# Stress–Strain Model of High-Strength Concrete Confined by Lateral Ties under Axial Compression

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Existing Models

Proposed by | Complete Curve Equation | Model Parameters |
---|---|---|

Cussion et al. [9] (1995) | $\left(1\right)\text{}0\le \epsilon \le {\epsilon}_{\mathit{cc}}$ ${\sigma =f}_{\mathit{cc}}\frac{k\left({\epsilon /\epsilon}_{\mathit{cc}}\right)}{k-1+{({\epsilon /\epsilon}_{\mathit{cc}})}^{k}}$ $\left(2\right)\text{}{\epsilon}_{\mathit{cc}}\le \epsilon $ ${\sigma =f}_{\mathit{cc}}\mathit{exp}{[{k}_{1}({\epsilon /\epsilon}_{\mathit{cc}})]}^{{k}_{2}}$ | ${f}_{l}{=f}_{\mathit{yv}}({\displaystyle \sum _{i=1}^{o}{A}_{\mathit{sbi}}}+{\displaystyle \sum _{i=1}^{p}{A}_{\mathit{sbi}}})/[s({b}_{c}{+h}_{c})]$$;\text{}{k=E}_{c}/({E}_{c}-{f}_{\mathit{cc}}{/\epsilon}_{\mathit{cc}})$ ${k}_{e,\mathit{mander}}=[1-{\displaystyle \sum _{i=1}^{n}{\omega}_{i}^{2}/(6}{b}_{c}{h}_{c})][1-({s}^{\prime}/2{b}_{c})][1-({s}^{\prime}/2{h}_{c})]/(1{-\rho}_{\mathit{cc}})$ ${f}_{\mathit{le}}{=k}_{e,\mathit{mander}}{f}_{l}$$;\text{}{f}_{\mathit{cc}}{/f}_{c}=1+2.1{({f}_{\mathit{le}}{/f}_{c})}^{0.7}$$;\text{}{\epsilon}_{\mathit{cc}}/{\epsilon}_{c}=1+0.21{({f}_{\mathit{le}}{/f}_{c})}^{1.7}$ $\mathrm{If}\text{}\mathrm{no}\text{}\mathrm{test}\text{}\mathrm{data}\text{}\mathrm{is}\text{}\mathrm{available},\text{}{\epsilon}_{c50}$$\mathrm{takes}\text{}0.004;\text{}{\epsilon}_{\mathit{cc}50}={\epsilon}_{c50}+0.15{({f}_{\mathit{le}}{/f}_{c})}^{1.1}$ ${k}_{1}=\mathit{ln}(0.5)/{({\epsilon}_{\mathit{cc}50}{-\epsilon}_{\mathit{cc}})}^{{k}_{2}}$$;\text{}{k}_{2}=0.58+16{({f}_{\mathit{le}}{/f}_{c})}^{1.4}$ |

Razvi et al. [20] (1999) | $\left(1\right)\text{}0\le \epsilon \le {\epsilon}_{\mathit{cc}}$ Same as the ascending portion of the Cussion model. $\left(2\right)\text{}{\epsilon}_{\mathit{cc}}\le \epsilon \le {\epsilon}_{\mathit{cc}20}$ ${\sigma =f}_{\mathit{cc}}[1-0.15({\epsilon /\epsilon}_{\mathit{cc}})/({\epsilon}_{c85}{-\epsilon}_{\mathit{cc}})]$ $\left(3\right)\text{}{\epsilon}_{\mathit{cc}20}\le \epsilon $$\sigma =0.2{f}_{\mathit{cc}}$ | ${m}_{1}=6.7{({f}_{\mathit{le}})}^{0.17}$$;\text{}{m}_{2}=0.15\sqrt{({b}_{c}/s)({h}_{c}{/s}_{l})}$$;\text{}{m}_{3}=40{/f}_{c}\le 1.0$ ${m}_{4}{=f}_{\mathit{ys}}/500\ge 1.0$$;\text{}{f}_{\mathit{cc}}{=f}_{c}+0.5{m}_{1}{\rho}_{\mathit{sv}}{f}_{\mathit{ys}}$$;\text{}{\epsilon}_{c}=0.0028-0.008{m}_{3}$ ${\epsilon}_{c85}{=\epsilon}_{c}+0.0018{m}_{3}^{2}$$;\text{}{\epsilon}_{\mathit{cc}}{/\epsilon}_{c}=1+5{m}_{1}{m}_{3}{f}_{\mathit{le}}{/f}_{c}$ ${\epsilon}_{\mathit{cc}85}=260{m}_{3}{\rho}_{c}{\epsilon}_{\mathit{cc}}[1+0.5{m}_{2}({m}_{4}-1)]{+\epsilon}_{c85}$ ${E}_{c}=3320\sqrt{{f}_{c}}+6900$ ${f}_{\mathit{ys}}$$,{\rho}_{c}$ Calculation formulae are shown in Table 2 |

