- Email: [email protected]
An experimental study on the dynamic response of a hull girder subjected to near field underwater explosion
Marine Structures 44 (2015) 43e60
Contents lists available at ScienceDirect
Marine Structures journal homepage: www.elsevier.com/locate/ marstruc
An experimental study on the dynamic response of a hull girder subjected to near field underwater explosion Zhenhua Zhang*, Yuanxin Wang, Haifeng Zhao, Haifeng Qian, Jinlei Mou Department of Ship & Ocean Engineering, Naval University of Engineering, Wuhan 430033, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 January 2015 Received in revised form 4 June 2015 Accepted 23 July 2015 Available online xxx
An experiment of hull girder model subjected to near field underwater explosion at midship is implemented. High-speed photography is applied to achieve the time history of hog displacement of the hull girder model subjected to shock wave of undex. The determination method of hog distortion using these show-motion pictures is presented. The experiment also achieves the local plate distortion of the hull girder model. Based on these works, the damage mechanism and mode of hull girder subjected to near field undex at midship are discovered. Finally, the coupling effect between whole motion of hull girder and distortion of local structure is discussed. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Underwater explosion Shock wave Near field Hull girder Hog damage
1. Introduction Warships might be hit by all kinds of anti-ship weapons during maritime operation, and near field explosion of torpedo is one of the most destructive means of attacking warships. Many scholars developed a lot of theoretical and experimental investigations to study the dynamic response of ship structure subjected to undex, and they usually treated ship structure as a hull girder. The previous researches mainly focused on the whipping response of hull girder subjected to far field undex [1e4], but there are few literatures mentioned in the overall dynamic response research of whole warship
* Corresponding author. E-mail address: [email protected] (Z. Zhang). http://dx.doi.org/10.1016/j.marstruc.2015.07.002 0951-8339/© 2015 Elsevier Ltd. All rights reserved.
44
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
subjected to near field attack of torpedo. Zhang carried out a systematic study on the dynamic response of ship structure subjected to far field and near field undex, and put forward a similarity prediction method by using Rn number [5,6]. Zhang built up the finite element model of a whole ship by ABAQUS, and calculated the overall damage response of the hull girder subjected to the explosion below the midship 0.04 times the ship's length by boundary element method [7]. Zong made a theoretical study of dynamic response of floating and submerged hull girder subjected to underwater explosion bubble [8,9], and it was found that for a floating beam, the energy used for plastic deformation should be greater than the energy used for rigid body motion, and three plastic hinges cannot happen. The dynamic response of the ship structure subjected to undex was numerically simulated by other groups. Jen, C. Y. applied a nonlinear FEM/DAA coupling to research transient response of multiple intersecting spheres of deep-submerged pressure hull subjected to underwater explosion. According to the numerical result, the shock wave caused by the detonation of a 782 kg TNT charge is detonated 25 m below the midship section, make the pressure hull yield, but not collapse [10]. Ucar, H. used ABAQUS to research the dynamic Response of a Catamaran-Hull Ship subjected to 3401.94 kg TNT 113 and 150.52 m far away from the ship [11]. Chung, J. also simulated the dynamic behavior of high-speed catamaran craft subjected to underwater explosion [12]. His simulation result predicted the shock response of different point on the ship. Liang, C. C Shock built an FEM model of 2000-ton patrol-boat and gave the responses of the ship when subjected to undex of 576 kg TNT which was 30 m far away from the target [13]. In the study of Gong, S.W., an explicit finite element approach interfaced with the boundary element method was used for analysis on attenuation of floating structures response to underwater shock, where the bulk cavitation induced by underwater shock near the free surface was considered [14]. Sprague, M.A. employed cavitating acoustic finite element (CAFE) method to simulate the response of a realistic ship-like structure to an underwater explosion [15]. Shin, Y. S. and Schneider, N. A. built three-dimensional ship shock modeling subject to far-field undex and simulation has been performed and the predicted results were compared with ship shock test data [16,17]. In impact dynamics research, Jones [18] put forward theoretical solution of dynamic response of uniform freeefree beam under triangle impact loading and the stepwise beam under uniform impact loading. It is found that for a freeefree beam, only 25% of external energy is used for the plastic deformation of the beam, and the other 75% generates rigid body acceleration. So compared with clamped beam, free beam is more difficult to destroy. As for the similarity problem of typical structures subjected to impact load, the Rn number considering geometric feature put forward by Zhao is a landmark work [19]. In this paper, the hull girder model which represents the whole structure of a warship is made to study the overall damage of hull girder subjected to near field explosion below the midship. Highspeed photography is applied to achieve the time history of hog displacement of the hull girder model, and the damage mode of overall structure and local plate of the hull girder is obtained to support further theoretical analysis of the damage mechanism. 2. Experiment set-up and explosion case As shown in Fig. 1, the length of the experimental hull girder model is 2.5 m. The width is 0.3 m and the molded depth is 0.12 m. The model is divided into 9 cabins, the length of each cabin is 278 mm, and the cabins are numbered from 1# to 9# along the hull girder. In order to simulate the deck opening and observe the deformation conveniently, each cabin has a rectangular opening on its deck. The width of the deck stringer is 70 mm; the width of the side plate is 120 mm; the width of the bottom plate is 300 mm; the cross section of the hull girder is shown in Fig. 1(c), and the area is 0.002 m2; the inertia moment of the cross section is 4.866 106 m4; the height of the centroid is 0.046 m. The thickness of all plates is 3 mm. The mass of the hull girder is 48.32 kg; the draught of the model is 0.06 m. The material of the model is mild steel whose yield strength is 250 MPa and the elasticity modulus is 2.1 105 MPa. The plastic limit bending moment of the midship section is 22,795 N$m. Due to a higher spray produced by near-surface undex, the rods with the height of 0.4 m are welded on each bulkhead of the hull girder in order to be easy to observe the motion of model. There is a square marker on each rod, and the rods are numbered from 1 to 10, as shown in Fig. 2. Because the light in explosion container is limited and high-speed photography needs much stronger light, fluorescent
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
45
Fig. 1. Three basic orthographic view of experiment model of hull girder.
plates are installed on each square marker to strengthen the reflection of light. In addition, 6 strain measuring points are set up on the deck stringer. They are: (1) the points s1 and s2 which are on the center of the port and starboard deck stringers of 6# cabin respectively; (2) the points s3 and s4 which are on the center of the port and starboard deck stringers of 5# cabin respectively; (3) the points s5 and s6 which are on the center of the port and starboard deck stringers of 4# cabin respectively. When a ship is subjected to undex of dynamite explode under the near field of midship, the whole structure would be lethally damaged. As a result, the TNT is placed under the midship of the experiment model. The undex experiment with 55 g TNT put 0.2 m below 5# cabin was carried out in an airtight container, which is shown in Fig. 3. The actual depth of water in the experiment is 2 m. The hull girder was in a floating state before the experiment, and was located in the center of the container as far as possible. There are many observation windows on the explosion container which can help observing the dynamic response of the model subjected to undex. Due to the critical demand of
Fig. 2. Displacement marker of experiment model of hull girder.
46
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Fig. 3. Sketch map of slow-motion system of undex.
underwater photography for light resource, the container is equipped with the underwater lights, the over water lights and the lights out of the container. The container is also equipped with an operating platform above the waterline. The movement of the structure was shot by Fastcam Ultima APXI2 highspeed camera with the image intensifier tube in the experiment. The camera can speed up to 120,000 per second. The highest resolution is 1024 1024 pixels and the maximum recording time is 3 s. Implement scene of hull girder subjected to undex is shown in Fig. 4. The parameters measured in the experiment include the following aspects: (1) the time history of overall deformation of the hull
Fig. 4. Implement of hull girder subjected to undex.
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
47
girder subjected to near field undex, recorded by high-speed photography; (2) the ultimate deformation of the hull girder model, including the overall deformation and the local deformation, measured by the dividing ruler; (3) the dynamic strain signal, recorded by TEST system with the sampling frequency of the acquisition collector set to 500 Hz and the sampling length is 126 K. After checking whether all equipments were in a state of trigger, the explosive was detonated. 3. High-speed photography and interpretation method Fig. 5 shows the initial static state of the hull girder observed from the observation window. Because of the limitation of the shooting angle, 1# rod cannot be seen in the picture. However, due to the symmetry of the hull girder, the analysis of the movement would not be affected. The pictures shot by the high-speed photography were input into graphics software for interpretation to obtain the deformation of the hull girder. The coordinate is established, among which the left side of the hull girder in visual field is the origin and the horizontal direction is the X-axis as well as the vertical direction is the Y-axis. Fig. 5 shows the initial coordinates of the markers on the hull girder. As shown in Fig. 5, there is a certain deviation between the imaging size and the real size of the hull girder, which is embodied in the following two aspects. At first, the total length of the hull girder which is divided into 9 cabins is 2.5 m, and the real size of each cabin is 278 mm. However, the distance between rods (i.e. the length of the cabin) shown in Fig. 5 gradually shortens from left to right. Secondly, the real height of the rods is 400 mm, but the height of the rods shown in Fig. 5 gradually shortens from left to right. This is because the direction of the camera shot and the model is not strictly vertical and the light refracted through the camera. In order to obtain the real displacement of the hull girder subjected to undex, according to the principle of optics, the ratio of the image height to the object height is equal to the ratio of the image distance to the object distance in a static state:
D o Lo ¼ ¼l Di Li
(1)
where Do and Di are the object height and the image height, Lo and Li are the object distance and the image distance, respectively. Since the explosive is below the hull girder, the hull girder moves in a vertical plane theoretically. Thus the object distance and the image distance of each square marker are constant. So the ratio of the image height to the object height is still equal to the ratio of the image distance to the object distance after the movement of the markers.
