EFFECT OF UNIFORMLY DISTRIBUTED THERMAL LOAD ON BENDING RESPONSE OF ISOTROPIC PLATES USING TWO VARIABLE PLATE THEORY

Int. J. of Appl. Math and Mech. 10 (5): 1-21, 2014.

EFFECT OF UNIFORMLY DISTRIBUTED THERMAL LOAD ON BENDING RESPONSE OF ISOTROPIC PLATES USING TWO

VARIABLE PLATE THEORY

B. A. Mhaske1, K. B. Ladhane2, V. R. Rathi2, and A. S. Sayyad1

1Department of Civil Engineering, SRES’s College of Engineering Kopargaon-423603, University of Pune, MS, India

2Department of Civil Engineering, Pravara Rural Engineering College, Loni-413736, University of Pune, MS, India

Email: [email protected]

Received 16 August 2013; accepted 4 July 2014

ABSTRACT This paper presents two variable plate theory for thermal analysis of isotropic plates which takes into account effect of transverse shear deformations. Unlike any other theory, the theory presented gives rise to only two governing equations, which are completely uncoupled for static analysis. Number of unknown functions involved is only two, as against three in case of simple shear deformation theories of Mindlin and Reissner. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying the zero traction boundary conditions on the top and bottom surfaces of the plate. Comparison studies are performed to verify the validity of the present results. Finally, the effect of aspect ratios on the deflection and stress of isotropic plates subjected to uniformly distributed thermal load are investigated and discussed. Keywords: Two Variable Plate Theory, Shear Deformation, Isotropic Plate Theory, Transverse Shear Stress, Thermal Loading 1 INTRODUCTION The isotropic and anisotropic structural elements are used in the construction of aeronautical and aerospace vehicles, construction of rockets, nuclear reactors and in a wide variety of civil and mechanical engineering structures. The temperature variations often represent a significant factor, and sometimes the predominant causes of failure of structures. The deflection and stress analysis of plates subjected to thermal load has been the subject of research interest in recent years. The simplest theory is the classical plate theory (CPT) which is based on Kirchhoff hypothesis that, straight lines normal to the mid-plane before deformation remain straight and normal to the mid-plane after deformation and do not undergo thickness stretching. This

B. A. Mhaske et al.


2

implies that the transverse shear strains vanish. Jones (1975) and Wu and Tauchert (1980) used classical plate theory for the analysis of plates subjected to thermal load. First order shear deformation theory (FSDT) is considered as an improvement over the classical plate theory. FSDT is based on hypothesis that, straight lines normal to the mid-plane before deformation remain straight but not necessarily normal to the mid-plane after deformation. This results the inclusion of effect of shear deformation. Mindlin (1951) has provided displacement based first order shear deformation theory whereas Reissner (1944, 1945) was the first to provide a consistent stress based first order shear theory. The first-order shear deformation theory needs a shear correction factor. Stress-free boundary conditions are not satisfied in first-order shear deformation theory. A shear correction factor of 5/6 is adopted in computed results. Thermal stresses in laminated plates using classical plate theory and first-order shear deformation theory subjected to single sinusoidal thermal load and thermal stresses in a laminated plate subjected to a uniformly distributed thermal load using classical plate theory are given by Reddy (1997). The limitations of classical plate theory and first order shear deformation theories forced the development of higher order shear deformation theories to avoid the use of shear correction factors, to include correct cross sectional warping and to get the realistic variation of the transverse shear strain and stresses through the thickness of plate. Chen and Chen (1987) have studied thermal buckling of laminated composite plates subjected to a temperature change. The global higher-order theory has been developed by Matsunaga (2004, 2009) for the analysis of plates subjected to a single sinusoidal thermal loading. A semi-analytical model for composite plates subjected to a single sinusoidal thermal load has been developed by Kant et al. (2008). The global–local higher order theory for the thermal analysis of plate is studied by Zhen and Wanji (2005) and Wu et al. (2007). Using unified shear deformation plate theory Zenkour (2004) presented analytical solution for bending of cross-ply laminated plates under thermo-mechanical single sinusoidal loading. Fares and Zenkour (1999) developed a mixed variational formula for the thermal bending and thermo-mechanical bending under combined sinusoidal thermo-mechanical load. Ghugal and Kulkarni (2011, 2013a, 2013b) have used trigonometric shear deformation theory for the thermal analysis of isotropic, orthotropic and laminated plates. Benli and Sayman (2011) have studied the effects of temperature and thermal stresses on impact damage in laminated composites. Shinde (2013) and Shinde et al. (2013) extended the thermal analysis of isotropic plate by hyperbolic shear deformation theory under sinosidal and uniformly distributed thermal load. A two variable refined plate theory (RPT) was first developed by Shimpi (2002). He applied the theory to flexure of shear-deformable isotropic plates and was extended by Shimpi and Patel (2006a, 2006b) to free vibration analysis and later for orthotropic plates. Recently, this theory was successfully extended to buckling analysis of isotropic, orthotropic and for laminated composite plates by Kim et al. (2009a, 2009b). Mechab et al. (2010) have developed this theory for bending analysis of FGM plates. The analytical solution for free vibration and bending analysis of orthotropic Levy-type plates developed by Thai and Kim (2012). The application of this theory is extended to the free vibration of FGM plates by Benachour et al. (2011) and for nanoplates by Malekzadeh and Shojaee (2013). The novel feature of the theory is that it does not require shear correction factor, satisfying the shear stress free boundary conditions at top and bottom of the plate and has strong similarities with

