 to the define the CDF of V (Equation
53) allows determination of an N year wind speed VN . The 50 year wind speed V5 0
with 2% likelihood of exceedance is found to equal 88mph. Given the sampling error
of the 42 year data set, this is statistically similar to the ASCE 90 mph wind zone
for Atlanta.
F (V ) = exp( e(x µ)
/ ) (53)
To execute the Monte Carlo method a random number generator was used to
produce a synthetic 1000 year set of extreme gust from the Gumbel distribution. The
synthetic set of extreme values, was found to be statistically similar to the 42 year
historical data set. Figure 61 of the synthetic set shows the probability of exceedance
Coupling the fragility model with the wind distribution, represented by the syn-
thetic wind data, produces a reliability model capable of estimating annual survival
experiment where for each sample the random variables distributions are sampled
and actual moment Ma . Using these random variables, the limit state function is
evaluated for each sample with the outcome equal to failure or survival. The annual
130
Figure 60: Gumbel theoretical vs empirical quartiles plot of Atlanta extreme 3-second
gust
Figure 61: Frequency plot of Synthetic set of extreme wind speeds for Atlanta
(n=1000)
131
resulted in a failure by the total number of samples. Finally Pf is used in the bi-
n!
Pf (x) = pxf (1 pf )(n x)
(54)
x!(n x)!
where x = the number of failures n = the operating life in years p = the probability
of failure
Similarly, the survival probability may be calculated for other periods. This ap-
proach assumes that the system is either operating with similar strength to its original
installation condition or that the system has an ultimate failure. This approach does
not allow for structural degradation due to fatigue from non-catastrophic wind events.
For failure modes that are susceptible to partial degradation, incorporation of dis-
crete event simulation into the reliability analysis has proven to adequately model
An experiment using the coupled reliability model was conducted for each of the ten
design case applied to three roof angles; 15 , 30 and 45 for 30 total experiments.
Each experiment consisted of 1000 Monte Carlo samples. Using the 1000 samples
from each experiment, an experimental standard deviation for G(X) was found to
range from 0.006 to 0.3 over the 30 experiments. With standard deviations in this
range and sample size = 1000 the model uncertainty given by Equation is estimated
Table 13 summarizes the results from each of 30 experiments with a 1 year and 25
signed PV racking design cases, using the high fidelity (”exact”) Monte Carlo method.
132
The reliability reported is only in regard LS #1 and does not quantify the likelihood
of failure due to other structural failure modes for which the proposed method may
be repeated.
The results indicate the 1 year survival probability for code engineered 0-D design
cases (18 101 18 104) range between 0.857 and 1.000 depending on roof zone and
slope.
In contrast, the minimum 1 year survival probability for code engineered 1-D
(90 101 and 90 102) and 2-D (180 101 270 102) system is 0.999 and 0.995 re-
spectively. Further, the experimental results suggest, each of the 1-D and 2-D design
cases have a 1.000 chance of survival, when installed on a 15 degree and 30 degree
roof.
Table 13: Survival probabilities for design cases engineered for 90 mph wind zone
133
Discussion of Results
the 25 year survival rates for alternative systems types with a fixed roof angle. A
T-Test is used with the null hypothesis that there is no di↵erence between sample
means. 1-D and 2-D systems have P > 0.05 indicating that the null hypothesis
can not be rejected and the survival rates for the two systems types are statistically
similar. Next, comparison is made between 0-D samples and the combined set of 1-D
and 2-D samples. For each roof angle, P < 0.05 (P=0.03 for 15 , P=0.02 for 30 ,
and P=0.002 for 45 ) indicating a statistical di↵erence between system types and
to varying roof angles, paired T-Test are used to compare the reliability of each design
case across roof angles with the null hypothesis indicating no statistical di↵erence in
means between roof angles. In comparison of the 25 year probability of survival for
design cases installed on a 15 roof to a 30 roof, the null hypothesis could not be
in p = 0.048 and p = 0.03 respectively. These results allow rejection of the null
hypothesis and suggest that code engineered PV systems have statistically di↵erent
These finding support the sub-hypothesis that a coupled reliability model provides
a platform for comparing alternative system types. Further, the findings suggest a
reliability target and calculation of the percent above or below the reliability target.
