Strain-path-dependent stress–strain model for ultrahigh-performance concrete columns constrained by stirrups

Abstract

The incorporation of steel fibers improves the microscopic ductility of ultrahigh-performance concrete (UHPC) materials, and their macroscopic ductility can also be improved by the restraining effect of composite stirrups. Two defects can be found in the stress–strain model of UHPC restrained by stirrups based on the design-oriented method: the restraint coefficient of the stirrup is the effective restraint coefficient at the peak state of the specimen, while the variation in the restraint stress of the stirrup with the axial–hoop strain relationship (strain path) is ignored. In this study, a strain-path-dependent stress–strain model for UHPC constrained by stirrups and fibers is proposed through a database with 10 sets of test results on UHPC columns, which alleviates the shortcomings of the design-oriented model. (1) On the basis of the axial–hoop strain relationship curve in the test results, an axial–hoop strain model of UHPC restrained by stirrups is established. (2) The variation in the restraint stress of the stirrup with strain path is considered. (3) On the basis of the stress–strain model of actively constrained concrete, an iterative method of subsections is adopted, and a UHPC stress–strain model restrained by stirrup based on strain path is established. Results show that the established axial–hoop strain equation can better predict the hoop strain of UHPC restrained by stirrups and steel fibers, the calculated value of peak stress of UHPC constrained by stirrups based on strain path is in good agreement with the test results, and the outcomes of the path-dependent stress–strain model are in better agreement with the experimental results than those of the design-oriented stress–strain model.

Keywords

ultrahigh-performance concrete composite stirrup hoop strain strain path stress–strain model

Introduction

Ultrahigh-performance concrete (UHPC) is a cement-based material made of quartz sand and ultrafine silica fume (Alsalman et al., 2019; Shi et al., 2015). It is featured by ultrahigh resistance strength, but the ductility of the UHPC matrix is poor. Method Ⅰ to improve the ductility of the UHPC matrix is to incorporate an appropriate amount of steel fiber into the matrix (Mcswain et al., 2018). Method Ⅱ is to provide more transverse reinforcement (Hosinieh and Aoude, 2015; Shin et al., 2014). Steel fibers can effectively inhibit the protective layer of UHPC columns from falling off, and decreasing the spacing of transverse reinforcement can effectively improve the ductility of UHPC columns after peak axial compression(Shin et al. (2015, 2017). Chang et al. (2021) found that the restraining effect of an end hook-shaped steel fiber prevents the protective layer from peeling off when the UHPC column is destroyed. The restraining effect of specimens with dense stirrups is significant, and the deformability is improved.

Scholars has developed the design oriented stress–strain model of laterally confined concrete. Chung et al. (2002) established a stress–strain model for concrete restrained by stirrups controlled by three coordinates. Mander et al. (1988a, b) proposed a calculation method for the effective restraint coefficient of stirrups with various stirrup forms. Cusson and Paultre (1994, 1995) focused on factors, such as concrete strength, stirrup yield strength, stirrup ratio, stirrup spacing, and longitudinal reinforcement ratio of confined high-strength concrete (HSC). While, Razvi and Saatcioglu (1999) focused on factors, such as spiral stirrup, linear stirrup, cross stirrup, and welded steel wire fabric. Chang et al. (2021) revised the effective restraint coefficient of stirrups proposed by Mander though considering the fiber restraint coefficient, and they established an axial stress–strain model of UHPC restrained by stirrup. The specimens experiencing stress reduction, or a parabolic first portion and an ascending linear second portion (Liao et al., 2022) were adopted to established a stress–strain model for UHPC restrained by fiber-reinforced polymer (FRP). There are two shortcomings in the design-oriented models, (1) the restraint coefficient of the stirrup is a constant value at the peak state of the specimen; (2) the variation in the restraint stress of the stirrup with the axial–hoop strain relationship (strain path) is ignored. The restraint stress of the stirrup cannot be increased with the increase in axial strain.

Unlike the design-oriented model, The axial stress–strain model of actively restrained concrete can be considered to obtain the axial stress–strain model of passively restrained concrete. The axial–hoop strain relationship curve of concrete restrained by FRP is nonlinear (Li and Wu, 2016). Lin et al. (2018) believed that the compressive strength of confined concrete is path-dependent. An analytical model was developed to predict the mechanical behavior of concrete columns confined by FRP or steel tubes (Chen et al., 2018). Lai et al. (2020a) established an axial stress–strain model of CFST columns based on the equation of axial–hoop strain and the interaction between steel tube and concrete in consideration of the debonding effect. Lai et al. (2020b) also established an axial stress–strain model of FRP-confined concrete columns by considering the hoop equation and the interaction between FRP and concrete. Ho et al. (2020) established an axial stress–strain model of concrete-filled FRP tube (CFFT) columns considering the strain path and the axial stress–strain relationship of FRP tubes. These path-dependent axial stress–strain models are for confined concrete, especially FRP, FRP tube, and steel tube confined concrete. These models match very well with the experimental curves of CFFT and CFST columns. Few have reported considering the research on a UHPC stress–strain model constrained by fiber and stirrup based on strain path.

In this study, in consideration of the axial–hoop strain equation and the interaction between stirrups and UHPC, a strain-path dependent stress–strain model for UHPC constrained by stirrups and fibers is established through the axial compression test of 10 sets of UHPC columns constrained by composite stirrup. Herein, to better understand and simulate the behaviour of UHPC columns constrained by stirrups and fibers, a semi-theoretical - empirical stress–strain model which consists of the following four crucial parts: (1) an axial–hoop strain equation for UHPC columns constrained by stirrups and fibers; (2) a stress–strain model for actively confined concrete considering path dependence; (3) a model for the interaction between hoops and UHPC; (4) the fiber constraint coefficient equation for steel fibers that considers their type, shape, and aspect ratio. A theoretical reference for the calculation analysis and engineering application of UHPC columns constrained by stirrups is provided.

