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Buckling load and deformations of a cylindrical shell under non-uniform belt tensions
Computers d Sfrucmres Vol. 37, No. 4* pp. 43-39, Printed in Great Britain.
0045.7949/90 $3.00 + 0.00 Pergamcm Press
1990
plc
BUCKLING LOAD AND DEFORMATIONS OF A CYLINDRICAL SHELL UNDER NON-UNIFORM BELT TENSIONS M. C. PAL Department of Mathematics, Regional Engineering College, Durgapur 713 209, West Bengal, India (Received
11 September 1989)
Abstract-The
stability of a cylindrical shell which is used in a conveyor power pulley subjected to exponential belt tensions is analysed. A set of differential equations for buckling of a cylindrical shell under non-uniform external pressure has been solved. The formulas and results, deduced for the first time, may be readily applied to determine the critical load and deformations of conveyor power pulley shells for optimum design or other applications.
INTRODUCTION
The problem of the stability of a cylindrical shell has arisen in recent years because of its importance in applications. Stability problems in cylindrical shells subjected to uniform loading have been studied by a number of authors [14]. However, very few papers, to the author’s knowledge, have been published so far on shell buckling under variable loadings. Shell buckling problems under exponential loading are studied for the first time in the present paper. Theoretical deformations of the cylindrical shell of a conveyor power pulley subjected to uniform and exponential belt tensions have been studied by Scholich [5] and Das and Pal [6]. An attempt has been made to establish a rational design procedure and the results of the theoretical analysis have been validated by experimental findings for steel conveyor pulleys by Das and Pal [6] without consideration of the buckling load. It is further noted that the shell’s thickness optimization is necessary for such a rational design procedure. The present work also for the first time derives the formulas for belt tension under buckling load and finds the shell’s deformation with the motivation that the results may be applied in the rational design of the power pulley shell structure or any other cylindrical structure in modern engineering practice. Comparisons of the results for uniform loading obtained in the limiting case when p -rO have been made with those by Flugge [4] and they are found to be in good agreement.
The shell is considered to be subjected to belt tensions T2. Taking non-uniform applied load along normal to the shell [6] in the form
= 0
for094
a2t4 i-va3t4
*+
2zp+
+K
a2v 3-v ---31 -vlar2 2
a3w ac2a4
-q,&"-o)
I-V
a%
a%
adw
aC2a4+ay'
adw adw a2w +2ac2a42+@i+2@+w
for a < d, < II,
a*0 aw 2 ata4 +‘ar
--I+V
I+~ a2u a% 1-~a2~ aw -2 aya4+ag'+--i+2 at: a4
3-v --2
BASIC EQUATIONS
P= Tze’@-@/al
where a is a wrap angle, the governing differential equations of the shell buckling problem derived by Flugge [4, p. 4221 are
(1) 435
a%
M. C. PAL
436
where
where
K=&
2 aJm= ;
x
= 6” - 4)cos rn+ cos jtp d4 s0
r=;
b/, = 1 O1 d(“- 0)sin rn4 sin j4 d#. x I0 4, =;
D=-
We substitute the expressions from eqns (3) and (4) in eqns (2) to obtain a set of an infinite number of homogeneous linear, algebraical equations for the co-efficients A,,,, B,,, and C,,, as
Eh l-9
T, = Dlq, = normal pressure or radial pressure = radius of the cylinder = length of the cylinder = angular distance at any point P(x) = Young’s modulus of elasticity = Poisson’s ratio = thickness of the shell. We assume the middle surface displacements and w in the form
a2LAm+a22Bm+a23G= -q1d2,
P a I 4 E v h
(5) where d, = f
u, u
(j2Aj + Izcj)~,,,~
j-0 mfj
d2 = i u=cosIt;
f
(j2Bj + jC’)b,
j-l
A,cosm&
m#i
m-0
(AA, + jBj + j2C,)um,.
d, = f v=sinl[
j-0 m-j
5 B,,,sinm$ m-l
w = sin At f C,cosm4, ,..,” ..~”
(3)
where 1 = na/l to satisfy simply supported conditions, and expand
where
his x3
e@@ - +)cos rn4
A =
e’(“- @sin rn4
A,=k$3.3
l-v
12+ --i-m2(l =
The co-efficient matrix A of eqns (5) may be written as A=A,+A,,
+ k)
1+v -+??I l-v m2+ -T-A2(l
1+v -2Im
-vl
+ 3k)
&-+) cos m$ =fajm cw%
m+-
3-v 2
kA2m
m2a_ 0
A2=q1 [
amm
0 m2b_ -,,
1
&WU mb,,,,,, . 2 ma,,
From the first two of eqns (5) we obtain
j-0 m
e’(“- @)sin rn4 = c b,,,,sin j4, /-I
i’-Tim’)
1 +k(14+212m2+m4-2m2+1)
m +qkl’m
in half range cosine and sine series, respectively, in 0 d r$ < a so that
-k
(4)
Bucklingload and deformationsof a cylindricalshell
i
C_+-$-
x’,c,.
