This paper addresses a simplified analytical method for evaluating the impact responses of the stiffeners in a ship side shell subjected to head-on collision by a bulbous bow. The stiffeners are classified as the “central stiffener” and the “lateral stiffener” according to their relative position to the bulbous bow. In analytical predictions, it is assumed that the flexural bending of the central stiffener and plate occurs simultaneously. However, the deformation mode of the central stiffener outside the indenter contact region is simplified as linear to derive its deformation resistance. The curved deformation mode of the lateral stiffener is proposed to calculate the deformation resistance and to consider the interaction effect with the plate, which can cause the plate to fracture earlier. Model tests with three specimens (one unstiffened plate for reference and two stiffened plates) quasistatically punched by a conical indenter are performed to validate the proposed analytical method. Resistance-penetration curves and damage shapes for the three specimens are obtained. The experimental results illustrate the effects of the stiffeners on the deformation resistance and fracture initiation of the stiffened plate and the influence of stiffener tripping on the lateral resistance. Moreover, the experimental and analytical predicted results correspond well, suggesting that the proposed analytical method can accurately predict the crashworthiness of a ship side shell subjected to bulbous bow collision.
Ship side shells are generally equipped with stiffened steel panels to simplify fabrication; additionally, these panels have an excellent strength-to-weight ratio. During the sailing life of a ship, the ship side may suffer various types of loads. Among the applied loadings, the load from a collision with another ship can lead to serious consequences, such as loss of structural integrity, flooding of the ship tank, and severe oil pollution. Therefore, accurate crashworthiness assessment of ship side shells in the predesign stage has been continuously studied by engineers.
The commonly used approaches in ship collision investigations are experiments, numerical simulations, and simplified analytical methods [
Different types of stiffener [
Unlike experiments, numerical simulations are low-cost and easily repeatable with the help of powerful computers. The numerical simulation method has the ability to predict the collapse mode and reaction force of structures subjected to collisions when provided with appropriate modeling parameters. Until now, numerous failure criteria considering different factors (stress state, loading path, mesh size, strain rate, etc.) that can influence plate fracture have been proposed to predict the initial fracture of ship structures in collision and grounding analysis [
Compared with the first two methods, the simplified analytical method is the preferred tool in the predesign stage because this method can most rapidly assess the crashworthiness of ship structures [
Moreover, fracture prediction of the ship side plate is crucial for estimating energy dissipation and structural resistance. Several analytical expressions have been proposed to obtain the critical penetration depth of an unstiffened plate indented by a sphere [
In this study, simplified analytical methods are proposed to predict the deformation resistance and the critical penetration depth of a stiffened plate punched by a bulbous bow. Deformation modes for the central and lateral stiffeners are proposed, and the resistance-penetration relations are derived by theoretical calculations. In addition, the reduction in the critical penetration depth with the stiffener is derived considering the interaction effect between the plate and the stiffener. Moreover, experimental tests are conducted on specimens with different numbers of stiffeners quasistatically punched by a conical indenter. The experimental results validate the feasibility of the proposed method. Finally, some conclusions are drawn.
This section presents analytical predictions for the large deformation resistances of the central and lateral stiffeners and the initial fracture of the stiffened plate in a typical ship bulbous bow-side collision scenario, as shown in Figure The ship side shell is assumed to undergo head-on collision by a bulbous bow at the midspan between the web girders. The web girders are assumed to be stiff enough to constrain the boundary of the outer side plate. The bulbous bow is assumed to be rigid, and the shape of the bulbous bow is simplified as conical. The residual stress and initial deflection of the stiffened plate are not considered.
Ship bow-side collision scenario.
Based on the assumptions, theoretical deformation modes and the derived formulae for the central stiffener and the lateral stiffener are described in detail. The deformation shape of the central stiffener is identical to that of the plate, but the region not in contact with the indenter is treated as linear to merely calculate the lateral resistance for simplicity. In particular, a curved deformation mode for the lateral stiffener is proposed to consider its interaction effect with the side plate, which can influence the initial fracture of the ship side plate.
The analytical method to predict the deformation resistances of the central and lateral stiffeners is presented in this section. In general, the stiffener used for ship construction is the bulb-bar stiffener. Nevertheless, the resistance of the flat-bar stiffener is analyzed for simplicity.
