MIT International Journal of Mechanical Engineering, Vol ...mitpublications.org/yellow_images/1378375235_logo_Static%20... · Timoshenko beam theory is used to capture the shear deformation

MIT International Journal of Mechanical Engineering, Vol. 3, No. 2, August 2013, pp. 80–85ISSN No. 2230–7680 © MIT Publications

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Static Response Analysis of a FGM Timoshenko’s Beam Subjected to Uniformly

Distributed Loading ConditionAkash Roy

Department of Mechanical EngineeringNational Institute of Technology Durgapur

Durgapur, Mahatma Gandhi AvenueWest Bengal (INDIA)

Email: [email protected]

Kallol KhanDepartment of Mechanical Engineering

National Institute of Technology Durgapur Durgapur, Mahatma Gandhi Avenue

West Bengal (INDIA) Email: [email protected]

ABSTRACTFunctionally graded material beams (FGMs) possess a smooth variation of material properties due to continuous change in micro structural details. The variation of material properties is along the beam thickness and assumed to follow the power-law. A cantilever beam subjected to uniformly distributed load has been chosen here for the analysis. For numerical implementation Galerkin’s weighted residual method (approximation) is used here within the framework of Timoshenko or first order shear deformation theory (FSDT). Timoshenko beam theory is used to capture the shear deformation. The governing equations and boundary conditions are derived from virtual work principle. In this study, the effect of various material distributions on the mid plane deflections and stresses distribution along the thickness of FGM beam are examined and compared with isotropic beam. This study can be helpful for those beam type structures where the need for minimum stresses and displacements are required. Keywords: FGM, Timoshenko’s beam, Power-law, Mechanical load, Galerkin’s weighted residual method.

1. INTRODUCTIONAs technology progresses at an ever increasing rate, the need for advanced capability materials becomes a priority in the engineering of more complex and higher performance systems. Pure metals are of little use in engineering applications because of the demand of conflicting property requirement. For example, an application may require a material that is hard as well as ductile, there is no such material existing in nature. To solve this problem, the term alloying is discovered in material science. Bronze, alloy of copper and tin, was the first alloy that appears in human history. This need for new materials different from the parent materials can be seen in many fields. Composite material is also a class of advanced material. It is made up of one or more materials combined in solid states with distinct physical and chemical properties. Composite material offers an excellent combination of properties which are different from the individual parent materials and are also lighter in weight. Wood is a composite material from nature which consists of cellulose in a matrix of lignin. But Composite materials will fail under extreme working conditions through a process called delamination (separation of fibers from the matrix) used by M. Mahamood

et al. [1]. To solve this problem, researchers in Japan in the mid 1980s, invented Functionally Graded Materials (FGMs).

In materials science functionally graded material (FGM) may be characterized by the variation in composition and structure gradually over volume, resulting in corresponding changes in the properties of the material. The materials can be designed for specific function and applications. In laminated composite materials provide the design flexibility to achieve desirable stiffness and strength through the choice of lamination scheme, the anisotropic constitution of laminated composite structures often result in stress concentrations near material and geometric discontinuities that can lead to damage in the form of delamination, matrix cracking, and adhesive bond separation. FGMs alleviate these problems because they consist of a continuous variation of material properties from one surface to the other. Also the smooth transition through the various material properties reduces both thermal and residual stresses. In most cases the material progresses from a metal on one surface to a ceramic or mostly ceramic on the opposite surface, with a smooth transition throughout the centre of the material. Also the material properties can change in any orientation across a material, but the majority of applications