431
W
33m j-0
A, are the co-factors of al and suffix in the bracket replaces m in the expression aq. j#m A,,
Z& =
=
A:,
+
a&
m*[ai,b_ A32
=
a:2
+ aLad +
j#m
%&
Equation (7) may be written as
K, = m[(a;* - a~,m)b,,,,,, - IaLa_]
4 +4 A,,=A:,+q,&
ti12 a2 +
:::
I1
$32
ti13
. . .
@23
. . .
a,+1,
...
&=[(la~2-m2s~2)a_-ma~Ib,].
= 0
(8)
... ...
Ah are the co-factors of ai
or Iz;+P,1;-‘+.**+P,=O.
(6b) Substituting the above expressions in the third equation of (5), with the necessary simplifications the infmite set of linear equations representing the buckling problem are obtained as (I, + a,)m*C,
+ g
,f
*;j2Cj
(9)
Coefficients P, may be obtained by Newton’s formula [I and the Newton-Raphson method may be applied to find the eigenvalue. Equation (9) may also be written as l+P, +=O
neglecting 0 $
1
0
I
or
= 0,
1,=-P,=
1 J-1
-
f
a,.
(10)
m-1
j#m
m=l,2...,
(6c)
where
+ij=2{$($+),+($)j+
l}hj+$j
Formula (10) may be used to obtain the eigenvalue to the first-order approximation and consequently the buckling load or any other analytical method combined with numerical method may be used to solve eqn (7) for eigenvalues and eigenvectors. However, in order to find the largest eigenvalue and eigenvector with a greater rate of convergency, the power method is used [8] and eqn (7) is written in matrix form as
A,=’ 41
VW = Au”‘,
Al=IA,I,A2=K,at,+K2a12+K3af3
a = I(Vn/ = my]V(k)J,
For the convergence requirement of eqn (6) we replace m*C,,, and j’C, by C,,,and C,, respectively. The final forms of the buckling equations are Therefore, i tiWC,=O, I-1 lfm
m=l,2...
a
where
+a,,,,,,(m*A~, +nA!,).
&+a,)&+
U’k+ 1)= 1 I/W
(7)
’
(11)
438
M. c. PAL
AK= 1 x 10d AK=6x
lOa
AK= 2 x IO8 - AK= 1 x lo8
x10
'1
2
4
10
20
40
loo
I/a
Fig. 1. Buckling diagram for non-uniform external pressure P.
from eqn (11) gives the minimum
q,
2
Eigenvector
of q, and
Ci =
u
(12)
T, = Dlq, ,
gives the formula for the buckling load due to non-uniform belt tension. Formula (12), when p + 0 in eqn (7), gives the critical load due to uniform belt tension. (a)
Vi’) gives I/w, ,
i=l,2
,....
Equation (7a) with eqn (13) gives Ai and Bi, i = 1,2. . . . Substituting the values of Ai, Bi, Ci in eqn (3), we obtain the middle surface displacements at any point in the power pulley shell due to buckling load.
b)
h=O.O05m
T, = 612.95 kN
(13)
h = 0.012 m T, = 5901.50kN
h = 0.001 m T, = 9.71 kN
Fig. 2. Buckling mode of a circular cylindrical shell under exponential belt tension at its mid-length. I = 0.712 m; p = 0.25; a = 180”.
Buckling load and deformations of a cylindrical shell
(a)
439
CONCLUSIONS The results and formulas deduced in the present paper may be applied to determine the critical belt tension for an isotropic power pulley shell acting on a certain part of the cylinder. It is noted that tension T2 due to buckling becomes 5910,612 and 12 kN for shells of thicknesses h = 12, 5 and 1 mm, respectively. There is a very sharp drop-off in load due to the drop in the shells’ thickness. This fact regarding critical load should be taken into consideration in the optimum design of a power pulley shell structure. It is further observed that there is a decrease in the critical load q, (Fig. 1) when the co-efficient of frictional force increases considerably. Radial deflection W is found to be maximum on the line 4 = 0, which is expected in our cases. The result may be used to study the power pulley shell for optimum design. The present method may also be used for determining critical load under non-uniform normal pressure for the optimum design of any other circular cylindrical structure.