Test results of UHPC columns

Specimen design

In the experiment, 10 sets of constrained UHPC columns with geometric dimensions of 200 mm × 200 mm × 600 mm were designed and denoted as UHPC-1–UHPC-10. The geometric dimensions and reinforcement of the specimens are shown in Figure 1, and the major design parameters and partial test results are listed in Table 1. Steel pipe hoops with a height of 180 mm were used at the upper and lower ends of a specimen, and concrete with a height of 100 mm was poured inside to ensure that failure will occur in the middle of the column.

Figure 1.

Geometric dimensions and reinforcement of the specimens (mm).

Table 1.

Test design parameters and partial test results.

Code	Stirrup parameters			V_f/%	f_cu/MPa	σ_ccf,z/MPa	E_cc,f/GPa	λ _v	σ_ccsf,z/MPa	σ_ccsf,z/σ_ccf,z	ε_ccf,z/10^–3	ε_ccsf,z/10^–3	ε _ccsf,z/ ε _ccf,z
Code	s/mm	ρ_v/%	Type	V_f/%	f_cu/MPa	σ_ccf,z/MPa	E_cc,f/GPa	λ _v	σ_ccsf,z/MPa	σ_ccsf,z/σ_ccf,z	ε_ccf,z/10^–3	ε_ccsf,z/10^–3	ε _ccsf,z/ ε _ccf,z
UHPC-1	40	3.07	DC	2	117.5	91.4	39.50	0.17	122.5	1.34	2.63	3.87	1.47
UHPC-2	60	2.05	DC	2	117.5	91.4	39.50	0.11	109.8	1.20	2.63	3.46	1.32
UHPC-3	80	1.54	DC	2	117.5	91.4	39.50	0.09	103.3	1.13	2.63	3.21	1.22
UHPC-4	100	1.23	DC	2	117.5	91.4	39.50	0.07	99.6	1.09	2.63	3.05	1.16
UHPC-5	120	1.02	DC	2	117.5	91.4	39.50	0.06	95.9	1.05	2.63	2.92	1.11
UHPC-6	60	2.05	DC	1.5	114.8	89.2	39.04	0.12	108.1	1.21	2.55	3.32	1.30
UHPC-7	60	2.05	DC	1	110.2	85.3	38.72	0.12	103.9	1.22	2.45	3.14	1.28
UHPC-8	60	2.05	DC	0	92.5	71.7	32.52	0.14	88.9	1.24	2.26	2.85	1.26
UHPC-9	40	2.61	CC	2	117.5	91.4	39.50	0.14	116.9	1.28	2.63	3.69	1.40
UHPC-10	60	1.74	CC	2	117.5	91.4	39.50	0.10	106.9	1.17	2.63	3.36	1.28

Note. s is the stirrup spacing; ρ_v is the volumetric stirrup ratio; CC and DC are the cross- and diamond-shaped stirrups, respectively; V_f is the volume of steel fiber; f_cu is the compressive strength of fiber UHPC cube; E_cc,f is the elastic modulus of the fiber-constrained UHPC; σ_ccf,z and ε_ccf,z are the peak stress and corresponding strain of UHPC without stirrups, respectively; λ_v is the stirrup characteristic value; and σ_ccsf,z and ε_ccsf,z are the UHPC peak stress and corresponding strain under the combined stress of stirrups and fibers, respectively.

Test loading was performed using a 30,000 kN electrohydraulic servo testing machine (Figure 2a). Constant displacement was adopted in the experiment for continuous loading at a speed of 0.3 mm/min; a similar method was adopted by Wu et al. (2018). Two vertical displacement meters (D1, D2) were arranged in the middle of two opposite sides of a specimen, while two transverse displacement meters (D3, D4) were arranged in the two remaining opposite sides. Two vertical strain gauges (S1, S2) and two transverse strain gauges (S3, S4) were arranged in opposite sides (Figure 2b). The strains of the outer and inner stirrups were marked as W1–W3 and N1–N3, respectively (Figure 2c).

Figure 2.

Test loading device and layout of measuring points. (a) Photo of setup. (b) Layout of UHPC measuring points. (c) Layout of reinforcement points.

Failure mode of UHPC columns

Changes in the volume stirrup ratio, stirrup form, stirrup spacing, and fiber volume content result in different final failure modes, as shown in Figure 3. The UHPC-8 group without steel fibers peeled off in large pieces at the corner of the protective layer when it was damaged (Figure 3h) and exhibited serious failure. The number and width of cracks in the middle of specimens UHPC-1–UHPC-5 decreased during failure, and the deformation capability of the specimens deteriorated as the spacing between stirrups increased (Figure 3a–e). The protective layer of the specimens did not peel off due to the bridging effect of steel fibers at the crack. The angle between the main crack and the vertical line in the middle of specimens UHPC-2, UHPC-6, and UHPC-7 increased as the fiber content increased, and the deformability gradually increased (Figure 3b, f, and g). Similar results were obtained by EL-Attar et al. (2016). The diamond-shaped composite stirrups could restrain core UHPC more effectively than the cross-shaped composite stirrups for specimens UHPC-1, UHPC-2, UHPC-9, and UHPC-10. The diamond-shaped composite stirrup specimens had numerous small-width cracks, and their plastic deformation capability was better at failure than their cross-shaped counterparts.

Figure 3.

Failure mode of the specimens. (a) UHPC-1. (b) UHPC-2. (c) UHPC-3. (d) UHPC-4. (e) UHPC-5. (f) UHPC-6. (g) UHPC-7. (h) UHPC-8. (i) UHPC-9. (j) UHPC-10.

