Elasticity modulus calcs.pdf

MODULUS

OF

ELASTICITY

REPORT

September 24th, 2003

Report Prepared By:

Sean Mulligan

Testing Performed By:

Bob ThompsonChuck Cook

Donita Weatherspoon

Modulus of ElasticityReport

Page 1

PURPOSE

The purpose of this experiment was to determine the variation in the modulus of elasticity (E) of three (3)different classes of concrete (a normal concrete [Class S] and two high performance concretes [Class HP-3 and Class HP-4]). The chord modulus of elasticity (Ec) was determined for each class of concrete usingdata on cylinder deformation obtained during in-house experimentation. The value determined for Ec foreach class of concrete was then substituted into the equation for the secant modulus of elasticity (Es), andthe coefficient (C) for the secant modulus was calculated for each class of concrete for comparisonpurposes.

PROCEDURE

ASTM C 469-94 was used as the procedure for this experiment. In summary, numerous six inch by 12inch (6" x 12") pairs of concrete cylinders were tested as follows:

1) The first cylinder of each pair was tested to compressive failure and the load at failure wasrecorded (Cyl. #1 Failure).

2) The second cylinder of each pair was fitted with a compressometer such that the compressometerwas centered around the cylinder making the deformation experienced by the cylinder equal to ½the gage reading.

3) A load equal to approximately 40% of the load at failure of the first cylinder was applied to thesecond cylinder to “seat” the gage, then the load was released and adjustments were made ifnecessary to ensure the compressometer was properly seated.

4) A load was re-applied to the second cylinder until the compressometer gage read 0.0005 inchesand the corresponding load at a gage reading of 0.0005 inches was recorded (LL).

5) The load was increased to approximately 40% of the load at failure of the first cylinder and thecorresponding load (HL) and gage reading (HG) were recorded.

6) The load was released then the HL was re-applied to to check the accuracy of the HG reading.

7) The load was released and the compressometer was removed.

8) The second cylinder was tested to compressive failure and the load at failure was recorded (Cyl.


Page 2

#2 Failure).

CALCULATIONS

The method used in this experiment to calculate the modulus of elasticity is based upon deformation ofconcrete cylinders under a compressive load. This yields the chord modulus of elasticity (Ec). Theequation for the chord modulus of elasticity is [equation 1] Ec = (S2 - S1) / (e2 - e1), where:

Ec = chord modulus of elasticityS1 = stress corresponding to longitudinal strain of e1

S2 = stress at HL where HL is 40% of the load at compressive failure for cylinder #1e1 = longitudinal strain at S1

e2 = longitudinal strain at S2

Longitudinal strain is the deformation that the specimen is subjected to when under compression.According to the compressometer used, each longitudinal strain (e1, e2) is equivalent to ½ the gage readingdivided by the effective gage length where the effective gage length is 8 inches. The first gage reading foreach pair of cylinders was 0.0005 inches. The second gage reading (HG) varied among the cylinder pairsand was taken at 40% of Cyl. #1 Failure. This yields:

(e2 - e1) = [HG / (2 x 8')] - [0.0005 in. / (2 x 8")] = (HG - 0.0005 in.) / 16"

Stress is the force exerted on the cylinder in pounds per square inch. Stress is calculated by dividing theload (pounds) by the surface area (square inches) of the cross section of the specimen. Each specimenwas a 6" diameter cylinder which equates to a surface area of PI x (D)2 /4 = 3.142 x (6 inches)2 / 4 =28.27 square inches. The first stress (S1) is equivalent to the load at e1 divided by 28.27 square inches(LL). The second stress (S2) is equivalent to 40% of the ultimate load divided by 28.27 square inches(HL). This yields:

(S2 - S1) = [(0.4 x HL) / 28.27 in2]- (LL / 28.27 in2) = [(0.4 x HL) - LL)] / 28.27 in.2

Substituting stresses and longitudinal strains back into the original equation yields:

Ec = (S2 - S1) / (e2 - e1) = {[(0.4 x HL) - LL] / 28.27 in.2} / [(HG - 0.0005 in.) / 16]

=> Ec = (0.566 / in.) x [(0.4 x HL) - LL] / (HG - 0.0005 in.)where HL and LL are in pounds, HG is in inches, and HL, LL, and HG were recorded during the


Page 3

experiment.

