Clarkson University – ES222, Strength of Materials Final Exam – Formula Sheet

Axial Loading

Normal Stress: PA

σ = Splice joint: aveFA

τ = Single shear: aveFA

τ =

Double shear: 2aveFA

τ = Bearing stress: bPtd

σ =

2cos , sin coso o

P PA A

σ θ τ θ θ= =

Factor of Safety = F.S. = ultimate loadallowable load

Stress and Strain – Axial Loading

Normal strain: Lδε = Normal stress: Eσ ε= Shear stress: Gτ γ=

Elongation: PLAE

δ = Rods in series: i i

i i i

PLA E

δ =∑

Thermal elongation: ( )T T Lδ α= ∆ Thermal strain: ( )T Tε α= ∆

Poisson’s ratio: lateral strainaxial strain

ν = −

Generalized Hooke’s Law: yx zx E E E

νσσ νσε = − −

yx zy E E E

σνσ νσε = − + −

yx zz E E E

νσνσ σε = − − +

, ,xy yz xzxy yz xzG G G

τ τ τγ γ γ= = =

Units: k = 103 M = 106 G = 109 Pa = N/m2 psi = lb/in2 ksi = 103 lb/in2

Coordinates of the Centroid: i ii

ii

x Ax

A=∑∑

i ii

ii

y Ay

A=∑∑

Parallel Axis Theorem: 2'x xI I Ad= + , where d is the distance from the x–axis to the x’–axis

3112zI bh=

3112yI hb=

Torsion:

Lρφγ = max L

cφγ =

TJρτ = max

TcJ

τ = Gτ γ=

TLJG

φ = solid rod: 412J cπ=

hollow rod: ( )4 412 o iJ c cπ= −

Rods in Series: i i

i i i

T LJ G

φ =∑

Pure Bending:

xMyI

σ = − maxMc MI S

σ = =

xyερ

= − y z xε ε νε= = − Eσ ε= 1 MEIρ

=

ρ = radius of curvature General Eccentric Loading:

yzx

z y

M zP M yA I I

σ = − +

z yM d P= ×!! !

y zM d P= ×!! !

Shear and Bending Moment Diagrams

T

y

z

b

h

x

y

MM

x

y

zC

PP

dy

dz

d

c

x

D C x

dV w V V wdxdx

= − → − = − = −∫ (area under load curve between C and D)

d

c

x

D C x

dM V M M Vdxdx

= → − = =∫ +(area under shear curve between C and D)

Shear Stress in Beams

aveVQIt

τ = VQqI

= = shear per unit length Q Ay=

Stress Transformation

Principal stresses: ( )2

2

max,min 2 2x y x y

xy

σ σ σ σσ τ

+ − = ± +

Principal planes: 2

tan 2 xyp

x y

τθ

σ σ=

−

Planes of maximum in-plane shear stress: tan 22x y

sxy

σ σθ

τ−

= −

Maximum in-plane shear stress: ( )2

2

max 2x y

xy Rσ σ

τ τ− = + =

Corresponding normal stress: '2

x yave

σ σσ σ

+= =

Thin Walled Pressure Vessels

Cylindrical: Hoop stress = 1prt

σ = Longitudinal stress = 2 2prt

σ =

Maximum shear stress (out of plane) = max 2 2prt

τ σ= =

Spherical: Principal stresses = 1 2 2prt

σ σ= =

Maximum shear stress (out of plane) = 2max 2 4

prt

στ = =

Deflections of Beams ( ) 2

2

1 M x d yEI dxρ

= = slope = ( ) ( )1

M xdyx dx Cdx EI

θ = = +∫

deflection = ( ) ( ) 2y x x dx Cθ= +∫ = elastic curve Columns

2

2cre

EIPL

π=

For x > a, replace x with (L-x) and interchange a with b.