Axial load–axial/hoop strain curve of UHPC columns

Figure 4 shows the relationship curve of axial load (N)–axial/hoop (ε) strain of constrained UHPC specimens. In the figure, the negative and positive directions of the abscissa axis represent hoop and axial strains, respectively. The axial load was borne by the following three parts: (1) all longitudinal steel bars, (2) the UHPC protective layer, and (3) the UHPC constrained core area. Figure 4 illustrates that the hoop strain gradually increased with the increase in the axial strain of the specimens, and the restraining force of the stirrup on the UHPC also gradually increased. When the peak load reached 0.75 N_max (i.e. the maximum load), the descending section of the curve steepened, which then slightly slowed down and formed an inflection point at 0.75 N_max. This result was due to that new cracks appeared in the UHPC protective layer at this time, and the effective carrying capacity of the UHPC protective layer gradually decreased. After the curve approached 0.75 N_max, the UHPC protective layer could no longer bear the axial load, although the UHPC protective layer did not feel off; this phenomenon was also observed by Wu et al. (2018).

Figure 4.

Axial load (N)–axial/hoop (ε) strain curve. (a) The effect of stirrup spacing. (b) The effect of volume of steel fiber. (c) The effect of volume of steel fiber.

Stress–strain curve of steel reinforcement

The material performance indicators of steel bars are listed in Table 2.

Table 2.

Material performance indicators of steel bars.

Rebar grade	d/mm	f_y/MPa	f_u/MPa	E_s1/MPa	E_s2/MPa	δ/%	ε _fs	ε _fsu
HRB400	6	444.5	566.5	224.1	9.125	19.5	0.00198	0.01536
HRB400	10	425.8	533.6	222.2	7.760	17.7	0.00191	0.01582

Note. d is the diameter of steel bars; f_y and f_u are the yield strength and ultimate strength of steel bars, respectively; E_s1 is the elastic modulus of bars in the linear elastic section; E_s2 is the elastic modulus of bars in the yield section; δ is the elongation of steel bars; ε_fs is the yield strain of longitudinal bars or stirrups; and ε_fsu is the ultimate strain of longitudinal bars or stirrups.

An enhanced elastic–plastic model was adopted to simplify the calculation of the stress–strain relationship of steel reinforcement. Therefore, the relationship of steel reinforcement can be obtained as follows:

ε_{cz} = ε_{sz} = ε_{z}

(1)

ε_{c θ} = ε_{s θ} = ε_{θ}

(2)

{\begin{cases} σ_{s} = E_{s 1} ε_{s} ε_{s} \leq ε_{fs} \\ σ_{s} = f_{y} + E_{s 2} (ε_{s} - ε_{fs}) ε_{fs} < ε_{s} \leq ε_{fsu} \end{cases}

(3)

where ε_cz and ε_cθ are the axial and hoop strains of UHPC, respectively; ε_sz and ε_sθ are the axial strain of longitudinal bars and the hoop strain of stirrups, respectively; and ε_z and ε_θ are the axial and hoop strains of UHPC columns, respectively.

UHPC protection layer

To simplify the calculation, the assumptions were as follows: (1) the UHPC protective layer and the core UHPC shared the axial force before the peak load, (2) the axial force assumed by the UHPC protective layer was 0 when the specimen approached 0.75 N_max, and (3) the UHPC protective layer showed a linear relationship with the axial strain and lost its bearing capacity during the decrease from N_max value to 0.75 N_max. Therefore, the effective pressure-bearing area of the UHPC protective layer was as follows:

{\begin{cases} A_{cov,e} = (A - A_{cor}) ε_{cz} \leq ε_{ccsf, z} \\ A_{cov,e} = (A - A_{cor}) \frac{ε_{0.75, z} - ε_{cz}}{ε_{0.75, z} - ε_{ccsf, z}} ε_{ccsf, z} < ε_{cz} \leq ε_{0 .75, z} \\ A_{cov,e} = 0 ε_{cz} > ε_{0 .75, z} \end{cases}

(4)

where A is the cross-sectional area of UHPC columns, A_cov,e is the effective area of the UHPC protective layer, A_cor is the core area of UHPC, and ε_0.75,z is the corresponding strain when the bearing capacity of the specimen drops 0.75 times of the peak stress.

Axial stress–strain curve of restrained UHPC

After deducting the axial force borne by the longitudinal bars and the UHPC protective layer from the total axial force of the UHPC column, the stress of constrained UHPC in the core area can be obtained as follows; a similar method was adopted by Wu et al. (2018):

σ_{cc} (ε_{cz}) = \frac{N_{u, t} (ε_{cz}) - σ_{s} (ε_{sz}) A_{s}}{A_{cor} + A_{cov,e}}

(5)

where σ_cc(ε_cz) is the stress of constrained UHPC when the axial strain is ε_cz, N_u,t(ε_cz) is the total axial force of the UHPC column when the strain is ε_cz, σ_s(ε_sz) is the stress of longitudinal bars when the axial strain is ε_sz, and A_s is the cross-sectional area of all longitudinal bars.

UHPC stress–strain model constrained by stirrups based on strain path

The following four components should be considered for the compressive stress–strain model of UHPC constrained by stirrups based on strain path, which was adopted by Lai et al. (2020a): (1) a restraint stress equation generated by the stirrup, (2) a fiber constraint coefficient for steel fiber, (3) a hoop–axial strain model for UHPC column, and (4) an iterative method of four sections is adopted on the basis of the actively restrained concrete stress–strain model incorporating path dependence. Although the model is complicated, it can actively characterize the constraint state of the composite stirrups and fibers in UHPC.

