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Unformatted text preview: D. L. Shelleman,1 O. M. Jadacm,2 J. C. Conway, Jr.,3 and]. J. Mecholsky, Jr.4 Prediction of the Strength of Ceramic Tubular Components:
Part I—'—Experimental Verification REFERENCE: Shelleman, D. L., Jadaan, O. M., Conway, J. 0, Jr.,
and Mecholsky, J. J ., Jr., “Prediction of the Strength of Ceramic
Tubular Components: Part II—Experimental Veriﬁcation,” Journal of Testing and Evaluation, JTEVA, Vol. 19, No. 3, May 1991, pp.
192—200. ABSTRACT: The strength distribution of reactionbonded silicon car
bide tubes that failed by internal pressurization was predicted from
strength distributions obtained from simple laboratory test specimens
at room temperature. The strength distributions of flexure bars,
Crings tested in tension, Crings tested in compression, diametrally
compressed Orings, and internally pressurized short tubes were com—
pared with the strength distribution of internally pressurized long
tubes. The methodology involved application of Weibull statistics
using elasticity theory to define the stress distributions in the simple
specimens. The ﬂexural specimens did not yield acceptable results,
since they were ground before testing, thereby altering their ﬂaw
population in comparison with the processinginduced ﬂaw popula«
tions of the tubular specimens. However, the short tube internal
pressure test, the C—ring tested in tension, and the diametrally com
pressed O—ring test conﬁgurations yielded accurate strength predic
tions of full—scale tubular components, since these specimens more accurately represent the strength~limiting flaw population in the long
tubes. KEY WORDS: failure probability, failure statistics, Weibull statistics,
Weibull modulus, C—ring, Oring Because of their better strength and corrosion resistance as
compared to metals, ceramics such as silicon carbide and silicon
nitride are being contemplated for structural applications at high
temperatures in natural gas environments. Examples may include
recuperative heat exchangers and radiant tube heaters. Due to
their low strain tolerance, low fracture toughness, and large vari
ations in strength that are attributed to the variation in inherent
microscopic ﬂaws or defects that may be introduced during pro
cessing, handling, or service, ceramics must be designed using
statistics and reliability analysis for failure predictions [1—6]. The objective of this paper is to define and develop test meth
odologies applicable to the use and design of tubular compo—
nents. A companion paper [7] presented the stress distributions Manuscript received 3/23/90; accepted for publication 11/6/90. ‘Graduate Assistant, Center for Advanced Materials, The Pennsyl
vania State University, University Park, PA 16802. 2Assistant Professor, University of WisconsinPlatteville, Platteville',
WI 53818. 3Professor, Department of Engineering Science and Mechanics, The
Pennsylvania State University, University Park, PA 16802. 4Professor, Department of Materials Science and Engineering, Uni«
versity of Florida, Gainesville, FL 32611. 192 '  ' t’. and the relationships between the fracture stress and the failure
probability for the specimen configurations used in this study,
The selection of specimen configurations (Crings, Orings, and
ﬂexure bars) was based on the following criteria: simple geom.
etry, easy machining, easy loading, and unaltered ﬂaw popula.
tion in comparison with the tubular sections from which the
specimens were cut. The exception, flexure bars, were machined
and ground from plates of the same material before testing. The
ﬂexure bar specimen was selected because of its widespread use
and to check for the effects of grinding on the strength prediction
of tubular components. I
The most widely accepted statistical method for characterizing
the strength behavior of brittle materials is Weibull analysis,
which is based on the weakest link theory [8—11]. However, other
statistical methods, which also incorporate shear stresses and/or
multiaxial stress states, have been proposed [2, 1218]. Due to
its wide acceptability and previous successes in characterizing
strength distributions, the scaling ability of Weibull statistical
methods for predicting the strengths of large industrial tubular
components from strength distributions obtained from laboratory
size specimens will be assessed. Although previous investigators
have attempted to predict the mechanical behavior of one simple
specimen configuration from another using Weibull statistics, no
comprehensive reports were found in the literature discussing the strength prediction of fullscale tubular components from
simple specimens. The reader is referred to the companion paper [7] for a dis
cussion of Weibull statistical analysis and for the derivation of
the stress analysis of the specimen conﬁgurations used in this
study. Reference [7] used computergenerated data to demon
strate how the strength of largetubular components can be pre
dicted from the Oring strength distribution. This paper will apply
the analysis described in Part I to scale Weibull strength distri—
butions obtained from simple specimen configurations (e.g., C
ring, Oring‘, and ﬂexure bar) to predict the strength distribution
of large tubular specimens. The remainder of this paper will , discuss the experimental procedure, the strength results, and the applicability of the analysis presented in Ref 7 for predicting the
strength of large tubular components. Experimental Procedure A siliconized silicon carbide material (SCRB210)5 was used in
this study. SCRB210 has a bimodal SiC grain size distribution 5Reactionbonded silicon carbide, Coors Ceramics Company, Golden,
Colorado. © 1991 by the American Society for Testing and Materials » , 4 1
rigﬁzkm ., .rmgzr a ,,v,_._.. .—m_ where exhibl‘
dark P urathI
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I
l SHELLEMAN ET AL. ON CERAMIC TUBULAR COMPONENTS: PART II where the small grain sizes range from 2 to 5 pm and the large grain sizes range from 25 to 75 pm. This material contains 19 , vol% of free silicon which completely infiltrates the silicon car
} bide compact. An optical micrograph of the microstructure is, t exhibited in Fig. 1, where the light phase is the silicon and the l. l
I, a I s and the failure ' :d in this study.