Another method that can be used to calculate the modulus of elasticity is based upon the ultimate strengthof concrete cylinders. This yields the secant modulus of elasticity (Es). The secant modulus of elasticityis equivalent to a factor multiplied by the square root of the ultimate strength of the concrete. The equationfor the secant modulus of elasticity is [equation 2] Es = C x (F’c)1/2 where:

Es = Secant Modulus of ElasticityC = coefficient that is dependent upon density where the equation is [equation 3] CEXP = 33 x(concrete density)1.5 F’c = ultimate strength of the concrete (average of a pair of cylinders at failure)

When Ec is substituted into equation 2 for Es and the equation is rearranged to solve for C, the equationbecomes [equation 4] C = Ec / (F’c)1/2.

Determining the numerical values of C between the different classes of concrete was the main intent of thisexperiment. A value for C was determined for each set of cylinders tested using equation 4. CAVG wasdetermined for each class of concrete by averaging the individual results for C. CCALC was determined byusing equation 4 and the average values of Ec and F’c for that class of concrete. CEXP was determinedusing equation 3 and the average density for each class of concrete. For normal density concrete, thegenerally accepted value for CEXP is approximately 57,000.

RESULTS

Appendix A shows all results in tabular and graphic format:

Table A on page A-1 and Graph A on page A-4 shows all results for the class S concretes.

Table B on page A-2 and Graph B on page A-5 shows all results for the class HP-3 concretes..

Table C on page A-3 and Graph C on page A-6 shows all results for the class HP-4 concretes.

A summary of the results is shown below (Table 1). These results are shown graphically on page 4 (Graph1).

TABLE 1 - E, F’c, and C by Class


Page 4

Classof

Concrete

# ofCylinder

PairsTested

AverageModulus

of ElasticityE (PSI)

AverageStrengthF’c (PSI)

AverageCoefficient

CAVG

CalculatedCoefficient

CCALC

AverageConcreteDensity(lbs./cf)

ExpectedCoefficient

CEXP

S 26 4.05E+06 5,615 54,380 54,057 140.48 54,946

HP-3 21 3.81E+06 6,228 48,375 48,293 140.39 54,893

HP-4 25 3.89E+06 7,065 46,462 46,227 138.89 54,015

GRAPH 1 - E, F’c, and C by Class


Page 5

0 1000 2000 3000 4000 5000 6000 7000 8000

HP-4

HP-3

S

F'c (PSI)

E (PSI) / 1,000

Cavg / 10

Ccalc / 10

E (PSI) / 1,000

E (PSI) / 1,000

Cavg / 10

Cavg / 10

Ccalc / 10

Ccalc / 10

F'c (PSI)

F'c (PSI)

INTERPRETATION OF RESULTS

Table 1 and Graph 1 indicate the following:

1) Class HP-4 concrete has the highest strength (F’c = 7,065 PSI), Class HP-3 had the secondhighest strength (F’c = 6,228 PSI), and Class S had the lowest strength (F’c = 5,615 PSI)


Page 6

2) Class S concrete has the highest modulus of elasticity (E = 4.05E+06 PSI) and the highestcoefficient (CCALC = 54,057 and CAVG = 54,380).

3) Class HP-3 concrete has the lowest modulus of elasticity (E = 3.81+06 PSI) and secondhighest coefficient (CCALC = 48,293 and CAVG = 48,375).

4) Class HP-4 concrete has the second highest modulus of elasticity (E = 3.89+06 PSI) andlowest coefficient (CCALC = 46,227 and CAVG = 46,442).

5) The modulus of elasticity for the high performance concretes are similar (E = 3.81+06 for classHP-3 and E = 3.89+06 for Class HP-4).

6) The calculated coefficient for the Class S concrete (CCALC = 54,057 for Class S) is close tothe expected coefficient (CEXP = 54,946) which is also reasonably close to the generallyaccepted value of 57,000, but the calculated coefficients for the high performance concretesare not close to their expected coefficients (CCALC = 48,293, CEXP = 54,893 for Class HP-3and CCALC = 46,227, CEXP = 54,015 for Class HP-4) or to the generally accepted value of57,000. All of the calculated coefficient values (CCALC) are very close to their respectiveaverage coefficient values (CAVG) indicating the calculated values (CCALC) are accurate.

Graphs A, B, and C in Appendix A indicate the following:

1) From Graph A (page A-4), the modulus of elasticity (E) and the coefficient (C) curves for theclass S concrete (normal concrete) both increase as the strength of the concrete increases, butthe increase in each of these is at a rate lower than the rate of strength increase. The modulusof elasticity (E) and the coefficient (C) curves have the same shape, but the modulus ofelasticity (E) curve increases at a slightly higher rate than the coefficient (C) curve.