D0o Lo ¼ ¼l D0i Li
(2)
Fig. 5. Initial coordinate of displacement marker of experiment model.
48
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
where D0o and D0i are the object height and the image height after the movement of the hull girder, respectively. The vertical displacement of the hull girder can be obtained from Eqs. (1) and (2):
D0o Do D0o Do ¼ ¼ 0 ¼l D0i Di Di Di
(3)
According to Eq. (3), the real vertical displacement of the hull girder can be calculated by the interpretation of the displacements of the markers in high-speed photography images, and this method can also be applied to the interpretation of the horizontal displacement. Table 1 shows the value of lx and ly which are the correction coefficients of the displacement of the markers in X-axis and Y-axis respectively. In this way, the real displacement of the marker can be calculated by measuring the displacement in the images in a certain time. Fig. 6 shows the scene of the hull girder model subjected to near field undex, and the interpretation result of the displacements of the markers in 5 ms, 13 ms, 21 ms and 29 ms. Since the distance between the explosive and the water surface is very short, the vertical movement of the water particles caused by the shock wave can be seen in 5 ms. In about 13 ms, it appeared an obvious spray on the water surface. The width of the spray almost covered all the ship length wholly and the maximum height is about 50 cm. It is found that the hull girder was obviously deformed, and the bottom of the hull girder between 3# rod and 8# rod was over the water surface. Because of the limitation of view field, only the displacements of the markers from 2# to 10# can be obtained from the images. However, some markers cannot be identified because the spray obstructed the view. According to the final deformation of the hull girder, the hull girder is rigid except the midship. Therefore, the displacement of the unseen markers can be obtained by stretching the real displacements of the left and right sides of the hull girder outwards linearly. The displacement of the markers on the hull girder model at different times could be calculated by the data of Fig. 2. The real displacements of the markers can be obtained by combining the correction coefficients in Table 1 and the result, listed in Table 2. The displacement of 1# marker can be obtained by stretching the real displacements of the left side of the hull girder outwards linearly. The displacements of the markers which are covered by the spray can be obtained by fitting the displacement data of the neighboring markers. In this way, the overall movement of the hull girder can be obtained. The hog distortion of the hull girder model is calculated by subtracting the average displacement of 1# and 10# marker from the displacement of the midship, as listed in Table 3. Fig. 7 shows the fitting curve and function which present the relationship between the hog distortion of hull girder model and the changing time. It is found in Fig. 7 that the hog distortion reaches a maximum of 355.1 mm at 29 ms. By measuring the experiment model, the actual final hog distortion of the hull girder model is 330.0 mm. The comparison result shows the high precision of this optical measurement method.
Table 1 Revise parameter of displacement marker between real and photograph. Rod
Dox
Dix
lx
D0oy
D0iy
ly
2 3 4 5 6 7 8 9 10
258 536 814 1091 1369 1647 1925 2202 2480
258 546 845 1118 1378 1635 1878 2108 2301
1.000 0.981 0.963 0.976 0.994 1.007 1.025 1.045 1.078
400 400 400 400 400 400 400 400 400
401 397 392 378 369 361 354 344 335
0.998 1.008 1.020 1.058 1.084 1.108 1.130 1.163 1.194
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Fig. 6. Coordinates of displacement marker of hull girder subjected to undex.
49
50
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Table 2 Real displacement of each marker (mm). Time
5 ms
Rod
x
y
x
13 ms y
x
21 ms y
29 ms x
y
2 3 4 5 6 7 8 9 10
3.0 3.9 3.9 1.0 1.0 1.0 5.1 5.2 0.0
39.9 51.4 75.5 109.0 100.8 82.0 55.4 23.3 1.2
29.0 24.5 15.4 25.4 11.9 16.1 20.5 1.0 18.3
30.9 93.7 174.5 233.9 247.2 186.1 111.9 68.6 26.3
113.0 107.0 71.2 69.3 77.5 112.8 e 55.4 57.1
78.8 178.3 270.4 340.7 358.8 298.1 e 86.0 19.1
142.0 149.2 108.8 128.9 e 106.8 e 65.8 61.4
125.7 228.7 323.5 405.3 e 331.3 e 114.0 13.1
Table 3 Hog displacement of experiment model of hull girder (mm). Time (ms) Hog displacement (mm)
5 103.9
13 276.2
21 349.9
29 355.1
4. Dynamic response of the hull girder model 4.1. Overall deformation The final deformation of the hull girder model subjected to undex is shown in Fig. 8. As shown in the figure, the model has a hog distortion. The obvious plastic hinge is produced at the midship of the hull girder, while the decks and side plates of the rest parts is in a rigid state without obvious plastic deformations. Fig. 9 shows the bottom deformation of the hull girder. It shows that the bottom of 5# cabin has a serious recess plastic deformation, which is much larger than those of 4# and 6# cabins. This reflects the coupling effect of the overall and local deformation of the hull girder obviously. On the one hand, the bilge at midship is subjected to great compressive stress as a result of the large hog deformation at
Fig. 7. Time history of hog distortion of experiment and regression result.
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
51
Fig. 8. Hog distortion of experiment model.
midship. On the other hand, an inward deformation is produced on the bottom of 5# cabin directly by the shock wave. Due to the above two factors, the bilge which is the longitudinal boundary of the bottom plate of 5# cabin produces the inward instability and severe creases, while the bulkheads which are the transverse boundary have small distortion. From the above analysis, the overall deformation of the hull girder destroyed the boundary restriction of the bottom plate at midship and aggravated the damage to the bottom; furthermore, the damage to the bottom reduced the stiffness of the midship section in turn, thereby enlarging the overall deformation of the hull girder. The coupling relationship between the overall and local deformation is the typical characteristic of the response of the hull girder subjected to near-field undex, which is worthy of further study. The measurement of the overall final deformation is shown in Fig. 10. Defining the line between the bottoms of the fore and aft bulkheads as the baseline, it is measured that the length of baseline is 2353.5 m; the distance between the baseline and the middle of the deck side line is 443.0 mm; the distance of the baseline and the intersection of the front bulkhead of 5# cabin and the bottom side line is 305.5 mm; the distance of the baseline and the intersection of the front bulkhead of 5# cabin and the deck side line is 417.5 mm; the distance of the baseline and the intersection of the rear bulkhead of 5# cabin and the bottom side line is 297.0 mm; the distance of the baseline and the intersection of the rear bulkhead of 5# cabin and the deck side line is 409.5 mm.
Fig. 9. Distortion of bottom plate of experiment model.
52
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Fig. 10. Measure result of final hog distortion of experiment model (mm).
From the data above, in terms of the bottom of the hull girder, the overall deformation of the hull girder is about 330.0 mm; in terms of the deck of the hull girder, the overall deformation of the hull girder is about 443.0 mm. The rotation angle of the plastic hinge at midship is about 16 . 4.2. Local deformation Fig. 11 shows the local deformation of the hull girder, and it could be found that the plastic deformations are mainly existed in 4#, 5# and 6# cabins. Fig. 11(a) and (b) show the deformation of the port and starboard side of 5# cabin at midship, respectively, and the side plate of 5# cabin lost its instability. Fig. 11(c) shows the deformation of the bottom of 5# cabin. Because the side boundary of the bottom lost its stability under the compressive stress produced by the overall deformation, the bottom lost the effective constraint imposed by the side boundary and resulted in the columnar deformation. Fig. 11(d) shows the deformation inside 4#, 5# and 6# cabins. Due to the release of the compressive stress on the side plate of 5# cabin, the deformations of the side plates of 4# and 6# cabins are much smaller. The bottoms of 4# and 6# cabins produce minor recessed deformations because of the
Fig. 11. Local distortions of No.4, 5 and 6 cabins of experiment model (unit: mm).
