- Email: [email protected]
Strength criterion for plain concrete under multiaxial stress based on damage poisson's ratio
Acta Mechanica Solida Sinica, Vol. 19, No. 4, December, 2006 Published by AMSS Press, Wuhan, China. DOI: 10.1007/s10338-006-0637-1
ISSN 0894-9166
STRENGTH CRITERION FOR PLAIN CONCRETE UNDER MULTIAXIAL STRESS BASED ON DAMAGE POISSON’S RATIO Ding Faxing
Yu Zhiwu
(School of Civil Engineering and Architecture, Central South University, Changsha 410075, China)
Received 26 July 2005; revision received 20 November 2006
ABSTRACT A new unified strength criterion in the principal stress space has been proposed for use with normal strength concrete (NC) and high strength concrete (HSC) in compressioncompression-tension, compression-tension-tension, triaxial tension, and biaxial stress states. The study covers concrete with strengths ranging from 20 to 130 MPa. The conception of damage Poisson’s ratio is defined and the expression for damage Poisson’s ratio is determined basically. The failure mechanism of concrete is illustrated, which points out that damage Poisson’s ratio is the key to determining the failure of concrete. Furthermore, for the concrete under biaxial stress conditions, the unified strength criterion is simplified and a simplified strength criterion in the form of curves is also proposed. The strength criterion is physically meaningful and easy to calculate, which can be applied to analytic solution and numerical solution of concrete structures.
KEY WORDS plain concrete, high strength concrete, multiaxial stress, strength criterion, meridian, deviatoric plan
I. INTRODUCTION Strength theory of concrete deals with the failure of concrete in a complex stress state. It is important and fundamental for the strength calculation and design of concrete structures. It is of great significance in theoretical research and engineering application, and is also very important for the effective utilization of materials. Some structures may be subjected to multiaxial loads, such as concrete-filled steel tubular columns, offshore oil platforms, nuclear containment vessels, and many other structures. Since the failure of concrete in a structure can occur differently in complex stress states, an understanding of the behavior of concrete is needed to develop the failure criteria for concrete. The compressive strength of concrete is the principal property employed in the design of reinforced concrete structures. After the pioneering investigation by Richart et al.[1] on triaxial compressive behavior of NC, many other researchers, such as Mills and Zimmerman[2] , Ottosen[3] , and Podgorski[4] have also conducted studies on the behavior of NC in multiaxial stress states. More recently, the investigations[5−−7] have also been extended to include HSC. Many failure criteria for concrete, typically fitted with different functions, have been proposed by researchers[3, 4, 8, 9] . These criteria are the Octahedral-Stress Strength theory (OSS theory). Guo[10] made a general survey of these criteria and indicated that five-parameter Guo-Wang criteria[8] , four-parameter Ottosen criteria[3] , and five-parameter Podgorski criteria[4] could fit the experimental data regarding the strength of concrete subjected to multiaxial stress quite well.
Project supported by the National Natural Science Foundation of China (Nos. 50438020 and 50578162).
· 308 ·
ACTA MECHANICA SOLIDA SINICA
2006
Yu[11, 12] proposed a twin-shear strength theory and a unified strength theory, which can be adapted to metal, concrete, and rock materials and have been widely used in the strength analysis of engineering structures. The Mohr-Coulomb theory and Tresca criterion are special cases of Yu’s theory. The twinshear strength theory is physically meaningful in itself, but for the multi-parameter twin-shear strength criterion, the main stream of unified strength theory, it is more suited to functions than physical meaning. Zhou[13] proposed a theory of minimum energy dissipation rate which is suitable for the nonlinear state of non-equilibrium systems, and based on the theory, the theoretic expression of strength criterion for concrete in multiaxial stress states was developed. The failure mechanism of concrete was also illustrated, which shows that the failure of concrete is a process of energy dissipation and the failure must be restricted by the minimum energy dissipation rate. The merit of Zhou criterion[13] is its physical meaning and relatively high precision; the merit its complex expression and the parameters must be determined by the experimental data with complex calculation. Based on two basic assumptions, the conception of damage Poisson’s ratio is defined and the theories of damage mechanics and minimum energy dissipation rate are applied in this paper. The general formula of strength criterion for plain concrete under multiaxial stress states is proposed and the failure mechanism of concrete is also illustrated. Based on the general shape of failure surface for concrete and the existing experimental data, the expression for damage Poisson’s ratio is determined basically. The strength criterion is physically meaningful and easy for calculation, which can be applied to analytic and numerical solution of concrete structures under compression-compression-tension, compression-tensiontension, triaxial tension, and biaxial stress states with concrete strengths ranging from 20 to 130 MPa.
II. BASIC ASSUMPTION
Many experimental achievements[10] show that: for the concrete prisms or cylinder specimens under uniaxial compression, the Poisson’s ratio of concrete increases gradually as the axial strain increases; for low strain, the value of the Poisson’s ratio for concrete is about 0.2, and for the axial strain larger than the peak-strain, the value will be greater than 0.5; thereafter, the Poisson’s ratio increases continuously, even greater than 1.0 when it is loaded to failure. For the concrete specimens under uniaxial tension, the Poisson’s ratio of concrete decreases Fig. 1 Calculation model of strain for concrete. gradually as the axial strain increases. For low strain, the value of the Poisson’s ratio for concrete is about 0.2, but for the axial strain larger than the peakstrain, the value will be less than 0.2. Based on the above cognition, seen from Fig.1, the following assumption can be presented: (1) The total principal strain (ε) of concrete may be classified into elastic principal strain (εe ) and inelastic principal strain (εI ), that is ε = εe + εI
(1)
(2) The elastic strain follows the Hooke’s law, and the increase or decrease of the Poisson’s ratio is mainly relevant to damage Poisson’s ratio νD . For the Poisson’s effect, the inelastic principal strain under one axis caused by the stress ratio will result in the inelastic principal strains at the other two axes, and the inelastic principal strain ratio is defined as damage Poisson’s ratio. The stress ratio is defined as the ratio of the principal stress in the axis to the uniaxial (compressive or tensile) strength of the same stress state.
