Response of round-hole tubes submitted to pure bending creep and pure bending relaxation

This paper presents experimental study on the response of 6061-T6 aluminum alloy round-hole tubes with five different hole diameters of 2, 4, 6, 8, and 10 mm and four different diameter-to-thickness ratios of 30, 40, 50, and 60 submitted to pure bending creep and pure bending relaxation. Pure bending creep or relaxation is defined as bending the tube to the required moment or curvature and maintaining that moment or curvature for a period of time. The experimental results of pure bending creep show that the curvature increases with time. In addition, larger holding moment, diameter-to-thickness ratio, or hole diameter results in larger creep curvature. As the curvature continues to increase, the round-hole tube eventually breaks. The experimental results of pure bending relaxation show that the relaxation moment decreases sharply with time and tends to a stable value. In addition, larger holding curvature, diameter-to-thickness ratio, or hole diameter results in larger drop of the relaxation moment. Due to fixed curvature, the round-hole tube does not break. Finally, formulas proposed by the research team of Pan et al. were respectively improved to simulate the creep curvature-time relationship for pure bending creep in the initial and the secondary stages and the relaxation moment-time for pure bending relaxation. After comparing with the experimental results, it is found that theoretical analysis can reproduce the experimental results reasonably.


Introduction
Round-hole tube (abbreviation: RHT) refers to a tube with a round hole, as shown in Figure 1. RHT is usually used for connecting components of bicycles, motorcycles, or automobiles. When RHT submits to a bending load, the bending rigidity of RHT will progressively reduce as the amount of bending increases, which is called a phenomenon of deterioration. RHT will encounter fracture when the curvature reaches a certain critical value.
The study of smooth tube under bending has achieved plentiful results. In 1987, Kyriakides et al. set up a mechanical device that can perform cyclic bending tests on tubes of various materials (1020 steel, 1018 steel, 304 stainless steel, 6061-T6 aluminum alloy, and NiTi). They have carried out many experimental and theoretical studies on bending with or without internal or external pressure. [1][2][3][4][5][6][7][8][9] Several other researchers have also published related results. Elchalakani et al. 10 determined the slenderness limits of cold-formed CHS for fully ductile section by conducting cyclic bending tests with variable amplitudes. Mathon and Liman 11 experimentally studied the pure bending collapse of cylindrical thin shells. Elchalakani and Zhao 12 investigated the cyclic bending behavior of cold-formed steel tubes with concrete filled. Yazdani and Nayebi 13 examined the damage of pipes that undergo periodic bending with a stable internal pressure. Shariati et al. 14 experimentally discussed the cyclic bending behavior of SS316L cantilevered cylindrical shells. Elchalakani et al. 15 used the strain detected in bending test to find a new ductility slender limit for the CFT structural plastic designs. Li and Wang 16 investigated the stability of single-layer cable-stiffened latticed shells submitted to earthquake motions. Chegeni et al. 17 studied the influence of the shape and depth of corrosion on the performance of a pipe that is bent under a certain internal pressure.