gs, Orings, and
a: simple geom_
ed ﬂaw popu1a_ from which the ‘1‘ were machined
ore testing The
widespread use
:ngth prediction r characterizin
eibull analysis,
However, other
stresses and/or
12—18]. Due to
characterizing
ibull statistical
lustrial tubular
rom laboratory
is investigators
r of one simple
Ill statistics, no
:ure discussing
iponents from ' for a dis
derivation of
5 used in this
ita to demon
ts can be pre
iper will apply
:rength distri—
ions (e.g., G
h distribution
is paper will
suits, and the
)redicting the ‘5 was used in
: distribution pany, Golden, and Materials 1 '5 l u l
l l I i a; 4 i 1 ill. dark phase is the bimodally sized silicon carbide grains, Four simple specimen configurations and two tubular config
“rations were selected for this study. The simple specimen con
ﬁgurations were: (1) fourpoint bend bars (V4 point loading), (2)
C—rings tested in compression, (3) Crings tested in tension, and
(4) diametrally compressed Orings (Fig. 1, [7]). The tubular
configuratiOns were used for: (1) a short tube internal pressure
test in which a ~101 mm long tube was filled with an RTV
polyurethane rubber6 that was then axially compressed, resulting
in a uniform radial pressure being applied to the inner walls of
the tube (see [7] for analysis); and (2) a long tube internal pres—
sure test in which a ~660 mm long tube was internally pressurized
with argon gas to failure. The dimensions of the simple and
tubular specimens are listed in Table 1. The ﬂexure bars were cut from plate stock in directions lon
gitudinal and transverse to the axis of the plate, respectively, to
check for strength anisotrophy due to the fabrication process.
Both the plate stock and the tubes, from which the other speci
mens were machined, were manufactured via a slip casting pro—
cess. The tensile surfaces of the flexure bars were subsequently g H ground to within 0.076 to 0.127 mm (0.003 to 0.005 in.) oversize using a 220 grit wheel and lapped to final size (Table 1) using a
15 um lapping medium. The edges were rounded to 0.15 mm
radius. The grinding direction was always parallel to the long
axis of the ﬂexure specimens. The Cring and O—ring specimens were cut from the same lot
of tubes to be tested in the form of short and long tubular con
ﬁgurations to ensure that the study was not biased by testing
specimens from different lots. Both sides (surfaces where the
rings were sectioned from the tubes) of the Crings and Oring
specimens were ground to within 0.0762 to 0.127 mm (0.003 to , 0.005 in.) oversize and lapped using 15 um lapping medium to the final dimensions (Table 1). While the sides of the C~ring and
Oring specimens were machined and polished, the inner and
outer surfaces of these specimens (test surfaces) remained in the
as—received condition,‘ of the original tubes. Both the short tube
(~101 mm length) and long tube (~660 mm length) specimens
were machined from 1.83 m (6 ft) long tubes. Unlike the case
for the short tubes, additional machining was required for the
long tubes on the inside surface to remove residual silicon nod
ules and to accommodate the end seals of the internal pressure
tester [19]. The radial edges on the ends of the tubes were cham—
fered, and the inside diameters on both ends of the tubes were
machined to roundness to a distance of 50.8 mm (2 in.) from the
ends. ' A universal testing machine7 operating at a crosshead speed
0f 0.508 mm/min (0.02 in./min) was used to test all uniaxially
loaded specimens. To redistribute the load and reduce the fric
tion, a 1.5 mm thick piece of alumina felt was placed between
the specimen and the alumina loading rams. The short tube
Specimens were tested on another universal testing machine.8
The long tube specimens were tested in a tube burst tester where P “Devcon Flexane 80 Liquid, Norcen Industries, Inc., Jersey Shore,
enn. 7Instron Model 4202, Instron Corporation, Canton, Mass. 81lnstron Model 4606, Instron Corporation, Canton, Mass. 198 FIG. 1—0ptical micrograph showing the microstructure of the sili—
conized silicon carbide material (SCRBZJO). the tube specimen was mounted vertically using internal pressure
seals. The pressure seals consisted of a compression viton O—ring
seal; this design prevented the tube from experiencing axial con
straint. Thus the tube was essentially free ﬂoating on the O—ring
seals. The tube specimen was contained in a steel test chamber
and was internally pressurized with argon gas until failure oc«
curred. The tube fragments were contained by insulation which
served as a witness shield; thus the fragments could be collected
for fractographic examination. Results and Discussion Results are presented in the form of unadjusted and adjusted
Weibull strength distributions, using volume and area analyses.
