Document Type : Research Paper
Authors
Department of Civil Engineering, National Institute of Technology, Kurukshetra, 136119, India
Abstract
Keywords
Buckling Analysis of Functionally Graded Sandwich Beam Based on ThirdOrder Zigzag Theory
Department of Civil Engineering, National Institute of Technology, Kurukshetra, 136119, India
KEYWORDS 

ABSTRACT 
Buckling analysis; Zigzag theory; Power law; Exponential law; Functionally graded material. 
In this paper buckling response of a sandwich (SW) beam containing functionally graded skins and metal (TypeS) or ceramic core (TypeH) is investigated using a thirdorder zigzag theory. The variation of material properties in functionally graded (FG) layers is quantified through exponential and power laws. The displacements are assumed using higherorder terms along with the zigzag factors to evaluate the effect of shear deformation. Inplane loads are considered. The governing equations are derived using the principle of virtual work. The model achieves stressfree boundaries unlike higherorder shear deformation theories and is C0 continuous so, does not require any postprocessing method. The present model shows an accurate variation of transverse stresses in thickness direction due to the inclusion zigzag factor in assumed displacements and is independent of the number of layers in computing the results. Numerical solutions are arrived at by using three noded finite elements with 7DOF/node for sandwich beams. The novelty of the paper lies in presenting a zigzag buckling analysis for the FGSW beam with thickness stretching. This paper presents the effects of the power law factor, end conditions, aspect ratio, and lamination schemes on the buckling response of FGM sandwich beams. The numerical results are found to be in accordance with the existing results. The buckling strength was improved by increasing the power law factor for Type S beams while the opposite behavior was seen in type H beams for all types of end conditions. The end conditions played a major role in deciding the buckling response of FGSW beams. Exponential law governed FGSW beam exhibited a little higher buckling resistance for Type S beams, while a little lower buckling resistance was found for Type S beams for almost all lamination schemes and end conditions. Some new results are also presented which will serve as a benchmark for future research in a parallel direction. 
The Buckling phenomenon is a very different structural response than inplane compression and can lead to catastrophic failure at critical load. It is also a principal mode of failure for slender components like laminated sandwich beams. Recently sandwich (SW) structures having a threelayered architecture are used in abundance because of their attractive properties of high strengthtoweight ratios, high energy absorption, etc [1, 2]. SW structures use the advantages of two or more distinct materials at a time and in one place. For example, the most common SW structure comprising metal and ceramic exhibits both strength and temperature resistance properties. The main drawback of using SW laminated structures is related to the interface of the layers like debonding and stress concentration [3, 4]. A proper solution to these problems is using functionally graded material (FGM) having smoothly varied properties in between two widely varied properties of constituted layers. FGM layer(s) can be used as middle or edge layers of SW beams to produce functionally graded sandwich (FGSW) beams.
FGMs are a new class of composites and have found numerous applications in many engineering and biomedical fields such as nuclear projects, the aviation industry, the aerospace sector, defense, the automobile industry, electronics, manufacturing, the energy sector, dentistry, orthopedics [5], etc. As FGM is a combination of distinct materials so, through suitable tailoring, a designer can draw the most complicated requisite properties from them. It provides a better option than conventional composites. There are numerous examples of graded structures present in nature such as bones, teeth, skin, leaves, etc [6]. It is a wellknown fact that a structural element built by nature carry out all its functions effectively and is irreplaceable in any manner. So, it can be said that FGM is an ideal material for assigning a structure. Although FGM is ideally suited material for all requirements, can’t be used everywhere because of the difficulty and cost of production. Earlier it was impossible to attain the variable microstructure, but now with the advent of highedge technologies, it is possible to generate an FGM. Mostly volume gradient structure is made for an FGM.
The highly heterogeneous structure of FGM causes inflation of shear deformation effects in the thickness direction of SW beams. So its behaviour becomes complex which creates reliability issues in practical applications. So, response data of FGSW structures should be generated and compiled to help in increasing the practical application in various fields.
Many theories are developed for getting the true behaviour of SW beams. An elaborate review of different theories used for analyzing FGSW structures is given by [7]. Elasticity theories [812] provide benchmark data by achieving the highest degree of accuracy but they are cumbersome, so many simplified theories based on some assumptions are developed by researchers. When going through the literature two broad categories can be identified for the theories developed. They are equivalent single layer (ESL) and layerwise theories (LWT). ESL theory assumes the primary variable with reference to the central/neutral layer while LWT assumes the same layerwise. The order of displacement in independent variable assumption creates different theories like classical plate theories CPT [1316], First order shear deformation theory FSDT [1719], thirdorder shear deformation theory TOT [20], higherorder shear deformation theory HOT [2123], refined HOT [2430] and quasi3D theory [31]. CPT ignores shear deformation effects and overestimates the buckling loads. FSDT requires a correction factor for satisfying parabolic variation of shear deformation. TOT and HOT do not give tractionfree shear stresses. These displacementbased theories are applied using ESL or LWT approach. The ESL approach is simple so, is mostly used by researchers.
