Local Buckling of Locally Sharp-Notched C2700 Brass Circular Tubes Subjected to Cyclic Bending

Abstract

This paper aims to investigate the response and local buckling of locally sharp-notched C2700 brass circular tubes (LSN C2700 brass circular tubes) under cyclic bending loads. The study considers four different notch orientations (0°, 30°, 60°, and 90°) and five distinct notch depths (0.2, 0.4, 0.6, 0.8, and 1.0 mm). The results reveal that notch orientation and depth exert minimal impact on the moment–curvature relationship, leading to the formation of stable loops. The ovalization–curvature graphs demonstrate a trend of symmetry, serration, and growth with an increasing number of bending cycles. Additionally, larger notch orientations or smaller notch depths result in reduced ovalization. Furthermore, the double logarithmic coordinates of controlled curvature–number of cycles necessary to induce local buckling reveal five non-parallel lines representing different notch depths when the notch orientation is fixed. Finally, by adopting the formulas for smooth tubes and for locally sharp-notched 304 stainless steel circular tubes (LSN SS304 circular tubes), this study adjusts the related material parameters accordingly. These modifications effectively describe the controlled curvature–number of cycles necessary to induce local buckling for LSN C2700 brass circular tubes with different notch orientations and depths under cyclic bending, demonstrating reasonable agreement with the experimental results.

1. Introduction

Tubes are commonly subjected to prolonged cyclic bending loads, causing a gradual decrease in their strength as the cycle count increases. Simultaneously, these tubes undergo ovalization, defined as the reduction in the outer diameter (∆D) divided by the original outer diameter (D_o), indicating irreversible deformation. When ΔD/D_o reaches a critical value, the tube may undergo local buckling, leading to structural failure. Such failure modes can result in the tube’s inability to withstand loads, ultimately resulting in scenarios such as leakage or blockage of transported substances.

Numerous researchers have investigated the bending behavior of smooth tubes. Kyriakides and Shaw [1] developed a device capable of performing both monotonic and cyclic bending tests on tubes, with or without external pressure. This equipment has been extensively utilized in the study of tubes made from various materials, including 6061-T6 aluminum alloy, 304 stainless steel, 1018 steel, 1020 steel, and NiTi smooth tubes [1,2,3,4,5,6,7,8,9].

Several researchers have made notable contributions to the study of tube behavior under various bending conditions. For instance, Yuan and Mirmiran [10] focused on the buckling behavior of concrete-filled fiber-reinforced plastic tubes subjected to bending. Elchalakani et al. [11] conducted explorations into the slenderness limit of cold-formed hollow sections through cyclic bending tests. Additionally, Elchalakani and Zhao [12] studied the cyclic bending performance of concrete-filled cold-formed steel tubes, while Zhi et al. [13] examined the stability and instability of cylindrical shells under earthquake conditions. In a separate study, Yazdani and Nayebi [14] analyzed the damage sustained by pipelines when bending under stable internal pressure. Guo et al. [15] focused on studying the bending behavior of thin-walled hollow tubes, whereas Elchalakani et al. [16] established novel ductile slenderness limits based on strains for concrete-filled tubes through bending tests.

Other researchers have explored various aspects of tube behavior, including instability following non-proportional paths [17], the effect of corrosion on the bending performance of curved tubes [18], and local buckling behavior under bending and axial loads [19]. Additionally, Silveira et al. [20] demonstrated the applicability of the constructal law in evaluating the mechanical geometry of materials engineering systems. Montes [21] focused on the experimental study on the bending of a polyamide tube using high-power ultrasound. Shi et al. [22] found that the fracture of steel wires leads to the failure of the tube, and higher temperature has a negative effect on the tubes’ bending capacity. Chen et al. [23] experimentally and theoretically examined the flexural behavior of hollow section concrete-filled GFRP tubular (HS-CFGT) members. Lavayen-Farfan et al. [24] proposed the theoretical approach by validating with three-point bending experimental tests and an adequate agreement with experiments. He et al. [25] proposed the relationship between the bending angle and the length of the transition section formation zone through experimental and simulation verification. Wang et al. [26] conducted a four-point bending experimental study on the bending response of square steel tube concrete reinforced by internal angle steel. Meanwhile, Wu et al. [27] explored the response and damage of stacked sequentially braided composite pipes with 45° and 60° braids under three-point bending. Duda et al. [28] proposed a discrete model that can be applied to finite element analysis to model and predict the mechanical behavior of thin-walled CFRP tubes.