Equation generated by stirrups

Figure 5 is a schematic of the cross-sectional stress of UHPC constrained by stirrups. The restraining force generated by stirrups on UHPC is assumed to be uniformly distributed on the cross section; this method is different from that of Chang et al. (2021). Moreover, the hoop strains of UHPC materials on the cross section are equal. The UHPC hoop restraint force produced by the diamond- and cross-shaped composite stirrups can be obtained as follows:

b_{c} s f_{l, DC} + (2 + \sqrt{2}) A_{s v 1} σ_{s} = 0 \Rightarrow f_{l, DC} = - \frac{(2 + \sqrt{2}) A_{s v 1}}{b_{c} s} σ_{s}

(6)

b_{c} s f_{l,CC} + 3 A_{s v 1} σ_{s} = 0 \Rightarrow f_{l,CC} = - \frac{3 A_{s v 1}}{b_{c} s} σ_{s}

(7)

where b_c is the distance between the axis of the stirrup restraining the short and long sides of the UHPC cross section; s is the distance between stirrups; f_l,DC and f_l,CC are the hoop stress of UHPC in diamond and cross-shaped composite stirrups, respectively; σ_s is the stress of the stirrup; and A_sv1 is the area of a single stirrup section.

Figure 5.

Stress of stirrups and UHPC. (a) DC. (b) CC.

The volumetric stirrup ratio of the UHPC core area is provided as follows:

ρ_{sv, DC} = \frac{(4 + 2 \sqrt{2}) A_{s v 1}}{b_{c} s}

(8)

ρ_{sv, CC} = \frac{6 A_{s v 1}}{b_{c} s}

(9)

where ρ_{sv, DC} and ρ_{sv, CC} are the volumetric ratios of diamond- and cross-shaped composite stirrups, respectively.

Substituting equations (8) and (9) into equations (6) and (7), respectively, yields

f_{l} = - \frac{ρ_{sv}}{2} σ_{s}

(10)

where f_l is the restraint stress generated by the stirrup, and ρ_sv is the volumetric ratio of composite stirrups.

Fiber reinforcement

UHPC with fiber needs to consider the influence of the fiber constraint coefficient on its compression performance. The fiber constraint coefficient considers the effects of fiber type, fiber shape, fiber aspect ratio, and other parameters (JGJ/T 465-2019), as expressed below:

Ω_{f} = \sum_{i = 1}^{n} α_{c, i} V_{f, i} l_{f, i} / d_{f, i}

(11)

where Ω_f is the fiber constraint coefficient; α_c,i is the influence indicator of fiber type on confined fiber concrete, in which the steel fiber is 1.0; V_f,i is the fiber volume content; l_f,i is the fiber length; and d_f,i is the fiber diameter.

Axial–hoop strain equation

Figure 6 shows the axial–hoop strain relationship curve of the specimens. The figure shows that as the axial strain is gradually increased, the hoop strain is gradually increased, which results in a gradual increase in the restraining force of the stirrup. The restraint stress generated by the stirrup changes as the axial strain increases. When the axial strain is the same, hoop strain is generated when the stirrup spacing becomes increasingly larger (Figure 6a). Equation (10) reveals that as the stirrup spacing increases, the hoop restraint generated by the stirrups decreases, which causes an increase in the hoop strain. Therefore, the hoop strain should be related to the value of f_l. Equation (11) demonstrates that when the stirrup configuration is the same, the difference in hoop strain is smaller with the increase in fiber content (Figure 6b). Thus, fiber content plays a small role in the hoop strain of UHPC. This finding is due to that the compressive strength of the UHPC matrix is larger when the fiber content is larger (Ho et al., 2020; Lai et al., 2020b). The compressive strength of concrete exerts a small effect on the hoop strain. When the axial strain is of a certain value, the hoop strain differs when the form of the stirrup varies (Figure 6c), even though the difference is small. Equations (6) and (7) can also confirm this phenomenon.

Figure 6.

UHPC axial–hoop strain curve. (a) The effect of stirrup spacing. (b) The effect of volume of steel fiber. (c) The effect of volume of steel fiber.

Axial–hoop strain models of CFST and CFFT columns are established on the basis of the works of the literature in (Lai et al., 2020a, 2020b), and a piecewise formula is used to fit the UHPC material constrained by composite stirrups:

{\begin{cases} ε_{z} = L S {(\frac{σ_{ccf, z}}{30})}^{- 0 . 05} {ε_{ccf,z} {[1 + 0.75 (\frac{- ε_{θ}}{ε_{ccf,z}})]}^{0.7} - ε_{ccf,z} exp [7 (\frac{- ε_{θ}}{ε_{ccf,z}})] \\ + 0.07 {(- ε_{θ})}^{0.7} [1 + 26.8 (\frac{f_{l}}{σ_{ccf, z}})]} ε_{θ} \leq ε_{θ, y} \\ ε_{z} = L S {(\frac{σ_{ccf, z}}{30})}^{- 0 . 05} {ε_{ccf,z} {[1 + 0.75 (\frac{- ε_{θ}}{ε_{ccf,z}})]}^{0.7} - ε_{ccf,z} exp [7 (\frac{- ε_{θ}}{ε_{ccf,z}})] \\ + 0.07 {(- ε_{θ})}^{0.7} [1 + 26.8 \frac{[f_{l} - 0.5 ρ_{sv} ε_{yv} (E_{sv 2} - E_{sv 1})] \frac{E_{sv 1}}{E_{sv 2}}}{σ_{ccf, z}}]} ε_{θ} > ε_{θ, y} \end{cases}

(12)

where LS is a factor relating to the concrete crack effect; ε_θ,y is the yield strain of stirrups; and σ_c0,z and ε_c0,z are the peak stress and corresponding strain of UHPC without stirrups and fibers, respectively.

