Keywords

1 Introduction

Most ships needed to be composed of different materials for different missions to ensure tasks successful completion. A novel steel composed of Ni-Cr-Mn-Mo-N alloys had excellent magnetic stability and exhibited low magnetic conductivity of 1.001–1.003, which helped ships successfully complete their tasks. However, there were many differences in the ultimate bearing capacity between the novel steel and traditional steels. Therefore, the ultimate strength assessment of the ship made of the novel steel was urgently needed for guaranteeing life safety at sea [1].

Ultimate strength of hull structure was used to measure the ability to resist destruction, which reflected the reliability of ships [2]. Research about ultimate strength of hull structure was widely attracted by attention for a long time [3,4,5]. Most of experimental research was conducted on box beams, multi-span continuous beams and scaled modes instead of full ships [6, 7]. Theoretical calculation based on system theory and numerical analysis mainly contained progressive collapse methods and nonlinear finite element methods [8]. Wang compared the test results and the theoretical calculation, which suggested that the nonlinear finite element analysis was more accurate than the progressive collapse method [9].

Based on the certain ship, we carried out a four-point bending test to explore the ultimate bearing capacity of the typical cabin in the sagging state by a scaled model, and went on numerical simulation analysis by the nonlinear finite element method. We also investigated the effect of initial deformation and constitutive relation on ultimate strength. Our results helped the follow-up tests on the large-scale models and predicted the ultimate strength of the ship made of the novel steel.

2 Experimental Design

2.1 Experimental Model

Hulls had two states: sagging and hogging. According to different standards, the margin of ultimate strength of the certain ship in the sagging state was lower than that in the hogging state. Thus the sagging state was chosen as the hazardous state. For typical cabins, the number of spans had a great impact on the ultimate strength under the sagging condition. Based on our previous experiments, when the tests were carried out by the three-span continuous beams, the effect of the boundary constraints and the web frames could be considered, which facilitated the results close to the actuality. The web frames were designed as rigid support in order to stabilize structure. Considering the factors such as manufacture, transportation and loading capacity, we conducted the test by a scaled model of the typical cabin. The guidelines the scaled model (0.78 × 1.318 × 0.9 m) followed were as follows: (a) The structural geometry was designed with a certain scale ratio; (b) The scaled model was made of the novel steel; (c) The stress and strain of the scaled model was consistent with those of the full ship. Therefore, the destruction mode and order of the scaled model were consistent with those of the full ship. In addition, the longitudinals of the scaled model were used by flat steel for convenience.

2.2 Experimental Layout

In order to adapt the loading conditions in the laboratory, the scaled model was stretched as shown in Fig. 1. The two ends of the scaled model were placed on two supports to copy simple support. The extended sections were strong enough to ensure that the tested section was destroyed first. The high-strength transverse bulkheads were arranged at the loading and support locations to avoid structural instability. The connections between the tested section and the extended section were flanges by sixty-eight M24 screw-bolts. The brackets were added at the junction of the extended section and the flanges in order to ensure structural strength and avoid localized stress. We also processed the shear check of the screw-bolts and they were all satisfied test requirements. We made sure the loading capacity by designing the suitable force arms.

Fig. 1
A rectangular structure has 2 extended sides of 3413 and a tested section of 780 in the center. The tested section has two flanges on the sides and a loading of 3000 above it. The right side has an E deck, an F deck, and a bottom shell. A photo of an H-shaped structure is on the right side.

Experimental layout

Full size image

2.3 Experimental Points

Displacement test points. To monitor the overall deformation, five displacement test points were placed at the junction of the bottom shell and the central longitudinal section as shown in Fig. 2a. To monitor the out-of-plane deformation of the decks, four displacement test points were placed on the E deck at the B cross section as shown in Fig. 2b.

Fig. 2
A rectangular structure titled a is labeled number 1, 2, 3, 4, 5, A, B, C, loading, E deck, F deck, and bottom shelf. A table titled b has downward arrows labeled from numbers 6 to 9. A rectangle has 3 rows that are as follows. A, A 33, number 1. B, B 60, numbers 4 and 5. C, 86, number 7.

