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Multiaxial tensile–compressive strengths and failure criterion of plain high-performance concrete before and after high temperatures
Construction and Building Materials 24 (2010) 498–504
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Multiaxial tensile–compressive strengths and failure criterion of plain high-performance concrete before and after high temperatures He Zhen-jun a,*, Song Yu-pu b a b
Department of Civil Engineering, Tsinghua University, Key Laboratory of Structural Engineering and Vibration of China Education Ministry, Beijing 100084, PR China State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, PR China
a r t i c l e
i n f o
Article history: Received 8 June 2009 Received in revised form 13 October 2009 Accepted 15 October 2009 Available online 14 November 2009 Keywords: High-performance concrete (HPC) High temperatures Stress ratios Multiaxial tensile–compressive strengths Failure criterion
a b s t r a c t Multiaxial tensile–compressive tests were performed on 100 mm 100 mm 100 mm cubic specimens of plain high-performance concrete (HPC) at all kinds of stress ratios after exposure to normal and high temperatures of 20, 200, 300, 400, 500, and 600 °C, using a large static–dynamic true triaxial machine. Friction-reducing pads were three layers of plastic membrane with glycerine in-between for the compressive loading plane; the tensile loading planes of concrete samples were processed by attrition machine, and then the samples were glued-up with the loading plate with structural glue. The failure mode characteristic of specimens and the direction of the crack were observed and described, respectively. The three principally static strengths in the corresponding stress state were measured. The influence of the temperatures, stress ratios, and stress states on the triaxial strengths of HPC after exposure to high temperatures were also analyzed respectively. The experimental results showed that the uniaxial compressive strength of plain HPC after exposure to high temperatures does not decrease completely with the increase in temperature, the ratios of the triaxial to its uniaxial compressive strength depend on brittleness–stiffness of HPC after different high temperatures besides the stress states and stress ratios. On this basis, the formula of a new failure criterion with the temperature parameters under multiaxial tensile–compressive stress states for plain HPC is proposed. This study is helpful to reveal the multiaxial mechanical properties of HPC structure enduring high temperatures, and provides the experimental and theory foundations (testing data and correlated formula) for fire-resistant structural design, and for structural safety assessment and maintenance after fire. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Concrete has been the leading construction material for nearly a century. In recent years, high-performance concrete (HPC) is becoming an attractive alternative to traditional normal strength concrete (NSC). The alleged HPC is generally defined as highstrength, high fluidity, high durability concrete or to possess one of these performances; moreover, high-performance water reducer and superfine mineral admixtures are absolutely necessary ingredients. The dense microstructure of high-strength concrete (HSC) ensures high strength and very low permeability, so that it has a relatively low weak deformation capability or much higher ‘‘brittleness-stiffness” when compared to NSC. Recent fire test results show that there is a great difference between the properties of HSC and NSC after exposure to high temperatures [1–4]. It was observed that HSC is susceptible to spalling, or even explosive spalling when subjected to rapid temperature rise as in the case of a fire [5]. Hence, one of the main concerns for the usage of HSC in fire * Corresponding author. Tel.: +86 010 13811604337; fax: +86 010 62792450. E-mail address: [email protected] (Z.-j. He). 0950-0618/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2009.10.012
resistance applications is its performance under fire conditions. It is well known that the conventional analysis and design methods of reinforced concrete are generally still based upon material properties obtained from the basic uniaxial strength test, although we know that the true uniaxial condition in structures is extremely rare. In practice, many concrete structures such as shear wall, spiral columns and the node of building, nuclear reactor pressure container, etc are under multiaxial stress states. At the same time, with the wide use of computers, the finite element method and HPC in concrete structures, it has become increasingly evident that it is quite important and urgent for the experimental study on the mechanical property of HPC under multiaxial stress states, and for studies on the design and analysis of nonlinear behavior design of reinforced concrete structure based upon multiaxial mechanical behavior. So, many researchers have much paid attention to HPC at home and abroad. Late in the 1970s, some researchers [6] began to perform experimental study of ‘‘NSC” under multiaxial stress states in order to design for nuclear reactor container and so on. But, until now, most studies [2,7] have been carried out to characterize the mechanical behavior of HSC or HPC under uniaxial stress state, or NSC under multiaxial stress states. However, literature on the
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mechanical behavior of HSC or HPC under multiaxial loadings is extremely scarce; moreover, among them, the most tests of confined compression (with two equal stresses) behavior of concrete have nearly been performed under triaxial stress state using cylindrical specimens subjected to hydrostatic pressure in a triaxial cell and axial loading [8,9]. Lu and Thomas [9] studied the mechanical behavior of HSC and steel fiber reinforced high-strength concrete under uniaxial and triaxial compression, when testing U100 mm 200 mm and U100 mm 150 mm cylinder specimens lubricated with a combination of two 0.125 mm thick teflon sheets on top of one layer of 0.015 mm thick aluminum foil. The previous investigations conducted on the effect of high temperatures on properties of plain HSC or HPC have also merely focused on the behavior of strength and deformation under uniaxial loading. However, literature on the mechanical properties of plain HSC or HPC under multiaxial stress states after high temperatures, hasn’t been documented. Luo et al. [10] stated that the strength of the HPC degenerates more severely and has higher residual strength than that of NSC after exposure to high temperatures. Chan et al. [4] stated that the range between 400 °C and 800 °C is critical to the strength loss of concrete causing a large percentage of loss of strength. This paper presents the strength regularity and failure criterion of HPC at all kinds of stress ratios under multiaxial tensile–compressive stress states after exposure to six temperatures, using a large static–dynamic true triaxial machine. The multiaxial tensile–compressive tests were performed on 100 mm 100 mm 100 mm cubic concrete specimens. This study is helpful to reveal the multiaxial mechanical properties of HPC structure enduring high temperatures, and provides the experimental and theory foundations (testing data and correlated formula) for fire-resistant structural design, and for structural safety assessment and maintenance after fire.
The concrete specimens tested were in 100 mm 100 mm 100 mm, 150 mm 150 mm 150 mm or 150 mm 150 mm 300 mm. The 100 mm concrete cubes were used to measure the strength for multiaxial tests. For determination of the strength grade for the HPC and the strength of the prism, each batch was cast in six 150 mm cubic specimens, and six 150 mm 150 mm 300 mm prismatic specimens.
2.2.2. Apparatus and testing methods The tests of high temperatures and multiaxial mechanical properties were performed in State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The high temperatures testing apparatus and the multifunction triaxial experimental machine are shown in Figs. 1 and 2, respectively. The specimens in high temperatures test were 100 mm cubes. For each of stress ratio, six specimens were heated up at least. These 100 mm cubic specimens were elevated to the peak temperatures of 200, 300, 400, 500, and 600 °C at a heating rate of 10 °C/min, respectively. After the peak temperature was reached, it was maintained for 6 h; the cooling time in electric furnace was about 1 h; then, the specimens were took out and cooled naturally to room temperature. These specimens were performed in a multiaxial testing machine after 24 h. In this study, the surface
2. Materials and experimental procedures 2.1. Materials and mix proportions Fig. 1. The box-type electric furnace.
The cementitious materials used for this investigation were Chinese standard PI52.5R Portland cement (standard compressive strength of higher than 52.5 MPa at the age of 28 days) and one-level fly ash [11,12]. The coarse aggregate was crushed stone (diameter ranging from 5 mm to 20 mm) and the fine aggregate was natural river sand (fineness modulus of 2.7); water was tap-water. Table 1 shows the mix proportions by weight of the mixture and the major parameters of HPC (fc is the uniaxial compressive strength of 100 mm 100 mm 100 mm cubic HPC specimens with friction-reducing pads prior to high temperatures, and its strength value is about equal to that of 150 mm 150 mm 300 mm prism. fc shows average strength). 2.2. Samples and testing methods 2.2.1. Casting and curing of specimens The coarse aggregate and fine aggregate were mixed for about 1 minute when a portion of the proportional water was being sprayed on them, and then the cement and one-level fly ash were added in turn; after that, the residual proportional water with SiKa(R)NF-III superplasticizer (There are also the properties of increasing slump, durability and retardation besides ordinary water reducing agents.) was added slowly over a period of 1 min. Finally these ingredients were mixed for 2–3 min [12,13]. All specimens cast in steel molds and compacted slightly by vibrating table, were demoulded after 24 h of the casting and then cured in a condition of 20 ± 3 °C and 95% RH (relative humidity) for 28 days according to ‘‘GBJ82-85 The test method of long-term and durability on ordinary concrete [14]”, and then stored at a natural temperature outside. The age of the specimens tested was about a year.