Thermal Analysis of Isotropic plates


3

the CPT in some aspects such as governing equation, boundary condition and moment expressions. This paper extended the two variable plate theory developed by Shimpi (2002) and Shimpi and Patel (2006a, 2006b) for the bending analysis of thick isotropic plate under uniformly distributed thermal load. 2 THEORETICAL FORMULATION The theoretical formulation of a shear deformation theory for uniform thick plate based on certain kinematical and a physical assumption is presented. The variationally correct forms of differential equations and boundary conditions, based on assumed displacement field would be obtained using principal of virtual work. 2.1 Plate Under Consideration The plate (length ,a width ,b and thicknessh) under consideration occupies (in 0 x y z− − − right-handed Cartesian coordinate system) a region 0 ; 0 ; / 2 / 2x a y b h z h≤ ≤ ≤ ≤ − ≤ ≤ (1) 2.2 Assumptions Made In Theoretical Formulation 1. The displacements (u in x - direction, v in y -direction, w in z -direction,) are small in

comparison with the plate thickness h and, therefore, strains involved are infinitesimal.

2. The transverse displacement w has two components; bending component bw shear

component .sw both the components are functions of coordinates x and y only.

( , , ) ( , ) ( , )b sw x y z w x y w x y= + (2)

3. In general, transverse normal stress zσ is negligible in compression with inplane stresses

xσ and yσ .

4. The displacement u in x -direction consists of bending component bu and shear

component su similarly; the displacement v in y -direction consists of bending component

bv and shear component sv .

;b su u u= + b sv v v= + (3)

B. A. Mhaske et al.


4

5. The bending component bu of displacement u and bv of displacement v are assumed to be

similar, respectively, to the displacements u and v given by classical plate theory (CPT).

Therefore, the expressions for bu and bv can be given as

bb

wu z

x

∂= − ∂ (4)

bb

wv z

y

∂= − ∂ (5)

It may be noted that the displacement componentsbu , bv and bw together do not contribute

toward shear strainsyzγ and xzγ and therefore to shear stressesxzτ and .yzτ

6. The shear component su of displacement u and shear component sv of displacement v are

such that they give rise, in conjunction with ,sw to the parabolic variation of shear strains

xzγ and yzγ hence shear stressesxzτ and yzτ through the thickness of the plate h , in such a

way that shear stresses are zero at top and bottom faces of plate and their contribution

toward strains ,x yε ε and xyγ is such that in the moments ,x yM M and xyM there is no

contribution from the components su and .sv

31 5

4 3s

s

dwz zu h

h h dx

= − (6)

3

1 5

4 3s

s

dwz zv h

h h dy

= − (7)

7. The plate is made up of homogeneous, linearly elastic isotropic material. The plate material

obeys generalized Hooke's law. 2.3 The Displacement Field Using expression (2) and (3) – (7) one can write expression for displacement field u, v, and w as follows:

31 5

( , , )4 3

b sdw dwz zu x y z z h

dx h h dx

= − + − (8)



5

31 5

( , , )4 3

b sdw dwz zv x y z z h

dy h h dy

= − + − (9)

( ), , ( , ) ( , )b sw x y z w x y w x y= + (10)

2.4 Strain-Displacement Relationships Normal and shear strains are obtained within the framework of linear theory of elasticity using the displacement field given by Eqns. (8) - (10). These relationships are given as follows: Normal Strains

32 2

2 2

1 5

4 3b s

x

d w d wz zz h

dx h h dxε = − + −

32 2

2 2

1 5

4 3b s

y

d w d wz zz h

dy h h dyε =− + − (11)

