Document Type : Research Paper
Authors
^{1} Department of Civil Engineering, SRES’s College of Engineering, Savitribai Phule Pune University, Kopargaon, Maharashtra, India
^{2} Department of Applied Mechanics, Government College of Engineering, Karad, Maharashtra, India
Abstract
Keywords

Mechanics of Advanced Composite Structures 2 (2015) 4553


Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://macs.journals.semnan.ac.ir 
Static Flexure of Soft Core Sandwich Beams using Trigonometric Shear Deformation Theory
A.S. Sayyad^{a}^{*}, Y.M. Ghugal^{b}
^{a }Department of Civil Engineering, SRES’s College of Engineering, Savitribai Phule Pune University, Kopargaon, Maharashtra, India
^{b }Department of Applied Mechanics, Government College of Engineering, Karad, Maharashtra, India
Paper INFO 

ABSTRACT 
Paper history: Received 17 April 2015 Received in revised form 6 November 2015 Accepted 7 November 2015 
This study deals with the applications of a trigonometric shear deformation theory considering the effect of the transverse shear deformation on the static flexural analysis of the soft core sandwich beams. The theory gives realistic variation of the transverse shear stress through the thickness, and satisfies the transverse shear stress free conditions at the top and bottom surfaces of the beam. The theory does not require a problemdependent shear correction factor. The governing differential equations and the associated boundary conditions of the present theory are obtained using the principle of the virtual work. The closedform solutions for the beams with simply supported boundary conditions are obtained using Navier solution technique. Several types of sandwich beams are considered for the detailed numerical study. The axial displacement, transverse displacement, normal and transverse shear stresses are presented in a nondimensional form and are compared with the previously published results. The transverse shear stress continuity is maintained at the layer interface, using the equilibrium equations of elasticity theory. 



Keywords: Laminated beam Soft core Sandwich beam Flexure Trigonometric shear deformation theory



