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Strength of Materials Formula Sheet
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Strength of Materials Formula Sheet
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Clarkson University – ES222, Strength of Materials Final Exam – Formula Sheet Axial Loading Normal Stress: σ =
P A
Splice joint: τ ave =
F A F 2A
Double shear: τ ave =
σ=
Factor of Safety = F.S. =
F A P Bearing stress: σ b = td
Single shear: τ ave =
P P cos 2 θ , τ = sin θ cosθ Ao Ao
ultimate load allowable load
Stress and Strain – Axial Loading Normal strain: ε =
δ
Rods in series: δ = ∑ i
Thermal elongation: δ T = α ( ∆T ) L Poisson’s ratio: ν = −
PL i i Ai Ei
Thermal strain: ε T = α ( ∆T )
lateral strain axial strain
Generalized Hooke’s Law:
εx =
σ x νσ y νσ z
εz = − γ xy = M = 106
Coordinates of the Centroid: x =
−
E
εy = −
Units: k = 103
Shear stress: τ = Gγ
Normal stress: σ = Eε
L PL Elongation: δ = AE
E
νσ x
+
E
νσ x
−
E
τ xy G
−
E
σ y νσ z −
E
νσ y E
, γ yz =
+
i
i
i
i
i
σz
τ yz G
G = 109
∑ xA ∑A
E E , γ xz =
τ xz G
Pa = N/m2 y=
psi = lb/in2
ksi = 103 lb/in2
∑ yA ∑A i
i
i
i
i
Parallel Axis Theorem: I x ' = I x + Ad , where d is the distance from the x–axis to the x’–axis 2
y
1 3 bh 12 z 1 I y = hb3 12 Iz =
h
b
Torsion:
γ=
ρφ
L Tρ τ= J TL φ= JG
cφ L Tc = J
γ max = τ max
τ =γG
T
solid rod: J = 12 π c 4 hollow rod: J = 12 π ( co4 − ci4 )
Rods in Series: φ = ∑ i
Ti Li J i Gi
y
Pure Bending:
σx = −
εx = −
My I y
ρ
x
σ max =
Mc M = I S
ε y = ε z = −νε x
ρ = radius of curvature
M σ = εE
M 1
ρ
=
M EI y
General Eccentric Loading:
dy
P M y M z σx = − z + y A Iz Iy
! ! ! Mz = dy × P
dz
P
P
! ! ! M y = dz × P
Shear and Bending Moment Diagrams
C
z
x
xd dV = − w → VD − VC = − ∫ wdx = − (area under load curve between C and D) xc dx
dM =V dx
xd
→ M D − M C = ∫ Vdx = +(area under shear curve between C and D) xc
Shear Stress in Beams
τ ave =
VQ It
q=
VQ = shear per unit length I
Q = Ay
Stress Transformation
Principal stresses:
σ max,min =
Principal planes:
tan 2θ p =
σ x +σ y 2 2τ xy σ x −σ y
2 σ −σ y ± x + (τ xy ) 2
Planes of maximum in-plane shear stress:
2
tan 2θ s = −
σ x −σ y 2τ xy
2 σ −σ y τ max = x + (τ xy ) = R 2 σ +σ y σ ' = σ ave = x 2 2
Maximum in-plane shear stress: Corresponding normal stress:
Thin Walled Pressure Vessels Cylindrical:
Hoop stress = σ 1 =
pr t
Longitudinal stress = σ 2 =
Maximum shear stress (out of plane) = τ max = σ 2 = Spherical:
Principal stresses = σ 1 = σ 2 =
pr 2t
Maximum shear stress (out of plane) = τ max =
σ2 2
=
pr 2t
pr 4t
pr 2t
Deflections of Beams 1 M ( x) d 2 y = = 2 ρ EI dx
slope = θ ( x ) =
M ( x) dy =∫ dx + C1 dx EI
deflection = y ( x ) = ∫ θ ( x ) dx + C2 = elastic curve
Columns Pcr =
π 2 EI L2e
For x > a, replace x with (L-x) and interchange a with b.
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