The unadjusted distributions show the distributions obtained ex—
perimentally for each simple and tubular specimen configuration
investigated (see Appendix for sample calculation for Oring
specimen configuration). The adjusted distributions were ob
tained by using the following equations (see [7] for derivation): l/In (Tim _ K2V2 7 0”
llm at“ _ K2142 ' (Kai) (1b) where K V and KA are the effective volume and effective area,
respectively. As discussed in Part I [7], the KV and KA terms
are also a function of the Weibull modulus. The shift factor (the
ratio of the effective volumes (areas) raised to 1/m) was employed
to adjust the experimentally obtained distributions for the simple
and short tube test conﬁgurations to the same expected volume
(area) as that of the long tube, thereby accounting for differences
in stress distribution and volume (area) of material stressed in
tension. The long tube data were used as a baseline, since this
conﬁguration most closely resembles tubular components used
in industrial applications (e.g., radiant tubes and heat exchan
gers). A 95% confidence interval [20, 21] was constructed around
the strength distribution of the long tube specimens to justify 194 JOURNAL OF TESTING AND EVALUATION TABLE 1—Dimensions of simple and tubular specimen conﬁgurations. Specimen b h Flexure Bars 4.0 3.0
Cring (Tension) 9.5 C—ring (Compression) 9.5 O~ring 9.5 Short Tube Long Tube Dimensions* (mm) L ,. rol' ri‘l’
5.8 . . . . . .
21.9 i 0.4 17.1 i 0.4
21.9 i 0.4 17.1 i 0.4
... _ 21.9 i 0.4 17.1 i 0.4
101.6 21.9 i 0.4 17.1 i 0.4
660.4 21.9 t 0.4 17.1 i 0.4 *b = width, h = thickness, L = length, r0 = outer radius, ri = inner radius. T = as—received. STRENGTH (MP3) 20.1 54.6 148.4 403.4 1096.6
.999
.934
.632 + BEND LONG.
9"
LL
V
t o BENDTRANS.
:1
Q 308 E x CRING COMP.
LL
u “1'
v m
U
;: I:0: 0 came TENS.
V
In.
El .127 Lu
c a: D ORING
..J D
d
.x A SHORTTUBE
U. .049
0 LONG TUBE .018 .007 3 4 5 6 7
LN STRENGTH (MPa) FIG. Z—Weibull plot of unadjusted strength distributions for all spec
imen configurations tested at room temperature. the success of the analysis used to scale the strength distributions
of the simple specimen configurations. Figure 2 shows the unadjusted Weibull strength distributions
for the SCRB210 material. As expected, different strength dis
tributions were obtained for each specimen configuration due to
differences in stress distributions and volume of material stressed
in tension. The bend bars (lowest volume) exhibit the highest
strengths and show no evidence of strength anisotrophy due to
processing. The Cring and Oring conﬁgurations (intermediate
volumes) exhibit intermediate strengths and give somewhat sim . . . . . ,4 wmexxmmﬁsmga ..tﬁz‘zktﬁtﬂ ilar results at higher strengths. The tubular specimens (largest volumes) display the lowest strengths. Scaling the strength distribution of a smaller to larger specimen
conﬁguration is accomplished using Eq 1, where the ratio of the
effective volume (area) terms are raised to 1/m. In addition, both
effective volume (area) terms are evaluated as a function of the
same m. However, what is the proper value of m to use in
Eq‘ 1? Two methods for evaluating the shift factor will be dis
cussed below. The first assumes that the individual in value for
each conﬁguration is correct and the same as that of the long
tube. The second method uses an equivalent modulus obtained
by combining Weibull data. The latter method enables one to
sample more specimens and combines information from speci
mens of different sizes and stress states. This combined method
is acceptable, provided the fracture initiating flaws for the speci
mens are from the same population. V The results of the adjusted strength distributions for the simple
and short tube specimen conﬁgurations are illustrated in Fig. 3.
The adjusted strength distributions were obtained via Eq 1a (vol
ume analysis) by using a Weibull modulus obtained for each
conﬁguration. The adjusted strength distributions for the short
tube and Oring specimens accurately predict the strength dis
tribution of the long tube. The adjusted strength distribution for
the C—rings tested in compression predicted strengths higher than
the tubular distributions and did not compare well with the tensile
C—ring distribution. This lack of fit was expected because the
failures for C—rings tested in compression originated at the outer
surface unlike the failures for the Crings tested in tension and
the short and long tubes which originated from flaws at the inner
surface, thus yielding a different Weibull modulus and charac
teristic strength. The bend bars showed poor adjusted strength distributions because the surface grinding changed the ﬂaw pOP a: ulation and resulted in different Weibull moduli and character
istic strengths when compared with the as—received simple and
tubular configurations. To explain why the data for the C—rings tested in tension did
not overlap the O—ring and tubular data, the fracture origins for
the C—ring and Oring specimens were located and identified.
Figure 4 shows the unadjusted strength distributions for these
two specimen configurations, in which the fracture origins are
identified by the number 1 or 2, where 1 represents a failure
associated with residual silicon nodules on the inside surface and
2 represents a failure from a “natural” ﬂaw (e.g., processing,
handling, or machining flaws). In addition, specimens in which
failure occurred at the corner are indicated by the letter c. It 15
noted that these two distributions basically overlap except for 33!? V. 151332515 \ l I i v LN LN(1t(1F)‘j FIG.
unadju
95% cc the lo
eters test 21'
C—rinj . from deper
Eq 1)
result
noted
limite
specii
conve numt m,
ﬁfz‘tatatéanstmm )ecimens (largegf‘ 3 larger specimen ,
‘e the ratio of the i In addition, borh a function of the
of m to use in ictor will be dis; 7 dual m value for i that of the long
iodulus obtained
1 enables one to tion from speci ‘ )mbined method
,ws for the speci l.
