Behavior of ultra-high-performance concrete columns subjected to axial compressive load

Abstract

A constitutive model of confined ultra-high-performance concrete (UHPC) was developed based on the theoretical and regression analyses. This constitutive model could be applied to finite element analysis (FEA) according to the comparison of plastic damage analysis and load–displacement curves. A total of 25 stirrup-confined UHPC columns were created through FEA modeling. The variables included stirrup spacing, stirrup configuration, steel fiber volume, and longitudinal reinforcement ratio. The load–displacement curves and the plastic damage region of the UHPC columns were illustrated and analyzed. Moreover, parametric analysis was conducted to evaluate the effects of the aforementioned parameters. The improvement in the bearing capacity and ductility of the UHPC columns resulting from the reduction in stirrup spacing and increase in steel fiber volume indicated that the columns were significantly influenced by the stirrup spacing, stirrup configuration, and steel fiber volume.

Keywords

ultra-high-performance concrete columns axial load bearing capacity ductility constitutive model

Introduction

In recent years, a number of research works have been conducted to formulate techniques for avoiding the brittle failure of high-strength concrete (HSC) columns for high-rise building applications (Cusson and Paultre, 1994; Chen et al., 2018). Existing studies show that the application of stirrups with different spacing and cross-section details can effectively improve the ductility of concrete columns (Sharma et al., 2015; Shin et al., 2016). In addition, the incorporation of steel fibers can provide additional confinement, which delays the spalling of concrete and improves the axial bearing capacity and ductility of concrete columns (Paultre et al., 2010; Shin and Yoon, 2015; Shin et al., 2017; Ahmed and Hadi Muhammad, 2018). Ultra-high-performance concrete (UHPC) is one of the emerging cement-based composite materials with exceptional mechanical properties, such as superior compressive and tensile strengths and remarkable toughness (Xu et al., 2017; Marchand et al., 2019). The UHPC also exhibits excellent ductility, fatigue performance, and durability compared with ordinary concrete and HSC. Accordingly, the UHPC is an attractive alternative option for high-rise building applications. Numbers of studies have been conducted to investigate the axial compressive performance of UHPC columns. Eight UHPC columns were tested by Empelmann et al. (2008) to investigate the axial compressive performance of UHPC columns; all the columns were 600 mm long and had the same cross-sectional dimensions (200 mm in width and 200 mm in depth). It was observed that the ductility of the columns was significantly improved by providing adequate stirrup and steel fibers. Hosinieh et al. (2015) tested six large-scale columns under pure axial loading, and the results showed that the spacing and type of stirrups can substantially affect the performance of the columns. Xu et al. (2018) conducted numerical simulations to study the behavior of UHPC columns subjected to axial compressive loading and horizontal monotonic push loading. The variables considered mainly included steel fiber volume, axial compression ratio, stirrup ratio, and shear span ratio. The results showed that the finite element analysis (FEA) model accurately predicted the behavior and failure mode of the UHPC columns. To study the influence of the spiral reinforcement volumetric ratio, concrete compressive strength, and presence of steel fibers on the performance of UHPC columns, Shin et al. (2018) fabricated six large-scale UHPC columns confined by spiral stirrups, in which the volumetric ratio of the hybrid micro-steel fibers used was 1.5% and the compressive strengths varied in the range of 163–181 MPa. It was reported that the improvement in the ductility of the UHPC column resulted from the combined action of steel fibers and spiral stirrups.

Existing research works mainly focused on UHPC column tests. However, relatively few works utilized the FEA. This article proposes a constitutive model of the confined UHPC based on the theoretical and regression analyses. Considering the variation in stirrup spacing, stirrup configuration, steel fiber volume, and longitudinal reinforcement ratio, a total of 25 stirrup-confined UHPC columns were created through FEA with the application of the developed constitutive model, which subjected to an increasing axial compressive load. The axial compressive behavior, load–displacement curves, and plastic damage region of the columns were investigated.

Constitutive models of ultra-high-performance concrete

Typical constitutive models

Mander et al. (1988) developed a constitutive model of confined concrete by performing 31 full-scale axial compression tests. This model was suitable for concrete with rectangular hoops, spiral hoops, and round hoops. Cusson and Paultre (1995) proposed a constitutive model for the HSC according to the experimental results of 50 HSC columns subjected to axial loading. The concrete compressive strength of the columns varied in the range of 60–120 MPa, and the effects of the transverse configuration, stirrup spacing, and stirrup yield strength were considered in the model. The steel fibers were regarded as advantageous for controlling the spalling of unconfined concrete and improving its ductility and toughness. The expression provided by Aoude (2008) was the same as that of Cusson et al. However, the additional effect of steel fibers was considered in the prediction of peak stress. Saatcioglu and Razvi (1992) proposed a different model where the ascending branch was parabolic and the descending branch was linear up to 15% of the peak stress, and the main influencing factors considered in the model included stirrup configuration, volumetric ratio, stirrup spacing, stirrup yield strength, and section geometry. All the models are summarized in Table 1.

Table 1.

Typical constitutive models.

	Peak stress	Peak strain	Ascending branch	Descending branch
Mander et al. (1988)	$\frac{f_{c c}}{f_{c 0}^{'}} = - 1.254 + 2.254 \sqrt{1 + \frac{7.94 f_{l}}{f_{c 0}^{'}}} - 2 \frac{f_{l}}{f_{c 0}^{'}}$	$\frac{ε_{c c}}{ε_{c 0}^{'}} = 1 + 5 (\frac{f_{c c}}{f_{c 0}^{'}} - 1)$		—
Cusson and Paultre (1995)	$\frac{f_{c c}}{f_{c 0}^{'}} = 1.0 + 2.1 {(\frac{f_{l e}}{f_{c 0}^{'}})}^{0.7}$	$ε_{c c} = ε_{c 0}^{'} + 0.21 {(\frac{f_{l e}}{f_{c 0}^{'}})}^{1.7}$	$\begin{array}{l} σ = f_{c c} \frac{r ε / ε_{c c}}{r - 1 + {(ε / ε_{c c})}^{r}} \\ r = \frac{E_{c}}{E_{c} - f_{c c} / ε_{c c}} \end{array}$	$\begin{array}{l} σ = f_{c c} e^{k_{1} {(ε - ε_{c c})}^{k_{2}}} \\ k_{1} = \ln 0.5 / {(ε_{50} - ε_{c c})}^{k^{2}} \\ k_{2} = 0.58 + 16 {(f_{l e} / f_{c 0})}^{1.4} \end{array}$
Shin et al. (2018)
Aoude (2008)	$\frac{f_{c c}}{f_{c 0}^{'}} = 1 + 2.4 {(\frac{f_{l e}}{f_{c 0}^{'}})}^{0.7} + 4.1 (\frac{f_{l f}}{f_{c 0}^{'}})$	$ε_{c c} = ε_{c 0}^{'} + 0.21 {(\frac{f_{l e}}{f_{c 0}^{'}})}^{1.7}$
Saatcioglu and Razvi (1992)	$\begin{array}{l} f_{c c} = f_{c 0}^{'} + k_{1} f_{l e} \\ k_{1} = 6.7 f_{l e}^{- 0.17} \end{array}$	$\begin{array}{l} ε_{c c} = ε_{c 0}^{'} (1 + 5 k_{3} K) \\ k_{3} = 40 / f_{c 0}^{'}, K = k_{1} f_{l e} / f_{c 0}^{'} \end{array}$	$σ = f_{c c} {[2 (\frac{ε}{ε_{c}}) - {(\frac{ε}{ε_{c}})}^{2}]}^{1 / (1 + 2 K)}$	$σ = f_{c c} + 0.15 f_{c c} \frac{ε - ε_{c c}}{ε_{c c} - ε_{85}}$

