On the resistance of steel ring-stiffened cylinders subjected to low-velocity mass impact

International Journal of Impact Engineering 84 (2015) 108e123

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International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng

On the resistance of steel ring-stiffened cylinders subjected to lowvelocity mass impact Burak Can Cerik 1, Hyun Kyoung Shin, Sang-Rai Cho* School of Naval Architecture and Ocean Engineering, University of Ulsan, 93 Daehak-ro, Namgu, Ulsan, 680-749, Republic of Korea

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 October 2014 Received in revised form 18 April 2015 Accepted 25 April 2015 Available online 10 June 2015

This paper addresses the impact response of large-diameter thin-walled steel ring-stiffened cylinders subjected to low velocity mass impact and resulting local damage. Drop-weight impact tests with a striking mass, which had a knife-edge indenter, were conducted on two fabricated steel small-scale models. Details of the experiment setup, the procedure and the tests to obtain both quasi-static and dynamic material properties are described. With these observations, the experimental data, which include the final deformed shape, dynamic force-displacement curves and strain gauge measurements, are reported to be useful for future benchmark studies. The numerical prediction accuracy of the impact response of the test models were evaluated using the explicit solver of the finite element software package ABAQUS. The effect of the strain-rate hardening definition on the results is highlighted. Finally, the results that were obtained using a simplified analysis method based on smearing ring-stiffeners to obtain an equivalent circumferential bending strength were evaluated. The limitations of this simplified method were also discussed. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Ring-stiffened cylinder Dynamic tensile test Impact test Numerical simulation Simplified analysis method

1. Introduction Large diameter thin-walled cylinders are widely used in several industrial applications, such as the structural components of offshore platforms, submarine pressure hulls and pressure vessels. Cylindrical shell structures are prone to damage because of the impact loads, which may arise from such accidents as mass impact and impulsive pressure loading. Amongst these cases, the problem of low-velocity mass impact loading on a ring-stiffened cylindrical shell is of interest in this paper, such as collisions between ships and buoyancy columns of floating offshore installations [1] and dropped objects on submarine structures [2]. In the literature, many available studies are related to the deformation behaviour of offshore tubular members [3e11]. Offshore tubular members are relatively thick-walled unstiffened cylinders with large span and widely used as components of fixed offshore installations, such as jacket structures and braces of floating structures. Tubular members have different deformation

* Corresponding author. Tel.: þ82 52 259 2163; fax: þ82 52 259 2836. E-mail addresses: [email protected] (B.C. Cerik), [email protected] (H.K. Shin), [email protected] (S.-R. Cho). 1 Present affiliation: School of Marine Science and Technology, Newcastle University, Armstrong Building, Newcastle University, Newcastle upon Tyne, NE1 76U, UK. http://dx.doi.org/10.1016/j.ijimpeng.2015.04.011 0734-743X/© 2015 Elsevier Ltd. All rights reserved.

characteristics from large-diameter cylinders where the overall bending deformation is more dominant than the local shell denting. Only a small number of studies can be found on the behaviour of large-diameter unstiffened or stiffened thin-walled cylinders, which have relatively higher radius-to-thickness ratio and are used as columns or legs of offshore installations. Walker and Kwok [12] presented experimental and analytical work on quasi-static denting on cylinders. Harding and Onoufriou [13] and Karroum et al. [14] conducted quasi-static denting tests on small-scale ring-stiffened cylinder specimens. Walker et al. [15,16] reported quasi-static denting tests on both small-scale ring-stiffened and orthogonally stiffened cylinders. One advantage of imposing a specified damage using quasi-static denting is that it provides continuous recording of the damage process, which can be used to develop simplified analysis methods. However, dynamic effects, such as the strain-rate effect and inertial forces, are not taken into account. In fact, most experimental works that investigated the collision of marine structures at the structural component level follow the quasi-static approach and assume that the response of the structure under dynamic load caused by a low-velocity mass impact is similar to the static force-displacement response. For the local denting damage on cylinders due to collision, inertial forces may be neglected because the impact duration is usually longer than the natural period of the structure, but the strain-rate effect should be

B.C. Cerik et al. / International Journal of Impact Engineering 84 (2015) 108e123

considered to predict the impact response and permanent damage extents accurately. The latter has a practical importance because the design against accidental impact loading includes consideration of the residual strength in damaged condition. Recent experimental and numerical studies on basic ship structural components [17e20] that are subjected to dynamic mass impact loading show that an accurate prediction of the structural response using nonlinear finite element analysis is not straightforward unless the details of the experimental conditions and exact material characteristics are accounted for. In addition, Jones [21] noted that dynamic material properties particularly affect the credibility of numerical predictions. It is emphasised that there is a lack of understanding on many aspects of dynamic material properties and limited available data in this field. There are analytical studies on unstiffened cylinders subjected to concentrated load, which causes local denting deformation, such as [22e26]. Hoo Fatt and Wierzbicki [27] extended the work of Wierzbicki and Suh [10], which focused on the denting of cylindrical shell of ring-stiffened cylinders. Simplified analysis methods can rapidly assess collisions as long as they comply with the actual structural response. Because these studies are purely theoretical, extensive validation studies with reliable experimental data are required. In the given context, first, this paper describes the experiments on small-scale steel ring-stiffened experiments that were subjected to mass impact. The experimental work is of significant importance and can be used to validate any predictions for low-velocity mass impact loading. The tests were simulated using a nonlinear finite element analysis. The experimental study included an assessment of dynamic material properties, which was used to highlight the effect of the strain-rate hardening definition in numerical impact simulations. The effect of stiffening the cylinders in circumferential direction with rings was studied by evaluating the experimental and numerical results. Finally, an existing model for the local denting behaviour of unstiffened cylinders was revisited and the results obtained by modifying this model were compared with the experimental response.

2. Dynamic impact tests 2.1. Test models Two internally ring-stiffened steel cylinder models, which are denoted as RS-C-1 and RS-C-2, were tested. The material of the models is SS41 general-purpose structural steel. The model manufacturing followed the standard methods and techniques of full-scale structures of this type. The cylinder shell was cut from steel sheets of 4 mm thickness, cold-bent using rollers and welded to form a cylinder with an outer diameter of 800 mm. The ringstiffeners were 4-mm-thick flat-bars, which were cut from flat sheets and internally welded to the cylinder shell. The depth of the ring-stiffener web was 35 mm. The spacing of the stiffeners decreased towards the ends of the cylinder. In the middle three bays, the stiffener spacing was 200 mm whereas it was 150 mm in the next bays and 80 mm in the outmost bays. The cylinder was welded at one end to a circular plate of 20 mm thickness. At the other end, it was welded to a ring of 20 mm thickness. Similar to the

109

Fig. 1. Geometry of the ring-stiffened cylinder model (Unit: mm).

