Unified calculation method and its application in determining the uniaxial mechanical properties of concrete

Abstract

This paper presents a unified calculation method and its application in determining the uniaxial mechanical properties of concrete with concrete strengths ranging from 10 to 140 MPa. By analyzing a large collection of test results of the uniaxial mechanical properties of normal-strength, high-strength and super high-strength concrete in China and performing a regression analysis, unified calculation formulas for the mechanical indexes of concrete are proposed that can be applied to various grades of concrete for determining the size coefficient, uniaxial compressive strength, uniaxial tensile strength, elastic modulus, and strain at peak uniaxial compression and tension. Optimized mathematical equations for the nonlinear stress-strain relationship of concrete, including the ascending and descending branches under uniaxial stress, are also established. The elastic modulus is almost constant throughout the elastic stage for the ascending branches of the stress-strain relationship for concrete. The proposed stress-strain relationship of concrete was applied to the nonlinear finite element analysis of both a steel-concrete composite beam and a concrete-filled steel tubular stub column. The analytical results are in good agreement with the experiment results, indicating that the proposed stress-strain relationship of concrete is applicable. The achievements presented in this paper can be used as references for the design and nonlinear finite element analysis of concrete structures.

Keywords

concrete / mechanical properties / stress-strain relationship / uniaxial stress / application

Introduction

Over the past several decades, many experimental investigations of the mechanical properties of concrete under uniaxial stress have been reported in China and in other countries. However, researchers in China have adopted a piecewise description for the mechanical indexes of normal-strength concrete (with compressive cube strengths (f_cu) up to 60 MPa), high-strength concrete (with f_cu between 60 MPa and 100 MPa) and super high-strength concrete (with f_cu greater than 100 MPa). For each type of concrete, formulas for the mechanical indexes have been developed that lead to large errors in the calculation results at the divisional borderline. These not only cause confusion regarding this physical concept but also introduce errors into the analysis of concrete structures.

At present, numerous mathematical equations for the nonlinear stress-stain relationship of concrete under uniaxial stress have been developed in China and in other countries. However, the elastic modulus decays rapidly in the elastic stage of the ascending branches of these equations for the stress-strain relationship of concrete, which leads to an overestimate of the deformation in nonlinear structure analysis. Thus, the properties of concrete cannot be fully incorporated.

The research reported here was performed with the following objectives: 1) to propose unified calculation formulas that can be applied to various grades of concrete for determining the size coefficient, uniaxial compressive strength, uniaxial tensile strength, elastic modulus, and strain at peak uniaxial compression and tension through reanalyzing a large collection of test results of the mechanical indexes of various grades of concrete under uniaxial compression and tension in China; 2) to suggest generic forms for the mathematical equations for the ascending and descending branches of the stress-strain relationship of various grades of concrete under monotonic uniaxial compression and tension and to suggest formulas for the variables in the equations, where the elastic modulus for the ascending branches of the stress-strain relationship of concrete is almost constant when the stress ratio is less than 0.4; and 3) to apply this stress-strain relationship of concrete using the nonlinear finite element program of ABAQUS and, through structure analysis, to compare the analytical results and the experimental results of both a steel-concrete composite beam and a concrete-filled steel tubular column.

Mechanical indexes

Size coefficient

According to China code GB50010–2002 [1], the strength grades of concrete are determined using 150-mm cube specimens, where f_cu represents the compressive cube strength of the standard specimen. However, in engineering practice, 100-mm cube specimens are often adopted for testing the compressive cube strength (f_cu,10). There is a dimensional effect of the compressive cube strength that causes a difference between 100-mm cubes and 150-mm cubes because concrete is a highly complex composite material and because the axial stiffness of test machines is not sufficiently high. Therefore, the compressive cube strength of 100-mm cube specimens appears to be slightly higher than that of 150-mm cube specimens. With this increase in strength grade, the compressive cube strength ratio of 150-mm cubes decreases slightly in comparison to 100-mm cubes. For normal-strength concrete, the size coefficient of f_cu/f_cu,10 is 0.95 [2], and for high-strength concrete, Chen et al. [3] suggested:

(1){\displaystyle \frac{f_{\mathrm{c}\mathrm{u}}}{f_{\mathrm{c}\mathrm{u},\mathrm{10}}}}=\mathrm{0.91}+{\displaystyle \frac{\mathrm{1}}{f_{\mathrm{c}\mathrm{u},\mathrm{10}}}}.