C2700 brass circular tubes are widely used materials in various industries, valued for their good mechanical properties, corrosion resistance, excellent machinability, and high conductivity. They serve diverse purposes in industries such as manufacturing precision instruments, hip parts, gun shells, HVAC (Heating, Ventilation, and Air Conditioning), and construction. Examples of their applications include oil–water separators for ships, tubes for transporting liquids and gases in HVAC systems, roof drainage systems, and underwater cable structures. Due to the functional requirements in these applications, C2700 brass circular tubes often undergo various machining processes, including drilling, grooving, and circumferential cutting. Consequently, this study aims to investigate the response and local buckling of LSN C2700 brass circular tubes with different notch orientations and depths under cyclic bending.

Although Lee et al. [29] conducted relevant research on the response and local buckling of LSN SS304 circular tubes with different notch orientations and depths under cyclic bending loads, questions have arisen regarding the consistency of observed behaviors in LSN C2700 brass circular tubes. These questions include the relationships of moment–curvature, ovalization–curvature, and the controlled curvature–number of cycles necessary to induce local buckling. Additionally, can the empirical formulas proposed for the locally sharp-notched SUS304 stainless steel tubes be applied to the locally sharp-notched C2700 brass circular tubes? Furthermore, this paper will thoroughly discuss the applicability of the theory they proposed to other materials.

2. Experiments

2.1. Experimental Devices

In this study, cyclic bending tests were performed on LSN C2700 brass circular tubes using a pure bending (four-point bending) machine, illustrated in Figure 1. The test setup includes two sprockets mounted on beams and a strong chain running around them, capable of accommodating tube specimens up to 1 m in length. To facilitate testing, each tube is fitted with solid rod extensions. During the bending process, the interaction between the tube and the roller allows for free axial movement. The applied load is generated by two concentrated loads from a pair of rollers. Retraction of the upper or lower cylinder initiates sprocket rotation, enabling pure bending of the test tube. Reverse bending is achieved by reversing the hydraulic circuit’s flow direction. For a more comprehensive understanding of the machine, readers are directed to the detailed exposition by Pan et al. [30].

The Curvature and Ovalization Measurement Apparatus (COMA), developed by Pan et al. [30] and depicted in Figure 2, is a valuable tool for assessing tube curvature (κ) and ovalization (ΔD/D_o), crucial parameters in this study. It is a lightweight instrument that can be mounted close to the tube mid-span. The COMA includes three inclinometers. Two of them, referred to as side-inclinometers, are fixed on two holders. These holders are attached to the circular tube before testing begins. The distance between the two side-inclinometers is denoted as L_o.

Consider that the circular tube is subjected to pure bending, as shown in Figure 3. The angle changes detected by the two side-inclinometers are denoted as θ₁ and θ₂. From Figure 3, the value of L_o is determined as

L_o = ρ (θ₁ + θ₂),

(1)

where ρ is the radius of the curvature. The κ is

κ = 1/ρ = (θ₁ + θ₂)/L_o

(2)

2.2. Material and Specimens

In this study, tubes fabricated from C2700 brass were utilized. The chemical composition of the tubes includes Pb (0.05%), Fe (0.05%), Zn (32.0%), and the rest is Cu. The material properties are as follows: density of 8450 kg/m³, elastic modulus of 103 GPa, Poisson’s ratio of 0.31, yield strength of 275 MPa, and ultimate strength of 380 MPa.