effect of heavy bulkheads in actual structures, these end conditions ensured that the cylinder ends remained circular. The end plate and the end ring had extensions at the bottom, which were bolted to the support plates. The scantlings and configuration of these models were determined considering hydrostatic pressure testing to be conducted examining the damaged ring-stiffened cylinder behaviour. The hydrostatic pressure testing facility requires that one end is open and the other end is closed. The thickness of the cylinder shell and the ring-stiffeners were surveyed using an ultrasonic device and found to be 3.80 mm in average, which is less than the nominal value. The main dimensions of the models are summarised in Table 1. Fig. 1 shows the model geometry including the end plate and end ring. In Fig. 2, the detailed geometry of the end plate and end ring are shown. The four holes at the bottom of the end plate and the end ring are for bolting with the supports. The initial imperfections of the models were measured before mounting to the testing frame. On the inner and outer surface of the models longitudinal grid lines were drawn with 10 spacing. In the circumferential direction at each ring-stiffener location and in the middle part of each bay, grid lines were drawn. The out-ofroundness of the cylinders was evaluated at every crossing point of these lines based on the obtained measurements using a twopoint bridge gage as shown. This procedure corresponds to a radial measurement of the shape. The measurements were performed from the outside for 36 points at one time. The two-point bridge gage readings were converted to out-of-roundness values by performing Fourier series expansion. The imperfection profiles of each model are shown in Fig. 3. In these graphs, 0 corresponds to the longitudinal weld line. The magnitude of the imperfections is exaggerated by 10 times. Fig. 3 shows that the maximum out-of-roundness values are confined along the longitudinal weld line. There is an outward deviation at the weld line and inward deviation at the adjacent locations. The results of out-of-roundness measurements are summarised in Table 2. The maximum out-of-roundness values were lower than the upper limit of tolerable imperfection for ringstiffened cylinders according to PD5500 [28], which is 0.5% of the cylinder radius R.

Table 1 Nominal dimensions of ring-stiffened cylinder models used in dynamic impact tests. Outer diameter (mm)

Shell thickness (mm)

Inner bay length (mm)

Number of ring-stiffeners

Stiffener web height (mm)

Stiffener web thickness (mm)

800

4

200

6

35

4

110

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Fig. 2. Geometry of the end plate and the end ring (Unit: mm).

In addition, the initial tilting of the ring-stiffeners were checked and found to be negligibly small. Thus, the experimental models were considered suitable to obtain reliable results with regard to the effect of shape imperfection due to fabrication. However, the effect of residual stress has not been elucidated. The mechanical properties of SS41 structural steel were obtained by performing quasi-static tensile tests using the tensile test coupon dimensions and the procedures in Korean Standard [29]. From each parent plate of the cylinder shell and ring-stiffeners, five

flat coupons were cut and tested using the universal testing machine. The average mechanical properties are shown in Table 3. The average yield strength for all tested coupons is 306.2 MPa with a standard deviation of 7.6 MPa. This value is close to the mean yield strength of mild steel that is 310 MPa (the nominal yield strength of mild steel, which is widely used in engineering designs, is 235 MPa). In the scope of the experiments, tensile tests at high strain-rates were also conducted. The dynamic tensile stress-strain curves were obtained using Instron VHS-65/80-25 servo-hydraulic machine. The tests were conducted according to ISO 26203-2:2011 [30]. The test coupon dimensions for tensile tests at high strain-rates are shown in Fig. 4. The coupons were clamped at 40 mm distance from the ends of the coupon neck. The tests were aimed at to be conducted at three different strain-rates: 10 s1, 50 s1 and 100 s1. The dynamic tensile strain-stress curves are shown in Fig. 5. Fig. 5 shows that the upper and lower yield strengths considerably increase with increasing strain-rate. The ultimate tensile strength also increases but is not as much as the initial yield strength at larger strain rate. It is interesting to note that the strain hardening vanishes for high strain rates. It can be concluded that the material behaves perfectly plastic when the strain rate increases, which is an important point that should be considered in cases of large deformations. Using these test results, the coefficients associated with Cowper-Symonds equation were derived for SS41 based on the lower yield strength. Fig. 6 shows the ratio of the dynamic yield strength to the static yield strength (dynamic hardening factor) versus the strain-rate. Cowper-Symonds equation is also plotted with the derived coefficients as (C ¼ 310.17 s1 and q ¼ 3.22).

2.2. Experimental setup The experiment was conducted using the free-fall testing frame in Fig. 7. This testing frame was successfully used earlier in the impact tests for unstiffened tubular structures [11], beam

Table 2 Results of out-of-roundness measurements.

Fig. 3. Imperfection profiles of the test models.

Max. out-of-roundness Location

RS-C-1

RS-C-2

0.36% R 3rd bay, weld line (0 )

0.49% R 2nd bay, weld line (0 )

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111

Table 3 Mechanical properties of the test model components. RS-C-1

Yield strength (MPa) Ultimate tensile strength (MPa) Young's modulus (MPa) Yield strain Strain at the end of yield plateau Strain at ultimate tensile stress Fracture strain

RS-C-2

Cylinder shell

Ring-stiffener

Cylinder shell

Ring-stiffener

302.2 396.2 191,394 0.00159 0.0227 0.196 0.376

306.6 398.6 200,099 0.00154 0.0217 0.197 0.370

309.0 397.3 197,879 0.00157 0.0224 0.194 0.368

306.1 396.4 211,727 0.00147 0.0214 0.193 0.366

Fig. 4. Dimensions of the coupons for high strain-rate tensile tests (Unit: mm).

700

600

600

500

500

Stress (MPa)

700

400

Unfiltered test data - 11 per sec

300

Unfiltered test data - 11 per sec

400 300

Unfiltered test data - 49 per sec

200

200

Unfiltered test data - 48 per sec

100

100

Unfiltered test data - 50 per sec

0

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.05

0.1

0.15

Strain

0.2

Strain

900 800 700

Stress (MPa)

Stress (MPa)

structures [31] and small-scale tanker double-hull structures [20]. On the top of the tower of the frame, there is a pulley that guides and holds and an electromagnet attached to the striking mass. The height of the striking mass can be varied to achieve the desired impact energy. After the electromagnetic force is cut off, the striking mass falls down and accelerates with gravity. The velocity of the striking mass is simply determined by equating its potential energy to its kinetic energy prior to impact. The striking mass is 295 kg and has an indenter with a knifeedge tip. The tip is rounded with a radius of 5 mm. The

dimensions of the striking mass and its indenter surface are provided in Fig. 8. For both models, the identical striking mass was used; however, the drop height and impact energy were different. The experiment conditions for each model are listed in Table 4. With this experimental setup, the impact force and deflection throughout impact cannot be directly measured. Thus, an indirect measurement method was used. Four accelerometers were affixed on the upper surface of the striking mass to record the acceleration history. Using Newton's law of motion and double integration of the recorded acceleration time history during the impact, a dynamic force-displacement curve was obtained. After the impact, it was assumed that the indenter remained in contact with the struck model. Therefore, the indenter and impact region of the struck model had a common velocity and displacement throughout the entire impact before the separation. The permanent deflection of the model was obtained when both bodies were no longer in contact, i.e. when both bodies were separated. Strain gauges were placed at five positions to obtain more details about the impact response of the models. At each position, two

600

500 400

Unfiltered test data - 99 per sec

300

Unfiltered test data - 99 per sec

200

Unfiltered test data - 100 per sec 100 0 0

0.05

0.1

0.15

0.2

0.25

Strain Fig. 5. Dynamic tensile strain-stress curves.

