FE Review Mechanics of Materials - Auburn University

1 FE Review Mechanics of Materials FE Mechanics of Materials Review Stress F. V. M. N. N = internal normal force (or P). V = internal shear force M = internal moment Double Shear N P F/2. Normal Stress = = = F. A A F/2. V. Average Shear Stress = =. A. V = F/2 F. V = F/2. F. = 2. A. FE Mechanics of Materials Review Strain L L L0 . Normal Strain = = = Units of length/length L0 L0 L0. = normal strain L = change in length = . L0 = original length L = length after deformation (after axial load is applied).

2 Percent Elongation =. L. 100. L0. Ai A f Percent Reduction in Area = 100. Ai Ai = initial cross- sectional area Af = final cross-sectional area FE Mechanics of Materials Review Strain Shear Strain = change in angle , usually expressed in radians y B B'. xy . x FE Mechanics of Materials Review Stress-Strain Diagram for Normal Stress-Strain FE Mechanics of Materials Review FE Mechanics of Materials Review Hooke's Law (one-dimension). = E . = normal stress, force/length^2. E = modulus of elasticity, force/length^2.

3 = normal strain, length/length = G . = shear stress, force/length^2. G = shear modulus of rigidity, force/length^2. = shear strain, radians FE Mechanics of Materials Review E. G=. 2(1 + ). = Poisson's ratio = -(lateral strain)/(longitudinal strain). lat = . long '. lat = change in radius over original radius r . long = change in length over original length L. FE Mechanics of Materials Review Axial Load If A (cross-sectional area), E (modulus of elasticity), and P (load) are constant in a member (and L is its length): P A PL.

4 E= = = Change in length L AE. If A, E, or P change from one region to the next: PL. = Apply to each section where A, E, & P are constant AE. A / B = displacement of pt A relative to pt B. A = displacement of pt A relative to fixed end FE Mechanics of Materials Review -Remember principle of superposition used for indeterminate structures - equilibrium/compatibility FE Mechanics of Materials Review Thermal Deformations t = ( T ) L = (T T0 ) L. t = change in length due to temperature change, units of length = coefficient of thermal expansion, units of 1/.

5 T = final temperature, degrees T0 = initial temperature, degrees FE Mechanics of Materials Review Torsion Torque a moment that tends to twist a member about its longitudinal axis Shear stress, , and shear strain, , vary linearly from 0 at center to maximum at outside of shaft FE Mechanics of Materials Review T. Tr . = = shear stress, force/length^2. r J. T = applied torque, force length r = distance from center to point of interest in cross- section (maximum is the total radius dimension). J = polar moment of inertia (see table at end of STATICS.)

6 section in FE Review manual), length^4. = TL. JG = angle of twist, radians L = length of shaft G = shear modulus of rigidity, force/length^2. z = G z = Gr ( d / dz ). ( d / dz ) = twist per unit length, or rate of twist FE Mechanics of Materials Review Bending Positive Bending Makes compression in top fibers and tension in bottom fibers Negative Bending Makes tension in top fibers and compression in bottom fibers FE Mechanics of Materials Review dV. Slope of shear diagram = negative of distributed loading value = q( x ).

7 DM dx Slope of moment diagram = shear value =V. dx FE Mechanics of Materials Review x2. Change in shear between two points = neg. of area under V2 V1 = [ q( x )]dx distributed loading diagram between those two points .. x1. x2. Change in moment between two points = area under shear diagram between those two points M 2 M1 = [V ( x )]dx x1. FE Mechanics of Materials Review Stresses in Beams My . = = normal stress due to bending moment, force/length^2. I y = distance from neutral axis to the longitudinal fiber in question, length (y positive above NA, neg below).

8 I = moment of inertia of cross- section , length^4. Mc max = c = maximum value of y;. I distance from neutral axis to extreme fiber x = y = radius of curvature of deflected axis of the beam M. = E = E y . 1. From and =. EI. = My I. FE Mechanics of Materials Review S=I S = elastic section modulus of beam c Mc M. Then max = = . I S. VQ. Transverse Shear Stress: =. It Transverse Shear Flow: VQ. q=. I. Q = y ' A'. t = thickness of cross- section at point of interest t = b here FE Mechanics of Materials Review Thin-Walled Pressure Vessels (r/t >= 10).

9 Cylindrical Vessels pr t = = 1. t 1 = hoop stress in circumferential direction p = gage pressure, force/length^2. r = inner radius t = wall thickness pr a = = 2 = axial stress in longitudinal direction 2t See FE Review manual for thick-walled pressure vessel formulas. FE Mechanics of Materials Review 2-D State of Stress Stress Transformation x + y x y x' = + cos 2 + xy sin 2 . 2 2. x + y x y y' = cos 2 xy sin 2 . 2 2. x y x' y' = sin 2 + xy cos 2 . 2. Principal Stresses x + y x y 2.. 1, 2 = + ( xy ).

10 2. 2 2 . xy tan 2 p = No shear stress acts x y . on principal planes! 2 . FE Mechanics of Materials Review Maximum In-plane Shear Stress x y . 2. x + y in plane = + ( xy ) avg =. max 2. 2 2. x y . tan 2 s = / xy 2 . FE Mechanics of Materials Review Mohr's Circle Stress, 2D. , positive to the right x + y tau, positive downward! Center: Point C( avg = ,0). 2 . R = ( x avg ) + ( xy ). 2 2. 1 = avg + R = a 2 = avg R = b inmax plane = R. + . A rotation of to the x' axis on the element will correspond to a rotation of 2 on Mohr's circle!