- Email: [email protected]
New failure criterion models for concrete under multiaxial stress in compression
Construction and Building Materials 161 (2018) 432–441
Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
New failure criterion models for concrete under multiaxial stress in compression Chong Rong a, Qingxuan Shi a,⇑, Ting Zhang a, Hongchao Zhao b a b
School of Civil Engineering, Xi’an University of Architecture and Technology, Shaanxi, Xi’an 710055, China School of Geology and Mining Engineering, Xinjiang University, Urumqi 830000, China
h i g h l i g h t s The main factors that contribute to the failure under multiaxial stress are studied. The hydrostatic stress and shear stress are respectively as the main influencing factors to establish equations. The accuracy of each failure criterion model is verified. The applicability of each failure criterion model is discussed.
a r t i c l e
i n f o
Article history: Received 13 January 2017 Received in revised form 12 November 2017 Accepted 19 November 2017
Keywords: Biaxial and triaxial compression tests Twin shear strength theory Multiaxial stress Failure criterion models Boundary conditions
a b s t r a c t Biaxial and triaxial compression tests were performed on 100 100 100 mm cubic specimens of concrete under different stress loading rates. All the tests were performed using a true triaxial testing machine. The analysis of the data revealed that the intermediate principal stress, hydrostatic stress and shear stress are the main factors influencing the failure criterion for concrete under multiaxial stress. Based on the twin shear strength theory, three failure criterion models were developed. The fiveparameter model A considers the shear strength as the main factor, the five-parameter model B considers hydrostatic strength as the main factor, and the six-parameter model considers both the shear strength and hydrostatic strength. The parameters for these failure criterion models have clear physical significance and form the failure criterion in a theoretical analysis. Models A and B apply to different stress states and show similar results. The six-parameter model can apply to most stress states, but the computed results depend on the boundary conditions. This convenient model can also be extended for nonlinear analyses of concrete under multiaxial stress in compression. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Abbreviations: r1, the maximum principal stress; r1t, the maximum principal stress of each triaxial compressive ratio; r2, the intermediate principal stress; r2b, the intermediate principal stress of each biaxial compressive ratio; r2t, the intermediate principal stress of each triaxial compressive ratio; r3, the minimum principal stress; r3b, the minimum principal stress of each biaxial compressive ratio; r3t, the minimum principal stress of each triaxial compressive ratio; rm, the hydrostatic stress; roct, the octahedral principle stress; sij, the principle shear stress (i–j); sm, the shear stress corresponding to hydrostatic stress; soct, the octahedral shear stress; n, the height in cylindrical coordinate system; q, the radius in cylindrical coordinate system; u, the angle in cylindrical coordinate system; a, the tension–compression ratio; abc, the biaxial-axial compression ratio; acc, the triaxialaxial compression ratio; fbc, the biaxial compress strength; fcc, the triaxial compress strength; ft, the tense strength; fttt, the triaxial tense strength; I1, the first invariants of stress deviator; J2, the second invariants of stress deviator; J3, the third invariants of stress deviator. ⇑ Corresponding author. E-mail address: [email protected] (Q. Shi). https://doi.org/10.1016/j.conbuildmat.2017.11.106 0950-0618/Ó 2017 Elsevier Ltd. All rights reserved.
Concrete composite structures are a widely used structural form and are applied in the construction of high-rise buildings, arch dams, bridges, and other structures. In a composite structure, concrete is the most complicated component. Many researchers [1–6] have studied the behaviour of concrete under multiaxial stress. They found that the compressive strength of the concrete increases with increasing lateral compressive stress. In these previous studies, the strength of concrete under multiaxial stress is related to its uniaxial strength and lateral compressive strength, but the influences of these two strengths are different [7]. This difference is caused by several factors, for example, the strength [8] and material characteristics of the concrete [9,10], initial stress [5,11], and specimen size [12]. Many researchers have studied concrete using different methods and found that the change in the
433
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
behaviour of a composite material is systematic, indicating that different micro-characteristics can be expressed by a similar macroscopic pattern. In the micromechanical research of concrete [13–16], the basic characteristics of concrete under multiaxial stress can be formulated in terms of six mechanical effects: the tensile-compressive strength difference, the hydrostatic stress (rm), the shear stress (variation in the minimum stress), the normal stress, the intermediate principal stress (r2), and the variation in the intermediate principal stress. In concrete applications, the stress states are complex, and designing complex structures based on only the normal failure mode of concrete is unsafe. A failure criterion can be described by a mathematical function model to express the damage envelope surface. To quantify the mechanical properties of concrete under complex stress, many failure criteria have been proposed for a variety of concretes and stress states [17–23], such as the Mohr-Coulomb criterion, the Drucker-Prager criterion, the Willam-Warnker criterion, the Ottosen criterion, Asteris and Plevris and criterion, and the HsiehTing-Chen criterion. The original equations of tensile and compressive meridians in each failure criterion employ different stress parameters, such as the principal stress (r1, r2, r3), the stress invariant (I1, J2, J3), the octahedral stress (roct, soct, h) and the mean stress (rm, sm, h). However, the physical significance of the parameters in these model is ambiguous. There are few criteria that can precisely analyse the failure mechanism of concrete based on mechanical theory. Thus, it is necessary to explore a comprehensive failure criterion that has a clear physical significance for concrete under multiaxial stress. In this study, a scheme for biaxial and triaxial compression tests is proposed. In this scheme, a large number of stress states are tested. Using a large experimental dataset, the influencing factors of multiaxial strength are analysed, and the main factors are determined. Based on the twin shear strength theory, the concept of a multiparameter double-shear strength failure criterion is developed. Various strength characteristics are studied as boundary conditions. Finally, coordinate system transformations are used to solve the parameters, and a new failure criterion for concrete under multiaxial stress is established. 2. Experimental program
ually and placed in the mixer in that order, and the mixer was used to agitate the dry mixture until it became homogeneous (approximately 2 min). Then, the predetermined mass of water was slowly poured into the mixer as it was operating (approximately 5 min). The concrete mixtures were poured into steel formworks, and the formworks were placed on a vibration table. After slight vibration, the concrete mixture filled the formwork with a uniform density. After 24 h, the specimens were removed from the steel formwork and numbered. Next, specimens were immediately placed in standard conditions (temperature of 20 ± 2 °C and humidity >95%) for curing. After 28 days, the specimens were cured in the natural environment. The duration of natural curing exceeded three months, so the influence of different batches on the concrete strength was ignored.
2.3. The test procedure All specimens were tested in a large concrete triaxial testing machine, which can exert loads independently in three directions. For the multiaxial compressive experiment, the specimen size was 100 100 100 mm, and the three principal stresses were perpendicular to the specimen surfaces, as shown in Fig. 1(a)–(c). The multiaxial loading state is shown in Fig. 2. In the compressive direction, friction could be eliminated using three layers of plastic membrane with glycerin between each layer. The loading mode used was proportional monotonic loading, and the loading rate was 0.2–0.4 MPa/s. Different loading mechanisms were used in the biaxial compression tests and triaxial compression tests. In the biaxial compression experiments, five stress ratios were used (r2/r3 = 0.00, 0.25, 0.50, 0.75 and 1.00). In the triaxial compression experiments, 14 stress ratios were used (r1:r2: r3 = 0:0:1; 0:1:1; 1:1:1; 0.1:0.1:1; 0.1:0.25:1; 0.1:0.3:1; 0.1:0.5:1; 0.1:0.75:1; 0.1:1:1; 0.25:0.25:1; 0.25:0.3:1; 0.25:0.75:1; 0.25:1:1 and 0.3:1:1). All three principal stresses were negative because the concrete specimens were in a compressive stress state, and the principal stresses could be expressed as r1 r2 r3. Each stress ratio was tested with three specimens, and the results of similar tests were averaged to yield the experimental data. Discrete results should be removed.
2.1. The concrete mixture ratio Two kinds of concrete were prepared, one for biaxial compressive specimens (BCS) and another for triaxial compressive specimens (TCS). Table 1 shows the mix proportions of the concrete by weight. Ordinary Portland cement (grade of P.O32.5) with a 28-day compressive strength greater than 32.5 MPa was used. The coarse aggregate consisted of crushed limestone with a maximum size of less than 20 mm. The fine aggregate was mediumgrained sand from a natural river and had a fineness modulus of 3.0. Tap water was used to prepare the concrete samples. 2.2. Specimen preparation Concrete specimens were made in the sizes: 100 100 100 mm cubes. The aggregate, cement and sand were weighed individ-
3. Experimental results and discussion 3.1. Results of the experiments Table 2 shows the biaxial compression experimental results for the five BCS specimens under different stress ratios. Table 3 shows the triaxial compression experimental results for the 14 TCS specimens under different stress ratios and a uniaxial tensile strength (ft) of 2.07 MPa. Meanwhile, the concrete shows the splitting failure is changing into flow failure with the increasing triaxial compression. It reflects that the shear stress and hydrostatic stress should be seriously considered, they might take important effect on failure form.
Table 1 The concrete mixture ratios.
TCS BCS
Water (kg/m3)
Cement (kg/m3)
Sand (kg/m3)
Aggregate (kg/m3)
Water cement ratio
185 185
330 440
544 586
1250 1244
0.56 0.42
Note: TCS denotes the triaxial compressive strength mixture, and BCS denotes the biaxial compressive strength mixture.
434
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
(b) Biaxial loading direction
(a) Uniaxial loading direction
(c)Triaxial loading direction
Fig. 1. The loading and stress directions of specimens under multiaxial compression.
(a) The biaxial loading state
(b) The triaxial loading state
Fig. 2. Loading state of specimens under multiaxial compression.
3.2. Strength behaviour under multiaxial stress Table 2 Biaxial compressive strength of specimens under various stress ratios. Concrete type
Stress ratio (r2/r3)
r2b (MPa)
r3b (MPa)
BCS1 BCS2 BCS3 BCS4 BCS5
0.00 0.25 0.50 0.75 1.00
0 10.72 22.43 31.57 40.66
34.2 42.91 44.85 42.09 40.66
Note: r2b and r3b are the principal stresses of each of the biaxial compressive ratio at failure.
Table 2 shows the influence of shear stress (variation in the minimum stress), normal stress, and intermediate principal stress (r2). The maximum shear stress is not the only crucial factor. The maximum compressive stress increased with the intermediate principal stress. However, the maximum compressive stress did not always increase with a decrease in the intermediate principal stress; for example, Fig. 3 shows that the maximum compressive stress varied with the intermediate principal stress, and it shows quadratic trend. These results mean that the intermediate principal stress is an important factor in the failure of concrete.
Table 3 Triaxial compressive strength of specimens under various stress ratios. Concrete type
Stress ratio (r1:r2:r3)
r1t(MPa)
r2t (MPa)
r3t (MPa)
TCS1 TCS2 TCS3 TCS4 TCS5 TCS6 TCS7 TCS8 TCS9 TCS10 TCS11 TCS12 TCS13 TCS14
0.00:0.00:1.00 0.00:1.00:1.00 1.00:1.00:1.00 0.10:0.10:1.00 0.10:0.25:1.00 0.10:0.30:1.00 0.10:0.50:1.00 0.10:0.75:1.00 0.10:1.00:1.00 0.25:0.25:1.00 0.25:0.30:1.00 0.25:0.75:1.00 0.25:1.00:1.00 0.30:1.00:1.00
0 0 325.34 5.45 7.15 7.82 6.81 6.08 5.73 37.73 45.47 46.57 41.73 25.05
0 27.41 325.34 5.57 17.53 21.38 36.15 45.77 56.96 38.29 55.89 135.71 163.47 83.35
22.35 27.41 325.34 52.83 72.87 71.77 69.82 61.65 57.23 150.36 188.01 187.28 166.59 83.96
Note: r1t, r2t and r3t are the principal stresses of each triaxial compressive ratio at failure.
435
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
(a)
1t
=
2t
Fig. 3. The relation between r2b and r3b in the biaxial experiments.