The E.U. building code, EN1990, provide for a reliability approach with quantified
134
reliability targets. EN1990 establishes reliability targets for three Reliability class’
(RC) that correspond to three consequence classes (CC) (Table 14). Each consequence
class from CC1 to CC3 corresponds to increasingly higher risk to human life with
However, loss of life is not the only consequence of concern captured in the Eurocode,
each consequence class is also defined in terms of the capacity for economic, social or
environmental impact [3]. More recently, ASCE 7-10 published acceptable 1 year
values that may be used in lieu of the more established EN1990 values. For reference,
As shown, the reliability target is not based on the survival probability but rather
For this case there is limited ability to interpret how reliable the system is compared
graphically illustrated by Figure 62 which shows a clear failure region for 18 101
where the load probability distribution function (pdf) exceeds the resistance pdf; the
1 year survival rate for this case is 0.99. In contrast, there is no clear failure region
illustrated for 90 101 or 180 101 both of these have a 1 year survival rate of 1.000
yet there is a clear visual di↵erence in reliability because the standard deviation of
loads for 180 101 is less than the standard deviation of loads for 90 101. This case
illustrates the value of the the reliability index used to establish reliability perfor-
mance targets by accounting for the di↵erence in means and the standard deviations,
135
as shown in Equation 55 [29].
µ R µS
=p 2 2
(55)
R + S
To benchmark the reliability of each design case and roof angle calculation, 1
is calculated using equation 55 and compared against the target reliability index for
is restricted by code from having a reliability less than the target, and for economic
reasons may minimize the degree to which the target is exceeded. However, for the
limit and RC3 as an upper limit. These upper and lower specs are relevant in the
well as perspectives that account for actual pedestrian behavior during storm events.
Grouping of each roof angle for a design case clearly confirms the trends identified
by reviewing the survival rates. Further, Figure 63 shows that for all 0-D design
cases except a single 15 roof case, neither the target reliability, RC2 or the RC1
reliability are met. On the other hand, each of the 1-D and 2-D design case exceed
the target reliability for 15 and 30 roofs while 66% are below the target reliability
when installed on a 45 roof. Among the systems that exceed the target reliability,
for 2-D cases compared to the 1-D cases supports the conclusion that 2-D cases are
significantly over-engineered compared to both the target reliability and 1-D cases
(p = 0.0005). If the ASCE value is a preferred target, similar trends hold with 4
of the 12 O-D systems exceeding the target performance and all of the 1-D and 2-D
For benchmarking reliability over periods other than 1 year the target n for a n
year period can be calculated from the 1 year target, 1 according to Equation
56.
136
Figure 62: Frequency Distribution of Atlanta annual maximum 3-second gust
137
Figure 63: Reliability index, 1 for limit state #1 for each design case in the Atlanta
case study. Benchmark target reliability values are illustrated for reliability class,
RC, 2 with upper spec RC3 and lower spec RC1.
( n) = [ ( 1 )]n (56)
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CHAPTER VI
6.1 Summary
Recent investigations into wind loads on residential PV systems suggest actual wind
codified methods. In other words, the ASCE 7-2005 and ASCE 7-2010 codes does
not appear to be calibrated with actual pressure coefficient on currently system types.
