A stress function for 3D frames

This paper generalises Rankine diagrams for 3D trusses to be applicable to 3D frames. Rankine diagrams are a graphical representation of a state of self-stress in a 3D truss, with the area of reciprocal polygons representing the axial force in their corresponding original bars. Rankine diagrams are a polyhedral version of the continuous Maxwell-Rankine stress function. In this paper we present a new stress function. It is piecewise linear and discontinuous and it allows the analysis of 3D frames, giving all six stress resultants of axial and shear forces and bending and torsional moments in any member. A succinct statement of the stress function is given in terms of Clifford Algebra

A stress function for 3D frames

This paper generalises Rankine diagrams for 3D trusses to be applicable to 3D frames. Rankine diagrams are a graphical representation of a state of self-stress in a 3D truss, with the area of reciprocal polygons representing the axial force in their corresponding original bars. Rankine diagrams are a polyhedral version of the continuous Maxwell-Rankine stress function. In this paper we present a new stress function. It is piecewise linear and discontinuous and it allows the analysis of 3D frames, giving all six stress resultants of axial and shear forces and bending and torsional moments in any member. A succinct statement of the stress function is given in terms of Clifford Algebra

Numerical simulation of wave energy converters using Eulerian and Lagrangian CFD methods

During the last years many concepts of wave energy converters (WEC) have been proposed. All are designed to generate energy at competitive economic rates in average sea states and also to survive extreme wave conditions. Due to the complexity of most offshore wave energy devices and their motion response in different sea states, physical tank tests are common practice for WEC design. Full scale tests are also necessary, but are expensive and only considered once the design has been optimised. Computational Fluid Dynamics (CFD) is now recognised as an important complement to traditional physical testing techniques in offshore engineering. Once properly calibrated and validated to the problem, CFD offers a high density of test data and results in a reasonable timescale to assist with design changes and improvements to the device. Within the EPSRC funded research project "Extreme Wave Loading on Offshore Wave Energy Devices: a Hierarchical Team Approach" the two WECs Pelamis and the Manchester Bobber are investigated using different Eulerian and Lagrangian CFD techniques. Both devices float on the water surface and generate the electricity from the motion of the waves. Pelamis' overall movement is limited due to the mooring system, but the individual segments are allowed to move in 6 degrees of freedom and interact with the waves and the adjacent segments. The dynamics of the Manchester Bobber comprise the nominally vertical motion of the floats, which are arranged in an array, and the highly complex interactions between the floats and the waves. Two test cases leading towards simulation of the full dynamics of Pelamis and the Manchester Bobber have been modelled using different CFD techniques. The problems involve the interaction between regular waves and fixed horizontal cylinders of different levels of submergence. Results are compared with experimental data to calibrate the CFD codes. Furthermore, results for the fluid-structure interaction of an oscillating cone on the water surface are presented. The complexity of this problem is rather high, as it involves rigid body motion of an axisymmetric body. The motion is not linear, but is generated as a Gaussian focused wave packet. Complex jet-effects occur at the intersection of water and body surface. These and the forces on the structure are discussed. Four different CFD codes are applied to simulate the test cases: Smoothed Particle Hydrodynamics, a Cartesian Cut Cell method based on an artificial compressibility method with shock capturing for the interface, and two pressure-based Navier-Stokes codes, one using a Finite Volume and the other a control volume based Finite Element approach. © 2010 by The International Society of Offshore and Polar Engineers (ISOPE)

Defining fitness in an uncertain world

The recently elucidated definition of fitness employed by Fisher in his fundamental theorem of natural selection is combined with reproductive values as appropriately defined in the context of both random environments and continuing fluctuations in the distribution over classes in a class-structured population. We obtain astonishingly simple results, generalisations of the Price Equation and the fundamental theorem, that show natural selection acting only through the arithmetic expectation of fitness over all uncertainties, in contrast to previous studies with fluctuating demography, in which natural selection looks rather complicated. Furthermore, our setting permits each class to have its characteristic ploidy, thus covering haploidy, diploidy and haplodiploidy at the same time; and allows arbitrary classes, including continuous variables such as condition. The simplicity is achieved by focussing just on the effects of natural selection on genotype frequencies: while other causes are present in the model, and the effect of natural selection is assessed in their presence, these causes will have their own further effects on genoytpe frequencies that are not assessed here. Also, Fisher’s uses of reproductive value are shown to have two ambivalences, and a new axiomatic foundation for reproductive value is endorsed. The results continue the formal darwinism project, and extend support for the individual-as-maximising-agent analogy to finite populations with random environments and fluctuating class-distributions. The model may also lead to improved ways to measure fitness in real populations