Mech 360
Content
Mechanics of Materials Formula and Data Sheet
Some Engineering Material Properties: Property
Young’s Modulus, E Shear Modulus, G Poisson’s Ratio, ν Thermal Expansion, α Density, ρ
-6
Steel
210 GPa (30 psi x 106) 81 GPa (11.6 psi x 106) 0.30 11x10 /ºC (6x10 º/F) 7850 kg/m3 (0.28 lb/in3)
-6
Aluminum
70 GPa (10 psi x 106) 26 GPa (3.7 psi x 106) 0.33 22x10 /ºC (12x10 º/F) 2720 kg/m3 (0.10 lb/in3)
-6 -6 -6
Brass
105 GPa (15 psi x 106) 39 GPa (5.6 psi x 106) 0.35 20x10 /ºC (11x10-6 º/F) 8410 kg/m3 (0.30 lb/in3)
Moments of Area for Common Shapes: Shape solid circle thin-wall circle solid rectangle Elastic relationships: Cross-section A π c2 2πtc bh Ix π c4 / 4 π t c3 b h3 / 12 Iy π c4 / 4 π t c3 h b3 / 12 J = Ix + Iy π c4 / 2 2 π t c3 b h (b2 + h2) / 12
εx =
σy σx σ − ν − ν z + α ∆T E E E
γ =
τ G
G =
E 2(1 + ν)
K =
E 3(1 - 2ν)
Failure Criteria:
(for principal stresses σ1 ≥ σ2 ≥ σ3) Mises: (σ1 – σ2)2 + (σ2 – σ3)2 + (σ3 – σ1)2 = 2σY2 Mohr: σ1/σUT – σ3/σUC = 1
Tresca: σ1 – σ3 = σY
Thin-Walled Pressure Vessels:
Cylinder : σ θ =
Shaft Torsion:
pR t
σa =
pR 2t
σr ≈ 0
Sphere : σ θ = σ a =
pR 2t
σr ≈ 0
Beam Bending:
Parallel Axis Theorem:
T Gφ τ = = J r L
Power Transmission:
σ M E = = −Y I R
I(d) = I 0 + A d 2
2π f J τ max P = watts (metric units) c
Beam Formulas:
15.87 × 10 -6 rpm J τ max P = h.p. (lb.in units) c
V =
dM dx
w =
dV dx
d2y dx 2
=
M EI
q =
VAy VQ = I I
where Q =
∫
y dA
< x – a > = ( x – a ) if x – a > 0,
= 0 if x – a ≤ 0
Some Typical I-Beams:
Mohr’s Circle:
σy
τyx τxy σx
τ
σy, τyx
σ
σx, τxy
Use sign convention “in the kitchen, the clock is above and the counter is below” for the shear stresses. Then, rotations in the Mohr’s circle have the same direction and double the rotation angle of the physical stresses. For Mohr’s circle of strain, use ε and γ/2 in place of σ and τ.
