The Restraint Mechanism of High-Strength Rectangular Spiral Stirrup Confined Fiber Reinforced Concrete

Abstract

To reveal the confinement mechanism of high-strength rectangular spiral stirrups (HRSSs) on fiber-reinforced concrete, this study designed and conducted axial compression tests on 20 HRSS-confined fiber-reinforced concrete columns. The effects of stirrup spacing, stirrup strength, and concrete strength on the strength and ductility of the columns were analyzed. The experimental results demonstrate that HRSS can significantly improve the performance of fiber-reinforced concrete. The peak strength of concrete exhibited a maximum increase of 2.033 times, and the ductility ratio achieved a maximum increase of 2.588 times. Furthermore, the application of densely spaced high-strength spiral hoops to confine the core concrete not only effectively enhances its compressive strength but also markedly improves its deformability. Based on the stress distribution across the cross-section of HRSS-confined fiber-reinforced concrete columns, this paper proposes a method for delineating the effective confinement area, establishes calculation models for effective lateral confinement stress, effective confinement coefficient, peak stress, and peak strain, and develops a stress-strain constitutive relationship suitable for HRSS-confined fiber-reinforced concrete columns.

1. Introduction

China is situated at the intersection of the Circum-Pacific Seismic Belt and the Eurasian Seismic Belt, characterized by frequent tectonic activities. From 2000 to 2017, 46 earthquakes of magnitude 6 or above occurred in mainland China [1]. According to data from the China Earthquake Networks Center, direct economic losses caused by earthquakes in China exceeded CNY 120 billion in 2023, with building collapses accounting for over 75% of casualties. Although traditional reinforced concrete (RC) structures exhibit high load-bearing capacity, their brittle failure modes—such as shear failure and crushing of joint cores—often lead to catastrophic consequences under strong seismic actions. The 2023 Turkey earthquakes further highlighted this vulnerability, where insufficient hoop confinement caused plastic hinge failure at column ends in numerous RC structures [2,3], underscoring the urgent need to enhance the strength and ductility of confined concrete in engineering practice.

To address these challenges and overcome the performance limitations of conventional confined concrete, the international research community has proposed two primary technical pathways: (1) High-strength confinement systems: High-strength hoops (yield strength > 600 MPa) improve lateral confinement capacity. Spiral hoop confinement replaces conventional hoops to optimize confinement effectiveness. Studies show spiral hoops generate significantly higher lateral confinement stress than standard hoops at equivalent reinforcement ratios due to their continuous configuration [4]. (2) Fiber-reinforced concrete: Steel or synthetic fibers enhance concrete’s ultimate tensile/compressive strains through crack-bridging effects [5,6], mitigating brittle failure tendencies. The synergistic application of high-strength confinement and fiber reinforcement establishes a multi-level seismic defense system: external hoop confinement coupled with internal fiber toughening. High-strength rectangular spiral stirrup (HRSS)-confined fiber-reinforced concrete—a novel composite material combining high-strength materials, spiral hoop confinement, and fiber reinforcement—has demonstrated substantial research value and application potential in modern complex structural design contexts.

High-strength stirrups can effectively confine concrete, a mechanism critical for enhancing the load-bearing capacity and deformation performance of concrete columns [7]. Internationally, research on confined concrete dates back to the 1920s. Researchers worldwide have conducted extensive axial compression tests on stirrup-confined concrete specimens to investigate the strength and deformation properties of confined concrete. Scott and Park (1982) [8] proposed a constitutive model for confined concrete based on experimental studies and synthesis of prior research. This model postulated that stirrups exert no confining effect on concrete prior to reaching its compressive strength. However, due to the assumption that rectangular stirrups provide limited confining capacity, the model exclusively addressed circular stirrup-confined concrete and omitted considerations for rectangular stirrup confinement mechanisms. Sheikh et al. (1982) [9,10] conducted axial compression tests on 27 circular columns confined with spiral hoops and circular hoops, pioneering the concept of “effective confined concrete area”. Based on these findings, they developed a stress-strain constitutive model for confined concrete, which systematically quantified the relationship between confinement effectiveness and mechanical performance. The introduction of the “effective confined concrete area” marked a paradigm shift in confinement theory. This groundbreaking concept provided a critical analytical framework for subsequent researchers. Mander et al. (1982) [11] conducted experimental investigations on axially compressed members to systematically study the stress-strain behavior of confined concrete. Through these tests, they developed analytical formulations for calculating the effective confinement coefficient corresponding to various confinement configurations (e.g., spiral/rectangular hoops). This framework enabled the determination of the effective confined area of concrete, ultimately establishing a generalized stress-strain constitutive model for confined concrete under compression. The proposed model quantitatively characterized how confinement mechanisms enhance both the peak stress and ultimate strain of concrete, providing a theoretical basis for modern seismic design codes. Cusson and Paultre (1995) [12] conducted axial compression tests on 51 high-strength concrete columns subjected to confinement. Based on the effective confinement index, they classified the confinement zones into low-, medium-, and high-confinement regions and unconfined areas. This hierarchical confinement classification introduced a pioneering framework for subsequent theoretical advancements in confined concrete mechanics. Aoyama (2006) [13] conducted axial compression tests on circular and square columns confined with diverse hoop configurations. Based on experimental observations of confinement effectiveness and failure modes, he developed a constitutive model for confined concrete that explicitly accounted for the synergistic effects of hoop geometry and confinement stiffness. In China, Yang et al. [14] systematically summarized the axial compression, eccentric compression, and seismic performance of high-strength stirrup-confined concrete, identifying material strength enhancement and quantification of confinement efficiency as critical directions for future development. Deng [15] focused on the compressive behavior of stirrup-confined ultra-high-performance concrete (UHPC) columns, analyzing the synergistic effects between steel fibers and stirrups. Yan [16] and Sun [17] conducted in-depth studies on the seismic performance of high-strength stirrup-confined concrete columns. Their research demonstrated that such columns exhibit superior seismic resistance, including effective deformation resistance and energy absorption under earthquake loading. Shen et al. [18] established a calculation formula for the peak stress of confined concrete based on the confinement effect coefficient and the Guo-Wang [19] five-parameter failure criterion, further exploring the mechanical behavior of high-strength stirrup-confined concrete under axial compression. Zheng et al. [20] revealed the effects of concrete strength, stirrup strength, and stirrup spacing on the confinement efficiency of circular spiral stirrups through axial compression tests.

