CC BY
22Journal of Architectural and Engineering Research 2022-2 ISSN 2738-2656 DOI: https://doi.org/10.54338/27382656-2022.2-005
Lan Hoang That Ton
HCMC University of Architecture, HCMC, Vietnam
THREE KINDS OF POROSITY ON FUNCTIONALLY GRADED POROUS BEAMS
In this paper, the effects of three types ofporosity on bending behavior of functionally graded porous (FGP) beams are studied. The procedure of finite element method (FEM) is established and based on the simple Timoshenko beam theory. The results achieved in this paper are presented and compared with other results in the references to verify the feasibility of implementing the formula and writing the Matlab code. On the other hand, this paper can help researchers to have an overview of the bending behavior of the FGP beams.
Keywords: bending behavior, functionally graded porous (FGP) beam, transverse displacement, rotation, simple Timoshenko beam.
Introduction
Nowadays, functionally graded (FG) material has become one of the smart materials and it is used in many countries. From a mixture of ceramic and metal, it provided a continuous variation of material properties from the top surface to the bottom surface of a structure. For example, some structures like nuclear tanks, spacecraft, etc. are produced based on the above material [1-3]. Due to the high applicability of FG material, many studies related to various theories have been given to comment the mechanical behavior of FG structures as [4-9]. However, porosity of the material can occur during the manufacturing process [10-12]. So, for a good knowledge of porosity effect on bending behavior of FG structures, a study related to this issue must be considered as soon as possible. There are three types of structure, like beam, plate and shell, but researchers are usually interested in beam structures because of its wide applications. Furthermore, many different beam theories were used to analyze beam structures, like simple beam theory [13], classical beam theory [14, 15], first-order shear deformation theory [16-20] or higher-order shear deformation theory [21, 22]. However, using a simple Timoshenko beam model helps us to reduce the computational cost with the resulting error within the allowable range. On the other hand, beams made of FGP materials should be investigated as much as possible to help the designer have the right knowledge about the mechanical properties. The few published papers on static bending behavior of FG beams can be listed here. Author Chen and co-workers presented the Ritz method to obtain the transverse bending deflections and critical buckling loads, where the trial functions take the form of simple algebraic polynomials [23]. A novel model was introduced for bending of FGP cantilever beams by [24] related to shape memory alloy/poroelastic composite material. In this article, the authors verified the accuracy of the bending model by a three-dimensional (3D) finite element method (FEM). Another paper based on trigonometric shear deformation theory was used to analyze the bending, vibration and buckling characteristics of FGP graphene-reinforced nanocomposite curved beams from [25], and so on. From above reasons, this paper is given to investigate the bending behavior of FGP beams.
This paper has four parts. Part 1 gives the introduction as above. Part 2 presents the formulations as well as Part 3 shows some essential results. Finally, a few comments are also given in Part 4, respectively.
Formulations
A FGP beam with length L, width b and thickness h is considered. Three forms of porosity distributions are studied and shown in Figures 1 and 2, in which (1) is uniform porous distribution and (2) and (3) are nonuniform porous distributions respectively. The normalized Young's modulus E(z)/Ei is depicted in Fig. 2 (a) - (c) to clarify the influences of these three forms of porosity with the value of Young's modulus at top surface, E1.
Ei, G,
^ G2
(1) (2) (3)
Fig. 1. FGP beam with three types of porosity 1, 2 and 3 The material properties such as Young's modulus, E(z), and shear modulus, G(z), can be described as below:
\E( z) = E,(1 - e x) [G( z) = Gi(1 - ^o X)
E ( z) = Ei(1 - eo cos J) G(z) = Gi(1 - eocos I n I)
with x = 7"-7I 7 Vi -eo - ^ +1
nz h J)
nz h / )
nz / n
2h + 4
nz n
2h + 4
for type (1)
for type (2)
(1)
(2)
for type (3)
(3)
The porosity coefficient eo must satisfy 0 < eo < 1 and
1 E2 1 G2
en = 1 --r = 1 —z2 .
(4)
E Gi
Based on FEM, the degrees of freedom associated with a node of a simple Timoshenko beam element are a transverse displacement and a rotation as depicted in Fig. 3. Using the principles of simple beam theory, the
beam element stiffness matrix Kel will be derived
F J
_ FelJel
'■el
with
12/(1 + 5) 6LJJ /(1 + 5) -12/(1 + 5) 6LJJ ¡(1 + 5)
6LJI/(1 + 5) (4 + 5) L2J / (1 + 5) -6LJJ/(1 + 5) (2-5)L2J/(1 + 5)
-12/(1 + 5) -6LJJ/(1 + 5) 12/(1 + 5) -6LJJ/(1 + 5)
6Ljj/(1 + 5) (2-5)L2J/(1 + 5) -6LJJ/(1 + 5) (4 + 5)L2ej/(1 + 5)
5 = (12FejJei ) / (GejSAeiL2i ),
(5)
(6)
s =5/6 is called the shear correct factor; Eet, Gel and Iei are Young's modulus, shear modulus and second moment of area of element based on E(z), G(z), b and h; Lei is length of element and Aei is called the area of cross section.