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to deal with a material in which the properties change through the thickness of the material. Static and dynamic analyses of FGM structures have attracted increasing research effort in the last decade because of the wide application areas of FGMs. In many other static analyses the variation of poisson’s ratio is kept constant but in this study it is also varying along the beam thickness. For instance. Simsek [2], gave static analysis of a functionally graded simply-supported beam subjected to a uniformly distributed load with the help of Ritz method within the framework of Timoshenko and the higher order shear deformation beam theories. He examined the continuously variation of the material properties in the thickness direction according to the power-law form. In this paper it is expressed that the axial deflections and the rotation of the cross-section of the beam are expressed in trigonometric functions. B.V. Sankar [3], proposed an elasticity solution of a FG beam subjected to transverse loads. He assumed the Young’s modulus of the beam is varying exponentially in the thickness direction, and the Poisson ratio is kept constant. By using simple Euler-Bernoulli beam theory he also developed that stresses and displacements depend on a single non-dimensional parameter for a given variation of Young’s modulus. He found that the FG beam theory is valid for long, slender beams with slowly varying transverse loading. H.J. Ding et al. [4], investigate the elasticity solutions for plane anisotropic FG beams. They gave the concept of Airy stress function to solve the partial differential equation by considering the effect of body force. Zheng Zhong et al. [5], explained to get the analytical solution of a FG cantilever beam subjected to different loads. By assuming that all the elastic modulus of the beam material have the same variations along the beam thickness, they presented a two dimensional solution for a beam by using Airy stress function. In this paper the solution is obtained by means of the semi-inverse method. De-jin Huang et al. [6], studied the bending problem of a functionally graded anisotropic cantilever beam subjected to a linearly distributed load is investigated. The analysis is based on the exact elasticity equations for the plane stress problem. The stress function is introduced and assumed in the form of a polynomial of the longitudinal coordinate. Shyang-Ho Chi et al. (part-1) [7], investigate the mechanical behavior of an elastic, rectangular and simply supported FGM plate of medium thickness under transverse loading. They assumed Poisson’s ratio of the plate is constant, but Young’s modulus varies in the thickness direction according to the power law. According to their investigation the formulations of the solutions of FGM plates and homogeneous plates are similar, except the bending stiffness of plates. S.S. Alieldin et al. [8], investigate a first order shear deformation finite element model for elasto static analysis. They carried out the mechanical behavior of laminated composite and FG plates. They used three approaches to transform the laminated composite plates, with stepped material properties, to an equivalent FG plate with a continuous property function across the plate thickness. Trung-Kien Nguyen et al. [9], proposed the first order shear deformation plate models for modeling structures made of FGM. In this paper the transverse shear

stresses are derived from the expression of membrane stresses and equilibrium equations. Identification of transverse shear factors is investigated by energy equivalence. Chakraborty et al. [10], gave a new beam finite element method based on first order shear deformation theory to study the thermo elastic behavior of FGM beam structures. Here they used power law distribution of mechanical properties along the thickness and expressed mid plane deflections and rotation in terms of interpolating polynomials. C.M.Wang et al. [11], is used to find out differential governing equations for a beam by Timoshenko beam theory. [12], (Seshu) is used to calculate the residuals and weight functions by Galerkin’s method. [13], (Reddy) is used to find out the governing equations by virtual work principle.

As it is known, Timoshenko beam theory (TBT) or the first order shear deformation theory in which straight lines are perpendicular to the mid-plane before bending remain straight, but no longer remain perpendicular to the mid-plane after bending. In TBT, for isotropic material the distribution of the transverse shear stress with respect to the thickness coordinate is assumed constant. Thus, a shear correction factor (ks) is required to compensate for the error because of this assumption in TBT. However, studies show that TBT gives satisfactory results and it is very effective to investigate behavior of beams and plates.

2. THEORY AND FORMULATION

The term functionally graded materials (FGMs) refers to solid objects or parts that usually consist of multiple materials or embedded components, that is, they are materially heterogeneous. A FGM consists of a material whose properties change from one surface to another according to a smooth continuous function based on the position throughout the thickness of the material. Most often, this material consists of metallic and ceramic constituents. One surface is generally a pure metal while the opposite surface is usually pure ceramic or a majority ceramic [2]. The metal portion of the material acts in the role of a structural support while the ceramic provides thermal protection when subjected to harsh temperatures. Functionally graded materials (FGMs) are new advanced multifunctional composites where the volume fractions of the reinforcements phase(s) vary smoothly. This is achieved by using reinforcements with different properties, sizes, and shapes, as well as by interchanging the functions of the reinforcement and matrix phases in a continuous manner. The result is a microstructure bearing continuous changes in thermal and mechanical properties at the macroscopic or continuum scale. In other words, FGM is usually a combination of two materials or phases that show a gradual transition of properties from one side of sample to the other. This gradual transition allows the creation of superior and multiple properties without any mechanically weak interface. Moreover, the gradual change of properties can be tailored to different applications and service environments. This new concept of materials engineering hinges on materials science and mechanics due


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to the integration of the material and structural considerations into the final design of structural components.