III. THEORETIC ANALYSIS OF STRENGTH CRITERION FOR CONCRETE The constitutive model for concrete can be written as follows: ε=
σ g[D(t)]Ec
(2)
Vol. 19, No. 4
· 309 ·
Ding Faxing et al.: Strength Criterion for Plain Concrete
in which σ is the principal stress of concrete; Ec the initial elastic modulus of concrete; D damage variable; both g[D(t)] and D are the function of time t denoting the process of energy dissipation. For convenience, g[D(t)] can be abbreviated to g. g satisfies: at D=0, g=1; at D=Dr , g →0; ∀ 0 < D < Dr , the slope of the curve of g decreases monotonical and dg/dt = g˙ = 0. g and D are the functions of time t. Based on the theory of the minimum energy dissipation rate[13] , for the element of concrete, if the inelastic strains εI,i (t)(i=1,2,3) caused by loads are regarded as the only mechanism of energy dissipation, in which t is the time parameter, the ratio of energy dissipation rate ψi (t)|t=0 to fi2 can be expressed by 3 3 ψi (t) σi ε˙I,i (t) = (3) fi2 fi2 i=1
i=1
t=0
t=0
where fi (i=1,2,3) is the peak stress of concrete under uniaxial stress, for the uniaxial compressive case, fi is uniaxial compressive strength fc , for the uniaxial tensile case, fi is uniaxial tensile strength ft ; σi (i = 1, 2, 3) is the principal stress of concrete; ε˙I,i (t) is inelastic principal strain rate at any time t. Based on basic assumption (1), the constitutive model for concrete during energy dissipation can be expressed as ⎧ ⎫ ⎤⎧ ⎫ ⎤⎧ ⎫ ⎡ ⎡ 1 −ν −ν ⎨ σ1 ⎬ 1 −r2 −r3 ⎨ σ1 ⎬ ⎨ ε1 ⎬ 1 ⎣ 1 − g ⎣ −r1 1 −r3 ⎦ σ2 −ν 1 −ν ⎦ σ2 + ε2 = (4) ⎩ ⎭ Ec ⎩ ⎭ ⎩ ⎭ gE c −ν −ν 1 ε σ −r −r 1 σ 3
3
1
2
3
where ν is the initial Poisson’s ratio of concrete; ri (i=1,2,3) is the Poisson’s ratio caused by inelastic principal strain. The inelastic principal strains are expressed as ⎧ ⎫ ⎤⎧ ⎫ ⎡ 1 −r2 −r3 ⎨ σ1 ⎬ ⎨ εI,1 ⎬ 1 − g ⎣ −r1 1 −r3 ⎦ σ2 εI,2 = (5) ⎩ ⎭ ⎩ ⎭ gEc εI,3 −r1 −r2 1 σ3 For the concrete during energy dissipation, Eq.(5) can be written as σ2 σ3 ε˙I,1 (t) σ1 −g˙ = 2 − r2 − r3 f1 g Ec f1 f1 f1 ε˙I,2 (t) σ1 σ3 σ2 −g˙ = 2 − r1 − r3 f2 g Ec f2 f2 f2 σ1 σ2 ε˙I,3 (t) σ3 −g˙ = 2 − r1 − r2 f3 g Ec f3 f3 f3
(6)
Based on basic assumption (2), let f1 = lf2 , f1 = mf 3 , and f2 = nf 3 (l, m, and n are parameters). Thus Eq.(6) can be written as ε˙I,1 (t) σ2 σ3 σ1 −g˙ = 2 − νD,21 − νD,31 f1 g Ec f1 f2 f3 ε˙I,2 (t) σ1 σ3 σ2 −g˙ (7) = 2 − νD,12 − νD,32 f2 g Ec f2 f1 f3 σ1 σ2 ε˙I,3 (t) σ3 −g˙ = 2 − νD,13 − νD,23 f3 g Ec f3 f1 f2 where, νD,ij is the damage Poisson’s ratio in i-axial caused by j-axial, and i = j; νD,21 = r2 /l, νD,31 = r3 /m, νD,12 = r1 l, νD,32 = r3 /n, νD,13 = r1 m, νD,23 = r2 n. As the processes of energy dissipation and failure are nonreversible, the increase of inelastic strains of concrete is also nonreversible. By substituting Eq.(7) into Eq.(3), the ratio of energy dissipation rate to fi2 can be expressed by 3 ψi (t) −g˙ σ12 σ22 σ32 σ1 σ2 = + + 2 2 2 2 − (νD,12 + νD,21 ) f f 2E f g f f f c 1 2 1 2 3 i t=0 i=1 σ2 σ3 σ1 σ3 (8) −(νD,23 + νD,32 ) − (νD,13 + νD,31 ) f2 f3 f1 f3
· 310 ·
ACTA MECHANICA SOLIDA SINICA
2006
The failure criterion for concrete can be described in terms of the principal stresses in the following form F (σ1 , σ2 , σ3 ) = 0 (9) Equation (9) is the restrictive condition of Eq.(8) during energy dissipation at any time t. Based on the theory of minimum energy dissipation rate[13] , the following equation can be obtained as 3 ∂ ψi (t)/fi2 + λF i=1 =0 (i = 1, 2, 3) (10) ∂σi in which λ is Lagrange multiplicator. By substituting Eq.(8) into Eq.(10), we have σ1 ∂F g˙ σ2 σ3 2 = 2 − (νD,12 + νD,21 ) − (νD,13 + νD,31 ) ∂σ1 g Ec λ f12 f1 f2 f1 f3 ∂F σ2 g˙ σ1 σ3 2 2 − (νD,12 + νD,21 ) = 2 − (νD,23 + νD,32 ) (11) ∂σ2 g Ec λ f2 f1 f2 f2 f3 ∂F σ3 g˙ σ1 σ2 2 = 2 − (νD,13 + νD,31 ) − (νD,23 + νD,32 ) ∂σ3 g Ec λ f32 f1 f3 f2 f3 By substituting Eq.(11) into the following equation ∂F ∂F ∂F dσ1 + dσ2 + dσ3 dF (σ1 , σ2 , σ3 ) = ∂σ1 ∂σ2 ∂σ3 then through integral, we have 2 σ1 g˙ σ22 σ32 σ1 σ2 + + − (νD,12 + νD,21 ) F (σ1 , σ2 , σ3 ) = 2 2 2 2 g Ec λ f1 f2 f3 f1 f2 σ2 σ3 σ1 σ3 + C0 −(νD,23 + νD,32 ) − (νD,13 + νD,31 ) f2 f3 f1 f3
(12)
(13)
From Eqs.(9) and (13), we have σ12 σ22 σ32 σ1 σ2 σ2 σ3 σ1 σ3 + + − (νD,12 + νD,21 ) − (νD,23 + νD,32 ) − (νD,13 + νD,31 ) =C 2 2 2 f1 f2 f3 f1 f2 f2 f3 f1 f3
(14)
˙ For the concrete in uniaxial stress state, we can get C=1. Therefore, the in which C = −λEc g 2 C0 /g. general form of unified strength criterion for concrete in multiaxial stress state is obtained as σ12 σ22 σ32 σ1 σ2 σ2 σ3 σ1 σ3 + + − (νD,12 + νD,21 ) − (νD,23 + νD,32 ) − (νD,13 + νD,31 ) =1 2 2 2 f1 f2 f3 f1 f2 f2 f3 f1 f3
(15)
Seen from Eq.(15), it is indicated that the failure of concrete is only related to damage Poisson’s ratio. Especially, for νD,ij =0.5 and fi = fy , Eq.(15) is the famous von Mises yield criterion. Through the above analyses , the failure mechanism of concrete is illustrated, showing that the failure of concrete is a process of energy dissipation and must be restricted by the minimum energy dissipation rate; damage Poisson’s ratio plays a main role and is the key to determining the failure of concrete during energy dissipation.
