- Email: [email protected]
Torsion of solid and perforated semicircular cylinders
hr.
1. Solids Strurfurrs
Printed
Vol. 21. No. 2, pp. 135-139, VSS
0020-7683iss $3.M)+ .oo 0 1985 PergamonRess Ltd.
in the U.S.A.
TORSION OF SOLID AND PERFORATED SEMICIRCULAR CYLINDERS? A. K. NAGHDI Department of Ae~~utic~-As~~~uti~ E~n~~g and Ma#ema~i~ Sciences, Purdue Unjve~ity SchooI of Engineering and Technology and School of Science at Indianapolis, IN 46223, U.S.A. (Received 2 I
A ugw 1984)
INTRODUCTION
First, the solution of the problem of torsion of a solid semicircular cylinder is obtained in closed form. Next, employing a special set of multibipolar coordinate systems the problem of torsion of a perforated semicircular cylinder is formulated. In particular, numerical results for the cases of semicircular cylinders with one circular cavity are presented for various geometrical configurations. METHOD OF SOLUTION
In recent years torsion of multihole circular cylinders as well as torsion of prismatic bars with reinforced cavities have been investigated by Ling [ 11 and Kuo and Conway [Z-4]. These authors employed Howtand functions [S] in order to obtain the solutions of the aforementioned problems. Using another technique, the problem of torsion of a rectangular bar with two symmetrical circular cavities was recently solved by the author [6]. The technique employed in this investigation is quite different from those mentioned previously. Consider a prismatic bar whose cross-section is either a solid or a perforated semicircle as shown in Figs. I(a) and (b). The nondimensional polar coordinates p = r/i?, 0 are chosen for the first step of the analysis. According to the St, Venant’s theory for torsion of prismatic bars 171the equation p+
= -2,
a* 1 p=_+_.._+__ at+
(1)
a
I a*
P dP
P2 ae2
(for polar coordinates)
must be satisfied and the condition 5=0
(2)
on the outer boundary
has to be fulfilled. For the case of the perforated region, the following additional conditions also must be met: +
on the boundary of each inner cutout,
= K,,,
-2 x (nondimensional area of each cavity). Here in relations eqns (3) and (4) K,,, are constants, dS is the dimensionless element of arc length on the inner boundary C, and ii is the direction normal to that boundary. The closed form solution for u solid semicircular section
First, the right-hand side of eqn ( 1) is expanded in Fourier sine series to obtain a*V + I aF + --= 1 a*Q
ap* Pap
P2 de*
- 2 B_+3,5 2 (
>
sin ne.
(5)
t The author wishes to thank the Department of Compulins Services of WPlJf for providing computer time for this probject.
A. K. N.xittm
156
Fig,
t f Semicircular bars with solid and perforated cross-sections,
Next, the sofution of eqn (5) is sought in the form
The expression (6), which obviously satisfies the condition q = 0 along the diameter MAN [see Fig. l(a)], is substituted in eqn (5). The-integration of the resulting ordinary differential equations and the consideration that 9r = 0 at p = 1 finally lead to:
It is known [g-10] that FikJ, 4) =
F2(p,
$1
1 cash A -I- cos d, p” cos n+ = i: ‘;iln cash h - cas b, ’ n= 1.3.5 n
=
+ Arctan (G(h, 4)) + Arctan (GOI, n - 4))
(I+
1,
m
coshh)tan$
h = -In p, G(X, *) = cos g-z n=l.3.5
n$
n-
’
f>p>O,
1 i In cot $ , rt,#O,n (
n
sin n$ 5
sinh h
1
v
1.3.5 -=;i:*n
Empioying method of partial fractions and utilizing the relations given in eqn 18) the closed form solution for the torsion of a solid semicircular bar is derived. In the foi-
I57
Solid and perforatedsemi-ci~ula~cylinders
lowing the explicit expressions for vi1 warping function &, and shear stress ;izt,are given:
+ (P -k)
SinB}
- $p2(1 - ~0~2%
I >p>o,
V1=~{F,[2-(~+p’)c0~28]-~t(~-p’)sin28
+
ire=
(P
+
$)
--=Ml ap
COStI}
-
--
fp2sin28,
1 >p>O,
FAp, 0) cos 29
P sin 8 f
+ Fdp, 0) sin 28
1>
p >
0,
8
= - - sin 8, 3n p*a
;iLe Ip-4 = - -aT,
ap
- f [2 sin 9 - 2F,(l, 9) sin 281 +I
-
cos
28,
8 z 0, T.
It should be mentioned that the expression for - dvrlap ly-r~in eqn (9) has been directly obtained from the series solution eqn (7). The nondimensional torsional rigidity b is obtained from
(IO)
b = fo” 1,’ p2 $$ dp de.