Li et al. [15] (2001) | $\left(1\right)\text{}0\le \epsilon \le {\epsilon}_{c}$ ${\sigma =E}_{c}\epsilon +\frac{\left({f}_{c}{-E}_{c}{\epsilon}_{c}\right)\epsilon}{{\epsilon}_{c}^{2}}$ $\left(2\right)\text{}{\epsilon}_{c}\le \epsilon \le {\epsilon}_{\mathit{cc}}$ ${\sigma =f}_{\mathit{cc}}-\frac{({f}_{\mathit{cc}}{-f}_{c}\left)\right(\epsilon -{\epsilon}_{\mathit{cc}}{)}^{2}}{{({\epsilon}_{\mathit{cc}}-{\epsilon}_{c})}^{2}}$ $\left(3\right)\text{}{\epsilon}_{\mathit{cc}}\le \epsilon $ ${\sigma =f}_{\mathit{cc}}-\beta \frac{{f}_{\mathit{cc}}\left({\epsilon -\epsilon}_{\mathit{cc}}\right)}{{\epsilon}_{\mathit{cc}}}\ge 0.4{f}_{\mathit{cc}}$ | ${f}_{\mathit{le}}$$,{k}_{e,\mathit{mander}}$$\mathrm{are}\text{}\mathrm{the}\text{}\mathrm{same}\text{}\mathrm{as}\text{}\mathrm{the}\text{}\mathrm{Cussion}\text{}\mathrm{model};\text{}{E}_{c}=3320\sqrt{{f}_{c}}+6900$ ${f}_{\mathit{cc}}{=f}_{c}(-0.413+1.413\sqrt{1+11.4{f}_{\mathit{le}}{/f}_{c}}-2{f}_{\mathit{le}}{/f}_{c})$ (For high-strength concrete) ${\epsilon}_{\mathit{cc}}/{\epsilon}_{c}=1+11.3{f}_{\mathit{le}}{/f}_{c}{)}^{0.7}$$({f}_{\mathit{yv}}\le 550$$\mathrm{MPa});\text{}{\epsilon}_{c}=0.0007{({f}_{c})}^{0.3}$ $\beta =(0.048{f}_{c}-2.14)-(0.098{f}_{c}-4.57\left)\right({f}_{\mathit{le}}{/f}_{c}{)}^{1/3}$$\text{}({f}_{\mathit{yv}}\le 550$$\mathrm{MPa}\text{}\mathrm{and}\text{}{f}_{c}75$ MPa) |

Légeron et al. [21] (2003) | $\left(1\right)\text{}0\le \epsilon \le {\epsilon}_{c}$ Similar to the ascending portion of the Cussion model. $\left(2\right)\text{}{\epsilon}_{\mathit{cc}}\le \epsilon $ Same as the descending portion of the Cussion model. | ${\rho}_{\mathit{sey}}$, $\kappa $ Calculation formulae are shown in Table 2 ${k}_{e,\mathit{mander}}$$\text{}\mathrm{is}\text{}\mathrm{similar}\text{}\mathrm{to}\text{}\mathrm{the}\text{}\mathrm{Cussion}\text{}\mathrm{model};\text{}{\epsilon}_{c}=0.0005{({f}_{c})}^{0.4}$ ${\text{}I}_{e}^{\prime}{=f}_{\mathit{le}}{/f}_{c}$$;\text{}{I}_{e50}{=\rho}_{\mathit{sey}}{f}_{\mathit{ys}}{/f}_{c}$$;\text{}{f}_{\mathit{cc}}{/f}_{c}=1+2.4{({\text{}I}_{e}^{\prime})}^{0.7}$ ${\epsilon}_{\mathit{cc}}{/\epsilon}_{c}=1+35({{I}^{\prime}}_{e}{)}^{1.2}$$;\text{}{\epsilon}_{\mathit{cc}50}{/\epsilon}_{c50}=1+60{I}_{e50}$ ${k}_{1}$$\text{}\mathrm{is}\text{}\mathrm{the}\text{}\mathrm{same}\text{}\mathrm{as}\text{}\mathrm{the}\text{}\mathrm{Cussion}\text{}\mathrm{model};{k}_{2}=1+25({I}_{e50}{)}^{2}$ |