To study FGMs, a model must be created that describes the function of composition throughout the material. In Fig. 1, the volume fraction, Vc(z) , describes the volume of ceramic at any point z throughout the thickness h according to a parameter k (also called power law exponent) which controls the shape of the function .Vc(z) is given by

( ) 1

2

k

cz

V zh

Ê ˆ= +Á ˜Ë ¯ (1)

It follows that the volume fraction of metal, Vm(z) , in the FGM is 1-Vc (z). One of the most common methods to determine the effective properties of FGMs is the rule of mixtures, where the material properties through the thickness vary as a function of the volume fraction and are given by

( ) ( ) ( )m c c cP z P P V z P= - + (2)

Figure 1: Material properties throughout the FGM layer

A FGM cantilever beam of length L, width b and thickness h is shown in Figure 2. The beam is subjected to uniformly distributed load.

Figure 2: Cantilever beam subjected to U.D.L.

In this study, it is assumed that the FG beam is made of ceramic and metal, and the effective material properties of the FG beam, i.e., Young’s modulus E , Poisson’s ratio ν and shear modulus G vary continuously in the thickness direction (z axis direction) according to power-law form introduced by [2]

( ) 1( )

2

k

m c cz

E z E E Eh

Ê ˆ= - + +Á ˜Ë ¯ (1)

( ) 1( )

2

k

m c cz

zh

n n n nÊ ˆ= - + +Á ˜Ë ¯ (2)

( ) 1( )

2

k

m c cz

G z G E Gh

Ê ˆ= - + +Á ˜Ë ¯ (3)

where k is the non-negative variable parameter (power-law exponent) which dictates the material variation profile through the thickness of the beam, m and c stand for metal and ceramic constituents, respectively.

It is clear from Eqs. (1-3) that E = Ec, ν=νc, G = Gc at z = -h / 2 (4)E = Em, ν=νm, G =Gm at z = h / 2 (5)Based on the first order shear deformation theory, The axial

and transverse displacement using FSDT for a beam is given by u(x,y,z) = u0(x) + z ϕ(x) (6)w(x,y,z)= w0(x) (7)

where (x,y,z) are the coordinates of a point in the beam, (u,w) are the components of the displacement vector in the coordinate directions, ϕ(x) is the angle of rotation of the normal to the mid-surface of the beam, and u0(x), w0(x) are the displacement of the mid-surface in the axial and transverse directions.

Figure 3: Deformation of Timoshenko beam

from equations (6) and (7), the linear strains displacements are:

0 ( ) ( )x

du x d xz

dx dx

fe = + (8)

0 ( )( )xz

dw xx

dxg f= + (9)

According to the constitutive relation for FGM material:

( ) 0

0 ( )x x

xz xz

E z

G z

s et g

Ï ¸ Ï ¸È ˘=Ì ˝ Ì ˝Í ˙

Î ˚Ó ˛ Ó ˛ (10)

The virtual strain energy or internal virtual work [13] is given by:

( )0

L

x x xz xz

A

U dAdxd s de t dg= +Ú Ú (11)

The virtual potential energy of the beam is given by:

( ) ( )0

0

L

V q x w x dxd d= -Ú (12)


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The principle of virtual displacements states that if the beam is

in equilibrium it must satisfy: 0U Vd d+ = (13)

The boundary conditions of the cantilever beam are given by:u0(0) = 0, ϕ(0) = 0, w0(0) = 0 (Essential b. c) and Fx = 0, Mx =0, Vx = 0 at x = L (Natural b. c). where Fx is net axial force, Mx is bending moment and Vx is shear force respectively and are given by:

, ,X x X x X s xz

A A A

F dA M z dA V k dAs s t= = =Ú Ú ÚAfter solving equation (13):