Note: $σ$ and $ε$ are the stress and strain of confined concrete, respectively; $f_{c c}$ and $ε_{c c}$ are the peak stress and peak strain of confined concrete, respectively; $f_{c 0}^{'}$ and $ε_{c 0}^{'}$ are the peak stress and peak strain of unconfined concrete, respectively; $f_{l}$ is the average confining pressure provided by stirrups; $f_{l e}$ is the effective confining pressure provided by stirrups; $f_{l f}$ is the confining pressure provided by steel fibers; $E_{c}$ is the elastic modulus of concrete; $ε_{85}$ and $ε_{50}$ are the strains corresponding to the stresses drop to 85% and 50% of the peak stress, respectively.

Confining pressure analysis

Confining pressure provided by stirrups

The confined UHPC has a three-direction compressive stress because of the confinement provided by the stirrups. Horizontal bending would occur on the stirrups because of the lateral deformation of the confined concrete under the axial load. It is assumed that the pressure provided by the stirrups is evenly distributed along the stirrup side length. Accordingly, the lateral confining pressure can be obtained based on the principle of the equilibrium equation, as expressed in equation (1)

f_{l} = \frac{σ_{s v} A_{s v}}{s c}

(1)

where

σ_{s v}

is the stirrup stress when the confined UHPC stress reaches its peak;

A_{s v}

represents the sum of the projected areas of all stirrup along the column length;

s

denotes the stirrup spacing;

c

is the spacing between the central axes of edge stirrups.

Mander et al. (1988) modified the area of the effective confined concrete and proposed an expression for the effective confining pressure provided by stirrups, as expressed in equation (2)

f_{l e} = k_{e} f_{l} = \frac{k_{e} σ_{s v} A_{s v}}{s c} = k_{e} σ_{s v} ρ_{v}

(2)

where

f_{l e}

is the effective confining pressure provided by stirrups;

k_{e}

is the effective confining coefficient;

ρ_{v}

is the stirrup ratio.

The comprehensive influence of stirrup configuration, longitudinal reinforcement distribution, and sectional dimension are given in equations (3a)–(3c)

{confined by circular stirrup: k}_{e} = \frac{{(1 - \frac{s'}{2 d_{s}})}^{2}}{1 - ρ_{c}}

(3a)

{confined by circular spirals: k}_{e} = \frac{1 - \frac{s'}{2 d_{s}}}{1 - ρ_{c}}

(3b)

confined by rectangular stirrup:

k_{e} = \frac{(1 - \sum_{i = 1}^{n} \frac{ω_{i}^{2}}{6 c^{2}}) \cdot {(1 - 0.5 \frac{s^{'}}{c})}^{2}}{1 - ρ_{c}}

(3c)

where

d_{s}

is the diameter of spiral or hoops between bar centers;

\sum ω_{i}^{2}

is the sum of the squares of the clear spacing between adjacent longitudinal reinforcements;

s^{'}

is the clear spacing of stirrups;

ρ_{c}

refers to longitudinal reinforcement ratio.

Confining pressure provided by steel fibers

The additional confinement provided by steel fiber has a significant effect on the peak stress and peak strain (Hosinieh et al., 2015). Aoude (2008) proposed a fiber-reinforced concrete confinement model. This model modifies the expressions proposed by Cusson and Paultre (1995) and Légeron and Paultre (2003) to incorporate the confining pressure provided by steel fibers, where this confining pressure is obtained by multiplying the number of steel fibers and their average pull-out strength, as follows

f_{l f} = N_{f} F_{l f} = α_{1} V_{f} \frac{l_{f}}{d_{f}} τ_{b o n d}

(4)

where

f_{l f}

is the confining pressure provided by steel fibers;

N_{f}

is the number of fibers per unit area;

F_{l f}

refers to the average pullout strength of steel fibers;

α_{1}

is the fiber orientation factor, and its value can be taken as 3/8 (Hosinieh et al., 2015);

V_{f}

is the steel fiber volume;

l_{f}

is the length of steel fiber;

d_{f}

is the diameter of steel fibers;

τ_{b o n d}

is the bond strength.

Peak stress and peak strain

To consider the influence of steel fibers on the peak stress and strain, a comprehensive confining coefficient $(K)$ is proposed; this coefficient is the ratio of the combination of the confining pressure provided by stirrups and steel fibers to the axial compressive strength, calculated as: $K = (f_{l e} + f_{l f}) / f_{c 0}$ .

The experimental data of UHPC columns included steel fibers and stirrups is shown in Table 2. The following statistical regression equations are proposed for

f_{c c} / f_{c 0}

and

ε_{c c} / ε_{c 0}

based on these data, which are applicable to the concrete with confinement

\frac{f_{c c}}{f_{c 0}} = - 4.426 K^{2} + 3.732 K + 0.869

(5)

\frac{ε_{c c}}{ε_{c 0}} = 5.97 K^{2} - 4.44 K + 2.05

(6)

where

f_{c 0}

and

ε_{c 0}

are the axial compressive strength of unconfined concrete and the corresponding strain, respectively.

Table 2.

Experimental data.