However, current research on high-strength stirrup-confined concrete still exhibits several limitations. Existing studies [21,22,23,24] have yet to precisely define the boundary of the confined region or the internal stress distribution patterns. Research on the effective confinement coefficient also lacks sufficient accuracy. Furthermore, studies on how high-strength materials influence the constitutive relationships of confined concrete remain sparse, with a limited exploration into how material properties alter these relationships. In summary, to advance the theoretical framework of confined concrete and promote the practical application of high-strength stirrup-confined fiber-reinforced concrete in engineering, further detailed research on its mechanical properties and constitutive models is imperative.

This study will perform axial compression tests on 20 groups of HRSS-confined fiber-reinforced concrete columns. By integrating existing theoretical frameworks for confined concrete and experimental data, the research aims to investigate the confinement mechanisms of HRSS-confined fiber-reinforced concrete; propose a calculation formula for lateral confining stress in rectangular spiral stirrups; establish an effective confinement zone delineation method and peak parameter calculation method for HRSS-confined fiber-reinforced concrete; and develop a uniaxial compressive constitutive model for HRSS-confined fiber-reinforced concrete. The specific workflow is illustrated in Figure 1.

2. Test Overview

2.1. Specimen Design

To investigate the confinement mechanism of HRSS-confined fiber-reinforced concrete in depth, this study designed 20 HRSS-confined fiber-reinforced concrete columns for axial compression tests. The experiments aimed to analyze the influence of factors including stirrup spacing, stirrup strength, concrete strength, and stirrup diameter on the mechanical performance of HRSS-confined fiber-reinforced concrete. The concrete cube compressive strength grades ranged from CL80 to CL110, and the yield strengths of the stirrups were 511 MPa, 612 MPa, and 691 MPa. The configuration of the stirrups is illustrated in Figure 2. Detailed design parameters are provided in

Table 1. Design of specimen parameters.

Specimen Number	L × W × H/ mm × mm × mm	Stirrup				ρv/%	Concrete Grade
		Type	Strength/MPa	S/mm	D/mm
L-1	150 × 150 × 450	A	611.5	40	5	1.1	CL100
L-2	170 × 170 × 510	A	611.5	35	5	1.1	CL100
L-3	200 × 200 × 600	A	611.5	30	5	1.1	CL100
L-4	250 × 150 × 750	B	611.5	40	5	1.2	CL100
L-5	300 × 150 × 900	B	611.5	40	5	1.1	CL100
L-6	150 × 150 × 450	A	611.5	20	5	2.1	CL100
L-7	150 × 150 × 450	A	611.5	60	5	0.7	CL100
L-8	150 × 150 × 450	A	611.5	80	5	0.5	CL100
L-9	150 × 150 × 450	A	511.1	40	5	1.1	CL100
L-10	150 × 150 × 450	A	691.4	40	5	1.1	CL100
L-11	150 × 150 × 450	A	511.1	40	5	1.1	CL80
L-12	150 × 150 × 450	A	611.5	40	5	1.1	CL80
L-13	150 × 150 × 450	A	691.4	40	5	1.1	CL80
L-14	150 × 150 × 450	A	511.1	40	5	1.1	CL110
L-15	150 × 150 × 450	A	611.5	40	5	1.1	CL110
L-16	150 × 150 × 450	A	691.4	40	5	1.1	CL110
L-17	150 × 150 × 450	A	495.5	30	4	1.1	CL100
L-18	150 × 150 × 450	A	497.0	75	7	1.1	CL100
L-19	150 × 150 × 600	A	611.5	40	5	1.1	CL100
L-20	150 × 150 × 750	A	611.5	40	5	1.1	CL100

Note: L × W × H = Length × Width × Height, S = Stirrup Spacing, D = Diameter of Stirrups, ρ_v = Volumetric Stirrup Ratio.

.

2.2. Material Property Testing

2.2.1. Concrete

In the preparation of concrete, P.O 42.5R ordinary Portland cement, silica ash (enhance the density, strength, and durability of concrete), polycarboxylic acid high-performance water reducer, cement gypsum antifoam agent, 14 mm copper-coated steel fiber, and quartz sand were selected, as shown in Figure 3. Quartz sand includes 10–20 mesh, 20–40 mesh, 40–70 mesh, 70–120 mesh, and 325 mesh five particle sizes, the mass ratio of 1.1:0.6:1.5:1.3:1.4. The design mix ratio is shown in

Table 2. Concrete mix ratio.