\frac{σ_{ccf,z}}{σ_{c0,z}} = 1 + 0.28 Ω_{f} (1 - 0.2 Ω_{f})

(13)

\frac{ε_{ccf,z}}{ε_{c0,z}} = 1 + 0.85 Ω_{f} + 0.026 Ω_{f}^{2}

(14)

Based on the test results of axial–hoop strain curve, equations (1)–(3), and equations (10)–(12), linear regression analysis shows that the LS of the UHPC column constrained by composite stirrups is 0.681. If the fiber content is unknown, then the σ_ccf,z and ε_ccf,z of the UHPC matrix with different fiber contents can be obtained through equations (13) and (14), respectively.

Establishment of a restrained UHPC compression constitutive model based on strain path

Given that the stirrups arranged at equal intervals are directly embedded in the UHPC, the stirrups and the UHPC matrix have difficulty to slip and crack. The actively restrained stress–strain model is revised by adopting an iterative process using partitioned segments to establish an axial stress–strain model of the constrained UHPC column on the basis of strain path. In accordance with the results of the literature, the test results obtained by constrained UHPC columns with stirrups under the same axial strain are adjusted with the calculated results of the actively restrained concrete model (Lai and Ho 2016). The stress–strain formula of actively restrained concrete is as follows:

\frac{σ_{cc} (ε_{z})}{σ_{ccsf,z}} = \frac{A (ε_{z} / ε_{ccsf, z}) + B {(ε_{z} / ε_{ccsf, z})}^{2}}{1 + (A - 2) (ε_{z} / ε_{ccsf, z}) + (B + 1) {(ε_{z} / ε_{ccsf, z})}^{2}}

(15)

where σ_cc(ε_z) is the stress of UHPC constrained by stirrups when the axial stain is ε_z. The calculation process of the shape control parameters A and B can be referred to the literature (Attard and Setunge, 1996).

\frac{σ_{ccsf,z}}{σ_{ccf,z}} = {(1 + \frac{f_{l}}{0.56 \sqrt{σ_{ccf,z}}})}^{1.25 (1 + 0.062 f_{l} / σ_{ccf,z}) σ_{ccf,z} - 0.21}

(16)

\frac{ε_{ccsf,z}}{ε_{ccf,z}} = [1 + (17.0 - 0.06 σ_{ccf,z}) \frac{f_{l}}{σ_{ccf,z}}]

(17)

Two sections are used to obtain the stress–strain model of CFST by Lai et al. (2020a). On the basis of the strain path of UHPC constrained by stirrups and the stress–strain model of actively restrained concrete, four sections are used to obtain the stress–strain model of UHPC constrained by stirrups; that is, three intervals are adopted before the peak stress (section 1), two intervals are adopted from the peak stress to the stirrup yield section (section 2), three intervals are adopted after the stirrup reaches 1.232 times the stirrup yield strain section (section 3), and four intervals are adopted after the stirrup strain reaches 1.232 times the failure section of the specimen (section 4). For UHPC-1, when the hoop strain values are 0.000030, 0.000086, 0.000174, 0.000178, 0.000198, 0.000212, 0.000224, 0.000244, 0.000264, 0.000338, 0.000410, and 0.000476, the corresponding constrained force f_l values are 0.939, 2.679, 5.419, 5.607, 6.167, 6.195, 6.210, 6.235, 6.261, 6.335, 6.446, and 6.530 MPa, respectively, as obtained using equation (10). Then, the corresponding series of actively refined concrete stress–strain curves are drawn. The relative difference (Δ_i=(σ_cc-pas-σ_cc-act)/σ_cc-act×100%) between the axial stress obtained by actively restrained concrete and the test results of UHPC constrained by stirrups represents the axial stress deviation value of the same restraint stress. σ_cc-pas and σ_cc-act are the axial stress of UHPC constrained by stirrups and the axial stress of actively restrained concrete under equal restraining force, respectively. Table 3 shows the calculated results of UHPC-1 specimens based on strain path. Figure 7(a) shows the deviation in the axial stress–strain curves of actively restrained concrete and UHPC-1 restrained by stirrups. Before the peak load, the axial strain is small, and the axial stress of UHPC restrained by stirrups is in good agreement with the axial stress curve of actively restrained concrete. After the peak load, the Δ values are all negative with the increase in axial strain, and the deviations are larger. The reason for this phenomenon is that the actual lateral restraint stress fails to play its role in the weakly restrained area of the core UHPC, leading to higher calculated hoop stress than the actual hoop stress. The σ_ccsf,z value obtained through equation (16) should be reduced to decrease the Δ value of the descending section of the UHPC curve constrained by stirrups. This adjusted method was adopted by Lai et al. (2020b). Figure 7(b) shows the Δ value of the axial stress–strain curve of actively restrained concrete after stress adjustment and that of UHPC-1 restrained by stirrups. The Δ values in the ascending and descending sections are both within 4%.

Table 3.

Calculation results of UHPC-1 based on strain path.