0zε = Shear Strains:

32 21 52 2

4 3b s

xy

d w d wz zz h

dxdy h h dxdyγ = − + −

2

55

4s

yz

dwz

h dyγ = −

(12)

2

55

4s

zx

dwz

h dxγ = −

2.5 Constitutive Relationship For a plate of constant thickness, composed of isotropic material, the following stress-strain relationships (constitutive relations) are used to obtain normal and transverse shear stresses using strain expressions from (11) and (12):

B. A. Mhaske et al.


6

11 12

12 22

66

44

55

0 0 0

0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

x x x

y y y

z z

xy xy

yz yz

xz xz

TQ Q

TQ Q

Q

Q

Q

σ ε ασ ε ασ ετ γτ γτ γ

− − =

(13)

where ( , , , , , )x y z xy yz xzσ σ σ τ τ τ and( , , , , , )x y z xy yz xzε ε ε γ γ γ are the stress and strain components,

respectively. Using the material properties stiffness coefficients, can be expressed as

11 22 122 2

44 55 66

;1 1

2(1 )

E EQ Q Q

EQ Q Q G

µµ µ

µ

= = =− −= = = = +

Hear, E and G are the elastic constants of the plate material, µ is poisons ratio, T is thermal

load which consists of linear temperature distribution through the thickness, 1( , )T z T x y=

2.6 Moments and Shear Forces The moments and shear forces are defined as

/ 2

/ 2

bx x

z hby yz hb

xyxy

M z

M z dz

zM

σστ

==−

= ∫ (14)

( )( )( )( )( )

/2

/2

sx x

sy y

z hsxy xyz h

sxzxz

syzyz

M f z

M f z

M f z dz

g zS

g zS

σστττ

==−

=

∫ (15)

Using expressions of stresses and strains above the moments and shear forces are calculated as

/2

/2

2 2 2 2

11 11 11 1 12 12 12 12 2 2 2( , ) ( , )

z hbx xz h

b S b Sx y

M z dz

w w w wD B D T x y D B D T x y

x x y y

σα α

==−=

∂ ∂ ∂ ∂= − + − − + −∂ ∂ ∂ ∂∫

(16)



7

/2

/2

2 2 2 2

12 12 12 1 22 22 22 12 2 2 2( , ) ( , )

z hby yz h

b S b Sx y

M z dz

w w w wD B D T x y D B D T x y

x x y y

σα α

==−=

∂ ∂ ∂ ∂= − + − − + −∂ ∂ ∂ ∂∫

(17)

2 2

/2

66 66/22 2

z hb b Sxy xyz h

w wM zdz D B

x y x yτ=

=−∂ ∂= = − +∂ ∂ ∂ ∂∫ (18)

/2

/2

2 2 2 2

11 11 11 1 12 12 12 12 2 2 2

( )

( , ) ( , )

z hsx xz h

b S b Sx y

M f z dz

w w w wB C B T x y B C B T x y

x x y y

σα α

==−=

∂ ∂ ∂ ∂= − + − − + −∂ ∂ ∂ ∂∫

(19)

/2

/2

2 2 2 2

12 12 12 1 22 22 22 12 2 2 2

( )

( , ) ( , )

z hsy yz h

b S b Sx y

M f z dz

w w w wB C B T x y B C B T x y

x x y y

σα α

==−=

∂ ∂ ∂ ∂= − + − − + −∂ ∂ ∂ ∂∫

(20)

2 2

/2

66 66/2( ) 2 2

z hs b Sxy xyz h

w wM f z dz B C

x y x yτ=

=−∂ ∂= = − +∂ ∂ ∂ ∂∫ (21)

/2

55/2( )

z hs Sxz xzz h

wS g z dz A

xτ=

=−∂= = ∂∫ (22)

/2

44/2( )

z hs Syz yzz h

wS g z dz A

yτ=

=−∂= = ∂∫ (23)

where,

31 5

( )4 3

z zf z h

h h

= −

2

5( ) [1 '( )] 5

4

zg z f z

h

= − = − 2.7 Derivation of Governing Equations and Boundary Conditions Using Equations (16) through (23) and principle of virtual work, variationally consistent differential equations and boundary conditions for the plate under consideration are obtained. The principle of virtual work when applied to the plate can be written as

B. A. Mhaske et al.


8

( )

/2

/2 0 0

0 0

, 0

h b a

x x y y z z yz yz xz xz xy xy

h

b a

dxdydz

q x y wdxdy

σ δε σ δε σ δε τ δγ τ δγ τ δγδ

− + + + + +

− =∫ ∫ ∫

∫ ∫ (24)

where, symbol δ denotes the variation operator. Employing Green’s theorem in Equation (24) successively, we obtain the coupled Euler-Lagrange governing equations of the plate and the associated boundary conditions of the plate in terms of stress resultants.

