From the equation of motion of the differential element of the beam in Fig. 6.13 in
the x 3 direction and its moment equation, we obtain
Q ,1 þ f 3 x 1 , t
ð
Þ ¼ 2b ρ
1
ð Þ h þ ρ
2
ð Þ c
€ u 3 ,
M ,1 À Q ¼ I €
ψ,
ð6:117Þ
where f 3 (x 1 ,t) is the transverse load per unit length of the beam, and
I ¼
2
3
bc
3
ρ
1
ð Þ
þ
2
3
b c þ h
ð
Þ
3 À c
3
h
i
ρ
2
ð Þ ,
ð6:118Þ
which is the rotatory inertia of the cross section about the x 2 axis. The effects of the
rotatory inertia and shear deformation are more pronounced when the beams are not
very long and thin. Similarly, by considering the differential element in Fig. 6.13
under electric loads, the charge equation of electrostatics can be written as
b
D ,1 ¼ q Δp À Δn
ð
Þ A
2
ð Þ
:
ð6:119Þ
The one-dimensional conservation of holes and electrons are
q
∂
∂t
Δp
ð Þ ¼ ÀJ
p
1,1 ,
q
∂
∂t
Δn
ð Þ ¼ J
n
1,1 :
ð6:120Þ
For a cantilever beam under a constant end force F as shown in Fig. 6.12, the
boundary conditions are
x 1
dx 1
M
M+dM
Q
Q+dQ
f 3 dx 1
Fig. 6.13 A differential
element of the beam under
shear forces, bending
moments, and
transverse load
6.4 Bending of Beams with e 15
165