However, for super high-strength concrete, there have been few reports in China.

By analyzing the size coefficient results [2,4-6] of cubic specimens in China, a unified formula was developed:

(2)f_{\mathrm{c}\mathrm{u}}=\mathrm{1.17}{f}_{\mathrm{c}\mathrm{u},\mathrm{10}}^{\mathrm{0.95}}-\mathrm{0.7.}

A comparison between the predicted curve and the test results is shown in Fig. 1. The size coefficients of various grades of concrete calculated using Eqs. (1)–(2) are listed in Table 1. The results predicted by Eq. (2) show better agreement compared with both 0.95 and Eq. (1).

In the design codes of America, Europe and Japan, cylindrical specimens 150 mm in diameter and 300 mm in height are recommended as the standard, and f_{\mathrm{c}}^{\prime } represents the test strength of the standard cylindrical specimen. Cylindrical specimens 100 mm in diameter and 200 mm in height are often adopted in many research works and engineering practice. Similar size coefficients have been found for Φ100 × 200 mm to Φ150 mm × 300 mm specimens and 100-mm to 150-mm cubes as determined by analyzing the size coefficients of cylindrical specimens. Thus, a unified formula for the size coefficient of cylindrical specimens was also developed:

(3)f_{\mathrm{c}}^{\prime }=\mathrm{0.17}{(f_{\mathrm{c},\mathrm{10}}^{\prime })}^{\mathrm{0.95}}-\mathrm{0.7.}

A comparison of the coefficients from the test data and those predicted by Eq. (3) is shown in Fig. 2. The results show reasonably good agreement.

The coefficients for the conversion relations between f_{\mathrm{c}}^{\prime } and f_cu recommended in CEB-FIP-1990 [7], NS 3473 [8] and Chen et al. [3] are shown in Fig. 3. When f_cu = 40 ~ 80 MPa, the deviation of the conversion relations from these studies is relatively large. The CEB-FIP-1990 [7] value is small, while the value for NS 3473[8] is large. Moreover, the value presented by Chen et al. [3] is intermediate. For convenience of calculation, this paper suggests using the conversion relations between f_{\mathrm{c}}^{\prime } and f_cu presented by Chen et al. [3]:

(4)f_{\mathrm{c}}^{\prime }=\{\begin{array}{ll}\mathrm{0.8}{f}_{\mathrm{c}\mathrm{u}}\hfill & f_{\mathrm{c}\mathrm{u}}\le \mathrm{50}\mathrm{M}\mathrm{P}\mathrm{a}\hfill \\ f_{\mathrm{c}\mathrm{u}}-\mathrm{10}\hfill & f_{\mathrm{c}\mathrm{u}}>\mathrm{50}\mathrm{M}\mathrm{P}\mathrm{a}\hfill \end{array}.

Uniaxial compressive strength

Extensive research regarding the uniaxial compressive strength (f_c) of 100 mm × 100 mm × 300 mm or 150 mm × 150 mm × 450 mm prisms has been conducted in China. Figure 4 shows the measured uniaxial compressive strengths for various grades of concrete. The uniaxial compressive strength increases moderately with increasing compressive cube strength. The test data in the figure show that the increase in uniaxial compressive strength is not exactly proportional to the compressive cube strength. For normal-strength concrete, the China code [14] suggests the following relationship between the uniaxial compressive strength and the compressive cube strength:

(5)f_{\mathrm{c}}=\mathrm{0.76}{f}_{\mathrm{c}\mathrm{u}}.

For high-strength concrete, Li and Wang [15] proposed:

(6)f_{\mathrm{c}}=-\mathrm{12.31}+\mathrm{1.015}{f}_{\mathrm{c}\mathrm{u}}.

