7.6 Stability of Curvilinear Euler-Bernoulli Beams in Temperature Fields
249
Fig. 7.15 The investigated
beam [reprinted with
permission from
International the Journal of
Nonlinear Mechanics
publishers]
(i) Euler-Bernoulli hypothesis is taken [162];
(ii) nonlinear relation between deformations and displacements of von Kármán
form is introduced (see [163]);
(iii) curvature condition is employed owing to Vlasov’s theory [164];
(iv) elastic and isotropic beam satisfying Duhamel-Neuman principle is considered
(see [165]);
(v) heat transfer coefficient does not depend on the temperature;
(vi) beam material properties do not depend on the temperature;
(vii) the temperature field distribution along the beam thickness can be taken in an
arbitrary way.
Owing to the introduced hypotheses and assumptions, the mathematical model of
the beam is governed by the following PDEs with respect to the beam displacements:
∂ N x
∂ x
−
γ
g
h
∂
2 u
∂t 2 = 0,
(7.86)
∂
2 M x
∂ x 2 + k x N x +
∂
∂ x
N x
∂w
∂ x
+ q −
γ
g
h
∂
2 w
∂t 2 − ε
γ
g
h
∂w
∂t
= 0,
(7.87)
where
N x =
h
2
−
h
2
σ x dz =
h
2
−
h
2
Eε x dz −
h
2
−
h
2
EαT dz = Ehε x − N
T
x =
= Eh
∂u
∂ x
− k x w +
1
2
∂w
∂ x
2
− N
T
x ,
and
M x =
h
2
−
h
2
σ x zdz =
h
2
−
h
2
E
−z
∂
2 w
∂ x 2
zdz −
h
2
−
h
2
EαT zdz = −
Eh
3
12
∂
2 w
∂ x 2 − M
T
x