1 ns for the simple l strated in Fig. 3.
(1 via Eq 1a (vol
)tained for each
ms for the short
:he strength dis 1 distribution for ‘ gths higherthan l
l with the tensile ted because the e tted at the outer
i in tension and
aws at the inner
ilus and charac
ijusted strength ‘ 3d the ﬂaw pop~ _.
i and character '4 ved simple and l in tension did ’
ture origins for .
and identified.
tions for these a
are origins are a! :sents a failure
ide surface and ,
g., processing, l
mens in which °
.e letter c. It iS "lap except for I SHELLEMAN ET AL. ON CERAMIC TUBULAR COMPONENTS: PART II 195 STRENGTH (MP2) 7_4 20.1 54.6 148.4 403.4
.999
.934
+ BEND (Long)
.532 A
u.
‘'° BEND (Tran)
h g u CFlING (Comp.)
6: .308 m
I "I
5 m o C—HING (rem)
: 0
v a
E _127 Lu D ORING
I C:
J D A SHORTTUBE
d
4;
~049 “ 0 LONG TUBE
.018
.007 LN STRENGTH (MPa) FIG. 3—Weibull plot of adjusted strength distributions compared with
unadjusted long tube strength distribution. The curved lines represent the
95% conﬁdence bands constructed around long tube strength distribution. the lower strength specimens. Table 2 lists the Weibull param
eters for all specimen configurations. Based on a twotailed t
test at a 95% level of significance, the Weibull modulus for the
Crings tested in tension (4.3 i 0.7) is not significantly different
from that of the Orings (5.5 i 08). Because of the strong
dependence of the shift factor on the Weibull modulus (see
Eq 1), the lower Weibull modulus for the Crings tested in tension
resulted in a more conservative strength prediction. It should be
noted also that the number of specimens tested is somewhat
limited and that convergence to a single Weibull modulus for all
Specimens failing at the inner surface was not observed. Perhaps
Convergence to a single Weibull modulus would result if a large
number of specimens for each configuration were tested. STRENGTH (M Pa) 99.5 148.4 221.4 330.3 .999
El ORING 0 CRlNG TENS. .934
.632 .308 .127 Ln Ln (1t(1—F)) .049 1 Associated with St on tensile sudace
2 Natural Flaw .015
c corner failure FAILURE PROBABILITY (F) .007
4.5 4.8 5.0 5.2 5.4 5.6 5.8 6.0 LN STRENGTH (MPa) FIG. 4—Weibull plot of unadjusted strength distributions for Orings
and C—rings tested in tension with identiﬁcation of failure sources for all
specimens. Recently, Johnson and Tucker [22] discussed a method of
estimating Weibull parameters (i.e., m and 0'0) by combining
data from specimens of different loading factors and multiple
specimen sizes. This approach represents all the data from the
different Specimens on a plot of fracture stress versus the effective
volume. Figure 5 demonstrates the method used to determine
an equivalent or combined Weibull modulus. The open symbols
represent the original strength data as a function of that speci—
men’s effective volume (KV). The associated solid symbols
were obtained by transforming each original effective volume
(KV) to that effective volume (KV') that would be expected to
yield the observed strength at a particular probability level (e.g.,
F’ = 0.5). This transformation is accomplished via the following expression [22]: 7 _ ln (1 — F’)
KV — KV—————In (1 _ F) (2) From this procedure a linear regression fit to the data can be
used to estimate a combined or equivalent Weibull modulus,
since the slope of the line is equal to the negative of the inverse
of the Weibull modulus (i.e., —1/m). In addition, the position TABLE 2—Weibull parameters for simple and tubular specimens.  N0. of Weibull Modulus Characteristic Strength
Specimen Samples (m) (MPa) (0'0)
Flexure (Longitudinal) 17 26.9 t 5.1 280.4 : 13.1
Flexure (Transverse) 26 20.6 i 3.2 266.6 1— 14.5
Cring (Compression) 22 7.2 17 1.2 118.5 : 17.9
Cring (Tension) 26 4.3 t 0.7 75.8 i 17.9
O~ring ‘ 26 5.5 i— 0.8 86.1 i 158
Short Tube 10 6.0 1 1.5 85.4 i 14.5
Long Tube 9 4.5 t 1.2 ' 106.1 x 13.8 “WmammeMKMWFAM‘ NM mammwe "smears JWWAMWWWYMwWWWWk '  '. 196 JOURNAL OF TESTlNG AND EVALUATION Weibull Modulus=6.1 LO—RING
ORING
C~RING (Tens)
C—RING trans.)