However, for concrete with f_cu = 60 MPa, from Eq. (5), f_c = 45.6 MPa is calculated, and from Eq. (6), f_c = 48.6 MPa is calculated. The error between Eq. (4) and Eq. (6) is 6.6%, which is unreasonable. Based on the test results shown in Fig. 4, a unified formula of uniaxial compressive strength of concrete is proposed:

(7)f_{\mathrm{c}}=\mathrm{0.4}{f}_{\mathrm{c}\mathrm{u}}^{\mathrm{7}/\mathrm{6}}.

The strength ratios of f_c to f_cu of various grades of concrete as computed by Eq. (5), Eq. (6), and Eq. (7) are listed in Table 2. The results predicted by Eq. (7) show better agreement compared to Eqs. (5) and (6).

As presented in Fig. 4 and Table 3, for normal-strength concrete, Eq. (5) gives a good estimate of the uniaxial compressive strength, but for both high-strength and super high-strength concrete, it gives an underestimate; meanwhile, Eq. (6) shows the opposite trend. Moreover, Eq. (7) results in a good estimate of the uniaxial compressive strength for various grades of concrete.

Uniaxial tensile strength

Extensive research regarding the uniaxial tensile strength (f_t) has also been carried out in China. Figure 5 shows the measured uniaxial tensile strength of various grades of concrete from 217 groups. The uniaxial tensile strength increases with the compressive cube strength. As expected, the increase is not directly proportional to the compressive cube strength, and the scatter of the test results is high. For normal-strength concrete, the previous China code [14] suggests:

(8)f_{\mathrm{t}}=\mathrm{0.26}{f}_{\mathrm{c}\mathrm{u}}^{\mathrm{2}/\mathrm{3}}.

For high-strength concrete, Li and Wang [15] suggested:

(9)f_{\mathrm{t}}=\mathrm{0.2089}{f}_{\mathrm{c}\mathrm{u}}^{\mathrm{0.6737}}.

As shown in Fig. 5 and Table 3, for normal-strength concrete, Eq. (8) produces a good estimate of the uniaxial tensile strength, but for both high-strength and super high-strength concrete, it gives an overestimate. However, Eq. (9) underestimates the uniaxial tensile strength for various grades of concrete. Furthermore, the results predicted by Eq. (8) and Eq. (9) are discontinuous at f_cu = 60 MPa. Therefore, based on the test results shown in Fig. 5, a unified formula of uniaxial tensile strength was developed:

(10)f_{\mathrm{t}}=\mathrm{0.24}{f}_{\mathrm{c}\mathrm{u}}^{\mathrm{2}/\mathrm{3}}.

Equation (10) gives a reasonable estimate of the uniaxial tensile strength for various grades of concrete.

Elastic modulus

The elastic modulus of concrete (E_c) is defined as the secant modulus measured at a stress of 0.4f_c or 0.4f_t. Figure 6 shows test results for the elastic modulus related to the compressive cube strength for various grades of concrete. Most of the results were measured under uniaxial compression, as shown in Fig. 6(a). The elastic modulus increases with the compressive cube strength, but the scatter of the test results is relatively high because the elastic modulus of concrete is highly dependent on the type of aggregate used and its proportion within the mix. For normal-strength concrete and based on extensive work, the previous China code [14] suggests:

(11)E_{\mathrm{c}}={\displaystyle \frac{{\mathrm{10}}^{\mathrm{5}}}{\mathrm{2.2}+\mathrm{34.74}/f_{\mathrm{c}\mathrm{u}}}}.

For high-strength concrete, Li and Wang [15] suggested:

(12)E_{\mathrm{c}}=\mathrm{10002}{f}_{\mathrm{c}\mathrm{u}}^{\mathrm{0.3304}}.

Furthermore, Chen et al. [3] suggested:

(13)E_{\mathrm{c}}=(\mathrm{0.45}\sqrt{f_{\mathrm{c}\mathrm{u}}+\mathrm{0.5}})\times {\mathrm{10}}^{\mathrm{4}}.

Based on our analysis, the following unified equation of the elastic modulus for various grades of concrete is proposed:

(14)E_{\mathrm{c}}=\mathrm{9500}{f}_{\mathrm{c}\mathrm{u}}^{\mathrm{1}/\mathrm{3}}.