The comparison of the data from samples TCS4 to TCS9 (Table 3) also shows that the compressive stress varied with the intermediate principal stress. The results for the stress state in which r1t = r2t and r2t = r3t are shown in Fig. 4(a) and (b), respectively. The relationships in Fig. 4 show the same variations and clearly show that r3t increased rapidly as r1t increased during an early stage of the tests. A comparison of the results for samples TCS7 and TCS10 and for samples TCS9 and TCS11 (Table 3) shows that even when the intermediate principal stresses are equal, r3t increased with r1t. Thus, shear stress (variation in the minimum stress) is an important factor in the failure of concrete. Fig. 4(a) and (b) show that the rate at which r3t increased was very slow at the later stage of the tests, indicating that hydrostatic pressure ultimately became an important factor in the failure of the concrete. 4. The failure modes of concrete under multiaxial stress
(b) Based on the previous discussion, there are four main factors that contribute to the failure of concrete under multiaxial stress: the principal stress, the shear stress, the intermediate principal stress (r2) and the hydrostatic stress (rm). Of factors, the shear stress and hydrostatic stress are most important. In the following section, the twin shear strength theory is introduced for concrete under multiaxial stress. To cover the four main factors, the principal stress is added. Finally, shear stress and hydrostatic stress are respectively as the main influencing factors to establish three kinds of failure criterion equation, and Fig. 5 shows the solving process, the details are shown in Appendix. 4.1. The twin shear strength theory To investigate the four main factors, we used the twin shear strength theory [20]. This theory considers the influence of the intermediate principal stress and different tensile-compressive properties of materials and is suitable for various materials under complex stress states. The theory postulates that both the intermediate principal shear stress and the corresponding principal stress are important and that materials will be destroyed when the combination of these stresses reaches the extremum. The mathematical model describing the twin shear strength theory is shown in Eq. 1. In the theory, s13 + s12 and s13 + s23 can reflect the variation in the shear stress, and r13 + r12 and r12 + r23 can reflect the variation in the normal stress. Lastly, b is the effect coefficient of the principal stress.
2t
=
3t
Fig. 4. The relations between the maximum principal stress (r1t) and the minimum principal stress (r3t) in different stress states.
F ¼ s13 þ s12 þ bðr13 þ r12 Þ ¼ C;
F P F0
ð1aÞ
F 0 ¼ s13 þ s23 þ bðr13 þ r23 Þ ¼ C;
F < F0
ð1bÞ
where s12, s23 and s13 are the three principal shear stresses, for which s12 = (r1 r2)/2, s23 = (r2 r3)/2, and s13 = (r1 r3)/2, and r12, r23 and r13 are the corresponding principal stresses, such that r12 = (r1 + r2)/2, r23 = (r2 + r3)/2, and r23 = (r2 + r3)/2. Using these parameters, the twin shear strength model can fully balance the shear stress, the normal stress and the variations in the intermediate principal stress. The model is applied to determine the experimental characteristic point at which the plane stress, uniaxial tensile stress (ft) and biaxial compressive stress (fcc) can be computed using Eq. (1a) and the uniaxial compressive stress and biaxial tensile stress can be computed using Eq. (1b). However, the twin shear strength model is not applied for any uncharacteristic point in the plane stress state and space stress state of concrete. 4.2. Five-parameter failure criterion A (shear stress is the main stress) To consider the hydrostatic stress and shear stress, we developed a five-parameter failure criterion model based on the twin shear strength theory. The model is described by Eq. 2.
436
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
Fig. 5. The solving process.
F ¼ s13 þ s12 þ bðr13 þ r12 Þ þ Arm þ B1 ðs13 þ s12 Þ2 ¼ C;
F P F0 ð2aÞ
F 0 ¼ s13 þ s23 þ bðr13 þ r23 Þ þ Arm þ B2 ðs13 þ s23 Þ2 ¼ C;
F 6 F0
ð2bÞ
where rm is hydrostatic stress, rm = (r1 + r2 + r3)/3; A is the influence coefficient of hydrostatic stress; and B1 and B2 are influence coefficients of the shear stresses. We not only introduced hydrostatic stress into the failure criterion model but also focused on the influence of shear stress. To solve for the unknown coefficients (b, A, B1, B2 and C), Eq. 2 is expressed in a cylindrical coordinate system, as described by Eq. 3. The coordinate transformation method is described in Appendix A.
pffiffiffi pffiffiffi pffiffiffi pffiffiffi 3 3B1 3C 2b þ 3 2 n¼ q cos u ðq cos uÞ2 ; 0 6 u 6 ub 2A þ 4b A þ 2b 2A þ 4b
ð3aÞ
n¼
pffiffiffi pffiffiffi pffiffiffi 3C 3 2 2b q cosðu p=3Þ A þ 2b 2A þ 4b pffiffiffi 3 3B2 ½q cosðu p=3Þ2 ; ub 6 u 6 60 2A þ 4b
ð3bÞ
To solve for the five unknown quantities (b, A, B1, B2 and C), five experimental characteristic stresses should be used as boundary conditions. For convenient calculation, we selected the characteristic stresses from the tensile meridian (r1 r2 = r3, u = 0) and compression meridian (r1 = r2 r3, u = 60°). Therefore, Eq. 3 can be expressed as Eq. 4:
n ¼ a0 þ a1 q þ a2 q2 ; n ¼ b0 þ b1 q þ b2 q2 ;
u¼0 u ¼ 60
ð5bÞ
pffiffiffi 2aða þ 2abc Þ pffiffiffi a1 ¼ 6ða þ abc Þa2 =3 2abc 2a
ð5cÞ
b1 ¼
2n2
ð5eÞ
4.2.2. The simplified solution of failure criterion A In the physical experiments, the point in the tensile meridian (n1, q1) is difficult to test. Instead, we used the triaxial tensile stress (fttt) to replace this point. The tensile stress of concrete is very low, and the triaxial tensile stress is close to the uniaxial tensile stress (ft = fttt). In the cylindrical coordinate system (n, q, u), the new boundary conditions are uniaxial compressive stress (3½/3, 6½/3, 60°), uniaxial tensile stress (3½ a/3, 6½a/3, 0), biaxial compressive stress (2*3½ abc/3, 6½ abc/3, 0), triaxial tensile stress (3½ a, 0, 0) and a point in the compression meridian (n2, q2, 60°). Through boundary condition training, the calculations for the coefficients in Eq. 4 are described by Eq. 6.
pffiffiffi 3 3a a2 ¼ 2ðaabc a2bc Þ
a1 ¼ ð4bÞ
ð5aÞ
ð5dÞ
where a = ft/fc, abc = fcc/fc, ni = 3½rm/fc = (r1 + r2 + r2)/3½ and qi = 2sm/fc = [(r1r2)2 + (r2r3)2 + (r1r3)2]½/fc.
ð4aÞ
4.2.1. The solution of failure criterion A In the cylindrical coordinate system (n, q, u), the boundary conditions are uniaxial compressive stress (3½/3, 6½/3, 60°), uniaxial tensile stress (3½ a/3, 6½a/3, 0), biaxial compressive stress (2*3½ abc/3, 6½abc/3, 0), a point in the tensile meridian (n1, q1, 0) and a point in compression meridian (n2, q2, 60°). Through boundary condition training, the calculations for the coefficients in Eq. 4 are described by Eq. 5.
pffiffiffi 3q2 ð3q22 2Þa0 pffiffiffi 2 6q22 2q2
pffiffiffi pffiffiffi pffiffiffi ð 6n2 3q2 Þ ð3q2 6Þa0 pffiffiffi b2 ¼ 2q2 6q22
a0 ¼ b0 ¼
pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi 3B1 2þ 2b 2 2b 3C where a0 ¼ b0 ¼ Aþ2b , a1 ¼ 3 2Aþ4b ; a2 ¼ 32Aþ4b , b1 ¼ 3 2Aþ4b , and pffiffi pffiffi pffiffi 3 3B2 6b1 6a1 3 2 2 6a2 b2 ¼ 2Aþ4b. Then, we found that A ¼ , B1 ¼ 3ða1 þb1 Þ, a1 þb1 pffiffi pffiffi 3ða1 b1 Þ 2 6b2 6a0 B2 ¼ 3ða1 þb1 Þ, C ¼ a1 þb1 and b ¼ a1 þb1 .
pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 3að 2n1 þ q1 6abc Þ þ abc ð2 3q1 6n1 Þ pffiffiffi pffiffiffi a2 ¼ pffiffiffi ð 6a=3 q1 Þð 6aq1 þ 2a2bc 2aabc 6abc q1 Þ
pffiffiffi 3 3abc a þ 2a2 aabc ða abc Þ 3abc 3a
a0 ¼ b0 ¼
b1 ¼
b2 ¼
ð6aÞ
pffiffiffi 3a
ð6bÞ
3a2 þ 2abc a 2a2bc pffiffiffi 2ða2bc aabc Þ
ð6cÞ
pffiffiffi pffiffiffi 3q22 3að3q22 2Þ pffiffiffi 6q22 2q2
ð6dÞ
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ð 6n2 3q2 Þ 3að3q2 6Þ pffiffiffi 2q2 6q22
ð6eÞ
2n2
4.3. Five-parameter failure criterion B (hydrostatic stress is the main stress) To consider the hydrostatic stress as a dominating factor, we established a different five-parameter failure criterion model based on the twin shear strength theory. The model is described by Eq. 7.
F ¼ s13 þ s12 þ bðr13 þ r12 Þ þ Arm þ B1 ðrm Þ2 ¼ C;
F P F0
ð7aÞ
437
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
F 0 ¼ s13 þ s23 þ bðr13 þ r23 Þ þ Arm þ B2 ðrm Þ2 ¼ C;
F 6 F0
ð7bÞ
Eq. 7 can be expressed in a cylindrical coordinate system, as described by Eq. 8. The coordinate transformation method is shown in Appendix A.
q cos u ¼
pffiffiffi pffiffiffi pffiffiffi 6 2ðA1 þ 2bÞ 6B1 2 n n ; C 3þb 9 þ 3b 3þb
q cosðu p=3Þ ¼ ub 6 u 6 60
0 6 u 6 ub
ð8aÞ
pffiffiffi pffiffiffi pffiffiffi 6 2ðA2 þ 2bÞ 6B2 2 n n ; C 3b 9 3b 3b ð8bÞ
Five experimental characteristic stresses are used as the boundary conditions. For convenient calculation, we selected these characteristic stresses from the tensile meridian (r1 r2 = r3, u = 0) and compression meridian (r1 = r2 r3, u = 60°). In the cylindrical coordinate system (n, q, u), the boundary conditions are uniaxial compressive stress (3½/3, 6½/3, 60°), uniaxial tensile stress (3½ a/3, 6½a/3, 0), biaxial compressive stress (2*3½ abc/3, 6½ abc/3, 0), uniaxial tensile stress (3½ a, 0, 0) and a point in the compression meridian (n2, q2, 60°). Eq. 11 can be expressed as Eq. 9.
q ¼ a0 þ a1 n þ a2 n2 ; u ¼ 0
ð9aÞ
q ¼ b0 þ b1 n þ b2 n2 ; u ¼ 60
pffiffi 1 þ2bÞ where a0 ¼ a1 ¼ 2ðA3þb , pffiffi pffiffi 2ðA2 þ2bÞ 2 B 2 b1 ¼ 3b , and b2 ¼ pffiffi3ð3bÞ . pffiffi pffiffi 6a0 6b0 3 2a1 b0 6a0 6b0 3 2a0 b1 , A2 ¼ , A1 ¼ a0 þb0 a0 þb0 pffiffi 3ðb0 a0 Þ 6a0 b0 C ¼ a0 þb0 and b ¼ a0 þb0 . pffiffi 6C , 3þb
a2 ¼ Then,
pffiffi 2B1 pffiffi3ð3þbÞ ,
we pffiffi
ð9bÞ b0 ¼
pffiffi 6C , 3b
found
that pffiffi 3 6a2 b0 3 6a0 b2 B1 ¼ a0 þb0 , B2 ¼ a0 þb0 ,
Through boundary condition training, the calculations for the coefficients in Eq. 9 are described by Eq. 10.
a2 ¼
pffiffiffi 3 6a 2ð3a þ 2abc Þða þ 2abc Þ
pffiffiffi 6a þ 2a2 a2 a0 ¼ 2 a1 ¼
b2 ¼
pffiffiffi 6 þ 8aa2 pffiffiffi 2 3
pffiffiffi pffiffiffi pffiffiffi 6ðn2 þ 3aÞ 3q2 ð1 þ 3aÞ pffiffiffi pffiffiffi pffiffiffi ðn2 þ 3aÞð1 þ 3a 3 3n2 a 3n2 Þ
pffiffiffi pffiffiffi 3aq2 ð 3an22 þ 3a2 n2 Þb2 pffiffiffi b0 ¼ n2 þ 3a pffiffiffi ð1 9a Þb2 6 pffiffiffi pffiffiffi b1 ¼ 3 þ 3 3a 2
ð10aÞ
ð10bÞ ð10cÞ
ð10dÞ
ð10eÞ
F 0 ¼ s13 þ s23 þ bðr13 þ r23 Þ þ Arm þ Br2m þ C 2 ðs13 þ s23 Þ2 ¼ D; F 6 F 0 ð11bÞ Eq. 11 can be expressed in a cylindrical coordinate system as Eq. 12. The coordinate transform method is shown in Appendix A.