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
53
surrounding effective constraints. In terms of the deformations of the decks of the cabins, both of the port and starboard decks of 5# cabin produced the permanent tensile plastic deformation. While the decks of 4# and 6# cabins were still in elastic range without any plastic deformation which can be proved by the strain signal in Fig. 13. Because the bottom plate is subjected to the underwater explosion shock wave directly, the deformation of the bottom plate is the problem in which most scholars are very interested. Many scholars have done experimental and theoretical researches about the deformation of plate subjected to undex [20e29], among which the study of Veldman is related to the effect of the prestress on the shock response under undex [21]. However, most of the researches adopt the clamped boundary condition, and rarely refer to the effect of the overall movement of the hull girder on the deformation of the bottom plate. Fig. 12 shows the measure meshing of the bottom plates of 4#, 5# and 6# cabins. The cross range was divided equally into 5 parts, and the nodes are A-F, respectively. 4#, 5# and 6# cabins were divided equally into 6 parts along the hull girder, and the nodes are No.1e7, No.7e13 and No.13e19, respectively. The measurement result of the bottom plate distortion is listed in Table 4. It is found that the largest plastic deformations of the bottom plates of 4#, 5# and 6# cabins are 9.8 mm, 84.4 mm and 7.1 mm, which are located on C4, C9 and D15, respectively. Fig. 13 shows the dynamic strain signal of the measuring points where horizontal ordinate is time and the unit is second. Because the elastic modulus of model steel is 2.1 105 MPa and the yield stress is 250 MPa, the plate reaches into plastic range when the strain reaches 1200 mε at least. According to the measurement result, the strain signals of the points s3 and s4 which are on the deck of 5# cabin are very different from other points in both value and waveform. The maximum dynamic strain of the s3 point reaches 2200 mε and reaches plastic range obviously. While the maximum strain of the 4 measuring points in rigid region is only 421.8 mε and these points are still in elastic range. This is because the points s3 and s4 are located in the plastic hinge region on midship, and the other 4 measuring points are located in the rigid region of the hull girder. It is shown in Fig. 13 that the maximum strain of the s3 point is 2200 mε while the maximum strain of the s4 point is 961.6 mε. The reason why the difference occurs is probably the asymmetric response caused by the boundary instability of the bottom plate of 5# cabin.
5. Coupling effect of the overall and local deformation of the hull girder 5.1. Effect of the local deformation on the overall movement According to the overall deformation mode of the hull girder model, the plastic limit bending moment of the plastic hinge at midship M0 is playing an important role. When the explosion happens
Fig. 12. Measured coordinate of bottom plate of hull girder.
54
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Table 4 The bottom plate distortions of No.4, 5 and 6 cabins (mm). A
B
C
D
E
F
#4 cabin
1 2 3 4 5 6 7
0 0 0 0 0 0 0
0 3.8 6 6.5 7.5 6.5 0
0 3.4 7.3 9.8 9.2 7.2 0
0 3.1 6.8 9.3 9.4 6.5 0
0 3.1 5.3 6.5 6.9 5.3 0
0 0 0 0 0 0 0
#5 cabin
8 9 10 11 12 13
e e e e e 0
55.9 77.2 83.2 59.4 83.3 0
60.2 84.4 80.7 55.0 23.9 0
63.6 84.4 84.0 60.7 36.0 0
69.7 62.4 60.3 47.9 31.1 0
e e e e e 0
#6 cabin
14 15 16 17 18 19
0 0 0 0 0 0
4.6 4.5 4.1 2.5 1.4 0
5.4 7 6.5 4.4 2.3 0
5.6 7.1 6.2 5.1 2.6 0
4.9 4.9 4.3 3.4 2.3 0
0 0 0 0 0 0
under the bottom of the midship closely, the change of M0 is very complex and M0 reflects the coupling relationship between the overall and local deformation of the hull girder. On the one hand, the bottom plate at midship subjected to undex can produce a plastic deformation, or even break, which can reduce the moment of inertia of the midship section and consequently reduce the bending moment of the midship section; on the other hand, the increase of the rotation of the plastic hinge at midship can increase compressive stress imposed on the bottom plates, thereby causing the instability of the
Fig. 13. Dynamic strain signal of each measure point.
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
55
longitudinal boundary of the bottom. As a result, the bending resistance of the midship section is reduced further. MSC.DYTRAN is used to simulate the phenomenon in order to research the influence of local deformation on the overall deformation of hull girder. Lagrange shell element is used to model ship model and Euler element is used to model water and air. The ship model floats on the water element and there is air element above the ship model. The size of water region and air region are 3.6 m 1 m 0.5 m and 3.6 m 1 m 0.3 m. Euler element is hexahedron and the length of edge is 0.05 m. The minimum mesh size of shell element in the midship of model is 0.014 m. General coupling method is applied to simulate the fluidestructure interaction via MSC.DYTRAN as shown in Fig. 14. Supplementary video related to this article can be found at http://dx.doi.org/10.1016/j.marstruc. 2015.07.002. The deformation and movement of amidships cross section in different moment is shown in Fig. 15. Obviously, the deformation of amidships cross section reflects the local deformation of hull girder, while the movement of amidships cross section reflects the overall deformation of hull girder. As shown in Fig. 15, the amidships bottom plate is concaved severely before 0.9 ms while the movement of amidships cross section is small. This implies that the local deformation of amidships bottom plate plays a leading role before 0.9 ms. However, after 0.9 ms, the amidships cross section moves upwards obviously while the deformation of bottom plate changes slowly. This means that the overall deformation of hull girder is dominant after 0.9 ms. According to the deformation shape in Fig. 15, the moment of inertia of amidships cross section in different moment could be calculated, which represents the bending resistance of hull girder. The time history of the moment of inertia of amidships cross section is shown in Fig. 16. In Fig. 16, the curve could be divided into two linear parts where the boundary moment is 0.9 ms. In the first linear part, the concave of bottom plate of 5# cabin is the main reason which reduces the moment of inertia. While in the second linear part, the moment of inertia is decreased principally by the overall hog deformation of hull girder. Obviously, the degradation of the moment of inertia in the first part is much greater than the second.
5.2. Effect of the overall movement on the local deformation It could be illustrated how the local deformation is affected by the overall deformation of hull girder by analyzing the bottom plate deformation of 5#, 4# and 6# cabin when it is hogging upwards. The influence could be explained from two aspects as follows. (1). Effect of overall movement on the deformation mode of amidships bottom plate The deformation mode of amidships bottom plate is quite different when it is affected by the large hog deformation of hull girder. The influences are reflected in two aspects. Firstly, the boundary constraints of amidships bottom plate are destroyed, as shown in Fig. 17. When the hog deformation of the hull girder is very large, the bottom plate of amid ship is subjected to the lateral impact of the shock wave and great axial compression at the same time. As a result, the bottom
Fig. 14. FEM model of ship model subjected to near field undex.
56
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Fig. 15. Deformation and movement of hull girder.
plate loses the constraint of the bilge and the deformation pattern changes obviously. This effect is extremely destructive to the hull girder. Secondly, the deformations shape of amidships bottom plate changes greatly when large hog deformation happens. Due to losing the lateral constraints, the deformation shape of amidships bottom plate looks like a cylinder surface more than a parabolic one when a full clamped plate is subjected to underwater explosion. Due to the two reasons presented above, the deformation of bottom plate of 5# cabin is much greater than the deformation of full clamped plate subjected to the same case of underwater explosion, which is 20.73 mm according to the formulas of Nurick [30]. According to the simulation result of MSC.DYTRAN, the time history of axial stress and strain of point 1 and point 2 are shown in Fig. 18. The location of point 1 is in the middle of amidships bottom
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
57
Fig. 16. Time history of the moment of inertia of amidships section.
plate while point 2 is in the bilge of 5# cabin, as shown in Fig. 1. CowperseSymond model is adopted as the strain rate model of the model material, where D and q are equal to 40 and 5 respectively. The time history of the axial stress of point 2 reflects the typical characters of the compression of the amidships bottom plate when the hull girder has great hog deformation. Although point 1 and point 2 are both in the bottom plate of 5# cabin, their time history curves of axial stress are quite different. The tension of point 1 is obviously caused by the large concave deformation of the bottom plate of 5# cabin, which reflects the character of local deformation of amidships bottom plate subjected to the high pressure of shock wave of near field underwater explosion. Supplementary video related to this article can be found at http://dx.doi.org/10.1016/j.marstruc. 2015.07.002. The axial strain of point 1 is also very different from point 2. The time history of axial strain of point 1 reach its maximum 0.021 in a short time and remain relatively constantly after that time. However, it
Fig. 17. Deformation of bottom plate of 5# cabin.
