Correlation Between Uniaxial and Biaxial Compensation Data for Architectural Fabrics
Department
of Technology
Buffalo
State College
Buffalo,
NY 14222
Email: gellins@buffalostate.edu
ABSTRACT
A
linear regression model was formulated to relate biaxial compensation data for
four architectural fabrics with uniaxial data from both the manufacturer and
the construction company employing the fabric in its tensioned structures.
Good correlation was generally found with the construction company
data. The results of the study
can be used as justification for reducing the amount of testing necessary to
determine biaxial compensations for these materials.
INTRODUCTION
The
design and manufacture of tensioned fabric structures have a welldocumented
history^{1}. Central
to the design of these structures are the material properties of the fabric
used. The fabric portion of the
structure is generally preloaded biaxially in tension in order to give it
stability of shape and resistance to live loads^{2}. The
process of taking flat panels of fabric that will later assume the
threedimensional shape of the structure (, known as patterning^{3},) must take include stretching the fabric during
installation and final assembly of the structure. This induced strain in the fabric is usually referred to as
compensation. In order to ensure
that the fabric will be properly prestressed when installed, the
compensations for the desired prestress must be determined reasonably
accurately.
The
most prevalent fabric in use today for nonpressurized tension fabric
structures is PTFE coated glass fibers. These
materials usually are sold in rolls of 12 feet in width and in lengths up to
several hundred yards. The glass
fibers in the warp or length direction are generally “straight,” while
those in the fill or width direction are woven in between the warp fibers.
This causes the fabric to be considerably stiffer in warp than in fill.
The
manufacturer of the fabric generally supplies uniaxial compensation data for
both warp and fill at a stress (actually, a stress resultant) of S_{U}
for each roll of fabric it delivers to a major construction company that
designs and builds tensioned fabric structures. The sample of
fabric used for the test comes from one end of the roll.
The construction company assumes that the loading conditions on the
fabric are different at the end of the roll than in the middle of the roll.
The construction company generally repeats the uniaxial test in the
middle of the roll and then does a biaxial test with a specimen adjoining
those used for its uniaxial tests. The
biaxial test is designed to approximate the conditions of initial
installation, settling under moderate live loads and future extreme live
loading^{4},
though it is not done under the auspices of any standard test method^{5}. Typical
results of the biaxial test are shown in Figure 1. From these results, an engineer can choose a value for warp
and fill compensations for the design prestress S_{P}.
Figure 1: Typical Biaxial Test Results
The
uniaxial tests generally require an hour to perform, while the biaxial tests
require about six hours to perform.
For a large project that may use dozens of rolls of fabric, the cost in
time and money can be prohibitive if each and every roll is tested this way.
Furthermore, the internal location of the specimens used in testing may
result in significant waste of fabric if the designed flat panels cannot be
efficiently nested in the length between the end of the roll and the specimen
location.
Over
time, the construction company developed several informal rules to limit its
need for testing on large jobs. Generally,
it took the set of rolls assigned to a project and calculated the sum of the
warp and fill uniaxial compensations as supplied by the manufacturer.
It then grouped the rolls in informal subgroups, usually three, based
on these values. It then took one
“typical” roll from each subgroup and performed the full biaxial testing
on that roll, and used the values of the biaxial warp and fill compensations
obtained from the test and applied them to all rolls in the subgroup.
No studies were performed to verify if this was a feasible strategy to
employ.
A
study was undertaken to determine empirical formulas for warp and fill biaxial
compensations in terms of uniaxial compensations determined by the manufacturer
and the construction company for four different PTFE coated glass fabrics
sold by the manufacturer. These
formulas were examined using methods of statistical inference in order to
determine their validity and their implications for possible substitution
for costly testing. This paper
reports on the results of that study.
METHODOLOGY
Formulas
in the form
are
derived for biaxial compensations, where y represents either a warp (w) or
fill (f) biaxial compensation; w_{U} and f_{U}
represent the uniaxial warp and fill compensations, respectively, with the
x subscript being either M for the manufacturer data or C for the construction
company data; and W_{y}, F_{y} and C_{y} represent
the coefficients of the linear (least squares error) fit for the quantity
y.
Each
formula will be evaluated for its feasibility in approximating the given
quantity y. The following
statistical measures will be calculated and evaluated for each of the
formulas:
This
process will be repeated for the four materials, labeled Material 1, 2, 3 and
4. The number of the material
determines its rank in importance (as far as usage is concerned) to the
construction company with Material 1 being the most used.
Data used in the study was limited to those rolls that have undergone
the standard biaxial test at the construction company test facility.
The choice of biaxial compensations assigned based on the tests was
determined on a consistent basis by the vice president of engineering for the
construction company, who has over 20 years experience in the tensioned fabric
structures business.
RESULTS
The
results of the study are presented in Tables 1 – 4 for Materials 1 – 4,
respectively. The tables list the
derived coefficients, their respective tstatistics,
as well as the other statistics associated with linear fit.
For example, in Table 1, the upper left section refers to the
relationship between the biaxial warp compensation vs. the uniaxial data
provided by the manufacturer. The formula derived is:
All
the tstatistics for the coefficients
(0.23, 0.06, 0.37) are below the threshold of 1.71, indicating that the biaxial
warp is not strongly dependent on either of the variables w_{UM}
or f_{UM}, nor is it strongly
dependent on the additive constant term.
The R^{2} value of
0.19 indicates that the entire formula is a relatively poor fit.
The Fstatistic provides
additional confirmation of the poor fit. The upper right section refers to the relationship between
the biaxial warp compensation vs. the uniaxial data based on construction
company tests. Here, the formula
derived is:
The
tstatistic on the first coefficient
(0.54) is quite high, indicating that w
is strongly dependent on w_{UC},
while the other tstatistics are
below the threshold of 1.71, indicating that their contribution to the above
formula is not as critical. This
indicates that a simpler formula for w,
perhaps only dependent on w_{UC},
may give comparable R^{2}
and Fstatistic values.
The lower portion of the table summarizes comparable results for the
biaxial fill compensation.
With
one exception, the results imply that the correlation of biaxial compensations
with construction company uniaxial data is clearly superior to the correlation
with manufacturer uniaxial data. While
it was expected that the construction company data would correlate better than
the manufacturer data, due to the fact that the samples for the tests were cut
from the same location on the roll, the difference in the statistical
measurements between the correlations was surprisingly large.
Table
1 is typical of the results obtained.
First, it is noted that for both sets of uniaxial data that the fill
correlation is better than the warp correlation.
This could be a direct result of a narrower
Table
1: Results for Material 1 