pffiffiffi pffiffiffi ! 6 6 6C 1 ðq cos uÞ þ þ b q cos u 2 6 pffiffiffi ! pffiffiffi 2 3 B 3 b n n2 ; 0 6 u 6 ub ¼D Aþ 3 3 3 2
pffiffiffi pffiffiffi ! 6 6 6C 2 ½q cosðu p=3Þ þ b q cosðu p=3Þ 2 6 ! pffiffiffi pffiffiffi 2 3 B 3 b n n2 ; ub 6 u 6 60 ¼D Aþ 3 3 3
ð12aÞ
2
ð12bÞ
To solve for the unknown quantities, six experimental characteristic stresses are used as boundary conditions. For convenient calculation, we selected the characteristic stresses from the tensile meridian (r1 r2 = r3, u = 0) and compression meridian (r1 = r2 r3, u = 60°). Therefore, Eq. 12 can be expressed as Eq. 13. We study two kinds of complex boundary conditions: one kind considers the triaxial compressive stress and the other considers the triaxial tensile stress.
a1 q2 þ q ¼ a2 þ a3 n þ a4 n2 ;
u¼0
ð13aÞ
b1 q2 þ q ¼ b2 þ b3 n þ b4 n2 ;
u ¼ 60
ð13bÞ
pffiffi pffiffi pffiffi pffiffi pffiffi ðAþ2bÞ 6C 1 6C 2 6D 6B where a1 ¼ 63þb , a2 ¼ 3þb , a3 ¼ 23þb , a4 ¼ 3ð3þbÞ , b1 ¼ 63b , ffiffi p pffiffi pffiffi ðAþ2bÞ 6D 6B b2 ¼ 3b , b3 ¼ 23b , and b4 ¼ 3ð3bÞ . Then, we found that pffiffi pffiffi pffiffi pffiffi pffiffi 2a3 b2 a4 b2 a2 b3 1 b2 2 b1 , B ¼ 3a26þb , C 1 ¼ 6ða62aþb , C 2 ¼ 6ða62aþb , D ¼ a62 þb A ¼ 6a2 6ba22 3 þb2 2 2Þ 2Þ 2 2 a2 Þ and b ¼ 3ðb . a2 þb2
4.4.1. The boundary conditions considering triaxial compressive stress In a cylindrical coordinate system (n, q, u), the boundary conditions are uniaxial compressive stress (3½/3, 6½/3, 60°), uniaxial tensile stress (3½ a/3, 6½a/3, 0), biaxial compressive stress (2*3½ abc/3, 6½ abc/3, 0), triaxial compressive stress (3½ acc, 0, 0), a point in the tensile meridian (n1, q1, 0) and a point in the compression meridian (n2, q2, 60°). Through boundary condition training, the calculations for the coefficients in Eq. 13 are described by Eq. 14.
a4 ¼
pffiffiffi pffiffiffi 6aðx1 y2 z1 x1 y1 z1 Þþ 6abc ðx21 z2 x1 x2 z1 Þþ3q1 ðx1 x2 y2 x21 y2 Þ 3ðx21 y3 z2 þx1 x2 y1 z3 þx1 x3 y2 z1 x21 y2 z3 x1 x2 y3 z1 x1 x3 y1 z2 Þ ð14aÞ
ð10fÞ
4.4. Six-parameter failure criterion To fully consider both the hydrostatic stress and shear stress as the two important factors in the failure of concrete, we established a six-parameter failure criterion model based on the twin shear strength theory. This model is described by Eq. 11. In this model, we not only introduced the hydrostatic stress but also focused on the influence of the shear stress.
F ¼ s13 þ s12 þ bðr13 þ r12 Þ þ Arm þ Br2m þ C 1 ðs13 þ s12 Þ2 ¼ D; F P F 0 ð11aÞ
pffiffiffi 6ðay1 abc x1 Þ 3a4 ðx3 y1 x1 y3 Þ a3 ¼ 3ðx2 y1 x1 y2 Þ
ð14bÞ
a2 ¼
pffiffiffi 3a3 acc 3a4 a2cc
ð14cÞ
a1 ¼
pffiffiffi 6ðay2 abc x2 Þ 3a4 ðx3 y2 x2 y3 Þ x2 y1 x1 y2
ð14dÞ
b2 ¼
pffiffiffi 6q2 2q2 p ffiffiffi 2 a3 2 3q2 2 a2 ð 3q22 2n2 Þ þ aa42 ðq22 2n22 Þ
ð14eÞ
pffiffiffi pffiffiffi a3 a4 =2 6=2 b1 ¼ b2 3 3 þ a2 a2
ð14fÞ
438
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
b3 ¼
a3 b2 a2
ð14gÞ
b4 ¼
a4 b2 a2
ð14hÞ
5.1. The five-parameter model A
pffiffiffi x1 ¼ 2a2 =3, x2 ¼ 3ða þ 3acc Þ=3, x3 ¼ ða2 9a2cc Þ=3, pffiffiffi 2 2 y1 ¼ 2abc =3, y2 ¼ 3ð3acc 2abc Þ=3, y3 ¼ ð4abc 9a2cc Þ=3, z1 ¼ q21 , pffiffiffi z2 ¼ 3acc n1 , and z3 ¼ n21 3a2cc . where
4.4.2. The boundary conditions considering triaxial tensile stress In a cylindrical coordinate system (n, q, u), the new boundary conditions are uniaxial compressive stress (3½/3, 6½/3, 60°), uniaxial tensile stress (3½a/3, 6½a/3, 0), biaxial compressive stress (2*3½ abc/3, 6½ abc/3, 0), triaxial tensile stress (3½ a, 0, 0), a point in the tensile meridian (n1, q1, 0) and a point in the compression meridian (n2, q2, 60°). All the characteristic stresses are from the tensile meridian (r1 r2 = r3, u = 0) and compression meridian (r1 = r2 r3, u = 60°). Through boundary condition training, the calculations for the coefficients in Eq. 13 are described by Eq. 15.
a4 ¼
pffiffiffi pffiffiffi 6aðx1 y2 z1 x1 y1 z1 Þþ 6abc ðx21 z2 x1 x2 z1 Þþ3q1 ðx1 x2 y2 x21 y2 Þ 3ðx21 y3 z2 þx1 x2 y1 z3 þx1 x3 y2 z1 x21 y2 z3 x1 x2 y3 z1 x1 x3 y1 z2 Þ ð15aÞ
a3 ¼
pffiffiffi 6ðay1 abc x1 Þ 3a4 ðx3 y1 x1 y3 Þ 3ðx2 y1 x1 y2 Þ
ð15bÞ
pffiffiffi a2 ¼ ð 3a3 a þ 3a4 a2 Þ
ð15cÞ
pffiffiffi 6ðay2 abc x2 Þ 3a4 ðx3 y2 x2 y3 Þ a1 ¼ x2 y 1 x 1 y 2
ð15dÞ
b2 ¼
5. Failure criteria validation
To verify the veracity and reliability of the failure criteria, we used r1 and r2 as the known data to compute r3, and we compared the model results against the experimental data. The method for completing the computations with the principal stress equation is shown in Appendix B. The results are shown in Table 4. In Table 4, rexp is measured experimentally, r0 31 is computed using Eqs. 2 ,4, 3 5, and r0 32 is computed using Eqs. 2, 4, 6. Ei, the relative error, reflects the difference between the computation data and the experimental data and was computed using Eq. (16).
Ei ¼
ðr03i rexp 3 Þ
rexp 3
ði ¼ 1; 2; 3:::Þ
ð16Þ
In Table 4, E1 and E2 for specimen TCS10 and E1 for specimen TCS13 are 0 because the data for these specimens were used to compute the coefficients of the model. There is little difference between r0 31 and r0 32, indicating that the simplified solution of the failure criterion model is practical. However, the results in Table 4 also show that E1 and E2 for several specimens (TCS3, TCS6, TCS7, TCS8, TCS14) were large (exceeding 15%), indicating that the model (in which shear stress is the main stress) cannot be applied to the concrete loading state in which hydrostatic stress has a dominant influence. If the stress state is r1 – r2 – r3 or approximately r1 = r2 = r3, the hydrostatic stress becomes more important. 5.2. The five-parameter model B
pffiffiffi 6q2 2q2 p ffiffiffi 2 a3 2 3q2 2 a2 ð 3q22 2n2 Þ þ aa42 ðq22 2n22 Þ
ð15eÞ
pffiffiffi pffiffiffi a3 a4 =2 6=2 b1 ¼ b2 3 3 þ a2 a2
ð15fÞ
b3 ¼
a3 b2 a2
ð15gÞ
b4 ¼
a4 b2 a2
ð15hÞ
pffiffiffi x1 ¼ 2a2 =3, x2 ¼ 2 3a=3, x3 ¼ 8a2 =3, y1 ¼ 2a2bc =3, pffiffiffi y3 ¼ ð4a2bc 9a2 Þ=3, z1 ¼ q21 , z2 ¼ y2 ¼ 3ð3a þ 2abc Þ=3, pffiffiffi 2 2 ð 3a þ n1 Þ and z3 ¼ n1 3a . where
To verify the veracity and reliability of the failure criterion model that considers hydrostatic stress as a dominating factor, Table 5 The validation of the five-parameter model B. Concrete
rexp (MPa) 3
r0 33 (MPa)
E3 (%)
TCS3 TCS4 TCS5 TCS6 TCS7 TCS8 TCS9 TCS10 TCS11 TCS12 TCS13 TCS14
325.34 52.83 72.87 71.77 69.82 61.65 57.23 150.36 188.01 187.28 166.69 83.96
83.15 58.93 90.91 95.03 74.55 60.57 49.91 150.36 166.45 98.05 60.06 90.91
74.44 11.55 24.76 32.41 6.77 1.75 12.79 0.00 11.46 47.65 63.95 8.28
Note: rexp is the experimental data, r0 33 is the computed data, and E3 is the relative 3 error.
Table 4 The validation of the five-parameter model A. Concrete
rexp (MPa) 3
r0 31 (MPa)
E1 (%)
r0 32 (MPa)
E2 (%)
TCS3 TCS4 TCS5 TCS6 TCS7 TCS8 TCS9 TCS10 TCS11 TCS12 TCS13 TCS14
325.34 52.83 72.87 71.77 69.82 61.65 57.23 150.36 188.01 187.28 166.69 83.96
725.67 51.81 77.91 85.46 89.30 73.58 59.40 150.36 180.25 217.35 166.70 161.15
123.05 1.93 6.92 19.07 27.90 20.37 3.79 0.00 4.12 16.06 0.00 101.73
655.06 51.89 77.92 85.47 88.3 72.67 58.52 150.36 180.24 213.76 158.71 137
101.35 1.78 6.93 19.09 26.46 18.88 2.25 0.00 4.13 14.14 4.79 71.49
Note: rexp is the experimental data, r0 31 and r0 32 are the computed data, and Ei is the relative error (i = 1, 2). 3
439
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441 Table 6 The validation of six-parameter model. Concrete
rexp (MPa) 3
r’34 (MPa)
E4 (%)
r’35 (MPa)
E5 (%)
TCS3 TCS4 TCS5 TCS6 TCS7 TCS8 TCS9 TCS10 TCS11 TCS12 TCS13 TCS14
325.34 52.83 72.87 71.77 69.82 61.65 57.23 150.36 188.01 187.28 166.69 83.96
325.34 55.61 80.11 90.74 78.69 65.98 52.27 150.36 174.56 165.56 124.83 92.68
0.00 5.26 14.05 26.43 12.70 7.02 8.67 0.00 7.15 11.89 25.07 10.39
325.34 56.26 79.94 79.93 59.66 56.41 43.43 149.45 172.8 96 166.67 83.67
0 6.49 9.70 11.37 14.55 8.50 24.11 0.60 8.09 48.74 0.05 0.01
Note: rexp is the experimental data, r0 34 and r0 35 are the computed data, and Ei is the relative error (i = 4, 5). 3
we used r1 and r2 as the known data to compute r3, following the method shown in Appendix C. The results are shown in Table 5. In Table 5, rexp is measured experimentally, and r0 33 is computed 3 using Eqs. 7, 9, 10. In Table 5, E3 for specimen TCS10 is 0 because the data for this specimen were used to compute the coefficients for the model. If the three principal stresses are different from each other or if they become similar, the model yields an accurate result. The results in Table 5 show that E3 for some specimens (TCS3, TCS5, TCS6, TCS12, TCS13) was large (exceeding 15%), indicating that the model (in which hydrostatic stress is the main stress) is not applicable for concrete loading states in which shear stress has a dominant influence. For the validation of r33, model A and model B show similar results.
