Response of sharp-notched circular tubes under bending

1Paper No.13-00142

© 2014 The Japan Society of Mechanical Engineers[DOI: 10.1299/mej.2014smm0005 ]

Vol.1, No.2, 2014Bulletin of the JSME

Mechanical Engineering Journal

* Department of Innovative Design and Entrepreneurship Management, Far East University No. 49 Chung Hua Rd., Hsin-Shih Dist., Tainan 744, Taiwan, R.O.C.

** Department of Interior Design, Tung Fang Design University No.110, Dongfang Rd., Hunei Dist., Kaohsiung 829, Taiwan, R.O.C.

*** Department of Engineering Science, National Cheng Kung University No.1, University Rd., East Dist.,Tainan 701, Taiwan, R.O.C.

E-mail: [email protected]

Abstract In this paper, the response of sharp-notched circular tubes with notch depths of 0.2, 0.4, 0.6, 0.8 and 1.0 mm subjected to bending creep and relaxation are investigated. The bending creep or relaxation is to bend the tube to a desired moment or curvature and hold that moment or curvature constant for a period of time. From the experimental result of bending creep, the creep curvature and ovalization increase with time. In addition, higher held moment leads to the higher creep curvature and ovalization of the tube’s cross-section. From the experimental result for bending relaxation, the bending moment rapidly decreases with time and becomes a steady value. As for the ovalization, the amount increases a little with time and gradually becomes a steady value. Due to the constant ovalization caused by the constant curvature under bending relaxation, the tube does not buckle. Finally, the formulation proposed by Lee and Pan (2002) is modified for simulating both the creep curvature–time relationship in the first stage under bending creep and the relaxation moment-time relationship under bending relaxation for sharp-notched circular tubes with different notch depths. Through comparing with the experimental finding, the theoretical analysis can reasonably describe the experimental result.

Keywords

1. Introduction

The circular tube components, which are used in several industrial applications such as offshore pipelines,

platforms in offshore deep water, tubular components in nuclear reactors, etc., are often subjected to bending. It is well known that the ovalization of the tube cross-section is observed when a circular tube is subjected to bending. If the bending moment or curvature increases, the ovalization also increases. Increase in ovalization causes a progressive reduction in the bending rigidity, which can result in buckling of the tube components. Therefore, the study of the response of circular tubes subjected to bending is of importance in many industrial applications.

In early stage, Kyriakides and his co-workers designed and constructed a tube cyclic bending machine, and conducted a series of experimental and theoretical investigations. Shaw and Kyriakides (1985) investigated the inelastic behavior of 6061-T6 aluminum and 1018 steel tubes subjected to cyclic bending. Kyriakides and Shaw (1987) extended the analysis of 6061-T6 aluminum and 1018 steel tubes to the stability conditions under cyclic bending. Corona and Kyriakides (1988) investigated the stability of 304 stainless steel tubes subjected to combined bending and external pressure. Corona and Vaze (1996) studied the response, buckling and collapse of long, thin-walled seamless steel square tubes under bending. Vaze and Corona (1998) experimentally investigated the elastic-plastic degradation and collapse of steel tubes with square cross-sections under cyclic bending. Corona et al. (2006) used a set of bending experiments to conduct on aluminum alloy tubes for investigating the yield anisotropy effects on the buckling.

Response of sharp-notched circular tubes under bending creep and relaxation

Received 12 July 2013

: Sharp-notched circular tubes, Bending creep, Creep curvature, Creep ovalization, Bending relaxation, Relaxation moment, Relaxation ovalization

Kuo-long LEE*, Chien-min HSU** and Wen-fung PAN***

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Lee, Hsu and Pan, Mechanical Engineering Journal, Vol.1, No.2 (2014)


Kyriakides et al. (2006) studied the plastic bending of steel tubes with the diameter-to-thickness ratio (Do/t ratio) of 18.8 exhibiting Lüders bands through the experiment. Limam et al. (2008) investigated the plastic buckling and collapse of 321 stainless steel tubes with the Do/t ratio of 52 under bending and internal pressure. Sakakibara et al. (2008) studied the effect of internal corrosion or erosion defects on the collapse of pipelines under external pressure.

