Section 6 Whipping

6.1.1 The effects of a non-contact underwater explosion are described in Vol 1, Pt 4, Ch 2, 5 Underwater explosion (shock). Whilst the initial shock wave described in that section initiates whipping to some degree it is the pulsation of the bubble which leads to the majority of damage to the hull. The initial shock wave causes local hull damage and shock damage to the vessels equipment. In the strain history shown in Figure 2.6.1 Deck strains from hull whipping the initial shock wave can be seen to be not just the free response of an elastic system to an impulse as the amplitude continues to increase. There is a typical second kick to the system which stems from the first bubble pulse and which increases the response for several more cycles.

Figure 2.6.1 Deck strains from hull whipping

6.1.2 The nature and behaviour of the gas bubble are dependent upon the warhead charge size, the explosive composition, the detonation depth and the influence of boundaries such as the sea bed.

6.1.3 The maximum radius of the bubble at the end of the first expansion phase is given by:

6.1.4 The period of duration of the first bubble pulse is given by:

t bub

=

2,108sec

where

W and H are defined in Vol 1, Pt 4, Ch 2, 6.1 General 6.1.3.

6.1.5 Even a relatively modest warhead charge size can produce a bubble which displaces a large mass of water in a very short time frame. The momentum associated with this rapid incompressible flow of a sizeable volume of water constitutes a major loading mechanism for any structure within its sphere of influence.

6.1.6 The effect on the hull is a large amplitude vertical bending and vibration. This first introduces high shear forces at the quarter points which may cause shear wrinkling, this damage will probably not be catastrophic and the hull will go on to develop high compressive forces in the keel. These may cause buckling especially as the bottom structure may already be damaged from the initial shock wave. For extreme cases whipping may lead to the ‘back breaking’ and total loss of the ship.

6.1.7 An estimate of the hull natural frequency for steel ships is given by:

where

6.1.8 The risk of a whipping response from a particular threat can be determined using the approximation for the natural frequency and the bubble characteristics of Vol 1, Pt 4, Ch 2, 6.1 General 6.1.4.

6.1.9 If the threat is closer to the hull than 2R _bub then the bubble loading is to be specially considered.

6.2.1 The level to which a ship will be expected to survive an attack scenario that excites hull whipping is to be specified by the Owner and will remain confidential to LR.

6.2.2 The whipping threat level may be defined for a range of warheads detonating at a given stand off distance and longitudinal (axial) location. The probability of weapon hit locations can be determined from threat analyses which can be used to select the appropriate charge locations for the assessment.

6.2.3 It is also possible to undertake a parametric study to establish the detonation location which will lead to the worst case loading scenario. In this case, all possible hit locations that will induce whipping are assessed and the worst case induced bending moments are compared with an appropriate acceptance criteria. Contours from the keel of maximum threat size to induce failure can also be determined.

6.2.4 Where a shock threat is also being assessed for whipping effects, the warhead stand off distance from the keel is set to be the same as that which induces the prescribed severity of shock. A series of axial locations are assessed to establish the worst case excitation which is compared to the appropriate acceptance criteria.

6.2.5 The non-dimensional measure of whipping severity, commonly referred to as Whipping Factor, is simply the ratio of maximum induced hull girder bending moment at a section to the critical bending moment for that section. Each threat location assessed will generate a whipping factor which can be assigned to that particular location. In this way a series of iso-Whipping Factor contours can be mapped in the fluid beneath the keel for a particular threat weapon. These contours define hit volume boundaries within which that particular weapon will induce a known level of whipping response.

6.3.1 Ships for which a whipping assessment is performed will be eligible for a WH1, WH2 or WH3 notation as defined in Vol 1, Pt 4, Ch 2, 6.3 Notation assessment levels and methodology 6.3.4

6.3.2 There are two types of assessment to determine the whipping response of the hull girder.

• Simple 2D beam model.
• Advanced 3D beam model.

6.3.3 For most ships a simple analysis will be sufficient to determine the whipping capability of the hull girder. An advanced analysis will be required when:

• more detailed information is required on areas of a ship which have been shown by simple analysis to be deficient under whipping loads, for example where there are large structural discontinuities variations;
• the ship design cannot be idealised as a 2D beam, for example when it has an unusual structural configuration or has low frequency modes of vibration in additional to its vertical flexural modes;
• there is a requirement to predict the extent of plastic deformation in a section;
• the whipping threats are assessed in a shallow water environment.