LWT is more accurate than ESL theory because it has a more realistic approach for multilayered beams having an uneven distribution of dependent variables resulting from a layerwise construction. LWT is further classified as discrete LWT and zigzag LWT. Discrete LWT [32] assumes the variables with the individual layers, so the solution becomes difficult to achieve for a larger number of layers. Also, most of the discrete LWT does not satisfy the interlaminar stress continuity, since two different values of stresses are built at the interface of these layers from the different elastic modulus of two layers. Zigzag LWT [3343] describes the nonuniform variation in the dependent variable by including an additional term in the displacement field while adopting the ESL assumption. The superiority of zigzag LWT in getting static and dynamic results over FSDT and TOT is listed by Kapuria et al. [37]. By comparing zigzag LWT with benchmark elasticity results for SW beams, he found it gives the least percentage of errors. The manner by which the zigzag LWT is able to satisfy the interlaminar displacement continuity of laminated structures is well cited by [41].
Pandey and Pradyumna [32] analyzed the FGSW plate by employing eight noded C^{0} isoparametric finite elements having 13DOF using a layerwise expansion for displacement fields. Chakrabarty et al [38] defined the issue of C1 continuity associated with implementing the finite element method using the zigzag theory and solved the buckling problem of the SW beam. Averil [33] developed firstorder zigzag formulations for laminated beams and employed a penalty function to alleviate the C1 requirement for two noded finite elements. Later Cho and Averill [34] used sublaminate approximation to avoid the C1 requirement for four noded elements. Vo et al [24] used a linear interpolation function in generalized displacements to solve the C1 continuity issue. Neves [36] used the meshless method in place of FEM to solve the buckling problem of FGM sandwich beams. Kapuria and Ahmed [42] used interdependent interpolation. In the present work problem of C0 continuity arising from the use of zigzag theory in combination with FEM for getting solutions is avoided by expressing the derivation terms of shear stress in terms of other variables and solving the equations. It does not require any postprocessing method.
Solving a problem involves two main steps: consideration of a theory and the type of solution method. Until now most researchers have used ESL theories for simplicity. This paper uses an advanced theory that overcomes all the deficiencies of earlier proposed theories as overestimating the buckling loads (CPT); requiring a shear correction factor for correctly assessing the variation of shear stresses (FSDT); providing the actual parabolic variation of transverse shear stresses, but not fulfilling stressfree boundary conditions (HOT); providing a solution which is dependent on the number of layers and giving two values of shear stress at the interface because of the two different displacement equations in adjacent layers (LWT). The zigzag theory used in this work is supreme to all the abovestated theories as it does not need any shear correction factors, provides the actual parabolic variation of transverse shear stresses, fulfills stressfree boundary conditions, and is not dependent on the number of layers for the solution. The only issue in using Zigzag theory is the problem of C0 continuity due to the inclusion of additional terms in displacement approximation, which is avoided here as discussed earlier.
Based on the literature review, authors found that plenty of study is available for the structural response of FG sandwich beams based on analytical approach such as Navier’s solution, because of its accuracy and ease in finding solutions, but has restraint in terms of boundary conditions, material law, loading conditions, etc. These restraints are overcome by using numerical methods like FEM, meshfree method, etc. In this study a numerical method: FEM is employed to study the response of symmetric and asymmetric FGSW beams subjected to various end conditions and material laws. Sayyad and Ghugal [43] reported that an ample amount of research is available for the analysis of plates and bending and vibration response SW structures while buckling analysis of FGSW beams is very less studied. Although a sufficient amount of literature is present on the zigzag analysis of SW structures; but till now very less authors have taken up the zigzag method for analyzing FGSW beams because of the complexity in arriving at the solutions due to the inclusion of the zigzag factor along with FGM layer(s). Among the zigzag analysisbased studies, to the author’s best knowledge, buckling analysis is not available and the bending and vibration studies do not adopt the numerical solution method, nor take into account the thickness stretching effects. The novelty of the present paper is presenting the buckling analysis results for the FGSW beam using a zigzag theory with thickness stretching through a numerical method (FEM). Present paper deals with finding buckling responses of FGSW beams for various end conditions, aspect ratios, and homogenization laws based on the recently proposed zigzag theory [38]. The present theory satisfies interlaminar stress continuity. A C0 continuous FEM formulation of 3 noded elements with 7DOF/node is used. The present paper also gives new results which will serve as a benchmark for future works with a similar vision.
A threelayered SW beams (Fig 1) of two types: Type P and Type E are synthesized. These are further classified according to the core material: Type PH, EH (hardcore) (Fig 2a), and Type PS, ES (softcore) (Fig 2b) for analysis. These beams have FGM as face sheets and cores built as ceramic (Type H) and metallic (Type S). As it is established that the material property consideration has a great effect on the analysis results, so this study uses two types of material modeling (power law and exponential law) and the results were compared. Material property variation is modeled in two ways:
Garg et al. [44] presented a review of the analysis of FGM sandwich structures and prepared a list of literature based on the type of material law used. They found that more than 90% of the FGMrelated literature used power law for estimating the material properties as it is simplest as per the ease of use. A similar observation was made by Swaminathan et al. [45]. In this paper, material properties are graded in the thickness direction according to the powerlaw distribution in terms of the volume fractions of the constituents of the material, and the effective material properties are estimated on the basis of the Voigt model as the homogenization method. Although the Voigt rule does not consider the interaction among adjacent inclusions, Mori–Tanaka method considered these interactions of neighboring phases at the microscopic level as done in [46,47]. Using Voigt model material property in FGM, P(z) is expressed as:

(1) 
where and are material properties of metal and ceramic and their variation is shown in Fig. 3 for a lamination scheme of 221. is the volume fraction of the ceramic part which is written as: (For Type P beams)

for 

for 

for 
The FGM part is made to obey exponential law and the variation of material property by this law is shown in Fig. 4 for a lamination scheme 221and is written as:

(2) 
Fig. 1. Geometry of sandwich plate in the Cartesian coordinate system


Fig. 2. Layer configurations of FGSW beam of various types (a) Type PH and Type EH (b) Type PS and Type ES
where is given as: (For Type E beams)

for 

for 

for 
Consider a beam in the assumed coordinate system shown in Fig 1. The displacement in x ( ) direction is assumed as:

(3) 
where is midplane displacement along the xaxis for any point in the SW beam, is the rotation of normal to midplane and depict the number of upper and lower layers, respectively. , are higherorder unknowns, and , are the slope of ath and bth layers corresponding to the upper and lower layers respectively. denotes unit step function. The displacement in zdirection ( ) is assumed as taken in [38] as:
for core for upper face layer for lower face layer 
(4) 
where , and are the values of the transverse displacement at the top, middle and bottom lamina of the core, respectively, and , and are Lagrangian interpolation functions in the zdirection. By taking the reference of [38], the constitutive relation for stress in local coordinates ( ) of kth lamina is given as:

(5) 
where is transformed rigidity matrix of kth lamina and appeared in the above equation can be converted into a global coordinate system by using the transformed compliance matrix as:

(6) 
Now using the conditions of zero transverse shear stress at z=h/2 and z=h/2 and transverse shear stress continuity at interfaces of layers with at z=h/2, at z=h/2, .
The terms , and are expressed in terms of displacement and as:

Fig. 3. Variation of material property across the thickness of 221 FGSW beam (a) TypePH (b) Type PS

Fig. 4. Variation of material property across the thickness of 221 FGSW beam (a) Type EH (b) Type ES

(7) 
where ,
and material properties determine elements of . Through Equation (7) we are able to write the differentiation of transverse displacements in terms of other unknowns, thus avoiding the C1 continuity issue. Now Equation (3) is rewritten as:

(8) 
where the coefficients of s are determined by values of z, H, and material properties. Now, when all the coefficients of higher order terms of Equation (3) are eliminated so, we can write the generalized displacement with the help of Equations (4) and (8) as:

(9) 
Writing strain field in terms of unknowns as a combination of linear and nonlinear parts by using straindisplacement connection and Equations (36) as:

(10) 
where the linear part of the strain is
or

(11) 
and nonlinear part of strain is:

(12) 
where, or
and
And the elements of matrices , and are dependent on z and unit step functions. The data and the elements of can be accessed by the corresponding author through the mail.
Now applying the virtual work principle on the same lines as done in [35], the total potential energy of the system is given as :

(13) 
where is the strain energy and is the energy due to externally applied load. Utilizing equations (5) and (9), the strain energy is

(14) 
where

(15) 
is computed as:

(16) 
where and is the stress matrix of the ith layer generated from the external inplane loads which is given as
A numerical method i.e., FEM is used for the solution of buckling problems. A quadratic element with three nodes and seven degrees of freedom is considered.
The generalized displacement vector δ at any point can be expressed in terms of displacement and shape functions related to ith node.

(17) 
Here, n is no. of nodes in one element. From Equation (17), the strain vector used in Equation(11) is given as:

(18) 
where [B] is the strain displacement matrix.
The potential energy given in Equation (13) can be rewritten by using Equations (1416) as:

(19) 
where,

(20) 

