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Fundamental Mechanics of Materials Equations
Basic definitions
Average normal stress in an axial member
avg F A Average direct shear stress
avg (^) AV V Average bearing stress
b b
F
A
Average normal strain in an axial member
avg (^) L
Average normal strain caused by temperature change T (^) T Hooke’s Law (one-dimensional) E and G Poisson’s ratio
lat long Relationship between E , G , and ν G E 2 1( ) Defi nition of allowable stress (^) allow ^ failure^ (^) allow failure FS
or FS Factor of safety
FS ^ failureactual or FS failureactual
Axial deformation
Deformation in axial members
^ FL ∑
AE
F L
A E
i i i i i
or
Force-temperature-deformation relationship FL TL (^) AE
Torsion
Maximum torsion shear stress in a circular shaft max Tc J where the polar moment of inertia J is defined as J (^) ^ 2^ [ R^4 r^4 ] (^) 32 [ D^4 d^4 ]
Angle of twist in a circular shaft
^ TLJG ∑
T L
J G
i i i i i
or
Power transmission in a shaft P T
Six rules for constructing shear-force
and bending-moment diagrams
Rule Rule ( )
Rule
0 (^2 1 )
2
V P
V V V w x dx dV dx
x
x
w x
M M M V dx dM dx
V
x
x
Rule
Rule Rule
(^2 1 )
:^2
M M 0
Flexure
Flexure formula (^) x My I
Mc I
M
S
S I
c or^ max where Unsymmetric bending of arbitrary cross sections
x z^ yz y z yz
y
y yz y z y
I z I y I I I
M
I y I z I I I
zz 2 Mz
^ ⎥⎥
Unsymmetric bending of symmetric cross sections
x y y
z z
y z z y
M z I
M y I
M I
M I
^ tan
Horizontal shear stress associated with bending
H VQ It Shear fl ow formula q VQ I Shear fl ow, fastener spacing, and fastener shear relationship qs n Vf f n (^) f f Af For circular cross sections, Q (^) 1 d 12
(^3) (solid sections)
Q (^) 2 [ R (^) r ] (^) [ D (^) d ] 3
(^3 3 3 3) (hollow sections)
Beam defl ections
Elastic curve relations between w , V , M , θ , and v for constant EI Deflection Slope
Moment
Shear
v dv dx M EI d v dx V
2 2 ddM dx
EI d v dx w dV dx
EI d v dx
3 3 4 Load 4
𝑑 𝑜𝑟^
𝑤 𝑜𝑟^
or 𝑞 =
𝐼 =^
Plane stress transformations
Normal and shear stresses on an arbitrary plane
n x y xy nt x y
cos sin sin cos sin
ccos (^) xy ( cos 2 sin^2 ) or
Principal stress magnitudes
p 1 p 2 x^ y^ x^ y xy
2 2 , ^2
Orientation of principal planes
tan 2 2
p
xy x y
Maximum in-plane shear stress magnitude
max ^ ^ or max
x y^ xy
p p 2 2
2 2 1 2
avg ^
x y 2 Absolute maximum shear stress magnitude
(^) abs max ^ max^ min 2 Normal, stress invariance (^) x (^) y n t (^) p 1 p 2
Plane strain transformations
Normal and shear strain in arbitrary directions
n x y xy nt x y
cos sin sin cos ( )sin
2 2 2 ccos (^) xy (cos 2 sin 2 )
or
