1040
PROCEDURE FOR STANDARDIZING MASONRY FLEXURAL STRENGTHS
CLAYFORD T. GRIMM, P.E.
Department of Civil Engineering, ECJ 5.2
University of Texas, Austin, Texas 78712
ABSTRACT
Differing standard test methods for flexural strength of masonry produce
different results for the same materials. A method is presented for
relating the results from different tests to one another.
INTRODUCTION
Because masonry flexural test specimens are of various sizes and are loaded
in various ways, results from different tests are not directly comparable .
Standard test methods include those of the American Society for Testing and
Materials, i.e. ASTM E 72, Standard Methods of Conducting Strength Tests on
Panels for Building Construction; ASTM E 518, Standard Method of Test for
Flexural Bond Strength of Unreinforced Masonry Assemblages; and ASTM C
1072, Standard Method for Measurement of Masonry Flexural Bond Strength.
Since allowable stress in flexural tension in masonry permitted by the
building codes is based on ASTM E 72, it is desirable to relate all
flexural test data to that method. This paper provides a procedure for
doing 50, which is based on the method cited in Reference 2.
Mean Strength of Joints
When masonry with an even number of courses has a concentrated load at
center span or when the bond wrench is used to determine flexural strength,
maximum stress is placed on only one joint . When seven course prisms are
loaded at their third points , the two center joints are at maximum stress .
Other loading conditions and specimen sizes may also place more than one
joint at maximum stress. Test specimens whi0~ place N joints at maximum
1041
stress, test the strength of the weakest of N joints. lhe mean strength
of N joints is greater than the strength of the weakest joint.
Mn = Mw + (Sw Ks/Kv)
(1)
Where :
Mn = mean strength of N joints in test specimens with N joints at maximum
stress, psi (MPa)
Mw = mean strength of test specimens, i.e. mean strength of the weakest of
N joints, psi (MPa)
Sw = standard deviation in strength of test specimens, i.e. in the strength
of the weakest of N joints , psi (MPa)
Ks = joint strength coefficient for test specimen, dimensionless (See lable
1 or use the approximation in Eq. 2 where N > 4)
Ks
470 (NO . 001 - 1)
(2)
N = number of joints in test specimen at maximum stress, dimensionless
Kv = joint strength variance coefficient for test specimen, dimensionless
(See lable 1 or use approximation in Eq. 3 where N > 4)
Kv
(1.1252 * 10- 7 * N4) - (2.006 * 10- 5 * N3) +
(0.001272 * N2) - (0.03465 * N) + 0.83536
Sn
=
Sw/Kv
(3)
(4)
Where :
Sn = standard deviation in mean strength of N jo i nts, psi (MPa).
Example: Where two joints are tested at maximum stress (N = 2), from
lable 1 Ks = 0.5591 and Kv = 0.8211 . If the mean strength of the test
specimens is 494 psi (3.4 MPa) with a standard deviation of 56 psi(0.4 MPa)
Mw = 494 (3.4 MPa) and Sw = 56 (0 . 4 MPa) . lhen from Eq . 1:
Mn = 494 + (56 X 0.5591/0.8211)
Mn = 532 psi (3 . 7 MPa)
From Eq. 4: Sn = 56/0.8211 or Sn
68 psí (0.5 MPa)
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Strength of Standard Wall
F = Mn - n Sn
(5)
Where:
F = estimated f1exura1 strength of ASTM E 72, uniform1y 10aded, standard
wa11 bui1t of the same materia1s and workmanship as test specimens .
