7.6 Stability of Curvilinear Euler-Bernoulli Beams in Temperature Fields
255
stability of that method have been proved in numerous works devoted to numerical
analysis. The determination of a suitable step regarding the evolutionary coordinate
as well as examples of numerical experiments with respects to the results validity are
reported in Ref. [168]. Furthermore, estimation of the employed method accuracy has
been carried out with a help of the Runge method. Namely, two different computations
have been carried out with respect to time step t = 1/256, and with respect to
twicely decreased step t = 1/512. As the accuracy criterion, we have used a small
relative change of the obtained solution with respect to time step decrease.
Combinations of the boundary conditions (7.97)–(7.98) and the beam boundary allow to study different cases of the temperature loading. In this work, the
temperature field is employed through five methods reported in Table 7.14 together
with the graphical temperature field interpretation. The temperature intensity has
been changed in the interval 0 ≤ T ≤ 100
◦ C. Observe that here and further the
temperature T is taken in the non-dimensional form for steel, and for the studied
temperature interval, the physical beam material characteristics does not change. In
order to achieve dimensional values of the temperature, one may use the following
fixed data: α = 12.5 × 10
−6 and T = T + T 0 , where T 0 = 22
◦ C.
We have focused on the following investigations:
1. Influence of small beam curvature and small loading understood as initial imperfections influencing the beam stability loss;
2. Analysis of the temperature field type depending on T (w; 0.5);
3. Influence of temperature field type on the stationary loss of stability of curvilinear
beams.
4. Analysis of the temperature field intensity influence on the loss of stability of
curvilinear beams.
7.6.4 Influence of Imperfections
It is assumed that the beam with boundary conditions (7.91) and initial conditions
(7.92) is in a temperature field of type 1 (Table 7.14) and has small curvature parameter k x = 0.001. Investigation of the dependence T (w; 0.5) for the beam with
k x = 0 and k x = 0.001 is shown in Fig. 7.17 meaning that an action of small imperfections in the form of the introduced curvature yields a change in the beam deflection
sign. In the case of a straight beam k x = 0, stability loss occurs for T = 16
◦ C, while
the deflection increases and moves in the positive side. In the case of k x = 0.001,
stability loss occurs for T = 12
◦ C and the beam deflection is located on the negative side. Analogous results have been obtained for the beam with external load
imperfection.
Namely, for the load q = 0.1 and q = −0.1, loss of stability occurs for T = 6
◦ C
with an associated change in the deflection into positive and negative side, respectively. These investigations are within the physical expectations, which validates the
appropriate choice of both beam and the the temperature field models.
255
stability of that method have been proved in numerous works devoted to numerical
analysis. The determination of a suitable step regarding the evolutionary coordinate
as well as examples of numerical experiments with respects to the results validity are
reported in Ref. [168]. Furthermore, estimation of the employed method accuracy has
been carried out with a help of the Runge method. Namely, two different computations
have been carried out with respect to time step t = 1/256, and with respect to
twicely decreased step t = 1/512. As the accuracy criterion, we have used a small
relative change of the obtained solution with respect to time step decrease.
Combinations of the boundary conditions (7.97)–(7.98) and the beam boundary allow to study different cases of the temperature loading. In this work, the
temperature field is employed through five methods reported in Table 7.14 together
with the graphical temperature field interpretation. The temperature intensity has
been changed in the interval 0 ≤ T ≤ 100
◦ C. Observe that here and further the
temperature T is taken in the non-dimensional form for steel, and for the studied
temperature interval, the physical beam material characteristics does not change. In
order to achieve dimensional values of the temperature, one may use the following
fixed data: α = 12.5 × 10
−6 and T = T + T 0 , where T 0 = 22
◦ C.
We have focused on the following investigations:
1. Influence of small beam curvature and small loading understood as initial imperfections influencing the beam stability loss;
2. Analysis of the temperature field type depending on T (w; 0.5);
3. Influence of temperature field type on the stationary loss of stability of curvilinear
beams.
4. Analysis of the temperature field intensity influence on the loss of stability of
curvilinear beams.
7.6.4 Influence of Imperfections
It is assumed that the beam with boundary conditions (7.91) and initial conditions
(7.92) is in a temperature field of type 1 (Table 7.14) and has small curvature parameter k x = 0.001. Investigation of the dependence T (w; 0.5) for the beam with
k x = 0 and k x = 0.001 is shown in Fig. 7.17 meaning that an action of small imperfections in the form of the introduced curvature yields a change in the beam deflection
sign. In the case of a straight beam k x = 0, stability loss occurs for T = 16
◦ C, while
the deflection increases and moves in the positive side. In the case of k x = 0.001,
stability loss occurs for T = 12
◦ C and the beam deflection is located on the negative side. Analogous results have been obtained for the beam with external load
imperfection.
Namely, for the load q = 0.1 and q = −0.1, loss of stability occurs for T = 6
◦ C
with an associated change in the deflection into positive and negative side, respectively. These investigations are within the physical expectations, which validates the
appropriate choice of both beam and the the temperature field models.