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33 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 o , where ε = engineering strain (units per unit), L = change in length (units) of member, L o = original length (units) of member. Percent Elongation % Elongation = L L 100 o # D c m Percent Reduction in Area (RA) The % reduction in area from initial area, A i , to final area, A f , is: % RA = A A A 100 i i f # - e o Shear Stress-Strain γ = τ /G , where γ = shear strain, τ = shear stress, and G = shear modulus (constant in linear torsion-rotation relationship). , G v E 2 1 where = + ^ h E = modulus of elasticity v = Poisson’s ratio , and = – (lateral strain)/(longitudinal strain). STRESS, PSI STRESS, MPa MECHANICS OF MATERIALS 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. E L P A AE PL = = = v f d d True stress is load divided by actual cross-sectional area whereas engineering stress is load divided by the initial area. THERMAL DEFORMATIONS δ t = α L ( T T o ) , where δ t = deformation caused by a change in temperature, α = temperature coefficient of expansion, L = length of member, T = final temperature, and T o = initial temperature. CYLINDRICAL PRESSURE VESSEL Cylindrical Pressure Vessel For internal pressure only, the stresses at the inside wall are: P r r r r P and t i o i o i r i 2 2 2 2 = - + =- v v For external pressure only, the stresses at the outside wall are: , P r r r r P and where t o o i o i r o 2 2 2 2 =- - + =- v v 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: P r r r a i o i i 2 2 2 = - v σ t , σ r , and σ a are principal stresses. Flinn, Richard A. & Paul K. Trojan, Engineering Materials & Their Applications, 4th ed., Houghton Mifflin Co., Boston, 1990.
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MECHANICS OF MATERIALS 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 Pr t P r 2 and t i a i = = v v 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 , 2 2 0 a b x y x y xy c 2 2 ! = + - + = v v v v v v x v d n The two nonzero values calculated from this equation are temporarily labeled σ a and σ b and the third value σ 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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