2. Statement of the problem

(1)

International Journal of Advances in Applied Mathematics and Mechanics

Large deflection of a circular plate under non-uniform load pertaining to Aleph-Functions

Research Article

Vinod Gill∗, Kanak Modi

Department of Mathematics Amity University, Rajasthan, Jaipur-303002, India

Received 29 March 2016; accepted (in revised version) 30 April 2016

Abstract: The main object of the present paper is to obtain the large deflection and bending stresses for a clamped circular plate under non-uniform load by using Berger’s approximate method. The load shape considered here is an arbitrary function p(x) involving Jacobi polynomial, Fox-Wright function and Aleph-functions. The small deflection case is also considered as a particular case of large deflection. The obtained results of this paper provide an extension of the results given by the literature.

MSC: 33E20 • 33C45 • 33C60 • 33C70 • 34A12 • 34A30

Keywords:ℵ-function • I-function • Jacobi polynomial • Fox-Wright Function

1. Introduction

A lot of research work has been recently come up on the study and development of a function that is more general than I-function and Fox’s H-function, known as the Aleph (ℵ)-function. The Aleph (ℵ)-function introduced by S ¨ud l and et al. [1], however the notation and complete definition is presented here in the following manner in terms of the Mellin-Barnes type integrals (see also [2]):

ℵ[z] = ℵ^m,np_i,q_i,τi;r[Z ] = ℵ^m,np_i,q_i,τi;r

h

z|^(a_(b^j^,A^j⁾^1,n^,[^τⁱ^(a^{j i}^,A^{j i}^)]^{n+1,pi ;r}

j,Bj)1,m,[τi(bj i,Bj i)]_{m+1,qi ;r}

i

= 1 2πi

Z

LΩ^m,np_i,q_i,τi;r(ξ)z^−ξdξ (1) for all z 6= 0, where i =p

−1 and

Ω^m,np_i,qi,τi;r(ξ) =

m

Q

j =1Γ(bj+ Bjξ)Qⁿ

j =1Γ(1 − aj− Ajξ)

r

P

i =1τi pi

Q

j =n+1Γ(aj i+ Aj iξ) ^qQⁱ

j =m+1Γ(1 − bj i− Bj iξ)

(2)

the integration path L = Liγ∞,γ ∈ R,extended from γ−i∞ to γ+i∞, and is such that the poles, assumed to be simple of Γ(1−aj−Ajξ), j = 1,...,n do not coincide with the poles of Γ(bj+Bjξ) j = 1,...,m the parameter pi, q_iare non-negative integers satisfying 0 ≤ n ≤ pi, 1 ≤ m ≤ qi,τ > 0 for i = 1,...,r . The parameter Aj, B_j, A_{j i}, B_{j i}> 0 and aj, b_j, a_{j i}, b_{j i}∈ C . The empty product in (2) is interpreted as unity. The existence conditions for the defining integral (1) are given below:

φi> 0, |ar g (z)| <π

2φi, i = 1,...,r (3)

∗ Corresponding author.

E-mail addresses:[email protected](Vinod Gill), [email protected](Kanak Modi)

(2)

φi≥ 0, |ar g (z)| <π

2φi, and R(ξi) + 1 < 0 (4)

where

φi=

n

X

j =1

A_j+

m

X

j =1

B_j− τi

Ã _p

i

X

j =n+1

A_{j i}+

q_i

X

j =m+1

B_{j i}

!

(5)

and ξi=

m

X

j =1

b_j−

n

X

j =1

a_j+ τi

Ã _q

i

X

j =m+1

b_{j i}−

p_i

X

j =n+1

a_{j i}

! +1

2(p_i− qi), i = 1,...,r (6)

for detailed account of the Aleph (ℵ)-function see S ¨ud l and et al. [1,2]. Forτi= 1, ∀ i = 1, ..., r in (1), we get the I-function defined as follows (see Saxena [3]):

I [z] = ℵ^m,n_p_i_,q_i_,1;r[Z ] = ℵ^m,n_p_i_,q_i_,1;rh

z|^(a_(b^j^,A^j⁾^1,n^,(a^{j i}^,A^{j i}⁾^{n+1,pi ;r}

j,B_j)_1,m,(b_{j i},B_{j i})_{m+1,qi ;r}

i

= 1 2πi

Z

LΩ^m,n_p_i_,q_i_,1;r(ξ)z^−ξdξ (7) where the kernelΩ^m,n_p_i_,q_i_,1;r(ξ) is given in (2). The existence conditions for the integral in (7) are the same as given in (3) through (6) withτi= 1, i = 1, ..., r . If we set r = 1, then (7) reduces to the familiar H-function defined as follows (see [4]):