Experimental points. a Displacement test points at the bottom shell; b Displacement test points on the E deck; c Strain test points on the E deck

Full size image

Strain test points. Destruction locations were usually concentrated on decks or the side shell close to decks under sagging condition. There were eighty unidirectional strain gauges and eight triaxial strain gauges arranged the locations where damage might occur. The triaxial strain gauges were marked with circles as indicated in Fig. 2c.

2.4 Experimental Tests

The steel plates with the actual thickness of 3 and 5.3 mm were conducted uniaxial tensile tests for guiding numerical simulation. Due to various machining accuracy, various deformation was occurred, which affected ultimate bearing capacity of constructions. The initial deformation of localized deformation of plates, light side deformation of stringers and columnar deformation of plates could be calculated by the Smith method [10]. In order to obtain more precise estimate about initial deformation, the scaled model was tested by the handheld 3D laser scanner. The load was loaded through the multi-channel static loading system. The static strain instrument automatically recorded strain and displacement. The speed of loading was 50 kN/min. The frequency of the recording about loading was 10 Hz and the frequency of the recording about strain and displacement was 2 Hz. The initial load was 2 kN. When large nonlinear structural deformation occurred, the load was uninstalled until zero.

3 Test Results and Analysis

3.1 Constitutive Relation

As seen in Fig. 3, the constitutive relation of the two steel plates was not ideal elastic–plastic relationship. The specification required their yield strength to be no less than 400 MPa (ideal constitutive relation). The actual yield strength of processed steel was usually higher and they were 534 MPa (3 mm) and 491 MPa (5 mm) (actual constitutive relation). The following stress tested were all obtained through the ideal constitutive relation in terms of the actual strain.

Fig. 3
2 spline charts titled a and b plot stress versus strain. Chart a, (0.05, 650), (0.10, 710), (0.15, 790), (0.25, 815), (0.35, 850), (0.40, 780). Chart b, (0.00, 0), (0.05, 650), (0.10, 730), (0.20, 845), (0.30, 855), (0.40, 700). Values are estimated.

Constitutive relation of the two steel plates with different thickness. a 3 mm; b 5 mm

Full size image

3.2 Initial Deformation

The average initial deformation by the Smith method was ±2.2 mm (theoretical initial deformation). But the actual initial deformation of the decks was −4.2 ~ 4.2 mm and the actual initial deformation of the side shell was −3.1 ~ 5.3 mm by the handheld 3D laser scanner.

3.3 Stress and Displacement Results

Unidirectional stress on the E deck. The relationship between the stress on the three cross section and the test points was observed in Fig. 4. The A-33, the B-60 and the C-86 on the E deck were placed on the central longitudinal section and they were all the centres of the plates. When the load increased to 800 N, the stress on the A-33 and the C-86 transformed from tensile to compressive, however, the stress on the B-60 increased continuously, which manifested that the localized plates buckled and the buckling direction of the neighbouring localized plates was reverse.

Fig. 4
3 line graphs titled A, B, and C cross sections plot stress versus test points. A cross-section, 500 k N (number 35, 50). B cross-section, 925 k N (number 12, above 300). C cross-section, 250 k N (number 93, negative 400). Values are estimated.

Curves of stress on the three cross section

Full size image

Triaxial stress on the E deck. The B4-0°, the B4-45°and the B4-90° represented triaxial stress on the E deck as indicated in Fig. 5a. During sagging, the E deck was compressed. The B4-90° and B5-90° that stood for longitudinal stress were all compressed. But the B5-90° was higher than the B4-90° due to the reverse buckling direction among the neighbouring localized plates. When buckled, the localized plates were subjected to transverse stress. The B4-0° was tensile and the B5-0° was compressed because of the reverse buckling direction. Furthermore, the transverse stress started from beginning.

