MP S-74-30 'Concrete strength relationships · MISCELLANEOUS PAPER S-74-30 CONCRETE STRENGTH RELATIONSHIPS by George M. l-lammitt II Soils end Pavements Laboratory U.S. Army Engineer

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MISCELLANEOUS PAPER S-74-30

CONCRETE STRENGTH RELATIONSHIPS by

George M. l-lammitt II

Soils end Pavements Laboratory U.S. Army Engineer Waterways Experiment Station

P. 0. Box 631, Vicksburg, Miss. 39180

December 1974

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Concrete strength Concrete tests Load-carrying capacity Rigid pavements

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This paper presents a summary of five strength tests currently being used to evaluate the load-carrying capability of concrete pavements. The five tests were those for compression, flexure, splitting tensile, ring tensile, and shearing. The strength relationships between these tests reported by 13 sources are presented. A combined analysis was performed on test results from 3640 concrete samples and the following relationships were developed: compressive strength = 10.02 (flexural strength) - 2123; compressive

l Continued)

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20. ABSTRACT (Continued):

strength = 12.53 (splitting tensile strength) - 1275; compressive strength = 9.75 (ring tensile strength)- 1786; compressive strength= 7.39 (longitudinal shear strength) - 1578; and flexural strength = 1.02 (splitting tensile strength) + 210.5. In view of the numerous factors influencing the relationships of the strengths of concrete, it is not surprising that no simple exact relation is applicable. However, these correlations are felt to be representative in lieu of specific testing for concrete pavement design and evaluation.

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PREFACE

This study was accomplished while the author was attending Texas

A&M University under the Department of the Army's long-term training

and education for engineers and scientists. This study was conducted

during September and November 1971.

The investigation was under the general supervision of Dr. W. B.

Ledbetter of Texas A&M University and the report prepared by Mr. G. M.

Hammitt II of the Soils and Pavements Laboratory, U. S. Army Engineer

Waterways Experiment Station (WES). Messrs. J. P. Sale and R. G.

Ahlvin were Chief and Assistant Chief, respectively, of the Soils and

Pavements Laboratory.

BG E. D. Peixotto, CE, and COL G. H. Hilt, CE, were Directors of

WES during the investigation and the preparation and publication of

this report. Mr. F. R. Brown was Technical Director.

1

CONTENTS

PREFACE • • • . • . • • •

CONVERSION FACTORS, U. S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT. • • • • •

PART I: INTRODUCTION

General ••.• Purpose ••••

PART II: CURRENT METHODS OF DETERMINING STRENGTH OF FCC .•

Description of Tests •• Discussion. . • • • • •

PART III: DATA FROM PREVIOUS INVESTIGATIONS .•

Data Sampling • • • • • • Reference Analyses ••••

PART IV: CONSOLIDATED ANALYSIS .

PART V: CONCLUSIONS.

REFERENCES ..

BIBLIOGRAPHY.

TABLES 1-3

2

Page

1

3

4

4 6

7

7 11

14 14 14

15

19 21

24

CONVERSION FACTORS, U. S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT

U. S. customary units of measurement used in this report can be converted

to metric (SI) units as follows:

Multi:el;y By To Obtain

inches 2.54 centimeters

pounds (force) 4.448222 newtons

pounds per square inch 6894.757 pascals

3

CONCRETE STRENGTH RELATIONSHIPS

PART I: INTRODUCTION

General

1. For almost all concrete work, one or more of the three primary

strength parameters must be considered. The measurements of compression,

tension, and shear, however, yield only approximate values, as direct

tests for obtaining the true values of these parameters cannot be physi

cally performed in the laboratory. There is much discussion over the

effects of shape factors and secondary stresses induced in the ends of

test specimens. The so-called uniaxial "compressive" strength test is

standardized and widely used;1 it is interesting to note the misnomer of

this particular test in that concrete does not fail in true compression.

A failure, as evidenced by the compression test, is actually a shearing

or splitting failure as a result of shearing and tensile forces induced

by the compressive force. Actually, all concrete materials eventually

fail by one or another of modes of tension and shear.

2. Measurements of tension and shearing forces are not broadly

used in concrete work, but do have particular interest in specific prob

lem areas. A direct-pull tensile test would be the ideal test; however,

from a practical standpoint, true alignment during testing bas so far

presented problems. The problem in the direct-pull tests has been ef

fective clamping at the extremities of the test piece. Some success has

been noted recently by fastening the end pieces of the clamps to the

specimen with a strong glue such as an epoxy. This may prove to solve

the alignment problem, especially when applied to 20-in.*-diam specimens.

3. Although strength of concrete is commonly considered its most

valuable property, in many practical cases other characteristics such as

* A table of factors for converting U. s. customary units of measurement to metric (SI) units is given on page 3.

4

durability and impermeability may in fact be more important. Neverthe

less, strength usually gives an overall picture of the quality of con

crete, as strength is directly related to the structure of the hardened

cement paste. In engineering practice, the strength of the concrete at

a given age and cured at a prescribed temperature is assumed to depend

primarily on two factors only: the water/cement ratio and the degree

f t. 2 o compac 1on.

4. In the design and evaluation of airfield portland cement con

crete (PCC) pavements, the strength of the concrete is the primary input

parameter for the characterization of the concrete. The three basic

loading conditions of interior, corner, and edge loading are resisted in

part by the strength of the concrete. For all three loading conditions,

the load is assumed to ·be applied uniformly over an area equal in size

and shape to the footprint of the tire on the pavement. The interior

loading condition assumes the load to be applied at some distance from

an edge or corner of the slab, and the maximum tensile stress occurs on

the bottom of the slab directly under the center of the loaded area.