In addition, Elchalakani et al. (2002) experimentally conducted tests on the different Do/t ratios of grade C350 steel tubes under pure bending, and proposed two theoretical simulation models. Jiao and Zhao (2004) tested the bending behavior of very high strength (VHS) circular steel tubes, and proposed their plastic slenderness limit. Elchalakani et al. (2004) experimentally investigated the inelastic flexural behavior of concrete-filled tubular (CFT) beams, which were made of cold-formed circular hollow sections filled with normal concrete. Elchalakani et al. (2006) conducted the variable amplitude cyclic pure bending tests to determine fully ductile section slenderness limits for cold-formed circular hollow section (CHS) beams. Elchalakani and Zhao (2008) investigated the concrete-filled cold-formed circular steel tubes subjected to variable amplitude cyclic pure bending.

Recently, Pan and his co-workers also constructed a similar bending machine with a newly invented Curvature-Ovalization Measurement Apparatus (COMA), which was designed and set up by Pan et al. (1998) to study various kinds of tubes under different cyclic bending conditions. Such as Pan and Fan (1998) studied the effect of the curvature-rate at the preloading stage on the response of subsequent creep or relaxation, Pan and Her (1998) investigated the response and stability of circular tubes subjected to cyclic bending with different curvature-rates, Lee et al. (2001) studied the influence of the Do/t ratio on the response and stability of circular tubes subjected to symmetrical cyclic bending, Lee et al. (2004) experimentally explored the effect of the Do/t ratio and curvature-rate on the response and stability of circular tubes subjected to cyclic bending, Chang et al. (2008) studied the influence of the mean moment effect on circular thin-walled tubes under cyclic bending, Chang and Pan (2009) discussed the buckling life estimation of circular tubes subjected to cyclic bending.

In practical industrial applications, tubes are under the sea, so salt water can corrode the tube surface and produce notches. The mechanical behavior and buckling failure of a notched tube differs from that of a tube with a smooth surface. It is known that to understand the related material’s response, different loading cases have to be experimentally conducted on materials such as uniaxial loading, torsional loading, bending, creep, fatigue, …. Similarly, to completely understand the response of sharp-notched circular tube under bending, diverse loading paths have to be experimentally investigated. Beginning in 2010, Lee et al. (2010) studied the variation in ovalization of sharp-notched circular tubes subjected to cyclic bending. Lee (2010) investigated the mechanical behavior and buckling failure of sharp-notched circular tubes under cyclic bending. Lee et al. (2012) researched the influence of diameter-to-thickness ratios on the response and collapse of sharp-notched circular tubes under cyclic bending. Lee et al. (2013) experimentally discussed the viscoplastic response and collapse of sharp-notched circular tubes subjected to cyclic bending.

However, the response of notched circular tube subjected to bending creep or relaxation has not been investigated. The bending creep is to bend the tube to a desired moment and hold that moment constant for a period of time, and the bending relaxation is to bend the tube to a desired curvature and hold that curvature constant for a period of time. It is known that the bending creep leads to the collapse of the tube, but the bending relaxation will produce the only moment relaxation. In addition, their responses are time-related. Faced with such complex behaviors, experimental and theoretical analysis is necessary. Although Pan and Fan (1998) studied the effect of the prior curvature-rate at the preloading stage on the subsequent creep or relaxation behavior and Lee and Pan (2002) investigated the response of circular tubes with different Do/t ratios subjected to bending creep, all results are considered for the smooth circular tubes without any notch.

In this study, the response of sharp-notched circular tubes subjected to bending creep and relaxation are discussed. For experimental aspect, a four-point bending machine was used to conduct the experiment. For sharp-notched tubes, five different notch depths, 0.2, 0.4, 0.6, 0.8 and 1.0 mm, were considered. The magnitude of the bending moment was measured by two load cells mounted in the bending device, and the magnitudes of the curvature and ovalization of the tube’s cross-section were measured by COMA (Curvature- Ovalization Measurement Apparatus).

Although inelastic FEM analysis can be used for simulating the material’s creep behavior, it is very difficult to describe the creep curvature-time relationship for a tube subjected to bending creep. Because the bending creep is to hold a constant bending moment for a period of time, one has to calculate the stress distribution on the cross-section of the circular tube. Once the stress distribution is determined, according to the stress-strain relationship built by FEM software, the strain distribution on the cross-section of the circular tube can be obtained. Next, based on the strain

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distribution, the curvature can be calculated. However, the cross-section of the tube keeps changing during the bending creep (ovalization), thus, the aforementioned process has to repeat many small steps to obtain the complete creep curvature-time relationship.