Fig. 2. The triaxial testing machine.
Table 1 Mix proportions and major parameters of HPC. Concrete types
Water– cementitious ratio
Water (kg/m3)
Cement (kg/m3)
Fly ash (kg/m3)
Fine aggregate (kg/m3)
Coarse aggregate (kg/m3)
Superplasticizer (kg/m3)
Slump (cm)
Compressive strength (fC/MPa)
HPC
0.31
175
470
94.52
615.77
1094.71
6.77
24
60.16
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σ1
σ3
σ2
σ1
σ3
σ2
σ3
σ3
σ1
σ1
(a) Triaxial tensile-compressive loading direction
(b) Biaxial tensile-compressive loading direction
Spherichinge Antifriction pad s Specimen
Up-down loading platens
Specimen
(c) Uniaxial compression
Up-down left-right loading platens (d) Biaxial tension-compression
Fig. 3. States of the specimen in triaxial testing machine.
of all specimens was dried before being exposed to high temperatures. No explosive spalling was observed during the high temperature tests on HPC specimens with temperatures ranged from 200 °C to 600 °C. The triaxial tests were conducted in a triaxial testing machine that is capable of developing three independent compressive or tensile forces. The triaxial tests process was necessary to insure uniform dimensions for each cubic specimen. The major stress direction was always applied perpendicularly to the surface of specimens. The proportional loading mode was employed. Friction-reducing pads placed between the platens and the specimens were three layers of plastic membrane with glycerine in-between for the compressive loading plane; the tensile loading plane of concrete sample was processed by attrition machine, then, the samples were glued-up with the loading plate with structural glue. The type of structural glue is in JGN-II. Its value of the tensile, compressive and shear strength were 33 MPa, 62 MPa, 18Mpa respectively. The specimens under uniaxial compression were tested at a loading speed of 0.3–0.5 MPa/s in the direction of r3; but, those under uniaxial tensile, biaxial and triaxial tensile–compressive stress states were done at 0.03–0.05 MPa/s in the direction of r1. All kinds of stress ratios under multiaxial tensile–compressive loading after exposure to normal and high temperatures of 20, 200, 300, 400, 500, and 600 °C were tested. The principal stresses are expressed as r1 P r2 P r3 (compression denoted as negative and tension denoted as positive); three specimens were at least tested for each of stress ratio and the test values were averaged after the discrete ones were deleted. Fig. 3 shows the loading directions and the states of the specimen under multiaxial tensile–compressive stress states in triaxial experimental machine respectively.
3. Test results and discussion 3.1. Experimental results The experimental results of plain HPC under multiaxial tensile– compressive stress states after high temperatures are given in Table 2. 3.2. Failure modes The surface deterioration and the failure modes of the HPC specimens under multiaxial stress states subjected to the action of high temperatures are shown in Fig. 4a–d. From Fig. 4, it is obvious that the influence of high temperatures on HPC does not change the tensile splitting mode from occurring. In Fig. 4a and b, the HPC specimens under uniaxial compressive loading are split to multiple minor prisms (prism-type failure
modes). The tensile failures under biaxial and triaxial tensile–compressive loading are shown in Fig. 4c and d, respectively. Under multiaxial tension–compression after high temperature, sudden tensile failure appears for stress ratios less than or equal to a = r1/r3 = 0.10. In this case, the principal tensile stress is the key factor for the failure of concrete. There was no connection between the direction of the cracking and stress ratios. It was noticed that the cracks on the loading surface have a random direction because of the influence of coarse aggregates. The failure modes mentioned above demonstrates that providing confinement stress along r1, r2 and r3 directions will change the failure modes. Although the failure modes under multiaxial tensile–compressive stress states are different, the failure cause is that the splitting tensile strain along the unloading or less stress planes is greater than the ultimate tensile strain of HPC. 3.3. Strength characteristics As shown in Table 2 and Fig. 5, it is apparent that after 200 °C and 300 °C, the values of fcT =fc are greater than 1 at the stress ratio of a = 0; namely, the uniaxial compression strength fcT after 200 °C and 300 °C is increased, which differs from the conclusions drawn by the same research group for NSC [15]; but, when the suffered temperature is round 400 °C, it decreases gradually with the increase in temperature. As can be seen in Table 2 and Fig. 5a and b, the uniaxial compressive strengths (63.96, 61.64, 49.7, 35.72, 24.9 MPa) after being exposed to high temperatures of 200, 300, 400, 500, and 600 °C are 1.06, 1.02, 0.83, 0.59, 0.41 times the original strength at 20 °C (60.16 MPa), respectively; but, the uniaxial tensile strengths (4.81, 4.14, 2.99, 1.82, 1.00 MPa) are 0.95, 0.82, 0.59, 0.36, 0.20 times it (5.08 MPa) respectively. The discussion above indicates that in comparison with the other temperatures, the uniaxial compression strength fcT after 200 °C and 300 °C is increased; but, the uniaxial tensile strength ftT is gradually decreased with the increase in temperature. So, the temperatures around 400 °C are critical to the ultimate strength (fc, fcT is the uniaxial compression strength with friction-reducing pads at normal temperature and after different temperature levels, respectively).
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Z.-j. He, Y.-p. Song / Construction and Building Materials 24 (2010) 498–504 Table 2 The triaxial strength index of plain HPC under various stress states and stress ratios after high temperatures. Temperature (°C)
Stress ratio a = r1:r2:r3
r1f (MPa)
r2f (MPa)
r3f (MPa)
Temperature (°C)
Stress ratio a = r1:r2:r3
r1f (MPa)
r2f (MPa)
r3f (MPa)
20
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 5.08 2.20 3.23 4.28 4.73 4.86 2.23 3.02 3.76 4.11 4.19 4.22 4.23
0.00 0.00 0.00 0.00 0.00 0.00 0.00 44.64 30.18 15.05 11.73 8.37 5.63 4.23
60.16 0.00 43.92 32.26 17.11 9.45 6.48 44.64 30.18 15.05 11.73 8.37 5.63 4.23
400
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 2.99 1.59 2.08 2.19 2.26 2.50 1.07 1.59 1.84 2.09 2.22 2.48 2.52
0.00 0.00 0.00 0.00 0.00 0.00 0.00 21.48 15.91 7.34 5.98 4.44 3.30 2.52
49.70 0.00 31.80 20.79 8.75 4.52 3.33 21.48 15.91 7.34 5.98 4.44 3.30 2.52
200
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 4.81 2.07 2.98 3.70 4.27 4.57 1.94 2.75 3.53 3.55 3.98 3.94 4.19
0.00 0.00 0.00 0.00 0.00 0.00 0.00 38.71 27.52 14.12 10.15 7.96 5.25 4.19
63.96 0.00 41.33 29.75 14.81 8.54 6.09 38.71 27.52 14.12 10.15 7.96 5.25 4.19
500
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 1.82 0.72 1.05 1.26 1.33 1.02 0.66 0.90 1.06 1.17 1.43 1.49 1.52
0.00 0.00 0.00 0.00 0.00 0.00 0.00 13.11 9.04 4.22 3.34 2.86 1.98 1.52
35.72 0.00 14.30 10.49 5.04 2.65 1.36 13.11 9.04 4.22 3.34 2.86 1.98 1.52
300
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 4.14 1.86 2.34 3.11 3.48 3.81 1.68 2.22 2.70 3.15 3.42 3.56 3.47
0.00 0.00 0.00 0.00 0.00 0.00 0.00 33.51 22.22 10.80 9.00 6.85 4.74 3.47
61.64 0.00 37.22 23.41 12.43 6.97 5.08 33.51 22.22 10.80 9.00 6.85 4.74 3.47
600
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 1.00 0.39 0.52 0.58 0.99 0.76 0.35 0.46 0.61 0.68 0.73 0.77 0.81
0.00 0.00 0.00 0.00 0.00 0.00 0.00 7.00 4.65 2.42 1.93 1.46 1.02 0.81
24.90 0.00 7.82 5.20 2.33 1.98 1.01 7.00 4.65 2.42 1.93 1.46 1.02 0.81
Note: r1f, r2f and r3f are the triaxial strengths in the three principal directions respectively.