Specimens		$f_{c 0}$ (MPa)	$f_{c c}$ (MPa)	$ε_{c c}$ (%)	$ε_{85}$ (%)	$ε_{50}$ (%)>	$\frac{f_{c c}}{f_{c 0}}$	$\frac{ε_{c c}}{ε_{c 0}}$	$f_{l e}$ (MPa)	$f_{l f}$ (MPa)	$k_{1}$	$k_{2}$
Hosinieh et al. (2015)	C3-40	130.0	163.8	0.72	0.95	3.4	1.37	2.57	14.25	6.15	5.9	0.59
	C3-60	124.6	143.1	0.61	0.69	2.0	1.25	2.18	8.60	5.99	6.1	0.51
	C3-80	137.8	147.5	0.49	0.68	1.3	1.18	1.75	5.94	6.40	85.5	1.00
	C3-120	126.8	133.6	0.43	0.57	1.0	1.15	1.54	2.90	6.06	144.0	1.03
	C4-60	134.9	146.6	0.67	1.09	4.0	1.15	2.39	11.24	6.31	7.7	0.71
	C4-120	129.1	139.6	0.52	0.66	1.2	1.14	1.86	4.14	6.13	67.5	0.92
Shin et al. (2017)	180-A3.6	205.2	247.9	0.46	0.71	1.4	1.40	1.18	14.65	6.63	0.71	1.07
	180-C4.5	205.2	253.8	0.48	0.73	3.4	1.45	1.26	16.78	6.63	0.64	0.98
	180-C6.1	205.2	273.5	0.58	0.83	3.8	1.50	1.57	24.47	6.62	0.47	0.76
	180-C9.9	205.2	303.3	0.62	0.81	6.8	1.63	1.72	38.98	6.46	0.39	0.54
	150-A3.0	185.1	217.5	0.36	0.57	1.4	1.53	1.03	10.85	6.18	1.04	1.19
	150-C3.0	185.1	213.5	0.38	0.61	1.4	1.54	1.12	9.49	6.19	1.04	1.28
	150-C3.8	185.1	224.5	0.45	0.69	3.2	1.46	1.22	13.31	6.17	0.72	1.05
	150-C5.1	185.1	241.2	0.47	0.68	3.6	1.54	1.31	19.59	6.19	0.58	0.82
Shunsuke et al. (2007)	200FM2-35	222	305.5	1.39	2.5	4.1	1.38	3.48	21.74	12.37	236.4	1.61
	200FM2-45	222	288.6	0.81	3.10	—	1.30	2.03	15.85	12.37	—	—
	200FM2-55	222	275.7	0.61	1.68	4.4	1.24	1.53	12.31	12.37	29.5	1.15
	160FM2-35	181	218.7	0.50	0.83	3.3	1.21	1.19	8.46	10.80	7.8	0.68
	160FM2-45	181	224.1	0.51	1.04	2.3	1.24	1.21	6.15	10.80	83.6	1.19
	120FM2-35	159	200.0	0.97	1.06	2.5	1.26	2.51	8.46	9.91	5.9	0.51
	120FM2-45	159	205.4	0.53	1.03	2.7	1.29	1.38	6.15	9.91	30.5	0.99
	200NF-35	213	217.5	1.34	3.6	5.4	1.02	3.27	21.74	0	1926	2.48
	200NF-45	213	215.7	0.96	3.4	—	1.01	2.34	15.85	0	—	—
Empelmann et al. (2008)	S1	122.5	162.9	0.31	0.32	0.78	1.33	—	3.25	5.55	5.4	0.38
	S2	122.5	183.9	0.28	—	—	1.50	—	7.42	5.55	—	—
	S3	122.5	140.3	0.29	—	—	1.15	—	3.99	5.55	—	—
	S4	122.5	151.4	0.31	0.52	—	1.24	—	5.97	5.55	—	—
	S5	122.5	155.6	0.30	0.43	—	1.27	—	9.84	5.43	—	—
	S6	118.5	157.4	0.30	0.31	—	1.33	—	9.84	0	—	—
	VK1	126.4	180.6	0.34	0.35	0.7	1.43	—	5.76	9.63	6.6	0.4
	VK2	122.5	144.4	0.31	—	—	1.18	—	5.71	6.19	—	—

Note: $f_{c c}$ and $ε_{c c}$ are the peak stress and peak strain of confined concrete, respectively; $f_{c 0}$ and $ε_{c 0}$ are the axial compressive strength of unconfined concrete and the corresponding strain, respectively; $ε_{85}$ and $ε_{50}$ are the strains corresponding to the stresses drop to 85% and 50% of the peak stress, respectively; $f_{l e}$ is the effective confining pressure provided by stirrups; $f_{l f}$ is the confining pressure provided by steel fibers; $k_{1}$ and $k_{2}$ are the parameters affecting the descending branch of the stress–strain relationship of confined UHPC, as shown in equations (10) and (11).

The comparison between the results predicted by the models and those of the experiments is shown in Figure 1. The ratio of the predictions to experimental results is evaluated by the average and the coefficient of variation (COV). The proposed peak stress predictions are found to agree well with the test results, and the average and COV values of the ratios are 1.00 and 8.78%, respectively. In contrast, the peak stress predictions of the other models are overestimated. For the models proposed by Mander et al. (1988), Saatcioglu and Razvi (1992), Cusson and Paultre (1995), Aoude (2008), and Shin et al. (2018), the average values of the ratios are 1.16, 1.07, 1.13, 1.32, and 1.07, respectively. The strains predicted by the proposed models considerably differ from the experimental results, and the average and COV values of the ratio are 1.00 and 30.2%, respectively. Despite all this, the proposed expression has a better ability to predict the peak strain compared with the other models.

Figure 1.

Comparison between experimental and predicted results. (a) Peak stress (b) Peak strain

Formulation of constitutive model

The prediction results of the stress–strain relationship for the confined UHPC by different model are shown in Figure 3. At the ascending branch, the prediction results agree well with the test results. However, the curves obtained by different models are quite different at the descending branch. Hence, the function for the ascending branch in previous researches (as shown in Table 1) is chosen and modified to consider the effect of steel fiber. The function for the descending branch refers to the research done by Fafitis et al. (1985)

y = \frac{α x}{α - 1 + x} ε < ε_{c c}

(7)

y = \exp [- k_{1} ε_{c c} {(x - 1)}^{k_{2}}] ε \geq ε_{c c}

(8)

α = E_{c} / (E_{c} - f_{c c} / ε_{c c})

(9)

k_{1} = - \frac{\ln 0.85}{{(ε_{85} - ε_{c c})}^{k_{2}}}

(10)

k_{2} = 0.139 K^{- 0.9}

(11)

ε_{85} = 0.275 ε_{c c} K^{- 2.32 K - 0.521}

(12)

where

x = \frac{ε}{ε_{c c}}

;

y = \frac{σ}{f_{c c}}

;

E_{c}

is the elastic modulus of concrete;

α

is the parameter affecting the slope of the ascending branch;

k_{1}

and

k_{2}

are the parameters affecting the descending branch, which are summarized in Table 2. The relationship between

K

and

k_{2}

and the calculation method of the strain at the 85% peak stress can be obtained using the theoretical regression analysis, which is expressed as equations (11) and (12). The UHPC constitutive model graph is plotted in Figure 2.

