Experimental Study of Cyclic Bending Failure for Round-hole Tubes with a Redundant Round Hole in the Same Direction but Different Cross Sections

This paper presents the influence of a redundant round hole in the same direction but different cross sections on the response and failure of round-hole 6061-T6 aluminum alloy tubes subjected to cyclic bending. In this study, round-hole 6061-T6 aluminum alloy tubes with a constant hole diameter of 6 mm were drilled to obtain a redundant round hole in the same hole direction but different cross sections. The experimental results revealed that the moment–curvature relationship exhibited an almost steady loop from the beginning of the first cycle. The redundant round hole showed minimal influence on the moment–curvature relationship. However, the ovalization– curvature relationship demonstrated an asymmetrical, increasing, ratcheting and bow pattern along with the bending cycle, while the redundant round hole showed a significant influence on this relationship. In addition, six groups of round-hole 6061-T6 aluminum alloy tubes were tested, the controlled curvature-number of bending cycles required to ignite failure relationships on double logarithmic coordinates exhibited nonparallel straight lines. Finally, a theoretical model was proposed for simulating the controlled curvature–number of cycles to ignite failure. The simulation result was compared with experimental test data, which showed generally good agreement.


Introduction
A round-hole tube is a tube with a circular hole drilled through, as shown in Fig. 1. Round-hole tubes are often used as connections in parts of cars, locomotives, or bicycles. When a round-hole tube is submitted to a bending load, the circular cross section will gradually change to an elliptical shape with the increase of the bending degree. This will cause the bending rigidity of the tube to gradually decrease, which is called degradation phenomenon. A physical quantity called ovalization is employed to quantify the degradation phenomenon, which is the reduction of the outer diameter (ΔD o = D o −D) divided by the original outer diameter (D o ), as shown in Fig. 2. In cyclic bending loads, the ovalization increases with the number of bending cycles. Finally, when a certain number of bending cycles is reached and the ovalization reaches a certain critical value, the round-hole tube will undergo fracture failure.  In 1998, Pan et al. began a series of experimental and theoretical studies on tubes subjected to bending or cyclic bending with different loading and geometry conditions. Pan and Her [1] experimentally studied the curvature rate effect on the response and collapse of SUS304 stainless steel tubes submitted to cyclic bending and discovered that increasing the curvature rate increased the degree of hardening. Later, Pan and Fan [2] studied the effect of the prior curvature-rate at the preloading stage on subsequent creep (moment is kept constant for a period of time) or relaxation (curvature is kept constant for a period of time) behavior. Lee et al. [3] tested thin-walled tubes with different diameter-to-thickness ratios subjected to cyclic bending. Four groups of specimens, each with a distinct diameter-to-thickness ratio, were tested, and four parallel straight lines were obtained when the controlled curvature-number of bending cycles required to ignite buckling relationships were plotted on double logarithmic coordinates. Pan and Lee [4] examined the effect of the mean curvature on the response of tubes submitted to cyclic bending; this effect was investigated by considering three different curvature ratios (minimum curvature/ maximum curvature). Chang et al. [5] studied the viscoplastic cyclic bending response and collapse of 316L stainless steel tubes; in their research, the endochronic theory and the principle of virtual work were used to simulate the relevant behaviors. Chang and Pan [6] conducted cyclic bending experiments on the deterioration and buckling of circular tubes with different outer diameters. This investigation yielded a new formula for predicting the buckling life of circular tubes submitted to cyclic bending.
In 2010, Pan et al. began an experimental and analytical investigation on the cyclic bending response and failure of notched tubes. Lee et al. [7] examined the cyclic bending variations in ovalization of sharp-notched SUS304 stainless steel tubes. Later, Lee [8] investigated the cyclic bending collapse of circular tubes with different notch depths; this work proposed an empirical formula for predicting the number of bending cycles required to ignite buckling. Thereafter, Lee et al. [9] experimentally inspected the cyclic bending response and collapse of sharpnotched SUS304 stainless steel tubes subjected to cyclic bending with different curvature rates. Three different curvature rates, 0.003, 0.03 and 0.3 m -1 s -1 , were used to highlight the related behavior. Lee et al. [10] studied the pure bending creep and relaxation response of sharp-notched SUS304 stainless steel tubes. The Bailey-Norton law was modified for simulating the creep curvature-time relationships and relaxation moment-time relationships. Chung et al. [11] explored the cyclic bending response and buckling of sharp-notched 6061-T6 aluminum alloy tubes. An empirical formulation was proposed for describing the controlled curvature-number of bending cycles required to ignite buckling relationships. However, all investigations of notched tubes were focused on the circumferential sharp notch. Later, Lee et al. [12] examined the cyclic bending behavior of local sharp-notched 6061-T6 aluminum alloy tubes. The sharp notch was a local sharp cut. In addition, the moment-curvature and ovalization-curvature relationships were analyzed using finite-element ANSYS. Thereafter, Lee et al. [13] explored the cyclic bending response and failure of local sharp-notched SUS304 stainless steel tubes. The sharp notch was a local sharp groove. The moment-curvature relationships were similar to those tested by Lee et al. [9], however, the ovalization-curvature relationships were quite different.
In the present study, round-hole 6061-T6 aluminum alloy tubes with a redundant round hole in the same hole direction (x-direction) but different cross sections subjected to cyclic bending were experimentally investigated (Fig. 3). The round hole diameter of the round-hole 6061-T6 aluminum alloy tubes were fixed of 6 mm. The redundant round hole was drilled in the same direction but different cross section with a hole diameter of 2 mm. A tube-bending machine and curvature-ovalization measurement apparatus were employed to conduct curvaturecontrolled cyclic bending tests. The quantities of moment, curvature, and ovalization were measured by the testing devices. Additionally, the number of bending cycles required to ignite failure was also recorded.