SHORT TUBE SHORT TUBE LN STRENGTH (MPa) A LONG TUBE LONG TUBE LN EFFECTIVE VOLUME (KV) FIG. 5—Strength as a function of effective volume to determine an
equivalent Weibull modulus for SCRBZIO specimens tested at room tem
perature (based on analysis in Ref 22). of the line can also be used to estimate a combined or equivalent
characteristic strength (00). Batdorf and Sines [23] proposed an—
other method for combining data for estimating Weibull param
eters based on a least squares approach for obtaining a combined
m value. The Johnson and Tucker method has an advantage over
that proposed by Batdorf and Sines in that a single equivalent
on can be estimated. Although the latter method will produce
equivalent Weibull moduli, separate values of 00 are obtained
for each specimen distribution. The method proposed by Johnson and Tucker was used to
determine an equivalent Weibull modulus for all specimen con
figurations where fracture initiates at the inner surface (Crings
tested in tension, Orings, and both tubular configurations). Fig
ure 6 shows the adjusted volume strength distributions as a result
of using the equivalent Weibull modulus of 6.1 in Eq 1. Unlike
the data in Fig. 3, the data for the C—rings tested in tension tend
to overlap the other strength distributions and are within the
95% conﬁdence bounds. Thus the effective volume expression
for these specimens can be used to predict the strength distri
bution of industrial size components (e. g, open—ended tube). The discrepancy between the results for the Crings tested in
tension and C—rings tested in compression can be explained by
the presence of different ﬂaw distributions on the inner and outer
tube walls. As a result of the processing of these tubes, both the
inner and outer surfaces contain residual silicon nodules [24]. STRENGTH (MPa) 7.4 20.1 54.6 148.4 403.4 .999
SCRBZ10 .934
.632 .308 .127 LN LN(1t(1F)) .049 FAILURE PROBABILITY (F) 00 .018 Volume Weibull Analysls Room Temperature .007 2 a 4 5 5
LN STRENGTH, (MP3) FIG. 6— Weibull plot of adjusted volume strength distributions com
pared with unadjusted long tube strength distribution where the combined
Weibull modulus (m = 6.1) was used in Eq 14 to scale the strength data.
The curved lines represent the 95% conﬁdence bands constructed around
long tube strength distribution. The silicon nodules on the outer surface were removed via a
sandblasting procedure by Coors before shippings. However, the
inner tube wall, which did not receive the sandblasting treatment,
is often highly decorated with silicon nodules remaining from the
tube fabrication process. Figure 7 shows the fracture origin of a
C—ring tested in tension, in which the failure origin is associated
with a residual silicon nodule on the inside tensile surface. Local
residual stresses are most likely associated with these silicon
nodules due to thermal expansion mismatches during the pro
cessing of this material. Thus these silicon nodules can act as
local stress concentrators, thereby reducing the stress level nec—
essary to cause fracture. Failures associated with these silicon
nodules were also observed for some Cring and tubular (both
short and long) specimens. In fact, four out of the nine long tube
specimens had failures associated with these residual silicon nod
ules.‘ Figure 8 shows the‘reconstruction of a long tube specimen
which failed at a fracture stress of ~63 MPa. It is noted that this
tube‘ is decorated to a considerable extent with these residual
silicon nodules. Upon reassembling this tube, the crack paths
were used to located the fracture origin shown in Fig. 9. The
critical ﬂaw was measured to be ~2175 um in size. Upon inspection of Figs. 3 and 6, it is apparent that the tWO
lowest data points of the long tube specimens have deviated from
the rest of the strength distribution. Fractographic examination
after failure did not reveal the location of the fracture origin
within the unmachined section of the, tube. These two tubeS
apparently failed at the ends where the internal surfaces of the
tubes were machined before testing (to remove the silicon nod—
ules decorating the internal surfaces of the tubes) in order to fit .,.,_.~v_~ . we <~ ——. 1..— FI(
C rin;
Fuilu:
surfai ~ Arrot the t
devie
size. Th
the d
son [.
for t
more
crem éuwlmmuieniez 'l'l ' '3 0~RING O GRING (TENS) 0...... u...“ 0 LONG TUBE ttributionx Com.
re the combined
18 strength data.
structed around imoved via 3
However, the
ng treatment,
ning from the
re origin of a
, is associated
urface. Local
these silicon
ring the pro
es can act as
ess level nec~
these silicon
ubular (both
ine long tube
.1 silicon nod—
lbe specimen
)t'ed' that this
iese residual
crack paths
Fig. 9. The that the two
eviated from
examination
lCtLll‘C origin
6 two tubes
rfaces of the
silicon nod
1 order to ﬁt A SHORT TUBE x I : =(N+1)—n, I,
l, l 1) \ FIG. 7~Scanning electron micrograph: of the fracture surface for a
Cting specimen tested in tension tliatfai/cd at a stress of 180 MPa. Note:
y Failure origin is associated with a residual silicon nodule on the tensile
surface: (a) secondary electron image, (b) backseattered electron image.
~ Arrows indicate size of critical flaw (226 um). g the test fixture. This finding is a plausible explanation for the
' fifﬁViation of these two data points from the rest of the sample
. Size. The two outliers (outlying data points) were eliminated from
’ the distribution via the censored data method proposed by John—
5011 [25]. Essentially, this method determines the ranking number
, f0r the strengths by calculating a new increment when one or
w more censored strengths are encountered in the data. This in
Crement, A, is expressed as 4 A (3) 1+n,, SHELLEMAN ET AL. ON CERAMIC TUBULAR COMPONENTS: PART ll 197 where N is the total number of specimens, n, is the previous
ranking, and n, is the number of specimens beyond the present
censored set. The new ranking is obtained by adding the new
increment, A, to the previous ranking, n,. This new ranking is
then used to determine the failure probability using the same
Weibull statistical method (see Eq 2 in Since failures predominantly occurred at the surface, an area
Weibull analysis was performed. The adjusted strength distri—
butions are shown in Figs. 10 and 11 based on the individual
modulus and combined Weibull modulus methods, respectively.