(21) 
Finally minimizing with respect to {δ}

(22) 
where and are stiffness matrix and geometrical stiffness matrix and λ is the buckling load factor. A flow chart is made by incorporating all the steps needed to be followed for determining λ, which is given in the appendix. A code is written in FORTRAN for calculating the λ. A simultaneous iteration method is utilized for solving the buckling Equation (22).
Buckling analysis for four types of FGSW beams is presented here for the materials having properties: Ceramic (Al2O3) Ec=380GPA, µ=0.3 and Metal (Al) Ec=70 GPA, µ=0.3. The Nondimensional factor used in this study is given as:
where L and h are illustrated in Fig. 1 and is the transverse modulus of elasticity of the face sheet. Sixlayer configurations of FGSB are used in this study (Table 1), wherein h_{1,} h_{2,} etc are measured from the central layer of SW.
Table 1. Thickness coordinates of different lamination schemes
LS 
Thickness coordinates 
101 
h_{1}=h/2, h_{2}=0, h_{3}=0 and h_{4}=h/2 
212 
h_{1}=h/2, h_{2}=h/10, h_{3}=h/10 and h_{4}=h/2 
111 
h_{1}=h/2, h_{2}=h/6, h_{3}=h/6 and h_{4}=h/2 
121 
h_{1}=h/2, h_{2}=h/4, h_{3}=h/4 and h_{4}=h/2 
211 
h_{1}=h/2, h_{2}=h/4, h_{3}=0 and h_{4}=h/2 
221 
h_{1}=h/2, h_{2}=3/10, h_{3}=h/10 and h_{4}=h/2 
A convergence study was performed for present models (121), Type PH (Fig. 5), and Type PS (Fig. 6) at L/h=5 and n=2 for using different mesh divisions of 4, 8, 16, and 32. As the results converged at a mesh size of 16 so, it is adopted throughout this study. Table 2 presents the buckling response of the FGSW beam for the six lamination schemes and different power law factors. The present results are compared with those reported earlier: Kahya et al [17] used FSDT, Nguyen et al [19] used HOT, Vo et al [21] used refined HOT, Vo et al [28] used quasi3D theory for getting the responses and the present results are in good agreement with these.
Fig. 5. Variation of buckling load with mesh divisions for a SS 121 Type PH FGSW beam
Fig. 6. Variation of buckling load with mesh divisions for
a SS 121 Type PS FGSW beam
So, the present theory can be applied to buckling solution of FGSW beams. As expected, FSDT results [17] are yielding lower values of nondimensional buckling load in comparison to all other theories. Beams with lower powerlaw factors were found to withstand higher buckling loads irrespective of the lamination schemes for both TypeH beams owing to the variation of the material property of FGSW (Fig. 3a, 4a) in the thickness direction, while beams with higher powerlaw factor were found to withstand higher buckling loads irrespective of lamination schemes for both homogenization rule of Type S owing to the material property variation FGSW (Fig. 3b, 4b).
A high drift in buckling response is seen for a change of 1 unit (0 to 1) in the power law factor, which can be attributed to a change in material properties with a change of the powerlaw factor (Figs. 3a, 4a). However, for a change of 5 units (510) of the powerlaw, the change in the buckling resistance is found to be very less in comparison to that found an increase of 1 unit of power law factor from 0 to 1. The abovestated variation of buckling resistance with a change in power law factor is valid for both Type S and Type H beams and all lamination schemes considered in the present study. The 121 lamination scheme was found to have the highest buckling resistance for Type H beams for both homogenization schemes and for all power law factors, which can be attributed to the highest core thickness of the 121 scheme which is made up of ceramic material. End conditions were also found to have a significant effect on buckling strength.
Table 3 presents the buckling response of Type S FGSW beams for various lamination schemes and powerlaw factors. With an increase in the powerlaw factor, an enhanced buckling response is observed (Figs. 3b, 4b) owing to the dependency of the material property on the powerlaw factor. Lamination scheme 101 was found to have the highest buckling resistance for Type S beams for both homogenization rules and for all power law factors, which can be attributed to the lowest core thickness made of metal material (having lower strength in comparison to ceramic material). Again, lamination scheme 121 was found to have the highest buckling resistance among all lamination schemes for type H beams, for both homogenization rules and for all power law factors, which can be attributed to the highest core thickness made of ceramic material (having high strength in comparison to metallic material).
Tables 4 and 5 provide the variation of buckling response with an augment in the aspect ratio of the beam. Tables 4 and 5 are represented again in terms of graphs for greater clarity of the buckling strength variation. The effect of an increase in the lengthtoheight ratio on buckling response was found to be significant up to a value of 20, after which there was a little change in buckling response for both Type H and S FGSW beams (Figs. 712).
Fig. 7. Variation of buckling load with aspect ratio for Type PS FGSW beam with the CC end condition
Table 2. Variation of nondimensional buckling load for TypeH FGSW beam for SS end condition (L/h=5)
LC 
n 
Present models 
Reference solutions 

Type PH 
Type EH 
Ref. [17] 
Ref. [19] 
Ref. [21] 
Ref. [28] 

101 
0 
48.592 
48.592 
48.590 
48.596 
48.595 
49.590 

1 
19.658 
19.202 
19.485 
19.654 
19.652 
20.742 

2 
13.586 
13.171 
13.436 
13.582 
13.580 
13.883 

5 
10.240 
10.154 
10.012 
10.148 
10.146 
10.367 

10 
10.530 
10.414 
9.3292 
10.537 
9.4515 
9.6535 
121 
0 
48.592 
48.592 
47.969 
48.596 
48.595 
49.590 

1 
28.453 
28.062 
28.142 
28.444 
28.444 
29.075 

2 
22.791 
22.429 
22.571 
22.785 
22.786 
23.304 

5 
18.320 
18.004 
17.941 
18.091 
18.091 
18.509 

10 
16.562 
16.073 
16.244 
16.378 
16.378 
16.757 
111 
0 
48.592 
48.592 
48.152 
48.596 
48.595 
49.590 

