Bo Cerup Simonsen

Energy absorption and ductile failure in metal sheets under lateral indentation by a sphere

Abstract This paper is concerned with the mechanics of lateral indentation of a rigid sphere into a thin, ductile metal plate. The paper presents a study including experiments, analytical theories and finite element calculations. The focus is on the prediction plate failure and on the energy absorption up to this point. Load–displacement curves from experiments are presented for various plate geometries (circular, square, rectangular), indentor radii and locations of loading on the plate. The experiments show that the penetration to ductile fracture and the energy absorption is sensitive to both plate geometry, loading position and indentor geometry. The plate fails by localised necking followed closely by material fracture. Analytical theories are derived for the load–displacement behaviour of a plastic membrane up to failure. The point of plate failure is determined by a global stability criterion taking into account both the change of geometric and material stiffness during the indentation process. For the cases of axis-symmetric loading very good agreement between measured and theoretical load–displacement curves up to — and including — the point of initial plate failure is found. Curve fitting to the theoretical solutions produced the following expressions for the penetration and absorbed energy up to plate failure: δ f =1.41n 0.33 R 0.48 R b 0.52 , E= π C 0 t 0 RR b 0.318 R b R 0.607−0.387R b /R+1.20(R b /R) 2 +0.067(n−0.2) where C0 and n are the strength coefficient and the hardening exponent in the material power law, t0 is the initial plate thickness, Rb is the indentor radius and R is the plate radius. The tests were also modelled by the use of a commercially available finite element program. It is shown that the applied finite element method can accurately predict the response up to necking and fracture initiation for both symmetric and non-symmetric loading.

Experimental and Numerical Study of Interface Crack Propagation in Foam Cored Sandwich Beams

This article deals with the prediction of debonding between core and face sheet in foam-cored sandwich structures. It describes the development, validation, and application of a FEM-based numerical model for the prediction of the propagation of debond damage. The structural mechanics is considered to be geometrically nonlinear while the local fracture mechanics problem is assumed to be linear. The presented numerical procedure for the local fracture mechanics is a further development of the crack surface displacement method, here denoted as the crack surface displacement extrapolation method. The considered application example is to tear off one of the face laminates from the sandwich. This configuration can be found in many applications but is considered here to be occurring in a ship structure, particularly at the hard spot where the superstructure meets the deck. Face tearing experiments are carried out for structures with three different core densities, material tests are carried out and finally the face tearing tests are simulated with the developed procedure. It is shown that for low core densities, where the crack propagates in the interface immediately below the face sheet, there is fair agreement between experiments and theory. For cores with higher density, the crack tends to propagate in the laminate itself with extensive fiber bridging leading to rather conservative numerical predictions. However, for structural configurations where LEFM can be applied, the presented procedure is sufficiently robust and accurate to be used in a number of important engineering applications, for example risk-based inspection and repair schemes.

Ship grounding on rock-I. Theory

Abstract This paper presents a set of analytical expressions which can be used to calculate the reaction force on a ship bottom deformed by a conical rock with a rounded tip. Closed-form solutions are given for the resistance of inner and outer bottom plating, longitudinal stiffeners, girders and bulkheads and transverse frames, floors and bulkheads. The expressions are derived by use of an energy method or a type of ‘upper bound’ method which rigorously takes into account the effects of large plastic deformations, friction and fracture. A high level of generality for the methodology has been optained by postulating a global mode of deformation for the structure around the rock with one free parameter, the plate split angle, related to the shape of the deformation mode. It is assumed that intersections between structural components stay intact during the entire deformation process so the resistance of the individual structural members are derived according to the global deformation mode. The resistance of the entire structure is found by minimizing the energy dissipation from all the deformed members with respect to the plate split angle. In a subsequent paper (B.C. Simonsen, Ship grounding on rock—II. Validation and application, Marine structures 10 (1997) 563–584) it is shown that the theoretical model predicts the damage of four large scale tests and an accidental grounding with errors less than 10%. Moreover, it is illustrated by an example that the model evaluation of a grounding scenario is sufficiently fast to be used in a probabilistic framework in a Formal Safety Assessment.

Comparison of the crashworthiness of various bottom and side structures

The purpose of this work is to compare the resistance with damage of various types of double bottom structures in a stranding event. The comparative analyses are made by use of a commercial, explicit finite element program. The ship bottom is loaded with a conical indenter with a rounded tip, which is forced laterally into the structures in different positions. The aim is to compare resistance forces, energy absorption and penetration with fracture for four different structures. Those four structures are: a conventional double bottom, a structure (presently protected through a patent) with hat-profiles stiffened bottom plating, a structure where all-steel sandwich panel is used as outer shell and a bottom structure stiffened exclusively with hat-profiles. The paper shows that it is indeed possible to elevate the crashworthiness of side and bottom structures with regards to the loading considered here without increasing the structural weight.