IV. DETERMINATION OF νD AND VALIDATION OF STRENGTH CRITERION For the concrete in triaxial compression or tension stress state, Eq.(15) can be written briefly as follows: σ12 + σ22 + σ32 − 2νD (σ1 σ2 + σ2 σ3 + σ1 σ3 ) = fi2 (16) The three principal stresses can be expressed in terms of three invariants of the stress tensor: the octahedral normal stress, σoct ; octahedral-shear stress, τoct ; and the Lode angle, θ, defined as 1 σoct = (σ1 + σ2 + σ3 ) 3 1 (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 τoct = (17) 3 2σ1 − σ2 − σ3 √ cos θ = 3 2τoct
Vol. 19, No. 4
Ding Faxing et al.: Strength Criterion for Plain Concrete
Thus, Eq.(16) can be replaced as follows: 2 τ2 σoct − (6ν − 3) =1 (3νD + 3) oct D fi2 fi2
· 311 ·
(18)
The meridional curves expressed by Eq.(18) at various values of νD are seen in Fig.2, where both σoct /fi and τoct /fi are positive. Seen from Fig.2, with the increase of parameter νD , the meridional curves transit from elliptical curve to line, then from line to hyperbolic curve. From the experimenatal data on multiaxial strength of concrete, the general shape of failure surface for concrete is usually described as open-ended and has a convex polar figure which has triplicate symmetry with respect to the hydrostatic axis. The failure curve is nearly triangular for tensile and small comFig. 2 Influence of vD on meridian. pressive stresses, and becomes more circular corresponding to the increasing value of hydrostatic pressure. Seen from the general shape of failure surface for concrete, combined with Eq.(18), for concrete in the triaxial tension state, νD < 0.5 must be taken; seen from the behavior of concrete under unaxial tension, νD < 0.2 is appropriate. For concrete in triaxial compression state, νD > 0.5 must be taken; at the same value of hydrostatic pressure, octahedral normal stress σoct at compressive meridian is greater than that at tensile meridian, thus the value of νD at compressive meridian is greater than that at tensile meridian. For the failure surface satisfying the requirements of being convex, with the increasing value of hydrostatic pressure, νD has a certain trend of variety, but the variety is not monotonic and quite complex. At present, it is difficult to determine the expression of νD theoretically. For Eq.(15) satisfying the requirements of failure surface for concrete, it is indicated that νD is influenced by the Lode angle θ and hydrostatic pressure, and θ is the main factor. For convenience and simplicity, only θ is taken into account, and regression analysis is carried out based on 633 experimental data points of NC and HSC cubes and cylinders subjected to triaxial loads obtained by Ansariand Li[5] , Attard and Setunge[6] , Candappa et al.[14] , Guo[10] , Imran and Pantazopoulou[15] , Lai et al.[16] , Li and Guo[17] , Liang[18] , Liu[19] , Mills and Zimmerman[2] , Sfer et al.[7] , Song et al.[20] , Wang[21] , Xie et al.[22] , Xie et al.[23] and Ye[24] . Thus ⎧ 2 ⎪ ⎨ 3θ + 2 (σi < 0) π2 3 (19) νD,ij = 1 ⎪ ⎩ (σi > 0) 9 For the concrete in different stress state, based on Eq.(19), both the principal stresses form and the octahedral stresses form of Eq.(15) (where fc = 10ft , σ1 ≥ σ2 ≥ σ3 , compression negative), are listed in Table 1. For convenience of analysis and application, the strength criterion for concrete in Table 1 is used to generate strength curves in the tensile meridian, compressive meridian, and deviatoric planes (seen in Fig.3 and Fig.4). It can be seen from Fig.3 and Fig.4 that when the influence of hydrostatic pressure on νD is not considered, the failure surface for concrete proposed by the authors is different from that as is usually described. For concrete in the triaxial compressive stress state, the tensile and compressive meridians proposed by the authors are non-convex hyperbolic curves, and the deviatoric planes are also non-convex. Therefore, it is difficult to reflect the failure surface for concrete in the triaxial stress state through simple expression of νD . For concrete in the triaxial tension stress state, the tensile and compressive meridians proposed by the authors are convex elliptical curves, and the deviatoric planes are also convex circular curves. Due to limited reported data of concrete in the triaxial tension stress state, the failure surfaces by both tests and the authors need further confirmation. For concrete in the compression-compression-tension and compression-tension-tension stress state, the feature of the failure surface is almost the same as is usually described.
· 312 ·
ACTA MECHANICA SOLIDA SINICA
2006
Table 1. Strength criterion for plain concrete in the triaxial stress state
stress states T/T/T
T/T/C
T/C/C
C/C/C
principal stresses form 2 σ12 + σ22 + σ32 − (σ1 σ2 + σ2 σ3 + σ1 σ3 ) = ft2 (20) 9 2 2 2 σ σ 2 σ1 σ 2 σ1 + 2 + 32 − ft2 ft2 fc 9 ft2 7 σ1 σ3 7 σ2 σ3 3θ2 3θ2 + + + = 1 (21) + 2 π2 9 f π 9 ft fc t fc 3θ2 σ12 σ2 σ2 7 σ1 σ2 + 22 + 32 − + 2 ft fc fc π 2 9 ft fc 3θ2 3θ2 2 σ2 σ3 7 σ1 σ3 + + + = 1(22) +2 π2 3 f2 π2 9 ft fc c2 2 3θ + σ12 + σ22 + σ32 − 2 π2 3 ×(σ1 σ2 + σ2 σ3 + σ1 σ3 ) =
fc2
(23)
octahedral stresses form 2 2 10 τoct 7 σoct + =1 2 3 ft 3 ft2 (a)
(b)
5+
3θ π
2
− 1+2
9θ2 187 3θ2 √ 3θ2 − 1.485 sin2 θ + −1.134 + (a) + 3 sin θ cos θ − 0.35 + 2 2 20π 60 10π 10π 2 2 2 √ √ 3θ 3θ 2 + 0.918 + cos θ + 3 sin θ 2τoct σoct + 1.632 − σoct = ft2 20π 2 5π 2
2 9θ2 3θ2 2 (b) 0.147 + + 1.98 cos2 θ + 0.032 + cos θ − 3 sin2 θ τoct 2 2 20π 25π 3θ2 √ 33θ2 2 + 1.916 − 2 cos θτ σ + 0.851 − σoct = ft2 oct oct 25π 2 50π 2
2 τoct fc2
3θ π
2
2
2 σoct =1 fc2
2 cos θ τoct
Fig. 3. Comparison of proposed tensile and compressive meridians with experimental data.
The coordinate of the key point under tensile and compressive meridians (ftt and fcc are biaxial tensile and compressive strength of concrete under 1:1 load ratio, respectively, denoted as T=T/0 and C=C/0; fttt is the strength of concrete under equal triaxial tension, denoted as T=T=T): T=T=T, fttt = 0.655ft, τoct /fc =0, σoct /fc = 0.0655; T=T/0, ftt = 0.75ft , τoct /fc = 0.0354, σoct /fc =0.05; T/0/0, ft = 0.1fc , τoct /fc =0.0471, σoct /fc =0.0333; C/0/0, τoct /fc =0.471, σoct /fc = −0.333; C=C/0, fcc = −1.225fc, τoct /fc =0.577, σoct /fc = −0.817. Although the tensile and compressive meridians of the new strength criterion in the triaxial compressive stress state are quite different from those as is usually described, the comparisons of prediction and reported data in the tensile and compressive meridians are shown in Fig.3. Seen from Fig.3, (1) these experimental data are normalized with reference to the uniaxial compressive strength and the shape of tensile and compressive meridians appears to be independent of the strength of the concrete, which was drawn by Seow and Swaddiwudhipong[9] , and (2) for σ8 /fc > −8.23, new strength criterion of Eq.(17) is shown to give a relatively reasonable estimate of the test data at the tensile and compressive meridians. Through comparison of 399 experimental data points of NC and HSC cubes and cylinders in the triaxial or biaxial compressive stress state obtained by Ansari and Li[5] , Attard and Setunge[6] , Candappa
Vol. 19, No. 4
Ding Faxing et al.: Strength Criterion for Plain Concrete
· 313 ·
Fig. 4. Strength rules of concrete on the deviatoric stress planes.
et al.[14] , Guo[10] , Imran and Pantazopoulou[15] , Li and Guo[17] , Liu[19] , Mills and Zimmerman[2] , Sfer et al.[7] , Song et al.[20] , Xie et al.[22] , Xie et al.[23] , and Ye[24] with the octahedral-shear stress, the mean is 1.111, and the standard deviation is 0.105. Through comparison of 85 experimental data points of NC and HSC cubes in the triaxial compressioncompression-tension stress state obtained by Liu[19] , and Wang[21] with the octahedral-shear stress, the mean is 0.904, and the standard deviation is 0.097. Through comparison of 104 experimental data points of NC and HSC cubes in the triaxial compressiontension-tension or biaxial-tension stress state obtained by Guo[10] , Li and Guo[17] , Liang[18]] , Song et al.[20] , and Wang[21] with the octahedral-shear stress, the mean is 0.910, and the standard deviation is 0.171. Through comparison of 45 experimental data points of NC and HSC cubes in the triaxial or biaxial tensile stress state obtained by Guo[10] , Lai et al.[16] , Li and Guo[17] , Liang[18] and Song et al.[20] with the principal stress, the mean is 1.034, and the standard deviation is 0.113. All the 633 experimental data points of NC and HSC cubes and cylinders in the multiaxial stress state are compared, and the mean is 1.044, the standard deviation is 0.149. The new strength criterion is shown to give a relatively reasonable and mostly conservative estimate of the test results.