Proposed by | Empirical Formulae | Parameters |
---|---|---|

Razvi et al. [20] (1999) | ${f}_{y}{=E}_{s}(0.0025+0.04\sqrt[3]{{k}_{e}{\rho}_{c}{/f}_{c}})\le {f}_{y}$ | ${\rho}_{c}=\frac{{\displaystyle \sum _{i=1}^{n}{({A}_{\mathit{sb}})}_{i}}+{\displaystyle \sum _{i=1}^{n}{({A}_{\mathit{sh}})}_{i}}}{s({b}_{c}{+h}_{c})}$ |

Légeron et al. [21] (2003) | ${f}_{\mathit{ys}}=0.25{f}_{c}/[{\rho}_{\mathit{sey}}(\kappa -10)]\ge 0.43{\epsilon}_{c}{E}_{s}{f}_{\mathit{ys}}{,\text{}f}_{\mathit{ys}}\le {f}_{\mathit{yv}}$ | ${\rho}_{\mathit{sey}}{=k}_{e}{A}_{\mathit{sh}}/({\mathit{sh}}_{c})$, ${\kappa =f}_{c}/({\rho}_{\mathit{sey}}{E}_{s}{\epsilon}_{c})$ |

## 3. Details of the Proposed Model

#### 3.1. Normal Triaxial Compression Model

#### 3.2. Effective Confinement Coefficient

- (1)
- unconfined zone: the concrete is outside the ties without any confined effect in this region;
- (2)
- weakly confined zone: the concrete is near the inner side of the ties reinforcement with a weak triaxial compressive stress state in this region;
- (3)
- heavily confined zone: the remaining part of the concrete is in the core region with a strong triaxial compressive stress state in this region.

#### 3.3. Ties Strain at the Peak Stress in Confined Concrete

#### 3.4. Stress–Strain Model

- (1)
- $y\left(0\right)=0,\text{}y\left(1\right)=1$;
- (2)
- ${y}^{\prime}\left(0\right)=A,\text{}{y}^{\prime}\left(1\right)=0$;
- (3)
- When $x\to +\infty $, $y\to (B-1)/B$;
- (4)
- The ascending portion is a convex function, and the descending portion has an inflection point.

- (1)
- The peak stress and peak strain of unconfined concrete under compression are obtained with experiments or empirical formulae;
- (2)
- Equation (13) is used to calculate the ties strain when confined concrete experiences the peak stress;
- (3)
- Equation (5) is used to calculate the effective confinement coefficient based on the arrangement form of the ties reinforcement;
- (4)
- The equivalent uniform lateral confined stress is calculated; and Equations (1), (2), (14), (16), (18), and (19) are used to obtain the complete curve equation model parameters A and B.

## 4. Model Evaluation

## 5. Conclusions

- (1)
- The existing models of ties-confined concrete stress–strain were compared; the differences between different empirical models were evident, particularly because the dispersion of the descending portion was large.
- (2)
- The effective confinement coefficient and empirical formula for the ties strain when confined concrete experienced the peak stress were established. The stress–strain model was proposed using a continuous derivable function, which has fewer model parameters and facilitates numerical calculations.
- (3)
- The proposed model is in good agreement with the test curve, and the predicted peak stress is slightly lower than the test results. The relative error is within 10%, which accounts for 92% of the test data; overall, the prediction accuracy of the proposed model for the stress–strain relationship for the specimens with fewer parameters and simpler functional form is generally comparable to other models.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notations

${b}_{c}$, ${h}_{c}$ | core dimension measured center-to-center of perimeter ties |

$s$ | ties spacing |

${s}^{\prime}$ | clear spacing between ties |

${A}_{\mathit{sbi}}$, ${A}_{\mathit{shi}}$ | area of one leg of transverse reinforcement in b- and h-directions, respectively |