© 2015 Published by Semnan University Press. All rights reserved. 
Sandwich beam is a special form of laminated composite beam which has stiff face sheets and light weight but thick core. The modulus of the core material is significantly lower than that of the face sheets. The main benefit of using the sandwich concept in the structural components is its high bending stiffness and high strength to weight ratio. In addition, the sandwich constructions are much preferred to conventional materials because of their superior mechanical and durability properties. Due to these properties the composite sandwich structures have been widely used in the automotive, aerospace, marine and other industrial applications. Therefore, the analytical study of the sandwich beams becomes increasingly important.
Since the Classical Beam Theory (CBT) neglects the effect of the shear deformation and the Firstorder Shear Deformation Theory (FSDT) of Timoshenko [1] requires a shear correction factor, these theories are not suitable for the analysis of the laminated composite and the sandwich beams. These limitations of CBT and FSDT have led to the development of the Higherorder Shear Deformation theories (HSDTs) taking into account the effect of the transverse shear deformation, obviating the need of a shear correction factor.
The beam theories can be developed by expanding the displacements in power series of the coordinate normal to the neutral axis. In principle, the theories developed by this means can be made as accurate as desired simply by including the sufficient number of terms in the series. These higherorder theories are cumbersome and computationally more demanding, because with an additional power of the thickness coordinate, an additional dependent variable is introduced into the theory. It has been noted by Lo et al. [2, 3] that due to the higherorder terms included in their theory, it has become inconvenient to use. This observation is more or less true for many other higherorder theories as well. Thus, there is a wide scope to develop a simple to use higherorder beam or plate theory.
Several theories have been proposed by researchers in the last two decades. Among many theories, some of the wellknown theories are the parabolic shear deformation theories [45], the trigonometric shear deformation theory [6], the hyperbolic shear deformation theory [7] and the exponential shear deformation theory [8]. Recently, these theories are accounted into a unified shear deformation theory developed by Sayyad [9] and Sayyad et al. [10]. In accordance with Reddy’s thirdorder shear deformation theory, Sayyad [11] has developed the refined theories and applied them for the static and vibration analysis of the isotropic beams.
Mechab et al. [12] studied the deformations of the short composite beams using the refined theories. Carrera and Giunta [13] developed refined beam theories based on a unified formulation. Carrera et al. [14, 15] carried out the static and free vibration analysis of laminated beams using polynomial, trigonometric, exponential and zigzag theories. Giunta et al. [16] presented a thermomechanical analysis of isotropic and composite beams via collocating with radial basis functions. Chakrabarti et al. [17] and Chalak et al. [18] carried out a finite element analysis for the bending, buckling and free vibration of the soft core sandwich beams. Gherlone et al. [19] developed C^{0} beam elements based on the refined zigzag theory for the multilayered laminated composite and sandwich beams.
In the class of Trigonometric Shear Deformation Theories (TSDTs), the shear deformation is assumed to be trigonometric with respect to the thickness coordinate. These theories are accounted cosine distribution of transverse shear stress. The TSDTs are taking into account the kinematics of higherorder theories more effectively without loss of the physics of the problem. Some of the wellknown articles on trigonometric theories are published by Touratier [6], Shimpi and Ghugal [20], Ghugal and Shinde [21], Arya et al. [22], Sayyad and Ghugal [23], Mantari et al. [24], Ferreira et al. [25], Zenkour [26] and Sayyad et al. [27]. Recently, Dahake and Ghugal [28, 29] and Ghugal and Dahake [30] have applied the trigonometric shear deformation theory for the bending analysis of the singlelayer isotropic beams with various boundary conditions using general solution technique.
In the current study, a trigonometric shear deformation theory is applied for the bending analysis of the laminated composite and the soft core sandwich beams. The theory involves three unknowns. The theory satisfies the transverse shear stress free conditions at the top and bottom surfaces of the beam and does not require shear correction factor. The governing equations are obtained using the principle of the virtual work. The closedform solutions for the beam with simply supported boundary conditions are obtained using Navier solution technique. The displacements and stresses of three different types of lamination scheme are obtained.
The exact elasticity solution for the threelayered (0^{0}/90^{0}/0^{0}) laminated composite developed by Pagano [31] is used as a basis for the comparison of the present results. However, the exact elasticity solutions for the threelayered (0^{0}/core/0^{0}) and fivelayered (0^{0}/90^{0}/core/90^{0}/0^{0}) sandwich beams are not available in the literature. Authors have generated the numerical results using FSDT of Timoshenko [1], HSDT of Reddy [5] and CBT being not available. It is found that the present results are in excellent agreement with those of HSDT, FSDT, CBT and exact elasticity solution.
Consider a beam of length ‘L’ along x direction, width ‘b’ along y direction and thickness ‘h’ along z direction. The coordinate system and geometry of the beam under consideration are shown in Fig. 1. The beam consists of the face sheets at the top and bottom surfaces and the middle portion is made up of a soft core.
The beam is bounded in the region 0 ≤ x ≤ L, b/2 ≤ y ≤ b/2, h/2 ≤ z ≤ h/2 in Cartesian coordinate system. u and w are the displacements in x and z directions, respectively.
Figure 1. The beam geometry and the coordinate system