5.3. The six-parameter model To verify the veracity and reliability of the six-parameter failure criterion model, we computed r3 from the principal stress equation, as shown in Appendix D. These results are shown in Table 6. In Table 6, rexp is measured experimentally, r0 34 is computed using 3 Eqs. 11, 13, 14, and r0 35 is computed using Eqs. 11, 13, 15. The results in Table 6 show that the calculated data for most of the specimens were consistent with the experimental results. However, some computed data from the model under different boundary conditions were very different, indicating that the results from the six-parameter model mostly depend on the boundary conditions.
6. Conclusions Though biaxial and triaxial experiments on concrete specimens, six mechanical effects were examined. The intermediate principal stress, hydrostatic pressure and shear stress are the most important factors. Based on the twin shear strength theory, three kinds of multiparameter failure criteria models were developed for concrete under triaxial compression. The models have clear physical significance. The five-parameter failure criterion (model A) that considers shear stress as the main stress cannot be applied to the stress state in which r1 – r2 – r3 or approximately r1 = r2 = r3. However, this model can be applied to examine concrete structures under low confining pressure. The five-parameter failure criterion (model B) that considers hydrostatic pressure as the main stress can be applied to complicated stress states. Furthermore, the five-parameter failure criterion models A and B show similar results.
The six-parameter failure criterion model predicts stresses that closely match the experimental results. However, the model results are likely to depend on the boundary conditions. Acknowledgements The authors gratefully acknowledge the contributions of Maohong Yu (Xi’an Jiao Tong University) for the information on the twin shear strength theory. The first author acknowledges the China Scholarship Council, Xi’an University of Architecture and Technology and the University of Wollongong for supporting his Joint PhD programme. Funding sources This work was supported by the Major State Key Research Development Foundation of China – China [grant number 2017YFC0703400] and the National Science Foundation of China – China [grant number 51478382]. Conflict of interest The authors have no conflicts of interest to declare. Appendix A In this paper, we establish the model in the stress space in which the hydrostatic stress axis is the primary axis, and the coordinate in the p plane is defined using cylindrical coordinate system. The primary stress point P (r1, r2, r3) can be expressed as P (n, q, u) in a cylindrical coordinate system. The relation between the cylindrical coordinate system (n, q, u) and the primary stress system (r1, r2, r3) is shown below:
pffiffiffi n ¼ ðr1 þ r2 þ r3 Þ= 3
q¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ðr1 r2 Þ2 þ ðr2 r3 Þ2 þ ðr1 r3 Þ2 = 3
2r1 r2 r3 cos u ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ðr1 r2 Þ þ 2ðr2 r3 Þ2 þ 2ðr1 r3 Þ2 The principal stress expression can be derived as follows:
pffiffiffi
pffiffiffi
r1 ¼ ðn þ 2q cos uÞ= 3 pffiffiffi
pffiffiffi
r2 ¼ ½n þ 2q cosðu 2p=3Þ= 3 pffiffiffi pffiffiffi pffiffiffi ¼ ½2n þ 2qð 3 sin u cos uÞ= 3
440
r3
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
pffiffiffi pffiffiffi ¼ ½n þ 2q cosðu þ 2p=3Þ= 3 pffiffiffi pffiffiffi pffiffiffi ¼ ½2n 2qð 3 sin u þ cos uÞ= 3
a1 ¼ B1 =9; b1 ¼ b=2 þ A1 =3 þ 2B1 ðr1 þ r2 Þ=9 1=2; and c1 ¼ ð1 þ b þ A1 =3Þr1 þ ðb=2 þ A1 =3 1=2Þr2 þ B1 ð2r1 þ r2 Þ2 =9 C;
The parameters of the multiparameter failure criterion model can be expressed as follows:
pffiffiffi
s12 þ s13 ¼ 6q cos u=2 s13 þ s23
pffiffiffi ¼ 6q cosðu p=3Þ=2
r12 þ r13
pffiffiffi pffiffiffi ¼ ð4n þ 2q cos uÞ=2 3 pffiffiffi
a2 ¼ B2 =9; b2 ¼ b þ A2 =3 þ 2B2 ðr1 þ r2 Þ=9 1; and c2 ¼ ðb=2 þ A2 =3 þ 1=2Þðr1 þ r2 Þ þ B2 ð2r1 þ r2 Þ2 =9 C: The method to determine r3 is the same as that described in Appendix B.
pffiffiffi
r13 þ r23 ¼ ½4n 2q cosðu p=3Þ=2 3 pffiffiffi
Appendix D Eq. 11 can also be expressed as simplified monadic quadratic equations:
rm ¼ ðr1 þ r2 þ r3 Þ=3 ¼ 3n=3
a1 r23 þ b1 r3 þ c1 ¼ 0
Appendix B Eq. 2 can be translated into a monadic quadratic equation:
3B1 r23 þ r3 ½6b þ 4A 6 6B1 ð2r1 r2 Þ þ 6ð2r1 r2 Þ þ 6bð2r1 þ r2 Þ þ 4Aðr1 þ r2 Þ þ 3B1 ð2r1 r2 Þ2 12C ¼ 0 12B2 r23 þ r3 ½12b þ 4A 12 12B2 ðr1 þ r2 Þ þ 3B2 ðr1 þ r2 Þ2 þ ðr1 þ r2 Þð6 þ 6b þ 4AÞ 12C ¼ 0 The monadic quadratic equation can be expressed as simplified equations.
a1 r23 þ b1 r3 þ c1 ¼ 0
a2 r23 þ b2 r3 þ c2 ¼ 0: In the foregoing equations,
a1 ¼ B=9 þ C 1 ; b1 ¼ b=2 þ A=3 þ 2Bðr1 þ r2 Þ=9 2C 1 ð2r1 r2 Þ 1=2; and c1 ¼ ð2r1 r2 Þ=2 þ bð2r1 þ r2 Þ=2 þ Aðr1 þ r2 Þ=3 þ Bðr1 þ r2 Þ2 =9 þ C 1 ð2r1 r2 Þ2 D; a2 ¼ B=9 þ 4C 2 ; b2 ¼ b þ A=3 þ 2Bðr1 þ r2 Þ=9 4C 2 ðr1 þ r2 Þ 1; and
a2 r þ b2 r3 þ c2 ¼ 0 2 3
c2 ¼ ðb=2 þ A=3 þ 1=2Þðr1 þ r2 Þ þ ðB=9 þ C 2 Þð2r1 þ r2 Þ2 D:
In the foregoing equations,
a1 ¼ 3B1 ; b1 ¼ 6b þ 4A 6 6B1 ð2r1 r2 Þ; and
The method to determine r3 is the same as described in Appendix B.
c1 ¼ 6ð2r1 r2 Þ þ 6bð2r1 þ r2 Þ þ 4Aðr1 þ r2 Þ þ 3B1 ð2r1 r2 Þ2
References
12C; a2 ¼ 12B2 ; b2 ¼ 12b þ 4A 12 12B2 ðr1 þ r2 Þ; and c2 ¼ 3B2 ðr1 þ r2 Þ2 þ ðr1 þ r2 Þð6 þ 6b þ 4AÞ 12C: In this paper, r3 is compressive stress and is computed as follows:
r ¼ 0 3
b1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b1 4a1 c1 2a1
;r ¼ 00 3
b2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 4a2 c2 2a2
Because of the selection in Eq. 3, r3 should be greater than r03 and r003 , i.e.,
r3 ¼ maxðr03 ; r003 Þ: Appendix C Eq. 7 can be expressed as simplified monadic quadratic equations as follows:
a1 r23 þ b1 r3 þ c1 ¼ 0 a2 r23 þ b2 r3 þ c2 ¼ 0: In the foregoing equations,
[1] A. Kaya, M. Dawood, B. Gencturk, Repair of corroded and buckled short steel columns using concrete-filled GFRP jackets, Construct. Build. Mater. 94 (2015) 20–27. [2] J.A. Øverli, A density-dependent failure criterion for concrete, Construct. Build. Mater. 124 (2016) 566–574. [3] J. Wang, A.A.A. Molenaar, M.F.C.V.D. van de Ven, S. Wu, Behaviour of asphalt concrete mixtures under tri-axial compression, Construct. Build. Mater. 105 (2016) 269–274. [4] I.E. Shkolnik, Influence of high strain rates on stress–strain relationship, strength and elastic modulus of concrete, Cem. Concr. Compos. 30 (2008) 1000–1012. [5] D. Yan, G. Lin, Influence of initial static stress on the dynamic properties of concrete, Cem. Concr. Compos. 30 (2008) 327–333. [6] Z. Chen, Y. Hu, Q. Li, M. Sun, P. Lu, T. Liu, Behavior of concrete in water subjected to dynamic triaxial compression, J. Eng. Mech. 136 (2010) 379–389. [7] T. Ozbakkaloglu, Compressive behavior of concrete-filled FRP tube columns: assessment of critical column parameters, Eng. Struct. 51 (2013) 188–199. [8] Y.B. Wang, J.Y.R. Liew, S.Ch. Lee, D.X. Xiong, Experimental study of ultra-highstrength concrete under triaxial compression, ACI Mater. J. 113 (2016) 105– 112. [9] J. Zhou, X. Chen, L. Wu, X. Kan, Influence of free water content on the compressive mechanical behaviour of cement mortar under high strain rate, Sadhana 36 (2011) 357–369. [10] P. Rossi, J.L. Tailhan, F.L. Le Maou, Creep strain versus residual strain of a concrete loaded under various levels of compressive stress, Cem. Concr. Res. 51 (2013) 32–37. [11] D. Zheng, Q. Li, L. Wang, Rate effect of concrete strength under initial static loading, Eng. Fract. Mech. 74 (2007) 2311–2319. [12] H.W. Wang, Ch.Q. Wu, F.R. Zhang, Q. Fang, H. Xiang, P. Li, Z. Li, Y. Zhou, Y. Zhang, J. Li, Experimental study of large-sized concrete filled steel tube columns under blast load, Construct. Build. Mater. 134 (2017) 131–141. [13] X.W. Tang, C.H. Zhang, Study on concrete in macro-and meso-scale mechanical properties based on homogenization theory, Chin. J. Comp. Mech. 26 (2009) 876–881.
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441 [14] W.C. Zhu, C.A. Tang, S.Y. Wang, Numerical study on the influence of mesomechanical properties on macroscopic fracture of concrete, Struct. Eng. Mech. 19 (2005) 519–533. [15] J. Xiao, W. Li, Q. Liu, Meso-level numerical simulation on mechanical properties of modeled recycled concrete under uniaxial compression, J. Tongji Univ. 39 (2011) 791–797. [16] B. Zhang, Relationship between pore structure and mechanical properties of ordinary concrete under bending fatigue, Cem. Concr. Res. 28 (1998) 699–711. [17] P.G. Asteris, V. Plevris, Anisotropic masonry failure criterion using artificial neural networks, Neural. Comput. Appl. 28 (2016) 2207–2229. [18] C.A. Syrmakezis, P.G. Asteris, Masonry failure criterion under biaxial stress state, Cem. Concr. Res. 13 (2001) 58–64.
441
[19] S. Shang, Y. Song, Dynamic biaxial tensile-compressive strength and failure criterion of plain concrete, Construct. Build. Mater. 40 (2013) 322–329. [20] B. Piscesa, M.M. Attard, A.K. Samani, A Lateral strain plasticity model for FRP confined concrete, Compos. Struct. 158 (2016) 160–174. [21] V. Plevris, P.G. Asteris, Modeling of masonry failure surface under biaxial compressive stress using Neural Networks, Construct. Build. Mater. 55 (2014) 447–461. [22] M.H. Yu, New System of Strength Theory: Theory, Development and Applications, Xian Jiaotong University Press, Xi’an, 2011. [23] P. Menetrey, K.J. Willam, Triaxial failure criterion for concrete and its generalization, ACI Struct. J. 92 (1995) 311–318.
Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
New failure criterion models for concrete under multiaxial stress in compression Chong Rong a, Qingxuan Shi a,⇑, Ting Zhang a, Hongchao Zhao b a b
School of Civil Engineering, Xi’an University of Architecture and Technology, Shaanxi, Xi’an 710055, China School of Geology and Mining Engineering, Xinjiang University, Urumqi 830000, China
h i g h l i g h t s The main factors that contribute to the failure under multiaxial stress are studied. The hydrostatic stress and shear stress are respectively as the main influencing factors to establish equations. The accuracy of each failure criterion model is verified. The applicability of each failure criterion model is discussed.
a r t i c l e
i n f o
Article history: Received 13 January 2017 Received in revised form 12 November 2017 Accepted 19 November 2017
Keywords: Biaxial and triaxial compression tests Twin shear strength theory Multiaxial stress Failure criterion models Boundary conditions
a b s t r a c t Biaxial and triaxial compression tests were performed on 100 100 100 mm cubic specimens of concrete under different stress loading rates. All the tests were performed using a true triaxial testing machine. The analysis of the data revealed that the intermediate principal stress, hydrostatic stress and shear stress are the main factors influencing the failure criterion for concrete under multiaxial stress. Based on the twin shear strength theory, three failure criterion models were developed. The fiveparameter model A considers the shear strength as the main factor, the five-parameter model B considers hydrostatic strength as the main factor, and the six-parameter model considers both the shear strength and hydrostatic strength. The parameters for these failure criterion models have clear physical significance and form the failure criterion in a theoretical analysis. Models A and B apply to different stress states and show similar results. The six-parameter model can apply to most stress states, but the computed results depend on the boundary conditions. This convenient model can also be extended for nonlinear analyses of concrete under multiaxial stress in compression. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Abbreviations: r1, the maximum principal stress; r1t, the maximum principal stress of each triaxial compressive ratio; r2, the intermediate principal stress; r2b, the intermediate principal stress of each biaxial compressive ratio; r2t, the intermediate principal stress of each triaxial compressive ratio; r3, the minimum principal stress; r3b, the minimum principal stress of each biaxial compressive ratio; r3t, the minimum principal stress of each triaxial compressive ratio; rm, the hydrostatic stress; roct, the octahedral principle stress; sij, the principle shear stress (i–j); sm, the shear stress corresponding to hydrostatic stress; soct, the octahedral shear stress; n, the height in cylindrical coordinate system; q, the radius in cylindrical coordinate system; u, the angle in cylindrical coordinate system; a, the tension–compression ratio; abc, the biaxial-axial compression ratio; acc, the triaxialaxial compression ratio; fbc, the biaxial compress strength; fcc, the triaxial compress strength; ft, the tense strength; fttt, the triaxial tense strength; I1, the first invariants of stress deviator; J2, the second invariants of stress deviator; J3, the third invariants of stress deviator. ⇑ Corresponding author. E-mail address: [email protected] (Q. Shi). https://doi.org/10.1016/j.conbuildmat.2017.11.106 0950-0618/Ó 2017 Elsevier Ltd. All rights reserved.
Concrete composite structures are a widely used structural form and are applied in the construction of high-rise buildings, arch dams, bridges, and other structures. In a composite structure, concrete is the most complicated component. Many researchers [1–6] have studied the behaviour of concrete under multiaxial stress. They found that the compressive strength of the concrete increases with increasing lateral compressive stress. In these previous studies, the strength of concrete under multiaxial stress is related to its uniaxial strength and lateral compressive strength, but the influences of these two strengths are different [7]. This difference is caused by several factors, for example, the strength [8] and material characteristics of the concrete [9,10], initial stress [5,11], and specimen size [12]. Many researchers have studied concrete using different methods and found that the change in the
433
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
behaviour of a composite material is systematic, indicating that different micro-characteristics can be expressed by a similar macroscopic pattern. In the micromechanical research of concrete [13–16], the basic characteristics of concrete under multiaxial stress can be formulated in terms of six mechanical effects: the tensile-compressive strength difference, the hydrostatic stress (rm), the shear stress (variation in the minimum stress), the normal stress, the intermediate principal stress (r2), and the variation in the intermediate principal stress. In concrete applications, the stress states are complex, and designing complex structures based on only the normal failure mode of concrete is unsafe. A failure criterion can be described by a mathematical function model to express the damage envelope surface. To quantify the mechanical properties of concrete under complex stress, many failure criteria have been proposed for a variety of concretes and stress states [17–23], such as the Mohr-Coulomb criterion, the Drucker-Prager criterion, the Willam-Warnker criterion, the Ottosen criterion, Asteris and Plevris and criterion, and the HsiehTing-Chen criterion. The original equations of tensile and compressive meridians in each failure criterion employ different stress parameters, such as the principal stress (r1, r2, r3), the stress invariant (I1, J2, J3), the octahedral stress (roct, soct, h) and the mean stress (rm, sm, h). However, the physical significance of the parameters in these model is ambiguous. There are few criteria that can precisely analyse the failure mechanism of concrete based on mechanical theory. Thus, it is necessary to explore a comprehensive failure criterion that has a clear physical significance for concrete under multiaxial stress. In this study, a scheme for biaxial and triaxial compression tests is proposed. In this scheme, a large number of stress states are tested. Using a large experimental dataset, the influencing factors of multiaxial strength are analysed, and the main factors are determined. Based on the twin shear strength theory, the concept of a multiparameter double-shear strength failure criterion is developed. Various strength characteristics are studied as boundary conditions. Finally, coordinate system transformations are used to solve the parameters, and a new failure criterion for concrete under multiaxial stress is established. 2. Experimental program
ually and placed in the mixer in that order, and the mixer was used to agitate the dry mixture until it became homogeneous (approximately 2 min). Then, the predetermined mass of water was slowly poured into the mixer as it was operating (approximately 5 min). The concrete mixtures were poured into steel formworks, and the formworks were placed on a vibration table. After slight vibration, the concrete mixture filled the formwork with a uniform density. After 24 h, the specimens were removed from the steel formwork and numbered. Next, specimens were immediately placed in standard conditions (temperature of 20 ± 2 °C and humidity >95%) for curing. After 28 days, the specimens were cured in the natural environment. The duration of natural curing exceeded three months, so the influence of different batches on the concrete strength was ignored.
2.3. The test procedure All specimens were tested in a large concrete triaxial testing machine, which can exert loads independently in three directions. For the multiaxial compressive experiment, the specimen size was 100 100 100 mm, and the three principal stresses were perpendicular to the specimen surfaces, as shown in Fig. 1(a)–(c). The multiaxial loading state is shown in Fig. 2. In the compressive direction, friction could be eliminated using three layers of plastic membrane with glycerin between each layer. The loading mode used was proportional monotonic loading, and the loading rate was 0.2–0.4 MPa/s. Different loading mechanisms were used in the biaxial compression tests and triaxial compression tests. In the biaxial compression experiments, five stress ratios were used (r2/r3 = 0.00, 0.25, 0.50, 0.75 and 1.00). In the triaxial compression experiments, 14 stress ratios were used (r1:r2: r3 = 0:0:1; 0:1:1; 1:1:1; 0.1:0.1:1; 0.1:0.25:1; 0.1:0.3:1; 0.1:0.5:1; 0.1:0.75:1; 0.1:1:1; 0.25:0.25:1; 0.25:0.3:1; 0.25:0.75:1; 0.25:1:1 and 0.3:1:1). All three principal stresses were negative because the concrete specimens were in a compressive stress state, and the principal stresses could be expressed as r1 r2 r3. Each stress ratio was tested with three specimens, and the results of similar tests were averaged to yield the experimental data. Discrete results should be removed.
2.1. The concrete mixture ratio Two kinds of concrete were prepared, one for biaxial compressive specimens (BCS) and another for triaxial compressive specimens (TCS). Table 1 shows the mix proportions of the concrete by weight. Ordinary Portland cement (grade of P.O32.5) with a 28-day compressive strength greater than 32.5 MPa was used. The coarse aggregate consisted of crushed limestone with a maximum size of less than 20 mm. The fine aggregate was mediumgrained sand from a natural river and had a fineness modulus of 3.0. Tap water was used to prepare the concrete samples. 2.2. Specimen preparation Concrete specimens were made in the sizes: 100 100 100 mm cubes. The aggregate, cement and sand were weighed individ-
3. Experimental results and discussion 3.1. Results of the experiments Table 2 shows the biaxial compression experimental results for the five BCS specimens under different stress ratios. Table 3 shows the triaxial compression experimental results for the 14 TCS specimens under different stress ratios and a uniaxial tensile strength (ft) of 2.07 MPa. Meanwhile, the concrete shows the splitting failure is changing into flow failure with the increasing triaxial compression. It reflects that the shear stress and hydrostatic stress should be seriously considered, they might take important effect on failure form.
Table 1 The concrete mixture ratios.
TCS BCS
Water (kg/m3)
Cement (kg/m3)
Sand (kg/m3)
Aggregate (kg/m3)
Water cement ratio
185 185
330 440
544 586
1250 1244
0.56 0.42
Note: TCS denotes the triaxial compressive strength mixture, and BCS denotes the biaxial compressive strength mixture.
434
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
(b) Biaxial loading direction
(a) Uniaxial loading direction
(c)Triaxial loading direction
Fig. 1. The loading and stress directions of specimens under multiaxial compression.
(a) The biaxial loading state
(b) The triaxial loading state
Fig. 2. Loading state of specimens under multiaxial compression.
3.2. Strength behaviour under multiaxial stress Table 2 Biaxial compressive strength of specimens under various stress ratios. Concrete type
Stress ratio (r2/r3)
r2b (MPa)
r3b (MPa)
BCS1 BCS2 BCS3 BCS4 BCS5
0.00 0.25 0.50 0.75 1.00
0 10.72 22.43 31.57 40.66
34.2 42.91 44.85 42.09 40.66
Note: r2b and r3b are the principal stresses of each of the biaxial compressive ratio at failure.
Table 2 shows the influence of shear stress (variation in the minimum stress), normal stress, and intermediate principal stress (r2). The maximum shear stress is not the only crucial factor. The maximum compressive stress increased with the intermediate principal stress. However, the maximum compressive stress did not always increase with a decrease in the intermediate principal stress; for example, Fig. 3 shows that the maximum compressive stress varied with the intermediate principal stress, and it shows quadratic trend. These results mean that the intermediate principal stress is an important factor in the failure of concrete.
Table 3 Triaxial compressive strength of specimens under various stress ratios. Concrete type
Stress ratio (r1:r2:r3)
r1t(MPa)
r2t (MPa)
r3t (MPa)
TCS1 TCS2 TCS3 TCS4 TCS5 TCS6 TCS7 TCS8 TCS9 TCS10 TCS11 TCS12 TCS13 TCS14
0.00:0.00:1.00 0.00:1.00:1.00 1.00:1.00:1.00 0.10:0.10:1.00 0.10:0.25:1.00 0.10:0.30:1.00 0.10:0.50:1.00 0.10:0.75:1.00 0.10:1.00:1.00 0.25:0.25:1.00 0.25:0.30:1.00 0.25:0.75:1.00 0.25:1.00:1.00 0.30:1.00:1.00
0 0 325.34 5.45 7.15 7.82 6.81 6.08 5.73 37.73 45.47 46.57 41.73 25.05
0 27.41 325.34 5.57 17.53 21.38 36.15 45.77 56.96 38.29 55.89 135.71 163.47 83.35
22.35 27.41 325.34 52.83 72.87 71.77 69.82 61.65 57.23 150.36 188.01 187.28 166.59 83.96
Note: r1t, r2t and r3t are the principal stresses of each triaxial compressive ratio at failure.
435
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
(a)
1t
=
2t
Fig. 3. The relation between r2b and r3b in the biaxial experiments.
