ME 486 - Design of Mechanical SystemsQuiz and Exam Resource

Geometry

Rectangle: A=bh I= 112

bh3

Circle: A=π4d2 I= π

64d4 J= π

32d4

Stress Transformations

τ max, xy=√( σx−σ y

2 )2

+ τ xy2 ; σ Pxy=

σ x+σ y

2± τmax, xy ; σ 1≥σ2≥σ3 ; τ max=

σ1−σ32

Stresses and deformations

Tension, Compression: σ=P / A ; δ=PL /AE ; U= P2 L2 AE

Direct Shear: τ=F /A

Torsion: τ=Tr /J ; τ max=TcJ ; θ=TL /GJ ; U= T2 L

2GJ

Pure bending: σ=−MyI ; σ max=

McI ;

MEI

=d2 yd x2

U=∫ M 2

2 EIdx ; δ=

1EI∫M ∂M

∂ Pdx

Singularity functions: Concentrated moment at x=a: M (x )=−M 0< x−a¿0

Concentrated force at x=a: M (x )=P< x−a¿1

Uniformly distributed load starting at x=a: M (x )=12w<x−a¿2

Ramp starting at x=a: M (x )=16k<x−a ¿3

¿ x−a¿n=¿ if x ≥ a, = if x ≤ a

∫¿ x−a¿ndx= 1n+1

<x−a¿n+1 if n > 0

∫¿ x−a¿ndx=¿ x−a¿n+1 if n ≤ 0

Shear stress due to bending: τ=VQ¿

τ max=4 V3 A (circular section) ; τ max=

3V2 A (rectangular section) ; τ max=

VAweb

(structural shape)

Castigliano's Theorem: δ=∂U total

∂ F i

Columns

Critical slenderness ratio: ( lk )1=(2π2CES y )

1 /2

Euler columns: Pcr=C π2EI

l2 ;

Pcr

A=C π2E

(l /k )2 ; d=( 64 Pcr l

2

π 3CE )1 /4

for round columns

Johnson Columns: P cr

A=S y−( S y l

2πk )2 1CE

; d=2( Pcr

π S y+

S y l2

π2CE )1/2

for round columns

Failure Theories

Maximum Shear Stress Theory: τ maxall owable=Sy

2n

Distortion Energy Theory: σ maxallowable' =

S y

n

von Mises stress: σ '=√ (σ 1−σ2 )2+(σ2−σ3 )2+ (σ1−σ3 )2

2=√σ x

2−σ x σ y+σ y2+3 τxy

2

Brittle Coulomb-Mohr Theory:

σ A≥σB≥0 thenσ A=Sut

n

σ A≥0≥σ B thenσ A

Sut−

σB

Suc=1n

0≥σ A≥σ B thenσ B=−Sucn

Endurance limit: Se' =0.5 Sut if Sut≤200 kpsi (1400MPa ) ,100kpsi (1400MPa ) if not

Fatigue strength: Sf=a Nb ;a=(0.9Sut )

2

Se;b=−1

3 log (0.9 SutSe )Endurance limit: Se=Se

' ka kb k ckd ke

Surface finish factor: k a=a Sutb

Size factor: k b=0.879d−0.107 if 0.11≤d≤2∈;kb=0.910d

−0.157 if 2<d≤10∈¿k b=1.24d

−0.107 if 2.79≤d≤51mm; kb=1.51d−0.157 if 51<d≤254mm

de=0.370d for nonrotating round ,de=0.808√hb for nonrotating rectanglekb = 1 for axial loads

Loading factor: k c=1 for bending ,0.85 for axial loads ,0.59 for torsion

Fatigue stress concentration factor: k f=1+q(k t−1)

Mean stress, stress amplitude: σ m=σmax+σmin

2;σ a=

σmax−σ min

2

Modified Goodman criteria: σa

Se+σm

Sut=1n ;

σ rev=σ a

1−(σm

Sut)

Gerber criterion: nσa

Se+( nσ m

Sut)2

=1 ; σ rev=

σa

1−(σm

Sut)2

Static (Langer) failure: σ a+σm=S y

n

Fasteners

Power screws: raising: T R=F dm

2 ( l+πf dm sec απ dm−fl sec α ); lowering: T L=

F dm

2 ( πf dm sec α−lπ dm+fl sec α );

self-locking: πf dm> l

Bolt stiffness: k b=AEl

=Ad A t E

Ad lt+A t ld ; Member stiffness:

km=0.5774 πEd

ln( (1.155 t+D−d )(1.155 t+D+d )

(D+d )(D−d ) ) ;

C=kb

kb+km

Bolt torque: T = 0.2 Fi d

Load factor against overload: nL=Sp At−F i

CP ; Factor of safety against yielding: nP=

S p A t

CP+Fi=

Sp

σa+σm

Factor of safety against joint separation: no=F i

P (1−C)

Non-permanent connections: Fi = 0.75 AtSp ; permanent connections, Fi = 0.9 AtSp

Fatigue loading between no-load and max load P: σ a=CP2 A t

; σ m=CP2 A t

+Fi

At

Welds

Butt and fillet welds: τ= FA t

= F0.707hl

Joints in torsion: τ '= VAt

; τ' '=Mr

J= Mr0.707hJ u

Joints in bending: τ '= VAt

; τ' '=Mr

I= Mr0.707h Iu

Helical springs

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

Stress: τ max=K B8 FDπ d3

; K B=4C+24C−3 ;

Spring rate: k= d4G8D3 N

Absolute stability of steel compression springs: L0<2.63Dα

Spring materials: Sut=Adm ; Ssu = 0.67Sut ; Ssy = 0.45Sut

Critical frequency of helical springs: f= 12 √ kg

W

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)

τ a=KB

8 FaDπ d3

; τ m=KB

8 FmDπ d3

; r=τaτm

=Fa

Fm

Goodman: Sse=

Ssa

1−( Ssm

Ssu ) ; Ssa=r Sse Ssur Ssu+Sse

; n f=Ssa

τa

Gearing

Circular pitch = pDiametral pitch = P = N/d ; module = d/NpP = π

Train value: nlast=enfirst ; e=(−1 )¿ gear interfaces Product of driving toothnumbersProduct of driventoothnumbers

Planetary systems: e=nL−na

nF−na

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

W t=60000H

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

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