Based on the area analysis using the individual modulus for each
conﬁguration, the strength distributions for the C—rings tested in
tension, the O—rings, and the short tube fall Within the 95%
conﬁdence bounds. As was the case for the volume Weibull
analysis (Fig. 2), the C~rings tested in compression failed to
predict the long tube. A combined or equivalent Weibull modulus of 4.7 was ob
tained by the foregoing procedure, except that strength was plot—
ted as a function of area instead of volume (Fig. 12). The results
for the adjusted area strength distributions using the combined
modulus (Fig. 11) were analogous to those obtained using the
volume analysis. Namely, all specimen configurations that
sampled the flaw population on the inner surface of the tube
(C—ring tested in tension, Oring, and short tube) gave accurate
(within 95% conﬁdence) predictions of the long tube strength
distribution. Conclusions For the SCRB210 material, the Oring and short tube speci—
mens best predicted the strength distribution of the long tube
components using either the volume or area Weibull analysis.
The Crings tested in tension yielded accurate predictions based
on the area analysis, but gave conservative predictions based on
a volume analysis. With a combined Weibull modulus to scale
the strength distributions, analogous accurate strength predic
tions for the long tube strength distribution were obtained using
volume and area analyses. Thus the effective volume expressions
for the Crings tested in tension, Orings, and tubular specimen
conﬁgurations produce acceptable predictions (within a 95%
confidence interval) of the strength ofindustrial size components.
The Crings tested in compression overestimated the long tube
fracture strength because failures initiated at the outer surface,
where the flaw population was altered due to sandblasting. The
SCRBZIO tubes possessed a more severe flaw population at the
inner surface due the existence of silicon nodules, which cause
local stress concentrations due to residual stresses from thermal
expansion mismatch during processing. The bend bar specimens
resulted in poor predictions because they were ground before
testing, thus altering their inherent ﬂaw populationg Previously, a simpliﬁed stress solution for a diametrally com
pressed Oring was investigated to measure the strength of tubes
as a function of the lengthtoradius ratio [26]. The simplified
analysis used in Ref 26 applies only to thinwalled tubes and a
plane stress condition. As the lengthtoradius ratio is increased,
the stress condition changes from plane stress to plane strain.
These conditions were not incorporated into the stress analysis
used in Ref 26, and as such it was concluded that the analysis
used to evaluate the strength of ceramics subject to diametral
compression was inadequate. Based on the results of our study, 198 JOURNAL OF TESTING AND EVALUATION FIG. STReconsttuction of long tube specimen that failed at a stress of 63 MPa. Arrows indicate location of failure origin, which is associated with .
residual silicon nodules on the tnszde surfaCe of the tube. Cracks propagated in an axial direction, indicating that the hoop stress was the maximum " stress leading to failure of the tube. FIG. 9—Qplical photograph of the failure origin shown in Fig. 8; arrows indicate criticalflaw Size (2175 um) on fracture surface The silicon nmlll/L’
assoctated With the failure origin and the tensile surface of the tube fragment are indicated by labels and arrows. ‘ the analysis presented in Ref 7 can be used to predict the strength
of large tubular components. As discussed in Part I, as the wall
thickness of the ring decreases (ri/r0 increases), the exact stress
distributiOn (Eqs 5a to 8a, [7]) approaches the approximate so
lution obtained through strain energystraight beam theory con
siderations (Eq 9a [7]). ~ Based on this analysis and the results of this study, both the
C—ring Specimens tested in tension and the Oring specimens lead
to accurate strength predictions of fullscale tubular components
in which failures initiate on the inside surface. However, the
Oring specimen has two major advantages over the C—ring speci
men tested in tension; namely, less machining is required and a ‘J LN LN(1I(1—F)) M J
l
_, i t more stable loading configuration is obtained. Loading a SpCCi‘
men in compression is generally much easier than loading in
tension, which often requires complicated instrumentation 10
insure proper alignment of loading pins and/orgrips. Thus We »
recommend the Oring loaded in diametral compression as the
best configuration to evaluate the strength of large tubular com
ponents in which failure initiates at the inner tube surface. ,
For predicting the strengths of full—scale tubular componentS l
in which failure initiates on the outer surface, we would ref? ’
ommend the compressive Cring specimen configuration for 1‘5
easy machining and loading. Although fracture initiated on the inside surfaces of the long tubes in this study, actual in—servlce “AVluau.” ,~— g— . 
.s 1
r0 
U FIG.
individu
pared w.
The mm
long tub 7.
2
1
0
1 r
el
l
l LN LN(1!(1F)) 3 ,4 .sl 2 FIG.
Oring,
bined Vl
distribu ,
95% c0 1 ts associated with
was the maximum 2. The Silicon nodule Loading a Speci—
' than loading in
strumentation to
’r grips. Thus we
impression as the
irge tubular com
ZUbe surface.
rular components
3, We would rec
lfiguration for its
3 initiated on the
actual inservice “fl—r SHELLEMAN ET AL. ON CERAMIC TUBULAR COMPONENTS: PART II STRENGTH (MPG) 7.4 20.1 54.8 148.4 403.4
.999
SCRB210
Room Temperature °+
1; .934
§:
3* rs+ BEND (Lang)
g .632 E;
4» BEND .
gt E (Tran)
.4
E w 308 a n GRING(Oomp.)