58
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Fig. 18. Time history of axial stress and strain of point 1 and point 2.
is the axial strain of point 2 that should be paid more attention to. The bilge of 5# cabin continues to be compressed as the hog deformation of hull girder increases. As a result, the value of the axial strain of point 2 continually raises and finally reaches 0.047, which is much greater than point 1. (2). Effect of overall movement on the local deformation amplitude Nurick puts forward the formula of final plastic deformation of clamped rectangular plate subjected to undex [27]. The comparison between the theoretical prediction and the measured value of the maximum deformation of the amid ship bottom plate are shown in Table 5. The experimental maximum deformations of the bottom plates of 4# and 6# cabins are less than the theoretical values. In fact, the actual pressure called as Px(t) acting on the bottom of the hull girder subjected to the underwater explosion shock wave shall be obtained according to the coupling effect of the incident shock wave and the structure. If Pint(t) is the incident pressure of the shock wave, Pref(t) is the reflected pressure, and rwc is the impedance of the water, and mf is the added water mass. Px(t) satisfies:
8 > < Px ðtÞ ¼ Pint ðtÞ þ Pref ðtÞ Prad ðtÞ
dP ðtÞ d > : mf rad þ rw cPrad ðtÞ ¼ rw cmf vplate ðtÞ þ vbeam ðtÞ dx dt
(4)
where Prad(t) is the pressure of rarefaction wave which reflects into water because of the movement of structure. According to the theory of Geers, Prad(t) is obtained by the second formula in Eq. (4). Prad(t) is not only determined by the speed of the clamped plate vplate(t), but also influenced by the overall movement speed of the hull girder vbeam(t). Due to Eq. (4), this effect will reduce the actual wall pressure acting on the bottom. So this effect is beneficial to the structural protection of the hull girder. The distortions of the bottoms of 4# and 6# cabins are typical examples of this effect. If the total impulse acting on the bottom of 4# cabin subjected to undex is multiplied by 66%, the theoretical maximum deformation of the bottom is 9.78 mm, which is equal to the experiment result. Therefore, the total impulse acting on the bottom of 4# cabin is reduced by about 34% of it as a result of the overall rigid motion of the hull girder. Obviously, this effect cannot be ignored.
Table 5 Maximum distortions of bottom plate of hull girder compared with theoretical result.
#4 cabin bottom #6 cabin bottom
Blast distance (mm)
Total impulse (N.s)
Maximum deformation of clamped boundary (mm)
Real maximum deformation (mm)
342 342
50.20 50.20
13.70 13.70
9.8 7.1
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
59
6. Conclusions The experimental study about the hog damage of the hull girder model subjected to near field undex is carried out in this paper. The main conclusions are presented below: (1) The hull girder subjected to the shock wave of near filed undex produces a hog distortion and the plastic hinge is at midship and the rest parts are in a rigid state. High-speed photography is applied to achieve the time history of overall hog distortion of the hull girder model. The strain signal of the plastic hinge, whose waveform is in a shape of exponential decay, is very different from that of rigid parts whose waveform is in an elastic vibration state. (2) The effect of local deformation on overall distortion is mainly relative to the moment of inertia of midship. The amidships bottom subjected to undex would produce a recessed plastic deformation or even a crevasse, thereby reducing the moment of inertia of the amidships section and aggravating the overall distortion of the hull girder. The time history of the moment of inertia of amidships cross section could be divided into two linear parts. In the first linear part, the concave of amidships bottom plate is the main reason which reduces the moment of inertia. While in the second linear part, the moment of inertia is decreased principally by the overall hog deformation of hull girder. (3) The effect of the overall movement of hull girder on the local deformation can be considered as two aspects. One aspect is beneficial to the ship structure. The overall movement of the hull girder strengths the rarefaction wave in water, thereby reduces the wall pressure acting on the bottom plates. The other aspect is destructive and then causes a serious damage to hull girder. The hog distortion of the hull girder makes the amidships bottom under axial pressure. When the pressure reaches the bilge instability stress, the bilge will buckle and cannot provide clamped boundary for the bottom structure. The consequence of this effect is that the amidships bottom plate will produce a lateral deformation which is much larger than that of clamped plate.
Acknowledgments Support for this research was provided by a grant from the Naval Research Institute. The authors are also very grateful to Runquan Liu and Hailiang Hou for providing experimental support. References [1] Vernon TA. Whipping response of ship hulls from underwater explosion bubble loading. Report, AD-A178096. 1986. [2] Stettler JW. Damping mechanisms and their effects on the whipping response of a submerged submarine subjected to an underwater explosion [Dissertation for Master Degree]. Lehigh University; 1983. [3] Chen Y, Tong ZP, Hua HX, Wang Y, Gou HY. Experimental investigation on the dynamic response of scaled ship model with rubber sandwich coatings subjected to underwater explosion. Int J Impact Eng 2009;36(2):318e28. [4] Holloway DS, Thomas GA, Davis MR. Boundary element methods in the prediction of the acoustic damping of ship whipping vibrations. Aust N Z Ind Appl Math 2003;45:845e56. [5] Zhang ZH, Zhu X, Feng G, Sun XR, Mao LF. Numerical analysis of ship dynamic response due to shock waves induced by long-distance underwater explosion. Ship Build China 2003;44(4):36e42. [6] Zhang ZH, Wang Y, Zhang LJ, Yang JH, Zhao HF. Similarity research of anomalous dynamic response of ship girder subjected to near field underwater explosion. Appl Math Mech 2011;32(12):1491e504. [7] Zhang AM, Yao XL, Li J. The interaction of an underwater explosion bubble and an elastic-plastic structure. Appl Ocean Res 2008;30(3):159e71. [8] Zong Z. A hydroplastic analysis of a free-free beam floating on water subjected to an underwater bubble. J Fluids Struct 2005;20(3):359e72. [9] Zong Z. Dynamic plastic response of a submerged free-free beam to an underwater gas bubble. Acta Mech 2003;161: 179e94. [10] Jen CY, Lai WH. Transient response of multiple intersecting spheres of deep-submerged pressure hull subjected to underwater explosion. Theor Appl Fract Mech 2007;48(2):112e26. [11] Ucar H. Dynamic response of a catamaran-hull ship subjected to underwater explosions [Dissertation for Master Degree]. Naval Postgraduate School; 2006. [12] Chung J, Shin YS. Simulation of dynamic behaviour of high-speed catamaran craft subjected to underwater explosion. Ships Offshore Struct 2014;9(4):387e403.
60
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
[13] Liang CC, Tai YS. Shock responses of a surface ship subjected to noncontact underwater explosions. Ocean Eng 2006;33: 748e72. [14] Gong SW, Lam KY. On attenuation of floating structure response to underwater shock. Int J Impact Eng 2006;32(11): 1857e77. [15] Sprague MA, Geers TL. A spectral-element/finite-element analysis of a ship-like structure subjected to an underwater explosion. Comput Methods Appl Mech Eng 2005;195:2149e67. [16] Shin YS. Ship shock modeling and simulation for far-field underwater explosion. Comput Struct 2004;82:2211e9. [17] Schneider NA. Prediction of surface ship response to severe underwater explosions using a virtual underwater shock environment [Dissertation for Master Degree]. Naval Postgraduate School; 2003. [18] Jones N, Wierzbicki T. Dynamic plastic failure of a free-free beam. Int J Impact Eng 1987;6:225e40. [19] Zhao YP. Suggestion of a new dimensionless number for dynamic plastic response of beams and plates. Arch Appl Mech 1998;68:524e38. [20] Xie WF, Young YL, Liu TG, Khoo BC. Dynamic response of deformable structures subjected to shock load and cavitation reload. Comput Mech 2007;40(4):667e81. [21] Veldman RL, Ari-Gur J, Clum C, DeYoung A, Folkert J. Effects of pre-pressurization on blast response of clamped aluminum plates. Int J Impact Eng 2006;32:1678e95. [22] Rajendran R, Narasimhan K. Deformation and fracture behaviour of plate specimens subjected to underwater explosionea review. Int J Impact Eng 2006;32(12):1945e63. [23] Hung CF, Hsu PY, Hwang-Fuu JJ. Elastic shock response of an air-backed plate to underwater explosion. Int J Impact Eng 2005;31(2):151e68. [24] Ramajeyathilagam K, Vendhan CP. Deformation and rupture of thin rectangular plates subjected to underwater shock. Int J Impact Eng 2004;30:699e719. [25] Rajendran R, Narasimhan K. Damage prediction of clamped circular plates subjected to contact underwater explosion. Int J Impact Eng 2001;25(4):373e86. [26] Rajendran R, Narasimhan K. Linear elastic shock response of plane plates subjected to underwater explosion. Int J Impact Eng 2001;25(5):493e506. [27] Rajendran R. Reloading effects on plane plates subjected to non-contact underwater explosion. J Mater Process Technol 2008;206:275e81. [28] Rajendran R, Paik JK, Kim BJ. Design of warship plates against underwater explosions. Ships Offshore Struct 2006;1(4): 347e56. [29] Eslami-Majd A, Rahbar-Ranji A. Deformation behaviour of corroded plates subjected to blast loading. Ships Offshore Struct 2014;28(5):1683e90. [30] Nurick GN, Martin JB. Deformation of thin plates subjected to impulsive loadingda review, part II: experimental studies. Int J Impact Eng 1989;8(2):171e86.
