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Stresses and deformations of a power pulley shell with exponential belt tensions and variable thickness
Compurers & Strucfures Vol. 39, No. 5, pp. 425-430. Printed in Great Britam.
00457949/91 s3.00 + 0.w Pergamon Press plc
1991
OF A POWER PULLEY STRESSES AND DEFORMATIONS BELT TENSIONS AND SHELL WITH EXPONENTIAL VARIABLE THICKNESS B. B.
DHAL
PAL
and M. C.
Department of Mathematics, R.E. College, Durgapur 9, India (Received 8 March 1990) Abstract-A
set of differential equations of displacements of the cylindrical shell of the conveyor power pulley under exponential belt loading with variable thickness is derived and the analytical solution of the set of equations is determined. The present solution is found to be in good agreement with both the theoretical and the experimental results for the steel conveyor pulley, in particular when the thickness is considered to be uniform. Results using different parameters are computed and obtained in graphical form and are found to be important for thickness optimization.
INTRODUCTION
where a’ = half the radius of curvature at (l/2, h) so that h is the maximum thickness at the midpoint x = l/2 for withstanding maximum load. Using 5 = x/a in eqn (1) we obtain
Theoretical analysis of stresses and deformations of the cylindrical shell of a conveyor power pulley under various belt tensions with constant thickness has been studied by a number of authors, for example Scholich [l] and Das and Pal [2]. To the present authors’ knowledge, no other such work has so far been done which considers variable thickness. The present analysis is carried out with the distribution of belt loading in exponential form [2] and variable shell thickness. The present authors assume the thickness as parabolic to withstand maximum load on the top of the shell. Equations of equilibrium for the shell are then formulated considering the thickness of the shell as a variable [3]. Equilibrium equations together with boundary conditions [4] are solven by Galerkin technique assuming displacement components in double trigonometric series that satisfy the boundary conditions. It is found that the effects of variable thickness on the study of the numerical behaviour of a conveyor power pulley are very important. A general computer program is developed for computation of stresses and displacement components. The results of the theoretical analysis are also validated by experimental findings [5] for steel conveyor pulleys and it is found that they are important for the optimization of thickness of the above shell.
THICKNESS
MODELLING EQUILIBRiUM
The thickness form as
AND FORMULATION EQUATIONS
curve may be modelled
t
X-( 4a’
(2)
where
c,=-g C=12-
16a’h 4a2
’
a and 1 being
the radius and length of the cylinder, respectively. The shell is considered to be subjected to belt tension TE. Taking non-uniform applied load along the normal to the shell [2] in the form P = T, e”‘” _ $)/al
=o
forO
for c( < $J ,< X,
(3)
where LXis half the wrap angle and considering t as a function of 5, the Flugge differential equations for the equilibrium of the shell are formulated [3] as l-v
U” + -~+-
OF
2
1+v,. 2
o+v~+~(~+vti+vw)=O (4)
in parabolic
1 +v,.
t=h-
=c,(+1;* +c),
-L4+++ 2
I 2 2>
1 - v I, Tu+i
-k(iii
_;Lyii+$Ll
’ 425
+;)
(5)
B. B. DHAL and M. C. PAL
426
factors cos ~C#J cos Am5, sin n+ sin A,<, cos n@ sin I,<, respectively, from eqn (lo), we obtain
+++vti)-+,
AoTo+ A,T,
=0
(12)
(6)
A3T2-A;T;+A,T,C,+A,T,+R,I,,=O
(13)
where AgT2+A;,T;+A,Tq+AsT3-R,I,6=0,
(14)
where
k=$&=
192
a’2 To = I, - f I,, + CI, , a
Et’
T, = 21,, - f I2 a
K= 12(1 -v2)
, D=-
T; = kc;
I,,
Et
(7)
1-G’
T3=T
The derivatives with respect to the dimensionless co-ordinates 5 and 4 are indicated by primes and dots. Y and Z are the tension at any point on the cylinder per unit area along the y and z directions, respectively, and are given as
=o
for u < C$
(8)
and Z= TEeP(“-@)/al forO<4 =0
for N < f#~< 71,
(9)
where p = co-efficient of friction between pulley and belt. Assume the displacements U, v, w [6] to satisfy the simply supported conditions as
u=
‘f f
A,,cosn~cosI,~
m=,,'n=0,1
v=
f f B,,,,sinn$sin&,,~ VI=l,'n=0,1
M’=
f f C,, cos nf$ sin I,<, m=1.3n=O.l
(10)
where A,,,= trma/l. Substituting the above expressions in eqns (4)-(6) and applying Galerkin technique of the form
sss 2n
l/a
,
E?Ra d$ dc dt =O,
Q=O {=O 0
(11)
where E, = residue which is the left-hand side expressions of eqns (4) (5) and (6) and R = weightage
I,, =
STEP a&i2 + n2)
(n e’” - p sinncc -n
cosncr)
427
Stresses and deformations of a power pulley shell
Using the expressions of A,, A,, etc. from eqn (17) in eqns (13) we obtain
-j8 cosntr +n sinna)
- 3 f i?r, f CQ,
A,=
-(A:+J+)A*.