The comparison of the data from samples TCS4 to TCS9 (Table 3) also shows that the compressive stress varied with the intermediate principal stress. The results for the stress state in which r1t = r2t and r2t = r3t are shown in Fig. 4(a) and (b), respectively. The relationships in Fig. 4 show the same variations and clearly show that r3t increased rapidly as r1t increased during an early stage of the tests. A comparison of the results for samples TCS7 and TCS10 and for samples TCS9 and TCS11 (Table 3) shows that even when the intermediate principal stresses are equal, r3t increased with r1t. Thus, shear stress (variation in the minimum stress) is an important factor in the failure of concrete. Fig. 4(a) and (b) show that the rate at which r3t increased was very slow at the later stage of the tests, indicating that hydrostatic pressure ultimately became an important factor in the failure of the concrete. 4. The failure modes of concrete under multiaxial stress
(b) Based on the previous discussion, there are four main factors that contribute to the failure of concrete under multiaxial stress: the principal stress, the shear stress, the intermediate principal stress (r2) and the hydrostatic stress (rm). Of factors, the shear stress and hydrostatic stress are most important. In the following section, the twin shear strength theory is introduced for concrete under multiaxial stress. To cover the four main factors, the principal stress is added. Finally, shear stress and hydrostatic stress are respectively as the main influencing factors to establish three kinds of failure criterion equation, and Fig. 5 shows the solving process, the details are shown in Appendix. 4.1. The twin shear strength theory To investigate the four main factors, we used the twin shear strength theory [20]. This theory considers the influence of the intermediate principal stress and different tensile-compressive properties of materials and is suitable for various materials under complex stress states. The theory postulates that both the intermediate principal shear stress and the corresponding principal stress are important and that materials will be destroyed when the combination of these stresses reaches the extremum. The mathematical model describing the twin shear strength theory is shown in Eq. 1. In the theory, s13 + s12 and s13 + s23 can reflect the variation in the shear stress, and r13 + r12 and r12 + r23 can reflect the variation in the normal stress. Lastly, b is the effect coefficient of the principal stress.
2t
=
3t
Fig. 4. The relations between the maximum principal stress (r1t) and the minimum principal stress (r3t) in different stress states.
F ¼ s13 þ s12 þ bðr13 þ r12 Þ ¼ C;
F P F0
ð1aÞ
F 0 ¼ s13 þ s23 þ bðr13 þ r23 Þ ¼ C;
F < F0
ð1bÞ
where s12, s23 and s13 are the three principal shear stresses, for which s12 = (r1 r2)/2, s23 = (r2 r3)/2, and s13 = (r1 r3)/2, and r12, r23 and r13 are the corresponding principal stresses, such that r12 = (r1 + r2)/2, r23 = (r2 + r3)/2, and r23 = (r2 + r3)/2. Using these parameters, the twin shear strength model can fully balance the shear stress, the normal stress and the variations in the intermediate principal stress. The model is applied to determine the experimental characteristic point at which the plane stress, uniaxial tensile stress (ft) and biaxial compressive stress (fcc) can be computed using Eq. (1a) and the uniaxial compressive stress and biaxial tensile stress can be computed using Eq. (1b). However, the twin shear strength model is not applied for any uncharacteristic point in the plane stress state and space stress state of concrete. 4.2. Five-parameter failure criterion A (shear stress is the main stress) To consider the hydrostatic stress and shear stress, we developed a five-parameter failure criterion model based on the twin shear strength theory. The model is described by Eq. 2.
436
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
Fig. 5. The solving process.
F ¼ s13 þ s12 þ bðr13 þ r12 Þ þ Arm þ B1 ðs13 þ s12 Þ2 ¼ C;
F P F0 ð2aÞ
F 0 ¼ s13 þ s23 þ bðr13 þ r23 Þ þ Arm þ B2 ðs13 þ s23 Þ2 ¼ C;
F 6 F0
ð2bÞ
where rm is hydrostatic stress, rm = (r1 + r2 + r3)/3; A is the influence coefficient of hydrostatic stress; and B1 and B2 are influence coefficients of the shear stresses. We not only introduced hydrostatic stress into the failure criterion model but also focused on the influence of shear stress. To solve for the unknown coefficients (b, A, B1, B2 and C), Eq. 2 is expressed in a cylindrical coordinate system, as described by Eq. 3. The coordinate transformation method is described in Appendix A.
pffiffiffi pffiffiffi pffiffiffi pffiffiffi 3 3B1 3C 2b þ 3 2 n¼ q cos u ðq cos uÞ2 ; 0 6 u 6 ub 2A þ 4b A þ 2b 2A þ 4b
ð3aÞ
n¼
pffiffiffi pffiffiffi pffiffiffi 3C 3 2 2b q cosðu p=3Þ A þ 2b 2A þ 4b pffiffiffi 3 3B2 ½q cosðu p=3Þ2 ; ub 6 u 6 60 2A þ 4b
ð3bÞ
To solve for the five unknown quantities (b, A, B1, B2 and C), five experimental characteristic stresses should be used as boundary conditions. For convenient calculation, we selected the characteristic stresses from the tensile meridian (r1 r2 = r3, u = 0) and compression meridian (r1 = r2 r3, u = 60°). Therefore, Eq. 3 can be expressed as Eq. 4:
n ¼ a0 þ a1 q þ a2 q2 ; n ¼ b0 þ b1 q þ b2 q2 ;
u¼0 u ¼ 60
ð5bÞ
pffiffiffi 2aða þ 2abc Þ pffiffiffi a1 ¼ 6ða þ abc Þa2 =3 2abc 2a
ð5cÞ
b1 ¼
2n2
ð5eÞ
4.2.2. The simplified solution of failure criterion A In the physical experiments, the point in the tensile meridian (n1, q1) is difficult to test. Instead, we used the triaxial tensile stress (fttt) to replace this point. The tensile stress of concrete is very low, and the triaxial tensile stress is close to the uniaxial tensile stress (ft = fttt). In the cylindrical coordinate system (n, q, u), the new boundary conditions are uniaxial compressive stress (3½/3, 6½/3, 60°), uniaxial tensile stress (3½ a/3, 6½a/3, 0), biaxial compressive stress (2*3½ abc/3, 6½ abc/3, 0), triaxial tensile stress (3½ a, 0, 0) and a point in the compression meridian (n2, q2, 60°). Through boundary condition training, the calculations for the coefficients in Eq. 4 are described by Eq. 6.
pffiffiffi 3 3a a2 ¼ 2ðaabc a2bc Þ
a1 ¼ ð4bÞ
ð5aÞ
ð5dÞ
where a = ft/fc, abc = fcc/fc, ni = 3½rm/fc = (r1 + r2 + r2)/3½ and qi = 2sm/fc = [(r1r2)2 + (r2r3)2 + (r1r3)2]½/fc.
ð4aÞ
4.2.1. The solution of failure criterion A In the cylindrical coordinate system (n, q, u), the boundary conditions are uniaxial compressive stress (3½/3, 6½/3, 60°), uniaxial tensile stress (3½ a/3, 6½a/3, 0), biaxial compressive stress (2*3½ abc/3, 6½abc/3, 0), a point in the tensile meridian (n1, q1, 0) and a point in compression meridian (n2, q2, 60°). Through boundary condition training, the calculations for the coefficients in Eq. 4 are described by Eq. 5.
pffiffiffi 3q2 ð3q22 2Þa0 pffiffiffi 2 6q22 2q2
pffiffiffi pffiffiffi pffiffiffi ð 6n2 3q2 Þ ð3q2 6Þa0 pffiffiffi b2 ¼ 2q2 6q22
a0 ¼ b0 ¼
pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi 3B1 2þ 2b 2 2b 3C where a0 ¼ b0 ¼ Aþ2b , a1 ¼ 3 2Aþ4b ; a2 ¼ 32Aþ4b , b1 ¼ 3 2Aþ4b , and pffiffi pffiffi pffiffi 3 3B2 6b1 6a1 3 2 2 6a2 b2 ¼ 2Aþ4b. Then, we found that A ¼ , B1 ¼ 3ða1 þb1 Þ, a1 þb1 pffiffi pffiffi 3ða1 b1 Þ 2 6b2 6a0 B2 ¼ 3ða1 þb1 Þ, C ¼ a1 þb1 and b ¼ a1 þb1 .
pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 3að 2n1 þ q1 6abc Þ þ abc ð2 3q1 6n1 Þ pffiffiffi pffiffiffi a2 ¼ pffiffiffi ð 6a=3 q1 Þð 6aq1 þ 2a2bc 2aabc 6abc q1 Þ
pffiffiffi 3 3abc a þ 2a2 aabc ða abc Þ 3abc 3a
a0 ¼ b0 ¼
b1 ¼
b2 ¼
ð6aÞ
pffiffiffi 3a
ð6bÞ
3a2 þ 2abc a 2a2bc pffiffiffi 2ða2bc aabc Þ
ð6cÞ
pffiffiffi pffiffiffi 3q22 3að3q22 2Þ pffiffiffi 6q22 2q2
ð6dÞ
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ð 6n2 3q2 Þ 3að3q2 6Þ pffiffiffi 2q2 6q22
ð6eÞ
2n2
4.3. Five-parameter failure criterion B (hydrostatic stress is the main stress) To consider the hydrostatic stress as a dominating factor, we established a different five-parameter failure criterion model based on the twin shear strength theory. The model is described by Eq. 7.
F ¼ s13 þ s12 þ bðr13 þ r12 Þ þ Arm þ B1 ðrm Þ2 ¼ C;
F P F0
ð7aÞ
437
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
F 0 ¼ s13 þ s23 þ bðr13 þ r23 Þ þ Arm þ B2 ðrm Þ2 ¼ C;
F 6 F0
ð7bÞ
Eq. 7 can be expressed in a cylindrical coordinate system, as described by Eq. 8. The coordinate transformation method is shown in Appendix A.
q cos u ¼
pffiffiffi pffiffiffi pffiffiffi 6 2ðA1 þ 2bÞ 6B1 2 n n ; C 3þb 9 þ 3b 3þb
q cosðu p=3Þ ¼ ub 6 u 6 60
0 6 u 6 ub
ð8aÞ
pffiffiffi pffiffiffi pffiffiffi 6 2ðA2 þ 2bÞ 6B2 2 n n ; C 3b 9 3b 3b ð8bÞ
Five experimental characteristic stresses are used as the boundary conditions. For convenient calculation, we selected these characteristic stresses from the tensile meridian (r1 r2 = r3, u = 0) and compression meridian (r1 = r2 r3, u = 60°). In the cylindrical coordinate system (n, q, u), the boundary conditions are uniaxial compressive stress (3½/3, 6½/3, 60°), uniaxial tensile stress (3½ a/3, 6½a/3, 0), biaxial compressive stress (2*3½ abc/3, 6½ abc/3, 0), uniaxial tensile stress (3½ a, 0, 0) and a point in the compression meridian (n2, q2, 60°). Eq. 11 can be expressed as Eq. 9.
q ¼ a0 þ a1 n þ a2 n2 ; u ¼ 0
ð9aÞ
q ¼ b0 þ b1 n þ b2 n2 ; u ¼ 60
pffiffi 1 þ2bÞ where a0 ¼ a1 ¼ 2ðA3þb , pffiffi pffiffi 2ðA2 þ2bÞ 2 B 2 b1 ¼ 3b , and b2 ¼ pffiffi3ð3bÞ . pffiffi pffiffi 6a0 6b0 3 2a1 b0 6a0 6b0 3 2a0 b1 , A2 ¼ , A1 ¼ a0 þb0 a0 þb0 pffiffi 3ðb0 a0 Þ 6a0 b0 C ¼ a0 þb0 and b ¼ a0 þb0 . pffiffi 6C , 3þb
a2 ¼ Then,
pffiffi 2B1 pffiffi3ð3þbÞ ,
we pffiffi
ð9bÞ b0 ¼
pffiffi 6C , 3b
found
that pffiffi 3 6a2 b0 3 6a0 b2 B1 ¼ a0 þb0 , B2 ¼ a0 þb0 ,
Through boundary condition training, the calculations for the coefficients in Eq. 9 are described by Eq. 10.
a2 ¼
pffiffiffi 3 6a 2ð3a þ 2abc Þða þ 2abc Þ
pffiffiffi 6a þ 2a2 a2 a0 ¼ 2 a1 ¼
b2 ¼
pffiffiffi 6 þ 8aa2 pffiffiffi 2 3
pffiffiffi pffiffiffi pffiffiffi 6ðn2 þ 3aÞ 3q2 ð1 þ 3aÞ pffiffiffi pffiffiffi pffiffiffi ðn2 þ 3aÞð1 þ 3a 3 3n2 a 3n2 Þ
pffiffiffi pffiffiffi 3aq2 ð 3an22 þ 3a2 n2 Þb2 pffiffiffi b0 ¼ n2 þ 3a pffiffiffi ð1 9a Þb2 6 pffiffiffi pffiffiffi b1 ¼ 3 þ 3 3a 2
ð10aÞ
ð10bÞ ð10cÞ
ð10dÞ
ð10eÞ
F 0 ¼ s13 þ s23 þ bðr13 þ r23 Þ þ Arm þ Br2m þ C 2 ðs13 þ s23 Þ2 ¼ D; F 6 F 0 ð11bÞ Eq. 11 can be expressed in a cylindrical coordinate system as Eq. 12. The coordinate transform method is shown in Appendix A.