Contents lists available at ScienceDirect
Marine Structures journal homepage: www.elsevier.com/locate/ marstruc
An experimental study on the dynamic response of a hull girder subjected to near field underwater explosion Zhenhua Zhang*, Yuanxin Wang, Haifeng Zhao, Haifeng Qian, Jinlei Mou Department of Ship & Ocean Engineering, Naval University of Engineering, Wuhan 430033, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 January 2015 Received in revised form 4 June 2015 Accepted 23 July 2015 Available online xxx
An experiment of hull girder model subjected to near field underwater explosion at midship is implemented. High-speed photography is applied to achieve the time history of hog displacement of the hull girder model subjected to shock wave of undex. The determination method of hog distortion using these show-motion pictures is presented. The experiment also achieves the local plate distortion of the hull girder model. Based on these works, the damage mechanism and mode of hull girder subjected to near field undex at midship are discovered. Finally, the coupling effect between whole motion of hull girder and distortion of local structure is discussed. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Underwater explosion Shock wave Near field Hull girder Hog damage
1. Introduction Warships might be hit by all kinds of anti-ship weapons during maritime operation, and near field explosion of torpedo is one of the most destructive means of attacking warships. Many scholars developed a lot of theoretical and experimental investigations to study the dynamic response of ship structure subjected to undex, and they usually treated ship structure as a hull girder. The previous researches mainly focused on the whipping response of hull girder subjected to far field undex [1e4], but there are few literatures mentioned in the overall dynamic response research of whole warship
* Corresponding author. E-mail address: [email protected] (Z. Zhang). http://dx.doi.org/10.1016/j.marstruc.2015.07.002 0951-8339/© 2015 Elsevier Ltd. All rights reserved.
44
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
subjected to near field attack of torpedo. Zhang carried out a systematic study on the dynamic response of ship structure subjected to far field and near field undex, and put forward a similarity prediction method by using Rn number [5,6]. Zhang built up the finite element model of a whole ship by ABAQUS, and calculated the overall damage response of the hull girder subjected to the explosion below the midship 0.04 times the ship's length by boundary element method [7]. Zong made a theoretical study of dynamic response of floating and submerged hull girder subjected to underwater explosion bubble [8,9], and it was found that for a floating beam, the energy used for plastic deformation should be greater than the energy used for rigid body motion, and three plastic hinges cannot happen. The dynamic response of the ship structure subjected to undex was numerically simulated by other groups. Jen, C. Y. applied a nonlinear FEM/DAA coupling to research transient response of multiple intersecting spheres of deep-submerged pressure hull subjected to underwater explosion. According to the numerical result, the shock wave caused by the detonation of a 782 kg TNT charge is detonated 25 m below the midship section, make the pressure hull yield, but not collapse [10]. Ucar, H. used ABAQUS to research the dynamic Response of a Catamaran-Hull Ship subjected to 3401.94 kg TNT 113 and 150.52 m far away from the ship [11]. Chung, J. also simulated the dynamic behavior of high-speed catamaran craft subjected to underwater explosion [12]. His simulation result predicted the shock response of different point on the ship. Liang, C. C Shock built an FEM model of 2000-ton patrol-boat and gave the responses of the ship when subjected to undex of 576 kg TNT which was 30 m far away from the target [13]. In the study of Gong, S.W., an explicit finite element approach interfaced with the boundary element method was used for analysis on attenuation of floating structures response to underwater shock, where the bulk cavitation induced by underwater shock near the free surface was considered [14]. Sprague, M.A. employed cavitating acoustic finite element (CAFE) method to simulate the response of a realistic ship-like structure to an underwater explosion [15]. Shin, Y. S. and Schneider, N. A. built three-dimensional ship shock modeling subject to far-field undex and simulation has been performed and the predicted results were compared with ship shock test data [16,17]. In impact dynamics research, Jones [18] put forward theoretical solution of dynamic response of uniform freeefree beam under triangle impact loading and the stepwise beam under uniform impact loading. It is found that for a freeefree beam, only 25% of external energy is used for the plastic deformation of the beam, and the other 75% generates rigid body acceleration. So compared with clamped beam, free beam is more difficult to destroy. As for the similarity problem of typical structures subjected to impact load, the Rn number considering geometric feature put forward by Zhao is a landmark work [19]. In this paper, the hull girder model which represents the whole structure of a warship is made to study the overall damage of hull girder subjected to near field explosion below the midship. Highspeed photography is applied to achieve the time history of hog displacement of the hull girder model, and the damage mode of overall structure and local plate of the hull girder is obtained to support further theoretical analysis of the damage mechanism. 2. Experiment set-up and explosion case As shown in Fig. 1, the length of the experimental hull girder model is 2.5 m. The width is 0.3 m and the molded depth is 0.12 m. The model is divided into 9 cabins, the length of each cabin is 278 mm, and the cabins are numbered from 1# to 9# along the hull girder. In order to simulate the deck opening and observe the deformation conveniently, each cabin has a rectangular opening on its deck. The width of the deck stringer is 70 mm; the width of the side plate is 120 mm; the width of the bottom plate is 300 mm; the cross section of the hull girder is shown in Fig. 1(c), and the area is 0.002 m2; the inertia moment of the cross section is 4.866 106 m4; the height of the centroid is 0.046 m. The thickness of all plates is 3 mm. The mass of the hull girder is 48.32 kg; the draught of the model is 0.06 m. The material of the model is mild steel whose yield strength is 250 MPa and the elasticity modulus is 2.1 105 MPa. The plastic limit bending moment of the midship section is 22,795 N$m. Due to a higher spray produced by near-surface undex, the rods with the height of 0.4 m are welded on each bulkhead of the hull girder in order to be easy to observe the motion of model. There is a square marker on each rod, and the rods are numbered from 1 to 10, as shown in Fig. 2. Because the light in explosion container is limited and high-speed photography needs much stronger light, fluorescent
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
45
Fig. 1. Three basic orthographic view of experiment model of hull girder.
plates are installed on each square marker to strengthen the reflection of light. In addition, 6 strain measuring points are set up on the deck stringer. They are: (1) the points s1 and s2 which are on the center of the port and starboard deck stringers of 6# cabin respectively; (2) the points s3 and s4 which are on the center of the port and starboard deck stringers of 5# cabin respectively; (3) the points s5 and s6 which are on the center of the port and starboard deck stringers of 4# cabin respectively. When a ship is subjected to undex of dynamite explode under the near field of midship, the whole structure would be lethally damaged. As a result, the TNT is placed under the midship of the experiment model. The undex experiment with 55 g TNT put 0.2 m below 5# cabin was carried out in an airtight container, which is shown in Fig. 3. The actual depth of water in the experiment is 2 m. The hull girder was in a floating state before the experiment, and was located in the center of the container as far as possible. There are many observation windows on the explosion container which can help observing the dynamic response of the model subjected to undex. Due to the critical demand of
Fig. 2. Displacement marker of experiment model of hull girder.
46
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Fig. 3. Sketch map of slow-motion system of undex.
underwater photography for light resource, the container is equipped with the underwater lights, the over water lights and the lights out of the container. The container is also equipped with an operating platform above the waterline. The movement of the structure was shot by Fastcam Ultima APXI2 highspeed camera with the image intensifier tube in the experiment. The camera can speed up to 120,000 per second. The highest resolution is 1024 1024 pixels and the maximum recording time is 3 s. Implement scene of hull girder subjected to undex is shown in Fig. 4. The parameters measured in the experiment include the following aspects: (1) the time history of overall deformation of the hull
Fig. 4. Implement of hull girder subjected to undex.
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
47
girder subjected to near field undex, recorded by high-speed photography; (2) the ultimate deformation of the hull girder model, including the overall deformation and the local deformation, measured by the dividing ruler; (3) the dynamic strain signal, recorded by TEST system with the sampling frequency of the acquisition collector set to 500 Hz and the sampling length is 126 K. After checking whether all equipments were in a state of trigger, the explosive was detonated. 3. High-speed photography and interpretation method Fig. 5 shows the initial static state of the hull girder observed from the observation window. Because of the limitation of the shooting angle, 1# rod cannot be seen in the picture. However, due to the symmetry of the hull girder, the analysis of the movement would not be affected. The pictures shot by the high-speed photography were input into graphics software for interpretation to obtain the deformation of the hull girder. The coordinate is established, among which the left side of the hull girder in visual field is the origin and the horizontal direction is the X-axis as well as the vertical direction is the Y-axis. Fig. 5 shows the initial coordinates of the markers on the hull girder. As shown in Fig. 5, there is a certain deviation between the imaging size and the real size of the hull girder, which is embodied in the following two aspects. At first, the total length of the hull girder which is divided into 9 cabins is 2.5 m, and the real size of each cabin is 278 mm. However, the distance between rods (i.e. the length of the cabin) shown in Fig. 5 gradually shortens from left to right. Secondly, the real height of the rods is 400 mm, but the height of the rods shown in Fig. 5 gradually shortens from left to right. This is because the direction of the camera shot and the model is not strictly vertical and the light refracted through the camera. In order to obtain the real displacement of the hull girder subjected to undex, according to the principle of optics, the ratio of the image height to the object height is equal to the ratio of the image distance to the object distance in a static state:
D o Lo ¼ ¼l Di Li
(1)
where Do and Di are the object height and the image height, Lo and Li are the object distance and the image distance, respectively. Since the explosive is below the hull girder, the hull girder moves in a vertical plane theoretically. Thus the object distance and the image distance of each square marker are constant. So the ratio of the image height to the object height is still equal to the ratio of the image distance to the object distance after the movement of the markers.