V. SIMPLIFICATION OF STRENGTH CRITERION IN BIAXIAL STRESS STATE The three-dimensional strength criterion proposed for concrete is also degenerated and simplified into biaxial failure curves, as illustrated in Table 2 and Fig.5, and verified by the experimental data points[10, 16−−18] of NC and HSC cubes under biaxial loads. Although the predictions of the biaxial failure curves may not always be conservative, the proposed failure surface can still be used to predict the state of stress under failure for concrete under biaxial loads with reasonable accuracy.
Table 2. Strength criterion for plain concrete in the biaxial stress state
stress states T/T T/C C/C
unified strength criterion σ12 σ2 2 σ1 σ2 + 22 − =1 (20) 2 ft ft 9 ft2 3θ2 σ2 7 σ1 σ3 σ12 + 32 − + = 1 (21,22) 2 ft fc π2 9 ft fc 3θ2 σ2 2 σ2 σ3 σ22 + 32 − 2 + = 1(22,23) fc2 fc π2 3 fc2
simplified strength criterion σ12 σ2 2 σ1 σ 2 + 22 − =1 (20) 2 ft ft 9 ft2 σ12 σ2 10 σ1 σ3 + 32 − = 1 (24) 2 ft fc 9 ft fc σ22 σ2 4 σ2 σ3 + 32 − = 1 (25) fc2 fc 3 fc2
· 314 ·
ACTA MECHANICA SOLIDA SINICA
2006
Fig. 5. Strength criterion for plain concrete in the biaxial stress state.
VI. CONCLUSIONS Based on the authors’ investigation, the following main conclusions are drawn: (1) The conception of damage Poisson’s ratio is presented, and the general formula of strength criterion for plain concrete in the multiaxial stress state is proposed. Based on the general shape of failure surface for concrete and the existing experimental data, the expression for damage Poisson’s ratio is determined basically. (2) The general form of unified strength criterion shows that the failure of concrete is a process of energy dissipation and should be restricted by the minimum energy dissipation rate; damage Poisson’s ratio νD plays an important role and is the key to determining failure during energy dissipation. And the von Mises yield criterion is a special case of the unified strength criterion. (3) For the concrete under compression-compression-tension and compression-tension-tension stress states, the feature is almost the same as the usual description; for the concrete under triaxial tension stress states, the feature is just a little different from the usual description, but the influence is relatively small. The new strength criterion can be applied to analytic solution and numerical solution of concrete structures in these stress states with reasonable accuracy. (4) For the concrete under triaxial compressive stress, the feature is quite different from the usually given description, which indicates that the determination of νD needs further research. It is difficult to find the expression of νD by the conventional method, and artificial neural network principles may be a good method for predicting νD based on the previously mentioned experimental data. (5) The simplified strength criterion under biaxial stress states is also proposed and a simplified strength criterion in the form of curves is also proposed.
Vol. 19, No. 4
Ding Faxing et al.: Strength Criterion for Plain Concrete
· 315 ·
References [1] Richart,F.E., Brantzaeg,A. and Brown,R.L., Failure of plain and spirally reinforced concrete in compression, Bulletin 190, University of Illinois Engineering Experimental Station, Champaign, Illinois, 1929. [2] Mills,L.L. and Zimmerman,R.M., Compressive strength of plain concrete under multiaxial loading conditions, ACI Journal, Vol.67, No.10, 1970, 802-807. [3] Ottosen,N.S., A failure criterion for concrete, J. Eng. Mech. Div., Vol.103, No.4, 1977, 527-535. [4] Podgorski,J., General failure criterion for concrete, J. Eng. Mech. Div., Vol.111, No.2, 1985, 188-201. [5] Ansar,Farhad and Li,Qingbin, High-strength concrete subjected to triaxial compression, ACI Mater. J., Vol.95, No.6, 1998, 747-755. [6] Attard M.M. and Setunge,S., Stress-strain relationship of confined and unconfined concrete, ACI Material Journal, Vol.93, No.5, 1996, 432-442. [7] Sfer,D., Carol,I. and Gettu,R. et al, Study of the behavior of concrete under triaxial compression, J. Eng. Mech., Vol.128, No.2, 2002, 156-163. [8] Guo,Z.H. and Wang,C.Z., Investigation of strength and failure criterion of concrete under multi-axial stresses, China Civil Engineering Journal, Vol.24, No.3, 1991, 1-14 (in Chinese). [9] Seow,P.E.C. and Swaddiwudhipong,S., Failure surface for concrete under multiaxial load - A unified approach, Journal of Materials in Civil Engineering, Vol.17, No.2, 2005,219-228. [10] Guo,Z.H., The Strength and Deformation of Concrete-experimental Results and Constitutive Relationship, Tsinghua University Press, Beijing, 1997 (in Chinese). [11] Yu,M.H., Unified Strength Theory and Its Applications, Heidelberg-Berlin, Springer, 2000. [12] Yu,M.H., Advances in strength theories for materials under complex stress state in the 20th Century, American Society of Mechanical Engineers, Vol.55, No.3, 2002, 169-218. [13] Zhou,Z.B. and Lu,C.F., A new strength criterion of plain concrete under triaxial stresses conditions, Acta Mechanica Solida Sinica, Vol.20, No.3, 1999, 272-280 (in Chinese). [14] Candappa,D.P., Setunge,S. and Sanjayan,J.G., Stress versus strain relationship of high strength concrete under high lateral confinement, Cement and Concrete Research, Vol.29, No.12, 1999, 1977-1982. [15] Imran,I. and Pantazopoulou,S.J., Experimental study of plain concrete under triaxial stress, ACI Material Journal, Vol.93, No.6, 1996, 589-601. [16] Lai,W.D., Chen,J. and Li,G.L., Research on strength softening of concrete under biaxial tension, Concrete, Vol.24, No.6, 2002, 21-23 (in Chinese). [17] Li,W.Z. and Guo,Z.H., Experimental research of strength and deformation of concrete under biaxial tensioncompression, Shuili Xuebao, Vol.22, No.8, 1991, 51-56 (in Chinese). [18] Liang,W., Experimental research on strength and deformation of high strength concrete under triaxial stress states (T/T/C, T/T/T), Dissertation for the Master Degree in Engineering, Tsinghua University, Beijing, China, 1998, (in Chinese). [19] Liu,C.H., Experimental research on the behavior of high strength concrete under triaxial stress states (C/C/C, C/C/T), Dissertation for the Master Degree in Engineering, Tsinghua University, Beijing, China, 1997, (in Chinese). [20] Song,Y.P., Zhao,G.F. and Peng,F., Strength criterion of plain concrete under multiaxial stress conditions, Fifth Symposiam on Facture and Strength of Rock and Concrete, Science and Technical University for National Defense Press, Beijing, 1993, 121-129 (in Chinese). [21] Wang,J.Z., Experimental research on strength and criterion of concrete under compression-compressiontension and compression-tension-tension stress, Dissertation for the Master Degree in Engineering, Tsinghua University, Beijing, China, 1989 (in Chinese). [22] Xie,H.P., Dong,Y.L. and Li,S.P., Study of a constitutive model of elasto-plastic damage of concrete in axial compression test under different pressures, Journal of China Coal Society, Vol.21, No.3, 1996, 265-270 (in Chinese). [23] Xie,J., Elwi,A.E. and MacGregor,J.G., Mechanical properties of three high-strength concrete containing silica fume, ACI Material Journal, Vol.92, No.2, 1995, 135-145. [24] Ye,X.G., Researches on Deformation and Strength of Concrete under Triaxial Compression Stress, Dissertation for the Master Degree in Engineering, Tsinghua University, Beijing, China, 1987 (in Chinese).