Figure 2.

Comparison between test and prediction results of constitutive models. (a) C3-40, (b) C3-80, (c) 180-C4.5, (d) 150-C3.8, (e) 200FM2-55, (f) 160FM2-35.

Figure 3.

Compression constitutive model of UHPC.

Comparison with other constitutive models

The stress–strain relationship obtained by the proposed model is shown in Figure 3. It is found that the curve obtained by the proposed model agrees well with the test results for all the six randomly selected UHPC columns. The suboptimal models include those of Cusson and Paultre (1995) and Légeron and Paultre (2003), whereas the models of Mander et al. (1988), Shin et al. (2018), and Saatigule et al. (1992) cannot well predict the stress–strain relationship of UHPC.

Finite element analysis

Finite element analysis (FEA) is an important method for evaluating UHPC structures. The nonlinear behavior of UHPC columns was investigated by linking the research results of the concrete damaged plasticity model (CDP) in ABAQUS with the results of the UHPC constitutive model. Because the stirrups are also simulated in the models, the confining effect of stirrups is ignored in the constitutive model of UHPC (i.e., the comprehensive confining coefficient is calculated as $K = f_{l f} / f_{c 0}$ ).

Constitutive model of material

The compression constitutive model of the UHPC is obtained using equations (7) and (8). The tensile stress–strain relationship of the UHPC is shown in Figure 4. The tensile constitutive model of the UHPC is expressed as follows (Wille et al., 2014)

σ = f_{c t} / ε_{c a}, 0 < ε < ε_{c a}

(13)

σ = f_{c t}, ε_{c a} < ε < ε_{p c}

(14)

where

f_{c t}

is the tensile strength;

ε_{c a}

is the initial strain;

ε_{p c}

is the ultimate strain.

Figure 4.

Tensile constitutive model of UHPC.

The constitutive model of the stirrup and longitudinal reinforcement follows a bilinear model, as shown in Figure 5. The elastic-plastic model was used to simulate them.

Figure 5.

Behavior of steel reinforcing bars in tension.

Plastic damage model of concrete

The use of CDP in ABAQUS is suitable for reflecting the plastic deformation and elastic damage of the UHPC. It describes the inelastic behavior of materials by using the isotropic damage parameters as internal variables of the damage model characteristics and combining the elastic-plastic behaviors under tensile and compressive stress.

Plastic material

The critical stress state is defined by the yield surface, and the plastic deformation starts once the critical stress state is reached. The CDP model adopts the yield criterion proposed by Lubliner et al. (1988) and modified by Lee and Fenves (1998) to explain the different strength evolution rules under tension and compression. The two main parameters determining the shape of the yield surface are $σ_{b 0} / σ_{c 0}$ and K_c, where $σ_{b 0}$ and $σ_{c 0}$ are the initial biaxial and initial uniaxial compressive strengths, respectively; K_c refers to the ratio of the second stress invariant on the tensile meridian plane to that on the compressive meridian plane. The yield function is defined as follows

f = \frac{1}{1 - α} (\bar{q} - 3 α \bar{p} + β ({\bar{ε}}^{p l} 〈 {\bar{σ}}_{\max} 〉 - γ 〈 - {\bar{σ}}_{\max} 〉)) - {\bar{σ}}_{c} ({\bar{ε}}^{p l}) = 0

(15)

β ({\bar{ε}}^{p l}) = \frac{{\bar{σ}}_{c} ({\bar{ε}}^{p l})}{{\bar{σ}}_{t} ({\bar{ε}}^{p l})} (1 - α) - (1 + α)

(16)

α = \frac{(σ_{b 0} / σ_{c 0}) - 1}{2 (σ_{b 0} / σ_{c 0}) - 1}

(17)

γ = \frac{3 (1 - k_{c})}{2 k_{c} - 1}

(18)

where

< >

is defined as

< x > = (| x | + x) / 2

;

\bar{q}

is the Mises equivalent stress;

\bar{p}

is the hydrostatic pressure;

{\bar{σ}}_{\max}

is the maximum effective stress;

{\bar{σ}}_{c}

and

{\bar{σ}}_{t}

are the effective cohesive stresses under compression and tensile conditions, respectively.

In addition, the direction and size of the plastic deformation are determined according to the flow regularity controlled by the plastic flow potential function. The potential function is related to the expansion angle $(φ)$ and potential flow eccentricity $(ξ)$ . The function expression is as follows

G = \sqrt{{(ξ σ_{t 0} \tan φ)}^{2} + {\bar{q}}^{2}} - \bar{p} \tan φ

(19)

where

φ

is the expansion angle measured at a high confining pressure surface;

σ_{t 0}

is the uniaxial tensile stress;

ξ

is the potential eccentricity.

Material damage

For the damaged materials, the effective stress and strain are adopted to present the stress and strain fields corresponding to the damage state. The effective stress, which is the basic core of the continuous damage theory, has been successfully applied to the damage mechanics model.

In the case of the isotropic damage, the effective stress can be calculated as

\bar{σ} = \frac{σ}{1 - d}

(20)

where

d

is the damage factor, which quantitatively reflects the damage evolution degree of the material. The energy damage theory method solved by the Gaussian integral is applied to calculate the damage factor

(d)

d = \frac{A - \bar{A}}{A}

(21)

\bar{A} = \int f (ε) d ε

(22)

A = \frac{1}{2} E_{0} ε

(23)

where A and

\bar{A}

represent the strain energies in the lossless and loss states, respectively.

Stress–strain relationship of compression and tensile damage

The stress–strain curve of the compression damage is shown in Figure 6. According to the classic elastic-plastic theory, the total strain is divided into inelastic and elastic strains to reflect the nonlinearity and irreversibility of the concrete deformation, as expressed in equation (24)

ε_{c}^{i n} + ε_{0 c}^{e l} = ε_{c}

(24)

ε_{0 c}^{e l} = σ_{c} / E_{0}

(25)

where

ε_{c}^{i n}

and

ε_{0 c}^{e l}

are the inelastic and elastic compressive strains, respectively;

ε_{c}

is the total compressive strain;

E_{0}

is the initial stiffness.