In addition, Kyriakides and his co-workers used a group of Fourier series to describe the circumferential displacements of the tube (Shaw and Kyriakides, 1985; Kyriakides and Shaw, 1987; Corona and Kyriakides, 1988; Corona and Vaze, 1996). By combining the constitutive equations with the principle of virtual work, a system of nonlinear algebraic equations is determined. This system of equations can be solved by numerical method. Thus, the relationship among the bending moment, curvature and ovalization can be determined. After that, we employed the same theoretical process, but the constitutive equations were changed to the first-order ordinary differential constitutive equations of the endochronic theory. The response and collapse of tubes subjected to bending were investigated (Pan and Lee, 2002; Chang et al., 2005; Chang et al., 2008). However, those tubes studied were with a smooth surface. Once the sharp-notched circular tube is considered, the stress concentration will make the theoretical approach extremely difficult. Let alone the response and collapse for the bending creep or bending relaxation.

For theoretical aspect, due to the similar trend of the creep curvature–time relationship under bending creep for a sharp-notched circular tube and for a smooth circular tube, the formulation proposed by Lee and Pan (2002) is modified for simulating the aforementioned relationship. In addition, the relaxation moment-time relationship under bending relaxation is similar to the up-side-down creep curvature–time relationship under bending creep, and therefore, the formulation proposed by Lee and Pan (2002) is modified again for simulating the aforementioned relationship.

2. Experimental facility, material, specimens and test procedures

2.1 Bending device

Figure 1(a) shows a picture of the bending device and Fig. 1(b) is a schematic drawing of the bending device. It is designed as a four-point bending machine, capable of applying forward bending and reverse bending. The device consists of two rotating sprockets resting on two support beams. Heavy chains run around the sprockets and are connected to two hydraulic cylinders and load cells forming a closed loop. Each tube is tested and fitted with solid rod extension. The contact between tube and the rollers is free to move along axial direction during bending. The load transfer to the test specimen is in the form of a couple formed by concentrated loads from two of the rollers. Once either the top or bottom cylinder is contracted, the sprockets are rotated, and bending of the test specimen is achieved. Reverse bending can be achieved by reversing the direction of flow in the hydraulic circuit. Detail description of the bending device can be found in references of Kyriakides and Shaw (1985) and Lee et al. (2001).

(b)

(a)

(c) Fig. 1 (a) A picture of the bending device, (b) a schematic drawing of the bending device and (c) a schematic

drawing of the mechanism at the sprocket.

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The two sprockets (Fig. 1(c)) rest on two heavy support beams 1.25 m apart. This allows a maximum length of the test specimen to be 1 m. The bending capacity of the machine is 5300 N-m. Each tube is tested and fitted with a solid rod extension. The contact between the solid rod and the rollers is free to move along the axial direction during bending. The load transfer to the test specimen is a couple formed by concentrated loads from two of the rollers. The applied bending moment is directly proportional to the tension in the chains. Based on the signal from two load cells, the bending moment M exerted on the tube is calculated as

M = F R (1)

where F is the tension on the chain, which can be obtained from the pressure and area of the cylinder, and R is the radius of the sprocket (Fig. 1(c)). 2.2 Curvature-ovalization measurement apparatus (COMA)

The curvature-ovalization measurement apparatus (COMA), which was designed by Pan et al. (1998), is an instrument, which can be used to measure the tube curvature and ovalization of the tube cross-section. Figure 2(a) shows a picture of the COMA and Fig. 2(b) shows a schematic drawing of the COMA. It is a lightweight instrument, which can be mounted close to the tube mid-span. There are three inclinometers in the COMA. Two of them are fixed on two holders, which are denoted as side-inclinometers. The holders are fixed on the circular tube before the test begins. The distance between the two side-inclinometers is denoted as Lo. Let us now consider that the circular tube is subjected to bending, as shown in Fig. 2(c). The angle changes detected by two side-inclinometers are denoted as θ1 and θ2. From Fig. 2(c), the value of Lo is determined as

Lo = ρ (θ1 + θ2) (2)

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

κ = 1 / ρ = (θ1 + θ2) / Lo (3)

(a)

(b) (c)

Fig. 2 (a) A picture of the COMA, (b) a schematic drawing of the COMA and (c) the longitudinal deformation between two side-inclinometers under pure bending.