2) From Graph B (page A-5), the modulus of elasticity (E) and the coefficient (C) curves for theclass HP-3 concrete (high performance concrete) both increase as the strength of the concreteincreases, but the increase in each of these is at a rate lower than the rate of strength increase.The modulus of elasticity (E) and the coefficient (C) curves have the same shape, but themodulus of elasticity (E) curve increases at a slightly higher rate than the coefficient (C) curve.The increase in the coefficient (C) curve is small.

3) From Graph C (page A-6), the modulus of elasticity (E) and the coefficient (C) curves for theclass HP-4 concrete (high performance concrete) both stay flat as the strength of the concrete


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increases. The modulus of elasticity (E) and the coefficient (C) curves have the same shape,but the coefficient curve (C) curve decreases at a slightly greater rate than the modulus ofelasticity (E) curve. The decrease in the modulus of elasticity (E) curve is small.

4) Graphs A, B, and C show that the rates at which E and C increase with increasing strength arehigher for the class S concrete.

CONCLUSIONS

With respect to concrete, the modulus of elasticity (E) is an indication of a concrete’s ability to retain itsoriginal shape after being subjected to stresses and strains. E is calculated by dividing stress by strainwhere stress is an external force imposed upon the concrete and strain is the deformation of the concretecaused by the external force. In short, E is really the “sponginess” or “forgiveness” of a concrete.

Since E is stress over strain, one would expect concretes with a high E to be harder, or more brittle, thanconcretes with a low E. The reason for this is that strain is nothing more than deformation, and if thedeformation is low, E will be high because deformation is the denominator when calculating E. As thedeformation increases, E decreases.

It is logical to assume that a concrete with a higher strength will have less deformation, and therefore ahigher E, than a concrete with a lower strength which should have more deformation. This meansconcretes with high strengths should be more rigid or brittle, and less forgiving than concretes with lowstrengths. The basic premise behind this research project was to determine if higher strength concretes aremore rigid and brittle than lower strength concretes by evaluating E for a normal concrete and 2 highperformance concretes.

Test results indicated the following:

1) The Class S concrete had the lowest strength (f’c = 5,615 PSI), the Class HP-4 had the higheststrength (f’c = 7,065 PSI), and the Class HP-3 had the second highest strength (f’c = 6,228 PSI).

2) The lowest strength concrete (Class S) had the highest modulus of elasticty (E = 4.05E+06), thehighest strength concrete (Class HP-4) had the second highest modulus of elasticity (E =3.89E+06), and the second highest strength concrete (Class HP-3) had the lowest modulus ofelasticity (E = 3.89E+06).

3) The coefficient for the lowest strength concrete (CCALC = 54,057) determined by this researchproject was found to be accurate (CEXP = 54,946) and relatively close to the generally acceptedC value of 57,000.

4) The coefficients for the high performance concretes (CCALC = 48,293 for Class HP-3 and CCALC


Page 8

= 46,227 for Class HP-4) determined by this research project were not accurate (CEXP = 54,893for Class HP-3 and CEXP = 54,015 for Class HP-4), and neither result was very close to thegenerally accepted C value of 57,000.

In conclusion, modulus of elasticity generally increases with strength. The high performance concretestested did have higher strengths than the normal concrete tested, but their values for modulus of elasticitywere lower than the normal concrete tested. This means that the high performance concretes (HP-3 andHP-4) are stronger yet less brittle than the normal concrete (Class S), which is contrary to the belief thathigher strengths are less forgiving and produce higher values for modulus of elasticity.

In this case, the high performance concretes were able to have higher strengths and lower values formodulus of elasticity because the rates at which E increased with strength for the high performanceconcretes was less than the rate for the normal concrete. The normal concrete (Class S) showed a steadyincrease in E as strength increased while the first high performance concrete (Class HP-3) showed a smallincrease for E and the other high performance concrete (Class HP-4) showed no increase at all for E.

The equation [Es = C x (F’c)1/2 where C is 57,000] is valid for estimating the modulus of elasticity for classS concretes, but using a value of approximately 54,000 for C would yield better estimates. For the classHP-3 and HP-4 concretes, the equation [Es = C x (F’c)1/2 where C is 57,000] will not provide an accurateestimate unless a value of approximately 47,000 is used for C.