Figure 6.

Stress–strain relationship of compression plastic damage model.

The equivalent plastic strain, compressive stress, and effective compressive stress are obtained using equations (26)–(28)

ε_{c}^{p l} = ε_{c}^{i n} - (ε_{c}^{e l} - ε_{0 c}^{e l}) = ε_{c}^{i n} - \frac{d_{c}}{1 - d_{c}} \cdot \frac{σ_{c}}{E_{0}}

(26)

σ_{c} = (1 - d_{c}) E_{0} (ε_{c} - ε_{c}^{p l})

(27)

{\bar{σ}}_{c} = \frac{σ_{c}}{1 - d_{c}}

(28)

where

ε_{c}^{p l}

is the equivalent compressive plastic strain;

ε_{c}^{e l}

is the compressive elastic strain in a loss state;

d_{c}

is the compression damage factor;

σ_{c}

is the compressive stress;

{\bar{σ}}_{c}

is the effective compressive stress.

Similarly, the total tensile strain is divided into inelastic (cracking strain) and elastic strains, as shown in equations (29) and (30)

ε_{t}^{c k} + ε_{t}^{e l} = ε_{t}

(29)

ε_{t}^{e l} = σ_{t} / E_{0}

(30)

where

ε_{t}^{c k}

is the cracking strain; and

ε_{t}^{e l}

is the elastic tensile strain.

The equivalent plastic strain, tensile stress, and effective tensile stress were calculated using equations (31)–(33)

ε_{t}^{p l} = ε_{t}^{i n} - (ε_{t}^{e l} - ε_{0 t}^{e l}) = ε_{t}^{i n} - \frac{d_{t}}{1 - d_{t}} \cdot \frac{σ_{t}}{E_{0}}

(31)

σ_{t} = (1 - d_{t}) E_{0} (ε_{t} - ε_{t}^{p l})

(32)

{\bar{σ}}_{t} = \frac{σ_{t}}{1 - d_{t}}

(33)

where

ε_{t}^{p l}

is the equivalent tensile plastic strain;

ε_{t}^{i n}

is the inelastic tensile strain;

ε_{t}^{e l}

and

ε_{0 t}^{e l}

are the tensile elastic strains in the loss and lossless states, respectively;

d_{t}

is the tensile damage factor;

σ_{t}

is the tensile stress;

{\bar{σ}}_{t}

is the effective tensile stress.

Determination of model parameters

The parameter values of the yield criterion and flow rule in the plastic damage model are determined based on previous research (Curbach and Speck, 2008). The dilation angle parameter reflects the volume change observed in granular materials (e.g., concrete) when it is subjected to shear deformations (Othman and Marzouk, 2018). The dilation angle is in the range of 25–40° for traditional concrete (Genikomsou and Polak, 2015). However, the micro-structure of UHPC is different from other concrete classes, which of properties are improved due to the presence of steel fibers. Based on previous studies done by other researchers and considering the effect of confinement pressure (Curbach and Speck, 2008; Dogu and Menkulasi, 2020), the value of the dilation angle $(ψ)$ was determined as 42°. To control the deviation of the hyperbolic plastic potential from the asymptote, the potential flow eccentricity is determined to be 0.1. The ratio of the biaxial ultimate compressive strength to the uniaxial ultimate compressive strength is 1.16. To determine the failure surface shape of the skew plane, the value of $K_{c}$ is set to 0.667. The value of the viscosity parameter is 0.001. These parameter values have been found to provide satisfactory results in previous trials.

Model establishment

The C3D8R element type was chosen to model the UHPC. This element provides a reliable solution to both linear and complex non-linear analyses because it has eight nodes, each of which has three degrees of freedom. Consequently, it can accurately simulate the UHPC. The longitudinal and stirrups were modeled using the T3D2 element because they only sustain the axial forces applied to the UHPC columns.

An embedded region constraint was adopted to model the interaction between the steel bar and UHPC. The UHPC was selected as the host region, and the internal steel bars served as the embedded region. Then, a reference point couple was set at the top surface of the model. A fixed constraint was set at the bottom of the model to limit the translational and rotational degrees of freedom in three directions. In additional, it was observed that around 10 mm mesh provides accurate results within reasonable computational time through the mesh convergence analyses. Therefore, the mesh size of 10 mm is adopted.

Comparison with test results

The test results in previous studies done by Sugano et al. (2007) (specimen 160FM2-35), Hosinieh et al. (2015) (specimens C3-120 and C4-60), and Shin et al. (2017) (specimens 150-C3.0, 150-C5.1, 180-A3.6, 180-C4.5, and 150-A3.0) are used to verify the correctness of the modeling methods. The details of the models in this section including the reinforcement details, loading scheme, and boundary condition are the same as that of the specimens in these studies.

Load–displacement curve

The P-Δ curves of eight UHPC columns obtained by tests and FEA are shown in Figure 7. As shown in Figure 7, the FEA results were slightly overestimated compared to the test results in the descending segment. The bond performance between the reinforcement and concrete is accurately assumed in the FEA, whereas the bond performance does not reach the ideal state in the tests. However, the simulation curves do not exceed the test curves in specimens 150-C3.0 and 150-A3.0, which may be caused by the uneven distribution of steel fibers in these specimens. Generally, the FEA model can reasonably track the experimental behavior of the UHPC column.

Figure 7.

Comparison between simulation and test results. (a) 160FM2-35, (b) C3-120, (c) C4-60, (d) 150-A3.0, (e) 150-C3.0, (f) 150-C5.1, (g) 180-A3.6, (h) 180-C4.5.

Plastic damage

The proposed constitutive model is applied in the FEA. The plastic damage of two typical UHPC columns is shown in Figure 8. It is indicated that a certain amount of minor damage to the columns initiated after the yielding of the longitudinal reinforcement occurred, and the damage area gradually increased as the displacement continuously increased. Consistent with the test results, the specimens eventually failed in compression.

Figure 8.