NO.	Specimens	σ_ccf,z/MPa	ε_z/%	ε_θ/%	f_l/MPa	f_l/σ_ccf,z	σ_cc-pas/MPa	σ_cc-act/MPa	Δ_i/%	σ_ccsf,z-cal/MPa	γ	σ_cc-cal/MPa	Δ_i^′/%
1		91.4	0.130	0.030	0.939	0.010	49.68	52.00	−4.462	98.02	0.992	51.58	−3.684
2		91.4	0.251	0.086	2.679	0.029	95.22	94.54	0.719	110.21	0.992	93.78	1.536
3		91.4	0.380	0.174	5.419	0.059	122.51	124.38	−1.503	127.38	0.985	121.60	0.748
4		91.4	0.390	0.178	5.607	0.061	122.42	125.27	−2.275	126.54	0.977	122.42	0.000
5		91.4	0.415	0.198	6.167	0.067	120.57	130.54	−7.638	122.55	0.924	120.57	0.000
6	UHPC-1	91.4	0.430	0.212	6.195	0.068	119.39	131.18	−8.988	120.89	0.910	119.39	0.000
7		91.4	0.450	0.224	6.210	0.068	116.33	132.74	−12.363	116.49	0.876	116.12	0.181
8		91.4	0.480	0.244	6.235	0.068	112.34	132.91	−15.477	112.34	0.845	112.34	0.000
9		91.4	0.520	0.264	6.261	0.069	110.38	130.41	−15.359	112.75	0.846	110.67	−0.262
10		91.4	0.650	0.338	6.355	0.070	100.08	114.86	−12.868	116.52	0.871	100.73	−0.645
11		91.4	0.790	0.410	6.446	0.071	89.50	101.98	−12.238	117.81	0.878	89.76	−0.290
12		91.4	0.930	0.476	6.530	0.071	80.93	92.84	−12.829	117.42	0.872	81.10	−0.210

Note. σ_cc-pas, σ_cc-act, and σ_cc-cal are the test results of axial stress of UHPC constrained by stirrups, the calculated value of axial stress of actively restrained concrete, and the calculated value of the axial stress of UHPC constrained by stirrups under equal restraining force, respectively; σ_ccsf-cal is the calculated value of axial peak stress of UHPC constrained by stirrups; Δ_i=(σ_cc-pas-σ_cc-act)/σ_cc-act×100%; Δ_i^′=(σ_cc-pas-σ_cc-cal)/σ_cc-cal×100%; and γ=σ_cc-pas/σ_cc-act.

Figure 7.

Deviation in axial stress–strain curves of actively restrained concrete and UHPC-1 restrained by stirrups. (a) Before correction. (b) Revised.

Generation of an axial stress–strain model of restrained UHPC

Similarly, the deviation between the axial stress–strain curves of actively restrained concrete and CFST specimens is not perfectly linear, which can be obtained. Equation (18) is used to adjust the peak stress of CTSF columns by Lai et al. (2020a) through two sections.

{\begin{cases} \frac{σ_{ccsf,z - cal}}{σ_{ccf,z}} = 1 + 4.1 (\frac{f_{l}}{σ_{ccf,z}}) \frac{f_{l}}{σ_{ccf,z}} \leq 0.085 \\ \frac{σ_{ccsf,z - cal}}{σ_{ccf,z}} = [1 + 4.1 (\frac{f_{l}}{σ_{ccf,z}})] [0.71 - 0.12 {(\frac{f_{l}}{σ_{ccf,z}})}^{- 0.34}] {(\frac{f_{l}}{σ_{ccf,z}})}^{- 0.34} 0.085 < \frac{f_{l}}{σ_{ccf,z}} \leq 0.611 \end{cases}

(18)

Three main factors are used to obtain the model: (1) the axial-hoop strain equation of stirrup and fiber for UHPC columns, (2) At the same time change of constrained stress based on the strain equation, (3) At the same time change of the relationship between the peak stress of UHPC columns and the constrained stress of stirrups. For the UHPC specimens restrained by stirrups with the same restraint stress values, the Δ value of the descending section of the curve varies with the stirrup spacing and the fiber restraint coefficient. In the UHPC confined by stirrups and fibers, the increase of confining stress as the increase in the hoop strain, whereas the confined concrete confined by active stress. For achieving the same axial strain, lesser load is needed in passively confined concrete and the axial stress will become smaller. For section 1, the Δ values of UHPC-2–UHPC-10 specimens are less affected by the restraint stress, and the degree of agreement in the ascending section of the curve is higher. The reduction factor is used to modify the axial peak stress (equation (19)) under active restraint, and it is regarded as the average values of σ_cc-pas/σ_cc-act at three intervals in section 1. For section 2, the values of σ_ccsf,z-cal/σ_ccf at the three interval points of UHPC-2–UHPC-10 specimens change linearly with the values of f_l/σ_ccf. For sections 3 and 4, the σ_ccsf,z-cal/σ_ccf ratio at the four intervals and five intervals of UHPC-2–UHPC-10 specimens varies with f_l/σ_ccf in a curvilinear trend, respectively. Therefore, according to equation (20), the actively restrained peak stress in the descending section of the UHPC axial stress-strain curve under the stirrup constraint obtained using equation (16) is adjusted in this study through the four sections.

\frac{σ_{ccsf,z - cal}}{σ_{ccf,z}} = β (1 + 6.49 (\frac{f_{l}}{σ_{ccf,z}}))

(19)

{\begin{cases} β = 1 ε_{θ} \leq ε_{ccsf, θ} \\ β = a_{1} + k_{1} (\frac{f_{l}}{σ_{ccf,z}}) ε_{ccsf, θ} < ε_{θ} \leq ε_{σ s y, θ} \\ β = a_{2} + k_{2} (\frac{f_{l}}{σ_{ccf,z}}) + k_{3} {(\frac{f_{l}}{σ_{ccf,z}})}^{2} ε_{σ s y, θ} < ε_{θ} \leq 1.232 ε_{σ s y, θ} \\ β = a_{3} + k_{4} (\frac{f_{l}}{σ_{ccf,z}}) + k_{5} {(\frac{f_{l}}{σ_{ccf,z}})}^{2} 1.232 ε_{σ s y, θ} < ε_{θ} \leq ε_{r u p} \end{cases}

(20)

where k₁, k₂, k₃, k₄,a₁, a₂ and a₃ are all undetermined coefficients, and linear regression analyses are conducted. Their values are shown in Table 4. Using surface-fitting technique, the equation (21) is developed to estimate the passively-confined UHPC stress, whose volumetric stirrup ratio is from 0.0102 to 0.0307 and fiber constraint coefficient is from 0 to 1.34.