I ° '
 ¢+ 4:
3 9+ m o CRJNG crane.)
. o O
u + Cl:
1 ° .127 DEJ O—RING
—' Lu
5 g A SHORTTUBE
.1 °49 E o LONGTUBE(C) .018 .007 LN STRENGTH (MP3) . FIQ. lO—Weibull plot of the adjusted area strength distributions using
tndtvzdual Weibull modulus for the simple and short tube specimens com—
pared wtth unadjusted long tube strength distribution at room temperature.
The curved lines represent the 95% conﬁdence bands constructed around
long tube strength distribution. STRENGTH (MPa) 20.1 54.6 148.4 403.4 .999 .934 r.
.532 5;,
t D O»RING
9‘ a
.u': .306 g 0 Game (TENS.)
V
m
E» to: A SHORTTUBE
z 127 °
Tl ‘ Lu 0 LONG TUBE
2 cc
‘J D
.1
.049 '3;
LL .018 Area Weibull Analysls Room Temperature .007
2 3 4 5 B LN STRENGTH (MPa) FlG. ll—Weibull plot of the adjusted area strength distributions for
offing, Cring tested in tension, and short tube specimens where the com
b‘Pﬂ'i Weibull modulus (m = 4% was used in Eq 1b to scale the strength
ggttrtbutto'ns obtained at room temperature. The curved lines represent the % confidence bands constructed around long tube strength distribution. ‘ W WWWWWLHMWaMmi«m 1w ’
~ w .W ‘WW. ~ 199 Weibull Modulus  4.7 CI ORING
I O—RING o CRING (TENS) H «s o. a '. ‘CIRtN'G mNs.) I E .
,A T e z SHOR TU E E 1.. A SHORT ruse U1 2 .s 0 LONG TUBE
0 LONG ruse LN EFFECTIVE AREA (KA) FIG. 12—Strength as a function of effective area toldetermine an equiv
alent Weibull modulus (m = 4.7) for SCRBZJO specimens tested at room temperature (based on analysis in Ref 22). components such as radiant tubes and heat exchangers can ex—
perience complex stress states due to internal gas pressures as
well as thermal stresses which develop on transient heatup of
such components. In addition to thermal stresses, a constrained
tube can also develop large tensile stresses on the outer surface.
Therefore it is important to sample both the inner and outer flaw
distributions to evaluate thoroughly the strength behavior of tu—
bular components subjected to both thermal and mechanical' (internal gas pressure) stresses. Acknowledgments This research was supported by the Gas Research Institute.
The authors would like to than\k Darryl Butt, Dr. R. E. Tressler,
and Dr. J. R. Hellmann for their advice and assistance in de
signing the tube burst test apparatus used in this study. Appre
ciation is also extended to Dr. C. A. Johnson for helpful dis cussions throughout this study. APPENDIX Table 1 [7] lists the appropriate stress equations for calculating
the maximum tensile stresses for all specimen conﬁgurations used
in this study. A sample calculation for the O—ring specimen con
ﬁguration is given below. ' The maximum tensile stress (tangential stress, 0'0) is a function
of failure load (P), the width of the specimen (b), the outer
radius of the specimen (r0), and the tangential stress magnifi— ' ' W33}; 200 JOURNAL OF TESTING AND EVALUATION cation factor {Q(r, 111)} and is expressed as P bnro max _
0'3 .‘ Qtno) _ <4) The tangential stress magnification factor is given by Eq 22 in
Ref 7. In order to determine the maximum stress, L]! is set equal
to 0. This equation was programmed into a Vax 11/80 computer.
For P = 296.0 lb, b = 0.375 in., r0 = 0.866 in., and Q(r,0) =
143.0, the maximum tangential stress is calculated to be 37.8 ksi
or 260.5 MPa. References [1] Gyekenyesi, J. P., “SCARE: A Postprocessor Program to MSC/
NASTRAN for Reliability Analysis of Structural Ceramic Com
ponents,” Journal of Engineering for Gas Turbines and Power,
Transactions ofASME, Vol. 108, 1986, pp. 540—546. [2] Batdorf, S. B., “Fundamentals of the Statistical Theory of Failure,"
in Fracture Mechanics of Ceramics, Vol. 3, R. C. Bradt, D. P. H.
Hasselman, A. G. Evans, and F. F. Lange, Eds, Plenum Press,
New York, 1978, pp. 1—29. [3] Deslavo, G. J., “Theory and Structural Design Applications of
Weibull Statistics,” WANLTME2688, Westinghouse Electric Corp,
1970. [4] Shih, T., “An Evaluation of the Probabilistic Approach to Brittle ' Design,” Engineering Fracture Mechanics, Vol. 13, 1980, pp.
