The_Cambridge_Handbook_of_Physics_Formulas

82 Dynamics and mechanics
Bending beams a
Bending
moment
Light beam,
horizontal at
x =0, weight
at x = l
Heavy beam
Euler strut
failure
G b = E ∫
R c
ξ 2 ds (3.257)
= EI
R c
(3.258)
y = W (
l − x )
x 2 (3.259)
2EI 3
G b
E
R c
ds
ξ
I
y
W
l
x
bending moment
Young modulus
radius of curvature
area element
distance to neutral
surface from ds
moment of area
displacement from
horizontal
end-weight
beam length
distance along beam
EI d4 y
dx 4 = w(x) (3.260) w beam weight per
unit length
⎧
⎪⎨ π 2 EI/l 2 (free ends)
F c = 4π
⎪⎩
2 EI/l 2
F c critical compression
(fixed ends) force
π 2 EI/(4l 2 ) (1 free end) l strut length
(3.261)
a The radius of curvature is approximated by 1/R c ≃ d 2 y/dx 2 .
ds
ξ
neutral surface
(cross section)
x
y
W
free
F c
F c
fixed
Elastic wave velocities a
v t =(µ/ρ) 1/2 (3.262)
In an infinite v l =(M l /ρ) 1/2 (3.263)
isotropic solid b ( ) 1/2
v l 2−2σ
=
(3.264)
v t 1−2σ
v t speed of transverse wave
v l speed of longitudinal wave
µ shear modulus
ρ density
M l longitudinal ( modulus )
= E(1−σ)
(1+σ)(1−2σ)
In a fluid v l =(K/ρ) 1/2 (3.265) K bulk modulus
On a thin plate (wave travelling along x, plate thin in z)
y
z
k
x
[ ] 1/2
v (x) E
l
=
ρ(1−σ 2 )
(3.266)
v (y)
t =(µ/ρ) 1/2 (3.267)
v (z)
t = k
[
Et 2
12ρ(1−σ 2 )
] 1/2
(3.268)
v (i)
l
v (i)
t
speed of longitudinal
wave (displacement ‖ i)
speed of transverse wave
(displacement ‖ i)
E Young modulus
σ Poisson ratio
k wavenumber (= 2π/λ)
t plate thickness (in z, t ≪ λ)
In a thin circular
rod
v l =(E/ρ) 1/2 (3.269)
v φ =(µ/ρ) 1/2 (3.270)
v t = ka ( ) 1/2 E
(3.271)
2 ρ
v φ torsional wave velocity
a rod radius (≪ λ)
a Waves that produce “bending” are generally dispersive. Wave (phase) speeds are quoted throughout.
b Transverse waves are also known as shear waves, or S-waves. Longitudinal waves are also known as pressure
waves, or P-waves.