n = number of standard deviations be10w mean f1exura1 strength of test
joints coincident with mean f1exura1 strength of ASTM E 72, uniform1y
10aded, standard wa11, dimension1ess
p
n = T + 4.5 * 10- 4 - (2.51557 + 0.802853 *
T + 0.010328 * T2)/(1 + 1.432788 *
T + 0.189269 * T2 + 0.001308 * T3)
(6)
T = (ln p-2)-0.5
(7)
probabi1ity of no f1exura1 fai1ure in standard wa11, dimension1ess
p = p1 * p2
pn
Xn
Rn
(8)
probabi1ity of no f1exura1 fai1ure in n th joint in the standard wa11,
dimension1ess
pn = 0.5 * (1 + 0.049867347 * Xn + 0.0211410061 * Xn 2
+ 0.0032776263 * Xn 3 + 0.0000380036 * Xn 4
+ 0.0000488906 * Xn 5 + 0.000005383 * Xn 6 )-16
+ 1.5 * 10- 7
(9)
Xn = Mn * (1 - Rn)/Sn
(10)
number of standard deviations be10w mean strength of tested joints
coincident with stress in the n th joint of ASTM E 72, uniform1y
10aded, standard wa11, dimension1ess
ratio of stress in n th joint to maximum stress in ASTM E 72,
uniform1y 10aded, standard wa11 joints, dimension1ess
G = F/Mw
•
(11)
1043
~ere:
G = ratio of flexural strength of ASTM E 72, uniformly loaded, standard
wall to flexural strength of test specimen (F/Mw), dimensionless
Example
Problem: An ASTM E 72 standard, uniformly loaded, masonry wall has a
uniform lateral load applied over a span of 7.5 ft. (2.5 m). If the course
height is 8 in. (203 mm), the wall is 12 courses high, there are 11 joints
in the wall span, and the distance from the wall support to the first
interior joint is 5 in. (127 mm). See Figure 1 below.
Suppose that test specimens have a single load at midspan and an even
number of courses. If the mean modulus of rupture of the test specimens
is 30.06 psi (0.21 MPa) with a standard deviation of 10.43 psi (0.07 MPa),
what is the estimated strength of an ASTM E 72 standard, uniformly loaded,
masonry wall built of the same materials and workmanship as the test
specimens, and what is the estimated ratio of the ASTM E 72 standard wall
flexural strength to test specimen flexural strength?
Solution: Draw a bending moment diagram for the standard wall as in
Figure 1.
1.00
0.97
0.87
Relative
Stress,
Rn
0.72
0.49
0.21
0.00 --.--.--.--.--.--.--.--.-- .--.--.-I
Joint No. 11
Distance 15
I
2 3 4 5 6 7 8 9 10 11
13 21 29 37 45 53 61 69 77 85
in.
I
1<------------90 in.----------- >I
(2 . 29 m)
I
I
Figure 1.
Bending Moment Diagram For Standard Flexural Wall (ASTM E 72)
With Eight Inch Course Height .
Prepare a table showing the no failure probability in each joint in
the standard wall. If stress in the center joint equals the mean strength
1044
of all joints, the no failure probability in that joint is 50%. The no
failure probability in other joints increases as stress is reduced with an
inc rease in distance from the center of the panel. The no failure
probability of the panel is the product of the no failure probability of
all panel joints as indicated in Eq. 8.
Joint No.
1 and 11
2 and 10
3 and 9
4 and 8
5 and
6
7
Distance
from
Reaction
to Joint,
in.
5
13
21
29
37
and
and
and
and
and
45
85
77
69
61
53
Relative
Stress,
Rn
0.21
0.49
0.72
0.87
0.97
1.00
Xn in
Eq. 10
Pn in
Eq . 9
2.28
1. 46
0.82
0.36
0.09
0.00
0. 9887
0.9279
0.7939
0.6406
0.5359
0.5000
Therefore, from Eq. 8 the no failure probability of standard ASTM E 72
walls with 8 in. course height is determined as follows :
p = 0. 5 * (0.9887 * 0.9279 * 0.7939 * 0.6406 * 0.5359}2
P = 0. 0313
Therefore, from Eq. 6 and 7, n = 1.86.
For a single joint at maximum stress in test specimens from Table 1
Ks = O and Kv = 1. From Eq. 1 Mn = Mw = 30.06 (0.21 MPa), and from Eq. 4
Sn = Sw = 10.43 (0.072 MPa). Therefore, from Eq. 5 F = 30.06 - (1.86 *
10.43) or F = 10.7 psi (0.074 MPa).