Experiments
The tube-bending machine and the curvature-ovalization measurement apparatus were employed to conduct curvature-controlled cyclic bending tests on round-hole 6061-T6 aluminum alloy tubes with a redundant round hole. The bending device, measurement apparatus, materials, specimens, and test procedures are stated below.

Material and specimens
Round-hole 6061-T6 aluminum alloy tubes with a redundant round hole were used for the cyclic bending tests. The original 6061-T6 aluminum alloy tube with D o = 35.0 mm and t = 3.0 mm was drilled to obtain a roundhole tube with a hole diameter of 6 mm. Next, a redundant round hole with a hole diameter of 2 mm in the same direction (x-direction) but different cross section was drilled. Fig. 6 demonstrates the dimensional drawing of the round-hole tube with a redundant hole at different cross sections. The round hole of the round-hole tube is on the A-A cross section. The round hole direction is in y-direction (fixed). However, the redundant round hole is on the B-B cross section. The redundant round hole direction is in x-direction (fixed). In this study, six horizontal distances (X), 0, 10, 0, 30, 40, and 50 mm from the A-A cross section, were considered. Note that X = 0 mm means the redundant round hole is on the A-A cross section.

Test procedures
The experiments were curvature-controlled cyclic bending tests with a constant curvature rate of 0.03 m −1 s −1 . The moment was measured by two load cells settled in the tube-bending machine (Fig. 4). The curvature and ovalization were measured by the COMA in Fig. 5. Simultaneously, the number of cycles required to ignite failure was also recorded.   Fig. 9 shows the experimental result of normalized controlled curvature (κ c /κ o ) versus number of bending cycles required to ignite failure (N f ) for round-hole 6061-T6 aluminum alloy tube with a redundant round hole at X = 0, 10, 20, 30, 40, and 50 mm subjected to cyclic bending. Note that the definition of the κ o is t/D o 2 (Kyriakides and Shaw [15]). Subsequently, the experimental data in Fig. 9 were plotted on double logarithmic coordinates, and six nonparallel straight lines were observed in Fig. 10. Note that the straight lines were determined by the least square method.  Lee et al. [16] proposed an empirical formulation for simulating the relationship between κ c /κ o and N f of roundhole 6061-T6 aluminum alloy tubes subjected to cyclic bending to be:

Failure
where C and α are the material parameters. The C value is the quantity of κ c /κ o by letting N f = 1, while the α value is the slope of the line on double logarithmic coordinates. Fig. 11 demonstrates the six values of C for X = 0, 10, 20, 30, 40 and 50 mm in orange solid circles. It was observed that two linear relationships were respectively found for X = 0, 10, 20, 30 mm and X = 30, 40, 50 mm. Therefore, the C-X relationship was proposed as: And where c 1 , c 2 , c 3 and c 4 are materal constants. In this study, the magnitudes of c 1 , c 2 , c 3 and c 4 were respectively determined in Fig. 11 to be 0.0056, 0.8233, -0.0031 and 1.0927. that two linear relationships were respectively found for X = 0, 10, 20, 30 mm and X = 30, 40, 50 mm. Therefore, the α-X relationship was proposed as: and α = a 3 X + a 4 30 mm ≤ X ≤ 50 mm (5) where a 1 , a 2 , a 3 and a 4 are materal constants. In this study, the magnitude of a 1 , a 2 , a 3 and a 4 were respectively determined in Fig. 12

Conclusions
This study investigated the influence of a redundant round hole in the same direction but different cross sections on the response and failure of round-hole 6061-T6 aluminum alloy tubes subjected to cyclic bending. According to the experimental and theoretical results, the following conclusions were drawn: 1) The experimental moment-curvature relationship for round-hole 6061-T6 aluminum alloy tubes with a redundant round hole in the same direction but different cross sections subjected to cyclic bending displayed a closed and stable hysteresis loop from the first cycle. The redundant round hole was small and local. Hence, different cross sections had almost no influence on the moment-curvature relationship.  2) The experimental ovalization-curvature relationship for round-hole 6061-T6 aluminum alloy tubes with a redundant round hole in the same direction but different cross sections subjected to cyclic bending revealed an asymmetrical, increasing, ratcheting, and bow trend with the increase in the number of cycles. The redundant round hole showed a significant influence on the ovalization-curvature relationships. In addition, a redundant round hole at X = 0 mm had the largest ovalization. However, a redundant round hole at X = 40 mm had the smallest ovalization.
3) The empirical formula of Eq. (1) was used to simulate normalized controlled curvature-number of bending cycles required to ignite failure relationships for round-hole 6061-T6 aluminum alloy tubes with a redundant round hole in the same direction but different cross sections subjected to cyclic bending. According to the experimental data in Fig. 11. parameter C was proposed in Eqs. (2) and (3). In addition, according to the experimental data in

Acknowledgement
This study was conducted under the authority of the project MOST 108-2221-E-006-183, supported by the Ministry of Science and Technology. Here, we are very grateful for the support provided by the Ministry.