As shown in Fig. 6(a) and Table 4 with regard to the test results for uniaxial compression, for high-strength and super high-strength concrete, Eq. (11) slightly underestimates the elastic modulus, but Eqs. (12) and (13) give a slight overestimate. However, Eq. (14) gives a reasonable estimate for the elastic modulus for various grades of concrete.

The measured elastic modulus for various grades of concrete under uniaxial tension has also been tested by researchers in China, as shown in Fig. 6(b), revealing almost the same trends observed for uniaxial compression. Again, Eq. (14) gives a reasonable estimate for the elastic modulus of concrete under uniaxial tension, with a mean of 1.000 and a standard deviation of 0.118 from the test data.

Strain at peak uniaxial compression

Figure 7 shows that the strain at peak uniaxial compression (ϵ_c) increases as the uniaxial compressive strength of the concrete increases. For normal-strength concrete, the strain at peak uniaxial compression is less than 0.002. High-strength concretes have a larger range of linearity in the ascending branch, and the strain at peak stress is also larger than for normal-strength concrete. For uniaxial compressive strengths up to 120 MPa, Guo [19] suggested

(15){ϵ}_{\mathrm{c}}(\mathrm{700}+\mathrm{172}\sqrt{f_{\mathrm{c}}})\times {\mathrm{10}}^{-\mathrm{6}}.

Meanwhile, Xu [20] proposed

(16){ϵ}_{\mathrm{c}}(\mathrm{966}+\mathrm{155.64}\sqrt{f_{\mathrm{c}}-\mathrm{13.77}})\times {\mathrm{10}}^{-\mathrm{6}}.

Based on a regression analysis of the test results, a unified formula of the strain at peak uniaxial compression can be proposed:

(17){ϵ}_{\mathrm{c}}=\mathrm{520}{f}_{\mathrm{c}}^{\mathrm{1}/\mathrm{3}}\times {\mathrm{10}}^{-\mathrm{6}}.

As shown in Fig. 7 and Table 5, for various grades of concrete, Eqs. (15)–(17) give similar estimates, but Eq. (17) has the simplest expression.

In addition, following Eqs. (7), and (17) can be transformed as follows:

(18){ϵ}_{\mathrm{c}}=\mathrm{383}{f}_{\mathrm{c}\mathrm{u}}^{\mathrm{7}/\mathrm{18}}\times {\mathrm{10}}^{-\mathrm{6}}.

Equation (17) gives a mean of 1.007 and a standard deviation of 0.090 from the test results. A comparison of the test results and the curve predicted by Eq. (18) is given in Fig. 8.

Strain at peak uniaxial tension

Figure 9 shows that the strain at peak uniaxial tension (ϵ_t) increases as the uniaxial tensile strength of the concrete increases. The strain values at peak uniaxial tension are approximately 5% of the strain at peak uniaxial compression for the same grade of concrete. Guo [19] and Xu [20] suggested:

(19){ϵ}_{\mathrm{t}}=\mathrm{65}{f}_{\mathrm{t}}^{\mathrm{0.54}}\times {\mathrm{10}}^{-\mathrm{6}}.

Through regression analysis, the following unified equation for the strain at peak uniaxial tension for various grades of concrete is proposed:

(20){ϵ}_{\mathrm{t}}=\mathrm{67}{f}_{\mathrm{t}}^{\mathrm{1}/\mathrm{2}}\times {\mathrm{10}}^{-\mathrm{6}}.

In addition, according to Eq. (10), Eq. (20) can be transformed as follows:

(21){ϵ}_{\mathrm{t}}=\mathrm{33}{f}_{\mathrm{c}\mathrm{u}}^{\mathrm{1}/\mathrm{3}}\times {\mathrm{10}}^{-\mathrm{6}}.

A comparison of the test results and those calculated from Eqs. (19) and (20) is illustrated in Fig. 9; the means are 0.998 and 1.010, with standard deviations of 0.081 and 0.082, respectively. A comparison of the test results and those calculated from Eq. (21) is shown in Fig. 10, with a mean of 0.996 and a standard deviation of 0.132.

