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Summary Aerospace Mechanics of Materials
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This is a complete summary to the course Mechanics of Materials given in the first year of Aerospace Engineering at the TU Delft.
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AE1108-II Aerospace Mechanics of MaterialsAerospace Mechanics of
Materials
AE1108-II
Delft University of Technology
1
, AE1108-II Aerospace Mechanics of Materials
Table of Contents
Stress………………………………………….………………………………………………………………………………………………………3
Sign convention for stresses……….………………………………………………………………………………………………………3
Complementary stress……….……………………………………………………………………………………….………………………3
Strain…….……………………………………………………………………………………………………………………………………………4
Sign convention for strain……….…………………………………………………………………………………………….……………4
Material properties……….……………………………………………………………………………………………………..…………….4
Generalized Hooke’s Law……….…………………………………………………………………………………………..………………5
Four basic principles………….……………………………………………………………………………………………………………….5
Axial member deformation……….……………………………………………………………………………………………………….5
Saint Venant Principle……….…………………………………………………………………………………………………….…………6
Structural stiffness concept……….…………………………………………………………………………………………….…………6
Statically indeterminate problems……….……………………………………………………………………………….……………6
Sign convention for tosion……….………………………………………………………………………………………..……………….7
Stress-strain distribution in circular shafts……….…………………………………………………………………………….…..7
The torsion formula……….…………………………………………………………………………………………………………………..8
Angle of twist……….……………………………………………………………………………………………………………..……………..8
Torsion of non-circular shafts……….……………………………………………………………………………….……………………8
Stress-state simplifications due to thin-walled structures ……….……………………….………………………………..8
Bredt’s formula……….……………………………………………………………………………………….………………………………..9
Angle of twist……….…………………………………………………………………………………………………………………………….9
Sign convention for beams……….………………………………………………………………………………………………………..9
Deformation behavior of a beam……….…………………………………………………..………………………………………….9
The shear formula……….……………………………………………………………………………………………………………………10
Shear flow in thin-walled beam sections…….…………………………………..……………………………………………….11
The concept of a plane stress transformation……….………………………………………..………………………………..11
Mohr’s circle…….…………………………………………………………………………………..………………………………………….12
The moment-curvature relationship……….………………………………………………………………………………………..13
Macaulay step functions……….…………………………………………………………………………….……………………………14
Superposition……….…………………………………………………………………………………….……………………………………15
Statically indeterminate beams……….……………………………………………………………..………………………………..16
2
, AE1108-II Aerospace Mechanics of Materials
Stress
𝐹
If an object is in static equilibrium it will look like the object on the right.
In reality the force does not act in an infinitesimally small point, but it will be
distributed over the surface as shows underneath.
𝐹 =𝑚∙𝑔
𝐹 = ∫ 𝜎 ∙ 𝑑𝐴
𝐴
𝜎 is called the stress and it has units 𝑁/𝑚2 . It expresses the intensity of a force. An average stress
can be calculated by:
𝐹
𝜎=
𝐴
However, this only applies for a uniformly distributed load over the entire area.
Stress is a vectors, however scalars are often easier to work with. Therefore a force can be broken
down to different directions and thus the stress.
Perpendicular to a surface: normal stress
The normal stress is computed by:
∆𝐹𝑧 𝑧
𝜎𝑧𝑧 = lim
∆𝐴→0 ∆𝐴
Parallel to a surface: shear stress
∆𝐹𝑦
𝜎𝑧𝑥 = lim
∆𝐴→0 ∆𝐴
𝑦
Sign convention for stresses 𝑥
Stress magnitude
If we have a shear stress:
𝜏𝑥𝑦 → |𝜏| × 𝑥 × 𝑦
Resultant force direction (positive or negative)
Surface normal direction (positive or negative) 𝑦
Complementary stress
Equilibrium is required for the element shown on the right. 𝑥
We find, for normal stress equilibrium:
3
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