Post on 26-Dec-2014
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.