0.3

0.35

0.25

0.3

0.35

0.4

112

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450

25

2.5

270

Cowper-Symonds equation with C = 310.17 s-1 and q= 3.22

285

2

1.5

170

25

1

85

Dynamic yield strength / Static yield strength

3

0.5

10 400

0 0.0001

0.001

0.01

0.1

1

10

100

1000

Strain rate (s-1)

45°

R5

10

Fig. 8. Dimensions of the striking mass and its indenter (Unit: mm).

Fig. 6. Dynamic hardening factor plotted versus strain-rate for SS41.

strain gauges were affixed. The first two positions were 60 mm beside the impact point along the generator line. The strain gauges at these locations were on the inner surface of the cylinder. In order to understand the deformation behaviour of the ring-stiffeners, strain gauges were placed on the surface of the ring-stiffeners bordering the mid-bay. The last position was at the centre plane and on the generator line at 40 away from the top generator line on the inner surface of the cylinder. The arrangement of the strain gauges is sketched in Fig. 9.

It is important to describe the boundary conditions in the experiment in details because the impact response is highly sensitive to the provided restraints in some cases [17,21]. The models were firmly fixed to 10 mm thick support plates using 4 bolts. The support plates were bolted to the rigid foundation of the testing frame. Note that the support plates were in contact with only the end plate and the end ring. Therefore, some part of the bottom of the end plate and the end ring were fixed. On the upper part, the displacements and rotations were free. Consequently, the cylinder shell remained circular at both ends, but depending on the stiffness of the end plate and the end ring, the axial translation and rotations were partially restrained. The impact location was the mid-bay of the cylinder. After the striking mass hit the model, it rebounded, and its motion was not controlled during this time. In order to protect the model from further damage after the striking mass rebounded, the cylinder surface except the impact zone was covered with 5 mm thick rubber pad and the side surfaces were covered with polystyrene foam. These precautions do not affect the local damage process of the struck model.

2.3. Test results The deformed shapes of the models are shown in Fig. 10. After the damage profile was carefully checked, it was found that the

Table 4 Impact test conditions for ring-stiffened cylinder models.

Drop height (mm) Impact velocity (m/s) Striking mass (kg) Kinetic energy (J)

RS-C-1

RS-C-2

1600 5.602 295 4630.3

1200 4.852 295 3472.7

Ring stiffener

Ring stiffener

Pos. 3 & 4

y z x

Pos. 3

Pos. 1

Pos. 2

Pos. 4

Pos. 5 Mid-bay Fig. 7. Impact testing frame.

Fig. 9. Arrangement of the strain gauges.

y

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113

Fig. 10. Deformed shape of the models in top view: (a) RS-C-1 and (b) RS-C-2.

impact location shifted away from prescribed position towards the open end. The inevitable rotation of the striking mass when it leaves the electromagnet is likely the main cause. The damage can be described as denting at the impact region. The dent flattens the cross section with a length of the indenter width. At the ends of the flattened dent section, the shell bulges outward. The large dent is limited in the mid-bay. The boundaries of the dent are semi-ellipses. Nonetheless, the damage also spreads by flattening to the adjacent bays in the longitudinal direction. The longitudinal generators deflect inward and cause material stretching in the axial direction because of the end-plate resistance and membrane resistance of the shell. The remainder of the shell is unaffected. This damage shape follows the damage shape of unstiffened cylinders [12]. The end plate and the end ring were sufficiently stiff to maintain circular ends. However, a small degree of rotation was observed on the upper part of these end supports. The coating on the surface of cylinder shell provides information about the deformation behaviour and strain distribution. A close look into the damaged zone of RS-C-1, which is shown in Fig. 11, indicates that the shell adjacent to the contact line and at the locations of the ring-stiffeners have high surface strains. The elliptical boundaries mark the hinge edges of the dented zone. The shell inside these boundaries was stretched and flattened. Consequently, the coating was crazed and removed from the shell surface. At the position of the ring-stiffeners, the crazing of the coating on the cylinder surface marks the flattened section of the ring-stiffeners. In Fig. 12, the deformed shape of the ring-stiffeners is shown. The upper parts of the ring-stiffeners adjacent to the damage zone were completely flattened. At the ends of this flattened part, the strain concentration is noticeable where the coating was cracked. The ring-stiffeners bordering the bay adjacent to the mid-bay were also deformed as the cracked coating confirms. The flattened part was also tilted with a small degree of outward rotation. In addition to the flattened parts, the shape of the ring-stiffened cylinder remains unaffected. The damage profiles and extents were measured using CimCore portable measuring arm device. Thus, the exact hit location and

eccentricities in loading could be determined. The longitudinal and circumferential damage profiles for each model are shown in Figs. 13 and 14, respectively. In these figures, the origin is the centre of the circle associated with the end plate where 0 corresponds to the top generator of the cylinders. The circumferential damage profile of RS-C-1 clearly indicates that the dent line is not horizontal because of the large rotation of the striking mass before it hit the model. For RS-C-2, this rotation is small. For both models, the longitudinal damage profiles prove that the ring-stiffeners do not significantly limit the damage in the midbay. The outward bulging in the dented cross-section is noticeable.

Fig. 11. Damaged zone in RS-C-1.

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Fig. 12. Deformed shape of the ring-stiffeners: (a) RS-C-1 and (b) RS-C-2.

The tilting of ring-stiffeners bordering the mid-bay was also measured. The maximum out-of-plane displacement at the toe of the stiffener web was considered the reference. It was recognised that the dent line shift along the longitudinal axis significantly affected the deformation of the ring-stiffeners, where the tilting of the ring-stiffener closer to the dent line was much larger. The results of the damage extent measurements are listed in Table 5. In this table, the dent depth is shown as two values (maximum and minimum) because of the inclination of the dent line. The maximum dent depth corresponds to the highest distance from the dent line to the top generator of the undamaged cylinder. The minimum is the shortest one. The average of these two values is also provided for comparison with the accelerometer

450

500

Ring-stiffener

Ring-stiffener

RS-C-1

400

350 300

250

0 degree 10 degrees 20 degrees -10 degrees -20 degrees

200

150 100

300 250 200 150 100

0 -350

0 200

350

Dent line 730 mm 630 mm (Ring-stiffener) 330 mm 430 mm (Ring-stiffener)

50

50 0

RS-C-1

400

Vertical coordinate (mm)

450

Verrtical coordinate (mm)

measurements. The rotation of the dent line to the longitudinal (top generator) and vertical axes (centre plane) were also calculated and listed in this table. The strain measurements provided more details to understand the deformation process. The filtered strain histories are shown in Figs. 15 and 16 for the cylinder shell and ring-stiffeners, respectively. Note that the circumferential strain measurements from the strain gauge at position 3 in RS-C-1 failed because this gauge became separated from the ring-stiffener surface. According to the strain measurements at positions 1 and 2, the cylinder shell deformation was mainly in the longitudinal direction because the membrane stretching was marked with high tensile strains. In the

400

600

800

1000

-250

-50

50

150

450

500

Ring-stiffener

Ring-stiffener

RS-C-2

450 400

350 300

250

0 degree 10 degrees 20 degrees -10 degrees -20 degrees

200

150 100

0 200

350

RS-C-2

350 300 250 200 150 100

Dent line 730 mm 630 mm (Ring-stiffener) 330 mm 430 mm (Ring-stiffener)

50

50 0

250

400

Vertical coordinate (mm)

Verrtical coordinate (mm)

-150

Horizontal coordinate (mm)

Longitudinal coordinate (mm)

400

600

800

1000

Longitudinal coordinate (mm) Fig. 13. Measured longitudinal damage profiles of RS-C-1 and RS-C-2.