Uniaxial stress-strain relationship

Non-dimensional formula for the stress-strain relationship of concrete under uniaxial stress

Based on previous test results for the stress-strain relationship of concrete under uniaxial stress, the ascending branch boundary states have been previously defined and are the same for the uniaxial compressive and tensile cases. Compared with the descending branch of the uniaxial compression curve, the descending branch of the uniaxial tensile curve is much steeper. However, the influence of a precise estimation for the descending branch of the curve on the structural performance is very slight. Thus, the same mathematical equations for the descending part of the stress-strain curve were used.

The expression for the ascending branch of the stress-strain relationship of concrete suggested by Sargin [28] and that of the descending branch suggested by Guo [19] were selected. Therefore, the following non-dimensional mathematical form for the stress-stain relationship of concrete under uniaxial compression is proposed:

(22)y=\{\begin{array}{ll}{\displaystyle \frac{A_{n}x+(B_n-\mathrm{1})x^{\mathrm{2}}}{\mathrm{1}+(A_n-\mathrm{2})x+B_n{x}^{\mathrm{2}}}}\hfill & x\le \mathrm{1}\hfill \\ {\displaystyle \frac{x}{a_n{(x-\mathrm{1})}^{\mathrm{2}}+x}}\hfill & x>\mathrm{1}\hfill \end{array},

where A_n is the ratio of the initial tangent modulus to the secant modulus at peak stress, and the initial tangent modulus is equal to the elastic modulus.

Expressions of parameters for uniaxial compression

For the uniaxial compressive case, in Eq. (22), y = σ/f_c and x = ϵ/ϵ_c, where σ is the compressive stress measured in MPa and ϵ is the compressive strain. For n = 1, according to Eq. (7), Eq. (14) and Eq. (18), A₁ can be transformed as follows:

(23)A_{\mathrm{1}}=\mathrm{9.1}{f}_{\mathrm{c}\mathrm{u}}^{-\mathrm{4}/\mathrm{9}}.

B₁ is a parameter that controls the decrease in the elastic modulus along the ascending branch of the axial stress-strain relationship. According to the definition of the elastic modulus, when σ is within the ascending branch and is less than 0.4 f_c, the curve can be considered as a straight line. Thus, when x = 0.4/ A₁ and y = 0.4, the expression for B₁ can be obtained:

(24)B_{\mathrm{1}}=\mathrm{5}{(A_{\mathrm{1}}-\mathrm{1})}^{\mathrm{2}}/\mathrm{3}\approx \mathrm{1.6}{(A_{\mathrm{1}}-\mathrm{1})}^{\mathrm{2}}.

The variable α₁ was determined by regression analysis as:

(25){\alpha }_{\mathrm{1}}=\mathrm{2.5}\times {\mathrm{10}}^{-\mathrm{5}}{f}_{\mathrm{c}\mathrm{u}}^{\mathrm{3}}.

It should be noted that concrete is often confined by stirrups, carbon fiber polymers or steel tubes in engineering practice. The brittleness of unconfined concrete is much greater than that of confined concrete according to compressive failure tests. For the nonlinear finite element analysis of confined concrete structures, this paper suggests a constant value for α₁:

(26){\alpha }_{\mathrm{1}}=\mathrm{0.15.}

Comparisons between the curves predicted by theoretic stress-strain relationships of concrete under compression using Eq. (22) and Eqs. (7), (18), (23)–(25) and the experimental curves are shown in Fig. 11. The model gives a good estimate of the full stress-stain behavior of concrete under uniaxial compression.

Expressions of parameters at uniaxial tension

For the uniaxial tensile case, in Eq. (22), y = σ/f_t, x = ϵ/ϵ_t, and n = 2. According to Eqs. (10), (14) and (21), A₂ can be transformed as follows:

(27)A_{\mathrm{2}}=\mathrm{1.306.}

From Eqs. (24) and (27), the variable B₂ of the ascending branch can be obtained as:

(28)B_{\mathrm{2}}=\mathrm{5}{(A_{\mathrm{2}}-\mathrm{1})}^{\mathrm{2}}/\mathrm{3}=\mathrm{0.15.}

The variable α₂ was also determined by regression analysis as:

(29){\alpha }_{\mathrm{2}}=\mathrm{1}+\mathrm{3}{f}_{\mathrm{c}\mathrm{u}}^{\mathrm{2}}\times {\mathrm{10}}^{-\mathrm{4}}.