The corner loading condition assumes the load to be applied at the corner

of the slab with the loaded area tangent to the edges of the slab. The

maximum tensile stress occurs in the top of the slab at some distance

from the corner. The edge loading condition assumes the load to be ap

plied with one edge of the loaded area tangent to the edge of the slab

at some distance from the corner. The maximum tensile stress occurs at

the edge and is parallel to the slab edge. Tests have substantiated the

belief that edge loading produces greater stresses than interior or

corner loads. 3

5. Since the critical stress in the concrete slab is due to

bending of the slab under the applied wheel loading, the flexural

strength of the concrete has been adopted as being a representative

property of concrete for design purposes. For most airfield PCC pave

ments, the flexural strength of the concrete at an age of 90 days is

used for the determination of required thickness. However, the strength

at other ages can be used if it is anticipated that the pavement will be

subjected to loadings when the concrete is at this age. During

5

placement, the concrete mix is controlled on the basis of 7- and 28-day

tests, which, according to the strength-age relation established by the

laboratory mix design, will yield the 90-day design flexural strength.

Beams are also cast from the concrete mix being used for 90-day tests to

ensure that the design flexural strength is being attained.

6. The flexural strength for evaluation of in-place concrete

slabs must be determined, and it frequently has been necessary to re

move large slabs from the pavements to provide beam specimens for

flexural strength tests. In the removal of pavement slabs, sawing and

testing are time-consuming and costly operations. This large sampling

may also interfere with the use of the facility. To preclude this situa

tion and save money and time, correlations between more common concrete

strength tests are desired.4

These correlations with recognized advan

tages and limitations can be utilized to interrelate the measurements

of the three primary strength parameters of compression, tension, and

shear. The accuracies of these correlations are adequate for usage by

the designer or evaluator of concrete pavements.

Purpose

1. The purpose of this study was threefold: to perform a review

of the current methods being employed to test concrete strength; to

gather data from previous investigations and report their findings; and

to perform a consolidated analysis of' combined data. The resulting cor-

relations would provide a design engineer a tool for cross-checking

strength parameters and perhaps make unnecessary a series of testing

when preliminary estimates are desired.

6

PART II: CURRENT METHODS OF DETERMINING STRENGTH OF PCC

Description of Tests

8. The following failure modes and American Society for Testing

and Materials (ASTM) test procedures were correlated and will be dis

cussed in this report.

Failure Mode

Compression (mixed)

Tension

Shear or combination

Compression test

Type

Compressive

Flexure

Test

Splitting tensile Ring tensile

Shearing

ASTM Procedure

c 39-715

c 78-646 c 496-697

9. Three types of compression test specimens are used: cubes,

cylinders, and prisms. The general practice in the United States is to

use standard cylinders for testing. The cylinders are usually 6 in.

in diameter and 12 in. long and are cast in molds of steel or cast iron,

preferably with a clamped base (ASTM C 39-71).

10. Standard cylinders are of a height twice the diameter; but

sometimes other sizes are encountered, especially when cutting core

sample from existing concrete structures. The- relationship- of relative

strength to height-to-diameter ratio is shown in Reference 4. Gener

ally, the relative strength is a function of height-to-diameter ratio.

As this ratio decreases from 2.0 and the level of strength decreases,

the relative strength also decreases.

11. The procedure for preparation of satisfactory cylindrical

specimens, given in ASTM C 192-69,8 requires different methods of con

solidation, usually based on the slump. The ends of the cylinders

require further preparation after finishing; this is a big disadvantage

of using this type of specimen because test results may be affected.

Alternate means of casting have been suggested by various researchers,

7

and the best means is still under study. In addition to being plane,

the cylinder ends should be normal to the axis of the cylinder; this

guarantees that the end planes are parallel. A slight misalignment is

allowed since misalignment of 1/4 in. in 12 in. has been determined to

cause no loss of strength.9 The performance of a cylinder may also be

affected by the design of a particular testing machine, especially by

the energy stored in the machine. 9 These observations, combined with

the known start of columnar or shearing failures in the compressive test,

should cause one to be wary of interpreting this test as a true measure

of the compressive strength of concrete.

Flexure test

12. In many instances, the tensile capacity of a concrete struc

ture can control the performance of the structure. A good example is the

cracking of concrete pavements, which allows the infiltration of water

and the pumping of underlying soil. Because of the difficulties of

performing a direct tension test, it has been preferable to measure the

tensile strength of concrete by subjecting a concrete beam to flexure.

The flexure test (ASTM C 78-646) is an indicator of tensile capacity and

is also used extensively as a quality control test.

13. For the flexure test, a plain concrete beam of a given span

length, width, and depth is supported at its ends and loaded at either

its third points or at its center. 10 The third-point loading condition

is preferable due to the better reproducibility and the larger area of

specimen being tested. The flexural strength of a beam is represented

by a modulus of rupture R , which approximately equals the tension

stress in the outer fiber at failure, M /I (where M =moment, c

c =distance to outer fiber, and I= moment of inertia).

14. The modulus of rupture actually overestimates the tensile

strength of concrete and gives a higher value than would be obtained in

a true direct tension test. This approximate value is based on the as

sumption in the calculation of the modulus of rupture that stress is

linearly proportional to the distance from the neutral axis of the beam;

the shape of the actual stress block under loads nearing failure is

known not to be linear. A relationship between the modulus of rupture

8

and strength in direct tension shows a linear change as expressed by: 11

Modulus of rupture, psi = 100 + 1.4 tensile strength, psi

Actual values of this relationship may vary depending on the properties

of the mix employed.