2 22

2 22

b bbxy yx

b

M MMw q

x x y yδ ∂ ∂∂= + + +∂ ∂ ∂ ∂ (25)

( ) ( )4 4 4 4 4

11 11 12 66 12 66 224 4 2 2 2 2 4

4 2 22 21 1

22 11 12 12 22 55 444 2 2 2 2

2( 2 ) 2( 2 )

( , ) ( , )

b S b s b

S s sx y x y

w w w w wq B C B B C C B

x x x y x y y

w w wT x y T x yC B B B B A A

y x y x yα α α α

∂ ∂ ∂ ∂ ∂= − + + − + +∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂∂ ∂− + + + + − −∂ ∂ ∂ ∂ ∂

(26)

2 2 22 2

2 2 2 22

s s ss sxy y yzx xz

s

M M SM Sw q

x x y y x yδ ∂ ∂ ∂∂ ∂= + + + + +∂ ∂ ∂ ∂ ∂ ∂ (27)

( ) ( )4 4 4 4 4

11 11 12 66 12 66 224 4 2 2 2 2 4

4 2 21 1

22 11 12 12 224 2 2

2( 2 ) 2( 2 )

( , ) ( , )

b S b s b

Sx y x y

w w w w wq D B D D B B D

x x x y x y y

w T x y T x yB D D D D

y x yα α α α

∂ ∂ ∂ ∂ ∂= − + + − + +∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂− + + + +∂ ∂ ∂

(28)

The constants appear in the governing equations are the stiffness constants given as follows.

3/2 2

/2 12

z h

ij ij ijz h

hD Q z dz Q

==−= =∫

/2

/2( ) 0

z h

ij ij z hB Q f z z dz

==−= =∫

[ ] 3/2 2

/2( )

1008

z h

ij ij ijz h

hC Q f z dz Q

==−= =∫

[ ] 2/2

/2

5( )

6

z h

ij ij ijz h

hA Q g z dz Q

==−= =∫

The Boundary Conditions of plate are as follows: 1. At corners (x=0, y=0), (x=0, y=b), (x=a, y=0) and (x=a, y=b) the following conditions hold:



9

The conditions involving wb (i.e., bending component of transverse displacement) and ws (i.e., shear component of transverse displacement)

0b sxy xyM M= = (29)

2. On edges x=0 and x=a, the following conditions hold: The conditions involving wb (i.e., bending component of transverse displacement) and ws (i.e., shear component of transverse displacement)

0b sb s x xw w M M= = = = (30)

3. On edges y=0 and y=b, the following conditions hold: The conditions involving wb (i.e., bending component of transverse displacement) and ws (i.e., shear component of transverse displacement)

0b sb s y yw w M M= = = = (31)

Thus, the variationally consistent governing differential equations and boundary conditions are obtained. 2.8 Solution Scheme for Illustrative Examples The governing equations for thermal analysis of isotropic plate can be obtained by setting the applied transverse load equal to zero in Equations (26 and 28). A solution to resulting governing equations, when expressed in terms of displacement variables (wb, ws, T1), which satisfies the associated boundary conditions (time dependent), is of the following form:

1,3 1,3

sin sinb bmnm n

m x n yw w

a b

π π∞ ∞

= = = ∑ ∑ (32)

1,3 1,3

sin sins smnm n

m x n yw w

a b

π π∞ ∞

= = = ∑ ∑ (33)

1 11,3 1,3

sin sinmnm n

m x n yT T

a b

π π∞ ∞

= = = ∑ ∑ (34)

where, bmnw , smnw , mnT are constants associated with bw , sw and 1T respectively. Using

expressions in the governing Equations (32 - 34), one obtains two completely uncoupled equations which can be written in matrix form, as follows:

B. A. Mhaske et al.


10

11 12 1 1

21 22 2 1

mn mn mn mn

mn mn mn mn

b

s

K K w LT

K K w L T

= (35)

where, Kij are the stiffness given as follows

4 2 2 4

11 11 12 66 222( 2 )m m n n

K D D D Da a b b

π π π π = + + +

4 2 2 4

12 21 11 12 66 222( 2 )m m n n

K K B B B Ba a b b

π π π π = = + + +

4 2 2 4 2 2

22 11 12 66 22 55 442( 2 )m m n n m n

K C C C C A Aa a b b a b

π π π π π π = + + + − +

( ) ( )2 2

1 11 12 12 22x y x y

m nL D D D D

a b

π πα α α α = + + +

( ) ( )2 2

2 11 12 12 22x y x y

m nL B B B B

a b

π πα α α α = + + +

from Equation (41) the unknowns mnbw ,

mnsw can be readily determined using Cramer’s rule.

