θ p = deg ε n = με ε t = με γ nt = μrad

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1 IDE 110 S08 Test 7 Name: 1. The strain components ε x = 946 με, ε y = -294 με and γ xy = -362 με are given for a point in a body subjected to plane strain. Determine the strain components ε n, ε t, and γ nt at the point if the n t axes are rotated with respect to the x y axes by the amount and in the direction indicated by the angle θ = 12. (9 points) ε n = με ε t = με γ nt = μrad Sketch the deformed shape of the element. (3 points)

2. The strain components ε x = -800 με, ε y = 400 με and γ xy = -1350 με are given for a point in a body subjected to plane strain.

2 2. The strain components ε x = -800 με, ε y = 400 με and γ xy = με are given for a point in a body subjected to plane strain. Determine θ p and the normal and shear strains at the θ p orientation (i.e. the principals ). (12 points) θ p = deg ε n = με ε t = με γ nt = μrad Determine θ s and the normal and shear strains at the θ s orientation (i.e. the average normal strain and in-plane maximum shear strain ). (12 points) θ s = deg ε n = με ε t = με γ nt = μrad Show the angle θ p, the principal strain deformations, and the maximum in-plane shear strain distortion on a sketch. (4 points)

3 3. The strain components ε x = -200 με, ε y = -700 με and γ xy = 600 με are given for a point in a body subjected to plane strain. Draw Mohr s circle. (5 points) Using the circle, determine the principal strains, the maximum in-plane shear strain, and the absolute maximum shear strain at the point. (18 points) ε center = με radius = με ε 1 = με ε 2 = με γ in-plane max = μrad γ absolute max = μrad

4 4. The strain rosette shown in the figures was used to obtain normal strain data at a point on the free surface of a machine part. Determine the strain components ε x, ε y at the point. To save time, γ xy is not needed. (15 points) ε a = -600 με ε b = 250 με ε c = 800 με ε x = με ε y = με

(10 points) p = MPa 6. A closed cylindrical vessel contains a fluid at a pressure of 640 psi. The cylinder, which has an outside diameter of 72 in.

5 5. A strain gage is used to monitor the strain in a spherical steel tank (E = 210 GPa; ν = 0.32), which contains a fluid under pressure. The tank has an inside diameter of 2.50 m and a wall thickness of 100 mm. Determine the internal pressure in the tank when the strain gage reads 120 με. (10 points) p = MPa 6. A closed cylindrical vessel contains a fluid at a pressure of 640 psi. The cylinder, which has an outside diameter of 72 in. and a wall thickness of in., is fabricated from stainless steel [E = 28,000 ksi; ν = 0.32]. Determine the axial and hoop stresses and principal strains. (12 points) σ axial = psi σ hoop = psi ε 1 = με ε 2 = με

6 TRIGONOMETRY sin θ = opp / hyp cos θ = adj / hyp tan θ = opp / adj STATICS Symbol Meaning Equation Units x, y, z centroid position y = Σy i A i / ΣA i in, m I moment of inertia I = Σ(I i + d i 2 A i ) in 4, m 4 J N V M J solid circular shaft polar = πd 4 /32 moment J of inertia hollow circular shaft = π(d 4 o - d 4 i )/32 normal force shear force bending moment equilibrium V = -w(x) dx M = V(x) dx ΣF = 0 ΣM (any point) = 0 in 4, m 4 lb, N lb, N in-lb, Nm lb, N in-lb, Nm MECHANICS OF MATERIALS Topic Symbol Meaning Equation Units σ axial = N/A σ, sigma normal stress τ cutting = V/A σ bearing = F b /A b axial ε, epsilon normal strain ε axial = ΔL/L o = δ/l o ε transverse = Δd/d in/in, m/m γ, gamma shear strain γ = change in angle, γ = cθ rad E Young's modulus, modulus of elasticity σ = Eε (one-dimensional only) G shear modulus, modulus of rigidity G = τ/γ = E / 2(1+ν) ν, nu Poisson's ratio ν = -ε'/ε δ, delta deformation, elongation, deflection in, m δ = NL o /EA + αδtl o α, alpha coefficient of thermal expansion (CTE) in/inf, m/mc F.S. factor of safety F.S. = actual strength / design strength

7 Topic Symbol Meaning Equation Units τ, tau shear stress τ torsion = Tc/J torsion φ, phi angle of twist φ = TL/GJ rad, degrees θ, theta angle of twist per unit length, rate of twist θ = φ / L rad/in, rad/m P power P = Tω r 2 T 1 = r 1 T 2 watts = Nm/s hp=6600 in-lb/s ω, r 1 ω angular speed, speed of rotation 1 = r ω 2 2 rad/s omega f frequency ω = 2πf Hz = rev/s K stress concentration factor τ max = KTc/J flexure σ, sigma σ, sigma τ, tau normal stress σ beam = -My/I composite beams, n = E B /E A σ A = -My / I T σ B = -nmy / I T shear stress τ beam = VQ/Ib where Q = Σ(y bar i A i ) q shear flow q = V beam Q/I = nv fastener /s v or y beam deflection v = M(x) dx 2 / EI in, m Topic Equations Units stress transformation planar rotations σ u )/2 + (σ x )/2 cos(2θ) + τ xy sin(2θ) σ v )/2 - (σ x )/2 cos(2θ) - τ xy sin(2θ) τ uv = -(σ x )/2 sin(2θ) + τ xy cos(2θ) principals and max in-plane shear tan(2θ p ) = 2τ xy / (σ x ), θ s = θ p ± 45 o σ 1,2 )/2 ± sqrt { [ (σ x )/2 ] 2 + τ xy 2 } τ max = sqrt { [ (σ x )/2 ] 2 + τ xy 2 } = (σ 1 -σ 2 )/2 σ avg )/2 = (σ 1 +σ 2 )/2 strain transformation planar rotations ε u )/2 + (ε x )/2 cos(2θ) + γ xy /2 sin(2θ) ε v )/2 - (ε x )/2 cos(2θ) - γ xy /2 sin(2θ) γ uv /2 = -(ε x )/2 sin(2θ) + γ xy /2 cos(2θ) ε z = -ν (ε x + ε y ) / (1-ν) principals and max in-plane shear tan(2θ p ) = γ xy / (ε x ), θ s = θ p ± 45 o ε 1,2 )/2 ± sqrt { [ (ε x )/2 ] 2 + (γ xy /2) 2 } γ max /2 = sqrt { [ (ε x )/2 ] 2 + (γ xy /2) 2 } ε avg )/2 in/in, m/m Hooke's law 1D strain to stress σ = Eε 2D strain to stress σ x = E(ε x +νε y ) / (1-ν 2 ) σ y = E(ε y +νε x ) / (1-ν 2 ) τ xy = Gγ xy = Eγ xy / 2(1+ν) 2D stress to strain ε x -νσ y ) / E ε y = (σ y -νσ x ) / E ε z = -ν(ε x ) / (1-ν) γ xy = τ xy /G = 2(1+ν)τ xy / E in/in, m/m pressure σ spherical = pr/2t σ cylindrical, hoop = pr/t σ cylindrical, axial = pr/2t maximum principal stress theory σ 1,2 < σ yp σ radial, outside = 0 σ radial, inside = -p failure theories maximum shear stress theory τ max < 0.5 σ yp

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