( ) ( )2 2

01 22

0d u x d x

c cdxdx

f- - = (14)

( ) ( ) ( ) ( )2 20 0

2 3 42 20s

d u x d x dw xc c k c x

dxdx dx

ff

Ï ¸- - + ¥ + =Ì ˝

Ô ÔÓ ˛(15)

( ) ( ) ( )2

02

4

0s

d x d w x q x

dx k b cdx

f+ + =

¥ (16)

where ( ) ( )2 2

1 2

2 2

, ,

h h

h h

c b E z dz c b zE z dz- -

= =Ú Ú

( )2 2

23 4

2 2

, ( )

h h

h h

c b z E z dz c b G z dz- -

= =Ú Ú

Equations (14)–(16) are known as are known as governing differential equations of the beam.

These three governing equations are obtained in terms of assumed trial or guess solutions. Let the assumed solutions are [10]:

( ) 20 1 2 3u x a a x a x= + + (17)

( ) 2 30 4 5 6 7w x a a x a x a x= + + + (18)

( ) 28 9 10x a a x a xf = + + (19)

The order of interpolation of w0(x) is one order higher than slope ϕ(x). This is one of the requirements for the beam to be free of shear locking. The exact solutions for the displacements have a total ten constants and only six boundary conditions. But from essential boundary conditions a1, a4 and a8 = 0, so remaining constants are seven and boundary conditions are three. Hence, there are only three independent constants. The additional four dependent constants can be expressed in terms of three independent constants by applying Galerkin’s weighted residual method [12]. In general the guess solutions in equations (17-19) will satisfy neither the differential equation within the domain nor the boundary conditions. By substituting the assumed function in the differential equation and the boundary conditions of the problem, find the error

in satisfying these (also known as “domain residual” and “boundary residual”). The constants (in equations (17-19) are calculated by making these residuals as low as possible.

We see that the approximation functions are dependent not only on the beam length but also on its material properties and cross-sectional properties.

3. RESULT AND DISCUSSIONIn this section, the results are discussed for the FGM cantilever beam (L=100 mm, h=20 mm, b=1mm) subjected to a uniformly distributed load (q(x) =50 N/mm). The beam is composed of steel (E=210×103 N/mm2, G=80×103 N/mm2) and alumina (Al 2O3) (E=390×103 N/mm2, G=137×103 N/mm2) and its properties changes through the thickness of the beam according to the power-law [2]. The shear correction factor is taken as ks=5/6 for TBT. For the calculation of parameters c1 to c4 coding is done in FORTRAN 90. In this study, the compressive and tensile stresses are shown by positive and negative signs respectively.

Figure 4 show the variation of modulus of elasticity along the beam thickness for different values of k, as we know that modulus of elasticity for full metal (z/h =0.5) is 210x103 and for full ceramic (z/h = -0.5) is 210x103. Similarly the variation for shear modulus and poisson’s ratio along the thickness of the beam can be plotted.

Figure 4: Variation of modulus of elasticity along the thickness of the beam

Figures 5 and 6 show the mid plane transverse and mid plane axial (stretching) deflections of the beam along the beam length. In transverse deflection we see that for full metal (k=0) the deflection is more than the FGM composition and as the value of k increases, the deflections of the beam decreases. This is due to the fact that an increase in power-law exponent yields a decrease in the bending rigidity of the beam. In axial deflection, for full metal (k=0) the deflection is zero. Because in isotropic beams, there is no coupling between the bending and the stretching.

It is also to be noted that as the value of power-law exponent increases, the composition of the FG beam approaches to the composition of the full ceramic beam.