1

mnb

Dw

D= and 2

mns

Dw

D= (36)

where,

11 12

11 22 21 1221 22

K KD K K K K

K K= = −

1 12

1 1 22 2 122 22

L KD L K L K

L K= = −

11 1

2 11 2 21 121 2

K LD K L K L

K L= = −



11

After substituting obtained values ofmnbw ,

mnsw into Equations (32)-(34), the inplane and

transverse displacements are obtained from displacement field [Equations (8)-(10)] are as follows

( )( , , ) b sdw dwu x y z z f z

dx dx= − +

( )cos cosmn mnb s

m x m m x mz w f z w

a a a a

π π π π = − + (37)

( )( , , ) b sdw dwv x y z z f z

dy dy= − +

( )cos cosmn mnb s

n x n n x nz w f z w

b b b b

π π π π = − + (38)

( ), , ( , ) ( , )b sw x y z w x y w x y= + (39)

After substituting obtained values of displacements [Equations (37)-(39)] in strain displacement relationship [Equations (11)-(12)] the strains are calculated and stresses can be readily determined from stress-strain relationship [Equations (13)] ( ) ( )11 1 12 1[ , ] [ , ]x x x y yQ zT x y Q zT x yσ ε α ε α= − + −

( ) ( )( ) ( )

2 2

11 1

2 2

12 1

sin sin ,

sin sin ,

mn mn

mn mn

b s x

b s y

m x m m x mQ z w f z w zT x y

a a a a

n x n n x nQ z w f z w zT x y

b b b b

π π π π απ π π π α

= − − + − −

(40)

( ) ( )21 1 22 1[ , ] [ , ]y x x y yQ zT x y Q zT x yσ ε α ε α= − + −

( ) ( )( ) ( )

2 2

21 1

2 2

22 1

sin sin ,

sin sin ,

mn mn

mn mn

b s x

b s y

m x m m x mQ z w f z w zT x y

a a a a

n x n n x nQ z w f z w zT x y

b b b b

π π π π απ π π π α

= − − + − −

(41)

66xy xyQτ γ=

B. A. Mhaske et al.


12

( )

2

66

2

66

2 cos cos

2 cos cos

mn

mn

xy b

s

m x n x mnQ zw

a b ab

m x n x mnQ f z w

a b ab

π π πτπ π π

= − +

(42)

The transverse shear stressesxzτ , yzτ obtained by using the constitutive relations

2

44

55 cos

4 mnyz s

z n y nQ w

h b b

π πτ = − (43)

2

55

55 cos

4 mnxz s

z n x nQ w

h a a

π πτ = − (44)

The transverse shear stressesxzτ , yzτ can be obtained either by using the constitutive relations

or by equilibrium equations for isotropic plate with respect to the thickness coordinate. Equilibrium equations of three-dimensional elasticity, ignoring body forces, can be used to obtain transverse shear stress. These equations of equilibrium are:

0xyx xz

x y z

τσ τ∂∂ ∂+ + =∂ ∂ ∂ (45)

0xy y zy

x y z

τ σ τ∂ ∂ ∂+ + =∂ ∂ ∂ (46)

0yzxz z

x y z

ττ σ∂∂ ∂+ + =∂ ∂ ∂ (47)

Integrating Equation (45 and (47), both w.r.t the thickness coordinate z and imposing the following boundary conditions at top and bottom surfaces of the plate.

/2[ ] 0xz z hτ =± =

/2[ ] 0yz z hτ =± =

Expressions for xzτ , yzτ can be obtained satisfying the requirements of zero shear stress

conditions on the top and bottom surfaces of the plate. 2.9 Illustrative Example



13

To assess the performance of the present theory in the prediction of thermal response under linear thermal load, four examples are considered herein. Example: Simply supported square isotropic plate subjected to uniformly distributed Thermal loading. A plate of length a, width b, and thickness h is considered. The plate has simply supported boundary conditions at x = (0, a) and y = (0, b). The plate subjected to uniformly distributed thermal load T(x, y, z) = z T1(x, y) for the following material properties.

