1

BEHAVIOUR OF GLASS PLATES UNDER PRESSURE LOADING

Eri Iizumi∗, Gregory A. Kopp†

∗Department of Civil and Environmental Engineering, The University of Western Ontario, London, Ontario, N6A 5B9, Canada, +1-519-661-2111 ext 88144

e-mail: eiizumi@uwo.ca † Boundary Layer Wind Tunnel Labratory, The University of Western Ontario, London, Ontario, N6A

5B9, Canada, +1-519-661-3338 e-mail: gak@blwtl.uwo.ca

Abstract: Glass plate breakage tests under the realistic wind pressure loading, which have not been done anywhere else, are studied, a part of which are presented in this paper. Test results are reported with results of numerical simulation based on the fracture mechanics.

1 INTRODUCTION During strong windstorms (e.g., hurricanes), various cladding materials are often damaged.

Damage to windows and glazing has generally been outstanding in both residential structures and high-rise buildings. Although windborne debris has been considered as a significant factor of window damage [1], there is a potential of window glass breakage due to fluctuating wind pressures as well.

Whereas full-scale glass plate breakage tests have been conducted by several researchers ([2], [3]), the applied loads were ramp loading, step-up loading (hold the pressure for a certain time and linearly increase the pressure) and step-up cyclic loading. These loading do not accurately simulate real wind loading, which changes temporally because of turbulence. This has been mainly because of the limited technology and long experimental time.

In the current glass design in North America, the glass failure prediction model (GFPM) developed by Beason [4] is being utilized. This uses two interdependent surface flaw parameters, m~ and k~ , which are determined by testing for each geometric shape in quite a complex approach. In addition, several faults in this method have been brought up by other researchers ([3], [5]).

The primary purpose of the present study is to investigate the glass plate behavior under realistic, fluctuating wind loads, simulating windstorms. In order to understand glass plate behavior, preliminary tests with rather simple loading, ramp loading and saw-tooth loading, have been conducted in advance of fluctuating wind load. This paper presents the results of these preliminary tests, and the comparison of the test and numerical simulation results based on fracture mechanics.

Full-scale glass breakage tests are conducted with an application of a sophisticated loading device, Pressure Loading Actuator (PLA), developed for the “Three Little Pigs” Project [6] at the University of Western Ontario (UWO).

2 REVIEW OF PREVIOUS WORK

The reduction of glass plate strength has been evaluated based on the concept of damage accumulation since failure does not always occur at the maximum nominal stresses under constant or variable loads of a certain time duration. This is different from many other materials.

Brown [7] expressed the cumulative effect of loading on glass plate as

[ ] =∫ft

nt0

)(σ const (1)

2

where ft is the time over which the glass is stressed, )(tσ is the variation of stress at the critical flaw location with time, and n is a constant. This is called Load Duration Theory. This theory has been validated by glass plate breakage tests under static and cyclic loading, and utilized by many researchers for evaluation of the reduction of glass plate strength.

Besides Brown’s method based on the Load Duration Theory, there are two well-known glass failure models. One is the Crack Growth Model by Evans and Wiederhorn [8] which is based on crack propagation controlled fracture to predict component lifetimes after proof testing. The other is the GFPM suggested by Beason [4], which uses the finite different method and is recognized as the state-of-the-art in North America.

Fisher-Cripps and Collins [9] investigated all these models and concluded that none of the models can predict the failure probabilities for both short and long term stresses.

In Japan, Kawabata [3] conducted numerical simulation developed by Simiu [10] and verified it by comparing the results of full-scale glass plate breakage tests. In Simiu’s simulation, fracture mechanics were utilized and the probability of glass strength was assumed to follow the Weibul distribution.

3 FINITE ELEMENT ANALYSIS Finite element analysis on glass plates under uniform pressure is necessary to obtain input stress

data used in the numerical simulation as well as for confirmation of the simply-support condition of the experimental rig. This analysis was performed using commercial software SAP 2000. Since the deflection of the glass plate becomes much larger than the thickness of the plate, this needs to be treated using non-linear plate analysis. As validation of the use of SAP 2000, the FEA results (stress and deflection) were compared with those calculated using Seaman’s theoretical equation [11] and those obtained by Fisher-Cripps and Collins [9] in Table 1. The calculation was made for the case of )(004.011 m×× glass plate under the uniform pressure of kPa2.2 . Material properties used in the calculation were assumed as ( )GPaE 54.71= , .23.0=ν Considering the uncertainty in material properties, these data correspond well, and the use of SAP 2000 is deemed adequate for this analysis.