15. Limitations of this test consist of the stress measurement

being a maximum in the outer fiber and the questionability of such

measurements being representative. In addition, the coefficient of

variation in the field is approximately 15 to 20 percent, which is con

siderably higher than the coefficient of variation for compression

tests .12

Splitting tensile test

16. A different and slightly simpler method of measuring tension

is the indirect tensile test. This method was developed coincidentally

in Brazil and Japan in 1953 and has recently come into rather general

use. The test procedure has been standardized and is described in ASTM

test method C 496-69. 7

17. The specimen used for the test is a conventional cylinder

6 in. in diameter and 12 in. long, prepared and cured in the same manner

as that specified for cylinders for compression tests. The cylinder is

loaded in compression along two axial lines through bearing strips of

1/8-in.-thick, 1-in.-wide plywood. The 1-in.-wide plywood strips dis-

tribute the compressive load- o:v:er an_ area- which is sufficiently- narrow-

to avoid undue concentrations of stress. The strips compensate for

surface irregularities.

18. The application of a compressive force along the two lines

produces a biaxial stress distribution within the specimen, the hori

zontal stress being tension. 13 The cylinders fail suddenly along a

vertical plane in the center region. The almost constant tensile

stresses occur over approximately three-quarters of the vertical plane

between the two lines of load application.14

The magnitude of the aver

age tensile stresses along this plane at the time of failure is con

sidered to be the tensile stress.

9

19. According to the theory of elasticity, along the vertical

plane represented by the diameter of the cylinder, the principal tensile

stresses act for about 80 percent of the cylinder's length; and the

stresses are approximately constant for about 60 percent of the

length.15 The splitting tensile strength, if the compressive strength

is considered to be at least four times the tensile strength, is com

puted by means of the following equation, which was derived from the

theory of elasticity:

where

2P 0 = 1Tid

o = splitting tensile strength, psi

P = maximum applied load, lb

t = length of cylinder, in.

d = diameter of cylinder, in.

(1)

20. Although the splitting tensile test does not produce a uni

form tensile stress distribution, experimental evidence indicates that

the area not in tension is in a triaxial state of stress that is suffi

cient to retard compressive failure at the load points, allowing tensile

failure to prevail.

21. This test is more easily reproduced than the flexure tests

and more closely approximates the compressive test. Some investigators

have concluded that the splitting tensile test gives more uniform re

sults than other types of tension tests.16- 18 One possible reason for

this is that the failure point is usually through the center of the

specimen, which is least affected by the condition of the surface.

Ring tensile strength tests

22. One method of determination of the tensile strength of con

crete is to load ring-shaped specimens in a manner identical with that ]b

described for the splitting tensile tests.~~ The hole in the specimen

changes the uniform tensile stress field that occurs across the center

three-fourths of the failure plane in a standard cylindrical specimen.

The result is an increase in the unidirectional tensile stress component

10

at the edge of the hole, which causes fracture and thus represents the

tensile strength of the specimen.

23. Another method is to apply a hydrostatic pressure radially

against the inside periphery of a ring. 19 The application of this pres

sure produces tangential tensile stresses and radial compressive

stresses throughout the entire volume of the ring with a uniformly dis

tributed maximum tensile stress occurring along the entire internal

periphery of the ring. The magnitude of the radial compressive stress

is quite small when compared with tangential tensile stresses. Gener

ally the classic equations for determining stresses in thick-walled

cylinders due to internal hydrostatic pressure can be used to determine

tangential tensile stress at failure, but the equations for stresses in

thin-walled cylinders are more desirable because elastic equations are

unnecessary. However, thin-wall tests require a sample with a 15-in.

radius, which is harder to cast and handle.

Discussion

Shearing strength tests, shearing mode or combinationl,l9,20

24. Pure shear can be applied only through torsion of a cylindri

cal specimen, in which case the stresses are equal in primary shear,

secondary tension, and secondary compression. Testing to determine the

shearing strength directly is inconclusive because of the effects of

bending, friction, cutting, or lateral restraint of the sample during

testing. Results of early attempts reflect the discrepancies inherent 21 22

in direct testing for shear strength. ' Reference 21 reported that

shearing strength ranged from 46 to 97 percent of the compressive

strength and was 3 to 6 times as great as tensile strength. However,

Reference 22 gave results of shearing strength in the range of 10 to

18 percent of the compressive strength. Reference 23 notes that '~he

resistance of concrete to pure shearing stress has never been directly

determined •.. a reliable indication of the strength of concrete in pure

shear can be obtained only from tests under combined stresses."

11

25. The most widely used and accepted method of combined stress

t t . . t f f' d . t . 'al t 23 Th h es ~ng cons~s s o con ~ne compress~ve r~ax~ ests. e s ear-

ing strength determination obtained from Mohr's concept appears valid

and, therefore, makes for a convenient and encompassing method of ex

amining all strength parameters. There presently is no standardized

test for measuring the shear of concrete. The confined compressive

triaxial test was used to obtain the data presented for correlation of

longitudinal shear with compressive strength.

Compression test, mixed failure mode

26. All materials, especially the more common construction mate

rials, may fail in tension or shear. The term "compressive strength"

is actually incorrect since failure in the test results from a shearing

or splitting of the elements due to shearing and tensile forces induced.

The term "compressive strength" is quite commonly used and denotes a

performance index rather than an absolute quantitative testing.

Flexure, splitting tensile, and ring tensile tests, tension mode of failure

27. The term "tension" is used to describe each of the above test

procedures, but none of the tests measures the actual tension or tension

capacity of a material. All are indirect testing methods and are based

on the assumption of linear elastic distribution of stress-strain. This

assumption is completely invalid since the stress-strain for concrete is

curvilinear almost from the onset of loading. A more direct approach,

with less error in assumption, would be a direct tension method employ

ing different geometrical configurations and innovations in testing.

The test's primary advantage, especially for design work, continues to

be ease of performance and relatively high degree of reproducibility.