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In addition, a magnetic detector in the middle part of the COMA was used to measure the change in the outside diameter. For measuring the change in the outside diameter at the sharp-notched tube, the upper and lower magnetic blocks were set up with prominent and sharp tips on both magnetic blocks as shown in Fig. 3. The tips can touch the sharp- notched tube, thus, the change in the outside diameter at the sharp notch can be measured.

Fig. 3 Prominent and sharp tips on the upper and lower magnetic blocks.

2.3 Materials and specimens

Circular tubes made of SUS304 stainless steel were used in this study. Table 1 shows the proportion of the chemical composition. The ultimate stress, 0.2% strain offset the yield stress and the percent elongation are 626 MPa, 296 MPa and 35%, respectively. The raw smooth SUS 304 stainless steel tube had an outside diameter Do of 31.8 mm and wall-thickness t of 1.5 mm. The raw tubes were machined on the outside surface to obtain the desired notch depth a of 0.2, 0.4, 0.6, 0.8 and 1.0 mm. The notch width b was machined to be 1.0 mm for all notched tubes. The notch root radius for all tested tubes was controlled to be less than 1/100 mm and all tested tubes were carefully examined before the test. Figure 4 shows a schematic drawing of the sharp-notched tube.

Table 1 Chemical composition of SUS304 stainless steel.

Chemical Composition Fe Cr Ni Mn Si C P S

Proportion (%) 70.69 18.36 8.43 1.81 0.39 0.05 0.28 0.004

Fig. 4 A schematic drawing of the sharp-notched tube.

2.4 Test procedures The test involved bending creep and relaxation at room temperature. For each notch depth, different magnitudes of

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moment were held for bending creep and different magnitudes of curvature were held for bending relaxation. The curvature-rate of the bending stage was 0.003 m-1s-1. The magnitude of the bending moment was measured by two load cells mounted in the bending device. The magnitudes of the curvature and ovalization of the tube cross-section were measured by the COMA. In addition, the time for bending creep and relaxation was also recorded.

3. Experiment results and discussion 3.1 Experimental response of sharp-notched SUS304 stainless steel tubes under bending

Figure 5(a) shows the experimentally determined moment (M) - curvature (κ) curves for sharp-notched SUS304 stainless steel tubes with different a under bending. It can be seen that a tube with a higher value of a has a smaller wall-thickness. Thus, a lower magnitude of the moment is found when the tube bends to the maximum curvature. Figure 5(b) depicts the corresponding experimental ovalization of the tube’s cross-section (ΔDo/Do) as a function of the applied curvature (κ) for Fig. 5(a). The ovalization is defined as ΔDo/Do where Do is the outside diameter and ΔDo is the change in the outside diameter. It can be seen that the ovalization increases with the amount of curvature. In addition, higher a of the notch tube causes greater ovalization of the tube’s cross-section.

(a) (b)

Fig. 5 The experimental (a) moment (M) - curvature (κ) and (b) ovalization (ΔDo/Do) - curvature (κ) curves for sharp-notched SUS304 stainless steel tubes with different a under bending.

3.2 Experimental response of sharp-notched SUS304 stainless steel tubes under bending creep

Figure 6 depicts the experimental curvature (κ) – time ( t ) curves under bending and subsequent bending creep for sharp-notched SUS304 stainless steel tube with a = 0.2 mm. The starting and buckling points of the bending creep are marked by “●” and “×”, respectively. It can be seen that as soon as the creep starts the tube’s curvature quickly increases. Higher held moment leads to the higher creep curvature. Owing to the continuously increasing curvature, the tube buckles eventually. Note that the buckling time for M = 140 N-m is 1738 seconds.

We now consider the bending creep stage of Fig. 6 from the starting point “●” to the buckling point “×”. A typical result of the creep curvature (κc) – time ( t ) with the held moment of 155 N-m is shown in Fig. 7. It can be observed that the κc – t curve is similar to the creep strain-time curve for the uniaxial creep. However, the time for the bending creep is much shorter than that for the uniaxial creep. Therefore, the κc – t curve is now divided into two stages (first and second stages). The first stage is from the start of the bending creep to the end of the steadily increasing curve. The second stage is the final stage of the bending creep. In this stage, the κc increases drastically and buckling of the tube occurs. In addition, the starting point of the second stage is defined as the point which has 30% increase of the curvature-rate than that for the steadily increasing curve.