D0o Lo ¼ ¼l D0i Li
(2)
Fig. 5. Initial coordinate of displacement marker of experiment model.
48
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
where D0o and D0i are the object height and the image height after the movement of the hull girder, respectively. The vertical displacement of the hull girder can be obtained from Eqs. (1) and (2):
D0o Do D0o Do ¼ ¼ 0 ¼l D0i Di Di Di
(3)
According to Eq. (3), the real vertical displacement of the hull girder can be calculated by the interpretation of the displacements of the markers in high-speed photography images, and this method can also be applied to the interpretation of the horizontal displacement. Table 1 shows the value of lx and ly which are the correction coefficients of the displacement of the markers in X-axis and Y-axis respectively. In this way, the real displacement of the marker can be calculated by measuring the displacement in the images in a certain time. Fig. 6 shows the scene of the hull girder model subjected to near field undex, and the interpretation result of the displacements of the markers in 5 ms, 13 ms, 21 ms and 29 ms. Since the distance between the explosive and the water surface is very short, the vertical movement of the water particles caused by the shock wave can be seen in 5 ms. In about 13 ms, it appeared an obvious spray on the water surface. The width of the spray almost covered all the ship length wholly and the maximum height is about 50 cm. It is found that the hull girder was obviously deformed, and the bottom of the hull girder between 3# rod and 8# rod was over the water surface. Because of the limitation of view field, only the displacements of the markers from 2# to 10# can be obtained from the images. However, some markers cannot be identified because the spray obstructed the view. According to the final deformation of the hull girder, the hull girder is rigid except the midship. Therefore, the displacement of the unseen markers can be obtained by stretching the real displacements of the left and right sides of the hull girder outwards linearly. The displacement of the markers on the hull girder model at different times could be calculated by the data of Fig. 2. The real displacements of the markers can be obtained by combining the correction coefficients in Table 1 and the result, listed in Table 2. The displacement of 1# marker can be obtained by stretching the real displacements of the left side of the hull girder outwards linearly. The displacements of the markers which are covered by the spray can be obtained by fitting the displacement data of the neighboring markers. In this way, the overall movement of the hull girder can be obtained. The hog distortion of the hull girder model is calculated by subtracting the average displacement of 1# and 10# marker from the displacement of the midship, as listed in Table 3. Fig. 7 shows the fitting curve and function which present the relationship between the hog distortion of hull girder model and the changing time. It is found in Fig. 7 that the hog distortion reaches a maximum of 355.1 mm at 29 ms. By measuring the experiment model, the actual final hog distortion of the hull girder model is 330.0 mm. The comparison result shows the high precision of this optical measurement method.
Table 1 Revise parameter of displacement marker between real and photograph. Rod
Dox
Dix
lx
D0oy
D0iy
ly
2 3 4 5 6 7 8 9 10
258 536 814 1091 1369 1647 1925 2202 2480
258 546 845 1118 1378 1635 1878 2108 2301
1.000 0.981 0.963 0.976 0.994 1.007 1.025 1.045 1.078
400 400 400 400 400 400 400 400 400
401 397 392 378 369 361 354 344 335
0.998 1.008 1.020 1.058 1.084 1.108 1.130 1.163 1.194
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Fig. 6. Coordinates of displacement marker of hull girder subjected to undex.
49
50
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Table 2 Real displacement of each marker (mm). Time
5 ms
Rod
x
y
x
13 ms y
x
21 ms y
29 ms x
y
2 3 4 5 6 7 8 9 10
3.0 3.9 3.9 1.0 1.0 1.0 5.1 5.2 0.0
39.9 51.4 75.5 109.0 100.8 82.0 55.4 23.3 1.2
29.0 24.5 15.4 25.4 11.9 16.1 20.5 1.0 18.3
30.9 93.7 174.5 233.9 247.2 186.1 111.9 68.6 26.3
113.0 107.0 71.2 69.3 77.5 112.8 e 55.4 57.1
78.8 178.3 270.4 340.7 358.8 298.1 e 86.0 19.1
142.0 149.2 108.8 128.9 e 106.8 e 65.8 61.4
125.7 228.7 323.5 405.3 e 331.3 e 114.0 13.1
Table 3 Hog displacement of experiment model of hull girder (mm). Time (ms) Hog displacement (mm)
5 103.9
13 276.2
21 349.9
29 355.1
4. Dynamic response of the hull girder model 4.1. Overall deformation The final deformation of the hull girder model subjected to undex is shown in Fig. 8. As shown in the figure, the model has a hog distortion. The obvious plastic hinge is produced at the midship of the hull girder, while the decks and side plates of the rest parts is in a rigid state without obvious plastic deformations. Fig. 9 shows the bottom deformation of the hull girder. It shows that the bottom of 5# cabin has a serious recess plastic deformation, which is much larger than those of 4# and 6# cabins. This reflects the coupling effect of the overall and local deformation of the hull girder obviously. On the one hand, the bilge at midship is subjected to great compressive stress as a result of the large hog deformation at
Fig. 7. Time history of hog distortion of experiment and regression result.
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
51
Fig. 8. Hog distortion of experiment model.
midship. On the other hand, an inward deformation is produced on the bottom of 5# cabin directly by the shock wave. Due to the above two factors, the bilge which is the longitudinal boundary of the bottom plate of 5# cabin produces the inward instability and severe creases, while the bulkheads which are the transverse boundary have small distortion. From the above analysis, the overall deformation of the hull girder destroyed the boundary restriction of the bottom plate at midship and aggravated the damage to the bottom; furthermore, the damage to the bottom reduced the stiffness of the midship section in turn, thereby enlarging the overall deformation of the hull girder. The coupling relationship between the overall and local deformation is the typical characteristic of the response of the hull girder subjected to near-field undex, which is worthy of further study. The measurement of the overall final deformation is shown in Fig. 10. Defining the line between the bottoms of the fore and aft bulkheads as the baseline, it is measured that the length of baseline is 2353.5 m; the distance between the baseline and the middle of the deck side line is 443.0 mm; the distance of the baseline and the intersection of the front bulkhead of 5# cabin and the bottom side line is 305.5 mm; the distance of the baseline and the intersection of the front bulkhead of 5# cabin and the deck side line is 417.5 mm; the distance of the baseline and the intersection of the rear bulkhead of 5# cabin and the bottom side line is 297.0 mm; the distance of the baseline and the intersection of the rear bulkhead of 5# cabin and the deck side line is 409.5 mm.
Fig. 9. Distortion of bottom plate of experiment model.
52
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Fig. 10. Measure result of final hog distortion of experiment model (mm).
From the data above, in terms of the bottom of the hull girder, the overall deformation of the hull girder is about 330.0 mm; in terms of the deck of the hull girder, the overall deformation of the hull girder is about 443.0 mm. The rotation angle of the plastic hinge at midship is about 16 . 4.2. Local deformation Fig. 11 shows the local deformation of the hull girder, and it could be found that the plastic deformations are mainly existed in 4#, 5# and 6# cabins. Fig. 11(a) and (b) show the deformation of the port and starboard side of 5# cabin at midship, respectively, and the side plate of 5# cabin lost its instability. Fig. 11(c) shows the deformation of the bottom of 5# cabin. Because the side boundary of the bottom lost its stability under the compressive stress produced by the overall deformation, the bottom lost the effective constraint imposed by the side boundary and resulted in the columnar deformation. Fig. 11(d) shows the deformation inside 4#, 5# and 6# cabins. Due to the release of the compressive stress on the side plate of 5# cabin, the deformations of the side plates of 4# and 6# cabins are much smaller. The bottoms of 4# and 6# cabins produce minor recessed deformations because of the
Fig. 11. Local distortions of No.4, 5 and 6 cabins of experiment model (unit: mm).