${E}_{c}$ | modulus of elasticity of plain concrete |

${f}_{c}$ | compressive strength of unconfined concrete |

${\epsilon}_{c}$ | strain at maximum stress ${f}_{c}$ of unconfined concrete |

${\epsilon}_{c85}$ | strain corresponding to 85% of peak stress of unconfined concrete on descending branch |

${\epsilon}_{c50}$ | strain corresponding to 50% of peak stress of unconfined concrete on descending branch |

${f}_{\mathit{cc}}$ | compressive strength of confined concrete |

${\epsilon}_{\mathit{cc}}$ | strain at maximum stress ${f}_{\mathit{cc}}$ of confined concrete |

${\epsilon}_{\mathit{cc}85}$ | strain corresponding to 85% of peak stress of confined concrete on descending branch |

${\epsilon}_{\mathit{cc}50}$ | strain corresponding to 50% of peak stress of confined concrete on descending branch |

${\epsilon}_{\mathit{cc}20}$ | strain corresponding to 20% of peak stress of confined concrete on descending branch |

${\omega}_{i}$ | ith clear ties spacing between adjacent longitudinal bars |

$o$, $p$ | number of ties legs in b- and h-directions, respectively |

$n$ | total number of longitudinal bars |

${\rho}_{\mathit{cc}}$ | ratio of area of longitudinal steel to area of core of section |

${s}_{l}$ | spacing of longitudinal reinforcement, laterally supported by corner of ties or ties of crosstie |

${f}_{\mathit{yv}}$ | yield strength of ties reinforcement |

${f}_{\mathit{ys}}$ | tensile stress in transverse reinforcement at peak concrete stress |

${f}_{l}$ | average confinement pressure |

${f}_{le}$ | equivalent uniform lateral pressure that produces the same effect as nonuniform pressure |

${k}_{e,\mathit{mander}}$ | effective confinement coefficient proposed by Mander et al. [14] |

${E}_{s}$ | modulus of elasticity of ties |

${A}_{\mathit{ce}}$ | area of the weakest confining plane in the adjacent ties plane |

${A}_{\mathit{ce}0}$ | ties plane confining area |

${{f}^{\prime}}_{c}$ | cylindrical compressive strength |

B, H | cross-section width and height, respectively |

$d$ | diameter of the longitudinal bar |

${d}_{s}$ | diameter of the ties |

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**Figure 5.**Passive confined stress that develops in the sections: (

**a**) Rectangular or square cross-section; (

**b**) Circular cross-section.

**Figure 16.**Comparison of experimental curves and analytical models (Nagashima,1992): (

**a**) HH08LA and HH10LA; (

**b**) HH13LA and HH15LA; (

**c**) HH20LA and HL06LA; (

**d**) HL08LA.

**Figure 17.**Comparison of experimental curves and analytical models (Nishiyama,1993): (

**a**) 3, 4, and 7; (

**b**) 8 and 10; (

**c**) 11 and 12; (

**d**) 13 and 14.

**Figure 18.**Comparison of experimental curves and analytical models (Razvi,1999): (

**a**) CS-3 and CS-5; (

**b**) CS-4 and CS-8; (

**c**) CS-15 and CS-16; (

**d**) CS-20.

**Figure 20.**Frequency statistics of the relative error between model calculated ${f}_{\mathit{cc}}$ and tested ${f}_{\mathit{cc}}$.

Source | ID | Cross-Section | Longitudinal Bars | Ties | Concrete | ${{\mathit{f}}^{\prime}}_{\mathit{cc}}$ (MPa) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{B}$ (mm) | $\mathit{H}$ (mm) | $\mathit{n}$ | $\mathit{d}$ (mm) | ${\mathit{d}}_{\mathit{s}}$ (mm) | $\mathit{s}$ (mm) | ${\mathit{f}}_{\mathit{yv}}$ (MPa) | ${{\mathit{f}}^{\prime}}_{\mathit{c}}$(MPa) | ${\mathsf{\epsilon}}_{\mathit{c}}$$\left(\mathsf{\mu}\mathsf{\epsilon}\right)$ | ${\mathit{E}}_{\mathit{c}}$ (MPa) | |||

Nagashima et al. [7] (1992) | HH08LA | 225 | 225 | 12 | 10 | 5.1 | 55 | 1387 | 98.8 | 2459.1 | 47154.1 | 122.8 |