Surface under the action of σ 3
Surface under the action of σ 1
Surface under the action of σ 1 Surface under the action of σ 3
Surface under the action of σ 3
(a) prism-type failure after 20 ºC
(b) prism-type failure after 600 ºC
(c) tensile failure after 20 ºC
(d) tensile failure after 300 ºC
Fig. 4. Failure modes of plain HPC under multiaxial stress states at different temperatures.
It can be seen from Table 2 that the ultimate strength rT1f , rT3f of HPC under multiaxial tension–compression for all stress ratios are less than the corresponding uniaxial tensile and compressive strength for the same temperature, respectively. It is found that the tensile strength increases with the decrease in stress ratio for any given temperature, and reaches the greatest values under uniaxial tensile stress state. For example, for concrete at 600 °C, the
tensile strengths under biaxial and triaxial tension–compression for a = r1/r3 = 0.10 are 0.52, 0.46 MPa respectively, which are 52%, 46% of uniaxial tensile strength for the same endured temperature respectively, but for the stress ratio of a = r1/r3 = 0.50, the corresponding values are 0.99, 0.73 MPa and 99%, 73% respectively. In contrast, the absolute value of compressive strength drops with the decrease of stress ratio, and reaches the greatest value under
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1.2
1.2 HPC Trend curve
1
0.8
f tT / f t
f cT / f c
0.8 0.6
0.6
0.4
0.4
0.2
0.2
0
HPC Trend curve
1
0
200
400 T / °C
600
0
800
0
(a ) Under uniaxial compression
200
400 600 T / °C (b) Under uniaxial tension
800
Fig. 5. The influence of the different temperatures on the fcT =fc under uniaxial stress states.
0.8
0.8 20 ºC
0.7
20 ºC
0.7
200 ºC
200 ºC
300 ºC
300 ºC
0.6
400 ºC
σ 3Tf / f c
σ 3Tf / f c
0.6
500 ºC
0.5
600 ºC Failure criterion
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
-0.2
-0.4 σ 1 /σ
-0.6
-0.8
3
(a) Under biaxial tension-compression
400 ºC 500 ºC
0.5
0
600 ºC Failure criterion
0
-0.2
-0.4 σ
-0.6 /σ
1
-0.8
-1
-1.2
3
(b) Under triaxial tension-compression
Fig. 6. Comparison between test results and proposed failure criterion for HPC before and after high temperatures under multiaxial tension–compression.
uniaxial compressive stress state. This indicates that under biaxial and triaxial tension–compression, the tensile failure of concrete will take place at a greater stress ratio after high temperatures because the tensile stress is great. Fig. 6a and b demonstrate the influence of the stress ratios a = r1/r3 on the principal strength jrT3f j=fc of HPC prior to and after high temperatures under multiaxial tensile-compressive stress states, respectively. The decreasing percentage of the triaxial jrT3f j to uniaxial compressive strength depends on the stress states, the stress ratios, and the different temperatures. At the same temperatures and stress states, the change of jrT3f j depends on the stress ratio; in addition, the influence of the stress ratio on jrT3f j changes approximately by the exponential-like curve; with the increase in temperatures and stress ratios, especially when r1/ r3 = 0–0.2, the jrT3f j decreases quickly and inclines to the stress ratios axis, and their trend curves tends to be flat., as shown in Fig. 6. For example, when a = r1:r3 = 0.05, the values of rT1f , rT3f (2.07, 41.33 MPa and 1.94, 38.71 MPa) after 200 °C under biaxial and triaxial tension–compression decrease to 43.04%, 64.62% and 40.33%,60.52% of the uniaxial compressive strength (4.81, 63.96 MPa) at the corresponding temperature levels respectively; but, when a = r1:r3 = 0.75, they (4.57, 6.09 MPa and 3.94, 5.25 MPa) decrease to 95.01%, 9.52% and 81.91%, 8.21% of that, respectively. However, when a = r1:r3 = 0.05, the values of rT1f , rT3f (1.59, 31.80MP and 1.07, 21.48 MPa) after 400 °C under biaxial and triaxial tension–compression decrease to 53.18%, 63.98% and 35.79%, 43.22% of the uniaxial compressive strength (2.99, 49.7 MPa) at the corresponding temperatures respectively;
but, when a = r1:r3 = 0.75: 1, they (2.50, 3.33 MPa and 2.48, 3.30 MPa) decrease to 83.61%, 6.70% and 82.94%, 6.64% of that, respectively. As shown in Fig. 6a and b, the change trends under biaxial and triaxial tensile–compressive stress states are the similar; but, the decreasing percentage under triaxial tensile–compressive stress states is greater than that under biaxial ones; in addition, with the increase in temperatures and stress ratios, the decreasing percentage approaches gradually. For the same stress ratios, the peak stress values of rT1f , jrT2f j; jrT3f j in both principal compressive stress direction and principal tensile stress direction decrease with an increase in temperature. For example, when a = r1:r3 = 0.25:1, the values of rT1f , jrT3f j (0.58, 2.33 MPa) after 600 °C under biaxial tension–compression decrease to 11.42%, 3.87% of the uniaxial tensile and compressive strength (5.08, 60.16 MPa) prior to the high temperatures respectively; but, they (0.61, 2.42 MPa) under triaxial tension–compression decrease to 12.01%, 4.02% of that, respectively. When a = r1:r3 = 0.25:1, the values of rT1f , jrT3f j (4.28, 17.11 MPa) after 20 °C under biaxial tension–compression decrease to 84.25%, 28.44% of the uniaxial tensile and compressive strength (5.08,60.16 MPa), respectively; but, they (3.76, 15.05 MPa) under triaxial tension–compression decrease to 74.02%, 25.02% of that, respectively. This differs from the conclusions of the reference[16] that at a = 0.2 under biaxial tensile– compressive for ‘‘NSC” after the temperature of 600 °C, the value of rT1f is 0.74 MPa, namely, only 36% of that of normal temperature of 20 °C. While, at normal temperature and every stress ratios under multiaxial tensile–compressive stress states, the decreasing
Z.-j. He, Y.-p. Song / Construction and Building Materials 24 (2010) 498–504
percentage of |r3f/fc| in this paper is less than that of ‘‘NSC” in Refs. [6,7,17], etc. 3.4. Failure criterion Based on the strength characteristic of experimental results in Table 2 and the theoretical analysis of the failure envelope planes, the present paper proposes a new failure criterion. The tensile– compressive meridians are proposed as follows:
rT ð1Þ ¼ a c oct T fc sT rT The compressive meridians ð¼ 60 Þ : r Tc ¼ octc ð2Þ ¼ b c oct fcT fcT The tensile meridians ð¼ 0 Þ : r Tt ¼
r T ðhÞ ¼
sToctt fcT
4ððrTc Þ2 ðr Tt Þ2 Þ cos2 h þ ð2r Tt r Tc Þ2
rT1f þ rT2f þ rT3f
rToct ¼
sToct rT0 ¼
rToct fcT
;
; 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T r1f rT2f 2 þ rT2f rT3f 2 þ rT3f rT1f 2 ¼ 3
r Tt ¼
sToctt fcT
;
r Tc ¼
ð3Þ
sToctc
ð4Þ
fcT
Where a, b are the regression parameters on the tensile–compressive meridians; under equal tensile loading in the three direcsToctt ¼ sToctc ¼ 0), the tensile–compressive tions (namely, meridians(the failure envelope planes) and hydrostatic stress axis converge in one point; that is,
c¼
rTttt fcT
stress states after every high temperature level, the equations on the tensile meridians (Eq. (1)) are defined so that the values of the parameters a, c are obtained; namely, the parameter c is the values of the triaxial equal-tensile strengths (c ¼ rTttt =fcT ); and then the strength values of octahedral normal and shear stress for the triaxial equal-tensile strengths (c ¼ rTttt =fcT ) and uniaxial compressive strengths (h = 60°, fcT ) are regressed so that the equations on the compressive meridians (Eq. (2)) are determined so that the values of the parameter b are obtained. The calculated results in terms of the three parameters a, b, c are given in Table 3 after the different temperatures. Through repeated pilot calculation for strength values at different stress ratios of experimental results in Table 2, the equation of the deviatoric plane is given in the following form of Willam– Warnke [18].