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Figure 5: Mid plane transverse deflection along the beam length

Figure 6: Mid-plane axial deflection (stretching) along the beam length

Figures 7 and 8 shows the distribution of axial and shear stresses at the fixed end (origin) for different values of k. As seen from Fig. 7 (b), the axial stress distribution is linear only for full metal (k=0) and also the values of tensile and compressive stresses are equal for full metal (k=0). But for other values of k as shown in Fig.7 (a) the axial stress distribution is not linear and also the values of compressive stresses are greater than tensile stresses. Also from Fig.7 (b), the value of axial stress is zero at the mid-plane but it is clearly visible from Fig.7 (a) that the values of axial stresses are not zero at the mid-plane of the FG beam for the other values of k, it indicates that the neutral plane of the beam moves towards the lower side of the beam for FG beam. This is due to the variation of the modulus of elasticity through the thickness of the FG beam. It is seen from Fig. 8 (a) – (b) that the shear stress is constant for k=0 but for k=1 it is linear and for larger values of k shear stresses are neither linear nor constant. The shear stress distribution is greatly influenced by power-law exponent.

4. CONCLUSIONSStatic analysis of FG cantilever beam subjected to mechanical load shows that the material properties of the beam vary continuously in the thickness direction according to the power law form. Numerical results clearly show that the variation

Figure 7: (a) Axial stress distribution along the beam thickness for FGM (b) Axial stress distribution along the beam thickness for full metal

Figure 8: (a) Shear stress distribution along the beam thickness for FGM, (b) Shear stress distribution along the beam thickness for full metal


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of the modulus of elasticity plays a major role on the axial stress distribution and the displacements of the FG beam. Also for FGM beams as the value of power-law exponent increases, the bending rigidity decreases and the shear stress distributions are greatly influenced by the power law exponent. So, by concluding all of this it can be said that in the design of structures, by choosing a suitable power law exponent, the material properties of the FG beam can be tailored to meet the desired goals of minimizing stresses and the displacements in a beam-type structure.

REFERENCES

[1] Mahamood, M.R., Akinlabi, T.E., Shukla, M and Pityana, S., Functionally Graded Material: An Overview. Proceedings of the World Congress on Engineering, 2012 Vol III WCE 2012, July 4-6, 2012, London, U.K.

[2] Simsek, M., Static Analysis of a Functionally Graded Beam under a Uniformly Distributed Load by Ritz Method. International Journal of Engineering and Applied Sciences Vol. 1, 2009.

[3] Sankar, B.V., An elasticity solution for functionally graded beams. Composites Science and Technology, 61 (2001),pp. 689-696.

[4] Ding, H.J., Huang, D.J., Chen, W.Q., Elasticity Solutions for Plane Anisotropic Functionally Graded Beams. International Journal of Solids and Structures 44 (2007), pp. 176-196.

[5] Zhong, Z., Yu T., Analytical solution of a cantilever functionally graded beam. Composites Science and Technology, 67 (2007), pp. 481-488.

[6] Huang, De-jin, Ding Hao-jiang, Chenwei-qiu., Analytical solution for functionally graded anisotropic cantileverbeam subjected to linearly distributed load. Applied Mathematics and Mechanics (English Edition), 2007, 28(7): pp. 855–860.

[7] Chi .S-Ho., Chung .Y- Ling., Mechanical behavior of functionally graded material plates under transverse load- Part I: Analysis. International Journal of Solids and Structures 43 (2006) 3657-3674.

[8] Alieldin, S.S., Alshorbagy A.E., Shaat M., A first-order shear deformation finite element model for elastostatic analysis of laminated composite plates and the equivalent functionally graded plates. Ain Shams Engineering Journal (2011) 2, 53-62.

[9] Nguyen.T-Kien., Sab., K., Bonnet, G., First-order shear deformation plate models for functionally graded material. Composite Structures, 83 (2008), pp. 25-36.

[10] Chakraborty, A., Gopalakrishnan, S., and Reddy, J. N., 2003., A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences, 45(3), pp. 519-539.

[11] Wang, C.M., Reddy J.N., Lee K.H., Shear Deformable Beams And Plates Relationships With Classical Solutions. Elsevier UK.

[12] Seshu, P., Finite Element Analysis, PHI New Delhi.[13] Reddy, J.N., Mechanics of Laminated Composite Plates and

Shells, Theory and Analysis. CRC PRESS Washington.

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MIT International Journal of Mechanical Engineering, Vol ...mitpublications.org/yellow_images/1378375235_logo_Static%20... · Timoshenko beam theory is used to capture the shear deformation