ISSN 0894-9166
STRENGTH CRITERION FOR PLAIN CONCRETE UNDER MULTIAXIAL STRESS BASED ON DAMAGE POISSON’S RATIO Ding Faxing
Yu Zhiwu
(School of Civil Engineering and Architecture, Central South University, Changsha 410075, China)
Received 26 July 2005; revision received 20 November 2006
ABSTRACT A new unified strength criterion in the principal stress space has been proposed for use with normal strength concrete (NC) and high strength concrete (HSC) in compressioncompression-tension, compression-tension-tension, triaxial tension, and biaxial stress states. The study covers concrete with strengths ranging from 20 to 130 MPa. The conception of damage Poisson’s ratio is defined and the expression for damage Poisson’s ratio is determined basically. The failure mechanism of concrete is illustrated, which points out that damage Poisson’s ratio is the key to determining the failure of concrete. Furthermore, for the concrete under biaxial stress conditions, the unified strength criterion is simplified and a simplified strength criterion in the form of curves is also proposed. The strength criterion is physically meaningful and easy to calculate, which can be applied to analytic solution and numerical solution of concrete structures.
KEY WORDS plain concrete, high strength concrete, multiaxial stress, strength criterion, meridian, deviatoric plan
I. INTRODUCTION Strength theory of concrete deals with the failure of concrete in a complex stress state. It is important and fundamental for the strength calculation and design of concrete structures. It is of great significance in theoretical research and engineering application, and is also very important for the effective utilization of materials. Some structures may be subjected to multiaxial loads, such as concrete-filled steel tubular columns, offshore oil platforms, nuclear containment vessels, and many other structures. Since the failure of concrete in a structure can occur differently in complex stress states, an understanding of the behavior of concrete is needed to develop the failure criteria for concrete. The compressive strength of concrete is the principal property employed in the design of reinforced concrete structures. After the pioneering investigation by Richart et al.[1] on triaxial compressive behavior of NC, many other researchers, such as Mills and Zimmerman[2] , Ottosen[3] , and Podgorski[4] have also conducted studies on the behavior of NC in multiaxial stress states. More recently, the investigations[5−−7] have also been extended to include HSC. Many failure criteria for concrete, typically fitted with different functions, have been proposed by researchers[3, 4, 8, 9] . These criteria are the Octahedral-Stress Strength theory (OSS theory). Guo[10] made a general survey of these criteria and indicated that five-parameter Guo-Wang criteria[8] , four-parameter Ottosen criteria[3] , and five-parameter Podgorski criteria[4] could fit the experimental data regarding the strength of concrete subjected to multiaxial stress quite well.
Project supported by the National Natural Science Foundation of China (Nos. 50438020 and 50578162).
· 308 ·
ACTA MECHANICA SOLIDA SINICA
2006
Yu[11, 12] proposed a twin-shear strength theory and a unified strength theory, which can be adapted to metal, concrete, and rock materials and have been widely used in the strength analysis of engineering structures. The Mohr-Coulomb theory and Tresca criterion are special cases of Yu’s theory. The twinshear strength theory is physically meaningful in itself, but for the multi-parameter twin-shear strength criterion, the main stream of unified strength theory, it is more suited to functions than physical meaning. Zhou[13] proposed a theory of minimum energy dissipation rate which is suitable for the nonlinear state of non-equilibrium systems, and based on the theory, the theoretic expression of strength criterion for concrete in multiaxial stress states was developed. The failure mechanism of concrete was also illustrated, which shows that the failure of concrete is a process of energy dissipation and the failure must be restricted by the minimum energy dissipation rate. The merit of Zhou criterion[13] is its physical meaning and relatively high precision; the merit its complex expression and the parameters must be determined by the experimental data with complex calculation. Based on two basic assumptions, the conception of damage Poisson’s ratio is defined and the theories of damage mechanics and minimum energy dissipation rate are applied in this paper. The general formula of strength criterion for plain concrete under multiaxial stress states is proposed and the failure mechanism of concrete is also illustrated. Based on the general shape of failure surface for concrete and the existing experimental data, the expression for damage Poisson’s ratio is determined basically. The strength criterion is physically meaningful and easy for calculation, which can be applied to analytic and numerical solution of concrete structures under compression-compression-tension, compression-tensiontension, triaxial tension, and biaxial stress states with concrete strengths ranging from 20 to 130 MPa.
II. BASIC ASSUMPTION
Many experimental achievements[10] show that: for the concrete prisms or cylinder specimens under uniaxial compression, the Poisson’s ratio of concrete increases gradually as the axial strain increases; for low strain, the value of the Poisson’s ratio for concrete is about 0.2, and for the axial strain larger than the peak-strain, the value will be greater than 0.5; thereafter, the Poisson’s ratio increases continuously, even greater than 1.0 when it is loaded to failure. For the concrete specimens under uniaxial tension, the Poisson’s ratio of concrete decreases Fig. 1 Calculation model of strain for concrete. gradually as the axial strain increases. For low strain, the value of the Poisson’s ratio for concrete is about 0.2, but for the axial strain larger than the peakstrain, the value will be less than 0.2. Based on the above cognition, seen from Fig.1, the following assumption can be presented: (1) The total principal strain (ε) of concrete may be classified into elastic principal strain (εe ) and inelastic principal strain (εI ), that is ε = εe + εI
(1)
(2) The elastic strain follows the Hooke’s law, and the increase or decrease of the Poisson’s ratio is mainly relevant to damage Poisson’s ratio νD . For the Poisson’s effect, the inelastic principal strain under one axis caused by the stress ratio will result in the inelastic principal strains at the other two axes, and the inelastic principal strain ratio is defined as damage Poisson’s ratio. The stress ratio is defined as the ratio of the principal stress in the axis to the uniaxial (compressive or tensile) strength of the same stress state.