HH10LA | 225 | 225 | 12 | 10 | 5.1 | 45 | 1387 | 98.8 | 2459.1 | 47154.1 | 122.5 | |

HH13LA | 225 | 225 | 12 | 10 | 5.1 | 35 | 1387 | 98.8 | 2459.1 | 47154.1 | 131.5 | |

HH15LA | 225 | 225 | 12 | 10 | 6.4 | 45 | 1368 | 98.8 | 2459.1 | 47154.1 | 127.0 | |

HH20LA | 225 | 225 | 12 | 10 | 6.4 | 35 | 1368 | 100.4 | 2469.0 | 47407.2 | 148.2 | |

HL06LA | 225 | 225 | 12 | 10 | 5 | 45 | 807 | 100.4 | 2469.0 | 47407.2 | 118.2 | |

HL08LA | 225 | 225 | 12 | 10 | 5 | 35 | 807 | 100.4 | 2469.0 | 47407.2 | 133.2 | |

3 | 250 | 250 | 12 | 16 | 6 | 31 | 813 | 92.4 | 2418.3 | 46113.1 | 145.0 | |

4 | 250 | 250 | 12 | 16 | 6 | 45 | 813 | 92.4 | 2418.3 | 46113.1 | 122.0 | |

Nishiyama et al. [27] (1993) | 7 | 250 | 250 | 12 | 16 | 6 | 60 | 813 | 92.4 | 2418.3 | 46113.1 | 120.0 |

8 | 250 | 250 | 12 | 16 | 4 | 31 | 840 | 92.4 | 2418.3 | 46113.1 | 120.0 | |

10 | 250 | 250 | 12 | 16 | 6 | 31 | 462 | 96.2 | 2442.8 | 46736.8 | 133.0 | |

11 | 250 | 250 | 12 | 16 | 6 | 45 | 462 | 96.2 | 2442.8 | 46736.8 | 117.0 | |

12 | 250 | 250 | 12 | 16 | 6 | 60 | 462 | 96.2 | 2442.8 | 46736.8 | 115.0 | |

13 | 250 | 250 | 12 | 16 | 6 | 60 | 462 | 96.2 | 2442.8 | 46736.8 | 115.0 | |

Razvi et al. [11] (1999) | 14 | 250 | 250 | 12 | 16 | 4 | 31 | 481 | 96.2 | 2442.8 | 46736.8 | 115.0 |

CS-3 | 250 | 250 | 12 | 16 | 6.5 | 55 | 570 | 105.4 | 2499.2 | 48181.5 | 129.1 | |

CS-4 | 250 | 250 | 8 | 16 | 7.5 | 55 | 1000 | 105.4 | 2499.2 | 48181.5 | 123.4 | |

CS-5 | 250 | 250 | 12 | 16 | 7.5 | 120 | 1000 | 105.4 | 2499.2 | 48181.5 | 122.5 | |

CS-7 | 250 | 250 | 12 | 16 | 6.5 | 120 | 400 | 105.4 | 2499.2 | 48181.5 | 115.0 | |

CS-8 | 250 | 250 | 8 | 16 | 11.3 | 85 | 400 | 105.4 | 2499.2 | 48181.5 | 117.8 | |

CS-15 | 250 | 250 | 8 | 16 | 7.5 | 55 | 1000 | 68.9 | 2247.2 | 41815.8 | 95.5 | |

CS-16 | 250 | 250 | 12 | 16 | 7.5 | 85 | 1000 | 68.9 | 2247.2 | 41815.8 | 95.2 | |

CS-20 | 250 | 250 | 12 | 16 | 11.3 | 85 | 400 | 78.2 | 2319.5 | 43618.3 | 106.3 |

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## Share and Cite

**MDPI and ACS Style**

Wang, L.; Huang, X.; Xu, F. Stress–Strain Model of High-Strength Concrete Confined by Lateral Ties under Axial Compression. *Buildings* **2023**, *13*, 870.
https://doi.org/10.3390/buildings13040870

**AMA Style**

Wang L, Huang X, Xu F. Stress–Strain Model of High-Strength Concrete Confined by Lateral Ties under Axial Compression. *Buildings*. 2023; 13(4):870.
https://doi.org/10.3390/buildings13040870

**Chicago/Turabian Style**

Wang, Lei, Xiaokun Huang, and Fuquan Xu. 2023. "Stress–Strain Model of High-Strength Concrete Confined by Lateral Ties under Axial Compression" *Buildings* 13, no. 4: 870.
https://doi.org/10.3390/buildings13040870