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Fig. 6 The experimental curvature (κ) – time ( t ) curves under bending and subsequent bending creep for sharp-notched SUS304 stainless steel tube with a = 0.2 mm.

Fig. 7 The creep curvature (κc) – time ( t ) curve with the held moment of 155 N-m for sharp-notched SUS304 stainless steel tube with a = 0.2 mm.

Figure 8 shows the corresponding experimental ovalization (ΔDo/Do) – time ( t ) curves under bending and

subsequent bending creep for Fig. 6. The starting and buckling points of the bending creep are also marked by “●” and “×”, respectively. It can be seen that the curves are strongly influenced by the held moment for creep stage. The ovalization-rate for higher held moment is larger than that for lower held moment. In addition, the ovalizations at buckling for these four magnitudes of held moment are different. Higher held moment leads to the higher ovalization at buckling. This phenomenon is different from the result tested by Lee and Pan (2002) for smooth SUS304 stainless steel tubes under bending creep. To highlight the influence of the sharp-notched depth, tubes with a = 0.4, 0.6, 0.8 and 1.0 mm were also tested. The phenomenon of the κ- t and ΔDo/Do- t curves under bending creep was found to be very similar to the phenomenon of tubes with a = 0.2 mm.

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Fig. 8 The corresponding experimental ovalization (ΔDo/Do) – time ( t ) curves under bending and subsequent bending

creep for sharp-notched SUS304 stainless steel tube with a = 0.2 mm.

3.3 Theoretical formulation for sharp-notched SUS304 stainless steel tubes under bending creep Material creep behavior of 304 stainless steel at room temperature has been experimentally and theoretically

studied for many years (Krempl, 1979; Ohashi et al., 1982; Xia and Ellyin, 1993; Sasaki and Ishikawa, 1995; Wu and Ho, 1995). However, to simulate the uniaxial strain-time relationship, the Bailey-Nortan Law was used, formulated as

εc = Cσm

tn (4)

where εc is the creep strain, σ is the hold stress for uniaxial creep and C, m, n are material parameters. By using different values of m and n, the uniaxial strain-time relationship for the primary, secondary and tertiary creep stages can be obtained.

Lee and Pan (2002) tested SUS304 stainless steel tubes subjected to bending creep and discovered that the creep curvature-time relationship was similar to the uniaxial strain-time relationship. Therefore, Eq. (4) was modified by them to describe the aforementioned relationship to be

κc = C1Mm

1 tn

1 (5) where M is the held moment, C1, m1 and n1 are material parameters for bending creep. In their study, the parameter C1 was proposed as a function of Do/t ratio of the SUS304 stainless steel tubes. In this study, due to the similar trend of the creep curvature-time relationship for a sharp-notched and smooth SUS304 stainless steel tube, Eq. (5) is used as the theoretical formulation. It can be seen that most of the time for bending creep is spent in the first stage, and the theoretical formulation is only applicable in this stage. Figures 9(a)-(e) in dotted lines demonstrate the experimental κc- t curves for sharp-notched SUS304 stainless steel tubes with a = 0.2, 0.4, 0.6, 0.8 and 1.0 mm under bending creep in the first stage, respectively.

Due to the same material and same loading condition tested by Lee and Pan (2002), the material parameters m1 and n1 are the same amounts as they proposed to be 5.70 and 0.53, respectively. In this study, the parameter C1 is a function of notch depth a and wall-thickness t. By curve fitting with the experimental data, the amounts of C1 for a/t = 0.133, 0.267, 0.400, 0.533 and 0.667 are 1.028 × 10-15, 5.027 × 10-15, 4.712 × 10-14, 4.232 × 10-13 and 8.591 × 10-12, respectively. If we consider the relationship between logC1 and a/t, a straight line can be used to describe the relationship (see Fig. 10). Therefore, parameter C1 is proposed to be

logC1 = b1 (a/t) + b2 (6) where b1 and b2 are material parameters for bending creep which can be determined from Fig. 10 to be 7.012 and -16.133, respectively. The simulated results for sharp-notched SUS304 stainless steel tube for a = 0.2, 0.4, 0.6, 0.8 and 1.0 mm under bending creep in first stage are demonstrated in Figs. 9(a)-(e) in solid lines, respectively. Good

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agreement between the experimental and simulated results has been achieved.

Fig. 9 The experimental and simulated κc- t curves for sharp-notched SUS304 stainless steel tubes with a = (a) 0.2, (b) 0.4, (c) 0.6, (d) 0.8 and (e) 1.0 mm under bending creep in the first stage.