The movement of the central stiffener is driven by the indenter; thus, the deformation shape of the central stiffener is the same as that of the side plate. In the whole deformation process, the central stiffener is assumed to maintain an in-plane deformation process, i.e., tripping of the stiffener is ignored.
Initially, in the elastic stage, the central stiffener mainly exhibits a bending effect. Assuming that the central stiffener is placed in the
Deformation mode of the central stiffener.
Then, according to the energy method, the instantaneous force of the stiffener in the elastic stage can be obtained by integrating equations (
In the plastic stage, the stiffener will experience bending and tension simultaneously. This deformation mode is also shown in Figure
Moreover,
The actual deformation of the central stiffener illustrates that the central stiffener exhibits global bending and tension effects when punched by a sphere. However, the stiffener outside the contact region with the indenter is treated as linear in the current study for simplicity to obtain a large deformation resistance. Thus, the global bending effect of the stiffener will concentrate in the plastic hinges (see the dashed area in Figure
The rotation angle of the stiffener
Moreover,
Thus, the relation between the indentation velocities of the point C
Then, the angular velocity of rotational stiffener
The bending energy rate of the central stiffener can be expressed as
Moreover, the tension strain
Thus, the rate of membrane tension of the stiffener can be expressed as
According to the upper bound theorem, the equilibrium equation can be expressed as
Finally, the instantaneous resistance of the stiffener at the plastic stage can be derived by substituting (
The deformation of the lateral stiffener is driven by the deformed plate. It is assumed that the lateral stiffener deforms simultaneously with the side plate. Thus, the overall movement of the lateral stiffener is the superposition of the lateral deflection from the plate and rotation with the plate. Here, it is assumed that the stiffeners will remain perpendicular to the side plate until plate fracture occurs.
The deformation mode of the stiffener is shown in Figure
Deformation mode of the lateral stiffener.
As shown in Figure
According to (
Similar to the relation between
As the stiffener is assumed to be displaced vertically, the tension strain
The strain rate of the stiffener
The rate of membrane energy can be expressed as
According to the upper bound theorem, the rate of work by the external load is equal to the rate of internal energy dissipation. The equilibrium can be expressed as
Thus, the instantaneous resistance of the lateral stiffener
Analytical fracture prediction for the ship side plate under a bulbous bow striking scenario is crucial for estimating the critical energy dissipation. Several equations were proposed to calculate the critical penetration depth of an unstiffened plate. Recently, an expression that was validated by a number of experiments was proposed by Zhang et al. [
In particular, the effect of the lateral stiffener on the initial fracture of the ship side plate is considered in this section. An analytical method is proposed to predict the fracture initiation of a stiffened side shell.
Section
The cross section at the maximum deflection of the lateral stiffener is extracted to analyze the influence of the stiffener on the deflection of the plate. Figure
Load state for the components. (a) Plate. (b) Stiffener.
Thus, the vertical force
Moreover, the tension forces from the plate are assumed to be very similar and are expressed as
On the
Considering (
According to (
Based on (
Substituting (
The quasistatic indentation experiments were performed at Huazhong University of Science and Technology. The setup used in the experiments is presented in Figure
Designed penetration test. (a) Setup. (b) Dimensions of the clamping system. (c) End connections of the stiffeners.
Dimensions of the conical indenter (dimensions in mm).
The initial distance between the indenter and the specimen was approximately 20 mm. The deformation of the specimens was enforced at a rate of ∼10 mm/min [
Three specimens were designed at one-fourth scale from the ship side, as shown in Figure
Dimensions of the specimens (dimensions in mm). (a) Specimen US. (b) Specimen 2FB. (c) Specimen 3FB.
The material used for the plates and stiffeners is grade B normal structural steel qualified by the CCS (China Classification Society), considering the availability of the thin steel plate and the loading capacity of the hydraulic cylinder. These steel plates were from the same batch supplied by WISCO (Wuhan Iron and Steel (Group) Company) and were all 3.15 mm thick. To obtain the mechanical properties of the steel, quasistatic tensile tests are conducted using three standard tensile specimens and procedures. The dimensions of the machined tension test pieces are shown in Figure
Dimensions of the standard tension tested piece (ASTM, E8).
