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Mohrs Circle Lecture 2 ME EN 372 Andrew Ning aning@byu.edu - - PDF document
Mohrs Circle Lecture 2 ME EN 372 Andrew Ning aning@byu.edu - - PDF document
Mohrs Circle Lecture 2 ME EN 372 Andrew Ning aning@byu.edu Outline Material Properties Plane Stress Transformation Material Properties Isotropic materials: 2 constants to define. E : modulus of elasticity, or Youngs modulus :
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What is the modulus of elasticity?
Stress-strain curve
e p y u f
- σ
(strain) (stress)
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Stress-strain curve
e p y u f
- σ
(strain) (stress)
E
Stress-strain curve
e p y u f
- σ
(strain) (stress)
E σy, Sy σu, Su
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When stress levels rise, when distress appears, when tragedy strikes, too often we attempt to keep up the same frantic pace or even accelerate, thinking somehow that the more rushed our pace, the better off we will be... The wise...follow the advice “There is more to life than increasing its speed.” In short, they focus on the things that matter most... we learn over and over again the importance of four key relationships: with our God, with our families, with our fellowman, and with ourselves. – President Dieter F. Uchtdorf (Of Things That Matter Most) Boeing 777 Wing Test: https://youtu.be/Ai2HmvAXcU0
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What is Poisson’s ratio? What is the shear modulus of elasticity?
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Plane Stress Transformation
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Derive an expression for σ in terms of σx, σy, τxy. σ = σx cos2 φ + σy sin2 φ + 2τxy sin φ cos φ τ = (σy − σx) sin φ cos φ + τxy(cos2 φ − sin2 φ)
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Using a few trig identities, just for convenience later... σ = σx + σy 2
- +
σx − σy 2
- cos 2φ + τxy sin 2φ
τ = − σx − σy 2
- sin 2φ + τxy cos 2φ
Principal Stresses
φP = 1 2 tan−1 2τxy (σx − σy)
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σ1, σ2 = σx + σy 2 ± σx − σy 2 2 + τ 2
xy
τ = 0 The angles for maximum shear stress are ±45◦ from the principal stresses. τ1, τ2 = ± σx − σy 2 2 + τ 2
xy
σ = σx + σy 2
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Mohr’s Circle
C = σavg = (σx + σy) 2 R = σx − σy 2 2 + τ 2
xy