Shear test, shearing failure mode

28. Many discrepancies exist in the attempts to provide a direct

testing for shear strength. Since shear stress cannot exist without

accompanying tensile and compressive stresses, pure shear could be ap

plied only through torsion loading of a cylindrical specimen. The

shearing strength determination obtained from Mohr's concept of combined

12

stress testing seems valid for examining all strength parameters and,

therefore, shearing stress separately. The use of shearing stress in

structural design is justified only because it is a convenient measure

of diagonal tension.

13

PART III: DATA FROM PREVIOUS INVESTIGATIONS

Data Sampling

29. The data shown in Table 1 present a representative sampling

from which an independent analysis was performed to determine relation

ships among concrete strengths determined by various procedures. A

total of 280 observations were considered in the analysis.

Reference Analyses

30. A summary of reference analyses is shown in Table 2. Rela

tionships are shown for the compression test versus the flexure test;

compression test versus splitting tensile test; compression test versus

ring tensile test; compression test versus shear test (Mohr's circle);

and finally flexure test versus splitting tensile test. These relation

ships are expressed as approximations due to expressing the slight

nonlinearity as linear in the y = mx + b equation. Ranges of appli

cability are also given in certain of the relationships in order to

depict the findings of the referenced investigators.

14

PART IV: CONSOLIDATED ANALYSIS

31. The selected strength test results presented in Table 1 were

used as data input for a consolidated simple regression analysis. The

similarity of testing procedures and mixes allowed the analysis to be

performed without consideration of variables affecting each test. The

high degree of correlation indicated the validity of this assumption for

design or evaluation usage.

32. On the basis of the sample data, an estimation of one vari

able corresponding to a given value of another variable was determined.

This was accomplished by estimating the value of a variable from a least

square curve which fit the sample data. 'rhe resulting curve is called a

regression curve of Y on X , since Y is estimated from X •

33. The first problem to be considered was how well a straight

line defined the relationship between two methods of testing. To ac

complish this, the equations for the least square regression lines of one

testing method were compared with those of the other method as follows:

where y = dependent variable

a and ai = coefficients 0

X = independent variable

and a and ai are obtained from the normal equations: 0

where N = number of observations. Equations 3 and 4 yield:

a 0

= (~ Y)(~ x2) - (~ X)(E XY)

N ~ x2 - (E X)

2

15

(2)

( 3)

(4)

(5)

a = N E XY- (E X)(E Y) t N E x2

- (E X)2

(6)

34. Additional calculations are of interest in describing the

relationships between the different testing methods of concrete. The

first is the standard error of estimate, S , which has properties anal

ogous to those of the standard deviation and is computed using the fol

lowing equation:

(7)

35. The second is the coefficient of determination, defined as

the ratio of explained variation to the total variation. If there is

zero unexplained variation, i.e., the total variation is all explained,

the ratio is one. The quantity r is given by:

r = + explained variation = + total variation

(8)

For the case of linear correlation, the quantity r is the same regard

less of whether X or Y is considered the independent variable. Thus

r is a very good measure of the linear correlation between two testing

methods.

36. Several attempts were also made to express one method of

testing in various forms, such as exponential, logarithmic, etc., to

increase the coefficient of correlation. These attempts did not better

the relationships and are not reported. The simple regression analysis

computations yielded the results shown in Table 3, which can be compared

with the individual tests in Table 2.

37. The strength relationships used in the analysis were compres

sive versus flexural strength, compressive versus ring tensile strength,

compressive versus shear strength, and flexural versus splitting tensile

strength. These four relationships are shown graphically in Figures 1

and 2. Figure l gives a comparison of splitting tensile, flexural, ring

16

800 f------+-------+---iii Q.

VI l: tel z

700

w 600 Q: t-(1)

Q: <( w l: VI

0 500 z <(

w _J

iii z w

---1

.... 400 1-----

200~-------J--------~--------~--------~--------~--------~--------~ 0 1000 2000 3000 4000 5000 6000 7000

COMPRESSIVE STRENGTH, PSI

Figure 1. Tensile and shear strengths versus compressive strength relationships

17

tensile, and longitudinal shear strengths with the compressive strength.

The "direct tension" is shown only as an indicator and is the result of

work shown in Reference 24. Figure 2 shows a comparison of two tension

failure mode tests, flexural and splitting tensile.

IOOOr----------.----------~--------~----------~--~----~

in a. ....

z 800 0 Ill z w 1-

t!) z 1- 400 1-..J a.. Ill

200 400 800 800 1000 FLEXURAL STRENGTH, PSI

Figure 2. Splitting tension versus flexural strength relationships

18

PART V: CONCLUSIONS

38. The following developed relationships, based on approximately

190 series of testings, are representative of five test methods for

determining strength characteristics of concrete:

a. Compressive strength = -2123 + 10.02 flexural strength (r = 0.875).

b. Compressive strength = -1275 + 12.53 splitting tensile strength (r = 0.905).

c. Compressive strength = -1786 + 9.75 ring tensile strength (r = 0.826).

d. Compressive strength = -1'!78 + 7.39 longitudinal shear strength (r = 0.679).

e. Flexural strength = 210.5 + 1.02 splitting tensile strength (r = 0.857).

39. 'rhe relatively high coefficient of correlation r indicates

the measure of linear correlation for general design use. Emphasis

should be placed on the variability of data. These data should be used

as an indication and should in no way be interpreted to replace specific

testing. The actual number and magnitudes of variables involved may

allow extreme variations that could nullify these results. Some of the

variables that affect the tensile, shearing, and compression tests dif

ferently are:

a. Properties of fine aggregate.

b. Grading of the aggregate.

c. Age of the concrete.

d. Inadequate curing.

e. Air entrainment.

f. Degree of compaction.