In 1998, Pan et al. started a series of experimental and theoretical investigations on behavior of tubes under monotonic or cyclic bending with various loading paths. Pan et al. 18 invented and constructed a new measurement equipment to measure the ovalization (change of the outer diameter divides by the outer diameter, DD o /D o ) and curvature (k) of tubes submitted to bending. They confirmed the function of the equipment by conducting tubes under cyclic bending. Lee et al. 19 tested the stability of 304 stainless steel tubes with four diameter-to-thickness ratios (D o /t ratios) submitted to cyclic bending. They found that the log k-log N b (number of cycles required to initiate buckling) relationships demonstrate four parallel lines. Lee and Pan 20 examined the pure bending creep behavior of 304 stainless steel tubes with four D o /t ratios. By modifying the Bailey-Norton law for uniaxial creep, the creep curvature-time relationships of the first two stages were simulated. Pan and Lee 21 explored the mean curvature effect on the behavior of tubes undertaken cyclic bending. Different curvature ratios of 21.0, 20.5, and 0 were considered in their experimental investigation. Chang and Pan 22 experimentally explored cyclic bending instability of tubes with four D o /t ratios. They introduced a new formula for estimating tube's buckling life.
In 2010, Pan et al. started to experimentally and analytically investigate the behavior of tubes with a circumferential notch submitted to bending. Lee et al. 23 examined the changes of DD o /D o for circumferential sharp-notched tubes submitted to cyclic bending. The DD o /D o -N (number of cycles) curve was identified into three different growing stages. Later, Lee 24 investigated cyclic bending response of circumferentially sharp grooved tubes with different depths. A theoretical formula for estimating N b was proposed. Thereafter, Lee et al. 25 examined the stability of circumferential sharpnotched 304 stainless steel tubes submitted to cyclic bending at different curvature rates (viscoplastic response). They employed three curvature rates of 0.35, 0.035, and 0.0035 m 21 /s to demonstrate the viscoplastic response. Lee et al. 26 inspected the pure bending creep behavior and pure bending relaxation behavior of tubes with a circumferential sharp notch. The formula proposed by Lee and Pan 20 was improved for describing the creep curvature-time and relaxation moment-time relationships. Chung et al. 27 investigated the cyclic bending collapse of circumferential sharp-notched 6061-T6 aluminum alloy tubes. The k-N b relationship exhibited considerable differences from that discovered by Lee 24 in circumferential sharp-notched 304 stainless steel tubes subjected to cyclic bending.
In 2016, Lee et al. 28 explored the mechanical behavior of local sharp-cut 6061-T6 aluminum alloy tubes subjected to cyclic bending. In their study, the ANSYS was employed to simulate the DD o /D o -k and M (moment)-k relationships. After that, Lee et al. 29 researched the failure of local sharp-grooved 304 stainless steel tubes submitted to cyclic bending. The M-k relationships were very similar to that obtained by Lee et al. 25 ; but, the contours of the DD o /D o -k relationships were completely different. Lee et al. 30 inspected the deterioration and failure of local sharp-dented 6061-T6 aluminum alloys tubes submitted to cyclic bending. The dent was created by contacting the mold on the tube's surface. In addition, a theoretical formula was put forward to express the k-N f (number of cycles required to initiate failure) relationship.
In this study, the response of 6061-T6 aluminum alloy round-hole tubes (abbreviation: Al-RHTs) with different hole diameters of 2, 4, 6, 8, and 10 mm, and different diameter-to-thickness ratios of 30, 40, 50, and 60 submitted to pure bending creep and pure bending relaxation are examined. In the experimental part, a bending machine was employed to perform relative experiments on Al-RHTs. The magnitudes of M and k were obtained by the measuring devices on the machine. Additionally, time ( t) was also recorded. In the theoretical part, the formulas proposed by Lee and Pan 20 and Lee et al. 26 were respectively improved to simulate the creep curvature (k c )t relationships for pure bending creep in the initial and secondary stages and the relaxation moment (M r )t relationships for pure bending relaxation.

