structures under shock & impact vi, c.a. brebbia & n ... · concrete materials subjected to high...

Dynamic properties of concrete materials with

high rates of tri-axial compressive loads

K. Fujikake', K. Mori', K. Uebayashi', T. Ohno' & J. Mizuncr1 Department of Civil Engineering, National Defense Academy, Japan2 Nuclear Structures Department, Kajima Corporation, Japan

Abstract

This study is to find the dynamic properties of concrete materials under both high strain-rates and triaxial stress states and to formulate the dynamic constitutive model of concrete.Thus, triaxial rapid compressive loading tests for concrete specimens were executed. In tests,the compressive strength of concrete, the confining pressure and the loading rates werechosen as test parameters. Based on test results, the influence of these parameters on themechanical properties of concrete are examined, and then the constitutive model of concretewith the strain-rate effects is proposed.

1 Introduction

To assess properly the crashworthiness and the safety of reinforced concrete (RC) structuressubjected to severe impact/impulsive loads, it is important to find precisely the dynamicmechanical properties of concrete materials. Over the last two decades many studies havebeen undertaken on the behavior of concrete under uniaxial rapid loading [1,2,3]. However,the actual concrete materials comprising of RC structures may be generally under triaxialstress states due to the complex loadings and the confining effect of the transversereinforcement. For this reason, to predict analytically the behavior of RC structuressubjected to impact/impulsive loads, the dynamic mechanical properties of concretematerials under high strain-rates and triaxial stress states should be evaluated precisely andalso the dynamic constitutive model of concrete with both high-strain rates and multi-stressstates is essential.

Many studies on the behavior of concrete materials subjected to triaxial staticcompressive loading have been reported [4,5,6]. As a result, it has been well known thatconfinement greatly improves the maximum strength and the ductility. A lot of constitutivemodels for concrete materials under triaxial static compressive stress states have been

Structures under Shock & Impact VI, C.A. Brebbia & N. Jones (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-820-1

512 Structures Under Shock and Impact VI

developed [4,7,8]. On the other hand, a less information on the dynamic behavior ofconcrete materials subjected to high strain-rates and triaxial stresses is available.

Thus, the apparatus for testing concrete under triaxial rapid compressive stress stateswas newly developed, and the rapid loading tests for cylindrical concrete specimens wereexecuted. In tests, the compressive strength of concrete, the confining pressure and theloading rates were chosen as test parameters. From test results, the influence of theseparameters on the mechanical properties of concrete are examined. Furthermore, based onthe concept of the equivalent uniaxial strain, the orthotropic constitutive model of concretewith the strain-rate effects is proposed.

In this study, compression is considered positive, a set of principal axes of stress andstrain are configured to a cylindrical specimen as shown in Figure 1.

2 Outline of triaxial rapid compressive loading tests

2.1 Test parameters

In tests, the compressive strength of concrete, the confining pressure and the loading rateswere chosen as test parameters. Three different compressive strengths as normal-, medium-and high-strength concrete were employed in this study. Each compressive strength (//) is37.4, 46.2 and 85.6 (N/mnf), respectively. Six different confinement levels for eachcompressive strength concrete as shown in Table 1 were applied. Besides, the level-0 meanswithout the confining pressure, i.e., corresponding to the uniaxial compressive loading.Axial loads were applied to specimens by 4 sorts of rate as static-, low-, medium- and high-rate loadings. In this study, the strain-rate was used as an index of loading rate. The strain-rate was defined as the average rate of strain e i in axial direction from the start of loadingto the maximum triaxial compressive strength. Their loading rates were designated by thestrain-rate as 1.2x 10̂ (I/sec) for static, 3.Ox 10̂ (I/sec) for low-rate, 3.Ox 10'* (I/sec) formedium-rate and 2.0 x 10° (I/sec) for high-rate. Total numbers of test cases were (types ofconcrete: 3)x (types of confinement: 6)x (types of loading-rate: 4)=72 cases. Tests weredone 3 times for each test case.

2.2 Triaxial loading apparatus

The triaxial loading apparatus was newly developed to apply axial stress (

Structures Under Shock and Impact VI 513

diameter x 100mm height) with three different compressive strengths. Table 2 details mixproportions used in this study. Mechanical properties of each concrete series in uniaxialstatic compression tests are given in Table 3.