{\begin{cases} a_{1} = 2.535 - 82.131 ρ_{s v} - 0.386 Ω_{f} + 2311.35 ρ_{s v}^{2} + 0.062 Ω_{f}^{2} \\ a_{2} = - 9.13 \times 10^{9} + 1.27 \times 10^{10} \cos (ρ_{s v}) - 7.27 \times 10^{9} \cos (Ω_{f}) - 3.6 \times 10^{9} \cos (2 ρ_{s v}) \\ + 7.27 \times 10^{9} \cos (ρ_{s v}) \cos (Ω_{f}) - 23.49 \cos (2 Ω_{f}) \\ a_{2} = 1.76 \times 10^{10} - 2.45 \times 10^{10} \cos (ρ_{s v}) + 1.40 \times 10^{10} \cos (Ω_{f}) + 6.927 \times 10^{9} \cos (2 ρ_{s v}) \\ - 1.40 \times 10^{9} \cos (ρ_{s v}) \cos (Ω_{f}) + 381.13 \cos (2 Ω_{f}) \\ k_{1} = - 84.85 + 6126.28 ρ_{s v} + 4.55 Ω_{f} - 128076.62 ρ_{s v}^{2} + 1.935 Ω_{f}^{2} \\ k_{2} = 5.89 \times 10^{11} - 8.2 \times 10^{11} \cos (ρ_{s v}) + 4.69 \times 10^{11} \cos (Ω_{f}) + 2.32 \times 10^{9} \cos (2 ρ_{s v}) \\ - 4.69 \times 10^{11} \cos (ρ_{s v}) \cos (Ω_{f}) - 3132.85 \cos (2 Ω_{f}) \\ k_{3} = - 8.75 \times 10^{12} + 1.22 \times 10^{13} \cos (ρ_{s v}) - 6.96 \times 10^{12} \cos (Ω_{f}) - 3.45 \times 10^{12} \cos (2 ρ_{s v}) \\ + 6.96 \times 10^{11} \cos (ρ_{s v}) \cos (Ω_{f}) + 68529.16 \cos (2 Ω_{f}) \\ k_{4} = - 1.53 \times 10^{12} + 2.13 \times 10^{12} \cos (ρ_{s v}) - 1.22 \times 10^{12} \cos (Ω_{f}) - 6.03 \times 10^{11} \cos (2 ρ_{s v}) \\ + 1.22 \times 10^{12} \cos (ρ_{s v}) \cos (Ω_{f}) - 2222 \cos (2 Ω_{f}) \\ k_{5} = 3.3 \times 10^{13} - 4.6 \times 10^{13} \cos (ρ_{s v}) - 2.63 \times 10^{13} \cos (Ω_{f}) - 1.3 \times 10^{13} \cos (2 ρ_{s v}) \\ - 2.63 \times 10^{12} \cos (ρ_{s v}) \cos (Ω_{f}) - 186046.2 \cos (2 Ω_{f}) \end{cases}

(21)

Table 4.

Correction parameters of peak stress of UHPC columns restrained by stirrups.

Code	a ₁	a ₂	a ₃	k ₁	k ₂	k ₃	k ₄	k ₅
UHPC-1	1.787	−570.995	−46.515	−6.610	17027.559	126643.632	1350.333	−9536.069
UHPC-2	1.443	−8.60	−155.71	−5.634	573.00	−7919.18	6988.45	−77848.45
UHPC-3	1.429	−466.37	629.51	−10.290	27987.68	−418917.47	−36487.36	529410.08
UHPC-4	1.467	123.78	327.34	−17.632	−8905.60	161294.60	−23543.89	424210.89
UHPC-5	1.534	−556.55	1136.09	−28.400	49906.50	−1117000	−99930.84	2199000
UHPC-6	1.510	−191.936	66.185	−7.013	8549.307	−94599.815	−2663.685	27171.304
UHPC-7	1.581	−174.495	431.249	−8.813	7487.477	−79756.690	−17496.169	177852.77
UHPC-8	1.824	−241.524	609.028	−13.151	8722.875	−78364.906	−20909.751	179725.040
UHPC-9	1.552	−216.862	73.382	−4.489	7441.653	−63456.804	−2396.55	19891.782
UHPC-10	1.358	−309.145	−15.689	−5.067	15882.958	−203237.490	950.808	−13311.484

Figure 8 is a flowchart of the UHPC axial stress–strain model of stirrup-restrained UHPC based on strain path. The current hoop strain is assigned as ε_θⁱ. The specific process is as follows: (1) The transverse strain (ε_θ¹) is initially assumed as 0. (2) ε_θ is substituted into equation (3) to obtain the stress generated by the stirrup (σ_s), and then σ_s is substituted into equation (10) to obtain the hoop restrained force generated by the stirrup (f_l). (3) ε_θ and f_l are substituted into equation (12) to obtain the axis strain (ε_z) of the UHPC constrained by stirrup. (4) The axial peak stress and corresponding strain of the UHPC constrained by stirrups are calculated through equations (19) and (17), and then ε_z is substituted into equation (15) to obtain the axial stress of constrained UHPC (σ_cc). (5) If ε_θⁱ⁺¹ <ε_rup (hoop strain at specimen failure), then the abovementioned iterative process is repeated. If ε_θⁱ⁺¹>ε_rup (hoop strain at specimen failure), then the iteration is finished (Lai et al., 2020a, 2020b).