Table 1: Comparison in stress and deflection

deflection (mm)center max center

non-linear (-) 24.07 11.1FEA linear 37.5 37.5 22.0FEA non-linear 17.3 20.8 10.5FEA linear 37.4 37.4 22.2FEA non-linear 17.9 22.75 10.5Present

stress (MPa)

Seaman's equation [11]

Fisher-Cripps & Collins [9]

4 FULL SCALE TESTING

4.1 Pressure Loading Actuator (PLA) The PLA, which can replicate real, full-scale, temporally-varying wind pressures, was employed

to apply both static and dynamic wind loads. This device uses an advanced, adaptive control system to replicate target pressure time histories. For details, refer to [6].

3

4.2 Pressure box

The glass specimen is mounted on a pressure box ( )(16.07.12 m×× ) made from steel. On the front face of this box, a plywood panel is fastened with several bolts, and the glass specimen is placed on the plywood panel with aluminum frames. The glass specimen is supported by rubber tubes in order to create the simply-support condition. The hose from the PLA is connected to the pressure box, and positive and negative (suction) pressures are applied inside the pressure box. Note only negative pressures were applied in the current tests for safety reasons. All the broken glass pieces cannot be sucked into the Pressure Box and some of them are reflected at the back side of the Pressure Box and come out in front. For this reason, the Pressure Box was covered by the Safety Guard (three layers tarp, Figure 1(b)).

PLAPLA

Pressure box

Glass plate

Safety Guard

(a) Set up without Safety Guard (b) Set up with Safety Guard

Screen

PLAPLA

Pressure box

Glass plate

Safety GuardSafety Guard

(a) Set up without Safety Guard (b) Set up with Safety Guard

Screen

Figure 1: Test set up

5 GLASS FAILURE PREDICTION SIMULATION The numerical simulation conducted by Kawabata [3] was replicated and its results were compared with full-scale test results. The failure is defined as the shortest time until the nominal stress of the crack on glass panel )(tnσ becomes larger than the glass strength )(tS : )()( tStn ≥σ (2) The time varying glass strength )(tS is expressed as [3]

2

1

0

221

2 )(2

2)(−

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−= ∫

ntnn

nIC

ni

f

dttKYAnStS σ (3)

where iS is the initial glass strength induced by the Monte Carlo simulation with concentric ring-on-ring tests results, ICKYAn ,,),16( 1= are coefficients and ft is the failure time. Note that this simulation has been verified only with ramp loading full-scale test results by Kawabata.

6 RESULTS AND DISCUSSION

In all the testing shown in this paper, single annealed glass plates of size )(006.011 m×× were utilized. A minimum of fifteen glass plates were broken in each test configuration in order to have statistically significant data. After breaking fifteen glass plates, the data were analyzed and

4

additional trials were conducted if the results were skewed. Although the applied pressures were negative (suction), the pressures described below are positive.

6.1 RAMP LOADING TEST

Two different rates of ramp loading (16.4 secPa , 6.5 seckPa ) were conducted. The first ramp rate was chosen to compare with the results as Kawabata to ensure the appropriateness of the experimental rig. The second ramp rate was chosen in order to break within a short time, considering the limitations of the PLAs. Figure 2 shows the comparison of failure pressure test data on the same glass geometry and same loading test by Kawabata. In the figure, “F” stands for the failure probability. Both test results are fit to Weibul distributions, which is generally regarded as the best model for failure pressure. Although the number of trials was small in the present test, the data corresponded quite well, implying that the present test data and the testing setup are appropriate. Figure 3 shows the probability density function of failure pressure approximated by a Weibul distribution. The variations of the both test results are small (the standard deviations of both test results are 1.92 and 2.81, respectively). Based on these results, it is concluded that the test results are statistically appropriate.

