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

2. 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.

3. 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

4. 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

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

6. 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

7. 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

10. 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

11. 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

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

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

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

15. 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

16. 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

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

18. Shaped Materials on Materials Property Charts - Bending Failure