The ends of specimen were accurately plane and parallel by grinding them beforetesting. The cylinder faces were sandblasted before testing in order to expose possible airvoids near the surface. Then all voids were filled with cement paste in order to avoidpuncture of the rubber membranes and damage of the strain gages glued on the cylinder.The cylindrical specimens were jacketed with a rubber membrane to prevent penetration ofhydraulic fluid into the specimen during a test.

2.4 Test procedure

The triaxial rapid compressive loading tests were performed in a triaxial rapid compressiveloading machine, as shown in Figure 3, consisting of a newly developed triaxial loadingapparatus to apply the confining pressure and an existing servo-controlled rapid loadingmachine (max. load capacity of 980kN, max. loading speed of 4m/sec) to apply the axialload.

In tests, at first the hydrostatic loading was applied to specimen at static loading rateby means of the triaxial loading machine. Once the confining pressure was reached to aspecified value, under the confining pressure to be maintained constant, the axial load wasapplied to specimen at specified loading rate by means of the servo-controlled rapid loadingmachine.

Axial loads were measured by a load cell installed in the lower unit cell. Confiningpressures were measured by means of a pressure transducer mounted at the main chamber.The deformational response of the cylindrical specimens was measured in the central zoneof the specimens by using strain gages with a gage length of 30mm attached in the axial andcircumferential directions.

3 Test results

3.1 Failure modes

In the case of no-confinement, the mixed failure modes accompanied with the prominentsplitting failure mode and the shear failure mode were observed regardless of loading rates.Figure 4 shows typical failure modes of medium-strength concrete formed under triaxialcompression tests at both static and high-rate of loading. Under comparatively lower levelsof confining pressure, the shear slip failure modes were observed. Under higher levels ofconfining pressure, the bulging failure modes were observed at either ends of a specimen.

The angle of shear slip plane is about 62 deg with the horizontal regardless ofcompressive strength and loading rate. However, in case of specified confining pressure78.5(N/mnf) of high-strength concrete series, the angle of shear slip plane became smallwith the increasing of loading rate.

3.2 Stress path

Figure 5 shows the actual stress paths applied to specimens in triaxial rapid compression



tests of the medium-strength concrete series. As shown, in the case of static loading, theconfining pressures were maintained constant at the specified values while the axial stressesapplied to specimens. On the other hand, the confining pressures decreased from thespecified values with increasing axial stress in case of low-, medium- and high-rate ofloading. This may be due to the performance of oil pump used to maintain the confiningpressure.

3.3 Failure criterion for concrete materials under high strain-rates and triaxialstress states

3.3. 1 Failure criterion for static loadingFigure 6 shows the maximum strength versus the confining pressifre obtained from triaxialstatic compression tests which is plotted on Rendulic plane in normalized principal stressspace. Typical test results reported by other investigators [4,5,6] are also shown in Figure 6.This figure indicates that the failure envelope is essentially independent of the uniaxial staticcompressive strength. Test results in this study coincide well with those one reported byKotsovos.

To express the nonlinear relationship between the maximum strength and theconfining pressure, the Leon model is employed [7]. The Leon model is expressed in termsof the major and minor principal stress invariants as:

in which h is a constant given as the ratio of compressive strength and tensile strengthunder uniaxial static loading (h = fjf' ). The parameter h , determined by regressionanalysis of the test results, is 0.08. The failure criterion for static loading is plotted by a solidline in Figure 6. The main reason for adopting the Leon model on the static failure criterionis its simplicity with extending to dynamic failure criterion in next section.

3.3.2 Strain-rate effect on maximum strength under triaxial stress statesFigure 7 shows the maximum strength versus confining pressure obtained from triaxialrapid compression tests on the medium-strength series which is plotted on Rendulic planein normalized principal stress space. The static failure criterion with the reference toexamine the influence of confining stress level on the strain-rate effect in the maximumstrength is also plotted by a solid line in Figure 7. From this figure, the strain-rate effect onmaximum strength under the triaxial stress states decreases with increasing the confiningstress. Therefore, it is concluded that the strain-rate effect on the triaxial maximum strengthis not only the function of strain-rate, but also the function of confining stress.