III. THEORETIC ANALYSIS OF STRENGTH CRITERION FOR CONCRETE The constitutive model for concrete can be written as follows: ε=
σ g[D(t)]Ec
(2)
Vol. 19, No. 4
· 309 ·
Ding Faxing et al.: Strength Criterion for Plain Concrete
in which σ is the principal stress of concrete; Ec the initial elastic modulus of concrete; D damage variable; both g[D(t)] and D are the function of time t denoting the process of energy dissipation. For convenience, g[D(t)] can be abbreviated to g. g satisfies: at D=0, g=1; at D=Dr , g →0; ∀ 0 < D < Dr , the slope of the curve of g decreases monotonical and dg/dt = g˙ = 0. g and D are the functions of time t. Based on the theory of the minimum energy dissipation rate[13] , for the element of concrete, if the inelastic strains εI,i (t)(i=1,2,3) caused by loads are regarded as the only mechanism of energy dissipation, in which t is the time parameter, the ratio of energy dissipation rate ψi (t)|t=0 to fi2 can be expressed by 3 3 ψi (t) σi ε˙I,i (t) = (3) fi2 fi2 i=1
i=1
t=0
t=0
where fi (i=1,2,3) is the peak stress of concrete under uniaxial stress, for the uniaxial compressive case, fi is uniaxial compressive strength fc , for the uniaxial tensile case, fi is uniaxial tensile strength ft ; σi (i = 1, 2, 3) is the principal stress of concrete; ε˙I,i (t) is inelastic principal strain rate at any time t. Based on basic assumption (1), the constitutive model for concrete during energy dissipation can be expressed as ⎧ ⎫ ⎤⎧ ⎫ ⎤⎧ ⎫ ⎡ ⎡ 1 −ν −ν ⎨ σ1 ⎬ 1 −r2 −r3 ⎨ σ1 ⎬ ⎨ ε1 ⎬ 1 ⎣ 1 − g ⎣ −r1 1 −r3 ⎦ σ2 −ν 1 −ν ⎦ σ2 + ε2 = (4) ⎩ ⎭ Ec ⎩ ⎭ ⎩ ⎭ gE c −ν −ν 1 ε σ −r −r 1 σ 3
3
1
2
3
where ν is the initial Poisson’s ratio of concrete; ri (i=1,2,3) is the Poisson’s ratio caused by inelastic principal strain. The inelastic principal strains are expressed as ⎧ ⎫ ⎤⎧ ⎫ ⎡ 1 −r2 −r3 ⎨ σ1 ⎬ ⎨ εI,1 ⎬ 1 − g ⎣ −r1 1 −r3 ⎦ σ2 εI,2 = (5) ⎩ ⎭ ⎩ ⎭ gEc εI,3 −r1 −r2 1 σ3 For the concrete during energy dissipation, Eq.(5) can be written as σ2 σ3 ε˙I,1 (t) σ1 −g˙ = 2 − r2 − r3 f1 g Ec f1 f1 f1 ε˙I,2 (t) σ1 σ3 σ2 −g˙ = 2 − r1 − r3 f2 g Ec f2 f2 f2 σ1 σ2 ε˙I,3 (t) σ3 −g˙ = 2 − r1 − r2 f3 g Ec f3 f3 f3
(6)
Based on basic assumption (2), let f1 = lf2 , f1 = mf 3 , and f2 = nf 3 (l, m, and n are parameters). Thus Eq.(6) can be written as ε˙I,1 (t) σ2 σ3 σ1 −g˙ = 2 − νD,21 − νD,31 f1 g Ec f1 f2 f3 ε˙I,2 (t) σ1 σ3 σ2 −g˙ (7) = 2 − νD,12 − νD,32 f2 g Ec f2 f1 f3 σ1 σ2 ε˙I,3 (t) σ3 −g˙ = 2 − νD,13 − νD,23 f3 g Ec f3 f1 f2 where, νD,ij is the damage Poisson’s ratio in i-axial caused by j-axial, and i = j; νD,21 = r2 /l, νD,31 = r3 /m, νD,12 = r1 l, νD,32 = r3 /n, νD,13 = r1 m, νD,23 = r2 n. As the processes of energy dissipation and failure are nonreversible, the increase of inelastic strains of concrete is also nonreversible. By substituting Eq.(7) into Eq.(3), the ratio of energy dissipation rate to fi2 can be expressed by 3 ψi (t) −g˙ σ12 σ22 σ32 σ1 σ2 = + + 2 2 2 2 − (νD,12 + νD,21 ) f f 2E f g f f f c 1 2 1 2 3 i t=0 i=1 σ2 σ3 σ1 σ3 (8) −(νD,23 + νD,32 ) − (νD,13 + νD,31 ) f2 f3 f1 f3
· 310 ·
ACTA MECHANICA SOLIDA SINICA
2006
The failure criterion for concrete can be described in terms of the principal stresses in the following form F (σ1 , σ2 , σ3 ) = 0 (9) Equation (9) is the restrictive condition of Eq.(8) during energy dissipation at any time t. Based on the theory of minimum energy dissipation rate[13] , the following equation can be obtained as 3 ∂ ψi (t)/fi2 + λF i=1 =0 (i = 1, 2, 3) (10) ∂σi in which λ is Lagrange multiplicator. By substituting Eq.(8) into Eq.(10), we have σ1 ∂F g˙ σ2 σ3 2 = 2 − (νD,12 + νD,21 ) − (νD,13 + νD,31 ) ∂σ1 g Ec λ f12 f1 f2 f1 f3 ∂F σ2 g˙ σ1 σ3 2 2 − (νD,12 + νD,21 ) = 2 − (νD,23 + νD,32 ) (11) ∂σ2 g Ec λ f2 f1 f2 f2 f3 ∂F σ3 g˙ σ1 σ2 2 = 2 − (νD,13 + νD,31 ) − (νD,23 + νD,32 ) ∂σ3 g Ec λ f32 f1 f3 f2 f3 By substituting Eq.(11) into the following equation ∂F ∂F ∂F dσ1 + dσ2 + dσ3 dF (σ1 , σ2 , σ3 ) = ∂σ1 ∂σ2 ∂σ3 then through integral, we have 2 σ1 g˙ σ22 σ32 σ1 σ2 + + − (νD,12 + νD,21 ) F (σ1 , σ2 , σ3 ) = 2 2 2 2 g Ec λ f1 f2 f3 f1 f2 σ2 σ3 σ1 σ3 + C0 −(νD,23 + νD,32 ) − (νD,13 + νD,31 ) f2 f3 f1 f3
(12)
(13)
From Eqs.(9) and (13), we have σ12 σ22 σ32 σ1 σ2 σ2 σ3 σ1 σ3 + + − (νD,12 + νD,21 ) − (νD,23 + νD,32 ) − (νD,13 + νD,31 ) =C 2 2 2 f1 f2 f3 f1 f2 f2 f3 f1 f3
(14)
˙ For the concrete in uniaxial stress state, we can get C=1. Therefore, the in which C = −λEc g 2 C0 /g. general form of unified strength criterion for concrete in multiaxial stress state is obtained as σ12 σ22 σ32 σ1 σ2 σ2 σ3 σ1 σ3 + + − (νD,12 + νD,21 ) − (νD,23 + νD,32 ) − (νD,13 + νD,31 ) =1 2 2 2 f1 f2 f3 f1 f2 f2 f3 f1 f3
(15)
Seen from Eq.(15), it is indicated that the failure of concrete is only related to damage Poisson’s ratio. Especially, for νD,ij =0.5 and fi = fy , Eq.(15) is the famous von Mises yield criterion. Through the above analyses , the failure mechanism of concrete is illustrated, showing that the failure of concrete is a process of energy dissipation and must be restricted by the minimum energy dissipation rate; damage Poisson’s ratio plays a main role and is the key to determining the failure of concrete during energy dissipation.
