Finite element flexural analysis of sandwich plates under various types of loading conditions

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Volume 3, Issue 5, May 2015 Page 27

ABSTRACT The flexure analysis of Isotropic, Orthotropic and Sandwich plates using Ansys is presented using solid 46 element. Numerical results for simply supported isotropic, orthotropic and sandwich plate are presented. The results are compared with other available results from literature. Convergence for each plate is carried out to obtain stable results. The effects of various types of loadings, fiber orientation,aspect ratio and boundary conditions is presented. Keywords: FEM, sandwich plate, line load, point load

1. INTRODUCTION Fiber reinforced composite materials are unique in application because the use of long fibers results in a material which has a higher strength-to-density ratio and/or stiffness-to-density ratio than any other material system at moderate temperatures, and there exists the opportunity to uniquely tailor the fiber orientations to a given geometry, applied load and environment. Composite laminates have been used increasingly in a variety of industrial areas due to their high stiffness and strength-to weight ratios, long fatigue life, resistance to electro chemical corrosion, and other superior material properties of composites. Sandwich composite is a special form of laminated composite composed of two thin stiff face sheets and a thick lightweight core bonded between them. A sandwich structure offers different mechanical properties using different types of materials as performance of sandwich structures depends on the properties of the constituents. In a sandwich structure generally the bending loads are carried by the force couple formed by the face sheets and the shear loads are carried by the lightweight core material. The face sheets are strong and stiff both in tension and compression as compared to the low density core material whose primary purpose is to maintain a high moment of inertia.

2. FINITE ELEMENT MODELING Ansys is a general purpose software embedded with various disciplines like physics, static and dynamic structural, fluid, thermal and vibrations giving a wide platform for professionals as well common users to model and analyze products at their levels. The present analysis is carried out using Ansys GUI. A brief description of each finite element type has been given in Ansys Element Library [1].Solid 46 is designed to model layered thick shells or solids. The element allows up to 250 different material layers. If more than 250 layers are required, a user-input constitutive matrix option is available. The element may also be stacked as an alternative approach. The element has three degrees of freedom at each node: translations in the nodal x, y, and z directions.

Figure 1Solid 46 geometry [1]

Finite element flexural analysis of sandwich plates under various types of loading conditions

Bishwadeep pal1 , Jeeoot Singh2

Department of Mechanical Engineering, Birla Institute of Technology, Mesra, Ranchi, Jharkhand-835215




Boundary Conditions are as follows for solid 46 element type:

3. RESULTS AND DISCUSSION Using Solid 46 element is advantageous as it is a 3D element and supports element real constants. Hence, multiple elements can be created in through thickness direction to obtain deflection and stress values at desired locations along with different fiber orientations as well as thickness of each element. For Square Isotropic plate, 10 elements were created in through thickness direction. For Square 4-ply orthotropic plate, 5 elements were created in through thickness direction for each ply. Thus a total of 20 elements were created in through thickness direction for 4-Ply Orthotropic plate. For Square Sandwich Plate, 5 elements and 10 elements were created in through thickness direction for Skin and Core Respectively. To show the accuracy of current methodology, a detailed convergence study for simply supported square isotropic, orthotropic and sandwich plate under uniformly distributed load is carried out. The obtained results show a good convergence with other available results. 3.1 Isotropic Plate A simply supported square isotropic plate(E = 1 GPa, v = 0.3) has been analyzed and the results are shown in Table-1. The normalized deflection (wn) and stress (σxxn) are obtained using expression:

wn=100h3Ew/qoa4 σxxn=h2σxx/qa2

Table 1 Normalized maximum deflection and stress in Isotropic square plate under UDL (q= 1 Pa).

Edge For Simply Supported

For Clamped

X=0,a Uy=0; Uz=0 Ux=Uy=Uz=0

Y=0,b Ux=0; Uz=0 Ux=Uy=Uz=0




Figure 2 Convergence ofSimply supported Isotropic square plate under uniformly distributed load (a/h=10; q=1 Pa)

Table-1 shows the convergence for deflection and normal stress and same is depicted in Figure-2 for deflection. It can be seen that a better convergence is achieved for mesh size of 17x17. Present solution is in good agreement with the solutions obtained by Jeeootsingh et al [4] and Reddy[3]. 3.2 Orthotropic Plate A simply supported square 4-ply orthotropic plate (a/h=10) with stacking sequence [0/90/90/0] has been analyzed and

Material Properties:

E11 25e9

E22 1e9

E33 1e9

G12 0.5e9

G23 0.2e9

G13 0.5e9

v12 0.25

the results are shown in Table-2. The material properties are as follows: Table 2: Deflection (w) and stress (σx) of 4-ply orthotropic simply supported square plate (a/h=100) UDL (q=1Pa)

From Table-2, it can be seen that a better convergence is achieved for mesh size of 17x17. Present solution is in good agreement with the solutions obtained by Reddy[3].