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2rTc ððr Tc Þ2 ðr Tt Þ2 Þ cos h þ r Tc ð2rTt rTc Þ 4ððr Tc Þ2 ðr Tt Þ2 Þ cos2 h þ 5ðr Tt Þ2 4r Tt r Tc
where rToct is octahedral normal stress; sToctt and sToctc are the octahedral shear stress on tensile–compressive meridians for different temperatures, respectively. rToct , sToct , rT0 , rTt , rTc can be computed from:
¼
503
ð6Þ
where cT is deviatoric stress; moreover, the maximum cTc and minimum cTt occur at h = 60° and 0° respectively. When h = 0°, rT(0°) = r Tt ; h = 60°, rT (60°) =r Tc . When r Tt =rTc = 0.5 ? 1.0, the shape of the deviatoric plane change from triangle to ellipse enveloping curves so that it turns into the shape of a circle. These completely accord with the change rules of envelope curves in the deviatoric plane. Table 3 gives the regression coefficients from simple linear regression using the test data. Using regression analysis of the parameter values in Table 3 for plain HPC after the different temperatures, the following formula is obtained (20 °C 6 T 6 600 °C, and the correlation coefficient R):
9 8 T 2 T > a ¼ 0:0203 1000 0:0605 1000 þ 0:6503 ðR2 ¼ 0:9850Þ > > > = < T 2 T þ 1:1308 ðR2 ¼ 0:9994Þ b ¼ 0:2548 1000 þ 0:0760 1000 > > > > T 2 T ; : þ 0:0837 ðR2 ¼ 0:9999Þ c ¼ 0:0702 1000 0:0337 1000 ð7Þ
rTt gT
ð5Þ
fcT
where c is the ratios of the triaxial equal-tensile strengths to uniaxial compressive strength; namely, the values of crossing point coordinates between failure enveloping plane or tensile–compressive meridian and hydrostatic-pressure axis; rTt and rTttt represent strengths under uniaxial and triaxial-equal tension respectively; gT is proportionality coefficient between rTt and rTttt . In which a, b, c are the three parameters that need to be experimentally evaluated through the least square fit of the experimental results in Table 2 to Eqs. (1) and (2). According to the experimental results in Table 2, through linear regression on the values of octahedral normal and shear stress strengths(h = 0°) under uniaxial tensile and triaxial tensile–compressive–compressive
Table 3 The parameters values of failure criterion for plain HPC after different high temperatures. Temperature levels (°C)
a
b
c
20 200 300 400 500 600
0.6489 0.6388 0.6360 0.6276 0.6247 0.6219
1.1328 1.1542 1.1784 1.2019 1.2320 1.2681
0.0828 0.0751 0.0667 0.0589 0.0493 0.0384
Fig. 6 illustrates the comparison of the calculated results obtained from formula (1,2,6,7) and the test results. It can be seen from Fig. 6 that the model of failure envelopes in principal stress space for HPC under multiaxial tensile–compressive stress states after being subjected to high temperatures, is of better precision and applicability. Applicable conditions of the formula (1,2,6,7) are as follows: (1) The scope of the concrete class is C60. (2) The scope of stress states is under multiaxial tensile–compressive stress states. (3) The scope of high temperatures is between 20 °C (normal temperature) and 600 °C.
3.5. Discussion Concrete is a three-phase composite material at microscopic scale: a cementitious part, aggregate and the interfacial transition zone between the two. The micro-cracks will be formed due to drying and thermal shrinkage mismatch of aggregate particles and the cement part. The variation of temperature action can lead to the difference among the microstructures. The main reason for the change in the multiaxial tensile–compressive behavior of plain concrete after different temperatures was the formation and
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growth of micro-cracks during high temperatures. Micro-cracks mainly exist at cement paste-aggregate interfaces within concrete even prior to any load and environmental effects. The slight increase in HPC strength associated with a further increase in temperature (between 200 °C and 300 °C) is attributed to the general stiffening of the cement gel, or the increase in surface forces between gel particles, due to the removal of absorbed moisture. The temperature at which absorbed water is removed and the strength begins to increase depends on the porosity of the concrete. Above 400 °C, HPC loses their strength at a faster rate. At these temperatures, the dehydration of the cement paste results in its gradual disintegration so that it causes tension in the surrounding concrete. If the tensile stress exceeds the tensile strength of concrete, micro-cracks occur. The load carrying area will decrease with the initiation and growth of every new crack. On the other hand, considering the concrete specimen under multiaxial tensile–compressive stress states, the initiation and growth of every new crack will reduce the load carrying area. This reduction in load carrying area further causes an increase in the stress concentration at critical crack tips. So the tensile and compressive strength of plain concrete under compression with multiaxial stress decreased with the increase in temperature. This effect is evidently observed over the temperature 400 °C. When the temperature is between 400 and 600 °C, Ca(OH)2 decomposes; and above 600 °C, structural damage is influenced by CaCO3 dissociation. This phenomenon occurs because the greater the damage is, the lesser compression the elements carry. 4. Conclusions Based on the experimental work and the analysis of the test results, the following conclusions can be drawn: (1) No explosive spalling is observed during the high temperatures test on the 100 mm cubic specimens for HPC with temperatures ranged from 200 °C to 600 °C. (2) The effect of high temperatures on plain HPC does not change the failure modes. The failure modes of HPC prior to and after high temperatures under uniaxial compression are that of prism-type; but, they are tension failure under uniaxial tension, biaxial and triaxial tension–compression. The effect of confinement stress can change the failure modes. Under multiaxial tension–compression after high temperatures, sudden tensile failure appears for stress ratios less than or equal to a = r1/r3 = 0.10. In this case, the principal tensile stress is the key factor for the failure of concrete. (3) The ultimate strength r1f, r3f of HPC under multiaxial tension–compression for all stress ratios are less than the corresponding uniaxial tensile and compressive strength for the same temperature, respectively. The uniaxial compressive strength of plain HPC is not decreased after 200 and 300 °C. The temperature around 400 °C is critical to the ultimate strength that decreases rapidly. The decreasing per-
centage of the triaxial r3f to uniaxial compressive strength depends on the stress states, the stress ratios, and the different temperatures. (4) The formula of a new failure criterion with the temperature parameters under multiaxial tensile–compressive stress states for plain HPC is proposed. This study is helpful to reveal the multiaxial mechanical properties of HPC enduring high temperatures, and provides the experimental and theory foundations for fire-resistant structural design and for structural safety assessment after fire.
References [1] Husem Metin. The effects of high temperature on compressive and flexural strengths of ordinary and high-performance concrete. Fire Saf J 2006;41(2):155–63. [2] Husem Metin, Gozutok Serhat. The effects of low temperature curing on the compressive strength of ordinary and high performance concrete. Constr Build Mater 2005;19(1):49–53. [3] Li Min, Qian Chunxiang, Sun Wei. Mechanical properties of high-strength concrete after fire. Cem Concr Res 2004;34(6):1001–5. [4] Chan Sammy YN, Peng Gaifei, Chan John KW. Comparison between high strength concrete and normal strength concrete subjected to high temperature. Mater Struct/Matdriaux Construct 1996;29(12):616–9. [5] Peng Gaifei, Yang Wenwu, Zhao Jie, et al. Explosive spalling and residual mechanical properties of fiber-toughened high-performance concrete subjected to high temperatures. Cem Concr Res 2006;36(4):723–7. [6] Kupfer H. Behavior of concrete under biaxial stresses. ACI J 1969;66(8):656–66. [7] Lee Sangkeun, Song Youngchul, Han Sanghoon. Biaxial behavior of plain concrete of nuclear containment building. Nucl Eng Des 2004;227(2):143–53. [8] Mahboubi Ahmad, Ajorloo Ali. Experimental study of the mechanical behavior of plastic concrete in triaxial compression. Cem Concr Res 2005;5(2):412–9. [9] Lu Xiaobin, Thomas Hsu Cheng-Tzu. Behavior of high strength concrete with and without steel fiber reinforcement in triaxial compression. Cem Concr Res 2006;36(9):1679–85. [10] Luo X, Sun W, Chan YN. Residual compressive strength and microstructure of high performance concrete after exposure to high temperature. Mater Struct/ Materiaux Construct 2000;33(6):294–8. [11] The National Standards Compilation Group of the People’s Republic of China. GB175-1999 Portland cement and ordinary Portland cement. Beijing: China Standards Press; 1999 [in Chinese]. [12] The High-strength High-performance Concrete Committee of China Civil Engineering Society. Guide for structural design and construction of highstrength concrete. Beijing: China Architecture and Building Press; 2001 [in Chinese]. [13] The National Standards Compilation Group of the People’s Republic of China. GB/T 50081-2002 standard for test method of mechanical properties on ordinary concrete. Beijing: China Architecture and Building Press; 2003 [in Chinese]. [14] The National Standards Compilation Group of the People’s Republic of China. GBJ82-85 testing methods of long-term and long-lasting performance of ordinary concrete. Beijing: China Architecture and Building Press; 1985 [in Chinese]. [15] Qin Likun, Song Yupu, Zhang Zhong, et al. The research on strength and deformation of plain concrete under biaxial compression after high temperatures. J Dalian Univ Technol 2005;45(1):113–7 [in Chinese]. [16] Song Yupu, Zhang Zhong, Qin Likun, Yu Changjiang. Biaxial tensile– compressive experiment of concrete at high temperatures. Front Archit Civ Eng China 2007;1(1):94–8. [17] Tasuji ME, Nilson AH, Slate FO. Biaxial stress–strain relationships for concrete. Mag Concr Res 1979;31(109):217–24. [18] Willam KJ, Warnke EP. Constitutive models for the triaxial behavior of concrete. In: IABSE proceeding, Italy, vol. 19l; 1975. p. 1–30.
Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Multiaxial tensile–compressive strengths and failure criterion of plain high-performance concrete before and after high temperatures He Zhen-jun a,*, Song Yu-pu b a b
Department of Civil Engineering, Tsinghua University, Key Laboratory of Structural Engineering and Vibration of China Education Ministry, Beijing 100084, PR China State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, PR China
a r t i c l e
i n f o
Article history: Received 8 June 2009 Received in revised form 13 October 2009 Accepted 15 October 2009 Available online 14 November 2009 Keywords: High-performance concrete (HPC) High temperatures Stress ratios Multiaxial tensile–compressive strengths Failure criterion
a b s t r a c t Multiaxial tensile–compressive tests were performed on 100 mm 100 mm 100 mm cubic specimens of plain high-performance concrete (HPC) at all kinds of stress ratios after exposure to normal and high temperatures of 20, 200, 300, 400, 500, and 600 °C, using a large static–dynamic true triaxial machine. Friction-reducing pads were three layers of plastic membrane with glycerine in-between for the compressive loading plane; the tensile loading planes of concrete samples were processed by attrition machine, and then the samples were glued-up with the loading plate with structural glue. The failure mode characteristic of specimens and the direction of the crack were observed and described, respectively. The three principally static strengths in the corresponding stress state were measured. The influence of the temperatures, stress ratios, and stress states on the triaxial strengths of HPC after exposure to high temperatures were also analyzed respectively. The experimental results showed that the uniaxial compressive strength of plain HPC after exposure to high temperatures does not decrease completely with the increase in temperature, the ratios of the triaxial to its uniaxial compressive strength depend on brittleness–stiffness of HPC after different high temperatures besides the stress states and stress ratios. On this basis, the formula of a new failure criterion with the temperature parameters under multiaxial tensile–compressive stress states for plain HPC is proposed. This study is helpful to reveal the multiaxial mechanical properties of HPC structure enduring high temperatures, and provides the experimental and theory foundations (testing data and correlated formula) for fire-resistant structural design, and for structural safety assessment and maintenance after fire. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Concrete has been the leading construction material for nearly a century. In recent years, high-performance concrete (HPC) is becoming an attractive alternative to traditional normal strength concrete (NSC). The alleged HPC is generally defined as highstrength, high fluidity, high durability concrete or to possess one of these performances; moreover, high-performance water reducer and superfine mineral admixtures are absolutely necessary ingredients. The dense microstructure of high-strength concrete (HSC) ensures high strength and very low permeability, so that it has a relatively low weak deformation capability or much higher ‘‘brittleness-stiffness” when compared to NSC. Recent fire test results show that there is a great difference between the properties of HSC and NSC after exposure to high temperatures [1–4]. It was observed that HSC is susceptible to spalling, or even explosive spalling when subjected to rapid temperature rise as in the case of a fire [5]. Hence, one of the main concerns for the usage of HSC in fire * Corresponding author. Tel.: +86 010 13811604337; fax: +86 010 62792450. E-mail address: [email protected] (Z.-j. He). 0950-0618/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2009.10.012
resistance applications is its performance under fire conditions. It is well known that the conventional analysis and design methods of reinforced concrete are generally still based upon material properties obtained from the basic uniaxial strength test, although we know that the true uniaxial condition in structures is extremely rare. In practice, many concrete structures such as shear wall, spiral columns and the node of building, nuclear reactor pressure container, etc are under multiaxial stress states. At the same time, with the wide use of computers, the finite element method and HPC in concrete structures, it has become increasingly evident that it is quite important and urgent for the experimental study on the mechanical property of HPC under multiaxial stress states, and for studies on the design and analysis of nonlinear behavior design of reinforced concrete structure based upon multiaxial mechanical behavior. So, many researchers have much paid attention to HPC at home and abroad. Late in the 1970s, some researchers [6] began to perform experimental study of ‘‘NSC” under multiaxial stress states in order to design for nuclear reactor container and so on. But, until now, most studies [2,7] have been carried out to characterize the mechanical behavior of HSC or HPC under uniaxial stress state, or NSC under multiaxial stress states. However, literature on the
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mechanical behavior of HSC or HPC under multiaxial loadings is extremely scarce; moreover, among them, the most tests of confined compression (with two equal stresses) behavior of concrete have nearly been performed under triaxial stress state using cylindrical specimens subjected to hydrostatic pressure in a triaxial cell and axial loading [8,9]. Lu and Thomas [9] studied the mechanical behavior of HSC and steel fiber reinforced high-strength concrete under uniaxial and triaxial compression, when testing U100 mm 200 mm and U100 mm 150 mm cylinder specimens lubricated with a combination of two 0.125 mm thick teflon sheets on top of one layer of 0.015 mm thick aluminum foil. The previous investigations conducted on the effect of high temperatures on properties of plain HSC or HPC have also merely focused on the behavior of strength and deformation under uniaxial loading. However, literature on the mechanical properties of plain HSC or HPC under multiaxial stress states after high temperatures, hasn’t been documented. Luo et al. [10] stated that the strength of the HPC degenerates more severely and has higher residual strength than that of NSC after exposure to high temperatures. Chan et al. [4] stated that the range between 400 °C and 800 °C is critical to the strength loss of concrete causing a large percentage of loss of strength. This paper presents the strength regularity and failure criterion of HPC at all kinds of stress ratios under multiaxial tensile–compressive stress states after exposure to six temperatures, using a large static–dynamic true triaxial machine. The multiaxial tensile–compressive tests were performed on 100 mm 100 mm 100 mm cubic concrete specimens. This study is helpful to reveal the multiaxial mechanical properties of HPC structure enduring high temperatures, and provides the experimental and theory foundations (testing data and correlated formula) for fire-resistant structural design, and for structural safety assessment and maintenance after fire.
The concrete specimens tested were in 100 mm 100 mm 100 mm, 150 mm 150 mm 150 mm or 150 mm 150 mm 300 mm. The 100 mm concrete cubes were used to measure the strength for multiaxial tests. For determination of the strength grade for the HPC and the strength of the prism, each batch was cast in six 150 mm cubic specimens, and six 150 mm 150 mm 300 mm prismatic specimens.