Experimental equipment
The experiments executed by a tube-bending machine. Figure 2(a) and (b) respectively show a schematic diagram and a picture of the machine. It was designed to perform a pure bending test (four-point bending test), which can apply monotonic and cyclic bending. The machine consists of two rotating sprockets, placed symmetrically on two support beams. Two sprockets support two rollers, which apply point loads at each end of the tube. The chains run around these sprockets and are connected to two hydraulic cylinders and load cells to form a closed loop. Once the top or bottom cylinder is contracted, the sprocket will rotate to achieve pure bending of the tube. The load transferred to the tube is the moment formed by the concentrated load of the two rollers. For a detailed description of the tubebending machine, please refer to the following papers (Lee et al., 19 Lee and Pan, 20 and Pan and Lee 21 ).   18 When the tube is bent, the two side-inclinometers in the device is able to detect the tube's angle changes. The k can be easily calculated according to angle changes. In addition, the central part of the equipment can be used to monitor the change in the diameter of the tube cross section using a magnetic detector. Thus, the DD o /D o can be measured. For a detailed explanation of the apparatus, please refer to the Pan et al. 18 study.
Smooth 6061-T6 aluminum alloy tubes with D o = 36.0 mm and t = 3.0 mm were machined on the outer surface to acquire the desired D o /t ratios of 30, 40, 50, and 60, as shown in Figure 4. However, the inner radius (29.0 mm) of all tested Al-RHTs was unchanged. Next, the tube with every D o /t ratio was drilled again to obtain the Al-RHT with the required hole diameter d. In this study, five d values were considered, 2, 4, 6, 8, and 10 mm.
Because the hole is local, the hole direction f ( Figure 5) may also affect the RHT's behavior. Since the degradation and failure of the RHT undergoing bending is the most   severe when the bending moment direction (z direction) is perpendicular to the f. Therefore, this study only considers f = 0°(y direction).

Test procedures
The experiment includes pure bending creep (bend the tube to the required M and hold that M for a period of time) and pure bending relaxation (bend the tube to the required k and hold that k for a period of time). For each Al-RHT with certain D o /t and d, different M and k were respectively held for pure bending creep and pure bending relaxation. The M and k were measured and controlled by the related equipment. Furthermore, the t was also recorded.

Experimental results, simulated results, and discussion
Experimental response of Al-RHTs under pure bending   Figure 8 depicts the k ct relationship when the held moment is 150 N m. It can be seen that the k ct curve can be divided into three stages, which are called in this paper to be the initial stage, secondary stage, and tertiary stage. Once pure bending creep begins, the amount of k increases, which is called the initial stage. When _ k c gradually decreases, pure bending creep enters the second stage, which is called the secondary stage. The k c increases steadily with t and _ k c is approximate a constant in this stage. The third stage is called the tertiary stage. The _ k c increases sharply and the Al-RHT ruptures in this stage. In addition, most of pure bending creep time is spent in the initial and secondary stages.  Theoretical simulation for Al-RHTs under pure bending creep In 2002, Lee and Pan 20 tested smooth 304 stainless steel tubes with different D o /t ratios submitted to pure bending creep. They proposed a theoretical formulation to describe the k ct relationships in the initial and secondary stages as in which M is the held moment, C, m 1 , and n 1 are material parameters. In their study, m 1 and n 1 were respectively determined to be 5.70 and 0.53 and C was a function of the D o /t ratio. In 2014, Lee et al. 26 studied the k ct relationships for circumferential sharp-notched 304 stainless steel tubes submitted to pure bending creep. Due to the same material and similar trend of the k ct relationships, equation (1) was also used as the theoretical formulation. In their study, m 1 and n 1 were also used the same values, and C was proposed to be a function of circumferential sharp notch depth.
In this study, the k ct relationships for Al-RHTs with different D o /t ratios and different d under pure bending creep are similar to that of smooth 304 stainless steel tubes with different D o /t ratios submitted to pure bending creep tested by Lee and Pan. 20 Therefore, equation (1) is used in this investigation. However, the material parameter C is proposed to relate to D o /t ratios and d. As for material parameters m 1 and n 1 , they can be determined from smooth 6061-T6 aluminum alloy tubes subjected to pure bending creep. Figure 9 demonstrates the experimental k ct in the initial and secondary stages for smooth 6061-T6 aluminum alloy tubes subjected to pure bending creep. Note that the smooth 6061-T6 aluminum alloy tubes are with D o = 36.0 mm and t = 3.0 mm. By curve fitting with equation (1), the values of m 1 and n 1 are respectively obtained to be 1.086 and 0.046. Figure 10(a) to (e) respectively indicate the experimental k ct curves (dotted lines) of Al-RHTs with D o / t = 30 and d = 2, 4, 6, 8, and 10 mm under pure bending creep in the initial and secondary stages, Figure  11 Table 1.
When we consider the lnC and d/t relationship, four straight lines for D o /t = 30, 40, 50, and 60 are observed (see Figure 14). Thus, the parameter C can be proposed as where c 1 and c 2 are material parameters which are related to D o /t ratio. The amounts of c 1 and c 2 can be determined in Figure 14.
From Figure 15 where a 1 , a 2 , a 3, and a 4 are material parameters which are respectively determined from Figure 15  found that the moment of the tube drops sharply when the relaxation begins and approaches to a stable value. In addition, a larger held curvature results in a larger drop of the relaxation moment. Due to fixed curvature, the Al-RHT does not break.