Column Buckling:
PCR =
π 2 EI Le
2
=
π 2 EA (L e / r ) 2
where: Le = L (pinned-pinned) Le = 0.7 L (pinned-fixed)
⎞ ⎞ ⎟ − 1⎟ ⎟ ⎟ ⎠ ⎠
Le = 2 L (free-fixed) Le = 0.5 L (sliding-fixed)
⎛ ⎛π P Eccentric load : y max = e ⎜ sec ⎜ ⎜2 P ⎜ CR ⎝ ⎝
Initial curvature : y max =
aP PCR − P
For “short” columns:
Strain Energy:
rod : U =
Le < r
2π 2 E σY
⎛ σ 2 ⎞ ⎛ L ⎞2 σ CR = σ Y − ⎜ Y ⎟ ⎜ e ⎟ ⎜ 4π 2 E ⎟ ⎝ r ⎠ ⎝ ⎠
∫
P2 dx 2EA
beam :
U =
∫
M2 dx 2EI
shaft :
U =
∫
T2 dx 2GJ
Castigliano’s Theorem:
δj = ∂U ∂Pj θj = ∂U ∂M j
rod :
δj =
∫
P ∂P dx EA ∂Pj
beam :
δj =
∫
M ∂M dx EI ∂Pj
Some Engineering Material Properties: Property
Young’s Modulus, E Shear Modulus, G Poisson’s Ratio, ν Thermal Expansion, α Density, ρ
-6
Steel
210 GPa (30 psi x 106) 81 GPa (11.6 psi x 106) 0.30 11x10 /ºC (6x10 º/F) 7850 kg/m3 (0.28 lb/in3)
-6
Aluminum
70 GPa (10 psi x 106) 26 GPa (3.7 psi x 106) 0.33 22x10 /ºC (12x10 º/F) 2720 kg/m3 (0.10 lb/in3)
-6 -6 -6
Brass
105 GPa (15 psi x 106) 39 GPa (5.6 psi x 106) 0.35 20x10 /ºC (11x10-6 º/F) 8410 kg/m3 (0.30 lb/in3)
Moments of Area for Common Shapes: Shape solid circle thin-wall circle solid rectangle Elastic relationships: Cross-section A π c2 2πtc bh Ix π c4 / 4 π t c3 b h3 / 12 Iy π c4 / 4 π t c3 h b3 / 12 J = Ix + Iy π c4 / 2 2 π t c3 b h (b2 + h2) / 12
εx =
σy σx σ − ν − ν z + α ∆T E E E
γ =
τ G
G =
E 2(1 + ν)
K =
E 3(1 - 2ν)
Failure Criteria:
(for principal stresses σ1 ≥ σ2 ≥ σ3) Mises: (σ1 – σ2)2 + (σ2 – σ3)2 + (σ3 – σ1)2 = 2σY2 Mohr: σ1/σUT – σ3/σUC = 1
Tresca: σ1 – σ3 = σY
Thin-Walled Pressure Vessels:
Cylinder : σ θ =
Shaft Torsion:
pR t
σa =
pR 2t
σr ≈ 0
Sphere : σ θ = σ a =
pR 2t
σr ≈ 0
Beam Bending:
Parallel Axis Theorem:
T Gφ τ = = J r L
Power Transmission:
σ M E = = −Y I R
I(d) = I 0 + A d 2
2π f J τ max P = watts (metric units) c
Beam Formulas:
15.87 × 10 -6 rpm J τ max P = h.p. (lb.in units) c
V =
dM dx
w =
dV dx
d2y dx 2
=
M EI
q =
VAy VQ = I I
where Q =
∫
y dA
< x – a > = ( x – a ) if x – a > 0,
= 0 if x – a ≤ 0
Some Typical I-Beams:
Mohr’s Circle:
σy
τyx τxy σx
τ
σy, τyx
σ
σx, τxy
Use sign convention “in the kitchen, the clock is above and the counter is below” for the shear stresses. Then, rotations in the Mohr’s circle have the same direction and double the rotation angle of the physical stresses. For Mohr’s circle of strain, use ε and γ/2 in place of σ and τ.
Column Buckling:
PCR =
π 2 EI Le
2
=
π 2 EA (L e / r ) 2
where: Le = L (pinned-pinned) Le = 0.7 L (pinned-fixed)
⎞ ⎞ ⎟ − 1⎟ ⎟ ⎟ ⎠ ⎠
Le = 2 L (free-fixed) Le = 0.5 L (sliding-fixed)
⎛ ⎛π P Eccentric load : y max = e ⎜ sec ⎜ ⎜2 P ⎜ CR ⎝ ⎝
Initial curvature : y max =
aP PCR − P
For “short” columns:
Strain Energy:
rod : U =
Le < r
2π 2 E σY
⎛ σ 2 ⎞ ⎛ L ⎞2 σ CR = σ Y − ⎜ Y ⎟ ⎜ e ⎟ ⎜ 4π 2 E ⎟ ⎝ r ⎠ ⎝ ⎠
∫
P2 dx 2EA
beam :
U =
∫
M2 dx 2EI
shaft :
U =
∫
T2 dx 2GJ
Castigliano’s Theorem:
δj = ∂U ∂Pj θj = ∂U ∂M j
rod :
δj =
∫
P ∂P dx EA ∂Pj
beam :
δj =
∫
M ∂M dx EI ∂Pj