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
53
surrounding effective constraints. In terms of the deformations of the decks of the cabins, both of the port and starboard decks of 5# cabin produced the permanent tensile plastic deformation. While the decks of 4# and 6# cabins were still in elastic range without any plastic deformation which can be proved by the strain signal in Fig. 13. Because the bottom plate is subjected to the underwater explosion shock wave directly, the deformation of the bottom plate is the problem in which most scholars are very interested. Many scholars have done experimental and theoretical researches about the deformation of plate subjected to undex [20e29], among which the study of Veldman is related to the effect of the prestress on the shock response under undex [21]. However, most of the researches adopt the clamped boundary condition, and rarely refer to the effect of the overall movement of the hull girder on the deformation of the bottom plate. Fig. 12 shows the measure meshing of the bottom plates of 4#, 5# and 6# cabins. The cross range was divided equally into 5 parts, and the nodes are A-F, respectively. 4#, 5# and 6# cabins were divided equally into 6 parts along the hull girder, and the nodes are No.1e7, No.7e13 and No.13e19, respectively. The measurement result of the bottom plate distortion is listed in Table 4. It is found that the largest plastic deformations of the bottom plates of 4#, 5# and 6# cabins are 9.8 mm, 84.4 mm and 7.1 mm, which are located on C4, C9 and D15, respectively. Fig. 13 shows the dynamic strain signal of the measuring points where horizontal ordinate is time and the unit is second. Because the elastic modulus of model steel is 2.1 105 MPa and the yield stress is 250 MPa, the plate reaches into plastic range when the strain reaches 1200 mε at least. According to the measurement result, the strain signals of the points s3 and s4 which are on the deck of 5# cabin are very different from other points in both value and waveform. The maximum dynamic strain of the s3 point reaches 2200 mε and reaches plastic range obviously. While the maximum strain of the 4 measuring points in rigid region is only 421.8 mε and these points are still in elastic range. This is because the points s3 and s4 are located in the plastic hinge region on midship, and the other 4 measuring points are located in the rigid region of the hull girder. It is shown in Fig. 13 that the maximum strain of the s3 point is 2200 mε while the maximum strain of the s4 point is 961.6 mε. The reason why the difference occurs is probably the asymmetric response caused by the boundary instability of the bottom plate of 5# cabin.
5. Coupling effect of the overall and local deformation of the hull girder 5.1. Effect of the local deformation on the overall movement According to the overall deformation mode of the hull girder model, the plastic limit bending moment of the plastic hinge at midship M0 is playing an important role. When the explosion happens
Fig. 12. Measured coordinate of bottom plate of hull girder.
54
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Table 4 The bottom plate distortions of No.4, 5 and 6 cabins (mm). A
B
C
D
E
F
#4 cabin
1 2 3 4 5 6 7
0 0 0 0 0 0 0
0 3.8 6 6.5 7.5 6.5 0
0 3.4 7.3 9.8 9.2 7.2 0
0 3.1 6.8 9.3 9.4 6.5 0
0 3.1 5.3 6.5 6.9 5.3 0
0 0 0 0 0 0 0
#5 cabin
8 9 10 11 12 13
e e e e e 0
55.9 77.2 83.2 59.4 83.3 0
60.2 84.4 80.7 55.0 23.9 0
63.6 84.4 84.0 60.7 36.0 0
69.7 62.4 60.3 47.9 31.1 0
e e e e e 0
#6 cabin
14 15 16 17 18 19
0 0 0 0 0 0
4.6 4.5 4.1 2.5 1.4 0
5.4 7 6.5 4.4 2.3 0
5.6 7.1 6.2 5.1 2.6 0
4.9 4.9 4.3 3.4 2.3 0
0 0 0 0 0 0
under the bottom of the midship closely, the change of M0 is very complex and M0 reflects the coupling relationship between the overall and local deformation of the hull girder. On the one hand, the bottom plate at midship subjected to undex can produce a plastic deformation, or even break, which can reduce the moment of inertia of the midship section and consequently reduce the bending moment of the midship section; on the other hand, the increase of the rotation of the plastic hinge at midship can increase compressive stress imposed on the bottom plates, thereby causing the instability of the
Fig. 13. Dynamic strain signal of each measure point.
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
55
longitudinal boundary of the bottom. As a result, the bending resistance of the midship section is reduced further. MSC.DYTRAN is used to simulate the phenomenon in order to research the influence of local deformation on the overall deformation of hull girder. Lagrange shell element is used to model ship model and Euler element is used to model water and air. The ship model floats on the water element and there is air element above the ship model. The size of water region and air region are 3.6 m 1 m 0.5 m and 3.6 m 1 m 0.3 m. Euler element is hexahedron and the length of edge is 0.05 m. The minimum mesh size of shell element in the midship of model is 0.014 m. General coupling method is applied to simulate the fluidestructure interaction via MSC.DYTRAN as shown in Fig. 14. Supplementary video related to this article can be found at http://dx.doi.org/10.1016/j.marstruc. 2015.07.002. The deformation and movement of amidships cross section in different moment is shown in Fig. 15. Obviously, the deformation of amidships cross section reflects the local deformation of hull girder, while the movement of amidships cross section reflects the overall deformation of hull girder. As shown in Fig. 15, the amidships bottom plate is concaved severely before 0.9 ms while the movement of amidships cross section is small. This implies that the local deformation of amidships bottom plate plays a leading role before 0.9 ms. However, after 0.9 ms, the amidships cross section moves upwards obviously while the deformation of bottom plate changes slowly. This means that the overall deformation of hull girder is dominant after 0.9 ms. According to the deformation shape in Fig. 15, the moment of inertia of amidships cross section in different moment could be calculated, which represents the bending resistance of hull girder. The time history of the moment of inertia of amidships cross section is shown in Fig. 16. In Fig. 16, the curve could be divided into two linear parts where the boundary moment is 0.9 ms. In the first linear part, the concave of bottom plate of 5# cabin is the main reason which reduces the moment of inertia. While in the second linear part, the moment of inertia is decreased principally by the overall hog deformation of hull girder. Obviously, the degradation of the moment of inertia in the first part is much greater than the second.
5.2. Effect of the overall movement on the local deformation It could be illustrated how the local deformation is affected by the overall deformation of hull girder by analyzing the bottom plate deformation of 5#, 4# and 6# cabin when it is hogging upwards. The influence could be explained from two aspects as follows. (1). Effect of overall movement on the deformation mode of amidships bottom plate The deformation mode of amidships bottom plate is quite different when it is affected by the large hog deformation of hull girder. The influences are reflected in two aspects. Firstly, the boundary constraints of amidships bottom plate are destroyed, as shown in Fig. 17. When the hog deformation of the hull girder is very large, the bottom plate of amid ship is subjected to the lateral impact of the shock wave and great axial compression at the same time. As a result, the bottom
Fig. 14. FEM model of ship model subjected to near field undex.
56
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Fig. 15. Deformation and movement of hull girder.
plate loses the constraint of the bilge and the deformation pattern changes obviously. This effect is extremely destructive to the hull girder. Secondly, the deformations shape of amidships bottom plate changes greatly when large hog deformation happens. Due to losing the lateral constraints, the deformation shape of amidships bottom plate looks like a cylinder surface more than a parabolic one when a full clamped plate is subjected to underwater explosion. Due to the two reasons presented above, the deformation of bottom plate of 5# cabin is much greater than the deformation of full clamped plate subjected to the same case of underwater explosion, which is 20.73 mm according to the formulas of Nurick [30]. According to the simulation result of MSC.DYTRAN, the time history of axial stress and strain of point 1 and point 2 are shown in Fig. 18. The location of point 1 is in the middle of amidships bottom
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
57
Fig. 16. Time history of the moment of inertia of amidships section.
plate while point 2 is in the bilge of 5# cabin, as shown in Fig. 1. CowperseSymond model is adopted as the strain rate model of the model material, where D and q are equal to 40 and 5 respectively. The time history of the axial stress of point 2 reflects the typical characters of the compression of the amidships bottom plate when the hull girder has great hog deformation. Although point 1 and point 2 are both in the bottom plate of 5# cabin, their time history curves of axial stress are quite different. The tension of point 1 is obviously caused by the large concave deformation of the bottom plate of 5# cabin, which reflects the character of local deformation of amidships bottom plate subjected to the high pressure of shock wave of near field underwater explosion. Supplementary video related to this article can be found at http://dx.doi.org/10.1016/j.marstruc. 2015.07.002. The axial strain of point 1 is also very different from point 2. The time history of axial strain of point 1 reach its maximum 0.021 in a short time and remain relatively constantly after that time. However, it
Fig. 17. Deformation of bottom plate of 5# cabin.
