Peter Dewhurst

Analytical solutions and numerical procedures for minimum-weight Michell structures

A power-series method developed for plane-strain slip-line field theory is applied to the construction of minimum-weight Michell frameworks. The relationship between the space and force diagrams is defined as a basis for weight calculations. Analytical solutions obtained by the method are shown to agree with known solutions that were obtained through virtual displacement calculations. Framework boundary conditions are investigated, and matrix operators used in slip-line field theory are shown to apply to the force-free straight framework boundary-value problem. The matrix operator method is used to illustrate the transition from circular arc-based to cycloid-based Michell solutions. Finally, an example is given in the use of the method for evaluation of support boundary conditions.

An Investigaton of Minimum-Weight Dual-Material Symmetrically Loaded Wheels and Torsion Arms

A cylindrically symmetric layout of two opposite families of logarithmic spirals is shown to define the layout of minimum-weight, symmetrically loaded wheel structures, where different materials are used for the tension and compression members, respectively; referred to here as dual-material structures. Analytical solutions are obtained for both structure weight and deflection. The symmetric solutions are shown to form the basis for torsion arm structures, which when designed to accept the same total load, have identical weight and are subjected to identical deflections. The theoretical predictions of structure weight, deflection, and support reactions are shown to be in close agreement to the values obtained with truss designs, whose nodes are spaced along the theoretical spiral layout lines. The original Michell solution based on 45 deg equiangular spirals is shown to be in very close agreement with layout solutions designed to be kinematically compatible with the strain field required for an optimal dual-material design.

A theoretical and experimental investigation of a family of minimum-weight simply-supported beams

Space and force diagrams are presented for a family of simply-supported minimum-weight beam structures. The analytical methods, used to determine volume and displacements for a structure with different strength in tension and compression, are reviewed using the simplest circular arc beam example. The full design of a non-trivial symmetrical beam is described. The use of power series solutions and a matrix operator numerical method, developed for plane-strain slip-line field theory, are described to determine the beam layouts. The results of testing beam models, manufactured using both laser free-form sintering of nylon powder and CNC milling of aluminum alloy plate, are compared to the theoretical predictions of stiffness and maximum load capability. The theoretical predictions are shown to provide excellent agreement with both the isotropic aluminum test structure, and with the strongly bi-modulus nylon test structures.

Application of the PMR Topology Optimization Scheme to Dual Material Structures

An iterative finite element based topology optimization method based on prescribed material redistribution (PMR) has recently been demonstrated to effectively identify optimal topologies for single material structures. Through the application of a family of Beta probability density and cumulative distribution functions, the method provides a gradual transition from a unimodal distribution of uniform intermediate density to a bimodal distribution of void and fully dense regions. In this paper, the PMR method is extended to dual material structures in which the tensile member strength differs from the compressive member strength. Evolution to topologies which satisfy dual material optimality criteria is ensured through the introduction of local fictitious moduli based on the local stress state. For validation, both analytically derived solutions and a numerical dual material truss optimization procedure are applied. The truss optimization procedure is based on an assumed topology for which the optimal joint coordinates and member cross-sectional areas are determined using a quasi-Newton method and fully stressed design conditions. Validation problems considered include a two-bar dual material cantilever, a dual material truss structure subjected to combined loading, and dual material shear loaded frame structures with pre-existing members. For each case, validation is provided by correlating PMR results with results obtained from the dual material truss optimization procedure. It is demonstrated that for all cases considered, the PMR method provides a reliable tool for the identification of minimum weight dual material structures.Copyright