vs.
Manufacturer 
vs.
Company 

Data 
Data 
Warp
Compensation: 


W: 
0.23 
0.54 
(tStatistic) 
1.67 
8.43 



F: 
0.06 
0.04 
(tStatistic) 
1.12 
1.21 



C: 
0.37 
0.30 
(tStatistic) 
1.13 
1.63 



tStatistic
Threshold for 5% 
1.71 
1.71 



R^{2} 
0.19 
0.74 



Fstatistic 
3.08 
36.80 



Fstatistic
threshold for 5% 
3.37 
3.37 






Fill
Compensation: 


W: 
0.46 
0.08 
(tStatistic) 
1.37 
0.85 



F: 
0.44 
0.87 
(tStatistic) 
3.38 
16.54 



C: 
0.35 
1.19 
(tStatistic) 
0.43 
4.10 



tStatistic
Threshold for 5% 
1.71 
1.71 



R^{2} 
0.42 
0.92 



Fstatistic 
9.51 
158.11 



Fstatistic
threshold for 5% 
3.37 
3.37 



Table
2: Results for Material 2 


vs.
Manufacturer 
vs.
Company 

Data 
Data 
Warp
Compensation: 


W: 
0.74 
0.63 
(tStatistic) 
5.84 
17.25 



F: 
0.16 
0.05 
(tStatistic) 
2.70 
1.65 



C: 
1.97 
0.48 
(tStatistic) 
3.95 
2.54 



tStatistic
Threshold for 5% 
1.76 
1.76 



R^{2} 
0.71 
0.96 



Fstatistic 
17.14 
188.35 



Fstatistic
threshold for 5% 
3.74 
3.74 






Fill
Compensation: 


W: 
0.31 
0.23 
(tStatistic) 
1.10 
3.10 



F: 
0.17 
0.79 
(tStatistic) 
1.34 
12.82 



C: 
2.83 
1.26 
(tStatistic) 
2.54 
3.20 



tStatistic
Threshold for 5% 
1.76 
1.76 



R^{2} 
0.27 
0.92 



Fstatistic 
2.53 
84.10 



Fstatistic
threshold for 5% 
3.74 
3.74 



Table
3: Results for Material 3 


vs.
Manufacturer 
vs.
Company 

Data 
Data 
Warp
Compensation: 


W: 
0.09 
0.58 
(tStatistic) 
0.77 
7.47 



F: 
0.03 
0.07 
(tStatistic) 
0.47 
3.69 



C: 
0.28 
0.17 
(tStatistic) 
0.84 
1.04 



tStatistic
Threshold for 5% 
1.73 
1.73 



R^{2} 
0.04 
0.81 



Fstatistic 
0.38 
38.09 



Fstatistic
threshold for 5% 
3.55 
3.55 






Fill
Compensation: 