2.2.2. Apparatus and testing methods The tests of high temperatures and multiaxial mechanical properties were performed in State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The high temperatures testing apparatus and the multifunction triaxial experimental machine are shown in Figs. 1 and 2, respectively. The specimens in high temperatures test were 100 mm cubes. For each of stress ratio, six specimens were heated up at least. These 100 mm cubic specimens were elevated to the peak temperatures of 200, 300, 400, 500, and 600 °C at a heating rate of 10 °C/min, respectively. After the peak temperature was reached, it was maintained for 6 h; the cooling time in electric furnace was about 1 h; then, the specimens were took out and cooled naturally to room temperature. These specimens were performed in a multiaxial testing machine after 24 h. In this study, the surface
2. Materials and experimental procedures 2.1. Materials and mix proportions Fig. 1. The box-type electric furnace.
The cementitious materials used for this investigation were Chinese standard PI52.5R Portland cement (standard compressive strength of higher than 52.5 MPa at the age of 28 days) and one-level fly ash [11,12]. The coarse aggregate was crushed stone (diameter ranging from 5 mm to 20 mm) and the fine aggregate was natural river sand (fineness modulus of 2.7); water was tap-water. Table 1 shows the mix proportions by weight of the mixture and the major parameters of HPC (fc is the uniaxial compressive strength of 100 mm 100 mm 100 mm cubic HPC specimens with friction-reducing pads prior to high temperatures, and its strength value is about equal to that of 150 mm 150 mm 300 mm prism. fc shows average strength). 2.2. Samples and testing methods 2.2.1. Casting and curing of specimens The coarse aggregate and fine aggregate were mixed for about 1 minute when a portion of the proportional water was being sprayed on them, and then the cement and one-level fly ash were added in turn; after that, the residual proportional water with SiKa(R)NF-III superplasticizer (There are also the properties of increasing slump, durability and retardation besides ordinary water reducing agents.) was added slowly over a period of 1 min. Finally these ingredients were mixed for 2–3 min [12,13]. All specimens cast in steel molds and compacted slightly by vibrating table, were demoulded after 24 h of the casting and then cured in a condition of 20 ± 3 °C and 95% RH (relative humidity) for 28 days according to ‘‘GBJ82-85 The test method of long-term and durability on ordinary concrete [14]”, and then stored at a natural temperature outside. The age of the specimens tested was about a year.
Fig. 2. The triaxial testing machine.
Table 1 Mix proportions and major parameters of HPC. Concrete types
Water– cementitious ratio
Water (kg/m3)
Cement (kg/m3)
Fly ash (kg/m3)
Fine aggregate (kg/m3)
Coarse aggregate (kg/m3)
Superplasticizer (kg/m3)
Slump (cm)
Compressive strength (fC/MPa)
HPC
0.31
175
470
94.52
615.77
1094.71
6.77
24
60.16
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Z.-j. He, Y.-p. Song / Construction and Building Materials 24 (2010) 498–504
σ1
σ3
σ2
σ1
σ3
σ2
σ3
σ3
σ1
σ1
(a) Triaxial tensile-compressive loading direction
(b) Biaxial tensile-compressive loading direction
Spherichinge Antifriction pad s Specimen
Up-down loading platens
Specimen
(c) Uniaxial compression
Up-down left-right loading platens (d) Biaxial tension-compression
Fig. 3. States of the specimen in triaxial testing machine.
of all specimens was dried before being exposed to high temperatures. No explosive spalling was observed during the high temperature tests on HPC specimens with temperatures ranged from 200 °C to 600 °C. The triaxial tests were conducted in a triaxial testing machine that is capable of developing three independent compressive or tensile forces. The triaxial tests process was necessary to insure uniform dimensions for each cubic specimen. The major stress direction was always applied perpendicularly to the surface of specimens. The proportional loading mode was employed. Friction-reducing pads placed between the platens and the specimens were three layers of plastic membrane with glycerine in-between for the compressive loading plane; the tensile loading plane of concrete sample was processed by attrition machine, then, the samples were glued-up with the loading plate with structural glue. The type of structural glue is in JGN-II. Its value of the tensile, compressive and shear strength were 33 MPa, 62 MPa, 18Mpa respectively. The specimens under uniaxial compression were tested at a loading speed of 0.3–0.5 MPa/s in the direction of r3; but, those under uniaxial tensile, biaxial and triaxial tensile–compressive stress states were done at 0.03–0.05 MPa/s in the direction of r1. All kinds of stress ratios under multiaxial tensile–compressive loading after exposure to normal and high temperatures of 20, 200, 300, 400, 500, and 600 °C were tested. The principal stresses are expressed as r1 P r2 P r3 (compression denoted as negative and tension denoted as positive); three specimens were at least tested for each of stress ratio and the test values were averaged after the discrete ones were deleted. Fig. 3 shows the loading directions and the states of the specimen under multiaxial tensile–compressive stress states in triaxial experimental machine respectively.
3. Test results and discussion 3.1. Experimental results The experimental results of plain HPC under multiaxial tensile– compressive stress states after high temperatures are given in Table 2. 3.2. Failure modes The surface deterioration and the failure modes of the HPC specimens under multiaxial stress states subjected to the action of high temperatures are shown in Fig. 4a–d. From Fig. 4, it is obvious that the influence of high temperatures on HPC does not change the tensile splitting mode from occurring. In Fig. 4a and b, the HPC specimens under uniaxial compressive loading are split to multiple minor prisms (prism-type failure
modes). The tensile failures under biaxial and triaxial tensile–compressive loading are shown in Fig. 4c and d, respectively. Under multiaxial tension–compression after high temperature, sudden tensile failure appears for stress ratios less than or equal to a = r1/r3 = 0.10. In this case, the principal tensile stress is the key factor for the failure of concrete. There was no connection between the direction of the cracking and stress ratios. It was noticed that the cracks on the loading surface have a random direction because of the influence of coarse aggregates. The failure modes mentioned above demonstrates that providing confinement stress along r1, r2 and r3 directions will change the failure modes. Although the failure modes under multiaxial tensile–compressive stress states are different, the failure cause is that the splitting tensile strain along the unloading or less stress planes is greater than the ultimate tensile strain of HPC. 3.3. Strength characteristics As shown in Table 2 and Fig. 5, it is apparent that after 200 °C and 300 °C, the values of fcT =fc are greater than 1 at the stress ratio of a = 0; namely, the uniaxial compression strength fcT after 200 °C and 300 °C is increased, which differs from the conclusions drawn by the same research group for NSC [15]; but, when the suffered temperature is round 400 °C, it decreases gradually with the increase in temperature. As can be seen in Table 2 and Fig. 5a and b, the uniaxial compressive strengths (63.96, 61.64, 49.7, 35.72, 24.9 MPa) after being exposed to high temperatures of 200, 300, 400, 500, and 600 °C are 1.06, 1.02, 0.83, 0.59, 0.41 times the original strength at 20 °C (60.16 MPa), respectively; but, the uniaxial tensile strengths (4.81, 4.14, 2.99, 1.82, 1.00 MPa) are 0.95, 0.82, 0.59, 0.36, 0.20 times it (5.08 MPa) respectively. The discussion above indicates that in comparison with the other temperatures, the uniaxial compression strength fcT after 200 °C and 300 °C is increased; but, the uniaxial tensile strength ftT is gradually decreased with the increase in temperature. So, the temperatures around 400 °C are critical to the ultimate strength (fc, fcT is the uniaxial compression strength with friction-reducing pads at normal temperature and after different temperature levels, respectively).