0 -350

-250

-150

-50

50

150

250

Horizontal coordinate (mm) Fig. 14. Measured circumferential damage profiles of RS-C-1 and RS-C-2.

350

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115

Table 5 Results of damage extents measurements.

Max. dent depth (mm) Min. dent depth (mm) Average dent depth (mm) Longitudinal shift dent line (mm) Rotation of dent line about longitudinal axis (degree) Rotation of dent line about vertical axis (degree) Tilting of ring-stiffener closer to the dent line (degree) Tilting of the other ring-stiffener (degree)

RS-C-1

RS-C-2

54.15 41.86 48.01 27.15 2.65 0.87 6.98 2.72

36.71 33.93 35.32 16.3 0.59 1.67 5.34 1.33

circumferential direction, the strains were compressive. Note that the strains in the circumferential direction reached their permanent value earlier. Thus, the deformation starts when the circumferential curvature flattens and leads to compressive membrane strains. Because of the eccentricity of the loading and unsymmetrical end conditions, the measured strain levels at these two positions are different. Away from the dent zone the strain level at position 5 is relatively low (lower than the yield strain), which confirms that the shell was unaffected outside of the vicinity of the impact zone. The strain measurements support the comments on the behaviour of the ring-stiffeners. The circumferential strain in the ring-stiffener increases when the longitudinal damage profile reaches the ring-stiffener location. The strain is tensile because of the change in curvature of the ring. The ring-stiffeners laterally deform in the direction of impact. The flattened zone acts as a beam that is supported at the ends by the shell at the edges of the damaged zone. Further inward displacement of the cylindrical shell

Fig. 16. Strain time histories recorded by the strain gauges placed on ring-stiffeners.

provokes compression on ring-stiffeners and eventually leads tilting. Then, the damage longitudinally propagates. Finally, elastic spring-back occurs where the strain reaches its permanent value. The filtered average accelerometer measurements are shown in Fig. 17. The shape of the curve resembles a half-sine wave. As explained in the detailed description of the experimental setup, these data were used to obtain the velocity and the displacement histories in Fig. 18. The velocity history of the striking mass is similar to a cosine curve shifted to have initial amplitude as the prescribed impact velocity. The velocity decreases until all impact energy is dissipated by the struck model. After all energy is

500 450 RS-C-1

Acceleration (m/s2)

400

RS-C-2

350 300 250 200 150 100 50 0 0

5

10

15

20

25

Time (ms) Fig. 15. Strain time histories recorded by the strain gauges placed on cylinder shell.

Fig. 17. Acceleration time history of the striking mass.

30

116

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6

140

5 RS-C-2

100

Force (kN)

Velocity (m/s)

3 2 1

80 60 40

0 0

5

10

15

20

25

30

-1

20

-2

0 0

-3

10

20

Time (ms)

30

40

50

60

Displacement (mm)

60

Fig. 19. Force-displacement curves of the test models.

RS-C-1

50

Table 6 Summary of the test results.

RS-C-2

Displacement (mm)

RS-C-1 RS-C-2

120

RS-C-1 4

40 Peak force (kN) Deflection at peak force (mm) Energy at peak force (J) Permanent dent depth (mm) Final energy (J) Impact duration (ms)

30 20

10 0 0

5

10

15

20

25

30

Time (ms) Fig. 18. Velocity and displacement time history of the striking mass.

dissipated, the velocity becomes zero and the displacement reaches its maximum value. The required time to dissipate all impact energy is longer for RS-C-1, as expected. Elastic spring-back subsequently occurs. The striking mass moves back until it loses contact with the struck model. At the end of this process, it reaches a constant velocity. The permanent dent depth is equal to the displacement of the striking mass when it reaches this stage. By combining the calculated force history from the acceleration data and displacement history, the force-displacement curves were obtained as shown in Fig. 19. The impact response is similar to the response of unstiffened cylinders [12] and quasi-statically dented ring-stiffened cylinder models [13]. When the indentation increases, the slope of the forcedisplacement curve gradually decreases. In addition to the oscillations because of the dynamic nature of the problem, there are no sudden peak and trough, which indicate any clear effect of deformation of the ring-stiffeners on the impact resistance and the dent depth. The obtained experimental results of these measurements are listed in Table 6. It can be inferred that in both tests the absorbed energy is identical to the kinetic energy of the striking mass in Table 4. Thus, the accelerometer measurements and other derived quantities are reliable. The difference between the maximum absorbed energy and the final energy occurs because of elastic spring-back. 3. Numerical modelling of impact tests The numerical computations were performed using the nonlinear finite element software package ABAQUS. The finite

RS-C-1

RS-C-2

130.46 55.48 4630.3 45.92 4229.6 25.06

116.37 44.05 3472.7 34.14 3054.9 24.25

element model consists of full geometry of the test models, which includes the end plate and the end ring. Every component of the struck model was meshed with four-node shell element S4R from ABAQUS element library. Five integration points through the thickness and default hourglassing controls for this element were used. For the cylindrical shell and ring-stiffeners, the mesh is uniform and has elements with an edge size of 10 mm. High velocity impacts would cause localised damage and response, but as observed in the tests the low velocity mass impact considered in this study gives a rather global shell response. This mesh size is sufficiently fine to precisely capture the deformed shape of the cylinder shell and force-displacement response. It must be mentioned that a finer mesh will result in elements with smaller edge size than the shell thickness, which leads to problems in contact calculations. The end plate and end ring have a nonuniform mesh with 10 mm global edge size. The surfaces of the shell elements were outward upset to avoid overlapping of the material. The element thickness at the weld joints was not increased. It is believed that this assumption does not lead to significant discrepancies between the actual response and the simulated response. The engineering stress-strain data were obtained from quasistatic tensile tests and used to tabulate the true stress-equivalent plastic strain data, which are required for the plasticity definition in ABAQUS. In Fig. 20, the engineering stress-strain curve and plasticity data, which were defined in numerical simulations for the cylinder shell of RS-C-2, are shown. In the initial attempts to simulate the impact test, the strain levels do not exceed the strain associated with the ultimate tensile strength. Therefore, no concern was raised to extend the true stress-plastic strain curve after the initiation of necking when the impact tests were simulated. One of the most important aspects of this impact problem is the strain-rate effect. The current tests can be classified as a lowvelocity impact case where the impact velocity is lower than 20 m/s [32]. Nevertheless, the damage is local and high strain-rates

B.C. Cerik et al. / International Journal of Impact Engineering 84 (2015) 108e123

conditions. The indenting surface elements were restrained in all degrees of freedom except translation in the impact direction. The contact between the indenting surface and the struck model was defined using the general contact option in ABAQUS, which allows us to define the contact among all regions of the model with a single interaction. This contact algorithm uses a penalty method as the contact constraint. It also automatically solves initial overclosures. The finite element model was created with the previously provided definitions and is shown in Fig. 21.

