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Unformatted text preview: gitudinal stress
• Hoop stress:
∑ Fz = 0 = σ 1(2t Δx ) − p(2r Δx )
• Longitudinal stress:
∑ F x = 0 = σ 2 (2 π rt ) − p (π r 2 ) σ 1dA
p dA σ 1dA σ 2 dA p dA pr
σ 1 = 2σ σ 2 = 2 * Assume t/(2r) very small 14 7 D’ τ max E’ • Pts A & B correspond to hoop
= σ 2 stress, σ 1 , and longitudinal
stress, σ 2 .
• σmax = σ1 and σmin= σ2
• Max in-plane shearing stress: τ max(in − plane) = σ 2 / 2 = pr
4t • Max out-of-plane shearing
stress: τmax(out of plane) = σ2 = pr/(2t)
15 7.10 Transformation of Plane Strain
• Plane strain - deformations of
material take place in parallel
planes and are the same in each
of the planes.
• Plane strain occurs in a plate
subjected along its edges to a
uniformly distributed load and
restrained from expanding or
contracting laterally by smooth,
rigid and fixed supports
• Components of strains are: εx ε y γ xy (ε z = γ zx = γ zy = 0)
16 8 Conventions
• ε +ve when elongated • γ xy +ve when angle...
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