A, = - 1, A, + vnB,, -t- vc,,,,
A 3 =lfvnd 2
A Rtmn
+
T
+
2
(n2+ 43” ,z
T
2’
Ad= y(-nA,.+&Bm) Solving eqns (18) by Cramer’s rule we obtain A
5
=‘--’ 4 cm a2
A, = - 4, A,, + n3,, + C,, A, = -f
where
(A.:+ vn2)c,,
A,= --$(n2+1:)&C,,
where ffij, i = 1,2,3, j = 1,2,3, are co-factors of
R _ 2(1 - v3a* I&,,,nE . CAS 396-D
(17)
(19)
B. B. DHAL and M. C. PAL
428
Therefore the final solutions of the eqns (4)-(6) are given as follows:
7 6 5
u=
f
+cos.+sin1.,5
f’
In=I.3 n=cl,,
ii
14
mn
Ea u=
f
?sinnf$
5
m=l,3n=cl.I
E
sin&t
x2 s :: 5 1
rn”
cc w=
C
m 1
bnn
Fcosn+sinl,l.
m=l,3n=O.l
(21)
Inn
O -1
Applying a stress-strain displacement relation [3], stresses at any point on the cylinder are given by
1 -‘o
I
0.4
I
I
06
1.2
I
1.6
I
2.0
E Fig. 2. 4 = 0, a = 90, a’ = 5.28 m, E = 0.008.
where
Fig. 3. I$ = 0, a = 90, a’ = 100 m, E = 0.012.
x cos nb sin I,{
(
+;nl, ;+-
aft
t
>.1
sin n4 cos A,(.
(23)
5r
0
0.4
0.6
1.2
1.6
2.0
t
Fig. 1. 4 = 0, G(= 90, a’ = 5.28 m, E = 0.012.
Fig. 4. r#~= 0, a = 90, a’ = 2 m, E = 0.012.
429
Stresses and deformations of a power pulley shell 4.5
6
r
3.0 ^E 25 =L 1 2.0 1.5 1.0 0.5 I 0.4
0
I
I
I
I
0.8
1.2
1.6
2.0
<
Fig. 5. 4 = 0, h = 0.012, a’ = 5.28 m.
Fig. 7. E = 0.012, a’ = 5.28, TE = 13.223kN.
For uniform thickness t+h as a’-+co, and consequently from eqn (19)
Using the expressions aij from eqn (24) in eqn (18) we obtain the equations which are obtained by Das and Pal [2] for the uniform thickness case. COMPUTATIONAL
1+v
a,,+-
hl nL, -
2
Numerical calculations are carried out for a conveyor power pulley shell with various parameters. Results are obtained by programming the algorithm of the analytical solution for middle plane displacement components u, v, w and stress components OX,O$, r,4 at critical points. All the computational works to obtain the results in graphical form is done through the Departmental PC/AT micro-computer. Software for this problem is also designed for further applications. Numerical results in graphical form are given in the next section for various cases.
2a
hl
a,,+vA, -
2a
azl+--
1+v 2
-1
a23+ {
a,,+-VA,--
na I’ m
2a
+$(n2+“:)ng
I
GRAPHS
hl 2a
hl a3,+n
21;
a33+
l-~@~~+l~)~ $.