257—271. [5] Johnson, C. A., “Fracture Statistics in Design and Applications,”
General Electric Report No. 79CRD212, 1979. [6] Ferber, M. K., Tennery, V., Waters, 5., "and Ogle, J., “Fracture
Strength Characterization of Tubular Ceramic Materials Using a
Simple Cring Geometry,” Journal of Materials Science, Vol. 8,
1986, pp. 2628—2632. [7] Jadaan, O. M., Shelleman, D. L., Conway, J. C., Jr., Mecholsky,
J. J. , Jr., and Tressler, R. E., “Prediction of the Strength of Ceramic
Tubular Components: Part I—Analysis,” Journal of Testing and
Evaluation, Vol. 19, No. 3, May 1991, pp. 181—191. [8] Weibull, W., “A Statistical Distribution Function of Wide Appli
cability,” Journal of Applied Mechanics, Vol. 18, 1951, pp. 293—
297. [9] Shetty, D. K., Rosenﬁeld, A. R., and Duckworth, W. H., “Sta—
tistical Analysis of Size and Stress State Effects ‘on the Strength of
an Alumina Disk," in MethodsforAssessing the Structural Reliability
of Brittle Materials, ASTM STP 844, American Society for Testing
and Materials, Philadelphia, 1984, pp. 5780.  emunMrax waning was»: :3 21‘s." ' “3” "‘ﬁJL’HEL ' atria? animation ""21;‘iliﬁiiﬁﬁs‘i‘t‘ﬁﬁlﬁz‘fitﬁtEmits; ‘itéter’ifuﬂtﬁlbﬂ m: 2. [10] Bansal, G. K. and W. H. Duckworth, “Effect of Specimen Size on
Ceramic Strength,” in Fracture Mechanics of Ceramics, V01. 3, Plenum
Press, New York, 1978, pp. 189—204. [11] Weaver, G., “Engineering with Ceramics: Part I—The Weibun
Model,” Journal of Materials Education, Vol. 5, No. 5, 1983. [12] Evans, A. G., “AGeneral Approach for the Statistical Analysis of
Multiaxial Fracture,” Journal ofthe American Ceramic Society, Vol.
61, No. 7—8, 1978, pp: 302—308. [13] Giovan, M. N. and Sines, G., “Biaxial and Uniaxial Data for Sta.
tistical Comparisons of a Ceramic’s Strength,” Journal oft/1e Amer.
ican Ceramic Society, Vol. 62, No. 9—10, 1979, pp. 510—515. [14] Giovan, M. N. and Sines, G., “Strength of a Ceramic at High
Temperatures under Biaxial and Uniaxial Tension," Journal of the
American Ceramic Society, Vol. 64, N0. 2, 1981, pp. 68—73. [15] Petrovic, J. J. and Stout, M. G., “Fracture of A1203 in Combined
Tension/Torsion: II, Weibull Theory,” Journal of the American Ce.
ramic Society, Vol. 64, No. 11, 1981, pp. 661—666. [16] Stout, M. G. and Petrovic, J. J., “Multiaxial Loading Fracture of
A1203 Tubes: 1, Experiments," Journal of the American Ceramic
Society, Vol. 67, No. 1, 1934, pp. 14—23. [17] Lamon, J. and Evans, A. G., “Statistical Analysis of Bending Strengths
for Brittle Solids: A Multiaxial Fracture Problem,” Journal of the
American Ceramic Society, Vol. 66, No. 3, 1983, pp. 177—182. [18} Lamon, J., “Statistical Approaches to Failure for Ceramic Relia
bility Assessment,” Journal of the American Ceramic Society, Vol.
71, No. 2, 1988, pp. 106—112. [19] Shelleman, D. L., Jadaan, O. M., Mecholsky, J. J., Jr., and Com
way, J. C., Jr., “Tube Burst Test Apparatus for High Temperature
Strength Evaluation of Ceramics,” to be submitted to Journal of
Testing and Evaluation. 1 [20] Abernethy, R. B., Breneman, J. E., Hedlin, C. H., and Reinman,
G. L., Weibull Analysis Handbook, Final Report, AFWALTRv
2079, Pratt and Whitney Aircraft, West Palm Beach, Fla, 1983. [21] Srinivisan, R. and Wharton. R. M., “Confidence Bands for the
Weibull Distribution,” Technometrics, Vol. 17, N0. 3, 1975. [22] Johnson, C. A. and Tucker, W. T., “Advanced Statistical Concepts
of Fracture in Brittle Materials,” Ceram. Tech. Newsletter, No. 21,
1989. [23] Batdorf, S. B. and Sines, G., “Combining Data for Improved Wei
bull Parameter Estimation,” Journal of the American Ceramic So
ciety, V01. 63, No. 1—2, 1980, pp. 214—218. [24] Roy, D. W., Green, K. B., and Dobos, G. J., Ceramic Tube Ma
terials and Processing Development, Development of Large Reaction
Bonded Silicon Carbide Comporzentsfor GasFired Furnances, Top»
ical Report, Gas Research Institute, GRI88/0139, 1985—87. [25] Johnson, L. G., The Statistical Treatment of Fatigue Experiments,
Research Laboratories, General Motors Corp, 1959, pp. 44—50. [26] De With, G., “Note on the Use of the Diametral Compression Test , for the Strength Measurement of Ceramics," Journal of Materials
Science Letters, Vol. 3, No. 11, 1984, pp. 1000—1002. \ '" .ﬂkﬁiﬂ’ﬁkiﬁliﬁﬁlfdiﬂﬁ El L‘ARH‘E‘AEJ l
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 Fall '08
 Mecholsky

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