Selection of Materials and Shape … when choosing among different materials with different shapes and when the design is one in which shape matters…

Introduction The concept of performance indices extended to include SHAPE. Maximize the performance by choosing the materials and shape in which the materials are available or feasible The best material-shape combination depends on the mode of loading.

Mode of Loading and Shape In axial tension, shape not important In bending, I-beam or hollow box are better than solid sections In torsion, circular tube is the best A shape factor () is defined for each mode of loading

Shape Factor A shape factor is a dimensionless number which characterizes the efficiency of a section shape in a given mode of loading Shape factor does not depend on scale Be is the shape factor for elastic bending; Tf is for failure due to torsion. Two other shape factors Te and Bf are similarly defined. The factors reflect the load bearing capacity of the shaped material compared with a solid cylindrical bar with the same cross-sectional area They all equal to 1 for solid circular section

Derivation of Shape Factor for Elastic Bending Displacement for 3-point bending Stiffness, for shaped section and for solid circular section Shape factor

Derivation of Shape Factor for Bending Strength Failure load for bending of beam with shaped section Failure load for bending of beam with circular section Ratio of the two strengths: Shape factor

Other Shape Factors For elastic torsion where K is a modified polar moment of area (J) For torsional strength where Q = J/rm For buckling of column, the Euler load The shape factor is, again, the ratio between the moments of area, i.e. equal to Be

Performance Indices which include Shape The procedure is similar to that without consideration of shape, with one extra step to bring in the shape. Shape factor has no effect on the choice of materials for Axial tension loaded components

Performance Indices which include Shape - Elastic Bending Aim: Stiff light beam m = Al & Bending stiffness Eliminate A, gives Comparing beams with different sections and materials

Performance Indices which include Shape - Elastic Twisting Aim: light stiff shaft m = Al Torsional stiffness Eliminate A gives

Performance Indices which include Shape - Failure of Beam Aim: light strong beam Failure load & Replace (I/ym) by Bf Eliminate A gives

Performance Indices which include Shape - Failure of Shaft Aim: light strong shaft Failure load Eliminate A gives

Shaped Materials on Materials Property Charts By rearranging, The equation means the material, when structured, behaves like a material with modulus E* and density * Introducing shape moves the material along a line of slope 1 on the chart

Shaped Materials on Materials Property Charts - Elastic Bending The material move from below the selection criteria line to above The effective performance of the shaped materials is improved

Shaped Materials on Materials Property Charts - Bending Failure Similarly, by rearranging The shaped behaves like a material with

Shaped Materials on Materials Property Charts - Bending Failure