04
Acknowledgement-The
author gratefully acknowledges the support given by the Principal, Regional Engineering College, Durgapur for carrying out this research.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9
1
Fig. 3. Middle surface displacements for various cases. Poisson’s ratio = 0.3; material: steel; p = 0.25; I = 0.712 m; h = 0.005 m; a = 180”. (a) 4 = 0.0. (b) 4 = 90”.
REFERENCES 1. S. Cheng and F. B. Ho, Theory of orthotropic and composite cylindrical shells. Accurate and simple fourth order governing equations. ASME J. appl. Mech. 51, 736-744 (1984). 2. B. P. Ho and S. Cheng, Some problems in stability of
Figure 1 shows the buckling diagram for q, over for the cases p = 0.25 and p +O, respectively, in the logarithmic plot. Results in the diagram when the friction is not taken into consideration are, for uniform normal loading, in good agreement with those obtained by Flugge [4] for uniform external pressure. Figure 2 gives the buckling mode of the power pulley shell under exponential tension for various parameters. Figure 3 gives middle surface displacements for various cases which are similar to those obtained by Das and Pal [6]. All the computational work including graphics has been done either on the departmental computer PC/AT or on the computer HP 100/45 at C.M.E.R.I., Durgapur. I/a
heterogeneous aelotropic cylindrical shells under combined loading. AIAA Jnl 1, 1603-1607 (1963). 3. M. M. Lei and S. Cheng, Buckling of composite and homogeneous isotropic cylindrical shells under axial and radial loading. ASME J. appl Mech. 36, 791-798 (1969). 4. W. Flugge, Stress in Shells, pp. 422433.
5. 6.
7. 8.
Springer, Berlin (1967). S. Scholich, Spannungen and verformungen Gurtbandtrommeln. Bergbautechnik 15, 517-522 (1965). S. P. Das and M. C. Pal, Stresses and deformation of a conveyor power pulley shell under exponential belt tensions. Compur. Srract. 27, 787-795 (1987). V. N. Faddeeva, Computational Metho& of Linear Algebra, pp. 177-182. Dover, New York (1959). A. Jennings, Matrix Computation for Engineers and Scientists, pp. 289290. John Wiley, New York (1977).
0045.7949/90 $3.00 + 0.00 Pergamcm Press
1990
plc
BUCKLING LOAD AND DEFORMATIONS OF A CYLINDRICAL SHELL UNDER NON-UNIFORM BELT TENSIONS M. C. PAL Department of Mathematics, Regional Engineering College, Durgapur 713 209, West Bengal, India (Received
11 September 1989)
Abstract-The
stability of a cylindrical shell which is used in a conveyor power pulley subjected to exponential belt tensions is analysed. A set of differential equations for buckling of a cylindrical shell under non-uniform external pressure has been solved. The formulas and results, deduced for the first time, may be readily applied to determine the critical load and deformations of conveyor power pulley shells for optimum design or other applications.
INTRODUCTION
The problem of the stability of a cylindrical shell has arisen in recent years because of its importance in applications. Stability problems in cylindrical shells subjected to uniform loading have been studied by a number of authors [14]. However, very few papers, to the author’s knowledge, have been published so far on shell buckling under variable loadings. Shell buckling problems under exponential loading are studied for the first time in the present paper. Theoretical deformations of the cylindrical shell of a conveyor power pulley subjected to uniform and exponential belt tensions have been studied by Scholich [5] and Das and Pal [6]. An attempt has been made to establish a rational design procedure and the results of the theoretical analysis have been validated by experimental findings for steel conveyor pulleys by Das and Pal [6] without consideration of the buckling load. It is further noted that the shell’s thickness optimization is necessary for such a rational design procedure. The present work also for the first time derives the formulas for belt tension under buckling load and finds the shell’s deformation with the motivation that the results may be applied in the rational design of the power pulley shell structure or any other cylindrical structure in modern engineering practice. Comparisons of the results for uniform loading obtained in the limiting case when p -rO have been made with those by Flugge [4] and they are found to be in good agreement.
