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7 Mathematical Models of Functionally Graded Beams in Temperature Field
7.6 Stability of Curvilinear Euler-Bernoulli Beams in
Temperature Fields
In this section, stability of thin flexible Bernoulli-Euler beams by taking into account
a geometric nonlinearity as well as a type and intensity of the temperature field is
investigated. The applied temperature field T (x, z) is yielded by a solution to the
2D Laplace equation being solved for five kinds of the thermal boundary conditions,
and there are not any restriction on the temperature distribution along the beam
thickness. Action of the temperature field on the beam dynamics is studied with a
help of Duhamel theory, whereas the motion of the beam subjected to the thermal
load is yielded employing the variational principles [140].
The heat transfer Laplace equation is solved via the FDM of the third order with
respect to its accuracy, and the integrals along beam thickness defining the thermal
stress and moments are computed using Simpson’s method. PDEs governing beam
motion are reduced to Cauchy problem using FDM of the second-order accuracy. The
obtained ordinary differential equations are solved by the fourth-order Runge-Kutta
method.
Moreover, the problem of numerical results convergence versus a number of
beam partitions is investigated. A static solutions for a flexible Bernoulli-Euler beam
using the dynamic approach based on employment of the relaxation/set-up method
is obtained.
Novel stability loss beam phenomena under the thermal field versus the beam geometric parameters, boundary conditions and the temperature intensity are reported.
In particular, we have shown that a stability of the flexible beam while heating the
face beam surface essentially depends on its thickness.
7.6.1 Introduction
Although the classical models of beams based on either Euler-Bernoulli or Timoshenko theories and their assumptions have been successively used for many years,
there still are present open problems originated from both engineering and science.
This section is aimed at fulfilling gaps which still exist in the offered and available
results, putting emphasis on the interaction between vibrating Euler-Bernoulli beams
and thermal fields of different types as well as on the beam imperfections, which
usually cannot be neglected either during beam fabrication or in structural design
requirements.
Euler-Bernoulli and Timoshenko beams employing the geometrically nonlinear
theory and the thermally-induced post-buckling behaviour have been intensively
studied over the recent years [141–145]. Li et al. [142] have employed the accurate
geometrically nonlinear theory for Euler-Bernoulli beams, taking into account the
longitudinal and transverse motions of uniformly heated beams with/without static
7 Mathematical Models of Functionally Graded Beams in Temperature Field
7.6 Stability of Curvilinear Euler-Bernoulli Beams in
Temperature Fields
In this section, stability of thin flexible Bernoulli-Euler beams by taking into account
a geometric nonlinearity as well as a type and intensity of the temperature field is
investigated. The applied temperature field T (x, z) is yielded by a solution to the
2D Laplace equation being solved for five kinds of the thermal boundary conditions,
and there are not any restriction on the temperature distribution along the beam
thickness. Action of the temperature field on the beam dynamics is studied with a
help of Duhamel theory, whereas the motion of the beam subjected to the thermal
load is yielded employing the variational principles [140].
The heat transfer Laplace equation is solved via the FDM of the third order with
respect to its accuracy, and the integrals along beam thickness defining the thermal
stress and moments are computed using Simpson’s method. PDEs governing beam
motion are reduced to Cauchy problem using FDM of the second-order accuracy. The
obtained ordinary differential equations are solved by the fourth-order Runge-Kutta
method.
Moreover, the problem of numerical results convergence versus a number of
beam partitions is investigated. A static solutions for a flexible Bernoulli-Euler beam
using the dynamic approach based on employment of the relaxation/set-up method
is obtained.
Novel stability loss beam phenomena under the thermal field versus the beam geometric parameters, boundary conditions and the temperature intensity are reported.
In particular, we have shown that a stability of the flexible beam while heating the
face beam surface essentially depends on its thickness.
7.6.1 Introduction
Although the classical models of beams based on either Euler-Bernoulli or Timoshenko theories and their assumptions have been successively used for many years,
there still are present open problems originated from both engineering and science.
This section is aimed at fulfilling gaps which still exist in the offered and available
results, putting emphasis on the interaction between vibrating Euler-Bernoulli beams
and thermal fields of different types as well as on the beam imperfections, which
usually cannot be neglected either during beam fabrication or in structural design
requirements.
Euler-Bernoulli and Timoshenko beams employing the geometrically nonlinear
theory and the thermally-induced post-buckling behaviour have been intensively
studied over the recent years [141–145]. Li et al. [142] have employed the accurate
geometrically nonlinear theory for Euler-Bernoulli beams, taking into account the
longitudinal and transverse motions of uniformly heated beams with/without static