The initial, raw C2700 brass circular tubes had an outer diameter (D_o) of 31.8 mm and a wall thickness (t) of 1.5 mm. To introduce local sharp notches, the raw tubes underwent drilling on the outside surface to achieve expected notch depths (a) of 0.2, 0.4, 0.6, 0.8, and 1.0 mm. Figure 4 depicts the schematic diagram of the LSN tube. Based on the drill size used, the corresponding surface width (b) of the notches was 0.6, 1.2, 1.8, 2.4, and 3.0 mm, respectively. Figure 5 provides a visual representation of the LSN C2700 brass circular tubes, highlighting the appearance of the introduced sharp notches.

Since the sharp notch is local, the orientation (ϕ) of the sharp notch is expected to influence the response and local buckling of the LSN tubes. Therefore, ϕ must be taken into consideration. In Figure 6, orientations of ϕ at 0°, 30°, 60°, 90°, 120°, 150°, and 180° were examined. Since the bending moment is exerted in the z-direction, similar responses are expected to be observed for ϕ = 0° and ϕ = 180°, ϕ = 30° and ϕ = 150°, and ϕ = 60° and ϕ = 120° when subjected to cyclic bending loads. Consequently, this study focused solely on four orientations, specifically ϕ = 0°, 30°, 60°, and 90°.

2.3. Test Procedures

LSN C2700 brass circular tubes were subjected to curvature-controlled cyclic bending at a curvature rate of 0.01 m⁻¹ s⁻¹. The bending moment (M) was determined by using load cells in the bending machine, as depicted in Figure 1. The κ and ΔD/D_o were measured by using the COMA shown in Figure 2. The number of cycles necessary to induce local buckling (N_b) was recorded, which was defined as when the M decreased by 20%.

3. Experimental Results and Discussion

3.1. M-κ Relationships

Figure 7 illustrates the M-κ relationship of the LSN C2700 brass circular tube under cyclic bending with ϕ = 0° and a = 0.2 mm. The controlled curvature ranges from −0.5 m⁻¹ to 0.5 m⁻¹. The experimental results indicate that in the initial stage of loading, the LSN C2700 brass circular tube undergoes elastic deformation, exhibiting a linear trend in the M-κ relationship. However, upon entering the plastic region, the M-κ relationship becomes nonlinear, with the slope of the curve gradually decreasing, indicating permanent plastic deformation of the LSN C2700 brass circular tube. When the curvature reaches the control peak value of 0.5 m⁻¹, the bending test machine initiates reverse bending, reducing the curvature from the peak value of 0.5 m⁻¹ to −0.5 m⁻¹. In the initial unloading stage, the linear M-κ relationship is parallel to the linear M-κ relationship in the initial elastic range. Subsequently, the later part of the negative bending exhibits a similar behavior to the positive bending, with the slope of the curve gradually decreasing, indicating a nonlinear trend as it enters the plastic range. After completing the first cycle, the M-κ relationship graph shows a loop, and with an increase in cycles, the loops almost overlap. Due to the localized and small nature of the local sharp notches, the M-κ relationships for different values of ϕ and a are nearly identical. Therefore, only the M-κ relationship for the case of ϕ = 0° and a = 0.2 mm is presented in this study. This phenomenon aligns with the experimental findings of Lee et al. [29] on LSN SS304 circular tubes under cyclic bending loads, where stable loops remain unaffected by ϕ and a.