	Raw Materials	Cement kg/m3	Silica Fume kg/m3	Sand kg/m3	Water kg/m3	Water Reducer kg/m3	Steel Fiber Volume Fraction/%	Water- to-Binder Ratio
Strength Grade
CL80		1033	258	697	362	6.46	1	0.28
CL100		1074	269	725	282	6.71	1	0.21
CL110		990	330	713	317	6.60	1	0.24

. The concrete used in this experiment was cast in three batches with strength grades of CL80, CL100, and CL110, respectively. For each batch, two groups of concrete cube specimens were prepared and cured using the water immersion method. The compressive strengths at 7-day and 28-day curing ages were tested prior to the experiment (see

Table 3. Compressive strength of concrete cubes.

Strength Grade	CL80			CL100			CL110
	1	2	3	1	2	3	1	2	3
7-Day Compressive Strength/MPa	75.9	66.4	77.3	73.4	76.5	79.3	86.1	79.1	89.4
28-Day Compressive Strength/MPa	88.3	84.0	84.9	111.1	98.4	98.2	112.2	111.2	109.3

). The cube compressive strength (f_cu) was calculated in accordance with the current Code for Design of Concrete Structures [25]. The axial compressive strength (f_co) was derived using Equation (1):

$$f_{co}={\mathsf{\alpha }}_{c1}{\mathsf{\alpha }}_{c2}{f}_{cu},$$

(1)

where, α_c1 is the ratio of prismatic compressive strength to cube compressive strength; α_c2 is the reduction coefficient accounting for concrete brittleness. Both α_c1 and α_c2 are calculated according to the current code [25], with α_c1 = 0.95 and α_c2 = 0.8.

2.2.2. Steel Reinforcement

The longitudinal reinforcement in the specimens utilized HRB400-grade steel bars with a diameter of 16 mm. The mechanical properties of both the stirrups and longitudinal bars are detailed in

Table 4. Mechanical properties of rebar.

Steel Reinforcement Grade		d/mm	fyv/MPa	Es/MPa
HRB400		16	440.34	2.0 × 105
High-Strength Steel Wire	500	4	495.5	2.0 × 105
	500	5	511.1	2.0 × 105
	500	7	497.0	2.0 × 105
	600	5	611.5	2.0 × 105
	700	5	691.4	2.0 × 105

Note: d = Diameter of steel reinforcement, f_yv = Yield strength of steel, E_s = Elastic modulus of steel.

, and the stress-strain curve of the stirrups is illustrated in Figure 4.

2.3. Test Setup and Loading Protocol

The experiments were conducted on a 5000 kN hydraulic testing machine at the Structural and Seismic Engineering Laboratory of Xi’an University of Architecture and Technology. The loading setup and schematic diagram are illustrated in Figure 5. During testing, load values and displacement measurements were automatically recorded using a TDS-602 (Sansel Instruments and Controls, Chennai, India) data acquisition system, which features high sampling frequency, precision, and multi-channel capabilities, ensuring real-time and accurate data collection for comprehensive and reliable results.

The specimens were loaded under displacement-controlled mode. Prior to formal loading, geometric and physical alignment procedures were performed:

(1)

Geometric Alignment: The specimen was positioned to ensure its longitudinal axis coincided precisely with the loading direction;

(2)

Physical Alignment: Instrumentation (e.g., displacement transducers) was calibrated to eliminate initial measurement biases.

A constant loading rate of 0.2 mm/min was maintained until specimen failure. Throughout the process, the data acquisition system continuously recorded all parameters, providing a complete dataset for post-test analysis.

2.4. Test Observations

The testing process can be divided into four distinct stages.

(1)

Stage 1: Elastic Stage

The specimen exhibited linear elastic behavior with a nearly linear stress-strain relationship. Axial strain, stirrup strain, and surface concrete strain remained minimal, and the confining effect of the stirrups was not yet activated. As the load increased, fine vertical cracks initiated at the column corners. These cracks propagated slowly upward and downward due to the gradual exceedance of concrete’s tensile strength by internal tensile stresses (Figure 6a).

(2)

Stage 2: Yielding Stage

With continued loading, the specimen reached its yield state. Cracks multiplied, widened, and interconnected, forming a complex network on the column surface. Audible “cracking sounds” accompanied the spalling of the concrete cover, signaling microcrack coalescence and the transition from elastic to elastoplastic behavior (Figure 6b).

(3)

Stage 3: Peak Load and Cover Spalling

At peak load, the concrete cover spalled due to bond failure between the cover and core concrete under high stress. This resulted in a sharp drop in load-bearing capacity. However, high-strength stirrups (HRSSs) actively confined the core concrete, enabling it to sustain residual load (Figure 6c).

(4)

Stage 4: Post-Peak Ductility and Failure

When the load dropped to approximately 65% of the peak load, the decline stabilized temporarily. The stirrups’ lateral confinement enhanced the core concrete’s compressive strength and ductility by restricting transverse deformation. Eventually, cracks penetrated the entire column, leading to core concrete crushing, structural void expansion, and complete loss of load capacity (Figure 6d). The failure morphology highlighted the progressive damage mechanism of high-strength stirrup-confined fiber-reinforced concrete under axial compression.