H_p,q^m,n[z] = ℵ^m,n_p_i_,q_i_,1;1[Z ] = ℵ^m,n_p_i_,q_i_,1;1h z|^(a_(b^p^,A^p⁾

q,Bq)

i

= 1 2πi

Z

LΩ^m,n_p_i_,q_i_,1;1(ξ)z^−ξdξ (8) where the kernelΩ^m,n_p_i_,q_i_,1;1(ξ) can be obtained from (2).

The series representation of Aleph (ℵ)-function is given by [5]:

ℵ^m_p0⁰^,n⁰

i,q_i⁰,τ⁰_i;r⁰[Z ] = ℵ^m_p0⁰^,n⁰ i,q_i⁰,τ⁰_i;r⁰

· z|

(a⁰_j,A⁰_j)_1,n0,[τ⁰_i(a⁰_{j i},A⁰_{j i})]_n0+1,p0 i;r 0 (b⁰_j,B⁰_j)_1,m0,[τ⁰_i(b⁰_{j i},B⁰_{j i})]_m0+1,q0

i;r 0

¸

=

m⁰

X

h=1

X∞ k⁰=0

(−1)^k⁰φ⁰(s)

B_h⁰k⁰! z^−s (9)

where

φ⁰(s) =

m⁰

Q

j =1Γ(b⁰_j+ B⁰_js)

n⁰

Q

j =1Γ(1 − a⁰_j− A⁰_js)

r⁰

P

i =1τ⁰_i

q_i⁰

Q

j =m⁰+1Γ(1 − b⁰_{j i}− B⁰_{j i}s)

p⁰_i

Q

j =n⁰+1Γ(a⁰_{j i}+ A⁰_{j i}s)

(10)

and

s = ηh,k⁰=b⁰_h+ k⁰ B⁰

h

, p_i⁰< qi⁰, |z| < 1.

Also, the Fox-Wright’s function [6] is defined as

p⁰ψq⁰(z) =p⁰ψq⁰

h_(e_j_,E_j₎

1,p0;

( f_j,F_j)1,q⁰;zi

= X∞ l =0

p⁰

Q

j =1Γ(ej+ Ejl )

q⁰

Q

j =1Γ(fj+ Fjl ) z^l

l ! (11)

where Ej( j = 1,..., p⁰) and Fj( j = 1,..., q⁰) are real and positive and

1 +

q⁰

X

j =1

F_j−

p⁰

X

j =1

E_j> 0,

if we set, some suitable value to the parameters involved in (11) then its reduce into some useful functions as given in [7].

(3)

β

P_β^(a,b)(z) = Γ(a + β + 1) β!Γ(a + b + β + 1)

[β]

X

n⁰=0

³_β

n⁰

´Γ(a + b + β + n⁰+ 1) Γ(a + n⁰+ 1)

µz − 1 2

¶_β

(12)

for more hypergeometric series , see [9].

Plates are the flat structures whose thickness t is small compared to the other in- plane dimension is the radius ρ. Plate theories are classified in many ways. One of them is based on the thickness, that is, thin and thick-plate theories. Geometrically, a plate is said to be thin if its thickness ratio t/ρ is less than 1/20, otherwise the plate is known to be thick. The bending properties of a plate depend mainly on its thickness as compared with its other dimensions.

There are several theories for plates under large deflection; the most commonly used of them is the Von-Karman plate theory which is sometimes referred to as the Kirchoff-Foppel plate theory.

