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Unformatted text preview: 33 MECHANICS OF MATERIALS UNIAXIAL STRESSSTRAIN StressStrain 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 Fnal area, A f , is: % RA = A A A 100 i i f # e o Shear StressStrain γ = τ /G , where γ = shear strain, τ = shear stress, and G = shear modulus (constant in linear torsionrotation 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 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 = crosssectional 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 crosssectional 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 coefFcient of expansion, L = length of member, T = Fnal 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. ♦ ¡linn, Richard A. & Paul K. Trojan, Engineering Materials & Their Applications, 4th ed., Houghton Mif¢in Co., Boston, 1990. 34 MECHANICS OF MATERIALS When the thickness of the cylinder wall is about onetenth 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 twodimensional stress state, the equations for principal stress reduce to , 2 2 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 ....
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This document was uploaded on 11/04/2011 for the course MME 512 at Miami University.
 Fall '11
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