500

450 400

Stress (MPa)

350

300 250 Engineering stress-strain curve

200

117

True stress - Plastic strain curve

150

4. Numerical results and discussion

100 50

4.1. Force-displacement curves

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Strain Fig. 20. Engineering stress-strain curve and the plasticity data defined for the cylinder shell of RS-C-2 in numerical analysis.

develop at the impacted area. In ABAQUS, the strain-rate effect is usually considered using Cowper-Symonds constitutive equation. By assuming that the stress-strain dependence is similar at all strain-rate levels the strain-rate hardening behaviour is defined simply by scaling quasi-static plasticity data with a dynamic hardening factor, which is the ratio of the dynamic yield strength to the static yield strength. The dynamic hardening factor only depends on the strain-rate magnitude and can be expressed as a power-law function as proposed by Cowper and Symonds [33]:


s0o so ¼ 1 þ ð_ε=CÞ1=q

(1)

However, as observed in the dynamic tensile test curves, at least for SS41, the strain-rate hardening behaviour depends on the strain magnitude. In the current study, dynamic tensile test results were available and directly used to define the plasticity data for several strain-rate levels. In ABAQUS, for the strain-rate levels without the defined plasticity data, the information is obtained using interpolation. Therefore, the test data from closely spaced strain-rate levels are desirable to obtain more accurate results. This approach is more rational than using CowpereSymonds equation based on the initial yield strength for this particular problem and should yield more accurate results. The engineering stress-strain data of the dynamic tensile tests were filtered, averaged and used to tabulate the plasticity data for the corresponding strain-rate levels. The striking mass was simply modelled as an indenting rigid surface, whose dimensions are provided in Fig. 8. The mesh consists of four-node bilinear rigid quadrilateral element R3D4. The mesh size was chosen as 10 mm. All nodes of the indenter surface were tied to a reference node, to which the inertial properties of the striking mass were assigned. After the impact, the model inevitably vibrates elastically. Rayleigh damping was used to overcome these vibrations and quickly attain a static equilibrium state. Rayleigh damping consists of massproportional damping, which is associated with low-frequency oscillations and stiffness proportional damping, which is associated with high-frequency oscillations. The former is used to include a damping matrix in the dynamic analysis, which is obtained by multiplying the mass matrix of the system with the coefficient a. The coefficient a was set as the lowest natural frequency of the model which was obtained using a modal analysis. The elements at the bottom of the end plate and the end ring, which were fixed between the supporting plates, were restrained in all degrees of freedom as in the experimental boundary

Next, the numerically predicted responses were compared with the actual responses using the force-displacement curves. The numerical analysis setup was modified to include the eccentricities in the previously mentioned loading. The results are shown in Fig. 22. The displacement here is the indentation on the cylinder, which is equal to the lateral displacement of the striking mass. Because the accelerometer measurements were averaged, the displacement corresponds to the average dent depth. It is apparent from Fig. 22 that the numerical results are consistent with the test results. The overall tendencies and peak force magnitudes were accurately predicted in the numerical analyses. There is a small discrepancy in the permanent deformations, but considering the uncertainties and errors in the experiments, this discrepancy is expected. Particularly for RS-C-1, the force levels were slightly overestimated when the displacement increased, which resulted in lower permanent deformations than the actual ones. The major cause of these differences may be the impact location and hit angle, which cannot be exactly estimated in the experiments and included in the numerical analyses. After the numerical analysis methodology was validated, more information was obtained from the numerical results. The deformed shapes of the models as obtained from the numerical results are shown in Fig. 23. In these figures, the contours indicate equivalent plastic strain. Similar observations from the actual test results are apparent. The plastic strain is concentrated at the impact region between two stiffeners. The ring-stiffeners are flattened over a length that ends with two points with highly concentrated strain. At these two locations around the perimeter, the cylinder

Fig. 21. Finite element model of the test specimens.

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160

RS-C-1

140

Experimental result

Numerical result

Force (kN)

120 100 80

60 40 20 0 0

10

20

30

40

50

60

Displacement (mm) 140

RS-C-2

120

Experimental result

the longitudinal damage profile in the unstiffened model is shown. The end plate and the end ring are evidently much more pulled-in because of the membrane stretching of the cylinder shell upper surface. Based on these results the effect of stiffeners can be summarized as follows. The ring-stiffeners mainly act as point obstacles against circumferential bending. A ring-stiffener starts to deform when the length of the deformed zone reaches it. After the ring-stiffener becomes flattened, the resistance of cylinder shell increases primarily against circumferential bending until the ring-stiffener collapses by tripping. The membrane resistance should not be affected by the presence of ring-stiffeners. For the ring-stiffener cylinder geometries and impact conditions considered in the current study, contrary to the conclusion of Hoo-Fatt and Wierzbicki [27], high strain concentrations in the cylinder shell near the ringstiffeners are not at the level of the strain in the impact zone. However, the strain localisation and fracture at the base can be still a concern for thicker or larger ring-stiffeners. 4.4. Effect of boundary conditions

Numerical result

Force (kN)

100 80

60 40 20 0 0

10

20

30

40

50

60

Displacement (mm) Fig. 22. Experimentally and numerically obtained force-displacement curves.

shell also has high strains. The initiation of this deformation process for the next ring-stiffener pair is noticeable. In these figures, the locations with high plastic strain concentrations are consistent with the crazed coating locations in the actual tests, which further confirm the results of the numerical analysis. 4.2. Energy partition The energy partition throughout the impact provides more insights about the effect of the ring-stiffeners, which could not be easily detected in the force-displacement responses. In Fig. 24, the total internal and plastic strain energy histories for the entire struck model, cylinder shell and ring-stiffeners of both models are shown. The energy dissipated in the ring-stiffener deformation is considerably large for both models in this study. As shown in Table 7, approximately one third of the total impact energy is dissipated by the deformation of the ring-stiffeners. 4.3. Effect of ring-stiffeners Next, the response of the test models in the absence of ringstiffeners was assessed considering the impact conditions for RSC-2. In Fig. 25, the force-displacement curve for the unstiffened model is shown and compared with the response of RS-C-2. The results for the unstiffened case and differences with the ringstiffened case are summarised in Table 8. It is apparent that the permanent dent depth is almost doubled in the unstiffened model. The unstiffened model has a much lower resistance against denting than the ring-stiffened model. In Fig. 26,