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Z.-j. He, Y.-p. Song / Construction and Building Materials 24 (2010) 498–504 Table 2 The triaxial strength index of plain HPC under various stress states and stress ratios after high temperatures. Temperature (°C)
Stress ratio a = r1:r2:r3
r1f (MPa)
r2f (MPa)
r3f (MPa)
Temperature (°C)
Stress ratio a = r1:r2:r3
r1f (MPa)
r2f (MPa)
r3f (MPa)
20
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 5.08 2.20 3.23 4.28 4.73 4.86 2.23 3.02 3.76 4.11 4.19 4.22 4.23
0.00 0.00 0.00 0.00 0.00 0.00 0.00 44.64 30.18 15.05 11.73 8.37 5.63 4.23
60.16 0.00 43.92 32.26 17.11 9.45 6.48 44.64 30.18 15.05 11.73 8.37 5.63 4.23
400
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 2.99 1.59 2.08 2.19 2.26 2.50 1.07 1.59 1.84 2.09 2.22 2.48 2.52
0.00 0.00 0.00 0.00 0.00 0.00 0.00 21.48 15.91 7.34 5.98 4.44 3.30 2.52
49.70 0.00 31.80 20.79 8.75 4.52 3.33 21.48 15.91 7.34 5.98 4.44 3.30 2.52
200
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 4.81 2.07 2.98 3.70 4.27 4.57 1.94 2.75 3.53 3.55 3.98 3.94 4.19
0.00 0.00 0.00 0.00 0.00 0.00 0.00 38.71 27.52 14.12 10.15 7.96 5.25 4.19
63.96 0.00 41.33 29.75 14.81 8.54 6.09 38.71 27.52 14.12 10.15 7.96 5.25 4.19
500
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 1.82 0.72 1.05 1.26 1.33 1.02 0.66 0.90 1.06 1.17 1.43 1.49 1.52
0.00 0.00 0.00 0.00 0.00 0.00 0.00 13.11 9.04 4.22 3.34 2.86 1.98 1.52
35.72 0.00 14.30 10.49 5.04 2.65 1.36 13.11 9.04 4.22 3.34 2.86 1.98 1.52
300
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 4.14 1.86 2.34 3.11 3.48 3.81 1.68 2.22 2.70 3.15 3.42 3.56 3.47
0.00 0.00 0.00 0.00 0.00 0.00 0.00 33.51 22.22 10.80 9.00 6.85 4.74 3.47
61.64 0.00 37.22 23.41 12.43 6.97 5.08 33.51 22.22 10.80 9.00 6.85 4.74 3.47
600
0.00:0.00:1 1:0.00:0.00 0.05:0.00:1 0.10:0.00:1 0.25:0.00:1 0.50:0.00:1 0.75:0.00:1 0.05:1:1 0.10:1:1 0.25:1:1 0.35:1:1 0.50:1:1 0.75:1:1 1:1:1
0.00 1.00 0.39 0.52 0.58 0.99 0.76 0.35 0.46 0.61 0.68 0.73 0.77 0.81
0.00 0.00 0.00 0.00 0.00 0.00 0.00 7.00 4.65 2.42 1.93 1.46 1.02 0.81
24.90 0.00 7.82 5.20 2.33 1.98 1.01 7.00 4.65 2.42 1.93 1.46 1.02 0.81
Note: r1f, r2f and r3f are the triaxial strengths in the three principal directions respectively.
Surface under the action of σ 3
Surface under the action of σ 1
Surface under the action of σ 1 Surface under the action of σ 3
Surface under the action of σ 3
(a) prism-type failure after 20 ºC
(b) prism-type failure after 600 ºC
(c) tensile failure after 20 ºC
(d) tensile failure after 300 ºC
Fig. 4. Failure modes of plain HPC under multiaxial stress states at different temperatures.
It can be seen from Table 2 that the ultimate strength rT1f , rT3f of HPC under multiaxial tension–compression for all stress ratios are less than the corresponding uniaxial tensile and compressive strength for the same temperature, respectively. It is found that the tensile strength increases with the decrease in stress ratio for any given temperature, and reaches the greatest values under uniaxial tensile stress state. For example, for concrete at 600 °C, the
tensile strengths under biaxial and triaxial tension–compression for a = r1/r3 = 0.10 are 0.52, 0.46 MPa respectively, which are 52%, 46% of uniaxial tensile strength for the same endured temperature respectively, but for the stress ratio of a = r1/r3 = 0.50, the corresponding values are 0.99, 0.73 MPa and 99%, 73% respectively. In contrast, the absolute value of compressive strength drops with the decrease of stress ratio, and reaches the greatest value under
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Z.-j. He, Y.-p. Song / Construction and Building Materials 24 (2010) 498–504
1.2
1.2 HPC Trend curve
1
0.8
f tT / f t
f cT / f c
0.8 0.6
0.6
0.4
0.4
0.2
0.2
0
HPC Trend curve
1
0
200
400 T / °C
600
0
800
0
(a ) Under uniaxial compression
200
400 600 T / °C (b) Under uniaxial tension
800
Fig. 5. The influence of the different temperatures on the fcT =fc under uniaxial stress states.
0.8
0.8 20 ºC
0.7
20 ºC
0.7
200 ºC
200 ºC
300 ºC
300 ºC
0.6
400 ºC
σ 3Tf / f c
σ 3Tf / f c
0.6
500 ºC
0.5
600 ºC Failure criterion
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
-0.2
-0.4 σ 1 /σ
-0.6
-0.8
3
(a) Under biaxial tension-compression
400 ºC 500 ºC
0.5
0
600 ºC Failure criterion
0
-0.2
-0.4 σ
-0.6 /σ
1
-0.8
-1
-1.2
3
(b) Under triaxial tension-compression
Fig. 6. Comparison between test results and proposed failure criterion for HPC before and after high temperatures under multiaxial tension–compression.
uniaxial compressive stress state. This indicates that under biaxial and triaxial tension–compression, the tensile failure of concrete will take place at a greater stress ratio after high temperatures because the tensile stress is great. Fig. 6a and b demonstrate the influence of the stress ratios a = r1/r3 on the principal strength jrT3f j=fc of HPC prior to and after high temperatures under multiaxial tensile-compressive stress states, respectively. The decreasing percentage of the triaxial jrT3f j to uniaxial compressive strength depends on the stress states, the stress ratios, and the different temperatures. At the same temperatures and stress states, the change of jrT3f j depends on the stress ratio; in addition, the influence of the stress ratio on jrT3f j changes approximately by the exponential-like curve; with the increase in temperatures and stress ratios, especially when r1/ r3 = 0–0.2, the jrT3f j decreases quickly and inclines to the stress ratios axis, and their trend curves tends to be flat., as shown in Fig. 6. For example, when a = r1:r3 = 0.05, the values of rT1f , rT3f (2.07, 41.33 MPa and 1.94, 38.71 MPa) after 200 °C under biaxial and triaxial tension–compression decrease to 43.04%, 64.62% and 40.33%,60.52% of the uniaxial compressive strength (4.81, 63.96 MPa) at the corresponding temperature levels respectively; but, when a = r1:r3 = 0.75, they (4.57, 6.09 MPa and 3.94, 5.25 MPa) decrease to 95.01%, 9.52% and 81.91%, 8.21% of that, respectively. However, when a = r1:r3 = 0.05, the values of rT1f , rT3f (1.59, 31.80MP and 1.07, 21.48 MPa) after 400 °C under biaxial and triaxial tension–compression decrease to 53.18%, 63.98% and 35.79%, 43.22% of the uniaxial compressive strength (2.99, 49.7 MPa) at the corresponding temperatures respectively;
but, when a = r1:r3 = 0.75: 1, they (2.50, 3.33 MPa and 2.48, 3.30 MPa) decrease to 83.61%, 6.70% and 82.94%, 6.64% of that, respectively. As shown in Fig. 6a and b, the change trends under biaxial and triaxial tensile–compressive stress states are the similar; but, the decreasing percentage under triaxial tensile–compressive stress states is greater than that under biaxial ones; in addition, with the increase in temperatures and stress ratios, the decreasing percentage approaches gradually. For the same stress ratios, the peak stress values of rT1f , jrT2f j; jrT3f j in both principal compressive stress direction and principal tensile stress direction decrease with an increase in temperature. For example, when a = r1:r3 = 0.25:1, the values of rT1f , jrT3f j (0.58, 2.33 MPa) after 600 °C under biaxial tension–compression decrease to 11.42%, 3.87% of the uniaxial tensile and compressive strength (5.08, 60.16 MPa) prior to the high temperatures respectively; but, they (0.61, 2.42 MPa) under triaxial tension–compression decrease to 12.01%, 4.02% of that, respectively. When a = r1:r3 = 0.25:1, the values of rT1f , jrT3f j (4.28, 17.11 MPa) after 20 °C under biaxial tension–compression decrease to 84.25%, 28.44% of the uniaxial tensile and compressive strength (5.08,60.16 MPa), respectively; but, they (3.76, 15.05 MPa) under triaxial tension–compression decrease to 74.02%, 25.02% of that, respectively. This differs from the conclusions of the reference[16] that at a = 0.2 under biaxial tensile– compressive for ‘‘NSC” after the temperature of 600 °C, the value of rT1f is 0.74 MPa, namely, only 36% of that of normal temperature of 20 °C. While, at normal temperature and every stress ratios under multiaxial tensile–compressive stress states, the decreasing
Z.-j. He, Y.-p. Song / Construction and Building Materials 24 (2010) 498–504
percentage of |r3f/fc| in this paper is less than that of ‘‘NSC” in Refs. [6,7,17], etc. 3.4. Failure criterion Based on the strength characteristic of experimental results in Table 2 and the theoretical analysis of the failure envelope planes, the present paper proposes a new failure criterion. The tensile– compressive meridians are proposed as follows:
rT ð1Þ ¼ a c oct T fc sT rT The compressive meridians ð¼ 60 Þ : r Tc ¼ octc ð2Þ ¼ b c oct fcT fcT The tensile meridians ð¼ 0 Þ : r Tt ¼
r T ðhÞ ¼
sToctt fcT
4ððrTc Þ2 ðr Tt Þ2 Þ cos2 h þ ð2r Tt r Tc Þ2
rT1f þ rT2f þ rT3f
rToct ¼
sToct rT0 ¼
rToct fcT
;
; 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T r1f rT2f 2 þ rT2f rT3f 2 þ rT3f rT1f 2 ¼ 3
r Tt ¼
sToctt fcT
;
r Tc ¼
ð3Þ
sToctc
ð4Þ
fcT
Where a, b are the regression parameters on the tensile–compressive meridians; under equal tensile loading in the three direcsToctt ¼ sToctc ¼ 0), the tensile–compressive tions (namely, meridians(the failure envelope planes) and hydrostatic stress axis converge in one point; that is,
c¼
rTttt fcT
stress states after every high temperature level, the equations on the tensile meridians (Eq. (1)) are defined so that the values of the parameters a, c are obtained; namely, the parameter c is the values of the triaxial equal-tensile strengths (c ¼ rTttt =fcT ); and then the strength values of octahedral normal and shear stress for the triaxial equal-tensile strengths (c ¼ rTttt =fcT ) and uniaxial compressive strengths (h = 60°, fcT ) are regressed so that the equations on the compressive meridians (Eq. (2)) are determined so that the values of the parameter b are obtained. The calculated results in terms of the three parameters a, b, c are given in Table 3 after the different temperatures. Through repeated pilot calculation for strength values at different stress ratios of experimental results in Table 2, the equation of the deviatoric plane is given in the following form of Willam– Warnke [18].