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 25 30Pf (kPa) Pf :failure pressure (kPa)

PD

F16.4Pa/sec_estimate16.4Pa/sec_test6.5Pa/sec_estimate6.5Pa/sec_test

-4

-3

-2

-1

0

1

2

1.5 2 2.5 3ln(Pf ) Pf :failure pressure (kPa)

ln(-l

n(1-

F))

Kaw abata result

Present result

Figure 2: Comparison with Kawabata results Figure 3: PDF of failure pressure

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 25 30Pf (kPa) Pf :failure pressure (kPa)

PD

F16.4Pa/sec_estimate16.4Pa/sec_test6.5Pa/sec_estimate6.5Pa/sec_test

-4

-3

-2

-1

0

1

2

1.5 2 2.5 3ln(Pf ) Pf :failure pressure (kPa)

ln(-l

n(1-

F))

Kaw abata result

Present result

Figure 2: Comparison with Kawabata results Figure 3: PDF of failure pressure The two ramp loading cases were replicated in the numerical simulations. The comparison with test results is shown in Figure 4. The simulation results for both failure pressure and failure time correspond well with test results.

-8-7-6-5-4-3-2-10123

5.5 6 6.5 7ln(tf ) tf :failure time (sec)

ln(-l

n(1-

F))

simulationtest

-8-7-6-5-4-3-2-10123

1 1.5 2 2.5 3ln(Pf ) Pf :failure pressure (kPa)

ln(-l

n(1-

F))

simulationtest

(a) Failure pressure (b) Failure time

-8-7-6-5-4-3-2-10123

5.5 6 6.5 7ln(tf ) tf :failure time (sec)

ln(-l

n(1-

F))

simulationtest

-8-7-6-5-4-3-2-10123

1 1.5 2 2.5 3ln(Pf ) Pf :failure pressure (kPa)

ln(-l

n(1-

F))

simulationtest

(a) Failure pressure (b) Failure time Figure 4: Comparison between test results and simulation results (ramp loading 16.4Pa/sec)

5

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1.5 2 2.5 3 3.5ln(Pf ) Pf :failure pressure (kPa)

ln(-l

n(1-

F))

simulationtest

-8-7-6-5-4-3-2-10123

0 0.5 1 1.5ln(tf ) tf :failure time (sec)

ln(-l

n(1-

F))

simulationtest

(a) Failure pressure (b) Failure time

-8-7-6-5-4-3-2-10123

1.5 2 2.5 3 3.5ln(Pf ) Pf :failure pressure (kPa)

ln(-l

n(1-

F))

simulationtest

-8-7-6-5-4-3-2-10123

0 0.5 1 1.5ln(tf ) tf :failure time (sec)

ln(-l

n(1-

F))

simulationtest

(a) Failure pressure (b) Failure time Figure 5: Comparison between test results and simulation results (ramp loading 6.5 kPa/sec)

6.2 SAW-TOOTH LOADING TEST

A saw-tooth loading test whose rate of 6.5 seckPa and amplitude of kPa6.14~3.0 was conducted to create a situation similar to wind loading but still similar to the ramp loading so that the effect due to constrained loading amplitude is characterized. The amplitude of kPa6.14 was chosen considering the performance of the PLA and the predicted failure pressure cumulative distribution function (CDF) based on the Load Duration Theory to cause a single failure in the first cycle of saw-tooth loading in twenty trials. Figure 6 shows a portion of the applied saw-tooth loading. Although the longest testing time of this loading was more than 2 hours, the accuracy of the applied loading was quite good until the failure. Table 2 shows the test results of failure pressure and failure time. The variation of failure pressure is small and the plate always broke close to the peak pressure. On the other hand, the variation of failure time is significant, from 2 seconds to 2 hours. Figure 7 shows the comparison between test results and simulation results. As for the failure pressure, simulation results correspond well with test results. On the other hand, the simulation results are quite disparate from the test results in failure time. Since the way the simulation works is that failure occurs when Eq (2) is satisfied, a large reduction in strength ( )tS occurs at the large pressure due to the integration part in Eq (3). Therefore, it is expected that the failure pressure does not vary notably and the simulation can predict relatively well. As for the failure time, there are two possible reasons for this mismatch. One is the initial strength of glass plate and the other is the way of reducing the strength, meaning the Load Duration Theory on saw-tooth loading. The inconsistency is currently under investigation.