40. It is also concluded that testing should be channelized

toward the area of direct measurements of capacity of tension and shear.

Recent tests indicate that the effect of using different ordinary port

land cements on medium- and high-strength concrete is small in compari

son with the effects of aggregate type and entrained air. 25

41. In view of the numerous factors influencing the relationships

of strengths, it is not surprising that no simple relation is generally

applicable. However, the relationships presented herein are felt to be

representative for a cursory examination in lieu of specific testing for

concrete pavement design and evaluation.

20

REFERENCES

l. Saucier, K. 1., "Correlation of Hardened Concrete Test Methods and Results," Technical Report C-69-2, Mar 1969, U. S. Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss.

2. Nelville, A. t-1., Properties of Concrete, Wiley, New York, 1963, pp 216-273 and 385-439.

3. Hutchinson, R. L., "Basis for Rigid Pavement Design for Military Airfields," Miscellaneous Paper 5-7, May 1966, U. S. Army Engineer Division Laboratory, Ohio River, CE, Cincinnati, Ohio.

4. Murdock, J. W. and Kesler, C. E., "Effect of Length to Diameter Ratio of Specimen on the Apparent Compressive Strength of Concrete," American Society for Testing and Materials Bulletin No. 221, Apr 1957, pp 68-73.

5. American Society for Testing and Materi nls, "Standard Method of Test for Compressive Strength of Cylindrical Concrete Specimens," Designation: C-39-71, 1972 Book of ASTI-1 Standards, Part 10, 1972, Philadelphia, Pa.

6. , "Standard Method of Test for Flexural Strength of Con-crete (Using Simple Beam with Third-Point Loading)," Designation: C-78-64, 1972 Book of ASTM Standards, Part 10, 1972, Philadelphia, Pa.

7. , "Standard Method of Test for Splitting Tensile Strength of Cylindrical Concrete Specimen," Designation: C-496-69, 1969 Book of ASTM Standards, Part 9, 1969, Philadelphia, Pa. ----

8. , "Standard Method of Making and Curing Concrete Test Specimens in the Laboratory," Designation: C-192-69, 1972 Book of ASTM Standards, Part 10, 1972, Philadelphia, Pa.

9. Nelville, A. M., "The Failure of Concrete Compression Test Specimens," Civil Engineering, Vol 52, No. 613, Jul 1957, pp 773-774.

10. U. S. Army Engineer Waterways Experiment Station, CE, Handbook for Concrete and Cement, Aug 1949 (with quarterly supplements), Vicksburg, Miss.

11. Gonnerman, H. F. and Shuman, E. C., "Compression, Flexure and Tension Tests of Plain Concrete," Proceedings, American Society for Testing and Materials, Vol 28, Part II, 1928, pp 527-573.

12. Wright, P. J. F., "Variations in the Strength of Portland Cement," Magazine of Concrete Research, Vol 10, No. 30, Nov 1958, pp 123-132.

13. McNeely, D. J. and Lash, S.D., "Tensile Strength of Concrete," Proceedings, American Concrete Institute, Vol 60, Jun 1963, pp 751-760.

21

14. Hoff, G. C., "Evaluation of a Ring Test for Determining the Tensile Strength of Mortars and Concrete," Miscellaneous Paper C-69-5, May 1969, U. S. Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss.

15. Hondros, G., "The Evaluation of Poisson's Ratio and the Modulus of Materials of Low Tensile Resistance by the Brazilian (Indirect Tensile) Test with Particular Reference to Concrete," Australian Journal of Applied Science, Vol 10, 1959, pp 243-268.

16. Wright, P. J. F., "Comments on the Indirect Tensile Test on Concrete Cylinders," Magazine of Concrete Research, Vol 7, No. 20, Jul 1955, pp 87-96.

17. Thaulow, S., "Tensile Splitting Tests and High Strength Concrete Test Cylinders," Proceedings, American Concrete Institute, Vol 53, Jan 1957, pp 699-706.

18. Grieb, W. E. and Weaner, G., "Comparison of Splitting Tensile Strength of Concrete with Flexural and Compressive Strengths," Proceedin s American Societ for Testin and Materials Vol 62, 19 2, pp 972-990.

19. Popovics, S., "Relations Between Various Strengths of Concrete," Highway Research Record No. 210, 1967, pp 67-90.

20. Troxell, G., Davis, H., and Kelly, J., Composition and Properties of Concrete, 2d ed., McGraw-Hill, New York, 1968.

21. Taylor, F. W. and Thompson, S. E., A Treatise on Concrete, Plain and Reinforced, Wiley, New York, 1916, p 136.

22. Falk, M.S., Cements, Mortars and Concrete, Their Physical Properties, M. C. Clark, New York, 1904, p 27.

23. Kesler, C. E., "Hardened Concrete; Strength," Significance of Tests and Properties of Concrete and Conc~ete-Making Materials, Special Technical Publication No. 169-A, American Society for Testing and Materials, 1966, Philadelphia, Pa., pp 144-159.

24. Malhotra, -V-. W. and Zoldner£, N. G. , "ComparisDn Df Ring-Tensile Strength of Concrete with Compressive, Flexural, and SplittingTensile Strengths," Journal of Materials, American Society of Testing and Materials, Vol 2, No. 1, Mar 1967, pp 160-199.

25. Graf, 0., Albrech, W., and Schaffer, H., The Properties of Concrete, 2d ed., Springer Verlag, Berlin, 1960, pp 215-217.

26. Price, W. H., "Factors Influencing Concrete Strength," Proceedings, American Concrete Institute, Vol 47, Feb 1951, pp 417-432.

27. Dunn, H., "Relationship Between the Strengths of Concrete," SSTP, National Science Foundation Engineering Science Program, Aug 1971.