3.2. D/D_o-κ Relationships

Figure 8a–e, respectively, depict the ΔD/D_o-κ relationships of LSN C2700 brass circular tubes under cyclic bending with a fixed ϕ = 0° and different a values: 0.2, 0.4, 0.6, 0.8, 1.0 mm. The controlled curvature ranges from −0.5 m⁻¹ to 0.5 m⁻¹. The experimental results show that regardless of deformation in the elastic or plastic range, the ΔD/D_o-κ curve exhibits a nonlinear trend. When the LSN C2700 brass circular tube is loaded, ΔD/D_o shows an increasing trend, reaching its peak when the curvature is 0.5 m⁻¹. Subsequently, in the unloading phase, ΔD/D_o shows a decreasing trend, and when the curvature reaches 0.0 m⁻¹, ΔD/D_o is not zero, indicating plastic deformation in the LSN C2700 brass circular tube. During the reverse loading phase, ΔD/D_o shows an increasing trend again, reaching its peak when the curvature is −0.5 m⁻¹. Then, in the reloading phase, when the curvature returns to 0.0 m⁻¹, ΔD/D_o increases slightly. Through repeated cyclic loading, the ΔD/D_o-κ curve exhibits an increasing, serrated, and symmetrical trend. It is observed that as a increases, the growth of ΔD/D_o also increases. This phenomenon is consistent with the experimental results of Lee et al. [29]. However, a less consistent observation is that when a is small, the ΔD/D_o-κ relationships are symmetrical, but when a is large, the ΔD/D_o-κ relationships are asymmetrical.

Figure 9a–d, respectively, illustrate the ΔD/D_o-κ relationships of LSN C2700 brass circular tubes under cyclic bending with different ϕ values: 0°, 30°, 60°, 90° and a fixed a = 1.0 mm. It can be observed that a smaller ϕ leads to a larger ΔD/D_o. This phenomenon is consistent with the experimental results of Lee et al. [29]. However, our study found that the ΔD/D_o-κ curve exhibits an increasing, serrated, and symmetrical trend for any ϕ. This differs from the experimental results of Lee et al. [29], where they observed asymmetrical ΔD/D_o-κ relationships for small values of ϕ but symmetrical relationships for large values of ϕ.

3.3. Controlled Curvature (κ_c/κ_o)-N_b Relationships

Table 1. N_b values for LSN C2700 brass circular tubes under cyclic bending with different a and κ_c/κ_o at different ϕ.

(a) ϕ = 0°
κc/κo	a = 0.2 mm	a = 0.4 mm	a = 0.6 mm	a = 0.8 mm	a = 1.0 mm
0.33	542	303	192	112	65
0.29	694	403	249	154	93
0.26	890	511	316	212	128
0.23	1125	703	432	292	196
0.20	1734	988	694	423	305
(b) ϕ = 30°
κc/κo	a = 0.2 mm	a = 0.4 mm	a = 0.6 mm	a = 0.8 mm	a = 1.0 mm
0.33	847	439	219	145	97
0.29	986	571	309	182	131
0.26	1270	765	413	273	175
0.23	1765	1042	512	314	260
0.20	2664	1347	698	456	314
(c) ϕ = 60°
κc/κo	a = 0.2 mm	a = 0.4 mm	a = 0.6 mm	a = 0.8 mm	a = 1.0 mm
0.33	1605	1028	657	442	298
0.29	2293	1476	1059	661	422
0.26	2958	1821	1293	913	567
0.23	4191	2155	1803	1201	704
0.20	5705	3792	2430	1644	1155
(d) ϕ = 90°
κc/κo	a = 0.2 mm	a = 0.4 mm	a = 0.6 mm	a = 0.8 mm	a = 1.0 mm
0.33	2876	2277	1687	1325	1141
0.29	3977	3374	2257	1888	1534
0.26	5528	4002	3532	2587	1898
0.23	7297	6339	4611	3722	2936
0.20	11,338	8867	6487	5481	4930