2.5. Test Results

The test results are summarized in Table 5, where f_cc and ε_cc represent the peak compressive strength and peak strain of the confined concrete, respectively; ε₈₅ denotes the ultimate strain when the compressive strength drops to 85% of the peak value; ε₆₅ is the strain corresponding to 65% of the peak strength. The strength enhancement ratio (f_cc/f_co) and ductility ratio (ε₈₅/ε_cc) serve as key indicators for evaluating the strength and ductility of confined concrete. Key findings include the following:

(1)

HRSS Effectiveness: Specimen L-13 achieved a strength enhancement ratio of 2.033, while L-14 exhibited a ductility ratio of 2.588, demonstrating that HRSS significantly improves both peak strength and deformability by providing effective lateral confinement to fiber-reinforced concrete columns;

(2)

Impact of Stirrup Spacing: For specimens L-1, L-6, L-7, and L-8 (with identical parameters), smaller stirrup spacings resulted in higher strength enhancement ratios but lower ductility ratios;

(3)

Effect of Stirrup Diameter and Spacing: Specimens L-9, L-17, and L-18 (with equal volumetric stirrup ratios) showed that finer stirrups with closer spacing provided superior performance, yielding higher strength enhancement and ductility ratios;

(4)

Influence of Concrete Strength: Comparisons among specimens L-11, L-12, L-13; L-1, L-9, L-10; and L-14, L-15, L-16 revealed that higher concrete strength grades led to lower strength enhancement ratios, indicating an inverse relationship between base concrete strength and confinement effectiveness.

Table 5. Test results.

Specimen Number	fcc/MPa	fco/MPa	εcc	ε65	ε85	fcc/fco	ε85/εcc
L-1	140.14	78.84	0.0063	0.0084	0.0076	1.777	1.199
L-2	126.41	78.84	0.0118	0.0202	0.0136	1.663	1.153
L-3	113.62	78.84	0.0091	0.0132	0.0102	1.441	1.124
L-4	138.27	78.84	0.0077	0.0124	0.0105	1.754	1.360
L-5	143.44	78.84	0.0078	0.0114	0.0100	1.614	1.277
L-6	140.81	78.84	0.0082	0.0246	0.0091	1.786	1.109
L-7	134.43	78.84	0.0062	0.0070	0.0059	1.705	1.212
L-8	131.59	78.84	0.0049	0.0124	0.0081	1.669	1.300
L-9	135.20	78.84	0.0057	0.0070	0.0089	1.715	1.555
L-10	142.43	78.84	0.0067	0.0200	0.0130	1.807	1.945
L-11	127.79	65.19	0.0053	0.0188	0.0086	1.960	1.629
L-12	114.58	65.19	0.0077	0.0184	0.0093	1.758	1.210
L-13	132.51	65.19	0.0092	0.0245	0.0118	2.033	1.278
L-14	133.88	88.69	0.0080	0.0114	0.0207	1.510	2.588
L-15	123.94	88.69	0.0074	0.0096	0.0084	1.397	1.141
L-16	147.11	88.69	0.0065	0.0184	0.0082	1.659	1.259
L-17	137.49	78.84	0.0071	0.0121	0.0090	1.744	1.265
L-18	131.22	78.84	0.0056	0.0080	0.0065	1.664	1.165
L-19	120.37	78.84	0.0066	0.0138	0.0092	1.527	1.401
L-20	140.14	78.84	0.0142	0.0279	0.0166	1.778	1.167

Table 5. Test results.

Specimen Number	fcc/MPa	fco/MPa	εcc	ε65	ε85	fcc/fco	ε85/εcc
L-1	140.14	78.84	0.0063	0.0084	0.0076	1.777	1.199
L-2	126.41	78.84	0.0118	0.0202	0.0136	1.663	1.153
L-3	113.62	78.84	0.0091	0.0132	0.0102	1.441	1.124
L-4	138.27	78.84	0.0077	0.0124	0.0105	1.754	1.360
L-5	143.44	78.84	0.0078	0.0114	0.0100	1.614	1.277
L-6	140.81	78.84	0.0082	0.0246	0.0091	1.786	1.109
L-7	134.43	78.84	0.0062	0.0070	0.0059	1.705	1.212
L-8	131.59	78.84	0.0049	0.0124	0.0081	1.669	1.300
L-9	135.20	78.84	0.0057	0.0070	0.0089	1.715	1.555
L-10	142.43	78.84	0.0067	0.0200	0.0130	1.807	1.945
L-11	127.79	65.19	0.0053	0.0188	0.0086	1.960	1.629
L-12	114.58	65.19	0.0077	0.0184	0.0093	1.758	1.210
L-13	132.51	65.19	0.0092	0.0245	0.0118	2.033	1.278
L-14	133.88	88.69	0.0080	0.0114	0.0207	1.510	2.588
L-15	123.94	88.69	0.0074	0.0096	0.0084	1.397	1.141
L-16	147.11	88.69	0.0065	0.0184	0.0082	1.659	1.259
L-17	137.49	78.84	0.0071	0.0121	0.0090	1.744	1.265
L-18	131.22	78.84	0.0056	0.0080	0.0065	1.664	1.165
L-19	120.37	78.84	0.0066	0.0138	0.0092	1.527	1.401
L-20	140.14	78.84	0.0142	0.0279	0.0166	1.778	1.167

2.6. Parametric Analysis

(1)

Cross-Section Aspect Ratio

As shown in Figure 7a, the peak stress decreases with an increasing aspect ratio (length-to-width ratio). For specimens with smaller aspect ratios, the stress drops rapidly after reaching the peak load. In contrast, specimens with larger aspect ratios exhibit a plateau phase after peak stress before rapid softening. This indicates that increasing the aspect ratio effectively mitigates the abrupt post-peak stress drop, enhancing the ductility of HRSS fiber-reinforced concrete columns.