Plastic damage. (a) Specimen C3-40 (b) Specimen 180-C4.5

Parametric analysis

Design specimens

To investigate the effect of different parameters on the bearing capacity and ductility of UHPC columns, 25 UHPC column models were established. The main variables examined included stirrup spacing, stirrup configuration, steel fiber volume, and longitudinal reinforcement ratio. All of the columns have the same dimensions (i.e., 200 mm × 200 mm × 800 mm) with a 10-mm-thick concrete cover. The diameter of all the longitudinal reinforcements is 16 mm (D16), and 10-mm diameter (D10) steel bars were used as stirrups with various spacings and configurations. The yield strengths of the longitudinal reinforcements and stirrups were 685 and 700 MPa, respectively. All specimen information is summarized in Table 3, and the stirrup configuration is shown in Figure 9. The UHPC specimen codes indicate the compressive strength, steel fiber volume, stirrup configuration, and stirrup spacing. For example, for the specimen code 150F2-A40², “150” is the UHPC compressive strength; “F” represents steel fibers, and “2” indicates their volume; “A” is the stirrup configuration type; “40” is the stirrup spacing. The stirrup spacing superscript (“2” in this example) indicates the ratio of the longitudinal reinforcement. All the specimens are subjected to pure axial loading at a rate of 0.002 mm/s.

Table 3.

Specimen details.

Specimens	$f_{c}$ _(MPa)	Longitudinal reinforcement			Stirrup					$V_{f}$ (%)
Specimens	$f_{c}$ _(MPa)	Number and diameter (mm)	$ρ_{c}$	$f_{y}$	Diameter (mm)	Arrangement	s (mm)	$ρ_{v}$	$f_{y h}$ _(MPa)	$V_{f}$ (%)
150F2-A40²	150	8D16	1.04	685	10	A	40	2.18	700	2
150F2-B40	150	8D16	1.04	685	10	B	40	3.27	700	2
180F2-B40	180	8D16	1.04	685	10	B	40	3.27	700	2
200F2-B40	200	8D16	1.04	685	10	B	40	3.27	700	2
150F2-C40	150	12D16	1.04	685	10	C	40	4.36	700	2
150F2-D40	150	8D16	1.56	685	10	D	40	4.36	700	2
130F2-B40	130	8D16	1.04	685	10	B	40	3.27	700	2
150F2-C60	150	12D16	1.04	685	10	C	60	2.91	700	2
150F2-C80	150	12D16	1.04	685	10	C	80	2.18	700	2
150F2-A40¹	150	4D16	0.52	685	10	A	40	2.18	700	2
150F1-B40	150	8D16	1.04	685	10	B	40	3.27	700	1
180NF-B40	180	8D16	1.04	685	10	B	40	3.27	700	0
180F1-B40	180	8D16	1.04	685	10	B	40	3.27	700	1
200NF-B40	200	8D16	1.04	685	10	B	40	3.27	700	0
150NF-B40	150	8D16	1.04	685	10	B	40	3.27	700	0
150F2-B60	150	8D16	1.04	685	10	B	60	2.18	700	2
150F2-B80	150	8D16	1.04	685	10	B	80	1.64	700	2
150F2-A40³	150	12D16	1.56	685	10	A	40	2.18	700	2
130NF-B40	130	8D16	1.04	685	10	B	40	3.27	700	0
150F2-D60	150	8D16	1.56	685	10	D	60	3.27	700	2
150F2-D80	150	8D16	1.56	685	10	D	80	2.18	700	2
130F1-B40	130	8D16	1.04	685	10	B	40	3.27	700	1
200F1-B40	200	8D16	1.04	685	10	B	40	3.27	700	1
150F2-A60	150	8D16	1.04	685	10	A	60	1.45	700	2
150F2-A80	150	8D16	1.04	685	10	A	80	1.09	700	2

Note: $f_{c}$ is the compressive strength of UHPC; $ρ_{c}$ is the longitudinal reinforcement ratio; $f_{y}$ is the yield strength of longitudinal reinforcement; $s$ is the stirrup spacing; $ρ_{v}$ is the stirrup ratio; $f_{y h}$ is the yield strength of stirrup; $V_{f}$ is the steel fiber volume.

Figure 9.

Stirrup configuration. (a) Type A, (b) Type B, (c) Type C, (d) Type D

Finite element analysis results

The load versus displacement (P-Δ) curves of all specimens obtained by the FEA are shown in Figure 10. The parameters reflecting the axial behavior of UHPC columns are calculated and summarized in Table 4. The table also lists the peak load, the axial loads borne by the UHPC and confined UHPC, and the corresponding strain and toughness index. The axial load borne by the UHPC is the difference between the load sustained by the longitudinal reinforcement and the peak load of the UHPC column. The definition and calculation method of the toughness index are shown in Figure 11. This index is the ratio of the area surrounded by the concrete load–strain relationship curve up to a strain of 0.02 to the area of an equivalent rigid material $(A_{R} = P_{c} \cdot ε_{A u})$ (Shin et al., 2017).

Figure 10.

P-Δ curves of UHPC column. (a) 130 series, (b) 180 series, (c) 200 series, (d) 150A series, (e) 150B series, (f) 150C/D series.

Table 4.

Simulated results.

Specimens	Axial load						Axial strain			Toughness
Specimens	$P_{yield}$	$P_{\max}$	$P_{c}$	$P_{c c}$	$\frac{P_{c}}{P_{\max}}$	$\frac{P_{c c}}{P_{\max}}$	$ε_{yield}$	$ε_{c}$	$ε_{c c}$	$A_{u}$	$T . I .$
150F2-A40²	6546	8581	7455	6092	0.87	0.71	0.003	0.0043	0.0045	85.4	0.57
150F2-B40	7480	9270	8136	6838	0.88	0.74	0.0034	0.0047	0.0049	118.8	0.73
180F2-B40	8597	10275	9150	8142	0.89	0.79	0.0042	00053	0.0084	125.7	0.69
200F2-B40	9553	10912	9786	8874	0.90	0.81	0.0045	0.0068	0.0081	141.2	0.72
150F2-C40	8063	9452	8327	7220	0.88	0.76	0.0036	0.0047	0.0056	126.3	0.76
150F2-D40	85555	10018	8227	7423	0.82	0.74	0.0038	0.0049	0.0057	134.0	0.81
130F2-B40	7301	8219	7093	6089	0.86	0.74	0.0033	0.0043	0.0054	106.2	0.75
150F2-C60	7375	9124	7998	7373	0.88	0.81	0.0036	0.0048	0.0097	110.8	0.69
150F2-C80	6875	8796	7670	6741	0.87	0.77	0.0034	0.0049	0.0052	98.1	0.64
150F2-A40¹	6119	8298	7735	6426	0.93	0.77	0.003	0.0047	0.005	91.9	0.59
150F1-B40	7210	8606	7481	6962	0.87	0.81	0.0032	0.006	0.0081	102.7	0.69
180NF-B40	8429	9303	8219	7416	0.88	0.80	0.0043	0.006	0.0097	106.5	0.65
180F1-B40	7600	9816	8690	7524	0.89	0.77	0.0035	0.006	0.01	114.3	0.66
200NF-B40	9022	10294	9169	8078	0.89	0.78	0.0046	0.006	0.012	107.9	0.59
150NF-B40	7420	8200	7075	6558	0.86	0.80	0.0037	0.0056	0.011	97.5	0.69
150F2-B60	6531	8435	7309	6775	0.85	0.80	0.0029	0.0049	0.0059	93.2	0.64
150F2-B80	6056	8001	6876	5975	0.87	0.78	0.0028	0.0043	0.005	80.9	0.59
150F2-A40³	7489	9193	7504	6875	0.82	0.75	0.0035	0.0048	0.0051	89.7	0.60
130NF-B40	6411	7343	6218	5745	0.85	0.78	0.0032	0.0043	0.013	90.5	0.73
150F2-D60	7864	9157	7468	6809	0.82	0.74	0.0035	0.0048	0.0055	106.2	0.71
150F2-D80	7273	8914	7225	6712	0.81	0.75	0.0033	0.0047	0.005	89.3	0.62
130F1-B40	6545	7939	6813	6328	0.86	0.80	0.003	0.0043	0.0082	99.2	0.73
200F1-B40	8334	10399	9183	8294	0.89	0.80	0.0036	0.0068	0.0099	119.1	0.65
150F2-A60	5548	7706	6580	5939	0.85	0.77	0.0024	0.0047	0.0053	68.4	0.52
150F2-A80	5922	7770	6645	6009	0.86	0.77	0.0026	0.0048	0.0056	64.7	0.49