Fig. 5
A line graph titled a plots stress versus load. A line graph titled b plots displacement versus load. A line graph titled c plots stress versus height of test points. C is labeled B cross-section and divided into port and starboard on the left and right sides, respectively.

Stress and displacement results. a Curves of stress on the E deck; b Curves of absolute displacement on the E deck; c Curves of stress on the side shell

Full size image

Displacement on the E deck. The B cross section moved down along with increasing loading. For analysis of out-of-plane deformation, the absolute displacement based on the junction of the B cross section and the central longitudinal section (No.7) increased during loading as shown in Fig. 5b. The results indicated that the out-of-plane deformation occurred from beginning. When the load increased to 800 kN, the absolute displacement increased or reduced quickly, suggesting that the buckling direction was reverse among the adjacent localized plates.

Unidirectional stress on the side shell. As indicated in Fig. 5c, the unidirectional stress on the side shell varied with the distance between the test points and the baseline. In the sagging state, the decks were compressed and the bottom shell was tensile. The stress was zero at the neutral axis. The height of the neutral axis was 435 mm in the designed model. The distribution of the stress at the port and the starboard was symmetric because of structural symmetry. The height of the neutral axis stabilized 413 mm during loading. When the load was high up to 850 kN, the neutral axis moved to the baseline rapidly and the relationship between the stress and the load transformed from linear to nonlinear.

Unidirectional stress on the bottom shell. Because the bottom shell was thick enough to resist stretching, it was still in an elastic state though the model was under the limit.

4 Numerical Simulation Analysis

4.1 Nonlinear Finite Element Method

When the hull structure generated large deformation or geometric nonlinearity due to buckling, they were in the plastic state and nonlinear behaviour occurred. This paper simulated the ultimate strength through the nonlinear finite element method by ABAQUS. For computational convergence, the theoretical initial deformation was introduced.

The finite element model was composed of quadrilateral shell elements and a small number of triangular shell elements as shown in Fig. 6. The constrains were imposed at the connection of the model and the support. On the one side: Ux ＝ 0, Uy ＝ 0, Uz ＝ 0, θy = 0 and θz = 0; On the other side: Ux ＝ 0, Uy ＝ 0, θy = 0 and θz = 0. The concentrated force was loaded at the position consistent with the test. Bolt joints were simulated by MPC constrains. Meshing is important for calculation accuracy and workload. Although increasing the numbers of the element is beneficial for improving computational accuracy, the computational workload expanded. Therefore, we designed different element sizes in terms of different sections. The element size of the tested section was 9 mm and the element size of the extended section was 20 mm.

Fig. 6
A rectangular 3 D model with a textured surface. The rectangle has a horizontal line and an erect horizontal structure on its right and left sides. The bottom left corner contains an x, y, and z-axis that indicate the model's orientation.

Finite element model

Full size image

In order to provide reference for future simulation, we designed a simplified finite element model, which facilitated to reduce modelling and computing burden. The simplified model was the tested section extracted from the complete model. The simplified model gradually reached its limit by imposing an angle around x-axis.

4.2 Comparison of Test and Numerical Simulation

Ultimate Strength

The comparison of the test result and the simulation result was observed in Fig. 7. The two curves had similar trend. In terms of the test, there was an inflection point and the displacement was 12.5 mm when the load was up to 850 kN. The model was at the limit state and the displacement was 23 mm when the load was high up to 944 kN. The displacement increased by 46% and the load only increased by 10% from the inflection point to the limit point, which manifested that the hull structure buckled and yielded. The model was subjected to the bending moment calculated as follows:

$$M = P \times 2.17m$$
(1)
Fig. 7
A line graph plots load versus displacement. Test (0, 0), (5, 455), (10, 790), (15, 900), (20, 940), (25, 920). Numerical simulation (0, 0), (5, 310), (10, 670), (15, 880), (20, 895), (25, 830). Values are estimated.