Principal strain magnitudes
p p
x y x y xy 1 2
2 , (^2 2 )
⎟ ^
2
Orientation of principal strains
tan 2
p
xy x y
Maximum in-plane shear strain
(^) max ^ ^ ^ 2 2 2 or ma
2 2
⎟ ^ ^
x y xy xx
avg ^
p p x y
1 2
Normal strain invariance x (^) y (^) n (^) t (^) (^) p 1 p 2
Generalized Hooke’s Law
Normal stress/normal strain relationships
x x y z
y y x z
z z x
E
E
E
[ ( )]
[ ( )]
[ ( (^) y )]
Shear stress/shear strain relationships (^) xy (^) xy (^) yz (^) yz (^) zx zx G G G
1 1 ^1
where
G E 2 1( ) Normal stress/normal strain relationships for plane stress
^
x x y
y y x
z x y
E
E
E
or
x x y
y y x
E
E
(^ )
(^ )
2
2
Shear stress/shear strain relationships for plane stress (^) xy (^) xy (^) xy xy G 1 or G
Pressure vessels
Axial stress in spherical pressure vessel a pr t
pd 2 4 t Longitudinal and hoop stresses in cylindrical pressure vessels (^) long ^ pr hoop t
pd t
pr t
pd 2 4 2 t
Failure theories
Mises equivalent stress for plane stress (^) M [ (^) p^21 (^) p 1 (^) p 2 (^) p^2 2 ] 1 2/^ [ (^) x^2^ x y (^) y^2^ 3 xy^2 ]^1 /2^2
Column buckling
Euler buckling load
P EI cr ^ KL
^2
( )^2
Euler buckling stress
(^) cr E KL r
2 ( )^2 Radius of gyration
r I A
Fundamental Mechanics of Materials Equations
2 +^
2 cos 2𝜃^ +^ 𝜏𝑥𝑦^ sin 2𝜃 𝜎𝑡 =
2 cos 2𝜃 − 𝜏𝑥𝑦^ sin 2𝜃 𝜏𝑛𝑡 = −
2 sin 2𝜃^ +^ 𝜏𝑥𝑦^ cos 2𝜃
2 +^
2 cos 2𝜃^ + 𝜀𝑡 =
𝛾^2 cos 2𝜃 − 𝑛𝑡 2 =^ −
2 sin 2𝜃^ +^
2 cos 2𝜃
𝜀𝑧
𝑥^
2 sin 2𝜃 𝛾𝑥𝑦 2 sin 2𝜃
S
IMPLY
S
UPPORTED
B
EAMS
Beam
Slope
Deflection
Elastic Curve
PL 16
EI
θ
θ = −
= −
max
PL 48
v
EI
= −
(
4
)
48
for 0
2
Px
v
L
x
EI
L
x
= −
−
≤
≤
2
2
1
2
2
2
(^
P
b L
b
LEI
Pa L
a
LEI
θ θ
3
at
Pa b
v
LEI x^
a
= −
=
(^
)
6
for 0
Pbx
v
L
b
x
LEI
x
a
= −
−
− ≤
≤
1 2
M 3 6
L
EIMLEI
θ θ
2
max
at
ML
v^
EI
x^
L
⎛^
⎜^
⎜^
⎝^
(
3
)
M x 6
v
L
Lx
x
LEI
= −
−
wL 24
EI
θ
θ = −
= −
max
(^5384)
wL
v
EI
= −
(^
2
)
24
wx
v
L
Lx
x
EI
= −
−
2
2
1
2
2
2
2
wa 24
L
a
LEIwa
L
a
LEI
θ θ
3
2
2
at
wa
v^
L^
aL
a
LEI
x^
a
3
2
2
2
2
2
3
4 (^
4
2
4
24
4
)^
for 0
wx
v^
Lx
aLx
a x
a L
LEI
a L
a^
x^
a
= −
−^
+^
−^
+^
≤^
≤
2
3
2
2
2
2
(
6
4
)
24
for
wa
v^
x^
Lx
a x
L x
a L
LEI
a^
x^
L
= −
−^
−
≤^
≤
3 0
1
3 0
2
w L
EI
w L
EI
θ θ
max
at
w L
v
EI
x^
L
= −
=
(
10
3
)
360
w x
v
L
L x
x
LEI
= −
−
C
ANTILEVER
B
EAMS
Beam
Slope
Deflection
Elastic Curve
max
PL 2
EI
θ
= −
max
PL 3
v
EI
= −
(
)
Px 6
v
L
x
EI
= −
−
max
PL 8
EI
θ
= −
max
(^548)
PL
v
EI
= −
(
2 )
for 0
2
12
(
)^
for
2
Px 48
L
v
L
x
x
EI PL
L
v
x
L
x
L
EI
= −
−
≤
≤
= −
−
≤
≤
max
M
L EI
θ
= −
max
M^2
L
v^
EI
= −
M 2
x
v
EI
= −
max
wL 6
EI
θ
= −
max
wL 8
v
EI
= −
(
4
)
wx 24
v
L
Lx
x
EI
= −
−
max
w L 24
EI
θ
= −
max
w L 30
v
EI
= −
(
10
5
)
120
w x
v
L
L x
Lx
x
LEI
= −
−
−