It should be noted that concrete often works together with steel bars in engineering practice. The brittleness of plain concrete is much greater than that of reinforced concrete according to tensile failure tests. For the nonlinear finite element analysis of reinforced concrete structures, this paper suggests a constant for A₂ as follows:

(30)A_{\mathrm{2}}=\mathrm{0.8.}

Comparisons between the curves predicted by the theoretical stress-strain relationships of concrete under tension using Eq. (22) and Eqs. (10), (21), (27)–(29) and the experimental curves are displayed in Fig. 12. It can be seen that the model gives a good estimate of the full stress-stain behavior of concrete under uniaxial tension.

Application of the uniaxial stress-strain relationship of concrete in finite element analysis

Foundations of finite element models

Some important problems, such as those involving material constitutive models of concrete and steel, element selection, mesh division, boundary conditions, and interface contact between concrete and steel, are introduced as follows:

1) Material constitutive models

Equation (22) was used for the concrete constitutive. For α₁, Eq. (25) was used for a steel-concrete composite beam, and Eq. (26) was used for a concrete-filled steel tubular stub column. The Poisson ratio of concrete was assumed to be 0.2. The damage plasticity model defined in ABAQUS/Standard 6.8 [5] was used; the eccentricity is 0.1, the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress is 1.225 [30] , the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian is 2/3, the viscosity parameter is 0, and the dilation angle is 40°.

An elasto-plastic model with the Von-Mises yield criteria, associated with the Prandtl-Reuss flow rule and isotropic strain hardening, was used to describe the constitutive behavior of steel. The expression for the stress-strain relationship of steel is described as follows:

(31)

where σ_i is the equivalent stress of steel; f_s is the yield strength; f_u is the ultimate strength, f_u = 1.5f_s; E_s is the elastic modulus, E_s = 2.06×10⁵ MPa; E_st is the strengthening modulus, which is described by E_st = ζE_s; ϵ_i is the equivalent strain; ϵ_y is the yield strain; ϵ_st is the strengthening strain; and ϵ_u is the ultimate strain, which is described by ϵ_u = ϵ_st + 0.5f_s/(ζE_s), where ϵ_st = 12ϵ_y, ϵ_u = 120ϵ_y and ζ = 1/216.

2) Element type and mesh division

The steel tube and steel beam were simulated with 4-node shell elements with reduced integration (S4R). The concrete in the steel-concrete composite beam, the concrete in the concrete-filled steel tubular (CFST) stub column and the rigid end plate at the stub column were modeled using 8-node brick elements (C3D8R) with three translational degrees of freedom at each node. Two-node linear beam elements in space (B31) were used to model the studs, and 2-node linear truss elements (T3D2) were used to model the hoop reinforcement and longitudinal reinforcements in the reinforced concrete. The structured meshing technique was used to mesh the steel-concrete composite beam and the stub column.

3) Interface simulation

For the steel-concrete composite beam, studs were embedded into the concrete. The nodal translational degrees of freedom for the studs were coupled with the nodes of brick elements of concrete to allow the studs and brick elements of the concrete to work together. Reinforcement bars were also embedded into the concrete, and the bond-slip between the reinforcement bars and concrete was neglected. There was no contact between the steel beam and the concrete.

For the stub column, the surface-based interaction consisted of contact in the normal direction and bond-slip interactions in the directions tangential to the surface between the steel tube and concrete core. In the tangential directions, a friction coefficient of 0.5 was used in the analysis; the sliding formulation was finite sliding, and a hard contact was defined in the normal direction. The type of contact between the steel tube and concrete core was surface-to-surface, where the steel tube was considered the master face and the concrete core was the slave face. The constraint type between the concrete and the end plate was Tie, and the constraint type of shell-to-solid coupling was used to model the constraint between the steel tube and the end plate.

4) Boundary conditions and loading mode

The finite element model of both the steel-concrete composite beam and the CFST stub column were founded based on their test specimens. The boundary condition of the steel-concrete composite beam was taken to be that of a simply supported beam. The boundary condition of CFST stub column was axially loaded for the whole section. Figure 13 shows the mesh division and boundary conditions. To obtain the curves of the descending branch, using an incremental iteration method, the displacement loading mode was chosen to solve the nonlinear equations.