28. Ivenger, K. T., Sundara, R., and Sekhara, K. C., "On the Theory of !dentation Test for the Measurement of Tensile Strength of Brittle Materials," British Journal of Applied Physics, Vol 13, Oct 1962, pp 501-507.

22

29. ACI Committee 613, "Recommended Practice for Selecting Properties of Concrete," ACI 613-54, 1954, American Concrete Institute, Detroit, Mich., p 62.

30. Narrow, I. and Ullberg, E. , "Correlation Between 'l'ensile Splitting StrengLh and Flexural Strength of Concrete," Proceedings, American Concrete Institute, Vol 60, Jan 1963, pp 27-38.

31. Walker, S. and Bloem, D. L., "Effects of Aggregate Size on Properties of Concrete," Proceedings, American Concrete Institute, Vol 57, Sep 1960, pp 283-298.

32. Sozen, M.A., Zwoyer, E. M., and Siess, C. P., "Strength in Shear of Beams Without Web Reinforcement," Bulletin No. 452, Apr 1959, Engineering Experiment Station, University of Illinois, Urbana, Ill.

23

BIBLIOGRAPHY

Akroyd, T. N. W., "Concrete Under Triaxial Stress," Magazine of Concrete Research, Vol 13, No. 39, Nov 1961, pp 111-118.

Blanks, R. F. and McNamara, C. C., "Mass Concrete Tests in Large Cylinders," Proceedings, American Concrete Institute, Vol 31~ Jan-Feb 1935, pp 280-303.

Jones, R. and Kaplan, M. F., "The Effect of Coarse Aggregate on the Mode of Failure of Concrete in Compression and Flexure," Magazine of Concrete Research, Vol 9, No. 26, Mar-Nov 1957, pp 89-94.

Kesler, C. E., "Statistical Relation Between Cylinder, Modified Cube and Beam Strength of Plain Concrete," Proceedings, American Society for Testing and Materials, Vol 54, 1954, pp 1178-1187.

Malhotra, V. W., "Problems Associated with Determining the Tensile Strength of Concrete," Internal Report MPI, Jul 1966, Canadian Department of Mine and Technical Surveys, pp 12-14.

O'Cleary, D. P. and Byrne, J. G., "Testing Concrete and Mortar in Tension," Engineering, 18 Mar 1960, pp 384-385.

Office, Chief of Engineers, Headquarters, Department of the Army, "Standard Practice for Concrete," Engineer Manual EM 1110-2-2000, Dec 1965, Washington, D. C.

Spiegel, M. R., Schaum's Outline of Theory and Problems of Statistics, Schaum, New York, 1961, 241-268.

Talbot, A. N., "Tests on Concrete: I. Shear; II. Bond," Bulletin No.8, Sep 1906, Engineering Experiment Station, University of Illinois, Urbana, Ill.

24

T!ible 1

Sw-r.:or"'ry of 2elected Strer.s;tr. Test !'.CS'J.}ts

Str-2n h rsi StrEn tl::. si Sr1itting Rir.g Longi ·(udins.l Splitting P.ir.g Lo::.g::. tudinal

f'.curc:e of Data Co:s2re.3sion Flexural Iension Stre:;s - ;:::--_~~ CoMments Source of !hts. co.:::rression Flexars.1 T::-nsio::. 3tress S'H~ar Comments

Reference 24 1650 3\10 25C 2~' '~

3-) value:; seJ..ectea. 2-1 Series 7~TJ 9J3 670 1105 1680 390 255 280 !'ro!::. 629 test3 Referer,ce l~ 5330 2C7 5'3C 923 1520 425 245 275 ( C'ontir:ued) ~250 794 560 9<6 1600 445 245 280 5250 832 53J Boo 2000 485 290 405 5::L·:; 725 t!•C 81..5 1360 455 275 375 5!.t50 859 6•" ,~ 806 2675 605 335 400 5590 796 790 858 2670 600 350 405 5670 841 580 910 2630 615 36o 405 5890 774 610 2?7 3080 595 370 435 5730 '367 610

2800 565 370 420 5840 895 560 910 2655 590 360 410 5750 866 630 845 4210 650 455 520 5770 B12 570 910 4120 715 480 515 4410 764 470 728 4060 650 470 460 4380 715 430 793 4020 675 475 495 4460 650 460 793 4840 76o 475 575 4370 715 490 962 4930 765 525 585 4590 579 48o 715 4960 750 495 600 4460 630 500 845 4930 745 510 555 4300 606 450 806

5910 830 555 635 4680 670 460 754 5980 875 535 640 4250 616 510 5930 955 550 675 32' ' 61!. :;30 728 '" 6140 910 485 740 3240 6~5 L.-3·:) 676 6240 890 500 705 :)240 632 ~2C

6940 1000 550 758 :::2L.0 616 ~20 650 6590 1035 615 720 :;210 t~O 420 676 6510 1010 605 745 3110 579 420 624 6730 1020 600 710 3120 665 410 611 7630 1190 635 840 3170 046 400 624

3110 f42 360 598 C-1 Series 8820 1003 700 988 51 values selected

Reference 1.4 8590 1037 750 962 from 92 tests M Series 9590 1050 798 1001 54 values 8390 1045 770 975 Reference 14 9510 990 853 1019 8670 1013 800 897 C315 985 790 1079 8840 1033 850 806 3715 835 709 910 £510 968 520 845 3685 390 644 910 8630 1107 680 910 02~5 910 772 1019 8950 1045 630 923 9250 9~'5 872 832 8860 1092 720 936 913~ 930 8L.2 1118 8210 952 790 897 9700 870 859 1014

8450 1043 760 936 7820 755 644 871

8680 1047 690 845 7780 740 617 806 7140 1003 610 819 7265 780 650 897 7120 926 750 936 7865 875 710 876