GRACAT: software for grounding and collision risk analysis

From 1998 to 2000 an integrated software package for grounding and collision analysis was developed at the Technical University of Denmark within the project: Information technology for increased safety and efficiency in ship design and operation. The cost of the software development was 6 man-years. The software provides a toolbox for a multitude of analyses related to collision and grounding accidents. The software consists of three basic analysis modules and one risk mitigation module: (1) frequency, (2) damage, and (3) consequence. These modules can be used individually or in series and the analyses can be performed in deterministic or probabilistic mode. Finally, in the mitigation module risk profiles for the calculated consequences can be calculated and compared to alternative solutions by assignment of a cost function to the consequences. Thus, the possible analyses range from a deterministic crash analysis to a comparative risk analysis of two vessels operating on a specified route where the result is the probability density functions for the cost of oil outflow in a given area per year for the two vessels. In this paper, we describe the basic modelling principles and the capabilities of the software package. The software package can be downloaded for research purposes from www.ish.dtu.dk/GRACAT.

Plasticity, fracture and friction in steady state plate cutting

Abstract A closed form solution to the problem of steady-state wedge cutting through a ductile metal plate is presented. The considered problem is an idealization of a ship bottom raking process, i.e. a continuous cutting damage of a ship bottom by a hard knife-like rock in a grounding event. A new kinematic model is proposed for the strain and displacement fields and it is demonstrated that the analysis is greatly simplified if the strain field is assumed to be dominated by plastic shear strains and moving hinge lines. Also, it is shown that the present shear model offers the basis for a convenient extension of the presented plate model to include more structural members as for example the stiffeners attached to a ship bottom plating. The fracture process is discussed and the model is formulated partly on the basis of the material fracture toughness. The effect of friction and the reaction force perpendicular to the direction of motion is derived theoretically in a consistent manner. The perpendicular reaction force is of paramount importance for predicting the structural damage of a ship hull because it governs the vertical ship motion and rock penetration which is strongly coupled with the horizontal resistance and thus with the damaged length. The derived expressions are discussed and compared with previously published experimental results and formulas.

Experiments and theory on deck and girder crushing

Abstract This paper is concerned with theoretical and experimental analysis of deep plastic collapse of a deck or deep girder subjected to an in-plane, concentrated load. A theory is derived which is valid until initiation of fracture in the structure. The presented experimental results show load–deflection curves and modes of deformation for decks, stringer decks and deep thin-walled beams subjected to central or excentric point loads between transverse frames. Based on theory and experiments, various modelling aspects of the local/global failure of the beams are discussed. The agreement between the theoretical closed form solutions and the experimental results is good.

Ship grounding on rock—II. Validation and application

Abstract The primary purpose of this paper is to show examples of verification and application of the theory presented in B.C. Simonsen (Ship grounding on rock—I. Theory, Marine Structures 10 (1997) 519–562). Analysis of four largescale tests performed by the Naval Surface Warfare Centre (NSWC), USA, shows that the theory can predict the energy absorption of the four different ship bottoms with errors less than 10%. The rock penetration to fracture is predicted with errors of 10–15%. The sensitivity to uncertain input parameters is discussed. Analysis of an accidental grounding that was recorded in 1975 also shows that the theoretical model can reproduce the observed damage. Finally, it is demonstrated that the proposed methodology is sufficiently fast to be used in a probabilistic framework. Based on a set of stochastic input parameters, the probability density functions for the damage extents of a single hull VLCC were derived from simulations. Possible future applications of the methodology are discussed.

Dynamics of ships running aground

A comprehensive dynamic model is presented for analysis of the transient loads and responses of the hull girder of ships running aground on relatively plane sand, gravel, or rock sea bottoms. Depending on the seabed soil characteristics and the geometry of the ship bow, the bow will plow into the seabed to some extent. The soil forces are determined by a mathematical model based on a theory for frictional soils in rupture and dynamic equilibrium of the fluid phase in the saturated soil. The hydrodynamic pressure forces acting on the decelerated ship hull are determined by taking into account the effect of shallow water. Hydrodynamic memory effects on the transient hull motions are modeled by application of an impulse response technique. The ship hull is modeled as an elastic beam to determine the structural response in the form of flexural and longitudinal stress waves caused by the transient ground reaction and hydrodynamic forces. A number of numerical analysis results are presented for a VLCC running aground. The results include bow trajectory in the seabed, time variation of the grounding force, and the maximum values of the sectional shear forces and bending moments in the hull girder.

Plate tearing by a cone

The present paper is concerned with steady-state plate tearing by a cone. This is a scenario where a cone is forced through a ductile metal plate with a constant lateral tip penetration in a motion in the plane of the plate. The considered process could be an idealisation of the damage, which develops in a ship bottom raking accident or a collision with a floating object. The deformation involves a complex mixture of large plastic deformations, fracture and friction. The observed mode of deformation is idealised by a simplified, kinematically admissible deformation mode, and the rate of internal energy dissipation in plasticity, fracture and friction is quantified accordingly by analytical expressions. The idealised mode has two free parameters which are determined from the postulate that they adjust to give the least rate of energy dissipation. The theory is compared to a series of measurements. The coefficient of friction was not measured, so the calculations are presented for different realistic values and it is shown that, for a coefficient of friction of about 0.2, there is a reasonably good agreement between theory and measurements for the in-plane resistance force as well as for the out-of-plane reaction force.