PROCEDURE FOR STANDARDIZING MASONRY FLEXURAL STRENGTHS ABSTRACT INTRODUCTION

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1 1040 PROCEDURE FOR STANDARDIZING MASONRY FLEXURAL STRENGTHS CLAYFORD T. GRIMM, P.E. Department of Civil Engineering, ECJ 5.2 University of Texas, Austin, Texas 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

2 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 ) (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 ( * 10-7 * N4) - (2.006 * 10-5 * N3) + ( * N2) - ( * N) (3) Sn = Sw/Kv (4) Where : Sn = standard deviation in mean strength of N joints, psi (MPa). Example: Where two joints are tested at maximum stress (N = 2), from lable 1 Ks = and Kv = 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 = (56 X /0.8211) Mn = 532 psi (3. 7 MPa) From Eq. 4: Sn = 56/ or Sn 68 psí (0.5 MPa)

3 1042 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 n = T * ( * T * T2)/( * T * T * T3) T = (ln p-2)-0.5 (6) (7) p probabi1ity of no f1exura1 fai1ure in standard wa11, dimension1ess p = p1 * p2 (8) pn probabi1ity of no f1exura1 fai1ure in n th joint in the standard wa11, dimension1ess pn = 0.5 * ( * Xn * Xn * Xn * Xn * Xn * Xn 6) * 10-7 (9) Xn = Mn * (1 - Rn)/Sn (10) Xn 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 Rn ratio of stress in n th joint to maximum stress in ASTM E 72, uniform1y 10aded, standard wa11 joints, dimension1ess G = F/Mw (11)

4 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 psi (0.21 MPa) with a standard deviation of 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 Relative 0.72 Stress, Rn I Joint No. 11 Distance in. I I 1< in >i I (2. 29 m) 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

5 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. Distance from Reaction to Joint, in. Relative Stress, Rn Xn in Eq. 10 Pn in Eq. 9 1 and 11 2 and 10 3 and 9 4 and 8 5 and and and and and and 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 * ( * * * * }2 P = Therefore, from Eq. 6 and 7, n = For a single joint at maximum stress in test specimens from Table 1 Ks = O and Kv = 1. From Eq. 1 Mn = Mw = (0.21 MPa), and from Eq. 4 Sn = Sw = (0.072 MPa). Therefore, from Eq. 5 F = (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/ or G = 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.

6 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 (12) Nr required sample size (number of specimens) to next larger whole number. k 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 strength of field-built masonry is 0.25 for excellent control and workmanship, for good control and workmanship, and 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

7 1046 Example: If the probability is 90% of not exceeding a sampling error 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 Ks Kv

8 1047 TABLE 1 (continued) Number of Joints Ks Kv at r~aximum Stress in Test Specimen ll ll 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 Coefficient of Variation Course in Bond Wrench Strength Height in O. ll (a) Stress perpendicular to bed joints

9 1048 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 Coefficient of Variation Course in Bond Wrench Strength Height in (a) Stress perpendicular to bed joints TABLE 4 K Factors For Various Probabilities Of Not Exceeding Specified Sampling Error Confidence Leve 1, % Excessive Error Probabil ity REFERENCES Chance of Excessive Error 1 in in in 20 1 in 10 k Baker, L. R.: Design of r1asonry Panels to Resist Lateral Loads, Research Report No. CE 2/80, Deakin University, Victoria, Australia, 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 Natrella, r1ary Gibbons: Experimental Statistics, NBS Handbook 91, Supt. of Documents, U.S. Govt. Printing Office, Washington, D.C., 1966.

MENG 302L Lab 6: Stress Concentration

MENG 302L Lab 6: Stress Concentration Introduction 1 : The purpose of this experiment is to demonstrate the existence of stress and strain concentration in the vicinity of a geometric discontinuity in a cantilever beam, and to obtain an approximate