Engineering stress-strain curve.
Mechanical properties of material.
Property | Symbol | Units | Specimens |
---|---|---|---|
Young’s modulus |
|
GPa | 207 |
Poisson’s ratio |
|
— | 0.3 |
Mass density |
|
kg/m3 | 7850 |
Yield stress |
|
MPa | 302.8 |
Ultimate tensile strength |
|
MPa | 408.4 |
Fracture strain |
|
— | 0.306 |
Strain-hardening index |
|
— | 0.2 |
The experimentally measured resistance-penetration curves are shown in Figure
Experimental resistance-penetration responses.
Deformation shapes of specimens when plates are initially fractured. (a) Specimen US. (b) Specimen 2FB. (c) Specimen 3FB.
Moreover, the influence of stiffener tripping on the lateral deformation resistance is evaluated. Figure
Experimental results for specimen 3FB and the replicate test.
In this section, the proposed analytical method is verified with respect to the large deformation resistance and the critical penetration depth of the stiffened plate by comparing the analytically predicted resistance-penetration curves with the experimental curves, as shown in Figure
Comparison of analytical and experimental resistance-penetration responses. (a) Specimen 2FB. (b) Specimen 3FB.
Calculation of the resistance-penetration relation for the stiffened plate.
The current study proposes not only analytical predictions for the deformation resistance of the central stiffener and the lateral stiffener but also a method to obtain the critical penetration depth of the stiffened plate. With the analytical solutions for the large deformation resistance and the critical penetration depth of the unstiffened plate in Ref. [
In particular, the current study considers the influence of the stiffener on the critical penetration depth of the plate. Thus, the solution of the critical penetration depth for the stiffened plate is described in detail. First, the critical penetration depth for the plate can be calculated according to (
This paper assesses the effects of stiffeners on the crashworthiness of the ship side shell impacted by a bulbous bow. Analytical expressions are presented to calculate the large deformation resistances for the lateral and central flat-bar stiffeners and the critical penetration depth for the stiffened plate. The deformation shape of the central stiffener is identical to that of the plate. However, the deformation mode of the central stiffener outside the contact region with the indenter is treated as linear for simplicity to calculate the deformation resistance. In addition, the deformation mode of the lateral stiffener is treated as a sine curve to obtain its deformation resistance and the reduction in the critical penetration depth of the plate due to the tension effect.
Model tests with three specimens (one unstiffened plate for reference and two stiffened plates) quasistatically punched by a conical indenter are performed. The resistance-penetration curves and the damage shapes are obtained through the experiments. The experimental results illustrate that the improvement in resistance due to the central stiffener is more remarkable than that of the lateral stiffener. In addition, a smaller distance between the lateral stiffener and the indentation position can lead to a lower critical penetration depth of the ship side panel. Moreover, the resistance response influenced by the tripping of the stiffener web is small. Furthermore, the similarity of the experimental and analytical resistance-penetration curves demonstrates the reliability and accuracy of the proposed simplified analytical method.
Half-width of the rectangular plate
Penetration depth reduced by the lateral stiffener
Elastic modulus
Elastic resistance of the central stiffener
Resistance of the lateral stiffener
Resistance of the unstiffened plate
Plastic resistance of the central stiffener
Resistance of the stiffened plate
Stiffener height
Half-length of the stiffener
Work hardening exponent of the plate material
Radius of the spherical punch
Plate thickness
Stiffener thickness
Vertical distance from the point C to initial shape
Maximum deflection of the central stiffener
Deflection of the lateral stiffener
Maximum transverse deflection of the lateral stiffener
Critical penetration depth of the plate
Critical penetration depth of the stiffened plate
Indenter wrapping angle at the outermost contact point
Rotation angle of the central stiffener
Tension strain of the central stiffener
Tension strain of the lateral stiffener
Horizontal distance from the point C to plate edge
Initial horizontal distance between the stiffener and plate edge
Flow stress of the plate
Flow stress of the stiffener.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This study was supported by the Foundation of Wuhan Polytechnic University (grant no. 2020Y09) and National Natural Science Foundation of China (grant no. 51579110).