IV. DETERMINATION OF νD AND VALIDATION OF STRENGTH CRITERION For the concrete in triaxial compression or tension stress state, Eq.(15) can be written briefly as follows: σ12 + σ22 + σ32 − 2νD (σ1 σ2 + σ2 σ3 + σ1 σ3 ) = fi2 (16) The three principal stresses can be expressed in terms of three invariants of the stress tensor: the octahedral normal stress, σoct ; octahedral-shear stress, τoct ; and the Lode angle, θ, defined as 1 σoct = (σ1 + σ2 + σ3 ) 3 1 (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 τoct = (17) 3 2σ1 − σ2 − σ3 √ cos θ = 3 2τoct
Vol. 19, No. 4
Ding Faxing et al.: Strength Criterion for Plain Concrete
Thus, Eq.(16) can be replaced as follows: 2 τ2 σoct − (6ν − 3) =1 (3νD + 3) oct D fi2 fi2
· 311 ·
(18)
The meridional curves expressed by Eq.(18) at various values of νD are seen in Fig.2, where both σoct /fi and τoct /fi are positive. Seen from Fig.2, with the increase of parameter νD , the meridional curves transit from elliptical curve to line, then from line to hyperbolic curve. From the experimenatal data on multiaxial strength of concrete, the general shape of failure surface for concrete is usually described as open-ended and has a convex polar figure which has triplicate symmetry with respect to the hydrostatic axis. The failure curve is nearly triangular for tensile and small comFig. 2 Influence of vD on meridian. pressive stresses, and becomes more circular corresponding to the increasing value of hydrostatic pressure. Seen from the general shape of failure surface for concrete, combined with Eq.(18), for concrete in the triaxial tension state, νD < 0.5 must be taken; seen from the behavior of concrete under unaxial tension, νD < 0.2 is appropriate. For concrete in triaxial compression state, νD > 0.5 must be taken; at the same value of hydrostatic pressure, octahedral normal stress σoct at compressive meridian is greater than that at tensile meridian, thus the value of νD at compressive meridian is greater than that at tensile meridian. For the failure surface satisfying the requirements of being convex, with the increasing value of hydrostatic pressure, νD has a certain trend of variety, but the variety is not monotonic and quite complex. At present, it is difficult to determine the expression of νD theoretically. For Eq.(15) satisfying the requirements of failure surface for concrete, it is indicated that νD is influenced by the Lode angle θ and hydrostatic pressure, and θ is the main factor. For convenience and simplicity, only θ is taken into account, and regression analysis is carried out based on 633 experimental data points of NC and HSC cubes and cylinders subjected to triaxial loads obtained by Ansariand Li[5] , Attard and Setunge[6] , Candappa et al.[14] , Guo[10] , Imran and Pantazopoulou[15] , Lai et al.[16] , Li and Guo[17] , Liang[18] , Liu[19] , Mills and Zimmerman[2] , Sfer et al.[7] , Song et al.[20] , Wang[21] , Xie et al.[22] , Xie et al.[23] and Ye[24] . Thus ⎧ 2 ⎪ ⎨ 3θ + 2 (σi < 0) π2 3 (19) νD,ij = 1 ⎪ ⎩ (σi > 0) 9 For the concrete in different stress state, based on Eq.(19), both the principal stresses form and the octahedral stresses form of Eq.(15) (where fc = 10ft , σ1 ≥ σ2 ≥ σ3 , compression negative), are listed in Table 1. For convenience of analysis and application, the strength criterion for concrete in Table 1 is used to generate strength curves in the tensile meridian, compressive meridian, and deviatoric planes (seen in Fig.3 and Fig.4). It can be seen from Fig.3 and Fig.4 that when the influence of hydrostatic pressure on νD is not considered, the failure surface for concrete proposed by the authors is different from that as is usually described. For concrete in the triaxial compressive stress state, the tensile and compressive meridians proposed by the authors are non-convex hyperbolic curves, and the deviatoric planes are also non-convex. Therefore, it is difficult to reflect the failure surface for concrete in the triaxial stress state through simple expression of νD . For concrete in the triaxial tension stress state, the tensile and compressive meridians proposed by the authors are convex elliptical curves, and the deviatoric planes are also convex circular curves. Due to limited reported data of concrete in the triaxial tension stress state, the failure surfaces by both tests and the authors need further confirmation. For concrete in the compression-compression-tension and compression-tension-tension stress state, the feature of the failure surface is almost the same as is usually described.
· 312 ·
ACTA MECHANICA SOLIDA SINICA
2006
Table 1. Strength criterion for plain concrete in the triaxial stress state
stress states T/T/T
T/T/C
T/C/C
C/C/C
principal stresses form 2 σ12 + σ22 + σ32 − (σ1 σ2 + σ2 σ3 + σ1 σ3 ) = ft2 (20) 9 2 2 2 σ σ 2 σ1 σ 2 σ1 + 2 + 32 − ft2 ft2 fc 9 ft2 7 σ1 σ3 7 σ2 σ3 3θ2 3θ2 + + + = 1 (21) + 2 π2 9 f π 9 ft fc t fc 3θ2 σ12 σ2 σ2 7 σ1 σ2 + 22 + 32 − + 2 ft fc fc π 2 9 ft fc 3θ2 3θ2 2 σ2 σ3 7 σ1 σ3 + + + = 1(22) +2 π2 3 f2 π2 9 ft fc c2 2 3θ + σ12 + σ22 + σ32 − 2 π2 3 ×(σ1 σ2 + σ2 σ3 + σ1 σ3 ) =
fc2
(23)
octahedral stresses form 2 2 10 τoct 7 σoct + =1 2 3 ft 3 ft2 (a)
(b)
5+
3θ π
2
− 1+2
9θ2 187 3θ2 √ 3θ2 − 1.485 sin2 θ + −1.134 + (a) + 3 sin θ cos θ − 0.35 + 2 2 20π 60 10π 10π 2 2 2 √ √ 3θ 3θ 2 + 0.918 + cos θ + 3 sin θ 2τoct σoct + 1.632 − σoct = ft2 20π 2 5π 2
2 9θ2 3θ2 2 (b) 0.147 + + 1.98 cos2 θ + 0.032 + cos θ − 3 sin2 θ τoct 2 2 20π 25π 3θ2 √ 33θ2 2 + 1.916 − 2 cos θτ σ + 0.851 − σoct = ft2 oct oct 25π 2 50π 2
2 τoct fc2
3θ π
2
2
2 σoct =1 fc2
2 cos θ τoct
Fig. 3. Comparison of proposed tensile and compressive meridians with experimental data.
The coordinate of the key point under tensile and compressive meridians (ftt and fcc are biaxial tensile and compressive strength of concrete under 1:1 load ratio, respectively, denoted as T=T/0 and C=C/0; fttt is the strength of concrete under equal triaxial tension, denoted as T=T=T): T=T=T, fttt = 0.655ft, τoct /fc =0, σoct /fc = 0.0655; T=T/0, ftt = 0.75ft , τoct /fc = 0.0354, σoct /fc =0.05; T/0/0, ft = 0.1fc , τoct /fc =0.0471, σoct /fc =0.0333; C/0/0, τoct /fc =0.471, σoct /fc = −0.333; C=C/0, fcc = −1.225fc, τoct /fc =0.577, σoct /fc = −0.817. Although the tensile and compressive meridians of the new strength criterion in the triaxial compressive stress state are quite different from those as is usually described, the comparisons of prediction and reported data in the tensile and compressive meridians are shown in Fig.3. Seen from Fig.3, (1) these experimental data are normalized with reference to the uniaxial compressive strength and the shape of tensile and compressive meridians appears to be independent of the strength of the concrete, which was drawn by Seow and Swaddiwudhipong[9] , and (2) for σ8 /fc > −8.23, new strength criterion of Eq.(17) is shown to give a relatively reasonable estimate of the test data at the tensile and compressive meridians. Through comparison of 399 experimental data points of NC and HSC cubes and cylinders in the triaxial or biaxial compressive stress state obtained by Ansari and Li[5] , Attard and Setunge[6] , Candappa
Vol. 19, No. 4
Ding Faxing et al.: Strength Criterion for Plain Concrete
· 313 ·
Fig. 4. Strength rules of concrete on the deviatoric stress planes.
et al.[14] , Guo[10] , Imran and Pantazopoulou[15] , Li and Guo[17] , Liu[19] , Mills and Zimmerman[2] , Sfer et al.[7] , Song et al.[20] , Xie et al.[22] , Xie et al.[23] , and Ye[24] with the octahedral-shear stress, the mean is 1.111, and the standard deviation is 0.105. Through comparison of 85 experimental data points of NC and HSC cubes in the triaxial compressioncompression-tension stress state obtained by Liu[19] , and Wang[21] with the octahedral-shear stress, the mean is 0.904, and the standard deviation is 0.097. Through comparison of 104 experimental data points of NC and HSC cubes in the triaxial compressiontension-tension or biaxial-tension stress state obtained by Guo[10] , Li and Guo[17] , Liang[18]] , Song et al.[20] , and Wang[21] with the octahedral-shear stress, the mean is 0.910, and the standard deviation is 0.171. Through comparison of 45 experimental data points of NC and HSC cubes in the triaxial or biaxial tensile stress state obtained by Guo[10] , Lai et al.[16] , Li and Guo[17] , Liang[18] and Song et al.[20] with the principal stress, the mean is 1.034, and the standard deviation is 0.113. All the 633 experimental data points of NC and HSC cubes and cylinders in the multiaxial stress state are compared, and the mean is 1.044, the standard deviation is 0.149. The new strength criterion is shown to give a relatively reasonable and mostly conservative estimate of the test results.
