Strength of Materials Formula Sheet

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.