(2)

Stirrup Spacing

From Figure 7b,c, reducing the stirrup spacing increases both the peak stress and peak strain. This demonstrates that closely spaced, small-diameter, high-strength spiral stirrups effectively address the sharp post-peak stress decline, improving both strength and ductility by enhancing the confinement of the core concrete.

(3)

Stirrup Strength

As shown in Figure 7d–f, for CL100-grade concrete, the peak stress increases with higher stirrup strength. For CL80-grade and CL110-grade concrete, specimens with 600 MPa-grade stirrups exhibit the lowest peak stress, while those with 700 MPa-grade stirrups achieve the highest peak stress. Specimens with 500 MPa-grade stirrups show intermediate values. Notably, for CL80-grade concrete, the peak stresses of specimens with 500 MPa-grade and 700 MPa-grade stirrups are nearly identical. These show that low-strength concrete (CL80) exhibits insufficient lateral expansion to fully activate the confinement effect of medium-to-high-strength stirrups (600–700 MPa). Low-strength stirrups (500 MPa) are best suited for low-strength concrete (CL80) due to balanced deformation compatibility. Moreover, high-strength concrete (CL110) is overly brittle, limiting its ability to mobilize lateral confinement stresses from stirrups. High-strength stirrups (700 MPa) should pair with high-strength concrete (CL110) to maximize confinement effectiveness. Furthermore, the similar peak stresses between CL80 specimens with 500 MPa-grade and 700 MPa-grade stirrups suggest that excessive stirrup strength cannot compensate for the limited lateral deformation capacity of low-strength concrete, resulting in underutilized confinement.

(4)

Concrete Strength

As shown in Figure 7d–f, for 500 MPa-grade and 700 MPa-grade stirrups, peak stress increases with concrete strength. The sensitivity to concrete strength is higher for 700 MPa-grade stirrups. For 600 MPa-grade stirrups, the highest peak stress occurs in CL100-grade concrete, followed by CL110 and CL80. This suggests that 600 MPa-grade stirrups are optimally compatible with CL100-grade concrete, likely due to balanced lateral deformation and confinement stress development. These observations further validate the Material Compatibility Principle: Low-strength concrete (≤CL80) should be paired with low-strength stirrups, while high-strength concrete (≥CL110) requires high-strength stirrups to achieve optimal synergistic effects. When the strength disparity between concrete and stirrups is excessive, the passive confinement effectiveness is compromised due to insufficient lateral expansion in high-strength concrete (caused by brittleness) or inadequate plastic deformation capacity in low-strength concrete.

3. Theoretical Analysis

3.1. Lateral Confinement Stress

When concrete expands under compressive forces, the stirrups generate tensile forces in the horizontal direction to resist the lateral deformation of the concrete. Simultaneously, the concrete develops a corresponding confinement force due to the restraint provided by the stirrups. The product of the tensile strength of the stirrups and their cross-sectional area is equal to the product of the lateral confinement stress of the concrete and its cross-sectional area. The lateral confinement stress for ordinary rectangular stirrup-confined concrete is calculated as shown in Equation (2) [11]:

$$n{f}_{yv}{A}_{sp}=f_{l}sa,$$

(2)

where, n = number of stirrup legs within one stirrup spacing, f_yv = yield strength of the stirrups, f_l = lateral confinement stress acting on the concrete, A_sp = cross-sectional area of a single stirrup leg, s = center-to-center spacing of the stirrups, and a = width of the confined concrete region, as illustrated in Figure 8.

For spiral stirrup-confined concrete, the internal tensile force generated by the spiral stirrups in the horizontal direction is equal in magnitude to the lateral confinement force exerted on the concrete. The product of the horizontal tensile stress in the spiral stirrups and their cross-sectional area equals the product of the lateral confinement stress and the cross-sectional area of the concrete, as shown in Equation (3). The force equilibrium within any stirrup spacing interval is illustrated in Figure 9.

$$n{f}_1{A}_{sp}^{\prime }=f_{l}sa,$$

(3)

where, f₁ = lateral confinement stress exerted by the spiral stirrups in the horizontal direction, and $A_{sp}^{\prime }=$ cross-sectional area of a single spiral stirrup.

Due to the inclination angle between the spiral stirrups and the horizontal plane, the maximum stress provided internally by the stirrups is no longer solely their tensile strength. Both the axial stress and tangential stress of the stirrups contribute to the horizontal confinement force. Therefore, the magnitude of the maximum horizontal stress f₁ provided by the stirrups is determined by the sum of the horizontal components of the axial and tangential stresses in the stirrups, as shown in Figure 10.

Therefore, the formula to calculate the maximum horizontal stress provided by the stirrups is $f_1={\displaystyle \frac{f_{yv}}{\cos \mathsf{\theta }}}$, where the cross-sectional area of a single stirrup is $A_{sp}^{\prime }={\displaystyle \frac{A_{sp}}{\cos \mathsf{\theta }}}$. Consequently, the lateral confinement stress for rectangular spiral stirrup-confined concrete is expressed by Equation (4):

$$f_l={\displaystyle \frac{n{f}_{yv}{A}_{sp}}{sa {\cos }^2\mathsf{\theta }}}$$

(4)

3.2. Effective Confinement Zone (A_e) and Coefficient (k_e)

Research indicates [10] that when circular stirrups confine concrete, the passive confinement pressure provided by the stirrups is uniformly distributed around the circumference. In contrast, for rectangular stirrups, the passive confinement pressure is non-uniformly distributed, with the maximum pressure concentrated at the intersection points between stirrups and longitudinal bars. Away from these intersections, the lateral confinement effect diminishes rapidly. This occurs because of high axial stiffness at intersections: At stirrup-longitudinal bar intersections, the axial stiffness of the stirrups is high, resulting in minimal deformation under transverse concrete expansion, thereby enhancing confinement effectiveness; Reduced stiffness away from intersections. Away from intersections, the axial stiffness decreases, and the flexural rigidity of the stirrups dominates. Larger deformations under transverse expansion weaken the confinement effect. Similarly, along the longitudinal direction, the confinement effect is strongest at stirrup-longitudinal bar intersections and weaker elsewhere. Thus, the lateral confinement force of stirrups is only effective within the core concrete region subjected to active restraint, as illustrated in Figure 11.