The ratio of flexural strength of standard ASTM E 72 masonry walls
with 8 in. course height to flexural strength of these test specimens is
10 . 7/30 . 06 or G = 0.35 .
ASTM E 72 VS. ASTM C 1072 and ASTM E 518
The ratio of the mean flexural strength of uniformly loaded ASTM E 72 walls
to that of bond wrench tested specimens (ASTM C 1072) is a function of the
course height in the walls and coefficient of variation in the bond wrench
test data as shown in Table 2. The ratios of mean flexural strength of
uniformly loaded ASTM E 72 walls to the flexural strength of third point
loaded seven course prisms (ASTM E 518) is given in Table 3. The ratios of
mean flexural strength of uniformly loaded ASTM E 72 walls to the flexural
strength of third point and quarter point loaded walls are given in Ref. 2.
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1045
Sampl ing
A lot is a quantity of material which, insofar as is practical, consists of
a single type, grade, class, size, finish (texture), and composition
produced by a single source by the same process and under practically the
same conditions. A specimen is an individual piece of material from a loto
A sample is a number of specimens.
Valid conclusions about a lot can be drawn only by testing a
representative sample. The method of selecting specimens must be unbiased.
Bias is avoided by selecting a random sample , i.e . every specimen of the
sample must have an equal chance of being selected in every trial. A
sample selected without a conscious plan is a haphazard sample, not a
random sa~ple. It is virtually impossible to draw a sample at random by
exercise of human judgement alone. The proper use of an artificial or
mechanical device for selecting a random sample is necessary . Specimens
may be selected by assigning a number to each unit or small group of units
in the lot and using a table of random numbers or an electronic random
number generator to select a number of specimens. For any given set of
conditions there are usually several possible sampling plans, all valid,
but differing in speed, simplicity, and cost (3).
The number of test specimens required to determine the coefficient of
variation in test data should be based on ASTM E 122 as follows:
Nr = (k * v/e)2
Nr
k
(12)
required sample size (number of specimens) to next larger whole number.
a factor corresponding to a probability that the difference between
the sample mean and the lot mean will not exceed e, dimensionless.
See Table 4.
e
allowable difference between the sample mean and the lot mean , i . e.
the allowable sample error, %
v
coefficient of variation of the lot, %
Baker (1) suggests that the coefficient of variation in the flexural
st r ength of field-built masonry is 0.25 for excellent control and
workmanship, 0. 30 for good control and workmanship, and 0.35-0.4 for usual
control and workmanship. Laboratory-built specimens with close control and
supervision are said to have a coefficient of variation in flexural
strength of 0. 20 .
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Example: If the probability is 90% of not exceeding a sampling erro r
of 10% and the coefficient of variation is 20%, then e = 10, v = 20, from
Table 4 k = 1.645, and from Eq. 12 Nr = (1.645 * 20/10)2 or Nr = 11.
We conclude that if the testing of a representative sample of 11
specimens produces a coefficient of variation of 20 %, we can be 90%
confident that the mean strength of the sample does not differ from the
mean strength of the lot by more than 10%. Similarly, if the testing of a
representative sample of 20 specimens results in a coefficient of variation
of 17%, we can be 99% confident that the mean of the sample does not differ
from the mean of the lot by more than 9.8%.
Conclusion
Given the mean and standard deviation for the flexural strength of masonry
determined by a standard method, the strength which would have been
obtained by another standard method can be calculated by the procedure
presented. A technique is also provided by which to determine the size of
a representative sample required to achieve an allowable sampling error
with a given level of confidence .