Figure 8.

Flowchart of the UHPC axial stress–strain curve of stirrup-restrained UHPC based on strain path.

Verification of the constrained UHPC model

Axial–hoop strain path

The axial–hoop strain model directly determines the hoop restraint stress generated by stirrups, and then it exerts a significant effect on the axial stress–strain curve of constrained UHPC. Figure 9 shows the experimental and calculated values of axial–hoop strain curve of UHPC specimens constrained by stirrups. Figure 9(a)–(c) depicts that the calculated value of constrained UHPC specimens is in good agreement with the test value. The axial strain test value of the specimen in the same hoop direction is compared with the calculated value to verify the applicability of the proposed model. As shown in Figure 9(d), the mean, correlation coefficient, and coefficient of variation are 0.997, 0.995, and 0.028, respectively. Therefore, the proposed axial–hoop strain model can be efficiently applied to UHPC constrained by stirrups.

Figure 9.

Experimental and calculated values of axial–hoop strains of UHPC columns restrained by stirrups. (a) The effect of stirrup spacing. (b) The effect of volume of steel fiber. (c) The effect of volume of steel fiber. (d) Comparison of test and calculated values at the same hoop strain.

Peak stress of UHPC constrained by stirrups based on strain path

According to the strain path and the axial stress–strain model of actively restrained concrete, the peak stress of UHPC with stirrup restraint adjusted using equation (19) directly affects the axial stress–strain curve of UHPC with stirrup constraint. Figure 10 shows the experimental and calculated values of the peak stress of UHPC specimens with stirrup restraint based on strain path. Figure 10(a) illustrates that the calculated value of restrained UHPC specimens with different fiber and stirrup restraint coefficients is in good agreement with the test results. The experimental value of the peak stress in the same hoop strain is compared with the calculated value to verify the applicability of the calculation model. Figure 10(b) shows that the mean, correlation coefficient, and coefficient of variation are 1.001, 0.991, and 0.008, respectively. Therefore, the proposed calculation formula of peak stress of UHPC with stirrup restraint based on strain path can be efficiently applied to UHPC with stirrup constraints.

Figure 10.

Experimental and calculated values of peak stress of UHPC with stirrup restraint based on strain path. (a) Test and calculated values of peak stress. (b) Comparison of test and calculated values at the same hoop strain.

Axial stress–strain curve of UHPC restrained by stirrup

Figure 11 shows the axial stress–strain experimental curve and the established constitutive model of UHPC 1, UHPC 6, and UHPC 9 specimens restrained by stirrups, which includes a simple constitutive model based on the design-oriented method. From Figure 11(a)–(c), the compression constitutive model established on the basis of strain path can more accurately fit the test results. The experimental curve is in better agreement in the sections before and after the peak. The constitutive curves established using the two methods are in good agreement with the experimental curves. It can be seen from Figure 11 that the model in this paper is mainly improved in the descending section of UHPC constitutive model compared with the previous Chang model. At the end part of the descending section, the theoretical results are in good agreement with the test results. The following two reasons may be considered: (1) The adjusted actively restrained concrete stress–strain model suitable for UHPC constrained by stirrups can be used to better characterize the UHPC material compression performance under different fiber constraint coefficients. (2) Compared with the design-oriented model by Chang et al. (2021), the change in hoop restraint stress with strain path is considered.

Figure 11.

Test results and the proposed model of axial stress–strain curve of UHPC specimens. (a) UHPC 1. (b) UHPC 6. (c) UHPC 9. (d) The comparison of test and calculated values at the same axial strain.

To further evaluate the deviation in the axial stress–strain constitutive curve of the UHPC specimens restrained by stirrups between the proposed models and the test results, the compression constitutive model based on strain path and the design-oriented model are drawn in Figure 11(d). The figure shows that the mean, correlation coefficient, and coefficient of variation of the design-oriented model are 1.008, 0.990, and 0.055, respectively. The mean, correlation coefficient, and coefficient of variation of the model established on the basis of strain path are 0.998, 1.003, and 0.037, respectively. The increase of confining stress as the increase in the hoop strain, whereas the confined concrete confined by active stress. The new model considering dynamic confining stress increased gradually after the peak stress instead of the effective confining stress at the peak state. The compression constitutive model based on strain path can better predict the compression behavior of UHPC columns restrained by stirrups on the descending section. Therefore, the model presented in this paper can improve the accuracy of finite element nonlinear analysis of UHPC columns constrained by stirrups and fiber.

Conclusion

(1) Considering the influence of stirrup restraint stress and fiber restraint coefficient, the modified axial–hoop strain equation can well predict the hoop deformation of UHPC columns restrained by stirrups under axial stress.

(2) The proposed peak stress–hoop strain equation of UHPC restrained by stirrups can reasonably and accurately predict the peak stress of UHPC columns under stirrup restraint.

(3) Compared with the simple design-oriented model, the use of the axial–hoop strain equation, and a passively confined concrete axial stress–strain curve with adjusted σ_ccsf,z-cal, the axial stress–strain constitutive model of UHPC restrained by stirrups established on the basis of strain path can consider the dynamic change in the restraint stress of the stirrups with strain path and can better match the test curve sensitively and accurately.

(4) The reinforced concrete columns in the project are all in a passive restraint state of the stirrups, and the stress under the axial compression state is related to strain path. Therefore, though the focus of this paper is on UHPC columns restrained by stirrups and fibers, the methodology in this paper is helpful in predicting the influence of strain path under passive confinement stress of other reinforced UHPC columns.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No. 11872300) and Shaanxi Province Key Research and Development (No. 2020ZDLSF06-11).

ORCID iD

Junping Shi

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