1 
24.257 
23.847 
24.326 
24.560 
24.559 
25.107 

2 
18.389 
18.017 
18.190 
18.359 
18.358 
18.777 

5 
13.154 
13.018 
13.583 
13.722 
13.721 
14.035 

10 
12.658 
12.249 
12.112 
12.262 
12.260 
12.539 
212 
0 
48.592 
48.592 
48.333 
48.596 
48.595 
49.590 

1 
22.684 
22.023 
22.017 
22.212 
22.210 
22.706 

2 
15.417 
15.087 
15.762 
15.916 
15.915 
16.276 

5 
11.658 
11.149 
11.517 
11.669 
11.667 
11.930 

10 
10.581 
10.177 
10.354 
10.537 
10.534 
10.768 
211 
0 
48.592 
48.592 
48.277 
48.596 
48.595 
49.590 

1 
23.745 
23.129 
23.303 
23.525 
23.524 
24.083 

2 
17.778 
17.033 
17.144 
17.325 
17.324 
17.774 

5 
13.458 
13.171 
12.839 
13.027 
13.027 
13.392 

10 
11.814 
11.128 
11.606 
11.837 
11.838 
12.173 
221 
0 
48.592 
48.192 
48.130 
48.595 
48.596 
49.590 

1 
26.482 
26.029 
26.108 
26.361 
26.361 
26.976 

2 
20.462 
19.946 
20.186 
20.375 
20.375 
20.887 

5 
15.748 
15.249 
15.572 
15.730 
15.731 
16.160 

10 
14.413 
14.076 
14.027 
14.199 
14.200 
14.599 
Table 3. Variation of nondimensional buckling load for TypeS FGSW beam for SS end condition (L/h=5)
LC 
n 
Present models 
Reference solutions 

Type PS 
Type ES 
Ref. [21] 
CPT [38] 
FSDT [38] 
TOT [38] 
HBT*[38] 

101 
0 
8.9523 
8.9523 
8.9519 
9.869 
8.9508 
8.9533 
8.9579 

1 
36.227 
37.897 
36.210 
42.650 
38.252 
36.091 
35.624 

2 
41.86 
43.569 
42.450 
49.207 
44.415 
42.326 
41.293 

5 
46.750 
49.245 
46.650 
52.797 
48.105 
46.574 
45.022 

10 
46.487 
49.271 
47.782 
53.425 
48.918 
47.743 
46.043 
121 
0 
8.9523 
8.9523 
8.9519 
9.869 
8.9508 
8.9533 
8.9579 

1 
26.475 
27.513 
26.480 
33.089 
29.126 
26.369 
26.491 

2 
30.841 
32.410 
31.015 
39.372 
34.604 
30.793 
31.036 

5 
34.867 
35.487 
35.035 
44.504 
39.192 
34.693 
35.067 

10 
36.427 
37.691 
36.687 
46.356 
40.903 
36.302 
36.722 
111 
0 
8.9523 
8.9523 
8.9519 
9.869 
8.9508 
8.9533 
8.9579 

1 
30.379 
32.426 
30.244 
37.389 
33.063 
30.064 
30.262 

2 
35.512 
36.692 
35.705 
44.188 
39.139 
35.420 
35.732 

5 
40.216 
42.646 
40.323 
49.184 
43.790 
39.980 
40.354 

10 
42.165 
43.861 
42.069 
50.736 
45.326 
41.733 
42.098 
212 
0 
8.9523 
8.9523 
8.9519 
9.8696 
8.9508 
8.9533 
8.9579 

1 
32.912 
34.268 
32.897 
39.940 
35.506 
32.717 
32.914 

2 
38.714 
39.508 
38.858 
46.794 
41.757 
38.615 
38.881 

5 
43.476 
44.228 
43.533 
51.330 
46.137 
43.295 
43.555 

10 
45.253 
47.861 
45.114 
52.514 
47.403 
44.909 
45.132 
211 
0 
8.9523 
8.9523 
8.951 
 
 
 
 

1 
30.841 
32.629 
30.931 
 
 
 
 