TABLE 1
Joint Strength and Variance Coefficients
Number of Joints
at Maximum Stress
in Test Specimen
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Ks
0.0000
0.5591
0.8692
1. 0372
1.1982
1. 2641
1. 3823
1.4366
1.5278
1.5413
1. 6140
1.6310
1.6782
1. 7236
1. 7472
1. 7907
1.8085
1.8266
1. 8524
1.8729
Kv
1.0000
0. 8211
0. 7597
0. 7009
0. 6813
0. 6370
0. 6280
0.6095
0.6060
0. 5739
0. 5901
0. 5715
0.5507
0. 5542
0.5448
0.5579
0.5233
0.5401
0.5139
. 0.5091
1047
TABLE 1 (continued)
Number of Joints
at r~aximum Stress
in Test Specimen
22
24
26
28
30
32
34
36
38
40
42
44
46
Ks
Kv
1. 9074
1.9341
1.9882
2.0ll7
2.0632
2.0833
2.1029
2.ll33
2.1600
2.1580
2.1743
2.2041
2.2190
0.5096
0.5038
0.4999
0.4991
0.4943
0.4885
0.4834
0.4795
0.4886
0.4798
0. 4681
0.4823
0.4841
TABLE 2
Ratio of Flexural Strength Of Uniformly Loaded ~lasonry Wall Panel
(ASTM E 72) To Bond Wrench Flexural Strength (ASTM C 1072) (a)
Nominal
Course
Height
in .
Coefficient of Variation
in Bond Wrench Strength
0.05
0.10
0.87
0.68
2.00
0.73
2.67
0.90
0.75
3.00
0.91
4.00
0.93
0.80
0.85
5.33
0.95
0.86
6.00
0.95
0.90
8.00
0.97
0.94
12.00
0.99
0.98
16.00
1. 00
- -----------------------(a) Stress perpendicular to bed
0.15
0.20
0.25
0.30
0. 35
0.45
0.54
0.58
0.65
0.72
0.75
0.81
0.88
0.93
0. 19
0.32
0.38
0.49
0. 59
0.62
0.71
0.81
0.88
0.09
0.16
0. 31
0.44
0.49
0.60
0.73
0. 82
0.12
0.28
0.34
0. 47
0.64
0.75
O. ll
0. 18
0. 35
0. 55
0. 67
joints
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TABLE 3
Ratio Of Flexural Strength Of Uniformly Loaded r~asonry Wall Panel
(ASTM E 72) To Prism Flexural Strength (ASTM E 518) (a)
Nominal
Course
Height
in.
Coefficient of Variation
in Bond Wrench Strength
0.05
0.10
0.66
2.00
0.87
0.73
2.67
0.90
0.91
0.75
3.00
0.82
4.00
0.94
5.33
0.96
0.87
6.00
0.97
0.89
8.00
0.99
0.94
12.00
1.02
0.99
16.00
1. 03
1.03
------------------------(a) Stress perpendicular to bed
0.15
0.20
0.25
0.30
0.35
0.41
0.52
0.57
0.67
0.75
0.78
0.86
0. 95
1. 01
0.13
0.30
0.36
0.50
0.62
0.66
0.77
0.90
0.98
0.05
0.14
0.32
0.47
0.53
0. 67
0.84
0.94
0.12
0.32
0.39
0.56
0.77
0.89
0.16
0.25
0.45
0.69
0.85
joints
TABLE 4
K Factors For Various Probabilities Of Not Exceeding Specified
Sampling Error
Confidence
Leve 1 ,
%
99 . 9
99 . 0
95 . 0
90 . 0
Excessive
Error
Probabil ity
Chance
of
Excessive
Error
0.001
0.01
0.05
1 in 1000
1 in 100
1 in 20
1 in 10
0.10
k
3.00
2.576
1. 96
1.645
REFERENCES
1.
Baker, L. R.: Design of r1asonry Panels to Resist Lateral Loads,
Research Report No. CE 2/80, Deakin University, Victoria, Australia,
1980.
2.
Grimm, Clayford T. and Tucker , Richard L.: "Flexural Strength of
Masonry Prisms versus Wall Panels", Journal of Structural Engineering ,
American Society of Civil Engineers, New York, N.Y., Vol. 111, No. 9,
September, 1985, pp 2021-2032.
3.
Natrella, r1ary Gibbons: Experimental Statistics, NBS Handbook 91 ,
Supt . of Documents, U.S. Govt . Printing Office, Washington, D.C., 1966 .
•