Appendix A

Modulusof

ElasticityReport

Page A-1

TABLE A - CLASS S CONCRETE RESULTS

Sample

ID

Cyl. #1Failure

(pounds)

LowLoad

(pounds)

HighLoad

(pounds)

HighGage

(inches)

Cyl. #2Failure

(pounds)

Modulusof

Elasticity,Ec (PSI)

AverageStrength

F’c (PSI)

C

635645-01 164,500 3,800 65,900 0.0100 156,000 3.70E+06 5,669 49,139

629764-01 121,600 2,500 48,600 0.0085 184,900 3.26E+06 5,421 44,296

629764-01 114,900 3,000 47,000 0.0075 122,100 3.56E+06 4,192 54,948

634858-01 171,400 6,800 69,000 0.0095 172,600 3.91E+06 6,084 50,147

629764-01 119,200 4,600 47,400 0.0065 118,800 4.04E+06 4,209 62,227

631098-01 185,000 6,200 74,100 0.0085 188,000 4.80E+06 6,597 59,142

635741-01 250,300 5,300 100,000 0.0115 247,100 4.87E+06 8,797 51,949

628901-01 209,600 2,800 84,200 0.0090 247,400 5.42E+06 8,083 60,286

627560-01 176,200 11,600 70,800 0.0065 164,600 5.58E+06 6,028 71,927

631554-01 159,800 5,600 48,100 0.0085 131,900 3.01E+06 5,159 41,860

631124-01 135,400 3,300 54,200 0.0100 120,000 3.03E+06 4,517 45,119

635615-01 84,200 3,000 33,600 0.0060 77,100 3.15E+06 2,853 58,954

632030-01 153,300 4,800 61,900 0.0105 146,400 3.23E+06 5,301 44,388

615627-01 139,500 2,200 93,500 0.0155 153,500 3.44E+06 5,182 47,854

632642-01 166,400 4,850 66,500 0.0100 162,500 3.67E+06 5,817 48,156

633862-01 137,000 6,000 66,700 0.0095 128,200 3.82E+06 4,690 55,735

626187-01 137,000 1,000 57,400 0.0085 130,800 3.99E+06 4,736 57,977

625705-01 136,500 4,000 54,900 0.0075 140,300 4.12E+06 4,896 58,818

630994-01 158,900 5,000 63,300 0.0085 162,700 4.12E+06 5,688 54,688

625842-01 135,500 2,900 54,800 0.0075 144,100 4.20E+06 4,945 59,672

617131-01 159,800 3,500 63,900 0.0085 150,800 4.27E+06 5,493 57,653

629401-01 211,000 5,200 84,500 0.0110 125,400 4.27E+06 5,950 55,415

631723-01 190,600 5,700 76,200 0.0095 196,500 4.43E+06 6,846 53,581

628251-01 146,300 3,400 58,600 0.0075 160,200 4.46E+06 5,421 60,617

635143-01 200,600 5,000 80,100 0.0100 198,800 4.47E+06 7,064 53,233

632696-01 158,700 4,000 63,300 0.0080 201,000 4.47E+06 6,362 56,104

Page A-2

CCALC = 54,057 AVERAGE 4.05E+06 5,615 54,380

TABLE B - CLASS HP-3 CONCRETE RESULTS

SampleID

Cyl. #1Failure

(pounds)

LowLoad

(pounds)

HighLoad

(pounds)

HighGage

(inches)

Cyl. #2Failure

(pounds)

Modulusof

Elasticity,Ec (PSI)


