Ashby Method - 2.7 Materials selection and shape

Oct 29, 2026

Materials selection and shaping using Ashby’s approach focuses on how section geometry drives structural performance. It covers shape efficiency through shape factors and indices that include geometry, along with graphical ways to handle shape. The notes derive key mechanics relations for bending, torsion, and buckling, showing how area, moments of area, polar moments, stiffness, and critical loads depend on cross-sectional shape, including neutral axis and linear stress/strain variations.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages33

Reading Time1 h 23 min

AccessFree to download

2-Ashby Method

2.7 Materials selection and shape

Shape efficiency and shape indices

Bending mechanics (strain, stress, stiffness)

Torsion mechanics (shear strain, stress, stiffness)

Buckling and Euler critical load

| 2-Ashby Method | | --- | | 2.7-Materials selection and shape | | | | | | | --- | --- | | • Shape efficiency
• The shape factor, and shape limits
• Material indices that include shape
• Graphical ways of dealing with shape
Resources:
• M. F. Ashby, “Materials Selection in Mechanical Design” Butterworth Heinemann, 1999 Chapter 7
• W . C. Young, R. G. Budynas, “Roark’s Formulas for Stress and Strain” 7th ed, McGraw-Hill, 2002
• The Cambridge Material Selector (CES) software--Granta Design, Cambridge ([www.grantadesign.com](www.grantadesign.com)) | | | | | | --- | --- | | | | | | | | | | --- | --- | --- | --- | | | | | | | They depend on shapes | | | | | dA | Moment of area about axis x
Ixx = ∫A y 2 dA | | | | Moment of area about axis y | | | | | Iyy = ∫Ax 2 dA | | | | | Polar moment of area | | | | | J = ∫A r 2 dA | | | | | | | | | | | | --- | --- | --- | --- | --- | --- | |
| σ = | | FA | Stress
Strain | l): | | | ε = | δ L | | | | | | • From Hooke’s Law (linearly elastic materia σ = Eε ⇒ ε = σE = ~~ ~~FAE
• From the definition of strain:
ε = ~~ ~~δL~~ ~~ ⇒ δ = ~~ ~~FALE
S = F = AE Stiffness δ L | | | | | | | | | | --- | --- | --- | | | Pure Bending: Prismatic members subjected to couples acting in the longitudinal plane crossing one of the principal inertia axes
After deformation, the length of the neutral surface remains L. At other sections:
L′ = (ρ − y)θ
δ = L' − L = (ρ − y)θ − ρθ = −yθδ yθ y
ε z = L = − ρθ = − ρ Strain (varies linearly)ε max = ~~ ~~cρ~~ ~~ ⇒ ρ = ~~ ~~ε~~ ~~cmax~~ ~~
y ε z = − c ε max | | | | | | | --- | --- | --- | | | For a linearly elastic material:σ z = Eεz = − ~~ ~~yc~~ ~~ Eεmax
y
= − c σ max Stress (varies linearly)
σ max = ~~ ~~MZ = ~~ ~~IxMx/c~~ ~~ (c = ymax) Z = ~~ ~~I~~xx~~c~~ ~~ Bending strength modulus
σ = M IXX = moment of area
z Ixx /y about the bending axis | | |
Modes of loading: Bending | | | | | | | --- | --- | --- | --- | --- | --- | | | S = ~~ ~~Fδ = ~~ ~~C~~ ~~EIL3~~xx ~~ | | Stiffness | E Ixx d2~~ ~~ydz2~~ ~~ = M(z) | | | | IXX = moment of area about the bending axis | | | | | | C = constant (depending on the loading conditions) | | | | | | | | | | | | | | | | | | --- | --- | --- | |
T | Torsion: Prismatic members subjected to twisting couples or torques
Consider an interior section of the shaft. Asa torsional load is applied, the shear strain is equal to angle of twist.
) ) ρφ
AA' = Lγ = ρφ ⇒ γ = L
cφ ρ
γmax = L γ = c γmax
Shear strain ( ∝ twist angle and radius) | | | | | | | | --- | --- | --- | --- | | | | | | | | For a linearly elastic material:
τ = Gγ = ρ Gγ
c max ρ
= c τmax Shear stress (varies linearly)
τmax = ~~ ~~TQ = ~~ ~~TK/c~~ ~~ (c = ρmax) Q = ~~ ~~Kc~~ ~~ Twisting strength modulus
τ = ~~ ~~TK/ρ~~ ~~ K = torsional moment of area | | | |
Modes of loading: Torsion | | | | --- | --- | --- | |
T

| ST = T = KG Stiffness 􀀈􀀆 φ = Lγ = ~~ ~~L~~ ~~ τ = ~~ ~~L~~ ~~ Tρ􀀅􀀃
φ L 􀀇 ρ ρ G ρG K 􀀄
• Cross-sections of noncircular (non-axisymmetric) shafts are distorted when subjected to torsion.
• Cross-sections for hollow and solid circular shafts remain plain and undistorted because a circular shaft is axisymmetric.
K = J for circular sections only
τmax = ~~ ~~TQ = ~~ ~~TJ/c~~ ~~ (c = ρmax)
τ = ~~ ~~TJ/ρ~~ ~~ J = polar moment of area ST = ~~ ~~Tφ~~ ~~ = JGL~~ ~~ Stiffness | | | | | | | | --- | --- | --- | --- | | | | | | | | Buckling: Prismatic members subjected to compression in unstable equilibrium
• In the design of columns, cross-sectional area is selected such that
- allowable stress is not exceeded σ = ~~ ~~FA~~ ~~ ≤ σ y
- deformation falls within specifications δ = ~~ ~~FLAE ≤ δlim
• After these design calculations, may discover that the column is unstable under loading and that it suddenly buckles. | | | | |

FAQs

How is the critical buckling load estimated here?

It is calculated using Euler’s formula with EImin and the effective length Le. The corresponding critical stress relates to E and the slenderness parameter derived from L and the inertia radius r.

Oct 29, 2026

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