~ ,! 73°0 9(3 570 832 7695 855 742 962 7670 1001 730 793 7665 670 734 .,! 7710 930 710 832 8035 910 673

:.,~4. 7660 997 780 8'·5 8145 780 686 7710 ::.1 700 884 2430 810

~l 7730 926 700 1066 6310 839 609 949 6250 859 609 962

~:. ( Cor.tir.ue-j) (Sheet 1 of 3)

Table 1 (Continued)

Stren th si Stren th si Splitting Ring Longi tudin:ll Splitting Rir..g Longi tudina1

Source of r;.,_ta C2S:ression Flexural Tension Stress sr,ea.r __ Comments Source of Data Comoression Flexural Tension Stress Shear Comments

r-1 Series 6225 918 693 C-2 Series 7130 730 Reference 14 5600 795 435 319 Reference 14 7320 600

(Continued) 5630 745 560 806 (Continued) 5440 600 5910 655 471 858 5250 coo 5460 9l.6 593 819 5430 480 5740 784 575 858 5370 751 594 345 5370 520

4740 780 564 767 5770 520

4625 735 546 806 5520 500

4595 810 545 728 5910 590 5910 600

4750 719 422 767 5550 620 4670 628 474 689 4590 500 4610 684 427 754 4610 480 4570 616 526 715 4640 510 4525 557 530 676 4640 530 497 650 Reference 1. 5070 510 870 12 values 3310 545 350 637 4740 460 715 3340 545 372 663 4330 485 660 3395 640 385 2730 455 690 3170 580 294 533 2660 425 740

3480 560 333 663 2870 425 635

3280 550 403 507 4720 630 870

3340 570 443 611 4290 635 705

3170 490 419 624 4510 550 765

3140 495 400 629 2810 445 705

2660 498 172 535 2920 475 635

2630 453 327 624 2570 415 560

2885 396 157 494 Ref'erence 26 1000 230 9 values 2770 466 336 676 2915 525 393 663

2000 380 3000 480

2870 544 328 4000 600 2970 432 367 598 5000 700 2855 477 263 611 6000 780 2855 512 266 611 7000 840

8000 960 C-2 Series 9050 720 30 values 9000 990

Reference 14 8800 750 8910 740 Reference 27 3933 750 23 values 7790 760 4329 659 8210 700 4271 678 8430 760 3842 648 8660 800 4203 630 8210 640 4141 690 8270 800 3437 580 698o 620 3839 674

7140 590 3843 659

7170 580 3823 703

7300 650 4019 692 7140 630 3381 665 7180 630 3818 701 7o80 660 4142 726

(Continued) ( Steet 2 of 3)

T:;.Cle 1 ( Cor:2lude::i)

Stre h si Strer. h si Splitting Ring Longi tu,~r.al Splitting Ring Lcr.gi tudinal

Source of Data Co!.S(ression Flexural ~ Stress Shear __ Ccr:-.mcr.ts Source cf' T"'l'-.b. C,-:::rressicn F1Pxural TP~;:;ior.. ~ Shear Comments

Reference 27 3967 669 Reference 30 755 515 ( Contin:>ed) 4304 76R (Continued) 775 523

1.233 641 9C5 570 3502 6o7 6?5 445 2915 670 675 455 3259 619 755 485 36o3 527 735 500 3J55 554 79J 525 3288 657 730 500

800 540 Referenct:> 28 2200 266 174 6 v::o..l:.1es

920 560 3100 359 238 810 530 4100 472 312 830 590 5000 547 370 830 565 5350 613 420 6350 712 433 730 505

780 535

Reference 29 3500 555 6 values 870 590

35CD 555 865 610

3500 555 575 455 10 values, field 400 600 595 445 specimens taken

3200 525 595 385 from airfield 2600 450 580 420 pavement slabs

74'5 4?0 Reference 30 555 325 39 values selected 710 440

625 370 from results of 675 445 650 425 tests of 1560 tlC 365 655 410 bea:ns and 880 675 470 685 390 cylinders 635 415 5eo 340 615 375 Reference 13 oo 455 1 value 625 395 635 370 Reference 31 2900 450 410 10 values E55 425 3100 475 400

710 450 2900 450 400

735 460 5500 770 520

740 480 5200 630 510

63o 420 5250 690 530

630 405 5000 E25 500 2000 410 360

700 440 4300 610 450

705 465 4300 f90 490 700 480 640 390 705 480 Total 280 observations

(Sheet 3 of 3)

Table 2

Summary of Reference Analysis

Refer-ence Strength Relationship

Function No. si

Compressive strength ( f' c) versus 27 f'c "' l2.5MR - 4750 flexural strength modulus of rupture (MR) 3l f'c "' l3MR - 3100

20 f'c "' (~6l 28 f'c "' llMR

14 f'c "' l6.6MR - 6000

23 f'c "' 6.5MR + 125

2 f'c "'(~sf 32 f'c "' -3000 + 3000

4MR

24 f'c "' -6500 + l5MR

26 f'c ..::. 3000 ~ 5000

f'c "' MR

0.15

f'c > - 5000 ~ 7000

f'c MR , __ 0.13

25 f'I.! "' 3000

f'c "' 4MR + 400

f'c > 3000 ~ 6000 -f'c "' 8MR- 2300

Compressive strength (f'c) versus l f'c ST "' splitting tensile strength (ST) 0.11

24 f'c > 2000 < 4500 -f'c "'

ST- 170 0.07

f 1 c "' 4500 ~ 7000

f'c ST - 260 = 0.05

(Continued)

Table 2 (Concluded)

Function

Compressive strength (f'c) versus splitting tensile strength (ST) (Continued)

Compressive strength (f'c) versus ring tensile strength (RT)

Compressive strength (f'c) versus shear strength (Mohr's circle)

Flexural strength (MR) versus splitting tensile strength (ST)

Reference No.