In the classical theory of plates, small deflection and elastic behavior of the material are assumed. When the lateral deflection exceeds one half the plate thickness [10], the classical theory generally is not adequate and the sec- ond order effects of the vertical displacements on the membrane stresses need to be considered. Two-coupled non- linear partial differential equations considering these effects were given by [11]. Solutions based on these differential equations have been known as large deflection solutions. Berger [12] in 1955 proposed an approximate method for investigating the large deflection of initially flat isotropic plates. Here the large deflection of a clamped circular plate under non-uniform load has been calculated by using Berger’s approximate method. We consider the applied external pressure p(x) in the following form:

P (x) = K0

µ 1 −x²

ρ²

¶α

P_β^a,b µ

1 −2x² ρ²

¶

p⁰ψq⁰

½ K₁

µ 1 −x²

ρ²

¶¾

ℵ^m,np_i,q_i,τi;r

· K₂

µ 1 −x²

ρ²

¶¸

ℵ^m_p0⁰^,n⁰ i,q⁰_i,τ⁰_i;r⁰

· K₃

µ 1 −x²

ρ²

¶¸

(13) where K₀, K₁, K₂and K₃are constants. Recently, some research work based on modelling had done in [13].

2. Statement of the problem

Let us assume a clamped circular plate of thickness t, radiusρ and flexural rigidity R. Then by using Berger’s method, the approximate equations for a circular plate undergoing large deflections due to an externally applied load p(x) may be given as

µ d² d x²+1

x d d x

¶ µd²w d x² +1

x d w

d x − k²w

¶

=P

R= φ(x) (14)

where k is a normalized constant of integration given by the equation d y

d x+y x+1

2 µd w

d x

¶2

=k²t²

12 (15)

where w is the plate deflection, normal to the middle plane of the plate and y is the radial displacement.

The boundary condition of the problem are:

(i) w = 0 =d w

d x, at x = ρ (ii) y = 0, at x = ρ

Solution of the problem Let us consider

w =X

i

Gi£ J0(xti) − J0(ρti)¤

(16)

where t_iis the ith root of J₁(ρti) = 0.

It is clear that boundary conditions are satisfied by the above equation. Now using (16) in the Eq. (14), we find

X

i

Git_i²(k²+ t_i²)J0(xti) = φ(x) (17)

now expandingφ(x) in a series of Bessel’s function, we obtain on integration

ρ

Z

0

G_it_i²(k²+ t_i²)J₀²(xt_i)xd x =

ρ

Z

0

φ(x)J0(xt_i)ρdx (18)

(4)

now by left hand side of (18)

ρ

Z

0

x J²₀(xt_i)d x =ρ²

2 J₀²(ρti) (19)

and (18) becomes

G_it_i²(k²+ t_i²)ρ²

2 J²₀(ρti) =

ρ

Z

0

φ(x)J0(xt_i)xd x

or

Gi= 2

ρ

R

0

xφ(x)J0(xti)d x

ρ²t_i²(k²+ t_i²)J₀²(ρti). (20)

Now using [14], Eqs. (2), (9) and (10), the definition of Bessel function and interchanging the order of summa- tions and integration, we find

1

Z

0

θ²^λ+1(1 − θ²)^αP_β^a,b(1 − 2θ²)_p0ψq⁰[K₁(1 − θ²)]ℵ^m,npi,qi,τi;r[K₂(1 − θ²)]ℵ^m_p0⁰^,n⁰

i,q_i⁰,τ⁰_i;r⁰[K₃(1 − θ²)]J_µ(θτ)dθ

= X∞ l =0

β

X

n⁰=0

X∞ n⁰⁰=0

m⁰

X

h=1

X∞ k⁰=0

K₁^lK₃^−s(−1)^k⁰⁺ⁿ⁰⁰(−β)n⁰φ⁰(s)¡_τ

2

¢_µ+2n⁰⁰ 2l !n⁰!n⁰⁰!k⁰!B_h⁰

.

p⁰

Q

j =1Γ(ej+ Ejl )Γ(1 + a + β)(1 + a + b + β)n⁰Γ¡λ + n⁰+ n⁰⁰+^µ₂+ 1¢

q⁰

Q

j =1Γ(fj+ Fjn)Γ(1 + a + n⁰)Γ(1 + µ + n⁰⁰) .ℵ^m,n+1_p_i_+1,q_i_+1,τ_i_;r[K₂|(−α−l +s,1),(aj,Aj)1,n,[τi(aj i,Aj i)]_{n+1,pi ;r}

(b_j,B_j)_1,m,[τi(b_{j i},B_{j i})]_{m+1,qi ;r},(−1−λ−l −n⁰−n⁰⁰−α+s−^µ₂,1)] (21) where

Re(a) > −1,Re(b) > −1,Re(λ) > −1,Re(α) > −1,

Re(µ) > −1

2, Re(α +b_j

Bj) > 0,(j = 1,..., qi; her e i = 1,...,r ) and

Re(α +b⁰_j

B⁰_j) > 0,(j = 1,..., q_i⁰; her e i = 1,...,r⁰) .

