Beam-shaped components in large mechanical structures such as propellers, gas turbine blades, engine turbines, rotating railway bridges, and so on, when operating, usually engage in rotational movement around the fixed axis. Studying the mechanical behavior of these structures has great significance in engineering practice. Therefore, this paper is the first investigation on the static bending of rotating functionally graded material (FGM) beams with initial geometrical imperfections in thermal environments, where the higher-order shear deformation theory and the finite element method (FEM) are exercised. The material properties of beams are assumed to be varied only in the thickness direction and changed by the temperature effect, which increases the correctness and proximity to technical reality. The numerical results of this work are compared with those of other published papers to evaluate the accuracy of the proposed theory and mechanical model used in this paper. A series of parameter studies is carried out such as geometrical and material properties, especially the rotational speed and temperature, to evaluate their influences on the bending responses of structures.
Functionally graded materials (FGM) are made from two or more different materials, where the mechanical and physical properties vary smoothly from one surface to the other one. One common type of FG material is fabricated from ceramic and metal, where the mechanical properties change in one [
In technical operations, some mechanical components can be involved rotational movements such as rotor blades, turbine blades, rotating railway bridges, etc. Therefore, mechanical response investigations of these elements with rotational movements also play a very important role in computational design, which attracted scientists worldwide. Some plentiful publications can be counted as follows. Pradhan and Murmu [
In engineering practice, imperfection occurs as a result of the manufacturing, transporting, and handling processes. The imperfection appearance will affect the work of the structure. Research on the mechanical behavior of structures that account for initial geometrical imperfections has also been studied by scientists worldwide, for example, the problem of nonlinear static and dynamic buckling and vibration behavior of structures. The plate and shell structures made of FGM material taking into account the initial geometrical imperfections were also presented by Duc et al. in the works [
Based on the above review, it can be seen that there are no any publications dealing with the bending analysis of rotating (around one fixed axis) FG beams resting on elastic foundation in thermal environments. Therefore, this is truly a novel exploration, which has a significant meaning in engineering practices. As a result, this work aims to focus on the vibration response of the mentioned structures based on the higher-order shear deformation theory of Reddy and the finite element method.
The body of this paper is structured as follows. Section
Consider an FGM beam with the length
The model of a rotating FGM beam resting on a two-parameter elastic foundation.
Assume that the beam is made from ceramic (denoted by
The beam is placed in a thermal environment; therefore, the material properties are changed by the temperature as follows [
To describe exactly the mechanical responses of FGM beams, this work uses the third-order shear deformation theory of Reddy; thus, the displacements
The longitudinal and shear strains of the beam are calculated as follows:
The normal and shear stresses are expressed as follows:
The energy of the FGM beam has the following expression:
The energy of the elastic foundation has the following form:
For the FGM beam rotating around one axis
The work done by external uniformly distributed load
The beam is balanced by the external forces; the following equation is obtained by the minimizing the potential energy as
Herein, a two-node beam element is used, where each node has four degrees of freedom:
Equation (
Then, strain components are written according to the nodal displacement as
Therefore, the energy of the element FGM beam is expressed as follows:
Equation (
The energy of the elastic foundation and centrifugal inertia force has the following expression:
The work done by uniformly distributed load
Substituting equations (
One can see that the stiffness matrix of the element FGM beam includes the components, which are related to the rotational speed
The common boundary conditions are used in this paper as follows: Simply supported (denoted as S): Clamped (denoted as C): and One side is clamped; the other side is free: C–F Fully simply supported beam: S–S Fully clamped beam: C–C
To verify the theory and mechanical model used in this paper, this section carries out two examples to compare the results of the maximum deflection of the FGM beam.