2.1. The Assumptions made in the Theoretical Formulation
In the present equivalent singlelayer trigonometric shear deformation theory, the theoretical formulation is based on the six following assumptions:
1) The axial displacement u in x direction consists of two parts including (a) a displacement component analogous to the displacement in the classical beam theory and (b) a displacement component due to the shear deformation which is assumed to be sinusoidal in nature with respect to the thickness coordinate.
2) The transverse displacement w in the z direction is assumed to be a function of the x coordinate only.
3) The beam is made up of ‘N’ number of layers which are perfectly bonded together.
4) One dimensional Hooke’s law is used.
5) The beam is subjected to the lateral load only.
6) The body forces are ignored.
2.2. The Kinematics of the Present Theory
Based on the above mentioned assumptions, the displacement field of the present trigonometric shear deformation theory is written as:
(1)
where u and w are the displacements in x and z directions, respectively and is the thickness coordinate. , and are the unknown functions to be determined. The normal and shear strains obtained within the framework of the linear theory of elasticity are as follows:
(2)
where
and (3)
‘, _{x}’ represents the derivative with respect to x.
2.3. The constitutive relations
The normal and transverse shear stresses are obtained using onedimensional constitutive relations. These relations for the k^{th} layer of the beam are given by the following equations:
(4)
where and are the stiffness coefficients of the k^{th} layer of the beam and are defined as follows:
and
where is the Young’s modulus and is the shear modulus of kth layer of the beam.
In order to derive the governing equations, the principle of the virtual work is used.
(5)
Using the expressions for strains from Eq. (2) and stresses from Eq. (4), the Eq. (5) can be written as:
(6)
Integrating Eq. (6) with respect to the zdirection, Eq. (6) can be simplified as:
(8)
where A_{ij}, B_{ij}, etc. are the beam stiffnesses as defined below:
(9)
Integrating Eq. (8) by the parts and setting the coefficients of , and equal to zero, we obtain the coupled EulerLagrange equations which are the governing differential equations and associated boundary conditions of the beam. The governing equations of the beam are as follow:
(10) (11) (12)
The associated consistent natural boundary conditions at the ends x = 0 and x = L are as follows:
or is prescribed (13)
or is prescribed (14)
or is prescribed (15)
or is prescribed (16)
where the resultants are defined using the following equations:
(17)
(18)
(19)
Thus, the variationally consistent governing differential equations and boundary conditions are obtained.
The Navier solution satisfies the governing differential equation and boundary conditions when the beam is simply supported at the ends. Therefore, a static flexural analysis of the simply supported laminated composite and the soft core sandwich beams subjected to transverse distributed load has been carried out using Navier solution technique. According to this technique, the following solution form for unknown functions is assumed.
(20)
where and are the unknown Fourier coefficients to be determined for each m value. A beam of length L and thickness h is considered. The transverse load acting on the top surface of the beam is expanded in the following form:
(21)
(22)
where is the maximum intensity of the load. Substituting the solution form from Eq. (20) and transverse load from Eqs. (21) and (22) into the three governing Eqs. (10)(12), leads to the following set of simultaneous equatios.
(23)
Eq. (23) can be solved to obtain the Fourier coefficients , and . Further, the final expressions for displacements and stresses are obtained using Eqs. (1)(4).
To prove the efficacy of the present theory, it is applied to the flexural analysis of the following examples on the laminated composite and soft core sandwich beams.
Example 1:
A static flexure of the threelayered (0^{0}/90^{0}/0^{0}) laminated composite beams, as shown in Fig. 2 (a).
Example 2:
A static flexure of the threelayered (0^{0}/core/0^{0}) soft core sandwich beams, as shown in Fig. 2 (b).
Example 3:
A static flexure of the fivelayered (0^{0}/90^{0}/core/90^{0}/0^{0}) soft core sandwich beams, as shown in Fig. 2 (c).
The following material properties are used in the above examples:
Example 1:
0^{0} layer: Q_{11}= 25×10^{6} psi, Q_{55}= 0.5×10^{6} psi
90^{0} layer: Q_{11}= 1.0×10^{6} psi, Q_{55}= 0.2×10^{6} psi
Examples 2 and 3:
Face sheets (0^{0}): Q_{11}= 25×10^{6} psi, Q_{55}= 0.5×10^{6} psi
Face sheets (90^{0}): Q_{11}= 1.0×10^{6} psi, Q_{55}= 0.2×10^{6} psi
Core: Q_{11}= 0.04×10^{6} psi, Q_{55}= 0.06×10^{6} psi.
The numerical results obtained for the displacements and stresses at the critical points are presented in the following nondimensional form (Take E_{3} = 1).
(24)
Example 1:
In this example, the bending response of the simply supported threelayered (0°/90°/0°) laminated composite beam is investigated as shown in Fig. 2 (a). The numerical results for the nondimensional displacements and stresses are presented in Tables 1 and 2. For the comparison purpose, the numerical results are specially generated using the Higherorder Shear Deformation Theory (HSDT) of Reddy [5], the Firstorder Shear Deformation Theory (FSDT) of Timoshenko [1] and the Classical Beam Theory (CBT). The examination of Tables 1 and 2 reveals that when the laminated composite beam is subjected to the sinusoidal/uniform load, the displacements and normal stresses are in excellent agreement with those of Higherorder Shear Deformation Theory (HSDT) of Reddy [5]. The transverse shear stress is obtained using the equilibrium equation of the elasticity theory with the shear stress continuity at the layer interface.
Figure 2. The lamination scheme and the thickness coordinate for the simply supported beams
Table 1: The comparison of the axial displacement ( ), the transverse displacement ( ), the normal stress ( ), and the transverse shear stress (), for threelayered (0^{°}/90^{°}/0^{°}) laminated composite beam subjected to the sinusoidal load
L/h 
Theory 