2 
36.387 
37.816 
36.484 
 
 
 
 

5 
40.740 
42.486 
40.981 
 
 
 
 

10 
42.498 
43.826 
42.600 
 
 
 
 
221 
0 
8.9523 
8.9523 
8.952 
 
 
 
 

1 
27.554 
28.508 
27.887 
 
 
 
 

2 
32.482 
33.419 
32.790 
 
 
 
 

5 
36.785 
37.697 
37.035 
 
 
 
 

10 
38.617 
39.162 
38.701 
 
 
 
 
Table 4. Variation of nondimensional buckling load for TypeH FGSW beam at n=2
LC 
L/h 
CC 
CF 
SS 

Type PH 
Type EH 
Type PH 
Type EH 
Type PH 
Type EH 

101 
5 
47.725 
46.235 
3.514 
3.501 
13.487 
13.296 

10 
50.579 
49.527 
3.520 
3.516 
14.075 
13.952 

20 
56.283 
55.162 
3.562 
3.546 
14.201 
14.075 

50 
57.621 
55.862 
3.687 
3.636 
14.236 
14.086 

100 
57.694 
55.923 
3.692 
3.667 
14.238 
14.092 
121 
5 
78.562 
77.625 
5.946 
5.942 
22.768 
22.261 

10 
86.953 
85.361 
6.008 
5.998 
23.741 
23.420 

20 
94.865 
93.142 
6.026 
6.019 
23.974 
23.469 

50 
95.012 
94.267 
6.127 
6.113 
24.043 
23.895 

100 
95.167 
94.323 
6.137 
6.116 
24.057 
23.984 
111 
5 
64.427 
63.124 
4.761 
4.726 
18.372 
18.159 

10 
71.086 
70.268 
4.792 
4.781 
19.087 
18.946 

20 
76.124 
75.239 
4.831 
4.821 
19.211 
19.027 

50 
76.743 
75.563 
4.891 
4.861 
19.256 
19.082 

100 
76.886 
75.689 
4.885 
4.873 
19.254 
19.087 
212 
5 
56.241 
55.198 
4.125 
4.023 
15.931 
15.756 

10 
61.845 
60.271 
4.166 
4.110 
16.467 
16.004 

20 
65.861 
64.176 
4.183 
4.142 
16.602 
16.243 

50 
66.279 
65.142 
4.211 
4.189 
16.643 
16.281 

100 
66.386 
65.194 
4.237 
4.206 
16.651 
16.289 
211 
5 
60.621 
59.297 
4.498 
4.462 
17.324 
17.137 

10 
63.710 
62.581 
4.510 
4.496 
17.627 
17.340 

20 
71..987 
70.371 
4.531 
4.431 
18.142 
18.003 

50 
72.416 
71.892 
4.562 
4.452 
18.206 
18.129 

100 
72.449 
71.709 
4.569 
4.430 
18.427 
18.221 
221 
5 
70.756 
69.926 
5.296 
5.157 
20.374 
20.164 

10 
74.927 
73.804 
5.324 
5.234 
20.687 
20.531 

20 
84.847 
83.429 
5.368 
5.271 
21.396 
21.082 

50 
85.621 
84.155 
5.372 
5.293 
21.412 
21.210 

100 
85.699 
84.162 
5.376 
5.310 
21.345 
21.261 
Table 5. Variation of nondimensional buckling load for TypeS FGSW beam at n=2
LC 
L/h 
CC 
CF 
SS 

Type PH 
Type ES 
Type PS 
Type ES 
Type PH 
Type ES 

101 
5 
120.512 
122.629 
11.841 
11.986 
42.312 
43.260 

10 
156.219 
158.210 
12.069 
12.152 
47.351 
48.297 

20 
189.246 
190.856 
12.286 
12.298 
48.702 
49.106 

50 
192.458 
193.071 
12.349 
12.423 
49.165 
49.913 

100 
193.652 
193.303 
12.428 
12.520 
49.191 
50.097 
121 
5 
76.031 
77.527 
9.234 
9.356 
31.428 
32.568 