Model validation and analysis

To verify the rationality of the presented stress-strain relationship of concrete, experiments for a steel-concrete composite beam are reported in this paper. The main parameters of the steel-concrete composite beam are shown in Fig. 14, where the yield strength and ultimate tensile strength of the steel bar are 294 and 450 MPa, the yield strength and ultimate tensile strength of the steel beam are 304 and 436 MPa, and the yield strength and ultimate tensile strength of the studs are 410 and 475 MPa. The cubic compressive strength of concrete was taken to be 56.4 MPa in this test.

Using the calculation tool of ABAQUS/Standard 6.8 [31], a nonlinear finite element model of a steel-concrete composite beam was built as described above. The analytical results of the moment (M)-deflection (y_c) curve at the mid-span are in reasonable agreement with the experimental results, as shown in Fig. 15.

The analytical results of the load (P)-end slip (s) curve are in reasonable agreement with the experimental values, as shown in Fig. 16. A divergence between the predicted and experimental responses in the post-elastic region is noted, which may be due to an inaccurate representation of the material properties in the constitutive models or to an inaccurate representation of the element type of the studs in the finite element models. Figure 17 shows a comparison of the distribution of longitudinal strain (ϵ) along the sectional height (h) at mid-span between the analytical results and the test results, which indicates reasonable agreement.

Figure 18 shows the experimental failure mode of the steel-concrete composite beam. When the composite beam reaches its final state, at mid-span, the steel beam yields to the tension, and the concrete slab is crushed under compression. Figure 19 shows the numerical failure mode of the steel-concrete composite beam as determined by the nonlinear finite element model. The same failure mode of the composite beam was observed for both the test and the nonlinear finite element method.

Using the calculation tool of ABAQUS/Standard 6.8 [31], a nonlinear finite element model of an axially loaded CFST stub column was built as described above. Comparisons between the predicted curves and the test results or the curves predicted by Wang et al. [32] are shown in Fig. 20, where L is the length of the cylinder and D and t are the outer diameter and wall thickness of the steel tube, respectively.

Based on the assumed stress-strain relationship for the steel tube under two-dimensional stress, the axial stress (σ_L,s) and the perimeter stress (σ_q,s) versus the axial strain (ϵ_L) for the steel tube as calculated by Wang et al. [32] from test results, and thus the axial stress (σ_L,c) versus the axial strain (ϵ_L) for the concrete core, were separated. The results calculated by Wang et al. [32] are shown in Fig. 20(b). Good agreement is obtained by these predicted curves. The analytical results of the finite element method show that, for a concrete core confined by a steel tube, the axial stress increases greatly, the ductility is strengthened, the axial stress of the steel tube decreases rapidly, and the axial stress is transferred from the steel tube to the concrete core.

Figure 21 shows the numerical failure mode of the CFST stub column according to the nonlinear finite element model. When the stub column reaches its final state, a global outward bulge of the stub column is caused by expansion of the concrete under an axial load. However, the experimental failure mode reported by Wang et al. showed a shear failure plane between two local buckling occurrences of the stub column [32]. Thus, the failure mode of the CFST stub column was difficult to simulate, although the load-axial strain relationship could be simulated well.

Conclusions

1) The proposed mechanical indexes of various grades of concrete for the size coefficients of compressive cube and cylinder strength, uniaxial compressive strength, uniaxial tensile strength, elastic modulus, the strain at peak uniaxial stress and the uniaxial stress-strain relationship of concrete are all continuous.

2) Unified formulas that can be applied to various grades of concrete for the size coefficients of compressive cube and cylinder strength, uniaxial compressive strength, uniaxial tensile strength, elastic modulus, the strain at peak uniaxial stress and the uniaxial stress-strain relationship of concrete were suggested. These unified formulas predicted the test results well as a whole.

3) The mathematical equations proposed for the ascending and descending branches of the stress-strain relationship of concrete were applied to a steel-concrete composite beam and a concrete-filled steel tubular stub column using nonlinear finite element analysis. The predicted results for the beam and the column are in reasonable agreement with the experimental results. The results indicate that the uniaxial stress-strain relationship of concrete suggested by this paper has extensive applicability.

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