a–d present the N_b values for LSN C2700 brass circular tubes under cyclic bending with different a and κ_c/κ_o at ϕ values: 0°, 30°, 60°, and 90°. Additionally, Figure 10a–d illustrate the experimental κ_c/κ_o-N_b relationships for LSN C2700 brass circular tubes under cyclic bending with varying ϕ values: 0°, 30°, 60°, 90°, as well as different a. Here, κ_o is used to nondimensionalize κ_c, calculated as κ_o = t/D_o². The experimental results indicate that, for fixed values of ϕ and a, larger values of κ_c/κ_o correspond to a reduced N_b. Similarly, when ϕ and κ_c/κ_o are fixed, larger values of a result in a decreased N_b. Meanwhile, for fixed values of a and κ_c/κ_o, larger values of ϕ lead to an increased N_b. Figure 11a–d depict the same data as Figure 10a–d using double logarithmic coordinates, with straight lines representing the results of least squares fitting. This phenomenon is consistent with the experimental results of Lee et al. [29]. However, an inconsistency arises. For a certain ϕ, Lee et al. [29] showed five almost-parallel straight lines, while this study found five non-parallel straight lines. Note that for each κ_c/κ_o, two LSN C2700 brass circular tubes were tested, and there was no significant difference in the obtained N_b.

Shaw and Kyriakides [1] proposed the following κ_c/κ_o-N_b relationship for smooth tubes under cyclic bending in 1987:

κ_c/κ_o = C(N_b)^−α

(3a)

or

logκ_c/κ_o = logC − αlogN_b,

(3b)

in which C and α are material parameters associated with the mechanical properties of the material and the geometric shape. The value of α represents the slope of the double logarithmic coordinates line relationship between κ_c/κ_o and N_b, while C indicates the corresponding value of κ_c/κ_o when N_b is 1. However, this formula cannot cover the κ_c/κ_o-N_b relationship for LSN SS304 circular tubes under cyclic bending with different ϕ and a. Therefore, Lee et al. [29] proposed that the relationship between C and a is as follows:

logC = −m(a/t) + b,

(4)

in which m and b are material parameters. The quantity m represents the slope of the logC versus a/t relationship line, while b is the logC value when a/t is 0. Through experiments, the logC versus a/t relationships for different ϕ are shown in Figure 11a. It can be observed that Equation (4) is also applicable to LSN C2700 brass circular tubes. Four lines with different slopes are shown in Figure 12a. If these lines are extended to a/t = 0, they represent the scenario without notches. The logC values at this point converge to approximately 0.91; hence, b is 0.91. In addition, the values of m are determined for ϕ = 0°, 30°, 60°, and 90°, corresponding to 1.15, 0.82, 0.64, and 0.48, respectively, as shown in Figure 12b. Through curve fitting, a linear relationship is found; thus, this study proposes:

m = m₁ϕ + m₂,   0 ≤ ϕ ≤ π/2,

(5)

where m₁ and m₂ are material parameters, and they can be derived as −0.4196 and 1.1029, respectively. However, based on the experimental results of Lee et al. [29], it was found that the relationship between m and ϕ is described by the following equation:

(1.37 − m)^m₁ = m₂ϕ,   0 ≤ ϕ ≤ π/2.

(6)

Next, this paper continues to discuss the form of α. Lee et al. [29] found that for a specific ϕ, the κ_c/κ_o-N_b relationship on double logarithmic coordinates showed five almost-parallel straight lines. Based on this observation, they proposed the form of α to be:

α^1/3= a₁ϕ + a₂,   0 ≤ ϕ ≤ π/2,

(7)

where a₁ and a₂ are material parameters. However, the κ_c/κ_o-N_b relationship on double logarithmic coordinates showed five non-parallel straight lines in this study. Therefore, this study first attempts to establish the relationship between α and a/t. From Figure 11a–d, the α-a/t relationship for different ϕ can be obtained, as shown in Figure 13a. Thus, the following proposal is made:

α = −β(a/t) + γ,

(8)

where β and γ are material parameters. The quantity β represents the slope of the line in the α versus a/t relationship, while γ is the value of α when a/t is 0. The values of β for ϕ = 0°, 30°, 60°, and 90° are 0.213, 0.0338, 0.036, and 0.054, respectively. The relationship between the two is depicted in Figure 13b. Based on the results of curve fitting, this study proposes:

β = β₁ϕ³ + β₂ϕ² + β₃ϕ + β₄,   0 ≤ ϕ ≤ π/2

(9)

where β₁, β₂, β₃, and β₄ are material parameters, and their values are determined as β₁ = −0.1923, β₂ = 06329, β₃ = −0.6209, and β₄ = 0.2131. As for γ values, they are found to be 0.4864, 0.4507, 0.4134, and 0.3892 for ϕ = 0°, 30°, 60°, and 90°, respectively. The relationship between the two is shown in Figure 13c. Due to its linear nature, this study proposes:

γ = γ₁ϕ + γ₂,   0 ≤ ϕ ≤ π/2

(10)

where γ₁ and γ₂ are material parameters, and their values are determined as γ₁ = −0.0628 and γ₂ = 0.4843. Figure 14a–d illustrate the experimental and theoretical κ_c/κ_o-N_b relationships for LSN C2700 brass circular tubes under cyclic bending with varying ϕ values: 0°, 30°, 60°, 90°, and different a on double logarithmic coordinates. The comparison results demonstrate that the theoretical predictions reasonably describe the experimental results.

4. Conclusions

This study primarily investigates the response and local buckling of LSN C2700 brass circular tubes at different ϕ of 0°, 30°, 60°, and 90° and different a of 0.2, 0.4, 0.6, 0.8, and 1.0 mm under cyclic bending loads. The behavior of LSN SS304 circular tubes with different ϕ and a under cyclic bending tested by Lee et al. [29] is compared. Based on tested, analyzed, and compared data, the following conclusions can be drawn:

(1)

When local buckling has not yet occurred, it is observed from the experimental M-κ relationships that as κ increases, M also increases. Under cyclic bending loads, the M-κ relationship exhibits a stable loop until eventual local buckling failure. Additionally, results indicate that for different values of ϕ and a, the M-κ relationships show similar loops. This phenomenon is consistent with the experimental results reported by Lee et al. [29].

(2)

When local buckling has not yet occurred, it is observed from the experimental ΔD/D_o-κ relationships that higher κ leads to increased ΔD/D_o. Larger a or smaller ϕ values correspond to increased ΔD/D_o. This phenomenon is consistent with the experimental results of Lee et al. [29]. In addition, under cyclic bending loads, the ΔD/D_o-κ relationships exhibit a ratcheting, symmetrical, and growing trend. A less consistent observation with the experimental results of Lee et al. [29] is that when a is small, the ΔD/D_o-κ relationships are symmetrical, but when a is large, the ΔD/D_o-κ relationships are asymmetrical.

(3)

The experimental κ_c/κ_o-N_b relationships reveal that, when ϕ and a are fixed, larger κ_c/κ_o leads to fewer N_b. Fixing ϕ and κ_c/κ_o, larger a results in fewer N_b. Conversely, when fixing a and κ_c/κ_o, larger ϕ leads to more N_b. When ϕ is fixed, the relationships of κ_c/κ_o-N_b for five different a on double logarithmic coordinates exhibit nearly straight lines. This phenomenon is consistent with the experimental results of Lee et al. [29]. However, for a certain ϕ, they show almost-parallel lines, while we found non-parallel lines.

(4)

This study adopts the formula (Equation (3)) proposed by Kyriakides and Shaw [1] to describe the κ_c/κ_o-N_b relationships. The formulation of the material parameter C in Equation (4) proposed by Lee et al. [29] was used in this study. However, the material parameter m (Equation (5)) was proposed based on the experimental data. Additionally, a new equation for the material parameter α is proposed (Equation (8)), where material parameters β and γ are derived from Equations (9) and (10), respectively. Note that Equations (5), (9) and (10) are restricted to use between 0° and 90°. Finally, the theoretical Equations (Equations (3)–(5) and (7)–(10)) are employed to describe the κ_c/κ_o-N_b relationships for LSN C2700 brass circular tubes under cyclic bending with varying ϕ values: 0°, 30°, 60°, 90°, and different a (dashed lines in Figure 14a–d), and the results of experimental and theoretical analyses are reasonably consistent.