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