Using (21) in view of (13) and (14), we get

G_i= K0Γ(1 + a + β) R t_i²β!(k²+ t_i²)J₀²(ρti)

X∞ l =0

Xβ n⁰=0

X∞ n⁰⁰=0

m⁰

X

h=1

X∞ k⁰=0

K₁^lK₃^−s(−1)^k⁰⁺ⁿ⁰⁰(−β)n⁰

l !n⁰!n⁰⁰!k⁰!B_h⁰ φ⁰(s)

.

p⁰

Q

j =1Γ(ej+ Ejl )Γ(1 + n⁰+ n⁰⁰)(1 + a + b + β)n⁰ q⁰

Q

j =1Γ(fj+ Fjl )Γ(1 + a + n⁰)Γ(1 + n⁰⁰)

.ℵ^m,n+1_p_i_+1,q_i_+1,τ_i_;r[K2|(−α−l +s,1),(aj,A_j)_{l ,n},[τi(a_{j i},A_{j i})]_{n+1,pi ;r}

(b_j,B_j)_{l ,m},[τi(b_{j i},B_{j i})]_{m+1,qi ;r},(−1−l −n⁰−n⁰⁰−α+s,1)]. (22)

(5)

w = L1

X

i

L2

k²+ t_i²[J₀(xti) − J0(ρti)] (23)

where

L₁=K₀Γ(1 + a + β) Rβ!

and

L2= 1 t_i²J²₀(ρti)

X∞ l =0

Xβ n⁰=0

X∞ n⁰⁰=0

m⁰

X

h=1

X∞ k⁰=0

K₁^lK₃^−s(−1)^k⁰⁺ⁿ⁰⁰(−β)n⁰

l !n⁰!n⁰⁰!k⁰!B_h⁰ φ⁰(s)

.

p⁰

Q

j =1Γ(fj+ Fjl )Γ(1 + a + n⁰)Γ(1 + n⁰⁰)

.ℵ^m,n+1_p_i_+1,q_i_+1,τ_i_;r[K2|(−α−l +s,1),(aj,A_j)_{l ,n},[τi(a_{j i},A_{j i})]_{n+1,pi ;r}

(bj,Bj)_{l ,m},[τi(bj i,Bj i)]_{m+1,qi ;r},(−1−l −n⁰−n⁰⁰−α+s,1)].

Now the radial displacement y can be obtained by using Eqs. (15) and (16) as

y =k²t²x 24 −1

2 X∞ i =1

G_i²t_i²

"

x 2

(

J₁⁰²(xt_i) + Ã

1 − 1 x²t_i²

! J²₁(xt_i)

)#

−1 2

X∞ i =1

X∞ j =1

G_iG_jt_it_j" t_iJ₂(xt_i)J₁(xt_j) − tjJ₂(xt_j)J₁(xt_i) t_i²− t²_j

#

+C1, i 6= j (24)

where C₁is the constant of integration.

Applying the boundary condition y =0 at x = ρ and J1(ρti) = 0, we get

C₁= −k²t²ρ 24 +1

4 X∞ i =1

G²_it_i²ρJ₁⁰²(ρti) (25)

hence the radial displacement y is established as

y =k²t²(x − ρ)

24 −1

2 X∞ i =1

G_i²t_i²

"

x 2

(

J₁⁰²(xt_i) + Ã

1 − 1 x²t_i²

! J²₁(xt_i)

)#

−1 2

X∞ i =1

X∞ j =1

G_iG_jt_it_j" t_iJ₂(xt_i)J₁(xt_j) − tjJ₂(xt_j)J₁(xt_i) t_i²− t²_j

# +1

4 X∞ i =1

G_i²t_i²ρJ0²(ρti).

3. Applications

(3.A) The deflection given by Eq. (23) can be used to evaluate the boundary stresses at the surface of the plate which for the circular plate, are given by [12] as

σx=−6R t²

µd²w d x² +ν dw

x d x

¶

(26)

and

σ_θ=−6R t²

µ νd²w

d x² +1 d w x d x

¶

(27)

whereν is the Poisson’s ratio.