Curves of load and displacement

Full size image

The P represented the load. The ultimate bending moment was 2.05E + 09 N·mm. In terms of the simulation result, there was an inflection point and the displacement was 13 mm when the load was up to 850 kN. The model was at the limit state and the displacement was 24 mm when the load was high up to 893 kN, which was lower by 5.4% than the test result. The deviation was caused by the welding. The tested model had higher stiffness than the finite element model due to high strength of welding. The results also declared that the numerical simulation was more reliable.

As indicated in Table 1, the ultimate bending strength calculated by the simplified model was lower by 7.2% than that calculated by the complete model, which manifested that the simplified model simulated accurately enough to apply for subsequent numerical calculation.

Table 1 Comparison of complete model and simplified model
Full size table

Destruction Mode. At the limit state, the E deck farthest from the neutral axis exhibited significant buckling and generated large deformation at the middle span as seen in Fig. 8a, which agreed well with the test result in Fig. 8b. The buckling direction was inverse among the adjacent localized plates. The F deck was in the stage of plastic deformation and underwent yield failure under limit. The side shell between the E deck and the F deck was compressed during sagging. It exhibited significant buckling and generated large deformation at the middle span as seen in Fig. 8c, which agreed well with the test result in Fig. 8d. In general, the destruction mode calculated by the numerical simulation was all accordant to that observed by the test.

Fig. 8
Two 3-D models with their respective color scales are titled a and c. The color scales are titled S Meier S N E C and U U 1. A close-up of a rectangular structure with 2 rows of plastic and a rectangular structure divided into multiple numbered rows and columns are titled b and d.

Comparison of test and numerical simulation. a Distribution of stress of the simplified model; b Deformation of the E deck; c Distribution of displacement of the simplified model; d Deformation of the side shell

Full size image

4.3 Interfering Factors on Ultimate Strength

Initial deformation. Based on the aforementioned simplified model, the theoretical initial deformation was introduced and the comparison was exhibited in Table 2. The ultimate bending moment by the theoretical initial deformation was higher by 2.8% than that by the actual initial deformation. Although the actual initial deformation was almost double, the effect on the ultimate bearing capacity was little. So, the numerical simulation by the theoretical initial deformation was adopted to avoid obtaining the initial deformation through experiments.

Table 2 Comparison of theoretical initial deformation and actual initial deformation
Full size table

Constitutive relation. Based on the aforementioned simplified model, the actual constitutive relation was introduced and the comparison was exhibited in Table 3 and Fig. 9. The ultimate bending moment by the actual constitutive relation was higher by 16.7% than that by the ideal constitutive relation, which implied that the constitutive relation had great effect on the ultimate strength. However, the more conservative results were obtained by the ideal constitutive relation.

Table 3 Comparison of ideal constitutive relation and actual constitutive relation
Full size table
Fig. 9
A line graph plots the bending moment versus the rotation angle. Ideal consecutive relation (0.0, 0), (1.0, 1.4), (2.0, 1.75), (3.0, 1.58). Actual consecutive relation (0.0, 0), (1.0, 1.4), (2.0, 1.95), (3.0, 2.1). Values are estimated.

Numerical comparison of ideal constitutive relation and actual constitutive relation

Full size image

5 Conclusions

The typical cabin under the limit was taken as the research object. By comparing the test results and the numerical simulation analysis, the conclusions were as below:

  1. (1)

    By comparing with the test results, the numerical simulation successfully simulated the typical cabin at the limit. The ultimate bending strength calculated by the numerical simulation was 1.94E + 09 N mm, which was only lower by 5.4%.

  2. (2)

    During sagging, the decks were compressed and the bottom shell was tensile. Under the limit, the decks and the side shell between the two decks exhibited significant buckling and yielding, and generated large deformation. But the bottom shell was still in an elastic state.

  3. (3)

    The Initial deformation had little impact on the ultimate bearing capacity, but the constitutive relation had great effect on the ultimate strength. The theoretical initial deformation and the ideal constitutive relation were introduced into the numerical simulation for more conservative results.

This paper can provide technical support for the structural design of the ship made of the novel steel and offer important reference significance for the ships made of other steels.