Note: $P_{yield}$ and $ε_{yield}$ are the yield load and yield strain of the specimens, respectively; $P_{\max}$ is the peak load of the specimens; $P_{c}$ and $ε_{c}$ are the peak load and peak strain of the UHPC, respectively; $P_{c c}$ and $ε_{c c}$ are the peak load and peak strain of the confined UHPC, respectively; $T . I .$ is the toughness index.

Figure 11.

Definition of toughness parameters.

The maximum axial loads of all the specimens range from 7000 to 11,000 kN. Specimen 200F2-B40 has the highest axial bearing capacity because of its higher concrete strength, smaller stirrup spacing, and moderate steel fiber volume. In contrast, specimen 130NF-B40 has the lowest axial load. The axial compressive behavior of UHPC column is the combination of the contributions of the confined concrete, cover concrete, and longitudinal reinforcement. The average values of $P_{c} / P_{\max}$ and $P_{c c} / P_{\max}$ are 0.86 and 0.77, respectively. It is indicated that the UHPC, especially the confined UHPC, performed a key role in improving the performance of the UHPC columns.

Variable analysis

The parameters influencing the behavior of UHPC columns investigated in the article include the (1) stirrup spacing (s = 40–80 mm), (2) stirrup configuration (A, B, C, and D), (3) steel fiber volume (V_f = 0–2%), and (4) longitudinal reinforcement ratio (ρ_c = 0.52–1.54%). The control variable method is employed to investigate the influence of the parameters on the UHPC column performance.

(1) Influence of stirrup spacing

The influence of stirrup spacing on the load–displacement relationship curves of the UHPC column is shown in Table 4 and Figure 12. With the decrease in the stirrup spacing, the post-ductility of the UHPC columns significantly improves and the peak load also increases to a certain extent. The plastic damage is shown in Figure 13. A local failure resulting from the large spacing of the stirrup and weak protection of concrete could be observed in the middle of the column shown in Figure 13(c). By contrast, specimen 150F2-D40, which has closely spaced stirrup, exhibits better performance. This indicates that reducing the stirrup spacing significantly improves the UHPC column performance.

Figure 12.

Load–displacement responses corresponding to different stirrup spacings.

Figure 13.

Plastic damage of the UHPC columns with different stirrup spacings. (a) Specimen 150F2-D40, (b) Specimen 150F2-D60, (c) Specimen 150F2-D80.

(2) Influence of stirrup configurations

The influence of stirrup configuration on the load–displacement relationship curves of the UHPC column is shown in Table 4 and Figure 14. It is evident that by improving the cross-sectional details, the stirrup configurations change and the peak load correspondingly improves. In addition, the curves of the four specimens show that the strength degradation after the peak load becomes more moderate with the enhancement of the cross-sectional details, indicating that the ductility significantly increases. The plastic damage is shown in Figure 15. It is evident that the damage of these columns considerably decreases with the improvement of the cross-sectional details.

Figure 14.

Load–displacement responses corresponding to different stirrup configurations.

Figure 15.

Plastic damage of the UHPC columns with different stirrup configurations. (a) Specimen 150F2-A40², (b) Specimen 150F2-B40, (c) Specimen 150F2-C40, (d) Specimen 150F2-D40.

(3) Influence of steel fiber volume

The influence of steel fiber volume on the load–displacement relationship curves of the UHPC column is shown in Table 4 and Figure 16. The performances of these columns are improved because of the presence of steel fibers. The peak load gradually increases as the concrete compressive strength increases from 130 to 200 MPa. The plastic damage is shown in Figure 17. It is found that the damage area significantly decreases as the steel fiber volume increases. The improvement in the performance of the UHPC columns is more distinct when the steel fiber volume increases from 1 to 2%.

Figure 16.

Load–displacement responses corresponding to different steel fiber volumes.

Figure 17.

Plastic damage of the UHPC columns with different steel fiber volumes. (a) Specimen 180NF-B40, (b) Specimen 180F1-B40, (c) Specimen 180F2-B40.

(4) Influence of longitudinal reinforcement ratio

The influence of longitudinal reinforcement ratio on the load–displacement relationship curves of the UHPC column is shown in Table 4 and Figure 18. Increasing the longitudinal reinforcement ratio from 0.52 to 1.04 and 1.57% also increases the peak load by 3.4 and 10.8%, respectively. Although the toughness improved, the effect of longitudinal reinforcement ratio on ductility was not significant. The plastic damage is shown in Figure 19. Because the longitudinal reinforcement ratio decreases, the local failure of UHPC columns occurs and the embedded reinforcement in specimen 150F2-A40² buckled. Therefore, the minimum reinforcement ratio should be controlled during the design of UHPC columns to prevent the premature buckling of the longitudinal reinforcement.

Figure 18.

Load–displacement responses corresponding to different longitudinal reinforcement ratios.

Figure 19.

Plastic damage of the UHPC columns with different longitudinal reinforcement ratios. (a) Specimen 150F2-A40², (b) Specimen 150F2-A40¹, (c) Specimen 150F2-A40³.