3.3.3 Dynamic failure criterionThe static Mure criterion is extended to the dynamic failure criterion by incorporating thefollowing condition: the Mure criterion of concrete loaded by any strain-rate s to passthrough the both points of the uniaxial dynamic compressive strength ( /%, ) and the uniaxialdynamic tensile strength ( /, ). The dynamic failure criterion is given as follows:



/;

1-^ \1

v(2)

in which £ and 77 are the dynamic increase factors of compressive and tensile strengthrespectively. For £ and 77 the formulations proposed by Fujikake et al. [2] and Ross etal.[3] are adopted respectively.

0.006 Logl

(3)

Jt0,00126 Log —

8S,

(4)

where ^ = 1.2x 10^ (I/sec) and ̂ = 1.0 x 10^ (I/sec).The dynamic failure criterions calculated for low-rate and high-rate loadings are

plotted in Figure 8, together with the test results. It is found that this dynamic failurecriterion fits well to the test results at each loading rate.

3.4 Relation between maximum strength and corresponding axial strain

In order to describe the failure criterion for axial strain, normalized maximum strength

(PP / fed) and normalized axial strain (̂ / ê ) are employed here. Normalized maximumstrength and normalized axial strain are defined as the maximum strength (cp normalizedwith respect to the uniaxial dynamic compressive strength (/%/) and the axial straincorresponding to the maximum strength (ê normalized with respect to straincorresponding to uniaxial dynamic compressive strength (̂), respectively. From the testresults, the correlation relationship exists between the normalized axial strain and thenormalized maximum strength. Based on regression analysis in the test results, thefollowing equation is proposed as the failure criterion for axial strain:

\1.74ex/?(-0.05e(5)

The dynamic failure criterions of axial strain calculated from eqn (5) for static and high-rateloadings are plotted in Figure 9, together with the test results.


516 Structures Under Shock and Impact 17

4 Constitutive model with strain-rate effects

4.1 Model description

4.1.1 Basic conceptThe orthotropic constitutive model of concrete with the strain-rate effects, based on theconcept of equivalent uniaxial strain [8], is developed by incorporating the followingaspects: 1) strain-rate effect on initial tangent modulus, 2) strain-rate effects on themaximum strength and the corresponding axial strain, respectively.

Generally, the orthotropic incremental constitutive relation, when referred to theorthotropic principal axes, can be written as:

do-2

sym.

0

0

0

0

0

0

0

0

0

0

0

4/12

4/23

(6)

where ^ = 1 -

in which the subscripts 1, 2 and 3 stand for the axes of orthotropy; da and dr =incremental normal and shear stress, respectively; ds and dy = incremental normal andshear strain, respectively; E = the orthotropic moduli of elasticity; ju = equivalent

Poisson's ratio (=


4.1.2 Stress-equivalent uniaxial strain relationTo evaluate the incremental elastic moduli in eqn (6), the stress-equivalent uniaxial strainrelation [8] is introduced:

where

^ = ~T7 ; 7£~~T~' *if=**ic> °if=°icl

in which


4.2 Verification

The constitutive model proposed here is compared with the test results. For this purpose thestrain behaviors are calculated along each of the stress-paths obtained from the triaxial rapidcompressive loading tests. Typical examples in the normal-strength concrete series areshown in Figure 10. It can be seen that the constitutive model proposed here gives a goodmach with the test results at each loading rate.

5 Concluding remarks

The following concluding remarks are obtained from this study. ,1. The failure modes depend on the confinement level. However, the failure modes are notaffected noticeably by the difference of strain rate.

2. The strain-rate effect on maximum strength under triaxial stress states decreases withincreasing the confining stress.

3. Dynamic failure criterion to predict maximum strength of concrete subjected to highstrain-rates and triaxial stresses was formulated.

4. The constitutive model with the strain-rate effects was confirmed to be good agreementswith the test results.

References

[1] Bischof£ P. H. and Perry, S. H.: Compressive behaviour of concrete at high strain rates,Materials and Structures, pp.425-450,24,1991.