210E MPa= , 0.3µ = , 80.76 .2(1 )

EG GPaµ= =+ 1x

y

αα =

where xα is the coefficient of thermal expansion in the direction of fiber and

yα is the

coefficient of thermal expansion in transverse direction. 3 NUMERICAL RESULTS AND DISCUSSIONS The results obtained for isotropic plates are presented in Table 1 & 2. Because of unavailability of exact solution for the some loading cases considered, the results from Kirchhoff (CPT), Mindlins (FSDT), Ghugal & Kulkarni (TSDT) and Shinde et al (HYDT) are generated for the purpose of comparison of results obtained by present theory. The results obtained for displacements and stresses are presented in the following non-dimensional form.

( ) 2 20 0

( , ), ; ;

x x

u v wu v w

T hb T hbα α= =

( ) ( )0

, ,, , ;

x

x y xyx y xy T E

σ σ τσ σ τ α=

( ) ( )0

,,

x

yz xzyz xz T E

τ ττ τ α=

B. A. Mhaske et al.


14

Table 1: Comparison of displacements and stresses for the isotropic plate subjected to uniformly distributed thermal load.

b/h Source Model u v w xσ yσ xyτ

5

Present TVPT 0.0429 0.0429 0.4789 0.0525 0.0525 0.2406

Shinde et al. HYDT 0.0429 0.0429 0.4789 0.0525 0.0525 0.2406

Ghugal and Kulkarni TSDT 0.0429 0.0429 0.4789 0.0525 0.0525 0.2406

Mindlin FSDT 0.0429 0.0429 0.4789 0.0525 0.0525 0.2406

Kirchhoff CPT 0.0429 0.0429 0.4789 0.0525 0.0525 0.2406

10

Present TVPT 0.0215 0.0215 0.9579 0.0262 0.0262 0.1203

Shinde et al. HYDT 0.0215 0.0215 0.9579 0.0262 0.0262 0.1203


Mindlin FSDT 0.0215 0.0215 0.9579 0.0262 0.0262 0.1203

Kirchhoff CPT 0.0215 0.0215 0.9579 0.0262 0.0262 0.1203

20

Present TVPT 0.0107 0.0107 1.9157 0.0131 0.0131 0.0602

Shinde et al. HYDT 0.0107 0.0107 1.9157 0.0131 0.0131 0.0602


Mindlin FSDT 0.0107 0.0107 1.9157 0.0131 0.0131 0.0602

Kirchhoff CPT 0.0107 0.0107 1.9157 0.0131 0.0131 0.0602

100

Present TVPT 0.0021 0.0021 9.5786 0.0026 0.0026 0.0120

Shinde et al. HYDT 0.0021 0.0021 9.5786 0.0026 0.0026 0.0120


Mindlin FSDT 0.0021 0.0021 9.5786 0.0026 0.0026 0.0120

Kirchhoff CPT 0.0021 0.0021 9.5786 0.0026 0.0026 0.0120 Table 2: Comparison of transverse shear stresses for the isotropic plate subjected to uniformly

distributed thermal load.

Source Model b/h=5 b/h=10 b/h=20 b/h=100

xzτ yzτ xzτ yzτ xzτ yzτ xzτ yzτ

PresentCR TVPT

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

PresentEE 0.3805 0.3805 0.0951 0.0951 0.0238 0.0238 0.0010 0.0010

Shinde et al. HYDT 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Ghugal and Kulkarni

TSDT 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Mindlin FSDT 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Kirchhoff CPT 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 CR: Constitutive Relation, EE: Equilibrium Equation



15

Figure 1: Through thickness variation of u of Isotropic Plate at (x=0, y=b/2, z) subjected to

UDL for aspect ratio 10.

B. A. Mhaske et al.


16

Figure 2: Through thickness variation of xσ of Isotropic Plate at (x=a/2, y=b/2, z) subjected to


Figure 3: Through thickness variation of xyτ of Isotropic Plate at (x=0, y=0, z) subjected to




17

Figure 4: Through thickness variation of τ xz of Isotropic Plate at (x=a/2, y=0, z) subjected to

UDL for aspect ratio10.

Figure 5: Through thickness variation of τ yz of Isotropic Plate at (x=a/2, y=0, z) subjected to

UDL for aspect ratio10.

B. A. Mhaske et al.


18

Figure 6: Through thickness variation of w of Isotropic Plate at (x=a/2, y=b/2, z) subjected to UDL for various aspect ratios.