W: 
0.60 
0.07 
(tStatistic) 
1.60 
0.20 



F: 
0.61 
0.77 
(tStatistic) 
3.60 
15.24 



C: 
0.64 
0.65 
(tStatistic) 
0.62 
1.57 



tStatistic
Threshold for 5% 
1.73 
1.73 



R^{2} 
0.48 
0.93 



Fstatistic 
8.30 
116.97 



Fstatistic
threshold for 5% 
3.55 
3.55 



Table
4: Results for Material 4 


vs.
Manufacturer 
vs.
Company 

Data 
Data 
Warp
Compensation: 


W: 
0.02 
0.04 
(tStatistic) 
0.63 
0.28 



F: 
0.09 
0.06 
(tStatistic) 
3.78 
1.78 



C: 
0.13 
0.02 
(tStatistic) 
0.08 
0.12 



tStatistic
Threshold for 5% 
1.80 
1.80 



R^{2} 
0.60 
0.24 



Fstatistic 
8.08 
1.72 



Fstatistic
threshold for 5% 
3.98 
3.98 






Fill
Compensation: 


W: 
0.59 
0.37 
(tStatistic) 
2.24 
0.68 



F: 
0.44 
0.68 
(tStatistic) 
0.18 
6.16 



C: 
0.67 
0.18 
(tStatistic) 
1.14 
0.42 



tStatistic
Threshold for 5% 
1.80 
1.80 



R^{2} 
0.46 
0.79 



Fstatistic 
4.66 
20.12 



Fstatistic
threshold for 5% 
3.98 
3.98 



range
of warp compensations for the roll data, yet larger relative variance due to
rounding off to the nearest tenth of a percent. For installation purposes, it is probably more important to
have a good handle on the fill compensation, as it is easier to pull the fabric
in that direction. The values of R^{2} indicate a poor
correlation with manufacturer uniaxial data for both warp and fill biaxial
compensations, though it is interesting to note that the coefficients for fill
are nearly equal, lending some credence to the method of grouping the rolls
based on the sum of the uniaxial warp and fill values.
The values of R^{2}
for the construction company data indicate fair correlation in warp and
excellent in fill, with correspondingly large values of the Fstatistic.
Figure 2 is a plot of the
biaxial compensations for the rolls of Material 1 used in the data as well as
the predicted values using the formulas of Table 1. The plot graphically demonstrates the superiority of
correlation with construction company data.
It should be noted that the rolls are listed from left to right with
increasing sum of reported manufacturer uniaxial compensations.
Table
2 is similar to Table 1 in fill, but surprisingly good correlation is obtained
in warp for both sets of uniaxial data, including an incredible R^{2} value of 0.96 for the correlation
with the construction company data. Based on the tstatistics,
this formula can be simplified as:
Table
3 yields similar results to Table 1, though the warp correlation with
manufacturer data is virtually nonexistent.
Possible simplified formulas are:
Material
4 is unusual in that its biaxial warp compensation is usually negative.
As indicated in Table 4, this is the one instance in this study where
correlation with manufacturer data, though not great, is superior to that with
construction company data. The
biaxial fill compensations follow qualitatively the pattern of the other
materials, though the correlation with construction company data is not as
strong as with the other materials. This
formula could be simplified as:
Though
not reported in detail herein, it should be noted that correlations of the two
sets of uniaxial data with each other were generally poor for all materials.
CONCLUSIONS
The
uniaxial compensation data as provided by the manufacturer is of little value in
predicting biaxial behavior of PTFE coated glass fiber materials.
A decision to completely discard biaxial testing based on estimates
derived herein would be highly risky at best.
On
the other hand, the uniaxial compensation data as determined by construction
company testing generally correlates well in predicting biaxial behavior, with
the exception of the warp compensation for Material 4.
Performance of the less expensive uniaxial tests on each roll may be
adequate in lieu of both uniaxial and biaxial testing.
REFERENCES
1.
R.E. Shaeffer, ed. (1995) Tensioned Fabric Structures: A Practical
Introduction. Prepared by the Task
Committee on Tensioned Fabric Structures, Technical Committee on Special
Structures, and Technical Committee on Metals of the Structural Division of ASCE.
2.
M.R. Barnes, “Form Finding of Minimum Surface Membranes,” IASS
Conference, July 1976.
3.
S. Gellin, “Patterning of Tensioned Fabric Structures,” Proc. 5^{th}
International Conference of Engineering Design and Automation, Las Vegas, NV,
August 2001, pp. 482487.
4.
H. Minami and S. Motobayashi, “Biaxial Deformation Property of Coated
PlainWeave Fabrics,” Proc. Vol. 1, International Symposium on Architectural
Fabric Structures, Orlando, Nov. 1984.
5.
C.G. Huntington, “The Tensioned Fabric Roof,” ASCE Press (to be
published January, 2004).