Design of Mechanical Systems

Quiz and Exam Resource

Geometry
1 3
Rectangle: = =
12
2
Circle: = = 4 = 4
4 64 32

Stress Transformations

2 + 1 3
, = 2
+ = , 1 2 3 =
2 2 2

Stresses and deformations

2
Tension, Compression: = = =
2

Direct Shear: =

2
Torsion: = = = =
2
2 2 1
Pure bending: = = = 2 = =
2
Singularity functions:
Concentrated moment: () = 0 < >0
Concentrated force: () = < >1
1
Uniformly distributed load: () = < >2
2
1 3
Ramp: () = < >
6
< > = ( ) if x a, 0 otherwise
1
< > = +1 < >+1 if n > 0
< > = < >+1 if n 0

Shear stress due to bending: =

4 3
= (circular section) ; = (rectangular section) ; = (structural shape)
3 2

Castigliano's Theorem: =

Columns

1/2
22
Critical slenderness ratio: =
1
1/4
2 2 64 2
Euler columns: = ; = (/)2 ; =
2 3
1/2
2 1 2
Johnson Columns: = ; = 2 +
2 2
 Failure Theories

Maximum Shear Stress Theory: max =
2

Distortion Energy Theory: max =
(1 2 )2 +(2 3 )2 +(1 3 )2
von Mises stress: = = 2 + 2 + 3
2
2
Brittle Coulomb-Mohr Theory:

0 =

1
0 =


0 =

Endurance limit: = 0.5 200 (1400 ), 100 (1400 )

(0.9 )2 1 0.9
Fatigue strength: = ; = ; = log
3
Endurance limit: =

Surface finish factor: =
Size factor: = 0.8790.107 0.11 2 ; = 0.9100.157 2 < 10
= 1.240.107 2.79 51 ; = 1.51 0.157 51 < 254
= 0.370 , = 0.808
k b = 1 for axial loads
Loading factor: = 1 , 0.85 , 0.59
+ +
Mean stress, stress amplitude: = ; =
2 2
1
Modified Goodman criteria: + = ; =
1( )

2
Gerber criterion: + =1 ; =
1( )2


Static (Langer) failure: + =

 Fasteners

+ sec sec
Power screws: raising: = ; lowering: = ;
2 sec 2 + sec
self-locking: >

0.5774
Bolt stiffness: = = ; Member stiffness: = (1.155+)(+) ; =
+ (1.155++)() +

Bolt torque: T = 0.2 F i d

Load factor against overload: = ; Factor of safety against yielding: = =
+ +

Factor of safety against joint separation: =
(1)
Non-permanent connections: F i = 0.75 A t S p ; permanent connections, F i = 0.9 A t S p
Fatigue loading between no-load and max load P:
2 ( )
= ; = + ; Goodman: = ) with < (1 )
2 2 ( +

Welds

Butt and fillet welds: = =
0.707

Joints in torsion: = ;
= =
0.707

Joints in bending: = ; =
=
0.707

Helical springs

Mean diameter: D =OD d = ID + d ; Spring index: C = D/d

8 4+2
Stress: = 3 ; = ;
43
4
Spring rate: =
83

Absolute stability of steel compression springs: 0 < 2.63


Spring materials: = ; S su = 0.67S ut ; S sy = 0.45S ut

1
Critical frequency of helical springs: =
2
Fatigue loading: unpeened: S sa = 35 ksi (241 MPa), S sm = 55 ksi (379 MPa)
peened: S sa = 57.5 ksi (398 MPa), S sm = 77.5 ksi (534 MPa)
8 8
= 3 ; = 3 ; = =


Goodman: = ; = ; =
1 +

 Gearing

Circular pitch = p
Diametral pitch = P = N/d ; module = d/N
pP =

Train value: = ; = (1)#


Planetary systems: =

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

60000
= ; W t (kN), H (kW), d (mm), n (rpm)

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