Mander [11] proposed that the area of the non-effectively confined region is bounded by a standard parabola, where the base length W_i (i.e., the transverse spacing between adjacent longitudinal bars) is four times the height h of the parabolic arch. Based on the actual stress-strain curves of specimens L-1, L-2, and L-3 (Figure 12), it is observed that larger W_i results in smaller peak stress, indicating reduced effective confinement and a larger non-effectively confined area. Consequently, for HRSS-confined fiber-reinforced concrete, the height h of the parabolic arch in the non-effectively confined region increases with W_i. Assuming a proportional relationship between h/W_i and W_i, i.e., ${\displaystyle \frac{h}{W_i}}=k_1{W}_i=\mathsf{\mu }$, it follows that $h=k_1{W}_i^2$, where k₁ is a constant to be determined. Thus, the area of the non-effectively confined region in HRSS-confined fiber-reinforced concrete is calculated as shown in Equation (6):

$$A_i={\displaystyle \frac{{(W_i)}^2}{6}}={\displaystyle \frac{2}{3}}{W}_i({\displaystyle \frac{1}{4}}{W}_i)={\displaystyle \frac{2}{3}}{W}_{i}h$$

(5)

$$A_i={\displaystyle \frac{2}{3}}{W}_{i}h={\displaystyle \frac{2}{3}}{W}_i(k_1{W}_i^2)={\displaystyle \frac{2}{3}}\mathsf{\mu }{W}_i^2$$

(6)

For ordinary stirrup-confined rectangular concrete, the effectively confined area A_e is calculated by Equation (7), which accounts for the unconfined longitudinal regions and their relationship with stirrup spacing s. The first term of Equation (7) reflects the transverse confinement effect of the stirrups, while the second term represents the longitudinal confinement effect. For spiral stirrup-confined concrete, the longitudinal confinement is significantly enhanced due to the continuous helical reinforcement. Therefore, the value of the second term in the formula for the effectively confined area should be increased. The modified formula for spiral stirrup-confined concrete is defined in Equation (8), where k₂ is a constant greater than 0 and less than 1/2:

$$A_e=(ab-{\displaystyle \sum _{i=1}^n{A}_i)(1-\frac{s^{\prime }}{2a}})(1-{\displaystyle \frac{s^{\prime }}{2b}})=(ab-{\displaystyle \sum _{i=1}^n{A}_i)\frac{(a-\frac{s^{\prime }}{2})(b-\frac{s^{\prime }}{2})}{ab}}$$

(7)

$$A_e=(ab-{\displaystyle \sum _{i=1}^n{A}_i)\frac{(a-k_2{s}^{\prime })(b-k_2{s}^{\prime })}{ab}}$$

(8)

The effective confinement coefficient k_e is defined as the ratio of the effectively confined area to the difference between the cross-sectional area of concrete enclosed by stirrups and the total cross-sectional area of longitudinal reinforcement. Its calculation formula is given in Equation (9):

$$k_e={\displaystyle \frac{A_e}{A-A_s}},$$

(9)

where, A = cross-sectional area of the core region enclosed by stirrups, and A_s = total cross-sectional area of all longitudinal reinforcement in the section.

3.3. Peak Parameters

3.3.1. Peak Parameters for Square Cross-Section Columns

Compared with plain concrete, confined concrete exhibits enhanced peak stress and peak strain, primarily due to the passive confinement provided by stirrups. The effectiveness of confinement is directly quantified by the lateral confinement stress. Therefore, the peak stress and peak strain of confined concrete can be expressed by Equations (10) and (11):

$$f_{cc}=f_{co}[1+\mathsf{\alpha }({\displaystyle \frac{f_{le}}{f_{co}}}{)}^{\beta }]$$

(10)

$${\epsilon }_{cc}={\epsilon }_{co}[1+\gamma ({\displaystyle \frac{f_{le}}{f_{co}}}{)}^{\delta }]$$

(11)

$$f_{le}=k_e{f}_l,$$

(12)

where, $\mathsf{\alpha }$, $\beta$, $\gamma$, and $\delta$ are constants to be determined, and f_le represents the effective lateral confinement stress.

