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# fe_mechanics_of_materials - MECHANICS OF MATERIALS UNIAXIAL...

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38 MECHANICS OF MATERIALS UNIAXIAL STRESS-STRAIN Stress-Strain Curve for Mild Steel The slope of the linear portion of the curve equals the modulus of elasticity. DEFINITIONS Engineering Strain ε = L / L 0 , where ε = engineering strain (units per unit), L = change in length (units) of member, L 0 = original length (units) of member. Percent Elongation % Elongation = 100 o L L ⎛⎞ × ⎜⎟ ⎝⎠ Percent Reduction in Area (RA) The % reduction in area from initial area, A i , to final area, A f , is: %RA = 100 if i AA A × True Stress is load divided by actual cross-sectional area. Shear Stress-Strain γ = τ / G , where = shear strain, τ = shear stress, and G = shear modulus (constant in linear force-deformation relationship). () ν + = 1 2 E G , where E = modulus of elasticity v = Poisson's ratio , and = – (lateral strain)/(longitudinal strain). Uniaxial Loading and Deformation σ = P/A , where σ = stress on the cross section, P = loading, and A = cross-sectional area. ε = δ / L , where δ = elastic longitudinal deformation and L = length of member. AE PL L A P E = δ δ = ε σ = THERMAL DEFORMATIONS δ t = α L ( Τ o ), where δ t = deformation caused by a change in temperature, α = temperature coefficient of expansion, L = length of member, = final temperature, and o = initial temperature. CYLINDRICAL PRESSURE VESSEL Cylindrical Pressure Vessel For internal pressure only, the stresses at the inside wall are: i r i o i o i t P r r r r P > σ > + = σ 0 and 2 2 2 2 For external pressure only, the stresses at the outside wall are: o r i o i o o t P r r r r P > σ > + = σ 0 and 2 2 2 2 , where σ t = tangential (hoop) stress, σ r = radial stress, P i = internal pressure, P o = external pressure, r i = inside radius, and r o = outside radius. For vessels with end caps, the axial stress is: 2 2 2 i o i i a r r r P = σ These are principal stresses. Flinn, Richard A. & Paul K. Trojan, Engineering Materials & Their Applications, 4th ed., Houghton Mifflin Co., 1990.

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MECHANICS OF MATERIALS (continued) 39 When the thickness of the cylinder wall is about one-tenth or less, of inside radius, the cylinder can be considered as thin-walled. In which case, the internal pressure is resisted by the hoop stress and the axial stress. t r P and t r P i a i t 2 = σ = σ where t = wall thickness. STRESS AND STRAIN Principal Stresses For the special case of a two-dimensional stress state, the equations for principal stress reduce to 0 2 2 2 2 = σ τ + σ σ ± σ + σ = σ σ c xy y x y x b a , The two nonzero values calculated from this equation are temporarily labeled σ a and b and the third value c is always zero in this case. Depending on their values, the three roots are then labeled according to the convention: algebraically largest = σ 1 , algebraically smallest = σ 3 , other = σ 2 . A typical 2D stress element is shown below with all indicated components shown in their positive sense.
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fe_mechanics_of_materials - MECHANICS OF MATERIALS UNIAXIAL...

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