100 
Present 
8037.5 
0.5148 
6312.6 
44.22 

HSDT [5] 
8034.9 
0.5146 
6310.6 
44.27 

FSDT [1] 
8025.7 
0.5135 
6303.3 
44.22 

CBT 
8025.7 
0.5109 
6303.3 
44.22 

Exact [31] 
8040.0 
0.5153 
6315 
44.15 
10 
Present 
9.019 
0.8836 
70.853 
4.320 

HSDT [5] 
8.939 
0.8751 
70.212 
4.330 

TSDT [27] 
9.016 
0.8828 
70.836 
4.322 

FSDT [1] 
8.025 
0.8149 
63.033 
4.422 

CBT 
8.025 
0.5109 
63.033 
4.420 

Exact [31] 
9.105 
0.8800 
71.300 
4.200 
4 
Present 
0.892 
2.7340 
17.540 
1.532 

HSDT [5] 
0.865 
2.7000 
17.006 
1.557 

TSDT [27] 
0.891 
2.7252 
17.500 
1.528 

FSDT [1] 
0.514 
2.4107 
10.085 
1.769 

CBT 
0.514 
0.5109 
10.085 
1.769 

Exact [31] 
0.915 
2.8870 
17.880 
1.425 
The detailed procedure to obtain this stress using equilibrium equation is given by Sayyad et al. [27]. The through thickness distributions of axial displacement, normal stress and transverse shear stress are shown in Figs. 35.
Example 2:
This example investigates the bending response of the threelayered (0°/core/0°) soft core sandwich beams as shown in Fig. 2 (b). The beam has thin top and bottom face sheets of thickness 0.1h each and thick core of thickness 0.8h. The comparison of results for the beam subjected to the sinusoidal load is shown in Table 3.
Table 2: The comparison of the axial displacement ( ), the transverse displacement ( ), the normal stress ( ), and the transverse shear stress (), for the threelayered (0^{°}/90^{°}/0^{°}) laminated composite beam subjected to the uniform load
L/h 
Theory 

100 
Present 
10386.3 
0.6528 
7786.5 
68.239 

HSDT [5] 
10382.8 
0.6527 
7784.2 
68.243 

FSDT [1] 
10368.6 
0.6518 
7776.7 
68.387 

CBT 
10368.6 
0.6480 
7776.7 
68.387 
10 
Present 
11.844 
1.108 
85.68 
6.042 

HSDT [5] 
11.733 
1.098 
85.03 
6.090 

FSDT [1] 
10.368 
1.023 
77.76 
6.838 

CBT 
10.368 
0.648 
77.76 
6.838 
4 
Present 
1.195 
3.413 
20.30 
2.629 

HSDT [5] 
1.161 
3.368 
19.67 
2.795 

FSDT [1] 
0.663 
2.991 
12.44 
2.735 

CBT 
0.663 
0.648 
12.44 
2.735 
Table 3: The comparison of the axial displacement ( ), the transverse displacement ( ), the normal stress ( ), and the transverse shear stress (), for the threelayered (0^{°}/core/0^{°}) soft core sandwich beam subjected to the sinusoidal load
L/h 
Theory 