Further, recently developed racking system types have introduced an order of mag-
impact wind loads. The current mis-calibration does not appear to be uniform across
risk of a code engineered system is not adequately quantified to support risk based de-
cisions. 2) Actual structural risk may not be consistent with the level of risk intended
by code. 3) Structural risk may not be consistent across system types and building
assess fragility and structural reliability have been proposed for application as struc-
studies these two performance measures have been shown to work in concert by first
assessing the conditional failure probability and then incorporating site specific ex-
treme wind speed models. In order to demonstrate the fragility and reliability index
methodologies, initial limit states for performance based design have been drafted
even though understanding of structural supply is well established and not the focus
139
through the case studies presented in this thesis have quantified the probability of a
limit state (LS#1) failure for systems engineered according to the ASCE code 90mph
The Fragility analysis accounted for uncertainties in both the dynamic pressure
prediction and material strength prediction. Sources of model form uncertainty were
sistent with code prescribed values, wind tunnel testing was conducted to reduce epis-
system types and roof conditions. Subsequently, the fragility curves were coupled
with site specific extreme wind speed models in the reliability index . The pro-
wind speeds. The Gumbel distribution was then sampled to create a synthetic set of
6.2 Conclusions
limit state failure for a code engineered system across an array of wind speeds. Incor-
poration of uncertainties that influence the resistance and load e↵ects is critical for
the probability of failure for alternative code engineered system types given a design
wind speed event. However, fragility analysis has also shown that when any of the
system types are engineered according to code for 90 mph wind zone, the proba-
bility of LS#1 failure in a wind event 70 mph or less is near zero. These findings
140
failure during a systems service life. Treatment of the annual extreme wind speed as
ences in the 25 year probability of failure were found due to system type and roof
angle. Use of probability of failure as a metric was found to have limited capability
the reliability index was found to be a more descriptive metric capable of clearly
for each system type roof angle combination indicated significant di↵erences in relia-
bility predictions across system types and roof angles. Benchmarking the calculated
values against performance targets set by EN1990 and ASCE indicated poor preci-
sion of the current perspective code when implemented according to recent industry
less than the performance targets for all consequence classes suggesting these sys-
1 D and 2 D systems are predicted to have reliability greater than the target
spec for residential structures when installed on 15 and 30 roofs but less than the
motivate the proposed performance measures. They do not support analysis of any
The wind tunnel testing and analysis methods utilized recently developed methods
for small buildings. The wind tunnel method has been subject to only limited peer
review and may introduce additional uncertainty. Despite these significant limita-
tions, the results are adequately compelling to suggest that fragility curves and the
reliability index each plays a useful role in quantifying the structural performance of
141
residential PV systems.
Opportunities exist to both refine and expand upon this body of thesis. Refinements
construction quality, and model form uncertainty. Repetition should be made of the
wind tunnel testing in alternative wind tunnels with an expanded design of experiment
Limit state definitions may be advanced and additional limit states should be incor-
porated through systems reliability methods. Expanded limit states should include
serviceability limit states which address PV production. Also, PV racking system de-
or alternative measures or procedures such as testing are in place that may suggest
Together, these refinement activities may support a longer term e↵ort to update
future code revisions with more accurate wind pressure coefficients. However, because
the conclusions suggest that code has failed to provide a consistent level of reliability
across system types, code committee members may consider if a prescriptive code
methods should be promoted. Further, this thesis has assumed that structural re-
liability intended by the ASCE code and targeted in the E.U. code are appropriate
targets for power generation equipment. Currently, the switch to solar energy is being
motivated by financial performance and risk from existing power generation technolo-
gies. Future work should use a life cycle assessment methodology to compare the full
life cycle e↵ects of alternative power generation technologies [27], [61]. This analysis
should support determination of how much reduction in risk should code require of
solar when replacing a high risk alternative such as coal power generation. Ideally,
142
this decision should also incorporate the marginal cost of risk reduction and marginal
adoption rates. One possible scenario is that a higher level of risk from PV systems
may be accepted so that more severe global warming risk and air quality risk [68],
[69] may be mitigated faster. Already the metric lives lost per trillion kWh enables
consistent comparison of risk across system types. Another area of investigation re-
quired for improved risk assessment and risk based decision making is a more rational
mapping of limit states to damage levels. Speculation in the literature has suggested
lives lost due to structural failure of a solar panel system during an extreme wind
event may be low because people tend to go inside. Clearly in our present state
of increasingly complex hazards the role of the building safety code must evolve. A
143
CHAPTER VII
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Performance measures for residential PV structural response to wind e↵ects
153 Pages
At odds with this goal are existing codified practices which prescribe global factors
(allowable stress design) and partial factors (Load and resistance factor design) in-
When applied to solar this prescriptive approach has two flaws, (1) calibration e↵orts
needed to ensure consistency across structural system types have not kept up with
industry which, leads to (2) the actual expected reliability is not quantified. The reli-
ability performance measure applied include probability of failure and the reliability
index , both of which are commonly used in the application of performance based
design to other domains. The approach developed is based on the application of the
applied to 3 roof angles. In a case study, the design cases are utilized to demonstrate
how the existing prescriptive approach may lead to nonuniform reliability perfor-
mance measures. For each of the design cases on which the reliability measures are
demonstrated, a code compliant design is developed for three roof slopes and wind
coefficients. By applying the data collected, system specific fragility curves are gen-
that a non-site specific assessment of reliability may be made. But when dealing
with a specific site, a site or region specific wind model is applied to produce the
code designed systems. The case study indicates codified prescriptive method fail to
153