The shell is considered to be subjected to belt tensions T2. Taking non-uniform applied load along normal to the shell [6] in the form
= 0
for094
a2t4 i-va3t4
*+
2zp+
+K
a2v 3-v ---31 -vlar2 2
a3w ac2a4
-q,&"-o)
I-V
a%
a%
adw
aC2a4+ay'
adw adw a2w +2ac2a42+@i+2@+w
for a < d, < II,
a*0 aw 2 ata4 +‘ar
--I+V
I+~ a2u a% 1-~a2~ aw -2 aya4+ag'+--i+2 at: a4
3-v --2
BASIC EQUATIONS
P= Tze’@-@/al
where a is a wrap angle, the governing differential equations of the shell buckling problem derived by Flugge [4, p. 4221 are
(1) 435
a%
M. C. PAL
436
where
where
K=&
2 aJm= ;
x
= 6” - 4)cos rn+ cos jtp d4 s0
r=;
b/, = 1 O1 d(“- 0)sin rn4 sin j4 d#. x I0 4, =;
D=-
We substitute the expressions from eqns (3) and (4) in eqns (2) to obtain a set of an infinite number of homogeneous linear, algebraical equations for the co-efficients A,,,, B,,, and C,,, as
Eh l-9
T, = Dlq, = normal pressure or radial pressure = radius of the cylinder = length of the cylinder = angular distance at any point P(x) = Young’s modulus of elasticity = Poisson’s ratio = thickness of the shell. We assume the middle surface displacements and w in the form
a2LAm+a22Bm+a23G= -q1d2,
P a I 4 E v h
(5) where d, = f
u, u
(j2Aj + Izcj)~,,,~
j-0 mfj
d2 = i u=cosIt;
f
(j2Bj + jC’)b,
j-l
A,cosm&
m#i
m-0
(AA, + jBj + j2C,)um,.
d, = f v=sinl[
j-0 m-j
5 B,,,sinm$ m-l
w = sin At f C,cosm4, ,..,” ..~”
(3)
where 1 = na/l to satisfy simply supported conditions, and expand
where
his x3
e@@ - +)cos rn4
A =
e’(“- @sin rn4
A,=k$3.3
l-v
12+ --i-m2(l =
The co-efficient matrix A of eqns (5) may be written as A=A,+A,,
+ k)
1+v -+??I l-v m2+ -T-A2(l
1+v -2Im
-vl
+ 3k)
&-+) cos m$ =fajm cw%
m+-
3-v 2
kA2m
m2a_ 0
A2=q1 [
amm
0 m2b_ -,,
1
&WU mb,,,,,, . 2 ma,,
From the first two of eqns (5) we obtain
j-0 m
e’(“- @)sin rn4 = c b,,,,sin j4, /-I
i’-Tim’)
1 +k(14+212m2+m4-2m2+1)
m +qkl’m
in half range cosine and sine series, respectively, in 0 d r$ < a so that
-k
(4)
Bucklingload and deformationsof a cylindricalshell
i
C_+-$-
x’,c,.
431
W
33m j-0
A, are the co-factors of al and suffix in the bracket replaces m in the expression aq. j#m A,,
Z& =
=
A:,
+
a&
m*[ai,b_ A32
=
a:2
+ aLad +
j#m
%&
Equation (7) may be written as
K, = m[(a;* - a~,m)b,,,,,, - IaLa_]
4 +4 A,,=A:,+q,&
ti12 a2 +
:::
I1
$32
ti13
. . .
@23
. . .
a,+1,
...
&=[(la~2-m2s~2)a_-ma~Ib,].
= 0
(8)
... ...
Ah are the co-factors of ai
or Iz;+P,1;-‘+.**+P,=O.
(6b) Substituting the above expressions in the third equation of (5), with the necessary simplifications the infmite set of linear equations representing the buckling problem are obtained as (I, + a,)m*C,
+ g
,f
*;j2Cj
(9)
Coefficients P, may be obtained by Newton’s formula [I and the Newton-Raphson method may be applied to find the eigenvalue. Equation (9) may also be written as l+P, +=O
neglecting 0 $
1
0
I
or
= 0,
1,=-P,=
1 J-1
-
f
a,.
(10)
m-1
j#m
m=l,2...,
(6c)
where
+ij=2{$($+),+($)j+
l}hj+$j
Formula (10) may be used to obtain the eigenvalue to the first-order approximation and consequently the buckling load or any other analytical method combined with numerical method may be used to solve eqn (7) for eigenvalues and eigenvectors. However, in order to find the largest eigenvalue and eigenvector with a greater rate of convergency, the power method is used [8] and eqn (7) is written in matrix form as
A,=’ 41
VW = Au”‘,
Al=IA,I,A2=K,at,+K2a12+K3af3
a = I(Vn/ = my]V(k)J,
For the convergence requirement of eqn (6) we replace m*C,,, and j’C, by C,,,and C,, respectively. The final forms of the buckling equations are Therefore, i tiWC,=O, I-1 lfm
m=l,2...
a
where
+a,,,,,,(m*A~, +nA!,).