4 
Present 
1.7678 
10.091 
34.710 
1.3732 

HSDT [5] 
1.7413 
10.047 
34.189 
1.3681 

FSDT [1] 
1.0134 
5.2868 
19.898 
1.4106 

CBT 
1.0134 
1.0081 
19.898 
1.4106 
10 
Present 
17.758 
2.4887 
139.47 
3.5094 

HSDT [5] 
17.687 
2.4805 
138.91 
3.5091 

FSDT [1] 
15.834 
1.6927 
124.36 
3.5264 

CBT 
15.831 
1.0081 
124.36 
3.5264 
20 
Present 
130.55 
1.3794 
512.67 
7.0451 

HSDT [5] 
130.39 
1.3771 
512.05 
7.0441 

FSDT [1] 
126.67 
1.1792 
497.46 
7.0528 

CBT 
126.67 
1.0081 
497.46 
7.0528 
50 
Present 
1989.3 
1.0677 
3124.8 
17.6313 

HSDT [5] 
1988.6 
1.0672 
3123.7 
17.6285 

FSDT [1] 
1979.3 
1.0355 
3109.1 
17.6319 

CBT 
1979.3 
1.0081 
3109.1 
17.6319 
100 
Present 
15856 
1.0231 
12453 
35.2678 

HSDT [5] 
15853 
1.0229 
12451 
35.2624 

FSDT [1] 
15834 
1.0149 
12436 
35.2639 

CBT 
15834 
1.0081 
12436 
35.2639 
Figure 3. The through thickness variation of the axial displacement for the threelayered (0°/90°/0°) laminated composite beam subjected to the sinusoidal load
Figure 4. The through thickness variation of the normal stress for the threelayered (0°/90°/0°) laminated composite beam subjected to the sinusoidal load
Figure 5. The through thickness variation of the transverse shear stress for the threelayered (0°/90°/0°) laminated composite beam subjected to the sinusoidal load
It is observed from the results that the present theory is in excellent agreement with HSDT to predict the bending response of soft core sandwich beams. The through thickness distributions of displacement and stresses are shown in Figs. 68.
Example 3:
In this example, the bending response of the fivelayered (0°/90°/core/90°/0°) soft core sandwich beams is investigated as shown in Fig. 2 (c). The beam has two face sheets at the top and bottom and transversely flexible core at the center. The thickness of each face sheet is 0.05h each and thickness of core is 0.8h. The comparison of displacements and stresses for the beam subjected to the sinusoidal load is shown in Table 4 for various aspect ratios (L/h).
Figure 6. The through thickness variation of the axial displacement for the threelayered (0°/core/0°) soft core sandwich beam subjected to the sinusoidal load
Figure 7. The through thickness variation of the normal stress for the threelayered (0°/core/0°) soft core sandwich beam subjected to the sinusoidal load
Figure 8. The through thickness variation of the transverse shear stress for the threelayered (0°/core/0°) soft core sandwich beam subjected to the sinusoidal load
Table 4: The comparison of the axial displacement ( ), the transverse displacement ( ), the normal stress ( ), and the transverse shear stress (), for fivelayered (0°/90°/core/90°/0°) soft core sandwich beam subjected to the sinusoidal load
L/h 
Theory 

4 
Present 
2.2146 
10.929 
43.484 
1.3402 

HSDT [5] 
2.2070 
10.925 
43.334 
1.3376 

FSDT [1] 
1.7660 
7.1233 
34.676 
1.3469 

CBT 
1.7660 
1.7567 
34.676 
1.3469 
10 
Present 
28.718 
3.2141 
225.55 
3.3630 

HSDT [5] 
28.702 
3.2315 
225.42 
3.3623 

FSDT [1] 
27.594 
2.6154 
216.73 
3.3674 

CBT 
27.594 
1.7567 
216.73 
3.3674 
20 
Present 
223.00 
2.1212 
875.74 
6.7326 

HSDT [5] 
222.97 
2.1257 
875.62 
6.7322 

FSDT [1] 
220.75 
1.9714 
866.92 
6.7322 

CBT 
220.75 
1.7567 
866.92 
6.7347 
50 
Present 
3454.9 
1.8151 
5427.0 
16.8359 