6.3.4 A WH1 analysis method uses a 2D beam representation and a failure level criterion based on the bending moment to induce material yield.

6.3.5 A WH2 method of analysis uses a 2D beam representation and a failure level criterion based on the section ultimate bending moments. This will require assessment using ultimate strength calculations at each of the discrete sections of the hull girder beam model.

6.3.6 A WH3 method of analysis uses a 3D definition of a section of the hull girder and geometric and material failure criteria implicit in the chosen finite element code.

6.3.7 In each case, it is to be demonstrated that the hull section remains below the defined failure limits for all threat scenarios.

6.3.8 For certain ship types such as minesweepers, it will be necessary to carry out several levels of analysis. An elastic analysis is required for threat levels which are expected to be survived on a regular basis. An elasto-plastic analysis at a higher threat level for which the ship is expected to survive.

6.4.1 The modelling of ship interaction with explosion bubbles conveniently breaks down into a set of distinct sub-models.

6.4.2 The hull girder model is usually subdivided into at least twenty equal sections, each of which is assumed to form a ‘Timoshenko’ beam element. Since the stiffness and mass distributions may vary considerably along the length of a ship, a lumped mass/weightless beam representation is appropriate rather than a consistent mass model. The effect of shear deflection is to be included in the model.

6.4.3 The hull hydrodynamics may be modelled using standard strip theory to represent the effect of the inertia of surrounding water. At any lumped mass representing the hull girder, the added mass of water may be assumed using ‘Lewis’ forms coefficients. The added mass correction can be assumed to be constant for each mode of vibration.

6.4.4 For the bubble hydrodynamics it is assumed that the flow around the explosion bubble is inviscid and incompressible, that gaseous products obey ideal gas law, and that the bubble itself remains spherical. As a first approximation it may also be assumed that the bubble remains stationary but in general the migration is significant and should be considered. It is also assumed that the bubble motion is not modified by the presence of either the ship or the water surface. The loading model is to account for the dissipation of shock wave energy at the outset of detonation, generally achieved by using a modified initial radius for the bubble calculation.

6.4.5 The interaction hydrodynamics may also be assumed to be incompressible and inviscid consistent with the bubble hydrodynamics. The bubble radial flow may be resolved at the ship axis (the intersection line of the waterplane and the vertical centreline plane) at each lumped mass, into three components. Normally only the vertical z and athwartships y components need be considered as the bubble is assumed to be some distance from the ship. It may also be assumed that at each lumped mass, the transverse velocity around the whole section will be uniform in magnitude and direction.

6.4.6 The force acting on a strip is to account for this motion, plus the uniform pressure gradient assumed in the fluid which induces a buoyancy force proportional to the displaced volume of water. Wave generation and Bernoulli pressure effects may be neglected but the accelerations should account for the free surface reflection of the bubble.

6.4.7 Several assessment codes are available and calculation should be performed by a competent and experienced body with relevant experience and using recognised codes.

6.5.1 Advanced whipping assessments will normally be performed using a hybrid 3D/2D structural model for computational efficiency. However, care is to be exercised in the coupling of the 2D beam elements to the 3D section to ensure that this artificial boundary condition does not adversely influence the analysis. As an alternative, the ship may be defined as a full 3D shell model. In which case, it may be possible to invoke symmetry to reduce the problem size and reduce the computational burden.

6.5.2 More than one option exists for modelling the fluid domain. It may be modelled using a boundary element approach and coupled to the structural domain using a Doubly Asymptotic Approximation. Alternatively, a computationally intensive volume fluid element approach employing an Eulerian code may be used. This fluid domain model would have to be coupled to the Lagrangian structural domain through a general or arbitrary coupling scheme. The detonation process and the bubble development would be physically modelled in this approach. A combined approach would entail modelling an island of fluid around the ship, truncated by a boundary element surface on which a bubble loading model would be applied.

6.5.3 For surface ship problems, whichever solution strategy is adopted, the fluid solver must be able to cope with the proximity of the bubble to the free surface and where appropriate reflections from the sea bed. Analysis is to be undertaken by a competent and experienced body using recognised techniques and with the relevant expertise necessary to establish the correct interface strategy between structural and fluid element meshes.