{
RESULTS
I
(24)
For the computation of the above results, belt tensions T for a conveyor power pulley on the slack side and the tight side are considered to be. same. The mechanical behaviour of a shell for different thicknesses with thickness curves of different curvature is shown in Figs l-9. It is observed that transverse deflection is greater than other deflection components and it is not at a maximum at the middle, as in the case of uniform
2.5 z =L 2.0 1 1.5
a
0
I
I
I
I
I
0.4
0.8
1.2
1.6
2.0
0
0.4
0.8
E
Fig. 6.
K =
90, E = 0.008, a’ = 5.28m.
1.2
1.6
2.0
E
Fig. 8. 4 = 0,
K =
90, a’ = 5.28 m, h = 0.012.
B. B. DHAL and M. C. PAL
430
CONCLUSION
The stresses and deformations of a conveyor power pulley shell have been studied under exponential belt tensions and variable thickness. The present analysis aids in the rational design of the conveyor power pulley shell and is important in the study of the thickness optimization for such a structural design. Acknowledgement-The authors gratefully acknowledge the support given by the Principal, Regional Engineering College, Durgapur for carrying out this research. 0
0.4
0.6
1.2
1.6
2.0
-5 Fig. 9. E=0.012, a’=5.28m.
thickness. In particular, numerical results for transverse and longitude deflections are in good agreement with those obtained by Das and Pal [2] for the uniform case. Deformations are found to be greater on the top of the shell and they are symmetric about the 4 = 0 line.
REFERENCES
1. S. Scholich, Spannungen and Verformungen Gurtbandtrommelu. Bergbautechnik 15,10,517-522 (1965). 2. S. P. Das and M. C. Pal, Stresses and deformations of a conveyor power pulley shell under exponential belt tensions. Cornnut. Struct. 21. 787-795 (1987). 3. W. Flugge, Stresses in She& Springer; Berlin (1967). 4. S. Timoshinko and S. Woinowsky-Rrieger, Theory of Plates and Shells. McGraw-Hill. New York (1959). 5. S. P. Das, Studies on the rationahsation of design procedure of a typical power pulley used in conveyors. Ph.D. thesis, University of Burdwan (1988).
00457949/91 s3.00 + 0.w Pergamon Press plc
1991
OF A POWER PULLEY STRESSES AND DEFORMATIONS BELT TENSIONS AND SHELL WITH EXPONENTIAL VARIABLE THICKNESS B. B.
DHAL
PAL
and M. C.
Department of Mathematics, R.E. College, Durgapur 9, India (Received 8 March 1990) Abstract-A
set of differential equations of displacements of the cylindrical shell of the conveyor power pulley under exponential belt loading with variable thickness is derived and the analytical solution of the set of equations is determined. The present solution is found to be in good agreement with both the theoretical and the experimental results for the steel conveyor pulley, in particular when the thickness is considered to be uniform. Results using different parameters are computed and obtained in graphical form and are found to be important for thickness optimization.
INTRODUCTION
where a’ = half the radius of curvature at (l/2, h) so that h is the maximum thickness at the midpoint x = l/2 for withstanding maximum load. Using 5 = x/a in eqn (1) we obtain
Theoretical analysis of stresses and deformations of the cylindrical shell of a conveyor power pulley under various belt tensions with constant thickness has been studied by a number of authors, for example Scholich [l] and Das and Pal [2]. To the present authors’ knowledge, no other such work has so far been done which considers variable thickness. The present analysis is carried out with the distribution of belt loading in exponential form [2] and variable shell thickness. The present authors assume the thickness as parabolic to withstand maximum load on the top of the shell. Equations of equilibrium for the shell are then formulated considering the thickness of the shell as a variable [3]. Equilibrium equations together with boundary conditions [4] are solven by Galerkin technique assuming displacement components in double trigonometric series that satisfy the boundary conditions. It is found that the effects of variable thickness on the study of the numerical behaviour of a conveyor power pulley are very important. A general computer program is developed for computation of stresses and displacement components. The results of the theoretical analysis are also validated by experimental findings [5] for steel conveyor pulleys and it is found that they are important for the optimization of thickness of the above shell.
