Aircraft Structures-Mohr-circle-part II-2010

Aircraft Structures-Mohr-circle-part II-2010 - Mohrs...

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1 24 Mohr’s Circle: Derivation () θ σ + θ σ + σ = σ θ σ + θ σ + θ σ = σ 2 cos 2 sin 2 2 sin sin cos zs ss zz ' s ' z zs 2 ss 2 zz ' z ' z θ σ 2 sin 2 cos 2 1 2 1 ' ' zs ss zz ss zz z z + = + The stress transformation for 2D system is The first equation can be re-written as ( ) ( ) 2 sin 2 2 cos 1 2 2 cos 1 ' ' zs ss zz z z + + + = Squaring and adding the shear stress equation squared to this gives ( ) ( ) 2 zs 2 ss zz 2 ' s ' z 2 ss zz ' z ' z 2 2 σ + σ σ = σ + σ + σ σ or θ o dz ds dl σ zs σ ss σ zs σ zz σ z'z' σ z's' 25 Mohr’s Circle: Derivation ( ) 2 zs 2 ss zz 2 ' s ' z 2 ss zz ' z ' z 2 2 σ + σ σ = σ + σ + σ σ t tan cons a I 2 1 2 1 ss zz = = = σ + σ { ( ) t tan cons r I I 4 1 2 2 2 2 1 2 zs 2 ss zz = = = σ + σ σ [] 2 2 ' s ' z 2 ' z ' z r a = σ + σ [ ] 2 1 2 zs 2 ss zz 4 2 1 σ + σ σ ( ) 2 ss zz σ + σ The above equation can now be rewritten as This is the equation of a circle of radius
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Aircraft Structures-Mohr-circle-part II-2010 - Mohrs...

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