C

635035-01 166,200 5,700 66,600 0.0095 180,100 3.83E+06 6,125 48,935

629966-01 166,100 2,300 74,200 0.0105 178,000 4.07E+06 6,086 52,163

627875-01 188,500 5,400 75,300 0.0100 187,100 4.16E+06 6,643 51,093

635048-01 140,900 5,000 56,500 0.0075 200,300 4.16E+06 6,035 53,601

629966-01 166,100 4,000 66,700 0.0090 167,000 4.17E+06 5,891 54,392

627082-01 160,000 4,600 64,400 0.0080 160,400 4.51E+06 5,667 59,947

632010-02 123,600 4,850 49,600 0.0090 142,300 2.98E+06 4,703 43,450

635672-02 139,500 4,600 55,800 0.0100 143,400 3.05E+06 5,004 43,122

635672-01 148,800 4,400 59,400 0.0105 149,300 3.11E+06 5,272 42,870

632010-01 133,900 3,300 53,600 0.0095 145,100 3.16E+06 4,935 45,029

632010-01 134,300 2,800 53,600 0.0095 160,700 3.19E+06 5,218 44,226

617442-02 146,900 3,200 58,600 0.0095 152,500 3.48E+06 5,295 47,876

631131-02 186,900 4,000 74,900 0.0120 181,700 3.49E+06 6,519 43,216

631131-01 186,000 3,900 74,400 0.0115 178,500 3.63E+06 6,447 45,177

632413-01 166,000 4,000 66,300 0.0100 187,700 3.71E+06 6,243 46,973

628590-01 169,000 5,000 67,700 0.0100 174,700 3.74E+06 6,079 47,910

615512-01 181,400 5,400 72,800 0.0100 181,500 4.02E+06 6,418 50,120

635441-01 226,400 6,300 90,600 0.0120 230,000 4.15E+06 8,072 46,177

625875-01 200,700 3,300 80,900 0.0110 198,200 4.18E+06 7,055 49,798

615512-01 233,300 5,100 93,500 0.0120 243,600 4.35E+06 8,435 47,371

633697-01 240,400 6,000 96,400 0.0110 248,100 4.87E+06 8,640 52,423

Page A-3

CCALC = 48,293 AVERAGE 3.81E+06 6,228 48,375

TABLE C - CLASS HP-4 CONCRETE RESULTS

SampleID

Cyl. #1Failure

(pounds)

LowLoad

(pounds)

HighLoad

(pounds)

HighGage

(inches)

Cyl. #2Failure

(pounds)

Modulusof

Elasticity,Ec (PSI)


C

635216-01 194,900 4,800 78,100 0.0135 183,900 3.19E+06 6,700 38,988

632375-01 174,600 3,700 70,000 0.0120 175,400 3.26E+06 6,190 41,472

632379-01 210,700 4,000 84,200 0.0140 189,700 3.36E+06 7,082 39,954

627781-01 220,300 4,100 88,000 0.0145 231,700 3.39E+06 7,994 37,935

627555-01 199,800 3,100 81,900 0.0135 195,500 3.43E+06 6,992 41,029

632859-01 218,900 4,300 87,300 0.0140 171,800 3.48E+06 6,910 41,860

627555-02 220,100 3,900 87,900 0.0140 218,400 3.52E+06 7,756 39,988

627555-01 217,700 3,200 87,300 0.0140 216,200 3.53E+06 7,674 40,248

632313-01 188,300 3,000 75,500 0.0120 208,600 3.57E+06 7,020 42,586

632366-01 204,400 3,100 81,900 0.0125 178,600 3.72E+06 6,774 45,156

631298-01 248,500 3,700 99,600 0.0150 260,800 3.74E+06 9,008 39,440

631289-01 213,700 3,000 85,800 0.0130 212,400 3.75E+06 7,533 43,196

633397-01 165,900 5,600 66,300 0.0095 136,000 3.82E+06 5,340 52,238

626671-01 224,200 1,300 89,500 0.0135 231,800 3.84E+06 8,065 42,758

632304-01 217,400 5,300 86,800 0.0125 195,500 3.84E+06 7,303 44,981

632016-01 173,300 5,450 72,300 0.0100 178,300 3.98E+06 6,219 50,504

615547-01 169,300 7,300 67,600 0.0090 154,000 4.02E+06 5,718 53,097

615547-01 169,300 7,300 67,600 0.0090 203,300 4.02E+06 6,590 49,459

626402-01 187,700 4,000 75,000 0.0100 189,700 4.23E+06 6,675 51,773

633064-01 192,600 5,000 77,000 0.0100 204,900 4.29E+06 7,030 51,158

632379-01 211,100 5,800 84,300 0.0105 205,700 4.44E+06 7,372 51,746

634111-01 202,700 2,700 81,200 0.0105 202,400 4.44E+06 7,165 52,488

633064-01 198,600 4,800 79,400 0.0100 227,800 4.44E+06 7,542 51,177

615547-01 172,700 3,900 69,000 0.0085 154,000 4.61E+06 5,778 60,588

628189-01 232,800 5,600 93,300 0.0100 230,300 5.22E+06 8,191 57,731

CCALC = 46,227 AVERAGE 3.89E+06 7,065 46,462

GRAPH A

CLASS S CONCRETE RESULTS

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

Strength (PSI)

C / 10

E (PSI) / 1,000

Results shown in ascending order by strength

Page A-4

GRAP

H B

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

Strength (PSI)C / 10

E (PSI) / 1,000


Page A-5

CLASS HP-3 CONCRETE RESULTS

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

Strength (PSI)

C / 10

E (PSI) / 1,000


Page A-6

GRAPH C

CLASSHP-4

CONCRETE

RESULTS

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