1010



Fig. 10 The relationship between logC1 and a/t.

3.4 Experimental response of sharp-notched SUS304 stainless steel tubes under bending relaxation

The dotted lines in Figs. 11(a)-(e) depict the experimental moment (M) – time ( t ) curves under bending and subsequent bending relaxation for sharp-notched SUS304 stainless steel tube with a = 0.2, 0.4, 0.6, 0.8 and 1.0 mm, respectively. The starting point of the bending relaxation is marked by “●”. It can be seen that as soon as the relaxation starts the tube’s moment drastically decreases with time and gradually becomes a steady amount. Figure 12 shows the corresponding experimental ovalization (ΔDo/Do) – time ( t ) curves under bending and subsequent bending relaxation for a = 0.2 mm. It can be observed that the ovalization increases a little with time and gradually becomes a steady value. Due to the constant ovalization caused by the constant curvature under bending relaxation, the tube does not buckle. It is found that the ΔDo/Do– t curves for a = 0.4, 0.6, 0.8 and 1.0 mm under bending and subsequent bending relaxation are similar to the curves for a = 0.2 mm. Therefore, only the ΔDo/Do– t curves for a = 0.2 mm is demonstrated.

1111



Fig. 11 The experimental and simulated moment (M) – time ( t ) curves for sharp-notched SUS304 stainless steel tubes with a = (a) 0.2, (b) 0.4, (c) 0.6, (d) 0.8 and (e) 1.0 mm under bending and subsequent bending relaxation.

3.5 Theoretical formulation for sharp-notched SUS304 stainless steel tubes under bending relaxation

From Figs. 11(a)-(e), the experimental M- t curves under bending relaxation are similar to the up-side-down κc- t curves under bending creep in the first stage. Therefore, Eq. (5) is modified as

Mr = C2κm

2 tn

2 (7) where Mr is the relaxation moment, κ is the held curvature, and C2, m2 and n2 are material parameters for pure bending relaxation. By curve fitting, the material parameters m2 and n2 are determined to be 0.15 and 0.07, respectively, and the parameter C2 for a/t = 0.133, 0.2670.400, 0.533 and 0.667 are 85.42, 70.92, 53.94, 42.32 and 34.11, respectively. If we consider the relationship between C2

1/3 and a/t, a straight line can be used to describe the relationship (see Fig. 13). Therefore, parameter C2 is proposed to be

C21/3 = d1 (a/t) + d2 (8)

where d1 and d2 are material parameters for bending relaxation which can be determined from Fig. 13 to be -2.17 and 4.68, respectively. The simulated results for sharp-notched SUS304 stainless steel tube for a = 0.2, 0.4, 0.6, 0.8 and 1.0 mm under bending relaxation are demonstrated in Figs. 11(a)-(e), respectively, in solid lines. Good agreement between

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the experimental and theoretical results has been achieved.

Fig. 12 The experimental ovalization (ΔDo/Do) – time ( t ) curves for sharp-notched SUS304 stainless steel tubes with a = 0.2 mm under bending and subsequent bending relaxation.

Fig. 13 The relationship between C2

1/3 and a/t.

4. Conclusions In this study, the response of the sharp-notched SUS304 stainless steel tubes with different notch depths subjected to bending creep and relaxation are experimentally and theoretically investigated. Based on the experimental and theoretical results, the following important conclusions can be drawn: (1) For bending, it is found from the ΔDo/Do-κ curves that the ovalization increases with the amount of curvature.

Higher a of the notch tube causes greater ovalization of the tube’s cross-section. (2) For bending creep, it is seen that as soon as the creep starts the tube’s curvature quickly increases. In addition, the

κc- t curve can be divided into two stages: first and second stages. Most of the time for bending creep spends in the

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first stage. Due to increasing of the tube’s ovalization, the tube buckles eventually. (3) For bending relaxation, it is observed that as soon as the relaxation starts the tube’s moment drastically decreases

with time and gradually becomes a steady amount. The ovalization increases a little with time and gradually becomes a steady value. Due to the constant ovalization caused by the constant curvature, the tube does not buckle.

(4)

5. Acknowledgement

The work presented was carried out with the support of the National Science Council under grant NSC 100-2221-E-006-081. Its support is gratefully acknowledged.

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Response of sharp-notched circular tubes under bending