58
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
Fig. 18. Time history of axial stress and strain of point 1 and point 2.
is the axial strain of point 2 that should be paid more attention to. The bilge of 5# cabin continues to be compressed as the hog deformation of hull girder increases. As a result, the value of the axial strain of point 2 continually raises and finally reaches 0.047, which is much greater than point 1. (2). Effect of overall movement on the local deformation amplitude Nurick puts forward the formula of final plastic deformation of clamped rectangular plate subjected to undex [27]. The comparison between the theoretical prediction and the measured value of the maximum deformation of the amid ship bottom plate are shown in Table 5. The experimental maximum deformations of the bottom plates of 4# and 6# cabins are less than the theoretical values. In fact, the actual pressure called as Px(t) acting on the bottom of the hull girder subjected to the underwater explosion shock wave shall be obtained according to the coupling effect of the incident shock wave and the structure. If Pint(t) is the incident pressure of the shock wave, Pref(t) is the reflected pressure, and rwc is the impedance of the water, and mf is the added water mass. Px(t) satisfies:
8 > < Px ðtÞ ¼ Pint ðtÞ þ Pref ðtÞ Prad ðtÞ
dP ðtÞ d > : mf rad þ rw cPrad ðtÞ ¼ rw cmf vplate ðtÞ þ vbeam ðtÞ dx dt
(4)
where Prad(t) is the pressure of rarefaction wave which reflects into water because of the movement of structure. According to the theory of Geers, Prad(t) is obtained by the second formula in Eq. (4). Prad(t) is not only determined by the speed of the clamped plate vplate(t), but also influenced by the overall movement speed of the hull girder vbeam(t). Due to Eq. (4), this effect will reduce the actual wall pressure acting on the bottom. So this effect is beneficial to the structural protection of the hull girder. The distortions of the bottoms of 4# and 6# cabins are typical examples of this effect. If the total impulse acting on the bottom of 4# cabin subjected to undex is multiplied by 66%, the theoretical maximum deformation of the bottom is 9.78 mm, which is equal to the experiment result. Therefore, the total impulse acting on the bottom of 4# cabin is reduced by about 34% of it as a result of the overall rigid motion of the hull girder. Obviously, this effect cannot be ignored.
Table 5 Maximum distortions of bottom plate of hull girder compared with theoretical result.
#4 cabin bottom #6 cabin bottom
Blast distance (mm)
Total impulse (N.s)
Maximum deformation of clamped boundary (mm)
Real maximum deformation (mm)
342 342
50.20 50.20
13.70 13.70
9.8 7.1
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
59
6. Conclusions The experimental study about the hog damage of the hull girder model subjected to near field undex is carried out in this paper. The main conclusions are presented below: (1) The hull girder subjected to the shock wave of near filed undex produces a hog distortion and the plastic hinge is at midship and the rest parts are in a rigid state. High-speed photography is applied to achieve the time history of overall hog distortion of the hull girder model. The strain signal of the plastic hinge, whose waveform is in a shape of exponential decay, is very different from that of rigid parts whose waveform is in an elastic vibration state. (2) The effect of local deformation on overall distortion is mainly relative to the moment of inertia of midship. The amidships bottom subjected to undex would produce a recessed plastic deformation or even a crevasse, thereby reducing the moment of inertia of the amidships section and aggravating the overall distortion of the hull girder. The time history of the moment of inertia of amidships cross section could be divided into two linear parts. In the first linear part, the concave of amidships bottom plate is the main reason which reduces the moment of inertia. While in the second linear part, the moment of inertia is decreased principally by the overall hog deformation of hull girder. (3) The effect of the overall movement of hull girder on the local deformation can be considered as two aspects. One aspect is beneficial to the ship structure. The overall movement of the hull girder strengths the rarefaction wave in water, thereby reduces the wall pressure acting on the bottom plates. The other aspect is destructive and then causes a serious damage to hull girder. The hog distortion of the hull girder makes the amidships bottom under axial pressure. When the pressure reaches the bilge instability stress, the bilge will buckle and cannot provide clamped boundary for the bottom structure. The consequence of this effect is that the amidships bottom plate will produce a lateral deformation which is much larger than that of clamped plate.
Acknowledgments Support for this research was provided by a grant from the Naval Research Institute. The authors are also very grateful to Runquan Liu and Hailiang Hou for providing experimental support. References [1] Vernon TA. Whipping response of ship hulls from underwater explosion bubble loading. Report, AD-A178096. 1986. [2] Stettler JW. Damping mechanisms and their effects on the whipping response of a submerged submarine subjected to an underwater explosion [Dissertation for Master Degree]. Lehigh University; 1983. [3] Chen Y, Tong ZP, Hua HX, Wang Y, Gou HY. Experimental investigation on the dynamic response of scaled ship model with rubber sandwich coatings subjected to underwater explosion. Int J Impact Eng 2009;36(2):318e28. [4] Holloway DS, Thomas GA, Davis MR. Boundary element methods in the prediction of the acoustic damping of ship whipping vibrations. Aust N Z Ind Appl Math 2003;45:845e56. [5] Zhang ZH, Zhu X, Feng G, Sun XR, Mao LF. Numerical analysis of ship dynamic response due to shock waves induced by long-distance underwater explosion. Ship Build China 2003;44(4):36e42. [6] Zhang ZH, Wang Y, Zhang LJ, Yang JH, Zhao HF. Similarity research of anomalous dynamic response of ship girder subjected to near field underwater explosion. Appl Math Mech 2011;32(12):1491e504. [7] Zhang AM, Yao XL, Li J. The interaction of an underwater explosion bubble and an elastic-plastic structure. Appl Ocean Res 2008;30(3):159e71. [8] Zong Z. A hydroplastic analysis of a free-free beam floating on water subjected to an underwater bubble. J Fluids Struct 2005;20(3):359e72. [9] Zong Z. Dynamic plastic response of a submerged free-free beam to an underwater gas bubble. Acta Mech 2003;161: 179e94. [10] Jen CY, Lai WH. Transient response of multiple intersecting spheres of deep-submerged pressure hull subjected to underwater explosion. Theor Appl Fract Mech 2007;48(2):112e26. [11] Ucar H. Dynamic response of a catamaran-hull ship subjected to underwater explosions [Dissertation for Master Degree]. Naval Postgraduate School; 2006. [12] Chung J, Shin YS. Simulation of dynamic behaviour of high-speed catamaran craft subjected to underwater explosion. Ships Offshore Struct 2014;9(4):387e403.
60
Z. Zhang et al. / Marine Structures 44 (2015) 43e60
[13] Liang CC, Tai YS. Shock responses of a surface ship subjected to noncontact underwater explosions. Ocean Eng 2006;33: 748e72. [14] Gong SW, Lam KY. On attenuation of floating structure response to underwater shock. Int J Impact Eng 2006;32(11): 1857e77. [15] Sprague MA, Geers TL. A spectral-element/finite-element analysis of a ship-like structure subjected to an underwater explosion. Comput Methods Appl Mech Eng 2005;195:2149e67. [16] Shin YS. Ship shock modeling and simulation for far-field underwater explosion. Comput Struct 2004;82:2211e9. [17] Schneider NA. Prediction of surface ship response to severe underwater explosions using a virtual underwater shock environment [Dissertation for Master Degree]. Naval Postgraduate School; 2003. [18] Jones N, Wierzbicki T. Dynamic plastic failure of a free-free beam. Int J Impact Eng 1987;6:225e40. [19] Zhao YP. Suggestion of a new dimensionless number for dynamic plastic response of beams and plates. Arch Appl Mech 1998;68:524e38. [20] Xie WF, Young YL, Liu TG, Khoo BC. Dynamic response of deformable structures subjected to shock load and cavitation reload. Comput Mech 2007;40(4):667e81. [21] Veldman RL, Ari-Gur J, Clum C, DeYoung A, Folkert J. Effects of pre-pressurization on blast response of clamped aluminum plates. Int J Impact Eng 2006;32:1678e95. [22] Rajendran R, Narasimhan K. Deformation and fracture behaviour of plate specimens subjected to underwater explosionea review. Int J Impact Eng 2006;32(12):1945e63. [23] Hung CF, Hsu PY, Hwang-Fuu JJ. Elastic shock response of an air-backed plate to underwater explosion. Int J Impact Eng 2005;31(2):151e68. [24] Ramajeyathilagam K, Vendhan CP. Deformation and rupture of thin rectangular plates subjected to underwater shock. Int J Impact Eng 2004;30:699e719. [25] Rajendran R, Narasimhan K. Damage prediction of clamped circular plates subjected to contact underwater explosion. Int J Impact Eng 2001;25(4):373e86. [26] Rajendran R, Narasimhan K. Linear elastic shock response of plane plates subjected to underwater explosion. Int J Impact Eng 2001;25(5):493e506. [27] Rajendran R. Reloading effects on plane plates subjected to non-contact underwater explosion. J Mater Process Technol 2008;206:275e81. [28] Rajendran R, Paik JK, Kim BJ. Design of warship plates against underwater explosions. Ships Offshore Struct 2006;1(4): 347e56. [29] Eslami-Majd A, Rahbar-Ranji A. Deformation behaviour of corroded plates subjected to blast loading. Ships Offshore Struct 2014;28(5):1683e90. [30] Nurick GN, Martin JB. Deformation of thin plates subjected to impulsive loadingda review, part II: experimental studies. Int J Impact Eng 1989;8(2):171e86.