Mohrs Circle Lecture 2 ME EN 372 Andrew Ning aning@byu.edu - PDF document

Mohr’s 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 Young’s modulus ν : Poisson’s ratio. Recall: E G ≡ 2(1 + ν )

What is the modulus of elasticity? Stress-strain curve u y f e p σ (stress) � (strain)

Stress-strain curve u y f e p E σ (stress) � (strain) Stress-strain curve u σ u , S u y f σ y , S y e p E σ (stress) � (strain)

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What is Poisson’s ratio? What is the shear modulus of elasticity?

Plane Stress Transformation

Derive an expression for σ in terms of σ x , σ y , τ xy . σ = σ x cos 2 φ + σ y sin 2 φ + 2 τ xy sin φ cos φ τ = ( σ y − σ x ) sin φ cos φ + τ xy (cos 2 φ − sin 2 φ )

Using a few trig identities, just for convenience later... � σ x + σ y � � σ x − σ y � σ = + cos 2 φ + τ xy sin 2 φ 2 2 � σ x − σ y � τ = − sin 2 φ + τ xy cos 2 φ 2 Principal Stresses φ P = 1 2 τ xy 2 tan − 1 ( σ x − σ y )

�� σ x − σ y � 2 σ 1 , σ 2 = σ x + σ y + τ 2 ± xy 2 2 τ = 0 The angles for maximum shear stress are ± 45 ◦ from the principal stresses. �� σ x − σ y � 2 τ 1 , τ 2 = ± + τ 2 xy 2 σ = σ x + σ y 2

Mohr’s Circle C = σ avg = ( σ x + σ y ) 2 �� σ x − σ y � 2 R = + τ 2 xy 2 3D Mohr’s Circle Even in a biaxial stress state, remember that the stress state is actually three-dimensional. This is especially important for computing the maximum shear stress.