Table 3: Comparison of inplane displacements (u ,v ), transverse displacements (w ) shear

stress ( ,xσ yσ ), inplane shear stress (xyτ ), transverse shear stress ( xzτ , yzτ ) for the isotropic

plate subjected to uniformly distributed thermal load with respect to aspect ratio.

MODEL S

2 4 5 10 20 100

(u ) TVPT 0.1073 0.0536 0.0429 0.0215 0.0107 0.0021 HYDT 0.1073 0.0536 0.0429 0.0215 0.0107 0.0021 TSDT 0.1073 0.0536 0.0429 0.0215 0.0107 0.0021

(v ) TVPT 0.1073 0.0536 0.0429 0.0215 0.0107 0.0021 HYDT 0.1073 0.0536 0.0429 0.0215 0.0107 0.0021 TSDT 0.1073 0.0536 0.0429 0.0215 0.0107 0.0021

( w ) TVPT 0.1916 0.3831 0.4789 0.9579 1.9157 9.5787 HYDT 0.1916 0.3831 0.4789 0.9579 1.9157 9.5787 TSDT 0.1916 0.3831 0.4789 0.9579 1.9157 9.5787

( xσ ) TVPT 0.1312 0.0656 0.0525 0.0262 0.0131 0.0026 HYDT 0.1312 0.0656 0.0525 0.0262 0.0131 0.0026 TSDT 0.1312 0.0656 0.0525 0.0262 0.0131 0.0026

( yσ ) TVPT 0.1312 0.0656 0.0525 0.0262 0.0131 0.0026 HYDT 0.1312 0.0656 0.0525 0.0262 0.0131 0.0026 TSDT 0.1312 0.0656 0.0525 0.0262 0.0131 0.0026

( xyτ ) TVPT 0.6015 0.3008 0.2406 0.1203 0.0602 0.012 HYDT 0.6016 0.3048 0.2406 0.1203 0.0602 0.012 TSDT 0.6016 0.3048 0.2406 0.1203 0.0602 0.012

Table 4: Comparison of transverse shear stress (xzτ , yzτ ) for the isotropic plate subjected to

uniformly distributed thermal load with respect to aspect ratio.

MODEL S

2 4 5 10 20 100

( xzτ )

TVPTCR 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 TVPTEE 2.3782 0.5945 0.3805 0.0951 0.0238 0.0010 HYDT 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 TSDT 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

( yzτ )

TVPTCR 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 TVPTEE 2.3782 0.5945 0.3805 0.0951 0.0238 0.0010 HYDT 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 TSDT 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

The results obtained for displacement and stresses for simply supported isotropic plate subjected to uniformly distributed thermal load are presented in Tables 1 through 4. Through thickness variation of displacement and stresses for aspect ratio 10 are shown in Figures 1 through 6.



19

From Tables and Figures, it is observed that, the results obtained by present theory for inplane displacements (u, v), transverse displacement (w), inplane normal stresses, inplane shear stress and transverse shear stress, are identical with those obtained by Ghugal and Kulkarni’s TSDT theory, Shinde et al’s HYDT theory, Mindlin’s FSDT theory, Kirchhoff’s CPT theory. 4 CONCLUSIONS From the numerical results following conclusions are drawn. 1. The present theory is variationally consistent and does not require shear correction factor. 2. In the present theory number of unknown functions involved is only two. Even in the

Reissner’s and Mindlin’s theory (first order shear deformation theories), three unknown functions are involved.

3. Present theory gives accurate prediction of thermal response of isotropic plate with respect

of displacement and stresses.

4. Transverse shear stresses obtained by constitutive relation are zero and integrating equilibrium equations, stress variation is realistic (giving rise to shear stress free surfaces and parabolic variation of shear stress across the thickness).

5. The theory has strong similarity with the classical plate theory (CPT) in many aspects (in

respect of a governing equation, boundary conditions, and moment expressions). REFERENCES Benachour A, Tahar HD, Atmane HA, Tounsi A, and Ahmed MS (2011). A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient. Composites: Part B 42, pp. 1386–1394. Benli S and Sayman O (2011). The effects of temperature and thermal stresses on impact damage in laminated composites. Mathematical and Computational Applications 16(2), pp. 392-403. Chen LW and Chen LY (1987). Thermal buckling of laminated composite plates. Journal of Thermal Stresses 10, pp. 345-356. Fares ME and Zenkour AM (1999). Mixed variational formula for the thermal bending of laminated plates. Journal of Thermal Stresses 22, pp. 347–365. Ghugal YM and Kulkarni SK (2011). Thermal stress analysis of cross-ply laminated plates using refined shear deformation theory. Journal of Experimental and Applied Mechanics 2(1), pp. 47-66. Ghugal YM and Kulkarni SK (2013a). Flexural analysis of cross-ply laminated plates subjected to nonlinear thermal and mechanical loadings. Acta Mechanica 224, pp. 675–690.