Although the end plate and the end ring do not dissipate a significant amount of energy, their effect on restraining the axial translation of the cylinder ends affects the membrane resistance of the cylinder shell. In theoretical models the boundary conditions must be idealised by either imposing full restraints or allowing translations and rotations, which may not always represent the actual case. To clarify this issue, the effect of idealising the boundary conditions is assessed by comparing with fully modelling the experiment conditions. Three cases were considered: fully restrained translations and rotations; free axial translations, restrained rotations and radial translations; only restrained radial translations at the cylinder ends. A comparison is made for the impact conditions of RS-C-2. The resulting force-displacement curves are shown in Fig. 27. It is not surprising that the fully fixed condition does not well represent the actual case because the upper part of the end plate and the end ring were not restrained in the tests. The pull-in at the ends is sufficiently large to consider the axial restraint negligible. Fig. 27 shows that although the responses for each boundary condition are similar in the initial phase, there is significant resistance against denting in the moderate-to large-deformation range when the ends are not free to axially move. This further proves that the initial phase is the main flattening of the cross-section. The effect of the boundary conditions becomes clear when the longitudinal generators stretch and membrane resistance develops. 4.5. Effect of strain-rate hardening definition In addition to boundary conditions, it is important to assess the effect of other idealisations and uncertainties in numerical modelling to uncover any sensitivity of the numerical solutions to small changes in a parameter. As previously emphasised, the dynamic material characteristics can be a source of error in numerical predictions. The strain-rate hardening characteristics of various metals are listed in textbooks; for example, the coefficients in CowpereSymonds equation are often given as C ¼ 40.4 s1 and q ¼ 5 for mild steel. As remarked by Jones [21], these values significantly vary with the strain magnitude and plate thickness. When C increases, the sensitivity to the strain-rate decreases. As previously shown, the strain-rate sensitivity of the material used in the current tests depends on the strain magnitude. Accordingly, scaling quasi-static plasticity data with dynamic hardening factor may not be appropriate. The sensitivity of the impact response to the definition of strain rate hardening is

B.C. Cerik et al. / International Journal of Impact Engineering 84 (2015) 108e123

119

Fig. 23. Deformed shape of test models obtained numerically: (a) RS-C-1 and (b) RS-C-2.

demonstrated in the current study. For CowpereSymonds equation, the commonly used coefficients for mild steel (C ¼ 40.4 s1 and q ¼ 5) and high-tensile steel (C ¼ 3200 s1 and q ¼ 5) were considered with the derived coefficients. The resulting forcedisplacement curves are plotted for both test models in Fig. 28. The actual force-displacement responses that were obtained in the test are also shown for comparison. Fig. 28 evidently shows that the overall response tendencies are not affected by the strain-rate effect definition. The strain-rate effect definition slightly increases or decreases the stiffness of the struck model. Thus, the permanent deflection depends on this definition. It should be noted that for this particular case, negligence of the strain-rate effect results in 23.0% and 22.9% larger permanent deflection, respectively for RS-C-1 and RS-C-2, compared with the case of the dynamic tensile test data. When CowpereSymonds equation is used with the derived coefficients, the difference is 10.6% and 11.1%. It is interesting to note there is no significant difference when we used different coefficients in CowpereSymonds equation. The main difference is between the direct input of dynamic tensile test data and CowpereSymonds equation.

The strain-rate magnitudes increased when dynamic tensile test data were directly used. This effect resulted in larger strain-rate hardening and less permanent deformations. For larger impact velocities that result in higher strain-rates, the differences are obviously more apparent.

5. Comparison with simplified analysis results After experimentally and numerically assessing the effect of ring stiffeners on the local response of cylindrical shells, at this junction, a simplified analysis method is presented based on the developed insights. The aim is to modify the model proposed by Wierzbicki and Suh [10] for local denting for unstiffened cylindrical shells. The idea is not new, because Hoo Fatt and Wierzbicki [27] developed a model to extend Wierzbicki and Suh model by either treating the stiffeners as discrete entities or smearing them to obtain an equivalent thickness. In this study, a different smeared model that results in a closed-form solution was used and a good consistency with the experimental results in previous section was obtained.

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B.C. Cerik et al. / International Journal of Impact Engineering 84 (2015) 108e123 6 Tot. internal energy - whole model Tot. internal energy - ring stiff f eners Plast. strain energy - cylinder shell

5

Energy (kJ)

4

Tot. internal energy - cylinderr shell Plast. strain energy - whole model Plast. strain energy - ring-stiffeners

Table 8 Numerical analysis results for unstiffened cylinder model and comparison with ringstiffened case.

Peak force (kN) Deflection at peak force (mm) Permanent dent depth (mm) Final energy (J) Impact duration (ms)

RS-C-1

3

Value

Difference

78.75 78.69 68.24 3123.9 39.60

47.78% decrease 68.01% increase 78.63% increase 2.26% increase 63.30% increase

2

1

0 0

5

10

15

20

25

30

Time (ms) 4.5 Tot. internal energy - whole model Tot. internal energy - ring stiff f eners Plast. strain energy - cylinder shell

4 3.5

Energy (kJ)

3

Tot. internal energy - cylinderr shell Plast. strain energy - whole model Plast. strain energy - ring-stiffeners

RS-C-2

First, it is necessary to explain the model developed by Wierzbicki and Suh [10] for local denting of cylindrical shells. The problem formulation is briefly described. The model that defines the problem and all involved parameters is shown in Fig. 29. The cylindrical shell is considered a rigid-perfectly plastic material, and the elastic strains in the damage-affected zone are neglected. It is also assumed that the plastically deformed zone is localised and finite. All regions outside this zone are rigid. The length of the deformed zone is x, as shown in Fig. 29. Based on the principle of virtual work the equilibrium equation is expressed as

2.5

E_ ext ¼ E_ int

2 1.5

(2)

The rate of external work here is related to the lateral impact load:

1 0.5

E_ ext ¼ P d_

0 0

5

10

15

20

25

30

Time (ms) Fig. 24. Energy time histories for test models obtained numerically.

Table 7 Share of each structural component in energy absorption. Structural component

RS-C-1 Internal energy

Cylinder shell Ring-stiffeners End plate and ring

63.50% 35.51% 0.99%

RS-C-2 Plastic energy 64.60% 35.23% 0.17%

Internal energy 65.48% 33.74% 0.78%

Plastic energy 66.55% 33.29% 0.16%

140 120

Model w/ ring-stiffeners Model w/o ring stiffeners

Force (kN)

100

80

(3)

In this model, the shell is assumed to be inextensible in the circumferential direction. The bending in the axial direction is also neglected. These assumptions lead to the conclusion that the internal rate of energy dissipation is caused by membrane axial stretching and circumferential bending:

E_ int ¼

Z ðsxx ε_ xx þ zk_ qq ÞdV V

Now, the cylindrical shell can be considered a series of unconnected rings (slices) and a bundle of unconnected generators (beam-strings). The rings and generators are loosely connected so that the lateral deformations are compatible. For rings and generators, the bending and membrane energy can be calculated, respectively. The rings are inextensible and the energy is absorbed by circumferential bending in the stationary or moving plastic hinges. The generators are rigid-plastic beams that bend and stretch. Because the curvature in the axial direction is negligible when compared with the circumferential curvature, only the stretching energy is considered. The rate of internal energy is expressed as a sum of these two components

E_ int ¼ E_ ring þ E_ gen

60

(4)

(5)

Assuming a four-plastic hinge mechanism, Wierzbicki and Suh [10] expressed the ring energy dissipation rate as follows

40

8mo xd_ E_ ring ¼ R

20 0 0

20

40

60

80

100

Displacement (mm) Fig. 25. Force-displacement responses of models with and without ring-stiffeners.

