3. Triangle 8. Parabola
A bh=
36( ) ( 4 )x
3 2 2=
A
r
2
2π =
y
r4
3π =
( )2 2 2 2π π = − = −
I I R r
Common Greek letters
δ ρ ε σ θ τ κ φ λ ω
Σ
F
AV avgτ =
L
TTε α=
radγ π =
Poisson’s ratio lat
G E
2(1 )ν =
FL
Tc
where the polar moment of inertia J is defined as:
π π [ ][ ]= − = −J R r D d 2 32
4 4 4 4
JG φ = or
P Tω= Power units and conversion factors
1 W 1 N m
s 1 hp
550 lb ft
1 rpm 2 rad
AppendixE
828
T
R
T
R
R
D
D
N
A
B
A
B
A
B
Rule 1: V P0 =
x
x
y 1
xε ρ
c wheremaxσ = = =
Transformed-section method for beams of two materials [where material (2) is transformed into an equivalent amount of material (1)]
n E
E 2
A
My
I z I y
I I I M
I y I z
I I I Mx
σ = − −
I I I
I I I M I M I
M I M I
y z yz
M z
A dA
τ = = Σ VQ
Shear flow formula
q s n V n Af f f f fτ≤ =
For circular cross sections,
Q r d 2
[ ][ ]= − = −Q R r D d 2
3
1
12 3 3 3 3 (hollow sections)
Beam deflections Elastic curve relations between w, V, M, θ, and v for constant EI
v dv
Stresses on an arbitrary plane
cos sin 2 sin cosn x y xy 2 2σ σ θ σ θ τ θ θ= + +
sin cos 2 sin cost x y xy 2 2σ σ θ σ θ τ θ θ= + −
( )sin cos (cos sin )nt x y xy 2 2τ σ σ θ θ τ θ θ= − − + −
829
or
x y x y xyσ
σ σ σ σ θ τ θ=
+ +
− +
x y x y xyσ
σ σ σ σ θ τ θ=
+ −
− −
x y xyτ
− +
2 2σ
+ ±
−

+
2 or
2 abs max
Normal stress invariance
x y n t p p1 2σ σ σ σ σ σ+ = + = +
Plane strain transformations Strain in arbitrary directions
cos sin sin cosn x y xy 2 2ε ε θ ε θ γ θ θ= + +
sin cos sin cost x y xy 2 2ε ε θ ε θ γ θ θ= + −
2( )sin cos (cos sin )nt x y xy 2 2γ ε ε θ θ γ θ θ= − − + −
or
θ γ
θ γ
− −
−
( )sin 2 cos 2nt x y xyγ ε ε θ γ θ= − − +
Principal strain magnitudes
x y x y xy 1, 2
2 2
= +
± −

+

γ ε ε= ± −
Normal strain invariance
x y n t p p1 2ε ε ε ε ε ε+ = + = +
Generalized Hooke’s law Normal stress/normal strain relationships
E
E
E
1 [ ( )]
1 [ ( )]
1 [ ( )]
= − +
= − +
= − +
σ ν ν
σ ν ν
= + −
− + +
= + −
− + +
= + −
− + +
1 ;
1 ;
1 xy xy yz yz zx zxγ τ γ τ γ τ= = =
where
e V
V E
= + + =
− + +
E
E
E
1 ( )
1 ( )
G G
830
Thin-walled pressure vessels Tangential stress and strain in spherical pressure vessel
pr
t
pd
t
pr
pr
t
pd
t
pr
pr
t
pd
t
pr
Thick-walled pressure vessels Radial stress in thick-walled cylinder
a p b p
2 2
2 2
a p b p
− −θ
Longitudinal normal stress in closed cylinder a p b p
b a i o
δ ν ν[ ]= −
( ) (1 ) (1 )r
δ ν ν[ ]= − −
( ) (1 ) (1 )r
p r
−
Contact pressure for interference fit connection of thick cylinder onto a thick cylinder
δ ( )( ) ( )=
b c a2 c
2 2 2 2
3 2 2
Contact pressure for interference fit connection of thick cylinder onto a solid cylinder
δ ( ) =
− p
Failure theories Mises equivalent stress for plane stress
σ σ σ σ σ σ σ σ σ τ= − + = − + + 3M p p p p x x y y xy1 2
1 2 2 2 1/2 2 2 2 1/2
Column buckling Euler buckling load
P EI
KL( ) cr
822
823