ME 4300 - Design of Mechanical Systems Quiz and Exam Resource

Stress Transformations , = + ; , = ± , ; ≥ ≥ ; =

Stresses and deformations Tension, Compression: = / ; = / ; = 2 Direct Shear: = / Torsion: = / ; = ; = / ; = 2 Pure bending: = − ; = ; =

= 2 ; = Singularity functions: Concentrated moment at x=a: ( ) = − < − > Concentrated force at x=a: ( ) = < − > Uniformly distributed load starting at x=a: ( ) = < − > Ramp starting at x=a: ( ) = < − > < − > = ( − ) if x ≥ a, = 0 if x ≤ a < − > = < − > if n > 0 < − > = < − > if n ≤ 0 Shear stress due to bending: = = (circular section) ; = (rectangular section) ; = (structural shape) Castigliano's Theorem: = Columns Critical slenderness ratio: = / if l/k ≥ (l/k)1 use Euler, else Johnson Euler columns: = ; = ( / ) ; = / for round columns Johnson Columns: = − ; = 2 + / for round columns

Failure Theories Ductile materials - Maximum Shear Stress Theory: = Ductile materials - Distortion Energy Theory: =

von Mises stress: = ( ) ( ) ( ) = − + + 3 Brittle materials - Coulomb-Mohr Theory: ≥ ≥ 0 ℎ = ≥ 0 ≥ ℎ − =

0 ≥ ≥ ℎ = − Endurance limit: = 0.5 ≤ 200 (1400 ), 100 (700 ) Fatigue strength: = ; = ( ) ; = − log Endurance limit: = Surface finish factor: = Size factor: = 0.879 . 0.11 ≤ ≤ 2 ; = 0.910 . 2 < ≤ 10

= 1.24 . 2.79 ≤ ≤ 51 ; = 1.51 . 51 < ≤ 254 = 0.370 , = 0.808√ℎ kb = 1 for axial loads Loading factor: = 1 , 0.85 , 0.59 Fatigue stress concentration factor: = 1 + ( − 1)

Mean stress, stress amplitude: = ; = Modified Goodman criteria: + = ; = ( ) Gerber criterion: + = 1 ; = ( ) Static (Langer) failure: + | | =

Fasteners Power screws: raising: = ; lowering: = ;

self-locking: > Bolt stiffness: = = ; Member stiffness: = .

( . )( . )

( )( )

; = Bolt torque: T = K Fi d Load factor against overload: = ; Factor of safety against yielding: = = Factor of safety against joint separation: = ( ) Non-permanent connections: Fi = 0.75 AtSp ; permanent connections, Fi = 0.9 AtSp Fatigue loading between no-load and max load P: = ; = + Welds Butt and fillet welds: = = . Joints in torsion: = ; = = . Joints in bending: = ; = = . Helical springs Mean diameter: D =OD – d = ID + d ; Spring index: C = D/d Stress: = ; = ; Spring rate: = Absolute stability of steel compression springs: < 2.63 Spring materials: = ; Ssu = 0.67Sut ; Ssy = 0.45Sut Critical frequency of helical springs: = Fatigue loading: unpeened: Ssa = 35 ksi (241 MPa), Ssm = 55 ksi (379 MPa) peened: Ssa = 57.5 ksi (398 MPa), Ssm = 77.5 ksi (534 MPa) = ; = ; = = Goodman: = ; = ; =

Gears Circular pitch = p Diametral pitch = P = N/d ; module = d/N pP = π Train value: = ; = (−1)#

Planetary systems: =

= 33000 ; Wt (pounds), H (hp), = 12⁄ = pitch-line velocity in ft/min, d (in), n(rpm)

= , ; Wt (kN), H (kW), d (mm), n (rpm)

Notch-sensitivity for steel and aluminum – reversed bending and axial loads

Notch-sensitivity for steel and aluminum – reversed torsion

NOTE: The size range for Class 8.8 is M1.6 - M36