(6)

where mo is the fully plastic moment of the cylinder wall per unit width. The generator energy dissipation can be obtained for a cylinder that is free to axially move and rotate at both ends as follows:

B.C. Cerik et al. / International Journal of Impact Engineering 84 (2015) 108e123

121

Fig. 26. Longitudinal damage profile in the model without ring-stiffeners.

p dd_ E_ gen ¼ No R 4 x

(7)

where No is the plastic axial resistance of the cylinder wall per unit width. By inserting these two terms into Eq. (2) and expressing the external energy dissipation rate as Eq. (3), the force-displacement relation is obtained:

P ¼ 16mo

140

Now, the force depends on x and d. This equation should be minimised with respect to x to obtain a relationship between the dent depth and the length of the deformation zone:

120

rffiffiffiffiffiffi pd 8t

(9)

Force (kN)

(8)

x ¼ R

Experiment result Direct input of test data Cowper-Symonds with derived coefficients C = 40.4 per sec C = 3200 per sec w/o strain rate effect

100 80 60

20

(10)

0

0

5

10

15

20

140

200 Fully fixed

Experiment result

100

Force (kN)

Force (kN)

Free axial translations and rotations

140

30

35

40

45

50

55

Experiment result Direct input of test data Cowper-Symonds with derived coefficients C = 40.4 per sec C = 3200 per sec w/o strain rate effect

120

Free axial translations

160

25

60

65

70

Displacement (mm)

This equation is further simplified as follows:

180

RS-C-1

40

Eq. (9) is inserted into Eq. (8) to obtain the final formula:

rffiffiffiffiffiffi rffiffiffiffiffiffiffiffi pd ptd þ No P ¼ 8mo 8t 2

(11)

where D is the cylinder diameter. The first term in Eq. (10) corresponds to the circumferential bending resistance and is modified to

160

8mo x p d þ No R P¼ R 4 x

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p d D 16 R t

120

100 80

RS-C-2

80 60 40

60 40

20

20

0

0 0

10

20

30

40

50

Displacement (mm) Fig. 27. Force-displacement curves for idealized and actual boundary conditions.

0

5

10

15

20

25

30

35

40

45

50

55

Displacement (mm) Fig. 28. Force-displacement curves for different strain-rate hardening definitions.

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B.C. Cerik et al. / International Journal of Impact Engineering 84 (2015) 108e123

Fig. 29. Model describing denting of unstiffened cylindrical shells.

include the effect of the ring-stiffeners by replacing the fully plastic bending moment of the cylinder shell wall mo with an equivalent bending moment per length meq. The fully plastic moment of the ring-stiffener with attached plating is used to evaluate the equivalent bending moment:

Z M¼

zso dA

(12)

A

Here, A is the cross-sectional area, which includes the attached plating, and so is the average flow stress of the material. The effective width of the stiffener is assumed as the spacing of ringstiffeners lr. Then, the equivalent bending moment per length can be expressed as

meq ¼

M lr

(13)

Thus, the ring-stiffeners are smeared over a length that is equal to the stiffener spacing. An equivalent thickness is obtained as follows:

teq

sffiffiffiffiffiffiffiffiffiffiffi 4meq ¼ so

(14)

Hoo Fatt and Wierzbicki [27] proposed a smeared model based on this equivalent thickness. However, if the thickness of the cylinder shell is replaced with the equivalent thickness or if the fully plastic moment of cylinder shell wall is replaced with the equivalent bending moment per length, the membrane resistance also increases. In the current study only mo is replaced with meq:

rffiffiffiffiffiffi rffiffiffiffiffiffiffiffi pd ptd þ No P ¼ 8meq 8t 2

shortcomings. The flow stress was considered as the average of the initial yield strength and the ultimate tensile strength of SS41. The force-displacement response of the proposed model in this study was compared with experimental responses of the two models RSC-1 and RS-C-2 in Fig. 30. The proposed model yields a good estimation of the impact response for these particular test models although several important phenomena, such as the strain-rate effect and energy dissipated by shear deformation were not considered. In particular, the initial resistance was well predicted. It is believed that assuming an average flow stress may compensate the errors. When Eq. (16) is equated to the impact energy in the tests, maximum deflection at the peak force can be obtained. The resulting deflections are 57.92 and 47.40 mm for RS-C-1 and RS-C-2, respectively. Compared with the values in Table 5, the predicted values are 4.7% and 1.8% lower. This inaccuracy is considered acceptable for a simple tool. At this point, the presented method for densely or sparsely spaced stiffeners should be validated. It is expected that for densely spaced stiffeners, when the dent depth increases, the effect of the ring stiffeners diminishes because the stiffeners near the damaged zone will buckle earlier. For this type of cylinders, the resistance is overestimated. Consider a quasi-statically dented model that was reported by Harding and Onoufriou [13], CY-3, for which D/lr is 13.3. The results in Fig. 31 confirm that the presented method is not suitable to treat the denting problem of densely stiffened cylinders. However, for the practical range of stiffener spacing, e.g., D/lr < 5, the presented method should be adequate. For example, for the test model CY-4 in the quasi-static denting tests for which D/lr is 4, the present method provides notably consistent results as shown in Fig. 32. 6. Final remarks The tests and simulations presented in this study served to identify large deformation response of ring-stiffened cylinders subjected to low velocity mass impact. For the ring-stiffened cylinder models considered in this study, it was observed that the presence of ring-stiffeners significantly affects the resistance against denting and permanent deformation by increasing the circumferential bending strength. This fact was used when a simple formulation of the impact response of unstiffened cylinders was modified to include the ring-stiffeners effect. The simple formulation is shown to yield acceptable results when compared to the limited test results presented in this study. Further studies are

160 140

Consequently, only the circumferential bending stiffness of the cylindrical shell increases. One shortcoming of this approach is that when the stiffeners buckle, i.e. when they are tripped, this effect cannot be adequately reflected in the force-displacement response. The amount of absorbed energy is easily determined by integrating Eq. (15) with respect to d:

120

Ed ¼

16 meq 3

rffiffiffiffiffi rffiffiffiffiffi p 1:5 2 pt 1:5 d þ No d 8t 3 2

Force (kN)

(15)

Experimental result - RS-C-1 Experimental result - RS-C-2 Present formulation

100 80 60

40

(16)

The first term in this equation corresponds to the energy dissipated by circumferential bending and the second term corresponds to the membrane stretching energy. Finally, the proposed model was compared with the experimental results to assess the validity of the approach and any

20

0 0

10

20

30

40

50

60

Displacement (mm) Fig. 30. Comparison of proposed model with experimental results presented in this study.

B.C. Cerik et al. / International Journal of Impact Engineering 84 (2015) 108e123

8 Harding and Onoufriou [13] - Model CY-3

7 Experimental result

Force (kN)

6

Present formulation

5

4 3 2 1 0 0

1

2

3

4

5

6

7

8

Displacement (mm) Fig. 31. Comparison of proposed model with the experimental results reproduced from Ref. [13] for CY-3.