-5000

0

5000

Nor

mal

stre

ss

(σx)

z/h

σx

Figure 3 Variation of Normal stress (σx) along the thickness of 4-ply Orthotropic square plate (a/h=100) under UDL

(q=1Pa)




Figure 4 Variation of Normal stress (σy) along the thickness of 4-ply Orthotropic square plate (a/h=100) under UDL

(q=1Pa)

Figure 5 Variation of Shear stress (σxz)along the thickness of 4-ply Orthotropic square plate (a/h=100) UDL (q=1Pa)

3.3 Sandwich plate An example for analyzing sandwich plate is taken with following stacking sequence and dimensions. The meshed model and applied loads and boundary conditions are shown in Figure-6 and Figure-7 respectively. In this case, 2 elements were created in through thickness direction of each face-sheet of each ply while 10 elements were created in through thickness direction in case of core. Dimension: 0.8mx0.3mx0.0174m, Stacking sequence: [45/-45/0/90/90/0/-45/45], Load: 2 KN, Boundary Condition: Simply Supported

Figure 6 Top view of meshed sandwich plate

Figure 7 Isometric view of simply-supported sandwich plate showing loads and boundary conditions.




Table 3 Deflection of Simply supported sandwich plate.

From Table-3, it is observed that the present solution is closer to experimental values as compared to other value taken from literature [5].The variation of normal and shear stresses along the thickness are shown in figures below:

Figure 8 Variation of Normal stress (σx)along the thickness of simply supported sandwich plate under Double line

load.

Figure 9 Variation of Normal stress (σy)along the thickness of simply supported sandwich plate under Double line

load.

-400000

-200000

0

200000

400000

0.00

87

0.00

855

0.00

84

0.00

825

0.00

81

0.00

795

0.00

78

0.00

765

0.00

75

0.00

45

0.00

15

-0.0

015

-0.0

045

-0.0

075

-0.0

0765

-0.0

078

-0.0

0795

-0.0

081

-0.0

0825

-0.0

084

-0.0

0855

-0.0

087

Shea

r str

ess

(σxz

)

z/h

σxz

Figure 10 Variation of Shear stress (σxz)along the thickness of simply supported sandwich plate under Double line

load.




Some more results have been evaluated by applying different loading conditions, effect of fiber orientation and effect of varying aspect ratio with different boundary conditions (simply supported and clamped).

Table 4 Variation of transverse central deflection of simply-supported sandwich plate with different loading conditions

From Table 4, it can be seen that transverse central deflection is maximum when point load is applied. The deflection in double line load is higher than deflection due to UDL but less than centerline load.

Table 5 Deflection values for sandwich plate under UDL with different boundary conditions.

From Table 5, it is observed that transverse central deflection is maximumwhen all edges simply supported while minimum when clamped. When alternate boundary conditions are applied at edges, deflection is very close to that of deflection when all edges simply supported. Following figures show the variation of stresses along the thickness of sandwich plate with different loading conditions:

Figure 11 Variation of Normal stress (σx)along the thickness of simply supported sandwich plate under Center line

load.

-5.00E+07

0.00E+00

5.00E+07

0.00

87

0.00

8…

0.00

84

0.00

8…

0.00

81

0.00

7…

0.00

78

0.00

7…

0.00

75

0.00

45

0.00

15

-… -…

-0.0

0…

-…

-… -… -… -… -… -… -…

Nor

mal

stre

ss (σ

y)

z/h

σy

Figure 12 Variation of Normal stress (σy)along the thickness of simply supported sandwich plate under Center line

load.




Figure 13 Variation of Shear stress (σxz)along the thickness of simply supported sandwich plate under Center line load.