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2rTc ððr Tc Þ2 ðr Tt Þ2 Þ cos h þ r Tc ð2rTt rTc Þ 4ððr Tc Þ2 ðr Tt Þ2 Þ cos2 h þ 5ðr Tt Þ2 4r Tt r Tc
where rToct is octahedral normal stress; sToctt and sToctc are the octahedral shear stress on tensile–compressive meridians for different temperatures, respectively. rToct , sToct , rT0 , rTt , rTc can be computed from:
¼
503
ð6Þ
where cT is deviatoric stress; moreover, the maximum cTc and minimum cTt occur at h = 60° and 0° respectively. When h = 0°, rT(0°) = r Tt ; h = 60°, rT (60°) =r Tc . When r Tt =rTc = 0.5 ? 1.0, the shape of the deviatoric plane change from triangle to ellipse enveloping curves so that it turns into the shape of a circle. These completely accord with the change rules of envelope curves in the deviatoric plane. Table 3 gives the regression coefficients from simple linear regression using the test data. Using regression analysis of the parameter values in Table 3 for plain HPC after the different temperatures, the following formula is obtained (20 °C 6 T 6 600 °C, and the correlation coefficient R):
9 8 T 2 T > a ¼ 0:0203 1000 0:0605 1000 þ 0:6503 ðR2 ¼ 0:9850Þ > > > = < T 2 T þ 1:1308 ðR2 ¼ 0:9994Þ b ¼ 0:2548 1000 þ 0:0760 1000 > > > > T 2 T ; : þ 0:0837 ðR2 ¼ 0:9999Þ c ¼ 0:0702 1000 0:0337 1000 ð7Þ
rTt gT
ð5Þ
fcT
where c is the ratios of the triaxial equal-tensile strengths to uniaxial compressive strength; namely, the values of crossing point coordinates between failure enveloping plane or tensile–compressive meridian and hydrostatic-pressure axis; rTt and rTttt represent strengths under uniaxial and triaxial-equal tension respectively; gT is proportionality coefficient between rTt and rTttt . In which a, b, c are the three parameters that need to be experimentally evaluated through the least square fit of the experimental results in Table 2 to Eqs. (1) and (2). According to the experimental results in Table 2, through linear regression on the values of octahedral normal and shear stress strengths(h = 0°) under uniaxial tensile and triaxial tensile–compressive–compressive
Table 3 The parameters values of failure criterion for plain HPC after different high temperatures. Temperature levels (°C)
a
b
c
20 200 300 400 500 600
0.6489 0.6388 0.6360 0.6276 0.6247 0.6219
1.1328 1.1542 1.1784 1.2019 1.2320 1.2681
0.0828 0.0751 0.0667 0.0589 0.0493 0.0384
Fig. 6 illustrates the comparison of the calculated results obtained from formula (1,2,6,7) and the test results. It can be seen from Fig. 6 that the model of failure envelopes in principal stress space for HPC under multiaxial tensile–compressive stress states after being subjected to high temperatures, is of better precision and applicability. Applicable conditions of the formula (1,2,6,7) are as follows: (1) The scope of the concrete class is C60. (2) The scope of stress states is under multiaxial tensile–compressive stress states. (3) The scope of high temperatures is between 20 °C (normal temperature) and 600 °C.
3.5. Discussion Concrete is a three-phase composite material at microscopic scale: a cementitious part, aggregate and the interfacial transition zone between the two. The micro-cracks will be formed due to drying and thermal shrinkage mismatch of aggregate particles and the cement part. The variation of temperature action can lead to the difference among the microstructures. The main reason for the change in the multiaxial tensile–compressive behavior of plain concrete after different temperatures was the formation and
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Z.-j. He, Y.-p. Song / Construction and Building Materials 24 (2010) 498–504
growth of micro-cracks during high temperatures. Micro-cracks mainly exist at cement paste-aggregate interfaces within concrete even prior to any load and environmental effects. The slight increase in HPC strength associated with a further increase in temperature (between 200 °C and 300 °C) is attributed to the general stiffening of the cement gel, or the increase in surface forces between gel particles, due to the removal of absorbed moisture. The temperature at which absorbed water is removed and the strength begins to increase depends on the porosity of the concrete. Above 400 °C, HPC loses their strength at a faster rate. At these temperatures, the dehydration of the cement paste results in its gradual disintegration so that it causes tension in the surrounding concrete. If the tensile stress exceeds the tensile strength of concrete, micro-cracks occur. The load carrying area will decrease with the initiation and growth of every new crack. On the other hand, considering the concrete specimen under multiaxial tensile–compressive stress states, the initiation and growth of every new crack will reduce the load carrying area. This reduction in load carrying area further causes an increase in the stress concentration at critical crack tips. So the tensile and compressive strength of plain concrete under compression with multiaxial stress decreased with the increase in temperature. This effect is evidently observed over the temperature 400 °C. When the temperature is between 400 and 600 °C, Ca(OH)2 decomposes; and above 600 °C, structural damage is influenced by CaCO3 dissociation. This phenomenon occurs because the greater the damage is, the lesser compression the elements carry. 4. Conclusions Based on the experimental work and the analysis of the test results, the following conclusions can be drawn: (1) No explosive spalling is observed during the high temperatures test on the 100 mm cubic specimens for HPC with temperatures ranged from 200 °C to 600 °C. (2) The effect of high temperatures on plain HPC does not change the failure modes. The failure modes of HPC prior to and after high temperatures under uniaxial compression are that of prism-type; but, they are tension failure under uniaxial tension, biaxial and triaxial tension–compression. The effect of confinement stress can change the failure modes. Under multiaxial tension–compression after high temperatures, sudden tensile failure appears for stress ratios less than or equal to a = r1/r3 = 0.10. In this case, the principal tensile stress is the key factor for the failure of concrete. (3) The ultimate strength r1f, r3f of HPC under multiaxial tension–compression for all stress ratios are less than the corresponding uniaxial tensile and compressive strength for the same temperature, respectively. The uniaxial compressive strength of plain HPC is not decreased after 200 and 300 °C. The temperature around 400 °C is critical to the ultimate strength that decreases rapidly. The decreasing per-
centage of the triaxial r3f to uniaxial compressive strength depends on the stress states, the stress ratios, and the different temperatures. (4) The formula of a new failure criterion with the temperature parameters under multiaxial tensile–compressive stress states for plain HPC is proposed. This study is helpful to reveal the multiaxial mechanical properties of HPC enduring high temperatures, and provides the experimental and theory foundations for fire-resistant structural design and for structural safety assessment after fire.
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