5

Harding and Onoufriou [13] - Model CY-4

4.5

Experimental result

4

Present formulation

Force (kN)

3.5 3

2.5 2

1.5 1

0.5 0 0

2

4

6

8

10

12

Displacement (mm) Fig. 32. Comparison of proposed model with the experimental results reproduced from Ref. [13] for CY-4.

necessary to validate this approach for wide range geometries and use it for rapid assessment of the crashworthiness of a thin-walled ring-stiffened cylinder subjected to lateral impact load. The importance of strain-rate hardening is emphasised. The sensitivity of the accuracy of numerical simulations to the strainrate hardening definition is presented. It is suggested that the dynamic material properties should be used to define plasticity at high strain-rates whenever possible. In general, the quasi-static treatment of the collision problem is appropriate if the material hardening due to the strain-rate effect is reasonably considered. Acknowledgements The work was supported by the Human Resources Program in Energy Technology of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) granted financial resource from the Ministry of Trade, Industry & Energy, Republic of Korea (No. 20124030200110). This financial support is greatly acknowledged. References [1] Kvitrud A. Collision between platforms and ships in Norway in the period 2001e2010. In: 30th International Conference on Ocean, Offshore and Arctic engineering. Rotterdam, Netherlands: ASME; 2011. p. 637e41.

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[2] Stojko S, Shema B, Wreford J. Submarine hull impacts: validation of finite models from submarine impact trials. In: 79th Shock & Vibration Symposium,. Orlando, USA: SAVIAC; 2008. [3] Søreide TH, Amdahl J. Deformation characteristics of tubular members with reference to impact loads from collision and dropped objects. Nor Marit Res 1982;10(2):3e12. [4] Furnes O, Amdahl J. Ship collisions with offshore platforms. Hamburg, Germany: Intermaritec; 1980. p. 310e8. [5] Søreide TH, Moan T, Amdahl J, Taby J. Analysis of ship/platform impacts. In: Chryssostomidis C, Connor JJ, editors. International Conference on Behaviour of Offshore Structures. 3rd. Boston, USA: BOSS'82; 1982. p. 257e78. [6] Ellinas CP, Walker AC. Damage on tubular bracing members. In: Proceedings of IABSE Colloquium on Ship Collision with Bridges and Offshore Structures. Copenhagen, Denmark: IABSE; 1983. p. 253e61. [7] Frieze PA, Cho S-R. Impact damage and assessment of offshore tubulars. In: Offshore Technology Conference, OTC paper 7152. Houston, USA: OTC; 1993. p. 193e200. [8] de Oliveira JG. The behaviour of steel offshore structures under accidental collisions. In: Offshore Technology Conference, OTC paper 4136. Houston, USA: OTC; 1981. p. 187e98. [9] Guedes Soares C, Søreide TH. Plastic analysis of laterally loaded circular tubes. J Struct Eng 1983;109(2):451e67. [10] Wierzbicki T, Suh MS. Indentation of tubes under combined loading. Int J Mech Sci 1988;30(3e4):229e48. [11] Cho S-R, Seo BS, Cerik BC, Shin HK. Experimental and numerical investigations on the collision between offshore wind turbine support structures and service vessels. In: Amdahl, Ehlers, Leira, editors. Collision and Grounding of Ships and Offshore Structures. London, UK: CRC Press; 2013. p. 281e7. [12] Walker AC, Kwok M. Process of damage in thin-walled cylindrical shells. In: Smith, Clarke, editors. Advances in Marine Structures. London, UK: Elsevier Applied Science Publishers; 1986. p. 111e36. [13] Harding JE, Onoufriou A. Behaviour of ring-stiffened cylindrical members damaged by local denting. J Constr Steel Res 1995;33(3):237e57. [14] Karroum CG, Reid SR, Li S. Indentation of ring-stiffened cylinders by wedgeshaped indenters e Part 1: An experimental and finite element investigation. Int J Mech Sci 2007;49(1):13e38. [15] Walker AC, McCall S, Thorpe TW. Strength of damaged ring and orthogonally stiffened shells-part I: Plain ring stiffened shells. Thin Wall Struct 1987;5(6): 425e53. [16] Walker AC, McCall S, Thorpe TW. Strength of damaged ring and orthogonally stiffened shells-part II: T-ring and orthogonally stiffened shells. Thin Wall Struct 1988;6(1):19e50. [17] Villavicencio R, Guedes Soares C. Numerical modelling of the boundary conditions on beams stuck transversely by a mass. Int J Impact Eng 2011;38(5): 384e96. [18] Villavicencio R, Guedes Soares C. Numerical modelling of laterally impacted plates reinforced by free and end connected stiffeners. Eng Struct 2012;44: 46e62. [19] Villavicencio R, Guedes Soares C. Impact response of rectangular and square stiffened plates supported at two opposite edges. Thin Wall Struct 2013;68: 164e82. [20] Villavicencio R, Kim YH, Cho S-R, Guedes Soares C. Deformation process of web girders in small-scale tanker double hull structures subjected to lateral impact. Mar Struct 2013;32:84e112. [21] Jones N. The credibility of predictions for structural designs subjected to large dynamic loadings causing inelastic behaviour. Int J Impact Eng 2013;53: 106e14. [22] Hoo Fatt MS, Wierzbicki T. Damage of plastic cylinders under localized pressure loading. Int J Mech Sci 1991;33(12):999e1016. [23] Hoo Fatt MS, Wierzbicki T. Impact damage on long plastic cylinders. Int J Offshore Polar Eng 1992;2(2):147e56. [24] Wierzbicki T, Hoo Fatt MS. Impact response of a string-on-plastic foundation. Int J Impact Eng 1992;21(36):21e36. [25] Wierzbicki T, Hoo Fatt MS. Damage assessment of cylinders due to impact and explosive loading. Int J Impact Eng 1993;13(2):215e41. [26] Moussouros M, Hoo Fatt MS. Effect of shear on plastic denting of cylinders. Int J Mech Sci 1995;37(4):355e71. [27] Hoo Fatt MS, Wierzbicki T. Denting analysis of ring stiffened cylindrical shells. Int J Offshore Polar Eng 1991;1(2):137e45. [28] BSI, PD5500: Specification for unfired fusion welded pressure vessels. 2009. London, UK. [29] Korean Standard. Method of tensile test for metallic materials. KS B 0802. 2008. [30] ISO. ISO 26203-2:2011 Metallic materials e Tensile testing at high strain rates e Part 2: Servo-hydraulic and other test systems. 2011. [31] Cho S-R, Truong DD, Shin HK. Repeated lateral impacts on steel beams at room and sub-zero temperatures. Int J Impact Eng 2014;72:75e84. [32] Jones N, Paik JK. Impact perforation of steel plates. Ships Offshore Struct 2012;8(5):1e18. [33] Cowper GR, Symonds PS. Strain-hardening and strain-rate effects in the impact loading of cantilever beams. Technical Report, No. 28. Division of Applied Mathematics, Brown University; 1957.