This example considers the bending response of the S–S functionally graded Al/ZrO2 beam. The geometrical and material properties of the beam are
Nondimensional maximum deflections
This work | Ritz method [ | |||||
---|---|---|---|---|---|---|
6 elements | 8 elements | 10 elements | 12 elements | 14 elements | ||
0 | 1.0094 | 1.0094 | 1.0094 | 1.0094 | 1.0094 | 1.00975 |
0.2 | 0.7561 | 0.7563 | 0.7564 | 0.7564 | 0.7564 | 0.75737 |
0.5 | 0.6397 | 0.6398 | 0.6399 | 0.6399 | 0.6399 | 0.64065 |
1 | 0.5661 | 0.5663 | 0.5664 | 0.5664 | 0.5664 | 0.56699 |
2 | 0.5073 | 0.5074 | 0.5075 | 0.5075 | 0.5075 | 0.50780 |
5 | 0.4441 | 0.4442 | 0.4443 | 0.4443 | 0.4443 | 0.44442 |
This example continues to compare the results of nondimensional maximum deflections of the FGM beam resting on the two-parameter elastic foundation. The geometrical and material parameters of the structure are the length
Nondimensional maximum deflections
Foundation parameters | |||||
---|---|---|---|---|---|
DQM [ | Exact [ | Exact [ | This work | ||
0 | 0 | 1.302290 | 1.302290 | 1.3033 | 1.301692 |
10 | 0.644827 | 0.644827 | 0.6457 | 0.644679 | |
25 | 0.366111 | 0.366111 | 0.3671 | 0.366063 | |
10 | 0 | 1.180567 | 1.180567 | 1.1814 | 1.180075 |
10 | 0.613325 | 0.613326 | 0.6141 | 0.613192 | |
25 | 0.355668 | 0.355668 | 0.3566 | 0.355622 | |
100 | 0 | 0.640074 | 0.640074 | 0.6403 | 0.639927 |
10 | 0.425582 | 0.425582 | 0.4261 | 0.425517 | |
25 | 0.282846 | 0.282846 | 0.2836 | 0.282817 |
Nondimensional maximum deflections
Foundation parameters | ||||
---|---|---|---|---|
DQM [ | Exact [ | This work | ||
0 | 0 | 0.26064 | 0.2616 | 0.26033 |
10 | 0.20862 | 0.2095 | 0.20840 | |
25 | 0.16081 | 0.1617 | 0.16066 | |
10 | 0 | 0.25547 | 0.2565 | 0.25518 |
10 | 0.20528 | 0.2062 | 0.20507 | |
25 | 0.15880 | 0.1597 | 0.15865 | |
100 | 0 | 0.21670 | 0.2174 | 0.21649 |
10 | 0.17935 | 0.1800 | 0.17919 | |
25 | 0.14273 | 0.1435 | 0.14261 |
Now, the bending analysis of the rotating FGM beam with initial geometrical imperfection resting on the two-parameter elastic foundation in the thermal environment is carried out in this section. Two types (ZrO2/SUS304 and Si3N4/SUS304) of FGM material are considered, where the material properties are shown in Table
The dependence of material characteristics on the temperature [
Materials | P-1 | P (300 | ||||
---|---|---|---|---|---|---|
E (Pa) | 244.27e9 | 0 | −1.371e − 3 | 1.214e − 6 | −3.681e − 10 | 168.06e9 |
12.766e − 6 | 0 | −1.491e − 3 | 1.006e − 5 | −6.778e − 11 | 18.591e − 6 | |
0.288 | 0 | 1.133e − 4 | — | 0 | 0.298 | |
3657 | 0 | 0 | — | 0 | 3657 | |
E (Pa) | 348.43e9 | 0 | −3.070e − 4 | 2.160e − 7 | −8.946e − 11 | 322.27e9 |
5.8723e − 6 | 0 | 9.095e − 4 | 0 | 0 | 7.475e − 6 | |
0.24 | 0 | 0 | 0 | 0 | 0.240 | |
2370 | 0 | 0 | 0 | 0 | 2370 | |
E (Pa) | 201.04e9 | 0 | 3.079e − 4 | −6.534e − 7 | 0 | 207.79e9 |
12.330e − 6 | 0 | 8.086e − 4 | 0 | 0 | 15.321e − 6 | |
0.326 | 0 | −2.002e − 4 | 3.797e − 7 | 0 | 0.318 | |
8166 | 0 | 0 | 0 | 0 | 8166 |
The dependence of Young modulus of ceramic and metal on the temperature.