Theoretical simulation for Al-RHTs under pure bending relaxation
In 2014, Lee et al. 26 tested circumferential sharpnotched 304 stainless steel tubes submitted to pure bending relaxation. They proposed a theoretical formula to describe the relationship between relaxation moment (M r ) and t to be where k is the held curvature, R, m 2 , and n 2 are material parameters. In their study, m 2 and n 2 were respectively determined to be 0.15 and 20.07 and the parameter R was proposed as a function of notch depths for circumferential sharp-notched 304 stainless steel tubes. In this study, the M rt relationships for Al-RHTs with different D o /t ratios and different d under pure bending relaxation are similar to that of circumferential sharp-notched 304 stainless steel tubes subjected to pure bending relaxation tested by Lee et al. 26 Therefore, equation (5) is used in this investigation. However, the material parameter R is proposed to relate to the D o /t and d. As for material parameters m 2 and n 2 , they can be determined from smooth 6061-T6 aluminum alloy tubes subjected to pure bending relaxation. Figure 17 depicts experimental Mt curves (dotted lines) for smooth 6061-T6 aluminum alloy tubes under monotonic pure bending followed by pure bending relaxation. Note that the relaxation starting points are marked by '''' In addition, the smooth 6061-T6 aluminum alloy tubes are with D o = 36.0 mm and t = 3.0 mm. By curve fitting of equation (5) Table 2.
When we consider the R 1/3 -d/t relationship, four straight lines for D o /t = 30, 40, 50, and 60 are observed (see Figure 22). Therefore, parameter R is proposed to be where r 1 and r 2 are material parameters which are related to D o /t ratio. From Figure 23(a) and (b), linear relationships of lnr 1 -lnD o /t and lnr 2 -lnD o /t are observed. Therefore, parameters r 1 and r 2 are respectively proposed to be

Conclusions
This paper presents the experimental and theoretical research on the response of Al-RHTs with different D o / t ratios and different d submitted to pure bending creep and pure bending relaxation. Based on experimental and theoretical results, this paper draws the following important conclusions: (1) It is seen from the M-k curves for monotonic bending that once the deformation is in the plastic range, the M-k relationship becomes nonlinear. It can also be observed that the larger the D o /t ratio, the smaller the tube wall thickness. Therefore, a lower M value is found. In addition, for each D o /t ratio, the M-k curves are similar for different d. However, a smaller d leads to a larger breaking curvature. (2) It is found from pure bending creep that once the pure bending creep begins, the curvature increases quickly. In addition, the k ct relationship can divide into three stages. Most of the time for pure bending creep is spent in the initial and secondary stages. For a certain D o /t ratio under a constant moment for pure bending creep, a smaller d leads to a faster increasing and larger k c . As the curvature continues to increase, Al-RHT eventually breaks. (3) It is observed from pure bending relaxation that once the relaxation begins, the moment drops sharply and gradually becomes a stable value. In addition, the M rt relationship is similar to the reverse k ct relationship. For a certain D o /t ratio under a constant curvature for pure bending relaxation, a smaller d leads to a larger drop of the M r . Due to the fixed curvature, the Al-RHT does not break.

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 is funded by Ministry of Science and Technology under grant MOST 108-2221-E-006-183. We are very grateful for their support.