B. A. Mhaske et al.


20

Ghugal YM and Kulkarni SK (2013b) Thermal response of symmetric cross-ply laminated plates subjected to linear and non-linear thermo-mechanical loads. Journal of Thermal Stresses 36, pp. 466–479. Jones RM (1975). Mechanics of Composite Materials. McGraw Hill Kogakusha. Ltd. Tokyo. Kant T, Pendhari SS, and Desai YM (2008). An efficient semi-analytical model for composite and sandwich plates subjected to thermal load. Journal of Thermal Stresses 31, pp. 77–103. Kim SE, Thai HT, and Lee J (2009a). A two variable refined plate theory for laminated composite plates. Composite Structures 89, pp. 197–205. Kim SE, Thai HT, and Lee J (2009b). Buckling analysis of plates using the two variable refined plate theory. Thin-Walled Structures 47, pp. 455–462. Malekzadeh P and Shojaee M (2013). Free vibration of nanoplates based on a nonlocal two-variable refined plate theory. Composite Structures 95, pp. 443–452. Matsunaga H (2004). A Comparison between 2-D Single-Layer And 3-D Layerwise theories for computing interlaminar stresses of laminated composite and sandwich plates subjected to thermal loadings. Composite Structures 64, pp. 161-177. Matsunaga H (2009). Stress analysis of functionally graded plates subjected to thermal and mechanical loadings. Journal of Composite Structures 87, pp. 344-357. Mechab I, Atmane HA, Tounsi A, Belhadj HA, and Bedia EAA (2010). A two variable refined plate theory for the bending analysis of functionally graded plates. Acta Mechanica. 26, pp. 941–949. Mindlin RD (1951). Influence of rotary inertia and shear on flexural motions of Isotropic, elastic plates. ASME Journal of Applied Mechanics 18, pp. 31–38. Reddy JN (1997). Mechanics of laminated composite plates. CRC Press, Boca Raton. Reissner E (1944). On the theory of bending of elastic plates. Journal of Mathematics and Physics. 23, pp. 184-191. Reissner E (1945). The effect of transverse shear deformation on the bending of elastic plates. ASME Journal of Applied Mechanics 12, pp. 69-77. Shimpi RP (2002). Refined plate theory and its variants. AIAA Journal 40, pp. 137–146. Shimpi RP and Patel HG (2006a). A two variable refined plate theory for orthotropic plate analysis. International Journal of Solids and Structures 43, pp. 6783–6799. Shimpi RP and Patel HG (2006b). Free vibration of plate using two variable refined plate theory. Journal of Sound and Vibration 296, pp. 979-999.



21

Shinde BM (2013). Hyperbolic shear deformation theory for thermal analysis of composite plates. ME thesis, Department of Civil Engineering, Amrutvahini college of Engineering, Sangmner, University of Pune. Shinde BM, Kawade AB, and Sayyad AS (2013). Thermal response of isotropic plates using hyperbolic shear deformation theory. International Journal of Advanced Technology in Civil Engineering 2(1), pp. 140-145. Thai HT and Kim SE (2012). Analytical solution of a two variable refined plate theory for bending analysis of orthotropic Levy-type plates. International Journal of Mechanical Sciences 54, pp. 269–276. Wu CH and Tauchert TR (1980). Thermoelastic analysis of laminated plates. 1: Symmetric specially orthotropic laminates. Journal of Thermal Stress 3(2), pp. 247-259. Wu Z, Cheng YK, Lo SH, and Chen W (2007). Thermal stress analysis for laminated plates using actual temperature field. International Journal of Mechanical Science 49, pp. 1276–1288. Zenkour AM (2004). Analytical solution for bending of Cross-ply laminated plates under Thermo-mechanical loading. Composite Structures 65, pp. 367-379. Zhen W and Wanji C (2005). An efficient higher-order theory and finite element for laminated plates subjected to thermal loading. Journal of Composite Structures 73, pp. 99-109.

Documents

EFFECT OF UNIFORMLY DISTRIBUTED THERMAL LOAD ON BENDING RESPONSE OF ISOTROPIC PLATES USING TWO VARIABLE PLATE THEORY