Nondimensional maximum deflection of the FGM beam and other parameters are defined as
Consider an FGM beam structure with
The dependence of the nondimensional maximum deflection of the Si3N4/SUS304 beam on temperature and boundary conditions, (r)/(L) = 1,
The dependence of the nondimensional maximum deflection of the ZrO2/SUS304 beam on temperature and boundary conditions, (r)/(L) = 1,
The dependence of the nondimensional maximum deflection of the Si3N4/SUS304 beam on temperature, volume fraction exponent (n), and boundary conditions, (r)/(L) = 1,
The dependence of the nondimensional maximum deflection of the ZrO2/SUS304 beam on temperature, volume fraction exponent (n), and boundary conditions, (r)/(L) = 1,
The dependence of the nondimensional maximum deflection of the Si3N4/SUS304 beam on temperature and boundary conditions, (r)/(L) = 1, (n) = 0.2,
When increasing the temperature, the nondimensional maximum deflection of FGM beams depends on both material and boundary conditions. For the Si3N4/SUS304 beam, in the cases of S–S and C–C boundaries, when increasing the temperature, the nondimensional maximum deflection increases for all values of
On the other hand, due to the rotational movement of the beam, therefore, the nondimensional maximum deflection depends strongly on boundary conditions. This novel point is very different from the case of the without rotational movement phenomenon.
For the ZrO2/SUS304 beam, when the temperature increases, the deflection shape of the beam
Consider a rotating FGM beam with
The dependence of the nondimensional maximum deflection of the Si3N4/SUS304 beam on the rotational speed and boundary conditions, (r)/(L) = 1, (T) = 500 (K). (a) C–F. (b) S–S. (c) C–C.
The dependence of the nondimensional maximum deflection of the ZrO2/SUS304 beam on the rotational speed and boundary conditions, (r)/(L) = 1, (T) = 500 (K). (a) C–F. (b) S–S. (c) C–C.
The dependence of the nondimensional maximum deflection of the Si3N4/SUS304 beam on the rotational speed, volume fraction exponent (n), and boundary conditions, (r)/(L) = 1, (T) = 500 K. (a) C–F. (b) S–S. (c) C–C.
The dependence of the nondimensional maximum deflection of the ZrO2/SUS304 beam on the rotational speed, volume fraction exponent (n), and boundary conditions, (r)/(L) = 1, (T) = 500 K. (a) C–F. (b) S–S. (c) C–C.
The dependence of the nondimensional maximum deflection of the Si3N4/SUS304 beam on the rotational speed and boundary conditions, (r)/(L) = 1, (n) = 0.2, (T) = 500 K. (a) C–F. (b) S–S. (c) C–C.
When the rotational speed increases, due to the influence of centrifugal force, the nondimensional maximum deflection of the beam decreases. However, for different materials, the response of the beam is also different, which is the novel point in comparison with the case without rotational movement phenomenon. For the Si3N4/SUS304 beam, there is a value of the rotational speed for maximum deflection of this beam unchanged for all values of volume exponent
Figure
Consider an FGM beam resting on a two-parameter elastic foundation with
The dependence of the nondimensional maximum deflection of the FGM beam on
The dependence of the nondimensional maximum deflection of the beam on the thickness (h) and the geometrical imperfection coefficient
Based on the finite element method combining with the third-order shear deformation theory, finite element formulations are derived to carry out the static bending of the rotating FGM beams with initial geometrical imperfection resting on two-parameter elastic foundations in thermal environments, in which material properties are assumed to be varied by temperature. The novel explorations of this work can be drawn as follows: When increasing the value of temperature, material properties are changed; therefore, the nondimensional maximum deflection depends significantly on boundary conditions and the volume fraction exponent. Besides, the deflection shape of the beam is also affected by these parameters. When the rotational speed increases, the nondimensional maximum deflection decreases. For the Si3N4/SUS304 beam, there is a value of the rotational speed so that the nondimensional maximum deflection remains in all cases of the volume fraction exponent. Besides, the rotational speed has a strong effect on the deflection shape of the beam. Finally, when the initial geometrical imperfection coefficient increases, the nondimensional maximum deflection decreases; however, this change is not significant.
The data used to support the findings of this study are included within the article.
The author declares that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the University of Transport Technology Foundation for Science and Technology Development (Grant no. 1139/QD-DHCNGTVT).