-1.00E+08

-5.00E+07

0.00E+00

5.00E+07

1.00E+08

0.00

87

0.00

855

0.00

84

0.00

825

0.00

81

0.00

795

0.00

78

0.00

765

0.00

75

0.00

45

0.00

15

-0.0

015

-0.0

045

-0.0

075

-0.0

0765

-0.0

078

-0.0

0795

-0.0

081

-0.0

0825

-0.0

084

-0.0

0855

-0.0

087

Nor

mal

stre

ss (σ

x)

z/h

σx

Figure 14 Variation of Normal stress (σx)along the thickness of simply supported sandwich plate under uniformly

distributed load.

-2.00E+07

0.00E+00

2.00E+07

Nor

mal

stre

ss (σ

y)

z/h

σy

Figure 15 Variation of Normal stress (σy)along the thickness of simply supported sandwich plate under uniformly

distributed load.

-500000

0

500000

Shea

r str

ess

(σxz

)

z/h

σxz

Figure 16 Variation of Normal stress (σxz)along the thickness of simply supported sandwich plate under uniformly

distributed load.




3.4 Effect of fiber orientation

0.0048380.00484

0.0048420.0048440.0048460.004848

0.004850.0048520.0048540.004856

Def

orm

atio

n, w

Stacking sequence

Figure 17 Effect of fiber orientation on deflection(w) of Simply Supported Sandwich Plate

Figure 18 Effect of fiber orientation on Stress(σx) of Simply Supported Sandwich Plate

-1.00E+08

-5.00E+07

0.00E+00

5.00E+07

1.00E+08

Nor

mal

stre

ss (σ

y)

[45/-45/0/90/90/0/-45/45] [-45/45/0/90/90/0/45/-45]

[-45/0/45/90/90/45/0/-45] [-45/0/90/45/45/90/0/-45]

Figure 19 Effect of fiber orientation on Stress(σy) of Simply Supported Sandwich Plate




Figure 20 Effect of fiber orientation on Stress(σxz) of Simply Supported Sandwich Plate

From figures 18 and 19, it is observed that there is very slight change in deflection and stress values when top fiber oriented in angles 45o and -45o are interchanged. While the deflection decreases when the fiber oriented at 45o is in sequence with 90o oriented fibers. Also, the normal stresses (σx and σy) are maximum at 0o fiber orientations but mimimum at 90o fiber orientations. In case of shear stresses (σxz), it is observed that maximum shear stress occurs around the 45o fiber orientations. 3.5 Effect of Aspect ratio with different Boundary Conditions:

Table 6 Deflection (w) of sandwich plate for different aspect ratios and boundary conditions.

From Table 6, it is observed that the transverse central deflection increases with increase in aspect ratio of the sandwich plate. Also, deflection is higher when simply supported than clamped condition.

4. CONCLUSIONS Following conclusions are drawn from the present analysis of sandwich plates:

1. Solid 46 element is suitable for modeling sandwich composites. Results obtained using solid 46 element show good agreement with other available results from literature.

2. Effect of types of loads: The deflection in double line load is higher than deflection due to UDL but less than centerline load.

3. Effect of Boundary Conditions: Transverse central deflection is maximum when all edges simply supported while minimum when clamped.

4. Effect of Fiber orientations: Deflection and Stress values differ very slightly when top fibers stacked in sequence 45o or -45o. Maximum normal stress occur at fibers oriented at 0o while mimimum at 90o. Maximum shear stress occurs at 45o fiber orientation.

5. Effect of aspect ratio: The transverse central deflection increases with increase in aspect ratio of the sandwich plate.

REFERENCES [1]. Ansys Element Library. [2]. S. Xiang, W. Ke-ming, A. Yan-ting, S. Yun-dong, H. Shi, Analysis ofIsotropic, Sandwich and Laminated Plates by

aMeshlessMethod and VariousShear Deformation Theories, Composite Structures, vol. 91, pp. 31–37,2009




[3]. J.N.reddy; An Introduction to Finite Element Method (1993) [4]. Meshless analysis of laminated composite and sandwich plates SUBJECTED TO VARIOUS type of loads ,

International Journal for Computational Methods in Engineering Science and Mechanics, 15:158–171, 2014. [5]. Venugopal M.M., S K Maharana, K S Badarinarayan; Finite Element Evaluation of Composite Sandwich Panel

Under Static Four Point Bending Load; JEST-M, Vol. 2, Issue 1,2013

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Finite element flexural analysis of sandwich plates under various types of loading conditions