Through regression analysis of experimental data (excluding rectangular cross-section specimens L-4 and L-5) for Equations (10) and (11), the following coefficients are derived: k₁ = 0.002, k₂ = 0.3, $\mathsf{\alpha }$ = 2.817, $\beta$ = 0.418, $\gamma$ = 6.043, $\delta$ = 0.253. The formulas for the peak parameters, non-effectively confined lateral area, and effectively confined area of HRSS-confined fiber-reinforced concrete are as follows:

$$A_i={\displaystyle \frac{2}{3}}{W}_{i}h={\displaystyle \frac{2}{3}}{W}_i(k_1{W}_i^2)={\displaystyle \frac{W_i^3}{750}}$$

(13)

$$A_e=(ab-{\displaystyle \sum _{i=1}^n{A}_i)(1-}{\displaystyle \frac{3s\prime }{10a}})(1-{\displaystyle \frac{3s\prime }{10b}})$$

(14)

$$f_{cc}=f_{co}[1+2.817({\displaystyle \frac{f_{le}}{f_{co}}}{)}^{0.418}]$$

(15)

$${\epsilon }_{cc}={\epsilon }_{co}[1+6.043({\displaystyle \frac{f_{le}}{f_{co}}}{)}^{0.253}]$$

(16)

By applying the aforementioned calculation formulas, the theoretical values of the peak stress enhancement ratio (f_cc/f_co)_c and peak strain enhancement ratio (ε_cc/ε_co)_c for HRSS-confined fiber-reinforced concrete can be obtained, as illustrated in Figure 11. In the figure, (f_cc/f_co)_e and (ε_cc/ε_co)_e represent the experimental values of these enhancement ratios. As shown in Figure 13, the calculated enhancement ratios for both peak stress and peak strain exhibit good agreement with the experimental results, validating the accuracy of the theoretical model.

3.3.2. Peak Parameters for Rectangular Cross-Sections

For HRSS-confined fiber-reinforced concrete with rectangular cross-sections, the lateral confinement stresses differ along the width (x-direction) and length (y-direction). Therefore, the calculation method for peak parameters requires further refinement. Assuming the lateral confinement stress along the width (x-direction) is f_lx and along the length (y-direction) is f_ly, their formulas can be derived from Equation (3), as shown in Equations (17) and (18):

$$f_{lx}={\displaystyle \frac{n_1{f}_{yv}{A}_{sp}}{sa {\cos }^2\mathsf{\theta }}}$$

(17)

$$f_{ly}={\displaystyle \frac{n_2{f}_{yv}{A}_{sp}}{sb {\cos }^2\mathsf{\theta }}}$$

(18)

For effectively confined area and effective confinement coefficient, the calculation methods are identical for both square and rectangular confined concrete. Therefore, the formulas for the effective lateral confinement stress of HRSS-confined fiber-reinforced concrete are expressed as Equations (19) and (20):

$$f_{lxe}=k_e{f}_{lx}$$

(19)

$$f_{lye}=k_e{f}_{ly},$$

(20)

where, n₁ = number of stirrup legs along the x-direction within one stirrup spacing, n₂ = number of stirrup legs along the y-direction within one stirrup spacing, f_lxe = effective lateral confinement stress along the x-direction of the cross-section, and f_lye = effective lateral confinement stress along the y-direction of the cross-section. For ordinary rectangular spiral stirrup-confined concrete, the effective lateral confinement stresses differ in the x- and y-directions. Therefore, calculating the peak parameters requires comprehensive consideration of f_lxe, f_lxe, and their coupling effects. The formulas for peak stress and peak strain are given in Equations (21) and (22):

$$f_{cc}=f_{co}[1+2.817({\displaystyle \frac{x_1{f}_{lxe}+x_2{f}_{lye}+x_3\sqrt{f_{lxe}{f}_{lye}}}{f_{co}}}{)}^{0.418}]$$

(21)

$${\epsilon }_{cc}={\epsilon }_{co}[1+6.043({\displaystyle \frac{x_4{f}_{lxe}+x_5{f}_{lye}+x_6\sqrt{f_{lxe}{f}_{lye}}}{f_{co}}}{)}^{0.253}]$$

(22)

Regression analysis of the experimental data for all specimens yields the following coefficients: $x_1$ = 0.371, $x_2$ = 0.428, $x_3$ = 0.201, $x_4$ = 0.209, $x_5$ = 0.321, $x_6$ = 0.470. Therefore, the calculation formulas for peak parameters of HRSS-confined fiber-reinforced concrete in general rectangular sections are given in Equations (23) and (24):

$$f_{cc}=f_{co}[1+2.817({\displaystyle \frac{0.371f_{lxe}+0.428f_{lye}+0.201\sqrt{f_{lxe}{f}_{lye}}}{f_{co}}}{)}^{0.418}]$$

(23)

$${\epsilon }_{cc}={\epsilon }_{co}[1+6.043({\displaystyle \frac{0.209f_{lxe}+0.321f_{lye}+0.470\sqrt{f_{lxe}{f}_{lye}}}{f_{co}}}{)}^{0.253}]$$

(24)

4. Stress-Strain Constitutive Model

Building on experimental results, this study proposes a complete stress-strain curve for HRSS-confined fiber-reinforced concrete. The ascending branch adopts the Popovics [26] model, while the descending branch follows the Fafitis and Shah [27] model; the descending branch passes through the point where the concrete strength drops to 65% of the peak strength. The full stress-strain curve is expressed as Equation (25):

$$f_c=\left\{\begin{array}{c}{f}_{cc}[{\displaystyle \frac{r({\epsilon }_c/{\epsilon }_{cc})}{r-1+{({\epsilon }_c/{\epsilon }_{cc})}^r}}],({\epsilon }_c\le {\epsilon }_{cc})\\ f_{cc}\cdot \exp [k_1{({\epsilon }_c-{\epsilon }_{cc})}^{k_2}],({\epsilon }_c>{\epsilon }_{cc})\end{array}\right.$$