10 
124.091 
125.413 
9.532 
9.627 
36.834 
37.201 

20 
147.549 
148.109 
9.831 
9.916 
38.657 
39.644 

50 
152.365 
153.720 
9.843 
9.891 
39.237 
39.923 

100 
152.927 
153.806 
9.982 
9.993 
39.341 
40.108 
111 
5 
90.947 
91.743 
10.437 
10.861 
35.428 
36.638 

10 
126.879 
127.238 
10.854 
10.985 
41.612 
42.942 

20 
166.940 
167.280 
11.207 
11.305 
43.521 
44.054 

50 
173.830 
174.297 
11.304 
11.356 
44.097 
44.395 

100 
173.894 
174.309 
11.368 
11.413 
44.084 
44.409 
212 
5 
103.498 
104.986 
11.138 
11.206 
38.715 
39.264 

10 
138.496 
139.207 
11.349 
11.382 
44.537 
45.291 

20 
178.172 
179.206 
11.641 
11.753 
46.281 
47.059 

50 
183.607 
184.283 
11.726 
11.840 
46.725 
47.195 

100 
183.582 
184.782 
11.749 
11.851 
46.741 
47.238 
211 
5 
99.256 
100.231 
10.467 
10.561 
36.495 
37.951 

10 
130.719 
132.569 
10.561 
10.629 
39.259 
40.192 

20 
165.658 
166.217 
10.745 
10.861 
42.931 
43.187 

50 
168.265 
169.261 
10.835 
10.964 
43.380 
44.014 

100 
168.481 
169.445 
10.890 
10.983 
43.928 
44.464 
221 
5 
85.379 
86.120 
9.483 
9.496 
32.820 
33.165 

10 
126.843 
127.954 
9.641 
9.692 
35.619 
36.155 

20 
151.667 
152.294 
9.979 
9.986 
39.547 
40.127 

50 
154.831 
155.549 
9.986 
9.992 
39.803 
40.651 

100 
154.271 
155.982 
9.987 
10.107 
40.101 
40.756 
Fig. 8. Variation of buckling load with aspect ratio for Type PH FGSW beam with the CC end condition
Fig. 9. Variation of buckling load with aspect ratio for Type PS FGSW beam with CF end condition
Fig. 10. Variation of buckling load with aspect ratio for Type PH FGSW beam with CF end condition
Fig. 11. Variation of buckling load with aspect ratio for Type PS FGSW beam with the SS end condition
Fig. 12. Variation of buckling load with aspect ratio for Type PH FGSW beam with SS end condition
This paper presents buckling responses of the FGSW beams made of power law and exponential law using zigzag theory. Higherorder terms are assumed for displacement approximations. Numerical results are arrived at by using the FEM of three noded elements having 7DOF/node. The present model is C0 continuous and does not require any postprocessing method. The locking phenomenon which is associated with FEM is avoided here. Results of the present model are compared with the existing ones and are found to be consistent, which describes the suitability of the present model in deriving results for FGSW beams. It is found that buckling response is dependent on the powerlaw factor, aspect ratio, lamination schemes, and end conditions. Many new results are given which will pose as a benchmark for parallel studies. The main inferences drawn from the study are:
1) The buckling strength was improved by increasing the powerlaw factor for Type S beams while the opposite behavior was seen in type H beams for all types of lamination schemes and end conditions.
2) The end conditions played a major role in deciding the buckling response of FGSW beams. 3) Two types of laws were used in this paper to synthesize the FGM part of FGSW beams. The difference in buckling load resistance on using these two laws is small, but its trend is different for the two types: Type S and Type H.
4) It is found that exponential lawgoverned FGSW beams show a little higher buckling resistance behavior in comparison to power lawgoverned FGSW beams for Type S while the opposite behavior is seen for Type H beams for all types of end conditions and lamination schemes.
Acknowledgments
The first author of this paper was financially supported jointly by MHRD, GoI, and Director, NIT Kurukshetra, through a Ph.D. scholarship grant (2K18/NITK/PHD/6180093).
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this manuscript. In addition, the authors have entirely observed the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, and redundancy.
Appendix
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