Ultimate strength of steel brackets in ship structures
Introduction
Steel-plated structures are widely used in structural systems such as ships and offshore platforms. They are composed of plate panels supported by beam members such as longitudinal girders, transverse frames and stiffeners. As these supporting members tend to deform sideways, brackets are attached to prevent lateral-torsional buckling or ‘tripping’ (Paik and Thayamballi, 2003, Hughes and Paik, 2013).
The regulations established by various classification societies (DNVGL, 2012, IACS, 2006a, IACS, 2006b, LR, 2012) can be used to determine the scantling requirements for steel brackets at the structural-design stage. However, no detailed guidelines for predicting the strength performance of steel brackets are available.
It is noted that research efforts to investigate the strength of a steel bracket which has a triangular shape are relatively far less than rectangular plates such as plates and stiffened panels (Paik and Thayamballi, 2003, Vhanmane and Bhattacharya, 2008, Zhang et al., 2008, Paik and Seo, 2009a, Paik and Seo, 2009b, Shi and Wang, 2012, Paik et al., 2012). For over the last century, there were a number of researches related to buckling analysis of a equilateral triangular plate (Woinowsky-Krieger, 1933, Conway and Leissa, 1960, Wakasugi, 1960b, Wakasugi, 1961) and a isosceles triangular plate (Burchard, 1937, Wittrick, 1954, Li, 1959, Cox and Klein, 1955, Han, 1960, Wakasugi, 1960a, Salmon, 1962, Salmon et al., 1964, Ueda et al., 1976, Ueda et al., 1977, Tan et al., 1983, Krishnakumar, 1988, Wang and Liew, 1994, Vaaraniemi et al., 2003, Safar and Machaly, 2005, Aung, 2006) under compressive, shear force or combination of them.
In the early days, the buckling analysis of triangular plates analytically involved. In particular, Salmon (1962) studied the elastic stability characteristics of the connection utilising the Rayleigh–Ritz method under the assumption that the load is linearly applied on the loaded edge of the bracket with no horizontal displacement. Further, Salmon et al. (1964) conducted a series of laboratory tests to investigate the behaviour of eighteen connections with aspect ratio ranging from 0.75 to 2.0 for small and large plate dimensions to include plates where buckling and yielding are anticipated. It was confirmed that the maximum compressive stress takes place at the free edge which is found on his previous analytical work (Salmon, 1962). Wang and Liew (1994) utilised the pb-2 Rayleigh–Ritz method to investigate the triangular plates under isotropic in-plane compressive load. Further the study was extended to buckling of triangular thick plates based on the Mindlin plate theory (Xiang et al., 1994, Wang et al., 1994). Jaunky et al. (1995) studied the buckling of general triangular anisotropic plates subjected to combined in-plane loads utilising the Rayleigh–Ritz method. Xiang (2002) further investigated the buckling behaviour of triangular plates with both translational and rotational elastic edge constraints using the p-Ritz and presented extensive buckling factors for several selected isosceles and right-angled triangular plates at various edge boundary conditions under isotropic in-plane compressive load. Aung (2006) also used Mindlin plate theory to investigate the elastic–plastic buckling of various isosceles and right-angled triangular plates under combined compression and shear force.
As computing speeds and capabilities of numerical tools advance, it is expected that numerical simulations will play an important role and contribute to accelerate the level of researches higher than before. Particularly, the numerical simulations to examine the buckling analysis of triangular plates have been employed by lots of researchers (Ueda et al., 1976, Ueda et al., 1977, Tan et al., 1983, Vaaraniemi et al., 2003, Safar and Machaly, 2005).
The most distinguished numerical and experimental works are Ueda׳s series of studies (Ueda et al., 1976, Ueda et al., 1977, Ueda and Yao, 1987) for the triangular corner brackets using finite element method (FEM). A series of buckling analysis, elastic–plastic large deflection analysis and elastic–plastic thermal stress analysis were conducted for the triangular corner brackets subjected to compression to clarify the effects of initial imperfection due to welding (Ueda et al., 1977). It was observed that initial deflection decreases the rigidity and ultimate strength of a triangular corner bracket and this tendency is more remarkable when the side length to thickness ratio decreases. It was found that the welding residual stresses in the triangular corner bracket are usually in tension, and these stresses increase the buckling strength and the ultimate strength of the bracket. At last, a method was proposed to determine the optimum thickness of a corner bracket in relation to buckling and/or plastic strength (Ueda and Yao, 1987). The fundamental idea of the proposed method was that the collapse of a frame and a bracket takes place at the same time, since it is of no use for a bracket to carry more loads after the frame has collapse. However, in case of brackets attached to prevent lateral-torsional buckling or ‘tripping’ of stiffeners, it is not always true. If a bracket carries more loads, a stiffened-plate panel would stand more.
Furthermore, Safar and Machaly (2005) conducted experimental and analytical research work on triangular bracket plates considering both material and geometric nonlinearities. It was experimentally confirmed that yielding along the free edge usually takes place prior to buckling and the distribution of contact stresses between the beam and the bracket was triangular in shape with the peak stresses at free edge of bracket at buckling. It was concluded that the connection possesses a significant amount of post-buckling strength such that the limit load is almost twice the critical load. The results of all studies offer useful insights into the design of steel brackets. To the best of the authors׳ knowledge, however, a limited number of researches have been conducted on either the load-carrying capacity of brackets or their ultimate strength.
The aim of this study is to use nonlinear finite element analysis to examine the ultimate-strength characteristics of steel brackets. The structural features of steel brackets in real ships are investigated using data collected on six commercial ships currently in service: three tankers, two bulk carriers and one liquefied natural gas carrier (LNGC). Finite element modelling techniques are developed to calculate the ultimate-strength behaviour of steel brackets with a range of design variables, such as material type and breadth to height ratio. The validation of the developed modelling is conducted. Numerical computation is used to derive a plausible design formula that predicts the ultimate strength of a steel bracket.