2
Fig. 2. Normalized material property E(z)/Ei with (a) type 1, (b) type 2 and (c) type 3 of porosity According to the principle of minimum total potential energy, the element equation can be described as
Eel I el
12/(1 + E) 6Lel / (1 + E) -12/(1 + E) 6Lel / (1 + E)
6Lel/ (1 + E) (4 + E)L2i / {1 + E) -6Li/ (1 + E) {2-3)2/1 + E) -12/ (1 + E) -6Ln/ (1 + E) 12/ (1 + E) -6Ld/ (1 + E) 6Lei / (1 + E) (2-E)L2n/ (1 + E) -6Lii/ (1 + E) (4 + E)L2d / (1 + E)
After assembly, the bending variables can be obtained by solving the following equation:
Ktotal d total = F total
wi fi
(Pi mi
W ; f
1 1
Pi. mi
(7)
(8)
w\
Fig. 3. Two nodes i and j of a beam element
By using three letters C, S and F refer to the clamped, simply supported and free condition, all boundary conditions can be revealed as below (Fig.4).
C
t
S
F
Fig. 4. Three types of boundary condition
From Fig. 4, the boundary conditions of system can be presented:
w(0) = cp(0) = 0, w(L) = 0 for CS (9)
w(0) = cp(0) = 0 , w(L) = cp(L) = 0 for CC (10)
w(0) = (p(0) = 0 for CF (11)
More clearly, the finite element system of equations can be reached as below Input data: material and geometrical properties, Calculating constitutive matrix,
Loop over elements: calculating element stiffness matrix Ke7 and element force vector Fe7, Assembling all parts in the global coordinate system to have Ktotal and Ftotal, Applying boundary conditions CS, CC or CF, Solving equation to achieve &total,
Display transverse displacements w and rotations D at nodes of system. Numerical Examples
Firstly, the validity of the proposed model is checked for (CC) and (CS) isotropic beams under a uniform load q = 106 N/m2. The material and geometric properties are E = 1 GPa, v = 1/3, b = 0.1 m, h = 0.1 m and L = 10h. The maximum transverse displacement and rotation as in the Table below are calculated and compared with analytical solutions [26] as follows:
w = ■
(CC)
J_ EI
1 t2 2 1 T 3 1 4
— qL x--qLx +--qx
24 12 24
1
p = — EI
1 t2 1 T 2 1 3
— qLx--qLx + — qx
12 4 6
w = -
(CS)
[-3qL2x2 + 5qLx3 - 2qx4 ]
48EI
P =
1— ^-6qL2x + 15qLx2 - 8qx3 J
48EI
(12)
(13)
(14)
(15)
It can be seen that the results obtained from the paper are completely approximate with other results. The relative error among above results can be explained by using different approaches.
Table. The comparison of the maximum transverse displacements at position x = L/2 of (SS) isotropic beams with L/h = 5
w max Pmax
CC Analytical 0.3125 Paper 0.3126 Analytical 0.9375 Paper 0.9383
w max Pmax
CS Analytical 0.6480 Paper 0.6466 Analytical 1.7187 Paper 1.7002
Secondly, the material of the porous beam is assumed to be steel foam with Ei = 200 GPa, v = 1/3. The cross section of beam is h = 0.1 m, b = 0.1 m . The normalized maximum deflections w = wmax /h based on this study for two boundary conditions (CC) and (CS) are compared with other results of [23] as in Fig. 5. Again, their convergence proves the reliability of the proposed method in bending analysis of FGP beams.
Thirdly, by changing the boundary condition from (CC) to (CS) and (CF), the bending behaviors of FGP beams can be seen in Fig. 6 (a) - (c) for three types 1, 2 and 3. Once again, the effects of porosity on the bending behavior of this structure are clearly presented in these Figures. Furthermore, Fig. 7 depicts the influence of porosity on the deflections of (CF) porous beams for type 1, type 2 and type 3 respectively.
Finally, by varying the porosity coefficient eo, the length to thickness ratio L/h and three types 1, 2 and 3, the results of the normalized transverse displacement w = w(L/ 2)/ h at position L/2 of FGP beams with (CC)
boundary condition are plotted in Fig. 8 (a) - (c). As the
porosity value increases, the deflection of FGP beam
also increases and this statement holds for all cases. Fig. 5 Convergence of tlae deflectwia
Fig. 6. The influence of eo on the deflections ofporous beams with (a) type 1 / (b) type 2 / (c) type 3
for three boundary conditions
Fig. 7. The influence of eo on the deflection of (CF) porous beam with three types 1, 2 and 3
Fig. 8. The deflections of (a) type 1 / (b) type 2 / (c) type 3 (CC) porous beams by changing ratio L/h and porosity factor eo
Conclusion
The bending behaviors of FGP beams under three different types of boundary condition and three kinds of porosity are presented in this study. The verification results based on this Matlab code are in good agreement
with other results in reference and the main goal of the author is to demonstrate the applicability of simple
theory to analyzing the FGP beams with acceptable results.
Conflicts of Interest
No conflict of interest was declared by the author.
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Lan Hoang That Ton (Vietnam, HCMC) - HCMC University of Architecture, Faculty of Civil
Engineering, Assistant Professor, titechoniinesom@gmaii.com
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License
Received: 13.06.2022 Revised: 20.06.2022 Accepted: 21.06.2022 © The Author(s) 2022