Conclusion

A confined UHPC constitutive model was proposed based on the collated test data and statistical analysis. This model was further applied to FEA. It was indicated that the model accurately predicted the behavior of UHPC columns as evinced by the comparison with test results, as well as the plastic damage analysis and load–displacement response. Accordingly, this provides a reference for the subsequent FEA of UHPC columns. In addition, the influences of stirrup spacing, stirrup configuration, steel fiber volume, and longitudinal reinforcement ratio on the axial compressive behavior of UHPC columns were analyzed by FEA. The following conclusions were drawn.

(1). A confined UHPC constitutive with the consideration of the confinement provided by steel fiber and stirrup was developed. This model was then used in FEA, in which the plastic damage to concrete, material properties of steel bars, interaction between concrete and steel bars, and boundary conditions of the UHPC columns were considered. By comparing with the tested results, it was demonstrated that the FEA model reasonably predicted the axial compressive behavior of UHPC columns and the proposed constitutive model can be implemented in the FEA software.

(2). The response of UHPC columns subjected to axial loading can be recognized as the combination of the contributions of the confined concrete, cover concrete, and longitudinal reinforcement. The UHPC, especially the confined UHPC, performed a key role in improving the performance of the UHPC columns. The stirrup spacing, stirrup configuration, and steel fiber volume were important factors affecting the ductility of UHPC columns.

(3). The ductility of the UHPC columns was significantly improved by reducing the stirrup spacing and improving the cross-sectional details. The improvement in ductility was more considerable when the steel fiber volume was 1–2% than when it was 0–1%. Additionally, although the effect of the longitudinal reinforcement ratio on ductility was not significant, it was necessary to control the minimum reinforcement ratio in the design of 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 Foundations of China No. 51908041 and the Fundamental Research Funds for the Central Universities No. 31088171009.

ORCID iD

Kaize Ma

Guohua Xing

References

Ahmed

Hadi Muhammad

(2018) Influence of steel fibers on the behavior of RPC circular columns under different loading conditions. Structures 14: 111–123.

Aoude

(2008) Structural behavior of steel fiber reinforced concrete members. PhD Thesis, McGill University, Montreal, Canada.

Chen

Zhang

Jia

, et al. (2018) Structural behavior of UHPC filled steel tube columns under axial loading. Thin-Walled Structures 130(5): 550–563.

Curbach

Speck

(2008) Proceedings of Second International Symposium on Ultra-high-performance Concrete. Kassel, Germany: University of Kassel.

Cusson

Paultre

(1995) Stress-strain model for confined high-strength concrete. Journal of Surgical Education 121(3): 468–477.

Cusson

Paultre

(1994) High-strength concrete columns confined by rectangular ties. Journal of Surgical Education 120(3): 783–804.

Dogu

Menkulasi

(2020) A flexural design methodology for UHPC beams posttensioned with unbonded tendons. Engineering Structures 207(6): 110193.

Empelmann

Teutsch

Steven

(2008) Expanding the application range of RC-columns by the use of UHPC. In: Walraven JC and Stoelhorst D (eds) Tailor Made Concrete Structures. London: CRC Press, pp.461–468.

Genikomsou

Polak

(2015) Finite element analysis of punching shear of concrete slabs using damaged plasticity model in ABAQUS. Engineering Structures 98: 38–48.

10.

Hosinieh

Aoude

Cook

, et al. (2015) Behavior of ultra-high-performance fiber reinforced concrete columns under pure axial loading. Engineering Structures 15: 388–401.

11.

Lee

Fenves

(1998) Plastic-damage model for cyclic loading of concrete structures. Journal Engineering Mechanics 124(8): 892–900.

12.

Légeron

Paultre

(2003) Uniaxial confinement model for normal and high strength concrete columns. Journal of Surgical Education 129(2): 241–252.

13.

Lubliner

Oliver

Oller

, et al. (1988) A plastic-damage model for concrete. International Journal Solids and Structures 25(3): 299–326.

14.

Mander

Priestley

MJN

Park

(1988) Theoretical stress-strain model for confined concrete. Journal of Surgical Education 114(8): 1804–1826.

15.

Marchand

Baby

Khadour

(2019) Response of UHPFRC columns submitted to combined axial and alternate flexural loads. Journal of Surgical Education 145(1): 04018225.

16.

Othman

Marzouk

(2018) Applicability of damage plasticity constitutive model for ultra-high performance fibre-reinforced concrete under impact loads. International Journal of Impact Engineering 114: 20–31.

17.

Paultre

Eid

Langlois

(2010) Behavior of steel fiber-reinforced high-strength concrete columns under uniaxial compression. Journal of Surgical Education 136(10): 1225–1235.

18.

Saatcioglu

Razvi

(1992) Strength and ductility of confined concrete. Journal of Surgical Education 118(6): 1590–1607.

19.

Sharma

Bhargava

Kaushik

(2005) Behavior of confined high strength concrete columns under axial compression. Journal of Advanced Concrete Technology 3(2): 267–281.

20.

Shin

Yoon

(2015) Effect of steel fibers on the performance of ultra-high-strength concrete columns. Journal Materials in Civil Engineering 27(4): 04014142.

21.

Shin

Min

Mitchell

(2017) Confinement of ultra-high-performance fiber reinforced concrete columns. Composite Structures 176: 124–142.

22.

Shin

Min

Mitchell

(2018) Uniaxial behavior of circular ultra-high-performance fiber-reinforced concrete columns confined by spiral reinforcement. Construction and Building Materials 168: 379–393.

23.

Shin

Yoon

Cook

(2016) Axial load response of ultra-high-strength concrete columns and high-strength reinforcement. ACI Structure Journal 113(2): 325–336.

24.

Shunsuke

Hideki

Kazuyoshi

(2007) Study of new RC structures using ultra-high-strength fiber-reinforced concrete (UFC)—The challenge of applying 200 MPa UFC to earthquake resistant building structure. Journal Advanced Concrete Technology 5(2): 133–147.

25.

Wille

El-Tawil

Naaman

(2014) Properties of strain hardening ultra-high-performance fiber reinforced concrete (UHP-FRC) under direct tensile loading. Cement Concrete Composites 48(2): 53–66.

26.

Liu

(2017) Experimental investigation of seismic behavior of ultra-high-performance steel fiber reinforced concrete columns. Engineering Structures 152: 129–148.

27.

Liu

(2018) Numerical study of ultra-high-performance steel fiber-reinforced concrete columns under monotonic push loading. Advances in Structures Engineering 21(8): 1234–1248.