(6)

By using (23), we get

σx=−6R t² L₁X

i

L₂ (k²+ t_i²)

h

J₀⁰⁰(xt_i) +ν xJ₀⁰(xt_i)i

(28)

and σ_θ=−6R

t² L₁X

i

L₂ (k²+ t_i²)

·

νJ⁰⁰₀(xt_i) +1 xJ₀⁰(xt_i)

¸

(29)

Now, putting x = 0 in (28) and (29), we get the bending stresses at the centre of the plate as (σx)_x=0= (σ_θ)_x=0=3R

t²L1

X

i

L2

(k²+ t_i²)(ν + 1)t_i² (30)

also by putting x = ρ, the bending stresses at the edge of the plate are obtained as

(σx)_x=ρ=6R t²L₁X

i

L₂

(k²+ t_i²)t_i²J₀(ρti) (31)

and

(σ_θ)_x=ρ=6R t²L₁X

i

L2

(k²+ t_i²)νt_i²J₀(ρti) (32)

(3.B) When k = 0, the differential Eq. (14) corresponds to that of small deflection equation and then Eq. (23) leads to

w = L1

X

i

L₂

t_i²[J₀(xt_i) − J0(ρti)] (33)

(3.C) By using x = 0, we obtain the deflection w₀at the centre of the plate as w0= L1

X

i

L2

(k²+ t_i²)[1 − J0(ρti)] (34)

whereas the small deflection will be given by w₀= L1

X

i

L₂

t_i²[1 − J0(ρti)] (35)

4. Special cases

(i) By takingτi = 1 ∀ i = 1, ..., r and τ⁰i = 1 ∀ i = 1, ..., r⁰for ℵ^m,npi,qi,τi;r and ℵ^m_p0⁰^,n⁰

i,q⁰_i,τ⁰_i;r⁰ in the load p(x), both the Aleph-functions reduces to the I-function [3]. Then we obtain the deflection as

w = D1

X

i

D2

(k²+ t_i²)[J₀(xti) − J0(ρti)] (36)

where

D₁=K₀Γ(1 + a + β) Rβ!

and

D₂= 1 t_i²J²₀(ρti)

X∞ l =0

Xβ n⁰=0

X∞ n⁰⁰=0

m⁰

X

h=1

X∞ k⁰=0

K₁^lK₃^−s(−1)^k⁰⁺ⁿ⁰⁰(−β)⁰n

l !n⁰!n⁰⁰!k⁰!B_h⁰ φ⁰(s⁰)(ρti

2 )²ⁿ⁰⁰

.

p⁰

Q

j =1Γ(fj+ Fjl )Γ(1 + a + n⁰)Γ(1 + n⁰⁰) .I_p^m,n+1

i+1,qi+1,1;r[K2|^{(−α−l +s}

0,1),(a_j,A_j)_{l ,n},(a_{j i},A_{j i})_{n+1,pi ;r}

(b_j,B_j)_{l ,m},(b_{j i},B_{j i})_{m+1,qi ;r},(−1−l −n⁰−n⁰⁰−α+s⁰,1)]

(7)

w = D1

X

i

D2

t_i²[J₀(xti) − J0(ρti)] (37)

in this case, the deflection at the centre of the plate is given by

w₀= D1

X

i

D2

(k²+ t_i²)[1 − J0(ρti)]. (38)

(ii) By takingτi= 1 ∀ i = 1, ..., r, set r = 1 and τ⁰i= 1 ∀ i = 1, ..., r⁰set r⁰= 1 for ℵ^m,np_i,q_i,τi;rand ℵ^m_p0⁰^,n⁰

i,q⁰_i,τ⁰_i;r⁰in the load p(x), both the Aleph-functions reduces to the familiar H-function [4]. Then we reduces to known result obtained by Chaurasia and Arya [15].

Inview of the generality of the ℵ-function, on specializing the various parameters, we can obtain from our results, several results involving a remarkably wide variety of useful functions, which are expressible in terms of the Mittag- Leffler function ([4], p.25, Eq. (1.137)), the generalized Wright hypergeometric function ([4], p.25,Eq. (1.140)), the generalized Bessel-Maitland function ([4], p.25, Eq.(1.139)) and their various special cases. Thus, the results presented in this paper would at once yield a very large number of results involving a large variety of special functions occurring in the problems of science, engineering and mathematical physics etc.

Acknowledgements

The authors are grateful to Professor H.M. Srivastava, University of Victoria, Canada for his kind help and valu- able suggestions in the preparation of this paper.

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