pffiffiffi pffiffiffi ! 6 6 6C 1 ðq cos uÞ þ þ b q cos u 2 6 pffiffiffi ! pffiffiffi 2 3 B 3 b n n2 ; 0 6 u 6 ub ¼D Aþ 3 3 3 2
pffiffiffi pffiffiffi ! 6 6 6C 2 ½q cosðu p=3Þ þ b q cosðu p=3Þ 2 6 ! pffiffiffi pffiffiffi 2 3 B 3 b n n2 ; ub 6 u 6 60 ¼D Aþ 3 3 3
ð12aÞ
2
ð12bÞ
To solve for the unknown quantities, six experimental characteristic stresses are used as boundary conditions. For convenient calculation, we selected the characteristic stresses from the tensile meridian (r1 r2 = r3, u = 0) and compression meridian (r1 = r2 r3, u = 60°). Therefore, Eq. 12 can be expressed as Eq. 13. We study two kinds of complex boundary conditions: one kind considers the triaxial compressive stress and the other considers the triaxial tensile stress.
a1 q2 þ q ¼ a2 þ a3 n þ a4 n2 ;
u¼0
ð13aÞ
b1 q2 þ q ¼ b2 þ b3 n þ b4 n2 ;
u ¼ 60
ð13bÞ
pffiffi pffiffi pffiffi pffiffi pffiffi ðAþ2bÞ 6C 1 6C 2 6D 6B where a1 ¼ 63þb , a2 ¼ 3þb , a3 ¼ 23þb , a4 ¼ 3ð3þbÞ , b1 ¼ 63b , ffiffi p pffiffi pffiffi ðAþ2bÞ 6D 6B b2 ¼ 3b , b3 ¼ 23b , and b4 ¼ 3ð3bÞ . Then, we found that pffiffi pffiffi pffiffi pffiffi pffiffi 2a3 b2 a4 b2 a2 b3 1 b2 2 b1 , B ¼ 3a26þb , C 1 ¼ 6ða62aþb , C 2 ¼ 6ða62aþb , D ¼ a62 þb A ¼ 6a2 6ba22 3 þb2 2 2Þ 2Þ 2 2 a2 Þ and b ¼ 3ðb . a2 þb2
4.4.1. The boundary conditions considering triaxial compressive stress In a cylindrical coordinate system (n, q, u), the boundary conditions are uniaxial compressive stress (3½/3, 6½/3, 60°), uniaxial tensile stress (3½ a/3, 6½a/3, 0), biaxial compressive stress (2*3½ abc/3, 6½ abc/3, 0), triaxial compressive stress (3½ acc, 0, 0), a point in the tensile meridian (n1, q1, 0) and a point in the compression meridian (n2, q2, 60°). Through boundary condition training, the calculations for the coefficients in Eq. 13 are described by Eq. 14.
a4 ¼
pffiffiffi pffiffiffi 6aðx1 y2 z1 x1 y1 z1 Þþ 6abc ðx21 z2 x1 x2 z1 Þþ3q1 ðx1 x2 y2 x21 y2 Þ 3ðx21 y3 z2 þx1 x2 y1 z3 þx1 x3 y2 z1 x21 y2 z3 x1 x2 y3 z1 x1 x3 y1 z2 Þ ð14aÞ
ð10fÞ
4.4. Six-parameter failure criterion To fully consider both the hydrostatic stress and shear stress as the two important factors in the failure of concrete, we established a six-parameter failure criterion model based on the twin shear strength theory. This model is described by Eq. 11. In this model, we not only introduced the hydrostatic stress but also focused on the influence of the shear stress.
F ¼ s13 þ s12 þ bðr13 þ r12 Þ þ Arm þ Br2m þ C 1 ðs13 þ s12 Þ2 ¼ D; F P F 0 ð11aÞ
pffiffiffi 6ðay1 abc x1 Þ 3a4 ðx3 y1 x1 y3 Þ a3 ¼ 3ðx2 y1 x1 y2 Þ
ð14bÞ
a2 ¼
pffiffiffi 3a3 acc 3a4 a2cc
ð14cÞ
a1 ¼
pffiffiffi 6ðay2 abc x2 Þ 3a4 ðx3 y2 x2 y3 Þ x2 y1 x1 y2
ð14dÞ
b2 ¼
pffiffiffi 6q2 2q2 p ffiffiffi 2 a3 2 3q2 2 a2 ð 3q22 2n2 Þ þ aa42 ðq22 2n22 Þ
ð14eÞ
pffiffiffi pffiffiffi a3 a4 =2 6=2 b1 ¼ b2 3 3 þ a2 a2
ð14fÞ
438
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
b3 ¼
a3 b2 a2
ð14gÞ
b4 ¼
a4 b2 a2
ð14hÞ
5.1. The five-parameter model A
pffiffiffi x1 ¼ 2a2 =3, x2 ¼ 3ða þ 3acc Þ=3, x3 ¼ ða2 9a2cc Þ=3, pffiffiffi 2 2 y1 ¼ 2abc =3, y2 ¼ 3ð3acc 2abc Þ=3, y3 ¼ ð4abc 9a2cc Þ=3, z1 ¼ q21 , pffiffiffi z2 ¼ 3acc n1 , and z3 ¼ n21 3a2cc . where
4.4.2. The boundary conditions considering triaxial tensile stress In a cylindrical coordinate system (n, q, u), the new boundary conditions are uniaxial compressive stress (3½/3, 6½/3, 60°), uniaxial tensile stress (3½a/3, 6½a/3, 0), biaxial compressive stress (2*3½ abc/3, 6½ abc/3, 0), triaxial tensile stress (3½ a, 0, 0), a point in the tensile meridian (n1, q1, 0) and a point in the compression meridian (n2, q2, 60°). All the characteristic stresses are from the tensile meridian (r1 r2 = r3, u = 0) and compression meridian (r1 = r2 r3, u = 60°). Through boundary condition training, the calculations for the coefficients in Eq. 13 are described by Eq. 15.
a4 ¼
pffiffiffi pffiffiffi 6aðx1 y2 z1 x1 y1 z1 Þþ 6abc ðx21 z2 x1 x2 z1 Þþ3q1 ðx1 x2 y2 x21 y2 Þ 3ðx21 y3 z2 þx1 x2 y1 z3 þx1 x3 y2 z1 x21 y2 z3 x1 x2 y3 z1 x1 x3 y1 z2 Þ ð15aÞ
a3 ¼
pffiffiffi 6ðay1 abc x1 Þ 3a4 ðx3 y1 x1 y3 Þ 3ðx2 y1 x1 y2 Þ
ð15bÞ
pffiffiffi a2 ¼ ð 3a3 a þ 3a4 a2 Þ
ð15cÞ
pffiffiffi 6ðay2 abc x2 Þ 3a4 ðx3 y2 x2 y3 Þ a1 ¼ x2 y 1 x 1 y 2
ð15dÞ
b2 ¼
5. Failure criteria validation
To verify the veracity and reliability of the failure criteria, we used r1 and r2 as the known data to compute r3, and we compared the model results against the experimental data. The method for completing the computations with the principal stress equation is shown in Appendix B. The results are shown in Table 4. In Table 4, rexp is measured experimentally, r0 31 is computed using Eqs. 2 ,4, 3 5, and r0 32 is computed using Eqs. 2, 4, 6. Ei, the relative error, reflects the difference between the computation data and the experimental data and was computed using Eq. (16).
Ei ¼
ðr03i rexp 3 Þ
rexp 3
ði ¼ 1; 2; 3:::Þ
ð16Þ
In Table 4, E1 and E2 for specimen TCS10 and E1 for specimen TCS13 are 0 because the data for these specimens were used to compute the coefficients of the model. There is little difference between r0 31 and r0 32, indicating that the simplified solution of the failure criterion model is practical. However, the results in Table 4 also show that E1 and E2 for several specimens (TCS3, TCS6, TCS7, TCS8, TCS14) were large (exceeding 15%), indicating that the model (in which shear stress is the main stress) cannot be applied to the concrete loading state in which hydrostatic stress has a dominant influence. If the stress state is r1 – r2 – r3 or approximately r1 = r2 = r3, the hydrostatic stress becomes more important. 5.2. The five-parameter model B
pffiffiffi 6q2 2q2 p ffiffiffi 2 a3 2 3q2 2 a2 ð 3q22 2n2 Þ þ aa42 ðq22 2n22 Þ
ð15eÞ
pffiffiffi pffiffiffi a3 a4 =2 6=2 b1 ¼ b2 3 3 þ a2 a2
ð15fÞ
b3 ¼
a3 b2 a2
ð15gÞ
b4 ¼
a4 b2 a2
ð15hÞ
pffiffiffi x1 ¼ 2a2 =3, x2 ¼ 2 3a=3, x3 ¼ 8a2 =3, y1 ¼ 2a2bc =3, pffiffiffi y3 ¼ ð4a2bc 9a2 Þ=3, z1 ¼ q21 , z2 ¼ y2 ¼ 3ð3a þ 2abc Þ=3, pffiffiffi 2 2 ð 3a þ n1 Þ and z3 ¼ n1 3a . where
To verify the veracity and reliability of the failure criterion model that considers hydrostatic stress as a dominating factor, Table 5 The validation of the five-parameter model B. Concrete
rexp (MPa) 3
r0 33 (MPa)
E3 (%)
TCS3 TCS4 TCS5 TCS6 TCS7 TCS8 TCS9 TCS10 TCS11 TCS12 TCS13 TCS14
325.34 52.83 72.87 71.77 69.82 61.65 57.23 150.36 188.01 187.28 166.69 83.96
83.15 58.93 90.91 95.03 74.55 60.57 49.91 150.36 166.45 98.05 60.06 90.91
74.44 11.55 24.76 32.41 6.77 1.75 12.79 0.00 11.46 47.65 63.95 8.28
Note: rexp is the experimental data, r0 33 is the computed data, and E3 is the relative 3 error.
Table 4 The validation of the five-parameter model A. Concrete
rexp (MPa) 3
r0 31 (MPa)
E1 (%)
r0 32 (MPa)
E2 (%)
TCS3 TCS4 TCS5 TCS6 TCS7 TCS8 TCS9 TCS10 TCS11 TCS12 TCS13 TCS14
325.34 52.83 72.87 71.77 69.82 61.65 57.23 150.36 188.01 187.28 166.69 83.96
725.67 51.81 77.91 85.46 89.30 73.58 59.40 150.36 180.25 217.35 166.70 161.15
123.05 1.93 6.92 19.07 27.90 20.37 3.79 0.00 4.12 16.06 0.00 101.73
655.06 51.89 77.92 85.47 88.3 72.67 58.52 150.36 180.24 213.76 158.71 137
101.35 1.78 6.93 19.09 26.46 18.88 2.25 0.00 4.13 14.14 4.79 71.49
Note: rexp is the experimental data, r0 31 and r0 32 are the computed data, and Ei is the relative error (i = 1, 2). 3
439
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441 Table 6 The validation of six-parameter model. Concrete
rexp (MPa) 3
r’34 (MPa)
E4 (%)
r’35 (MPa)
E5 (%)
TCS3 TCS4 TCS5 TCS6 TCS7 TCS8 TCS9 TCS10 TCS11 TCS12 TCS13 TCS14
325.34 52.83 72.87 71.77 69.82 61.65 57.23 150.36 188.01 187.28 166.69 83.96
325.34 55.61 80.11 90.74 78.69 65.98 52.27 150.36 174.56 165.56 124.83 92.68
0.00 5.26 14.05 26.43 12.70 7.02 8.67 0.00 7.15 11.89 25.07 10.39
325.34 56.26 79.94 79.93 59.66 56.41 43.43 149.45 172.8 96 166.67 83.67
0 6.49 9.70 11.37 14.55 8.50 24.11 0.60 8.09 48.74 0.05 0.01
Note: rexp is the experimental data, r0 34 and r0 35 are the computed data, and Ei is the relative error (i = 4, 5). 3
we used r1 and r2 as the known data to compute r3, following the method shown in Appendix C. The results are shown in Table 5. In Table 5, rexp is measured experimentally, and r0 33 is computed 3 using Eqs. 7, 9, 10. In Table 5, E3 for specimen TCS10 is 0 because the data for this specimen were used to compute the coefficients for the model. If the three principal stresses are different from each other or if they become similar, the model yields an accurate result. The results in Table 5 show that E3 for some specimens (TCS3, TCS5, TCS6, TCS12, TCS13) was large (exceeding 15%), indicating that the model (in which hydrostatic stress is the main stress) is not applicable for concrete loading states in which shear stress has a dominant influence. For the validation of r33, model A and model B show similar results.