&+a,)&+
U’k+ 1)= 1 I/W
(7)
’
(11)
438
M. c. PAL
AK= 1 x 10d AK=6x
lOa
AK= 2 x IO8 - AK= 1 x lo8
x10
'1
2
4
10
20
40
loo
I/a
Fig. 1. Buckling diagram for non-uniform external pressure P.
from eqn (11) gives the minimum
q,
2
Eigenvector
of q, and
Ci =
u
(12)
T, = Dlq, ,
gives the formula for the buckling load due to non-uniform belt tension. Formula (12), when p + 0 in eqn (7), gives the critical load due to uniform belt tension. (a)
Vi’) gives I/w, ,
i=l,2
,....
Equation (7a) with eqn (13) gives Ai and Bi, i = 1,2. . . . Substituting the values of Ai, Bi, Ci in eqn (3), we obtain the middle surface displacements at any point in the power pulley shell due to buckling load.
b)
h=O.O05m
T, = 612.95 kN
(13)
h = 0.012 m T, = 5901.50kN
h = 0.001 m T, = 9.71 kN
Fig. 2. Buckling mode of a circular cylindrical shell under exponential belt tension at its mid-length. I = 0.712 m; p = 0.25; a = 180”.
Buckling load and deformations of a cylindrical shell
(a)
439
CONCLUSIONS The results and formulas deduced in the present paper may be applied to determine the critical belt tension for an isotropic power pulley shell acting on a certain part of the cylinder. It is noted that tension T2 due to buckling becomes 5910,612 and 12 kN for shells of thicknesses h = 12, 5 and 1 mm, respectively. There is a very sharp drop-off in load due to the drop in the shells’ thickness. This fact regarding critical load should be taken into consideration in the optimum design of a power pulley shell structure. It is further observed that there is a decrease in the critical load q, (Fig. 1) when the co-efficient of frictional force increases considerably. Radial deflection W is found to be maximum on the line 4 = 0, which is expected in our cases. The result may be used to study the power pulley shell for optimum design. The present method may also be used for determining critical load under non-uniform normal pressure for the optimum design of any other circular cylindrical structure.
04
Acknowledgement-The
author gratefully acknowledges the support given by the Principal, Regional Engineering College, Durgapur for carrying out this research.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9
1
Fig. 3. Middle surface displacements for various cases. Poisson’s ratio = 0.3; material: steel; p = 0.25; I = 0.712 m; h = 0.005 m; a = 180”. (a) 4 = 0.0. (b) 4 = 90”.
REFERENCES 1. S. Cheng and F. B. Ho, Theory of orthotropic and composite cylindrical shells. Accurate and simple fourth order governing equations. ASME J. appl. Mech. 51, 736-744 (1984). 2. B. P. Ho and S. Cheng, Some problems in stability of
Figure 1 shows the buckling diagram for q, over for the cases p = 0.25 and p +O, respectively, in the logarithmic plot. Results in the diagram when the friction is not taken into consideration are, for uniform normal loading, in good agreement with those obtained by Flugge [4] for uniform external pressure. Figure 2 gives the buckling mode of the power pulley shell under exponential tension for various parameters. Figure 3 gives middle surface displacements for various cases which are similar to those obtained by Das and Pal [6]. All the computational work including graphics has been done either on the departmental computer PC/AT or on the computer HP 100/45 at C.M.E.R.I., Durgapur. I/a
heterogeneous aelotropic cylindrical shells under combined loading. AIAA Jnl 1, 1603-1607 (1963). 3. M. M. Lei and S. Cheng, Buckling of composite and homogeneous isotropic cylindrical shells under axial and radial loading. ASME J. appl Mech. 36, 791-798 (1969). 4. W. Flugge, Stress in Shells, pp. 422433.
5. 6.
7. 8.
Springer, Berlin (1967). S. Scholich, Spannungen and verformungen Gurtbandtrommeln. Bergbautechnik 15, 517-522 (1965). S. P. Das and M. C. Pal, Stresses and deformation of a conveyor power pulley shell under exponential belt tensions. Compur. Srract. 27, 787-795 (1987). V. N. Faddeeva, Computational Metho& of Linear Algebra, pp. 177-182. Dover, New York (1959). A. Jennings, Matrix Computation for Engineers and Scientists, pp. 289290. John Wiley, New York (1977).