b
T ,3 þ f 3 x 3 , t
ð
Þ ¼ b ρ
1
ð Þ h 1 þ ρ
2
ð Þ h 2
€
w,
Q ,3 þ f 2 x 3 , t
ð
Þ ¼ b ρ
1
ð Þ h 1 þ ρ
2
ð Þ h 2
€ v,
M ,3 À Q À m 1 x 3 , t
ð
Þ ¼ 0,
ð7:118Þ
where f 2 (x 3 ,t) and f 3 (x 3 ,t) are the transverse and axial loads per unit length of the
beam at x 2 ¼ 0. m 1 (x 3 ,t) is the distributed moment per unit length of the beam.
Equation (7.118) 3 is the moment-shear force relationship needed. Equations (7.118) 3
and (7.116) imply that
Q ¼
c
1
ð Þ w ,33 þ e
1
ð Þ
φ ,33
2
h
2
1 b À
c
1
ð Þ v ,333
3
h
3
1 b
À
c
2
ð Þ w ,33
2
h
2
2 b À
c
2
ð Þ v ,333
3
h
3
2 b þ m 1 ,
ð7:119Þ
which serves as the constitutive relation for the shear force. Similar to the derivation
of Eq. (7.118), by considering the differential element in Fig. 7.19 under electrical
loads, the charge equation of electrostatics can be written as
b
D ,3 ¼ q Δp À Δn
ð
Þ A
2
ð Þ
:
ð7:120Þ
The one-dimensional conservation of holes and electrons are still given by
Eq. (7.90):
q
∂
∂t
Δp
ð Þ ¼ ÀJ
p
,3 ,
q
∂
∂t
Δn
ð Þ ¼ J
n
,3 :
ð7:121Þ
Fig. 7.19 A differential
element of the beam under
axial forces, shear forces,
bending moments, and
distributed mechanical loads
206
7 Thermal Effects