THICKNESS
MODELLING EQUILIBRiUM
The thickness form as
AND FORMULATION EQUATIONS
curve may be modelled
t
X-( 4a’
(2)
where
c,=-g C=12-
16a’h 4a2
’
a and 1 being
the radius and length of the cylinder, respectively. The shell is considered to be subjected to belt tension TE. Taking non-uniform applied load along the normal to the shell [2] in the form P = T, e”‘” _ $)/al
=o
forO
for c( < $J ,< X,
(3)
where LXis half the wrap angle and considering t as a function of 5, the Flugge differential equations for the equilibrium of the shell are formulated [3] as l-v
U” + -~+-
OF
2
1+v,. 2
o+v~+~(~+vti+vw)=O (4)
in parabolic
1 +v,.
t=h-
=c,(+1;* +c),
-L4+++ 2
I 2 2>
1 - v I, Tu+i
-k(iii
_;Lyii+$Ll
’ 425
+;)
(5)
B. B. DHAL and M. C. PAL
426
factors cos ~C#J cos Am5, sin n+ sin A,<, cos n@ sin I,<, respectively, from eqn (lo), we obtain
+++vti)-+,
AoTo+ A,T,
=0
(12)
(6)
A3T2-A;T;+A,T,C,+A,T,+R,I,,=O
(13)
where AgT2+A;,T;+A,Tq+AsT3-R,I,6=0,
(14)
where
k=$&=
192
a’2 To = I, - f I,, + CI, , a
Et’
T, = 21,, - f I2 a
K= 12(1 -v2)
, D=-
T; = kc;
I,,
Et
(7)
1-G’
T3=T
The derivatives with respect to the dimensionless co-ordinates 5 and 4 are indicated by primes and dots. Y and Z are the tension at any point on the cylinder per unit area along the y and z directions, respectively, and are given as
=o
for u < C$
(8)
and Z= TEeP(“-@)/al forO<4 =0
for N < f#~< 71,
(9)
where p = co-efficient of friction between pulley and belt. Assume the displacements U, v, w [6] to satisfy the simply supported conditions as
u=
‘f f
A,,cosn~cosI,~
m=,,'n=0,1
v=
f f B,,,,sinn$sin&,,~ VI=l,'n=0,1
M’=
f f C,, cos nf$ sin I,<, m=1.3n=O.l
(10)
where A,,,= trma/l. Substituting the above expressions in eqns (4)-(6) and applying Galerkin technique of the form
sss 2n
l/a
,
E?Ra d$ dc dt =O,
Q=O {=O 0
(11)
where E, = residue which is the left-hand side expressions of eqns (4) (5) and (6) and R = weightage
I,, =
STEP a&i2 + n2)
(n e’” - p sinncc -n
cosncr)
427
Stresses and deformations of a power pulley shell
Using the expressions of A,, A,, etc. from eqn (17) in eqns (13) we obtain
-j8 cosntr +n sinna)
- 3 f i?r, f CQ,
A,=
-(A:+J+)A*.
A, = - 1, A, + vnB,, -t- vc,,,,
A 3 =lfvnd 2
A Rtmn
+
T
+
2
(n2+ 43” ,z
T
2’
Ad= y(-nA,.+&Bm) Solving eqns (18) by Cramer’s rule we obtain A
5
=‘--’ 4 cm a2
A, = - 4, A,, + n3,, + C,, A, = -f
where
(A.:+ vn2)c,,
A,= --$(n2+1:)&C,,
where ffij, i = 1,2,3, j = 1,2,3, are co-factors of
R _ 2(1 - v3a* I&,,,nE . CAS 396-D
(17)
(19)
B. B. DHAL and M. C. PAL
428
Therefore the final solutions of the eqns (4)-(6) are given as follows:
7 6 5
u=
f
+cos.+sin1.,5
f’
In=I.3 n=cl,,
ii
14
mn
Ea u=
f
?sinnf$
5
m=l,3n=cl.I
E
sin&t
x2 s :: 5 1
rn”
cc w=
C
m 1
bnn
Fcosn+sinl,l.
m=l,3n=O.l
(21)
Inn
O -1
Applying a stress-strain displacement relation [3], stresses at any point on the cylinder are given by
1 -‘o
I
0.4
I
I
06
1.2
I
1.6
I
2.0
E Fig. 2. 4 = 0, a = 90, a’ = 5.28 m, E = 0.008.
where
Fig. 3. I$ = 0, a = 90, a’ = 100 m, E = 0.012.
x cos nb sin I,{
(
+;nl, ;+-
aft
t
>.1
sin n4 cos A,(.
