Bending of laminated beams using Timoshenko model via nonlocal elasticity

Bending of laminated beams using Timoshenko model via nonlocal elasticity

Abderrahmane Besseghier

Centre Universitaire El Wancharissi Tissemsilt Institut des Sciences et Technologies, B.P 182,

38000 Tissemsilt, Algeria

besseghier@yahoo.fr

Abdelouahed Tounsi , Adda Bedia El Abbas Laboratory of Materials and Hydrology University of Sidi Bel Abbes, BP 89 Cité Ben M'hidi

,

22000 Sidi Bel Abbés, Algérie

Abstract— In this paper, the bending of laminated beams using first order shear deformation theory via nonlocal elasticity is presented. The analysis is presented for static flexure of symmetric and anti-symmetric cross-ply laminated beams subjected to sinusoidal load.

Governing equations and boundary conditions are obtained by using the principle of virtual work.

Analytical solutions of bending for simply supported laminated beams are presented using this theory to illustrate the effect of nonlocal theory, and the influence of aspect ratio is examined.

Keywords— First-order shear deformation theory; bending;

laminated beam; non local

I. INTRODUCTION

In various engineering applications, composite structures, such as beams and plates, are experiencing widespread use, as in the aerospace, mechanical, and civil industries.

Composite beams are lightweight structures that can be found in many diverse applications including aerospace, submarine, medical equipment, automotive and construction industries. Buildings, steel framed structures and bridges are examples of beam applications in civil engineering.

Due to the outstanding mechanical properties such as high strength / stiffness to weight ratios, laminated composite beams present a remarkable role in the design of various engineering-type structures and replace in much conventional isotropic beam structure.

The classical theory of the beam called Euler-Bernoulli beam theory (EBT) predicts accurate bending behaviour of thin beams, but it is inaccurate while predicting bending behaviour of thick beams where shear deformation is significant. The Timoshenko beam theory takes into account shear deformation and rotational inertia effects. A shear correction factor is needed to appropriately represent the strain energy of the shear strain. A lot of models have been proposed based on Timoshenko theory.

Piska Raghu and all (2016) presented the analysis of laminated composite plates using a nonlocal third-order shear

deformation theory of Reddy considering the surface stress effects. A trigonometric beam theory (TBT) is developed for the bending analysis of laminated composite and sandwich beams considering the effect of transverse shear deformation (Sayyad and all 2015).

Huu-Tai Thai (2012) presented the analysis for bending, buckling, and vibration of nanobeams using Eringen’s nonlocal elasticity theory and Timoshenko beam theory.

Reddy presented (2007) the Euler–Bernoulli, Timoshenko, Reddy, and Levinson beam theories, using the nonlocal differential constitutive relations of Eringen.

In this paper, the analysis of laminated beams is presented on first-order shear deformation theory and applied to the investigation of bending by nonlocal elasticity. The results are obtained for two and three layered cross ply laminated beams subjected to sinusoidal load with simply supported. Hence, the influences of the nonlocal parameter, aspect ratio are discussed..

II. NONLOCAL ELASTICITY MODEL

Nonlocal theory states that the stress at a reference point x in an elastic continuous medium is not only dependent deformation at this point but also the deformation of all other points x 'in that domain (Eringen 2002).

Nonlocal constitutive equations for a problem with three- dimensional (3D) can be given as follows

(1)

Where σij and Cij are the stress tensor of the nonlocal elasticity and the classical stress tensor, which is related to the linear strain tensor εkl, respectively. τ= e0a/l is nonlocal parameter, which is a material constant that depends on a and l the internal characteristics length (such as the carbon–carbon bond length, lattice parameter and granular size) and external characteristics length (wave length, graphène sheet length and crack length), respectively. e0 is Eringen’s nonlocal elasticity constant appropriate to each material

III. FORMULATION

Before you begin to format your paper, first write and save The Timoshenko beam theory, which is based on the displacement field

(2) (u0, w0) denote the displacements of a point on the planez=0, and φ(x) represents the rotation of a transverse normal about theyaxe

We have only two nonzero strains

(3) The stress strain relationship reads as (4)

)

(5) (6) The shear stress in the layers can be deduced from the

equilibrium equations of two dimensional elasticity.

(7) The transverse shear stresses intensity is zero at the top and bottom, z = ± h/2, and satisfies continuity at layer interface when these stresses are obtained.

The coefficients Qij^((k) ) are known the reduced stiffness coefficients of thekth layer (Reddy 2004; M.Adnan Elshafei

2013)

where (8)

(10)(9) The principle of virtual displacements for laminated beams reads asdU+dV=0, wheredUis the virtual strain energy and dVis the virtual work done by applied forces and elastic foundations.

We have

Integrating the equation (6) by parts and collecting the(11) coefficients of δu , δw, and δφ the governing equations and boundary conditions can expressed in terms of displacement .

The governing equations are of the form:

Where (12)

μ=e

02

a

² (13)

Aijare the extensional stiffness,Dij the bending stiffnesses, andBij the bending extensional coupling stiffness

which are defined in terms of the lamina stifnesses as

(14)

(15)

(16) Example:

A simply supported laminated beam subjected to single sine load on surfacez=h/2, acting in the downwardzdirection.