5.3. The six-parameter model To verify the veracity and reliability of the six-parameter failure criterion model, we computed r3 from the principal stress equation, as shown in Appendix D. These results are shown in Table 6. In Table 6, rexp is measured experimentally, r0 34 is computed using 3 Eqs. 11, 13, 14, and r0 35 is computed using Eqs. 11, 13, 15. The results in Table 6 show that the calculated data for most of the specimens were consistent with the experimental results. However, some computed data from the model under different boundary conditions were very different, indicating that the results from the six-parameter model mostly depend on the boundary conditions.
6. Conclusions Though biaxial and triaxial experiments on concrete specimens, six mechanical effects were examined. The intermediate principal stress, hydrostatic pressure and shear stress are the most important factors. Based on the twin shear strength theory, three kinds of multiparameter failure criteria models were developed for concrete under triaxial compression. The models have clear physical significance. The five-parameter failure criterion (model A) that considers shear stress as the main stress cannot be applied to the stress state in which r1 – r2 – r3 or approximately r1 = r2 = r3. However, this model can be applied to examine concrete structures under low confining pressure. The five-parameter failure criterion (model B) that considers hydrostatic pressure as the main stress can be applied to complicated stress states. Furthermore, the five-parameter failure criterion models A and B show similar results.
The six-parameter failure criterion model predicts stresses that closely match the experimental results. However, the model results are likely to depend on the boundary conditions. Acknowledgements The authors gratefully acknowledge the contributions of Maohong Yu (Xi’an Jiao Tong University) for the information on the twin shear strength theory. The first author acknowledges the China Scholarship Council, Xi’an University of Architecture and Technology and the University of Wollongong for supporting his Joint PhD programme. Funding sources This work was supported by the Major State Key Research Development Foundation of China – China [grant number 2017YFC0703400] and the National Science Foundation of China – China [grant number 51478382]. Conflict of interest The authors have no conflicts of interest to declare. Appendix A In this paper, we establish the model in the stress space in which the hydrostatic stress axis is the primary axis, and the coordinate in the p plane is defined using cylindrical coordinate system. The primary stress point P (r1, r2, r3) can be expressed as P (n, q, u) in a cylindrical coordinate system. The relation between the cylindrical coordinate system (n, q, u) and the primary stress system (r1, r2, r3) is shown below:
pffiffiffi n ¼ ðr1 þ r2 þ r3 Þ= 3
q¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ðr1 r2 Þ2 þ ðr2 r3 Þ2 þ ðr1 r3 Þ2 = 3
2r1 r2 r3 cos u ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ðr1 r2 Þ þ 2ðr2 r3 Þ2 þ 2ðr1 r3 Þ2 The principal stress expression can be derived as follows:
pffiffiffi
pffiffiffi
r1 ¼ ðn þ 2q cos uÞ= 3 pffiffiffi
pffiffiffi
r2 ¼ ½n þ 2q cosðu 2p=3Þ= 3 pffiffiffi pffiffiffi pffiffiffi ¼ ½2n þ 2qð 3 sin u cos uÞ= 3
440
r3
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441
pffiffiffi pffiffiffi ¼ ½n þ 2q cosðu þ 2p=3Þ= 3 pffiffiffi pffiffiffi pffiffiffi ¼ ½2n 2qð 3 sin u þ cos uÞ= 3
a1 ¼ B1 =9; b1 ¼ b=2 þ A1 =3 þ 2B1 ðr1 þ r2 Þ=9 1=2; and c1 ¼ ð1 þ b þ A1 =3Þr1 þ ðb=2 þ A1 =3 1=2Þr2 þ B1 ð2r1 þ r2 Þ2 =9 C;
The parameters of the multiparameter failure criterion model can be expressed as follows:
pffiffiffi
s12 þ s13 ¼ 6q cos u=2 s13 þ s23
pffiffiffi ¼ 6q cosðu p=3Þ=2
r12 þ r13
pffiffiffi pffiffiffi ¼ ð4n þ 2q cos uÞ=2 3 pffiffiffi
a2 ¼ B2 =9; b2 ¼ b þ A2 =3 þ 2B2 ðr1 þ r2 Þ=9 1; and c2 ¼ ðb=2 þ A2 =3 þ 1=2Þðr1 þ r2 Þ þ B2 ð2r1 þ r2 Þ2 =9 C: The method to determine r3 is the same as that described in Appendix B.
pffiffiffi
r13 þ r23 ¼ ½4n 2q cosðu p=3Þ=2 3 pffiffiffi
Appendix D Eq. 11 can also be expressed as simplified monadic quadratic equations:
rm ¼ ðr1 þ r2 þ r3 Þ=3 ¼ 3n=3
a1 r23 þ b1 r3 þ c1 ¼ 0
Appendix B Eq. 2 can be translated into a monadic quadratic equation:
3B1 r23 þ r3 ½6b þ 4A 6 6B1 ð2r1 r2 Þ þ 6ð2r1 r2 Þ þ 6bð2r1 þ r2 Þ þ 4Aðr1 þ r2 Þ þ 3B1 ð2r1 r2 Þ2 12C ¼ 0 12B2 r23 þ r3 ½12b þ 4A 12 12B2 ðr1 þ r2 Þ þ 3B2 ðr1 þ r2 Þ2 þ ðr1 þ r2 Þð6 þ 6b þ 4AÞ 12C ¼ 0 The monadic quadratic equation can be expressed as simplified equations.
a1 r23 þ b1 r3 þ c1 ¼ 0
a2 r23 þ b2 r3 þ c2 ¼ 0: In the foregoing equations,
a1 ¼ B=9 þ C 1 ; b1 ¼ b=2 þ A=3 þ 2Bðr1 þ r2 Þ=9 2C 1 ð2r1 r2 Þ 1=2; and c1 ¼ ð2r1 r2 Þ=2 þ bð2r1 þ r2 Þ=2 þ Aðr1 þ r2 Þ=3 þ Bðr1 þ r2 Þ2 =9 þ C 1 ð2r1 r2 Þ2 D; a2 ¼ B=9 þ 4C 2 ; b2 ¼ b þ A=3 þ 2Bðr1 þ r2 Þ=9 4C 2 ðr1 þ r2 Þ 1; and
a2 r þ b2 r3 þ c2 ¼ 0 2 3
c2 ¼ ðb=2 þ A=3 þ 1=2Þðr1 þ r2 Þ þ ðB=9 þ C 2 Þð2r1 þ r2 Þ2 D:
In the foregoing equations,
a1 ¼ 3B1 ; b1 ¼ 6b þ 4A 6 6B1 ð2r1 r2 Þ; and
The method to determine r3 is the same as described in Appendix B.
c1 ¼ 6ð2r1 r2 Þ þ 6bð2r1 þ r2 Þ þ 4Aðr1 þ r2 Þ þ 3B1 ð2r1 r2 Þ2
References
12C; a2 ¼ 12B2 ; b2 ¼ 12b þ 4A 12 12B2 ðr1 þ r2 Þ; and c2 ¼ 3B2 ðr1 þ r2 Þ2 þ ðr1 þ r2 Þð6 þ 6b þ 4AÞ 12C: In this paper, r3 is compressive stress and is computed as follows:
r ¼ 0 3
b1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b1 4a1 c1 2a1
;r ¼ 00 3
b2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 4a2 c2 2a2
Because of the selection in Eq. 3, r3 should be greater than r03 and r003 , i.e.,
r3 ¼ maxðr03 ; r003 Þ: Appendix C Eq. 7 can be expressed as simplified monadic quadratic equations as follows:
a1 r23 þ b1 r3 þ c1 ¼ 0 a2 r23 þ b2 r3 þ c2 ¼ 0: In the foregoing equations,
[1] A. Kaya, M. Dawood, B. Gencturk, Repair of corroded and buckled short steel columns using concrete-filled GFRP jackets, Construct. Build. Mater. 94 (2015) 20–27. [2] J.A. Øverli, A density-dependent failure criterion for concrete, Construct. Build. Mater. 124 (2016) 566–574. [3] J. Wang, A.A.A. Molenaar, M.F.C.V.D. van de Ven, S. Wu, Behaviour of asphalt concrete mixtures under tri-axial compression, Construct. Build. Mater. 105 (2016) 269–274. [4] I.E. Shkolnik, Influence of high strain rates on stress–strain relationship, strength and elastic modulus of concrete, Cem. Concr. Compos. 30 (2008) 1000–1012. [5] D. Yan, G. Lin, Influence of initial static stress on the dynamic properties of concrete, Cem. Concr. Compos. 30 (2008) 327–333. [6] Z. Chen, Y. Hu, Q. Li, M. Sun, P. Lu, T. Liu, Behavior of concrete in water subjected to dynamic triaxial compression, J. Eng. Mech. 136 (2010) 379–389. [7] T. Ozbakkaloglu, Compressive behavior of concrete-filled FRP tube columns: assessment of critical column parameters, Eng. Struct. 51 (2013) 188–199. [8] Y.B. Wang, J.Y.R. Liew, S.Ch. Lee, D.X. Xiong, Experimental study of ultra-highstrength concrete under triaxial compression, ACI Mater. J. 113 (2016) 105– 112. [9] J. Zhou, X. Chen, L. Wu, X. Kan, Influence of free water content on the compressive mechanical behaviour of cement mortar under high strain rate, Sadhana 36 (2011) 357–369. [10] P. Rossi, J.L. Tailhan, F.L. Le Maou, Creep strain versus residual strain of a concrete loaded under various levels of compressive stress, Cem. Concr. Res. 51 (2013) 32–37. [11] D. Zheng, Q. Li, L. Wang, Rate effect of concrete strength under initial static loading, Eng. Fract. Mech. 74 (2007) 2311–2319. [12] H.W. Wang, Ch.Q. Wu, F.R. Zhang, Q. Fang, H. Xiang, P. Li, Z. Li, Y. Zhou, Y. Zhang, J. Li, Experimental study of large-sized concrete filled steel tube columns under blast load, Construct. Build. Mater. 134 (2017) 131–141. [13] X.W. Tang, C.H. Zhang, Study on concrete in macro-and meso-scale mechanical properties based on homogenization theory, Chin. J. Comp. Mech. 26 (2009) 876–881.
C. Rong et al. / Construction and Building Materials 161 (2018) 432–441 [14] W.C. Zhu, C.A. Tang, S.Y. Wang, Numerical study on the influence of mesomechanical properties on macroscopic fracture of concrete, Struct. Eng. Mech. 19 (2005) 519–533. [15] J. Xiao, W. Li, Q. Liu, Meso-level numerical simulation on mechanical properties of modeled recycled concrete under uniaxial compression, J. Tongji Univ. 39 (2011) 791–797. [16] B. Zhang, Relationship between pore structure and mechanical properties of ordinary concrete under bending fatigue, Cem. Concr. Res. 28 (1998) 699–711. [17] P.G. Asteris, V. Plevris, Anisotropic masonry failure criterion using artificial neural networks, Neural. Comput. Appl. 28 (2016) 2207–2229. [18] C.A. Syrmakezis, P.G. Asteris, Masonry failure criterion under biaxial stress state, Cem. Concr. Res. 13 (2001) 58–64.
441
[19] S. Shang, Y. Song, Dynamic biaxial tensile-compressive strength and failure criterion of plain concrete, Construct. Build. Mater. 40 (2013) 322–329. [20] B. Piscesa, M.M. Attard, A.K. Samani, A Lateral strain plasticity model for FRP confined concrete, Compos. Struct. 158 (2016) 160–174. [21] V. Plevris, P.G. Asteris, Modeling of masonry failure surface under biaxial compressive stress using Neural Networks, Construct. Build. Mater. 55 (2014) 447–461. [22] M.H. Yu, New System of Strength Theory: Theory, Development and Applications, Xian Jiaotong University Press, Xi’an, 2011. [23] P. Menetrey, K.J. Willam, Triaxial failure criterion for concrete and its generalization, ACI Struct. J. 92 (1995) 311–318.