79 80 81 82 83 84 85

-15

-10

-5

0

time (sec)

appl

ied

pres

sure

(kP

a)

Figure 6: Applied pressure trace

Table2: Saw-tooth loading test results

failure pressure (kPa)

failure time (sec)

failure pressure (kPa)

failure time (sec)

test1 14.89 85.78 test 12 14.22 23.48test2 10.43 2806.70 test 13 14.42 934.94test3 11.99 234.89 test 14 14.80 574.21test4 13.84 384.82 test 15 14.69 1132.90test5 14.68 4534.10 test 16 14.06 2.07test6 11.31 547.23 test 17 12.48 1986.20test7 12.76 969.85 test 18 14.86 11.23test8 13.59 235.18 test 19 14.42 327.73test9 12.50 50.21 test 20 13.67 318.80test10 13.89 6905.60 mean 13.55 1495.66test 11 13.50 7847.20 S.D 1.26 2306.88

79 80 81 82 83 84 85

-15

-10

-5

0

time (sec)

appl

ied

pres

sure

(kP

a)

Figure 6: Applied pressure trace

Table2: Saw-tooth loading test results

failure pressure (kPa)

failure time (sec)

failure pressure (kPa)

failure time (sec)

test1 14.89 85.78 test 12 14.22 23.48test2 10.43 2806.70 test 13 14.42 934.94test3 11.99 234.89 test 14 14.80 574.21test4 13.84 384.82 test 15 14.69 1132.90test5 14.68 4534.10 test 16 14.06 2.07test6 11.31 547.23 test 17 12.48 1986.20test7 12.76 969.85 test 18 14.86 11.23test8 13.59 235.18 test 19 14.42 327.73test9 12.50 50.21 test 20 13.67 318.80test10 13.89 6905.60 mean 13.55 1495.66test 11 13.50 7847.20 S.D 1.26 2306.88

6

-8-7-6-5-4-3-2-10123

-2 0 2 4 6 8 10ln(tf )

ln(-l

n(1-

F))

simulationtest results

(a) Failure pressure (b) Failure time

-8-7-6-5-4-3-2-10123

1.5 2 2.5 3ln(Pf )

ln(-l

n(1-

F))

simulationtest results

-8-7-6-5-4-3-2-10123

-2 0 2 4 6 8 10ln(tf )

ln(-l

n(1-

F))

simulationtest results

(a) Failure pressure (b) Failure time

-8-7-6-5-4-3-2-10123

1.5 2 2.5 3ln(Pf )

ln(-l

n(1-

F))

simulationtest results

Figure 7: Comparison between test results and simulation results (saw-tooth loading)

7 CONCLUSION

Test results of full-scale glass plate breakage test under ramp loading and saw-tooth loading were presented with a comparison of numerical simulation results. The numerical simulation can predict the failure pressure and time fairly well in ramp loading but cannot predict the failure time in saw-tooth loading. Modification of the numerical simulation is necessary in order to expand its use for realistic fluctuating wind pressure.

REFERENCE [1] J.E. Minor, Analysis of the window damaging mechanism in windstorms, PhD thesis, Texas Tech University, May 1974. [2] H. Ishizaki, S. Miyoshi, and T. Miura, On the design of glass pane against wind loading, Proc. 4th Int. Cof. on Wind Effects on Buildings and Structures, Heathrow(London), 1975. [3] S. Kawabata, Study on wind resistance design of glass plate for cladding, PhD thesis, Japan, August 1996. (in Japanese) [4] W. L. Beason and J.R. Morgan, Glass failgure prediction model, Journal of Structural Engineering, 110(2), 197-212, 1984. [5] S.G. Reid, Flaws in the failure prediction model of glass strength. Proc. 6th Int. Conf. on Applications of Statistics and Probability in Civil Engineering, June 1991. [6] G.A. Kopp, M.J. Morrison, E. Iizumi, D. Henderson. The “Three Little Pigs” Project: Integration of wind tunnel model scale tests and full-scale laboratory tests, The 3rd Int. Symp. on Wind Effects on Buildings and urban Environment, Japan, March 3-4, 2008. [7] W.G. Brown, A practicable formation for the strength of glass and its special application to large plates, National Research Council of Canada, No. NRC 14372, Ottawa, Nov 1974. [8] A.G. Evans and S.M. Wiederhorn, Proof testing of ceramic materials – an analytical basis for failure prediction, Int. Journal of Fracture, 10(3), 379-392, 1974 [9] A.C. Fisher-Cripps and R.E. Collins, Architectural glazings: Design standards and failure models, Building and Environment, 30(1), 29-40, 1995. [10] E. Simiu and D.A. Reed, Probabilistic design of cladding glass subjected to wind loads, 4th Int. Conf. on Applications of Statistics and Probability in Soil and Structural Engineering, Italy, 1983 [11] L. Seaman, Response of Windows to Sonic Booms, Technical Report to U.S. Air Force, Project ETU-5897, Stanford Research Institute, June 1967.