(25)

$$r={\displaystyle \frac{E_c}{E_c-(f_{cc}/{\epsilon }_{cc})}},$$

(26)

where, the values of f_cc and ε_cc are calculated using Equations (23) and (24) (Equations (15) and (16) for square columns), f_c and ε_c = current stress and strain of the confined concrete, E_c = elastic modulus of concrete, and r = coefficient controlling the initial stiffness and ascending branch shape. A larger r indicates that the ascending branch approaches a straight line with a higher elastic modulus, calculated by Equation (26). k₁ and k₂ = coefficients governing the descending branch. For highly confined concrete, the larger values of k₁ and k₂ result in a smooth, convex-shaped descending branch of the stress-strain curve. Conversely, for lightly confined concrete, the smaller values of k₁ and k₂ lead to a steep, concave-shaped descending branch. k₁ and k₂ are calculated using Equations (27) and (28), respectively.

$$k_1={\displaystyle \frac{\ln 0.65}{{({\epsilon }_{65}-{\epsilon }_{cc})}^{k_2}}}$$

(27)

$$k_2=5.198{(f_{le}/f_{co})}^{0.629}-0.262,$$

(28)

where, ε₆₅ = strain corresponding to the point where the confined concrete strength drops to 65% of the peak strength. Figure 14 shows the relationship between ε₆₅ and ρ_wε_cc, where ρ_w is the area-based stirrup ratio (ratio of stirrup area to confined concrete area). Through regression analysis, the formula for ε₆₅ is derived as Equation (29):

$${\epsilon }_{65}=706174{({\rho }_w{\epsilon }_{cc})}^2+6.105({\rho }_w{\epsilon }_{cc})+0.004$$

(29)

Figure 15 illustrates the relationship between the coefficient k₂ and the effective confinement index f_le/f_co. Based on this, the formula for calculating k₂ was derived through regression analysis and is given in Equation (28). The unknown coefficient k₂ was determined by developing a MATLAB 2024av24.1 program and performing computational iterations. The core methodology involved minimizing the area difference between the stress-strain curves calculated from the proposed constitutive model and the corresponding experimental curves for all specimens.

Several representative specimens were selected from the experimental tests. The stress-strain curves of these specimens were calculated using three models: the proposed model in this study (Proposed), the Mander [11] model, and the Cusson [12] model. The results were then compared against the experimental curves, as shown in Figure 16.

From the figures, it is evident that the complete stress-strain curve model for HRSS fiber-reinforced concrete proposed in this study aligns well with experimental results. Both the full-range stress-strain curves and the predictions of peak stress and peak strain are accurate. In contrast, the Mander [11] model and Cusson [12] model underestimate the peak stress and peak strain compared with actual test data primarily because they do not account for the enhanced confinement capability of spiral stirrups on core concrete. Furthermore, for specimens with small stirrup spacing (high confinement), the proposed model slightly overestimates peak stress; For specimens with large stirrup spacing (low confinement), the model underestimates peak stress (Figure 16a–c). The model performs less accurately for specimens with low stirrup strength (Figure 16a,d,e), showing deviations in the full stress-strain curve predictions. For ultra-high-strength concrete specimens, the model overestimates peak stress (Figure 16a,f,g).

5. Conclusions

Based on axial compression tests and theoretical analyses of 20 HRSS fiber-reinforced concrete columns, this study draws the following conclusions:

(1)

Superiority of Spiral Stirrups:

Experimental results indicate that HRSS-confined fiber-reinforced concrete exhibits an average peak strength enhancement factor of 1.70 and an average ductility ratio of 1.37. This improvement is attributed to the high-strength, continuous, and closed configuration of spiral stirrups, which provide uniform and sustained lateral confinement to the concrete columns. Such confinement effectively suppresses the lateral expansion of concrete under axial compression, significantly enhancing both compressive strength and load-bearing capacity. Compared with conventional hoops, spiral stirrups deliver more comprehensive and stable confinement effects, demonstrating superior performance in seismic design applications.

(2)

Enhanced Confinement with Small-Diameter, Closely Spaced Stirrups:

Analysis of specimens L-9, L-17, and L-18 reveals that under identical conditions (including equivalent volumetric stirrup ratios), smaller-diameter stirrups yield higher peak stresses. For instance, the peak stress of specimen L-17 is 1.05 times that of L-18, and L-17 exhibits gradual post-peak stress degradation rather than abrupt failure. This demonstrates that small-diameter, closely spaced, high-strength spiral stirrups provide more effective confinement. They not only enhance load-bearing capacity but also mitigate sudden stress drops after peak strength. When internal concrete damage initiates, these stirrups rapidly activate to restrain crack propagation and prevent concrete crushing, substantially improving the specimen’s overall bearing capacity and ductility.

(3)

Material Compatibility Principles:

The combination of low-strength stirrups with low-strength concrete effectively avoids premature failure caused by excessive material strength mismatch. Their synergistic interaction ensures stability during loading, as evidenced by specimen L-11 achieving 11.5% higher peak stress than L-12 (which used higher-strength stirrups). Conversely, pairing high-strength stirrups with high-strength concrete fully leverages their material advantages, providing robust confinement under heavy loads to ensure structural safety and stability (e.g., specimen L-16 reached a peak stress of 147.11 MPa). These findings underscore that rational material matching is critical for optimizing concrete structural performance and safety.

(4)

Proposed Constitutive Model:

This study introduced a refined confinement zone classification for spiral stirrup-confined concrete, establishing calculation methods for the effectively confined area and confinement coefficient. A constitutive model tailored for high-strength stirrup-confined high-strength concrete is proposed. Comparative analysis with experimental results and other constitutive models revealed that the proposed model more accurately reflects actual stress-strain curves and demonstrates close agreement with real mechanical behavior but exhibits deviations in predicting ultra-high-strength concrete.