HSDT [5] 
3454.9 
1.8150 
5426.9 
16.8359 

FSDT [1] 
3449.3 
1.7911 
5418.2 
16.8368 

CBT 
3449.3 
1.7567 
5418.2 
16.8368 
100 
Present 
27606 
1.7713 
21681 
33.6734 

HSDT [5] 
27606 
1.7714 
21681 
33.6734 

FSDT [1] 
27594 
1.7653 
21673 
33.6735 

CBT 
27594 
1.7567 
21673 
33.6735 
The results show that the axial displacement and stresses are increased with an increase in the aspect ratio, while the transverse displacement is decreased. Since an exact solution for this example is not available in the literature, the results of the present theory are compared with other theories and are found to agree well with each other. Table 5 shows the displacements and stresses for the fivelayered soft core sandwich beams subjected to the uniform load. The through thickness distributions of the axial displacement, the normal stress and the transverse shear stress via equilibrium equation are shown in Figs. 911.
Figure 9. The through thickness variation of the axial displacement for the fivelayered (0°/90°/core/90°/0°) soft core sandwich beam subjected to the sinusoidal load
Table 5: The comparison of the axial displacement ( ), the transverse displacement ( ), the normal stress ( ), and the transverse shear stress (), for fivelayered (0°/90°/core/90°/0°) soft core sandwich beam subjected to the uniform load
L/h 
Theory 

4 
Present 
2.9647 
13.449 
51.694 
2.2517 

HSDT [5] 
2.9438 
13.556 
51.532 
2.1906 

FSDT [1] 
2.2816 
8.8491 
42.782 
2.0828 

CBT 
2.2816 
2.2282 
42.782 
2.0828 
10 
Present 
37.382 
4.0263 
276.38 
5.1818 

HSDT [5] 
37.350 
4.0479 
276.22 
5.1653 

FSDT [1] 
35.650 
3.2875 
267.38 
5.2070 

CBT 
35.650 
2.2282 
267.38 
5.2070 
20 
Present 
288.67 
2.6778 
1078.58 
10.378 

HSDT [5] 
288.60 
2.6834 
1078.44 
10.375 

FSDT [1] 
285.20 
2.4930 
1069.55 
10.414 

CBT 
285.20 
2.2282 
1069.55 
10.414 
50 
Present 
4464.9 
2.3001 
6693.68 
26.020 

HSDT [5] 
4464.8 
2.3010 
6693.67 
26.019 

FSDT [1] 
4456.2 
2.2705 
6684.60 
26.035 

CBT 
4456.2 
2.2282 
6684.60 
26.035 
100 
Present 
35667.0 
2.2464 
26747.8 
52.062 

HSDT [5] 
35667.0 
2.2464 
26747.8 
52.062 

FSDT [1] 
35650.0 
2.2387 
26738.6 
52.069 

CBT 
35650.0 
2.2282 
26738.8 
52.070 
In this study, a trigonometric shear deformation theory has been presented for the bending analysis of the soft core sandwich beams. The theory is a displacementbased theory which includes the transverse shear deformation effect. The number of unknown variables is the same as that of the firstorder shear deformation theory. The theory satisfies zero shear stress conditions on the top and bottom surfaces of the beam perfectly. Hence, the theory obviates the need for the shear correction factor.
Figure 10. The through thickness variation of the normal stress for the fivelayered (0°/90°/core/90°/0°) soft core sandwich beam subjected to the sinusoidal load
Figure 11. The through thickness variation of the transverse shear stress for the fivelayered (0°/90°/core/90°/0°) soft core sandwich beam subjected to the sinusoidal load
From the numerical study and discussion it is concluded that the present theory is in an excellent agreement with other theories, while predicting the bending response of the laminated composite and soft core sandwich beams with transversely flexible core.
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