The load is expressed as Where

q

0is the magnitude of the load. (17)

The boundary conditions of simply supported beams are (18) The following expansions of the generalized displacementsw

andϕsatisfy the boundary conditions in Eq (18)

(19)

WhereWm,Umand

φ

^mare the unknown coefficients of the respective Fourier expansion andmis the positive integer. It is

assumed the integerm=1.

The following material properties for laminated beam are used

(20) (21) The following example of two-layered (0°/90°) and three - layered (0°/90°/0°) unsymmetric, symmetric cross-ply

laminated beams subjected to static flexure are considered.

IV. NUMERICAL RESULTS AND DISCUSSIONS The results obtained for displacements and stresses are presented in the following non-dimensional form.

(22)

Fig. 1.Variation of axial displacement through the thickness of (0°/90°) beam subjected to load at (x = 0, z) for aspect ratiol/h=4,l=10 with different μ.

Fig. 2.Variation of axial displacement through the thickness of (0°/90°/0°) beam subjected to load at (x = 0, z) for aspect ratiol/h=4,l=10 with different

μ.

Figs. 1 and 2 respectively shows the variation of axial displacement through the thickness of (0°/90°), and

(0°/90°/0°) beam subjected to load at (x= 0,z) for aspect ratio l/h=4,l=10 for different values of small scale parameter μ (i.e., μ= 0, 1, 3, 5 nm). The nonlocal parameter has a significant effect on the stresses.

The variation of bending stress through the thickness of (0°/90°), and (0°/90°/0°) beam subjected to load at (x = 0.5 l, z) for aspect ratiol/h=4,l=10 with different μ are illustrated in Fig.3 and Fig.4.

Figs.5 and 6 illustrate the variation of transverse shear stress through the thickness of (0°/90°), and (0°/90°/0°) beam subjected to load at (x= 0,z) for aspect ratiol/h=4,l=10 with different μ.

Figs.7 and 8 presents the effect of length-to-thickness ratio on nondimensionalized deflection of (0°/90°), and (0°/90°/0°) beam subjected to load at (x= 0.5 l,z) for aspect ratiol/h=4,l=10 for different μ.

Fig. 3.Variation of bending stress through the thickness of (0°/90°) beam subjected to load at (x = 0.5 l, z) for aspect ratio l/h=4.l=10 with different μ.

Fig. 4.Variation of bending stress through the thickness of (0°/90°/0°) beam subjected to load at (x = 0.5 l, z) for aspect ratiol/h=4,l=10 with different μ.

Fig. 5.Variation of transverse shear stress through the thickness of (0°/90°) beam subjected to load at (x = 0,z) for aspect ratiol/h=4,l=10 with different

μ.

Fig. 6.Variation of axial displacement through the thickness of (0°/90°/0°) beam subjected to load at (x = 0, z) for aspect ratio l/h=4,l=10 with different

μ.

Fig. 7.Effect of length-to-thickness ratio on nondimensionalized deflection of (0°/90°) beam subjected to load at (x = 0.5 l, z) for aspect ratiol/h=4,l=10 for

different μ.

Fig. 8.Effect of length-to-thickness ratio on nondimensionalized deflection of (0°/90°/0°) beam subjected to load at (x = 0.5 l, z) for aspect ratiol/h=4,l=10

for different μ.

The effect of shear deformation is more significant for laminated beams with a length-to- thickness ratios smaller than 10, the allure for μ=5, is elevated. Do not use hard tabs, and limit use of hard returns to only one return at the end of a paragraph. Do not add any kind of pagination anywhere in the paper. Do not number text heads-the template will do that for you.

V. CONCLUSION

In this paper, a nonlocal beam theory is developed for the bending of two-layer antisymmetric, and three-layer symmetric cross-ply laminated composite beams.

Based on the nonlocal differential constitutive relations of Eringen, the equations of motion are derived using Hamilton’s principle. The nonlocal parameter has a significant effect on the stresses. It is observed that inclusion of the nonlocal effect increases the deflections especially at high values of nonlocal parameter.

References

[1] Eringen.A.C,Nonlocal Continuum Field Theories, Springer-Verlag, New York ,2002.

[2] Huu-Tai Thai,A nonlocal beam theory for bending, buckling, and vibration of nanobeams. International Journal of Engineering Science 52, 56–64,2012.

[3] M. Adnan Elshafei,FE Modeling and Analysis of Isotropic and Orthotropic Beams Using First Order Shear Deformation Theory.

Materials Sciences and Applications, 4, 77-102,2013.

[4] Piska Raghu and all,Nonlocal third-order shear deformation theory for analysis of laminated plates considering surface stress effects.

Composite Structures 139 13–29,2016.

[5] Reddy.J.N,Mechanics of Laminated Composite Plates and Shells-Theory and Analysis,” 2nd Edition, CRC Press, USA,2004.

[6] Reddy.J.N, Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science 45, 288–307,2007.

[7] Sayyad A.S and Y. M. Ghugal,Effect of transverse shear and transverse normal strain on bending analysis of cross-ply laminated beams. Int. J.

of Appl. Math and Mech. 7 (12): 85-118, 2011.

[8] Sayyad A.S and all,Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory. Curved and Layer. Struct, 2:279–289, 2015.