Section snippets
Definition of geometrical parameters
The geometrical attributes of a typical steel bracket attached to a strong support member composed of plate panels are defined as in Fig. 1. The bracket is assumed to be attached to one side of the support member, i.e., on the positive side of the y-axis.
The following five parameters are considered: aspect ratio ; height to thickness ratio ; breadth to thickness ratio ; radius to height ratio ; and radius to breadth ratio .
Geometrical features of steel brackets
Data on 52 steel brackets
Finite-element model
A bracket with the standard parameters detailed in Section 2.2 is regarded as subject to in-plane bending moments arising from tripping or sideways deformation, as shown in Fig. 5(a). The structural response of the bracket is then as shown in Fig. 5(b or c). For simplicity, only the latter case is considered here. It is assumed that the vertical and horizontal edges of the steel bracket remain straight, and that the in-plane degrees of freedom at the horizontal edge are restrained.
Nonlinear
Effects of design variables
In this section, three sets of parameters are analysed and the results of the analysis are presented. First, to investigate the effects of the slenderness ratio on the ultimate strength, five radius ratios are considered. Second, the aspect ratio is taken as the design variable; its effect on ultimate strength is evaluated with reference to six aspect ratios. Finally, to consider possible dimensional variations in the full range of bracket characteristics, four aspect ratios, seven slenderness
Empirical formulations of the bracket ultimate strength
The results of the parametric analysis described in Section 4.3 are used to derive empirical formulations of predicting the ultimate strength of a steel bracket, as follows:
In the above, and are the non-dimensionalised ultimate bending moment for the simply supported and fixed boundary conditions, respectively, i is the number of fitting equations.
Fig. 25 reveals the
Discussion and usage of the design formulations
In the present study, the ultimate strength of steel brackets in ship structures is numerically investigated with varying parameters of influence such as radius ratio, aspect ratio and combination of them. Based on the computations, two sets of empirical formulae for predicting the ultimate strength are proposed for simply supported and fixed boundary conditions.
It is realized that the design formulations developed can be used for various purposes within the framework of multi-criteria
Conclusion
The aims of this study were to numerically examine the effects of various design variables on the ultimate-strength characteristics of steel brackets, and to propose a design formula for the ultimate strength of a steel bracket. A series of nonlinear finite element computations were undertaken to achieve these objectives. Several conclusions can be drawn from the results, as outlined below.
- (1)
First, the steel brackets used in real ship structures were investigated with reference to six types of
Acknowledgements
This work was supported by a 2-Year Research Grant of Pusan National University. This study was undertaken at The Lloyd׳s Register Foundation Research Centre of Excellence at Pusan National University. Lloyd׳s Register Foundation (LRF), a UK registered charity and sole shareholder of Lloyd׳s Register Group Ltd., invests in science, engineering and technology for public benefit, worldwide.
References (59)
Large deflection of simply supported isosceles triangular plates subjected to edge shear and compressive loading
Int. J. Eng. Sci.
(1988)- et al.
Nonlinear finite element method models for ultimate strength analysis of steel stiffened-plate structures under combined biaxial compression and lateral pressure actions – Part I: plate elements
Thin-Walled Struct.
(2009) - et al.
Elastic buckling of isotropic triangular flat plates by finite elements
Appl. Math. Model.
(1983) - et al.
Buckling of triangular plates under uniform compression
Eng. Struct.
(1994) - et al.
Buckling solutions for Mindlin plates of various shapes
Eng. Struct.
(1994) - et al.
Acoustic emission monitoring for detecting structural defects in vessels and offshore structures
Ships Offshore Struct.
(2009) User׳s Manual (Version 14.0)
(2012)Plastic Buckling of Mindlin Plates
(2006)Technical evolution in the ship structural analysis over the last 50 years
Ships Offshore Struct.
(2013)Beulspannungen der quadratischen Platte mit Schragsteife unter Druck bzw
Schub. Ing. Archiv.
(1937)
A method for investigating certain eigenvalue problems of the buckling and vibration of plates
J. Appl. Mech.
The buckling of isosceles triangular plates
J. Aeronaut. Sci.
Rules for classification of ships, Part 3.
Brittle fracture in ships – a lingering problem
Ships Offshore Struct.
Effect of weld shape imperfections on the structural hot-spot stress distribution
Ships Offshore Struct.
Reliability assessment of structures subjected to fatigue failure
Ships Offshore Struct.
The buckling and deflection of isosceles triangular plates
ASME J. Appl. Mech.
An analytical model to assess a ship side during a collision
Ships Offshore Struct.
Structural analysis and modifications – 2 tankers for offshore FPSO and FSO service
Ships Offshore Struct.
Ship Structural Analysis and Design
Detailed design and construction of the hull of a floating, production, storage and off-loading (FPSO) unit
Ships Offshore Struct.
Common Structural Rules for Double Hull Oil Tankers
Common Structural Rules for Bulk Carriers
Buckling analysis of general triangular anisotropic plates using polynomials
AIAA J.
Buckling problem of right-angled isosceles triangular plates
Publ. de l׳Inst. Math.
Recent advances on fatigue limit state design for FPSOs
Ships Offshore Struct.
Assessment of design criteria for fatigue cracking from weld toes subjected to proportional loading
Ships Offshore Struct.
A procedure for fatigue design of web-stiffened cruciform connections
Ships Offshore Struct.
Rules and Regulations for the Classification of Ships, Part 3. Ship Structures
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