(23)
5r
0
0.4
0.6
1.2
1.6
2.0
t
Fig. 1. 4 = 0, G(= 90, a’ = 5.28 m, E = 0.012.
Fig. 4. r#~= 0, a = 90, a’ = 2 m, E = 0.012.
429
Stresses and deformations of a power pulley shell 4.5
6
r
3.0 ^E 25 =L 1 2.0 1.5 1.0 0.5 I 0.4
0
I
I
I
I
0.8
1.2
1.6
2.0
<
Fig. 5. 4 = 0, h = 0.012, a’ = 5.28 m.
Fig. 7. E = 0.012, a’ = 5.28, TE = 13.223kN.
For uniform thickness t+h as a’-+co, and consequently from eqn (19)
Using the expressions aij from eqn (24) in eqn (18) we obtain the equations which are obtained by Das and Pal [2] for the uniform thickness case. COMPUTATIONAL
1+v
a,,+-
hl nL, -
2
Numerical calculations are carried out for a conveyor power pulley shell with various parameters. Results are obtained by programming the algorithm of the analytical solution for middle plane displacement components u, v, w and stress components OX,O$, r,4 at critical points. All the computational works to obtain the results in graphical form is done through the Departmental PC/AT micro-computer. Software for this problem is also designed for further applications. Numerical results in graphical form are given in the next section for various cases.
2a
hl
a,,+vA, -
2a
azl+--
1+v 2
-1
a23+ {
a,,+-VA,--
na I’ m
2a
+$(n2+“:)ng
I
GRAPHS
hl 2a
hl a3,+n
21;
a33+
l-~@~~+l~)~ $.
{
RESULTS
I
(24)
For the computation of the above results, belt tensions T for a conveyor power pulley on the slack side and the tight side are considered to be. same. The mechanical behaviour of a shell for different thicknesses with thickness curves of different curvature is shown in Figs l-9. It is observed that transverse deflection is greater than other deflection components and it is not at a maximum at the middle, as in the case of uniform
2.5 z =L 2.0 1 1.5
a
0
I
I
I
I
I
0.4
0.8
1.2
1.6
2.0
0
0.4
0.8
E
Fig. 6.
K =
90, E = 0.008, a’ = 5.28m.
1.2
1.6
2.0
E
Fig. 8. 4 = 0,
K =
90, a’ = 5.28 m, h = 0.012.
B. B. DHAL and M. C. PAL
430
CONCLUSION
The stresses and deformations of a conveyor power pulley shell have been studied under exponential belt tensions and variable thickness. The present analysis aids in the rational design of the conveyor power pulley shell and is important in the study of the thickness optimization for such a structural design. Acknowledgement-The authors gratefully acknowledge the support given by the Principal, Regional Engineering College, Durgapur for carrying out this research. 0
0.4
0.6
1.2
1.6
2.0
-5 Fig. 9. E=0.012, a’=5.28m.
thickness. In particular, numerical results for transverse and longitude deflections are in good agreement with those obtained by Das and Pal [2] for the uniform case. Deformations are found to be greater on the top of the shell and they are symmetric about the 4 = 0 line.
REFERENCES
1. S. Scholich, Spannungen and Verformungen Gurtbandtrommelu. Bergbautechnik 15,10,517-522 (1965). 2. S. P. Das and M. C. Pal, Stresses and deformations of a conveyor power pulley shell under exponential belt tensions. Cornnut. Struct. 21. 787-795 (1987). 3. W. Flugge, Stresses in She& Springer; Berlin (1967). 4. S. Timoshinko and S. Woinowsky-Rrieger, Theory of Plates and Shells. McGraw-Hill. New York (1959). 5. S. P. Das, Studies on the rationahsation of design procedure of a typical power pulley used in conveyors. Ph.D. thesis, University of Burdwan (1988).