14

31

17

28

20

14

24.

1

21

22

30

f'c

Strength Relationship si

"' (ST - 240)· 0.06

f'c > 2000 < 7000

f 1 c

f'c

f'c

f'c

f'c

"' (ST - 250) 0.045 .

ST "' 0.08*

ST "' 0.09*

ST "' 0.10*

ST "' 0.135

f'c "' ST where n

f'c "' RT - 550 0.05

n = 8%-14%

for

6-in.-ID specimen

f'c "' RT - 410

0.05 for

f'c

f'c

f'c

f'c

12-in.-ID specimen

"' ;:.;R.;;;.T --_;;:;1'-'-7.;;;..0 0.08

"' Shear strength 0.198



MR "' ST + 250

14 f'c > 3000 < 6000

20

* Average values for: f'c > 2000 < 6000

MR "' ST + 210

ST where MR:. n n = 50%-75%

l. Compressive strength

Number = 189 tests

Mean of X = 716.021

Estimating equation:

T- Test= 24.7388

Sum of Squares:

Table 3

Consolidated Analysis

as function of flexural strength:

Mean of Y = 5054.72

jY = -2122.99 + 10.0244 xl

Explained = 2.03025 E+8 Unexplained = 6.64452 E+8

Total = 8.67477 E+8

Standard Deviations:

Standard error of estimate = 1036.44 Standard deviation of X= 187.043 Standard deviation of Y = 2142.39

Standard Deviations Corrected for Degrees of Freedom:


Coefficients of Determination and Correlation:

R square= 0.765959 R = 0.875191

Therefore, compressive strength = 10.02 flexural strength - 2123.

2. Compressive strength as function of splitting tensile strength:

Number = 192 tests

Mean of X--= 536.839 Mean of Y = 5453.59 Estimating equation: IY = -1275.06 + 12.5338 xl T - Test = 29.3573

Sum of Squares:

Explained= 1.57179 E+8 Unexplained= 7.12974 E+8

Total= 8.70153 E+8


Standard error of estimate= 904.788 Standard deviation of X= 153.745 Standard deviation of Y = 2128.86

(Continued) (Sheet 1 of 4)

Table 3 (Continued)

2. Compressive strength as function of splittin tensile strength Continued :


Standard error of estimate = 909.537 Standard deviation of X = 154.147 Standard deviation of Y = 2134.43


R square = 0.819366 R = 0.905189

Therefore, compressive strength= 12.53 splitting tensile strength 1275.

3. Compressive strength as function of ring tensile strength:

Number = 125 tests ·

Mean of X = 740.88 Mean of Y = 5439.64

Estimating equation: IY = -1786.35 + 9.75325 xl T - Test = 16.2668

Sum of Squares:

Explained = 1.96885 E+8 Unexplained = 4.23558 E+8

Total = 6.20443 E+8


Standard error of estimate Standard deviation of X =

= 1255.02 188.735

= 2227.9 Standard deviation of Y




R square = 0.68267 R = 0.826238

Therefore, compressive strength= -1786 + 9.75 ring tensile strength.

4. Compressive strength as function of longitudinal shear strength (Mohr's circle):

Number = 12 tests

Mean of X= 712.5 Mean of Y = 3689.17

Estimating equation: IY = -1577.82 + 7.39226 xl T - Test = 2.931


Table 3 (Continued)

4. Co ressive stren th as function of lon itudinal shear stren h (Mohr's circle (Continued):

Sum of Squares:

Explained= 5.8378 E+6 Unexplained = 5.0151 E+6

Total = 1.08529 E+7


= 697.483 87.4524

= 951.003

Standard error of estimate Standard deviation of X = Standard deviation of Y




R square = 0.462098 R = 0.679778

Therefore, compressive strength = -1578 + 7.39 longitudinal shear strength.

5. Flexural strength as function of splitting tensile strength:

Number = 199 tests

Mean of X= 507.065 Mean of Y = 726.186

Estimating equation: jy = 210.456 + 1.01709 xl T - Test = 23.3319

Sum of Squares:

Explained = l. 59143 E+6 Unexplained = 4.39766 E+6

Total = 5.98909 E+6




Standard error of estimate = Standard deviation of X Standard deviation of Y

89.8796 = 146.528 = 173-919


Table 3 (Concluded)

5. Flexural strength as function of splitting tensile strength (Continued):


R square= 0.734278 R = 0.8569

Therefore, flexural strength = 210.5 + 1.02 splitting tensile strength.

(Sheet 4 of 4)

In accordance withER 70-2-3, paragraph 6c(l)(b), dated 15 February 1973, a facsimile catalog card in Library of Congress format is reproduced below.

Hammitt, George M Concrete strength relationships, by George M. Hammitt II.

Vicksburg, U. S. Army Engineer Waterways Experiment Station, 1974.

1 v. (various pagings) illus. 27 em. (U. S. Waterways Experiment Station. Miscellaneous paper S-74-30)

Prepared for Office, Chief of Engineers, U. S. Army, Washington, D. C.

Includes bibliography.

1. Concrete strength. 2. Concrete tests. 3. Load-carrying capacity. 4. Rigid pavements. I. U. S. Army. Corps of Engineers. (Series: U. S. Waterways Experiment Station, Vicksburg, Miss. Miscellaneous p-a-p-e-r- s---7-4~3-0-)-

TA7.W34m no.S-74-30

Documents

MP S-74-30 'Concrete strength relationships · MISCELLANEOUS PAPER S-74-30 CONCRETE STRENGTH RELATIONSHIPS by George M. l-lammitt II Soils end Pavements Laboratory U.S. Army Engineer