ENGR0135/0145 Statics and Mechanics of Materials 1 & 2Official Formula Sheet
δ =PLEA
δ =Pi Li
Ei Aii =1
n
∑ δ =Px
EAx
dx0
L
∫
Elastic Deformation of a Rod Under Axial Loading
δT = α ΔT( )L εT = α ΔT( )Thermal Changes
Stresses on an Inclined Plane
σ n =PA
cos2 θ =P
2A1+ cos2θ( )
τ n =PA
sinθ cosθ =P
2Asin2θ
G (giga) ⇒×109
M (mega) ⇒×106
k (kilo) ⇒×103
m (milli) ⇒×10−3
μ (micro) ⇒×10−6
γ ρ =ρθ
Lγ c =
cθL
Shearing Strain Due to Torsion
Shearing Stress Due to Elastic Torsion
Angle of Twist Due to Elastic Torsion
τ ρ =Tρ
Jτ c =
TcJ
θ =TLGJ
θ =Ti Li
Gi Jii =1
n
∑ θ =TdxGJ0
L
∫
J =12πc4
power = Tω
Polar Second Momentof a Circle of Radius c
Power Transmittedby a Rotating Shaft
Axial LoadingMultiples of
SI Units Torsional Loading
M M
V
VSign Convention
Load, Shear Force, andBending Moment RelationshipsdVdx
= w , V2 −V1 = wdxx1
x2
∫dMdx
= V , M 2 − M1 = V dxx1
x2
∫
Shearing Forces and Bending Moments in Beams
σ x = −MyI
σmax =McI
=MS
Elastic Flexure Formula
I ʹx = IxC + yC2 A
Parallel-Axis Theorem
τ =VQIt
Q = ydAʹA∫ = y ʹC ʹA
Shearing Stressesn.a
b
h 2
h 2I =
112
bh3
Flexural Loading: Stresses in Beams
University of PittsburghSchool of Engineering
EId 2 ydx2 = M x( )
EId 3ydx3 = V x( )
EId 4 ydx4 = w x( )
Equation of theElastic Curve
x − x0
n=
x − x0( )n when n > 0 and x ≥ x0
0 when n > 0 and x < x0
⎧⎨⎪
⎩⎪
x − x0
0=
1 when x ≥ x0
0 when x < x0
⎧⎨⎩
x − x0
n dx∫ =1
n +1x − x0
n +1+ C when n ≥ 0
ddx
x − x0
n= n x − x0
n −1 when n ≥ 1
Singularity Functions
Flexural Loading: Beam Deflections
σ n =σ x + σ y
2+σ x − σ y
2cos2θ + τ xy sin2θ
τ nt = −σ x − σ y
2sin2θ + τ xy cos2θ
σ p1, p 2 =σ x + σ y
2±
σ x − σ y
2⎛
⎝⎜⎞
⎠⎟
2
+ τ xy2 , tan2θ p =
2τ xy
σ x − σ y
τ p = ±σ x − σ y
2⎛
⎝⎜⎞
⎠⎟
2
+ τ xy2 , τmax =
σmax − σmin
2
Stress Transformation Equations for Plane Stress
Strain Transformation Equations for Plane Strain
εn =ε x + ε y
2+ε x − ε y
2cos2θ +
γ xy
2sin2θ
γ nt = − ε x − ε y( )sin2θ + γ xy cos2θ
ε p1, p 2 =ε x + ε y
2±
ε x − ε y
2⎛
⎝⎜⎞
⎠⎟
2
+γ xy
2⎛
⎝⎜⎞
⎠⎟
2
, tan2θ p =γ xy
ε x − ε y
γ p = ±2ε x − ε y
2⎛
⎝⎜⎞
⎠⎟
2
+γ xy
2⎛
⎝⎜⎞
⎠⎟
2
, γ max = εmax − εmin
yi =1EI
M∂M∂Pi
dx0
L
∫ , θi =1EI
M∂M∂Mi
dx0
L
∫
Castigliano s̓ Theorem for Beams
Pcr =π 2 EI
L2 , σ cr =Pcr
A=
π 2 EL r( )2
Euler Buckling Load
ε x =1E
σ x −ν σ y + σ z( )⎡⎣ ⎤⎦ , γ xy =τ xy
G
ε y =1E
σ y −ν σ z + σ x( )⎡⎣ ⎤⎦ , γ yz =τ yz
G
ε z =1E
σ z −ν σ x + σ y( )⎡⎣ ⎤⎦ , γ zx =τ zx
G
G =E
2 1+ν( )σ x
σ y
σ z
τ xy
τ xzτ zx
τ zy
τ yxτ yz
x
y
z
Generalized Hooke s̓ Law
σ a
σ aσ a
σ h
σ a =pr2t
, σ h =prt
Stresses in Thin-walled Pressure Vessels