[2] Fujikake, K., Shinozaki, Y, Ohno, T, Mizuno, J. and Suzuki, A.: Post-peak and strain-softening behaviors of concrete materials in compression under rapid loading, Proc. ofJSCE, No.627/V-44, pp.37-54,1999.8 (in Japanese).

[3] Ross, C. A., Thompson, P. Y. and Tedesco, J. W.: Split-hopkinson pressure-bar tests onconcrete and mortar in tension and compression, ACI Materials Journal, V.86, No.5,pp.475-481, Sep.-Oct, 1989.

[4] Chen, W. F.: Plasticity in reinforced concrete, McGraw-Hill Int. Book Company, NewYork, 1982.

[5] Ohnuma, H. and Aoyagi, Y. : Ultimate strength property of concrete under triaxialcompressive stresses, CRIEPI report, No.381021,1981.12 (in Japanese).

[6] Kotsovos, M. D. : A mathematical description of the strength properties of concreteunder generalized stress, Magazine of Concrete Research, Vol.31, No. 108, pp. 151-158,Sep., 1979.

[7] Pramono, E. and Willam, K. : Fracture energy-based plasticity formulation of plainconcrete, Journal of Engineering Mechanics, Vol. 115, No.6, pp.1183-1204, June, 1989.

[8] Elwi, A. A. and Murray, D. W. : A 3D hypoelastic concrete constitutive relationship,Journal of the Engineering Mechanics Division, ASCE, Vol.105, No.EM4, pp.623-640,August, 1979.


Structures Under Shock and Impact 17 519

Figure 1: Configuration ofprincipal axes.

Table 1. Specified confining pressures in tests.

Test Series

Normal-StrengthMedium-StrengthHigh-Strength

Confining Pressure (N/mnf)Level-0

0

0

0

Le\d-l

5.9

5.9

4.9

Le\d-2

11.8

11.8

9.8

Lewl-3

215

23.5

19.6

Lewl4

47.1

47.1

39.2

Level-5

94.1

94.1

78.5

(a) General view

Rapid loading direction

(D Retaining rod(D Retaining plate(D Upper unit cell@ Middle unit cell(D Lower unit cell(D Base plate(7) Loading platen


Servo valve unitMain actuator (980kN)

LVDT for servo-control

Accumulator

Figure 3: Set-up for triaxial rapid compression tests.

1) Static 2) High-rate 1) Static 2) High-rate 1) Static 2) High-rate(a) Level-1 (b) Level-3 (c) Level-5

Figure 4: Typical failure modes in medium-strength test series.

eBz

400

300

200

100

00

' StaticLow-rateMedium-rateHigh-rate

20 40 10060 800-2 = cr,3 (N/mnf)

Figure 5: Measured stress-paths in medium-strength test series


Structures Under Shock and Impact 17 521

20

15

"Z 10

J-Previous resarch results -Ottosen •• OttosenBalmerRichartOhnuma el al.Kotsovos

-Test results—I Normal-strengthI Medium-strengthk High-strength

Figure 6: Maximum strength versus confining pressure in static loading.

O Low-rate• Medium-rateffl High-rate

Static failure criterion [eqn (1)]

0-0.25 0 0.25 0.5 0.75 1

(a) All datas (b) Low confining pressure level enlargedFigure 7: Strain-rate effect on maximum strength in medium-strength test series.

10 r- . 10O Normal-strength• Medium-strengthffl High-strength

Dynamic failure criterion

O Normal-strength* Medium-strengthffl High-strength

Dynamic failure criterion

0 1 -1 0 1

(a) Low-rate loading (b) High-rate loadingFigure 8: Dynamic failure criterion.


522 Structures Under Shock and Impact 17

(a) Static loading (b) High-rate loadingFigure 9: Normalized axial strain versus normalized maximum strength.

400

350

300

250

200

150

100

50

-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1strain

(a) Static loading

-0.02 0 0.02 0.04 0.06 0.08 0.1strain

(b) Low-rate loading

-0.02 0 0.02 0.04 0.06 0.08 0.1strain

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06strain

(c) Medium-rate loading (d) High-rate loadingFigure 10: Verification of constitutive model in normal-strength test series.


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