V. SIMPLIFICATION OF STRENGTH CRITERION IN BIAXIAL STRESS STATE The three-dimensional strength criterion proposed for concrete is also degenerated and simplified into biaxial failure curves, as illustrated in Table 2 and Fig.5, and verified by the experimental data points[10, 16−−18] of NC and HSC cubes under biaxial loads. Although the predictions of the biaxial failure curves may not always be conservative, the proposed failure surface can still be used to predict the state of stress under failure for concrete under biaxial loads with reasonable accuracy.
Table 2. Strength criterion for plain concrete in the biaxial stress state
stress states T/T T/C C/C
unified strength criterion σ12 σ2 2 σ1 σ2 + 22 − =1 (20) 2 ft ft 9 ft2 3θ2 σ2 7 σ1 σ3 σ12 + 32 − + = 1 (21,22) 2 ft fc π2 9 ft fc 3θ2 σ2 2 σ2 σ3 σ22 + 32 − 2 + = 1(22,23) fc2 fc π2 3 fc2
simplified strength criterion σ12 σ2 2 σ1 σ 2 + 22 − =1 (20) 2 ft ft 9 ft2 σ12 σ2 10 σ1 σ3 + 32 − = 1 (24) 2 ft fc 9 ft fc σ22 σ2 4 σ2 σ3 + 32 − = 1 (25) fc2 fc 3 fc2
· 314 ·
ACTA MECHANICA SOLIDA SINICA
2006
Fig. 5. Strength criterion for plain concrete in the biaxial stress state.
VI. CONCLUSIONS Based on the authors’ investigation, the following main conclusions are drawn: (1) The conception of damage Poisson’s ratio is presented, and the general formula of strength criterion for plain concrete in the multiaxial stress state is proposed. Based on the general shape of failure surface for concrete and the existing experimental data, the expression for damage Poisson’s ratio is determined basically. (2) The general form of unified strength criterion shows that the failure of concrete is a process of energy dissipation and should be restricted by the minimum energy dissipation rate; damage Poisson’s ratio νD plays an important role and is the key to determining failure during energy dissipation. And the von Mises yield criterion is a special case of the unified strength criterion. (3) For the concrete under compression-compression-tension and compression-tension-tension stress states, the feature is almost the same as the usual description; for the concrete under triaxial tension stress states, the feature is just a little different from the usual description, but the influence is relatively small. The new strength criterion can be applied to analytic solution and numerical solution of concrete structures in these stress states with reasonable accuracy. (4) For the concrete under triaxial compressive stress, the feature is quite different from the usually given description, which indicates that the determination of νD needs further research. It is difficult to find the expression of νD by the conventional method, and artificial neural network principles may be a good method for predicting νD based on the previously mentioned experimental data. (5) The simplified strength criterion under biaxial stress states is also proposed and a simplified strength criterion in the form of curves is also proposed.
Vol. 19, No. 4
Ding Faxing et al.: Strength Criterion for Plain Concrete
· 315 ·
References [1] Richart,F.E., Brantzaeg,A. and Brown,R.L., Failure of plain and spirally reinforced concrete in compression, Bulletin 190, University of Illinois Engineering Experimental Station, Champaign, Illinois, 1929. [2] Mills,L.L. and Zimmerman,R.M., Compressive strength of plain concrete under multiaxial loading conditions, ACI Journal, Vol.67, No.10, 1970, 802-807. [3] Ottosen,N.S., A failure criterion for concrete, J. Eng. Mech. Div., Vol.103, No.4, 1977, 527-535. [4] Podgorski,J., General failure criterion for concrete, J. Eng. Mech. Div., Vol.111, No.2, 1985, 188-201. [5] Ansar,Farhad and Li,Qingbin, High-strength concrete subjected to triaxial compression, ACI Mater. J., Vol.95, No.6, 1998, 747-755. [6] Attard M.M. and Setunge,S., Stress-strain relationship of confined and unconfined concrete, ACI Material Journal, Vol.93, No.5, 1996, 432-442. [7] Sfer,D., Carol,I. and Gettu,R. et al, Study of the behavior of concrete under triaxial compression, J. Eng. Mech., Vol.128, No.2, 2002, 156-163. [8] Guo,Z.H. and Wang,C.Z., Investigation of strength and failure criterion of concrete under multi-axial stresses, China Civil Engineering Journal, Vol.24, No.3, 1991, 1-14 (in Chinese). [9] Seow,P.E.C. and Swaddiwudhipong,S., Failure surface for concrete under multiaxial load - A unified approach, Journal of Materials in Civil Engineering, Vol.17, No.2, 2005,219-228. [10] Guo,Z.H., The Strength and Deformation of Concrete-experimental Results and Constitutive Relationship, Tsinghua University Press, Beijing, 1997 (in Chinese). [11] Yu,M.H., Unified Strength Theory and Its Applications, Heidelberg-Berlin, Springer, 2000. [12] Yu,M.H., Advances in strength theories for materials under complex stress state in the 20th Century, American Society of Mechanical Engineers, Vol.55, No.3, 2002, 169-218. [13] Zhou,Z.B. and Lu,C.F., A new strength criterion of plain concrete under triaxial stresses conditions, Acta Mechanica Solida Sinica, Vol.20, No.3, 1999, 272-280 (in Chinese). [14] Candappa,D.P., Setunge,S. and Sanjayan,J.G., Stress versus strain relationship of high strength concrete under high lateral confinement, Cement and Concrete Research, Vol.29, No.12, 1999, 1977-1982. [15] Imran,I. and Pantazopoulou,S.J., Experimental study of plain concrete under triaxial stress, ACI Material Journal, Vol.93, No.6, 1996, 589-601. [16] Lai,W.D., Chen,J. and Li,G.L., Research on strength softening of concrete under biaxial tension, Concrete, Vol.24, No.6, 2002, 21-23 (in Chinese). [17] Li,W.Z. and Guo,Z.H., Experimental research of strength and deformation of concrete under biaxial tensioncompression, Shuili Xuebao, Vol.22, No.8, 1991, 51-56 (in Chinese). [18] Liang,W., Experimental research on strength and deformation of high strength concrete under triaxial stress states (T/T/C, T/T/T), Dissertation for the Master Degree in Engineering, Tsinghua University, Beijing, China, 1998, (in Chinese). [19] Liu,C.H., Experimental research on the behavior of high strength concrete under triaxial stress states (C/C/C, C/C/T), Dissertation for the Master Degree in Engineering, Tsinghua University, Beijing, China, 1997, (in Chinese). [20] Song,Y.P., Zhao,G.F. and Peng,F., Strength criterion of plain concrete under multiaxial stress conditions, Fifth Symposiam on Facture and Strength of Rock and Concrete, Science and Technical University for National Defense Press, Beijing, 1993, 121-129 (in Chinese). [21] Wang,J.Z., Experimental research on strength and criterion of concrete under compression-compressiontension and compression-tension-tension stress, Dissertation for the Master Degree in Engineering, Tsinghua University, Beijing, China, 1989 (in Chinese). [22] Xie,H.P., Dong,Y.L. and Li,S.P., Study of a constitutive model of elasto-plastic damage of concrete in axial compression test under different pressures, Journal of China Coal Society, Vol.21, No.3, 1996, 265-270 (in Chinese). [23] Xie,J., Elwi,A.E. and MacGregor,J.G., Mechanical properties of three high-strength concrete containing silica fume, ACI Material Journal, Vol.92, No.2, 1995, 135-145. [24] Ye,X.G., Researches on Deformation and Strength of Concrete under Triaxial Compression Stress, Dissertation for the Master Degree in Engineering, Tsinghua University, Beijing, China, 1987 (in Chinese).