Employing eqn (7) into eqn (10) and using a similar procedure for summing up the resulting series, it is found
(11) The actual value of shear stress Q, is obtained from Q = the appIied torque. In particular 72%
=
i&
. TIR3, in which T is
J-_w - 2) rr2 - 8 ’
R3
p-1 B-d2
T 728
=:T_=-
p-0
e-d2
R3
its 5
p-o
d-42
T = R3 3(m2’16-8) ’
(12)
Differentiating ;ite IPi in relation eqn (9) with respect to 8, and setting the result equal to zero, it is seen that the location of the maximum shear stress along p = 1 is at 0 = n/2. The highest value of shear stress in the semicircular cylinder occurs at p = 0, 8 = 42 as can be seen from relations eqn (12). It is interesting to note that the maximum
A. K.
158
NAGHDI
shear stress in a solid semicircular bar is approximately 4.48 times that of a solid circular bar of the same radius. It is aIso found that the shear stress T,~ becomes zero at p = 0.48022, 0 = nf2. Solution for- a perforated
senzicircular section
Consider a semicircular region with circular holes symmetrically located with respect to the y axis as shown in Fig. l(b). A set of complementary solutions of eqn ( I) in multibipolar coordinate systems are chosen as follows:
+
fe”“2
-
@n(28-rt2’]
c()s&
- f&6 - @*@+I)] cos&
+
.=* +
- [&
[em%
-
@(*@-TV)]
- @3-?;)]
cosfib
(13)
_ .*. _ [&
_
en(2P-?4},
in which 5, and rli are the bipolar coordinates measured with respect to rectangular coordinate system Xi and Y, [see Fig. I(b)] and are given by [ 1l] -5’=Arctan~2+*;2~~F2, 1
+yi=;, I
(14)
1 (xi f Z)2 + yf z - ’ = 5 sinb a , i=l,2,3 9i = - fn -E R 2 (Xi - Cl* + Yi!’
,a..
N.
Here in eqn (14) p is the common value of all r\is on the semicircular outer boundary of the bar. o! and p are obtained from the following relations [I 11:
P = cash - ‘(a cash a + U), ci = cash-’
1 _ if2 - p
(15) ) 5=4f.
2ZP
R’
z=Q
R’
It should be noted that the coefficient of each x,, in the complementary automatically satisfies the homogeneous condition on the semicircular should also be noted that the origins of the prime coordinates such as e;, the reflections of those of St, 31, (2,312. In fact, the combination of each such as
[em% - emc2~-VJ~0s nti - [em%_
&*e-$)I
cos
ny
solution (13) boundary. It qi, [i, -q; are pair of terms
(16)
produces an odd function withrespect to y having a zero value along the diameter of the semicircle. Adding 9, and Y2 in order to obtain the solution for a perforated semicircular bar, and employing the condition (4) it is found: A,, = 0.
(17)
The remaining condition to be_satisfied by3 = T, f q2 is (3). The constants Kt , Kz, * * * are evaluated along with A I, AZ) . . . A,, by satisfying the mentioned condition@) on the boundaries of the inner circular holes. In order to achieve this goal, p terms in the series solution (13) are retained and the boundary condition(s) are satisfied at q points (q > p) of the boundary (or boundaries) of the inner circular cutouts. This procedure leads to a set of q x p linear algebraic equations which are normalized and solved approximately by the technique of least square error [12]. For all the numerical results presented here q and p are chosen as 35 and 24 respectively. The obtained results are remarkably accurate. For example, for a case of a semicircular bar with one hole along the y axis the maximum value of relative error in satisfaction of
Solid and perforated semi-circular cylinders
159
Table 1. The values of dimensionless shear stress sze = T,s/E and dimensionless torsional rigidities E for
various F and ii
z
z
0.35 0.35
0.15
:: 0:so 0.50 0.50 :::
0.20 0.25 0.20 0.25 0.30 0.35 0.25 0.30
r:e at A
& at B
T:~ at C
$8 at E
Torsional rigidity D
- 2.8855 -3.021 I -2.9416 -2.7212 - 2.6988 - 2‘7293 - 2.8700 - 2.8266 - 2.88194
- I .a879
0.2287
- 2.3750 - 2.2184 -0.7551 - 1so876 - i .4586
0.4963 I.1620 1.6387 I .%32 2.3526 2.8781 2.8%7 3.7002
2.4150 2.4130 2.4795 2.5921 2.7245 2.9429 3.31IS 3.2488 3.9021
0.30057 0.30034 0.30091 0.29774 0.29430 0.28655 0.27213 0.27266 0.25395
- 1.9190 - 1.6311
- 0.4363
the inner boundag condition is of the order of IO”‘*. The values of dimensionless torsional rigidity D for a hollow bar is numerically determined by a highly accurate eight order polynomial approximation for numerical integration [ 121. In Table 1 the values of dimensionless shear stresses S;* = S&is at points A, B, C, E [see Fig. I(c)] as welt as the nondimensional torsional rigidities E are presented for various Z and i?. It is seen that for lower values of Z and B the maximum shear stress occurs at point A. However, for higher values of T and Z the maximum shear stress is shifted to point E. REFERENCES 1. C. 8. Ling, Torsion of a circular tube with logitudinal circular holes. Q. uppl. Muth. 5, 168-181 (1947). 2. Y-M. Kuo and H. D. Conway, Torsion of cylinders with multiple reinforcement. J. En~n# Meclr. Dip. ASCE 100, EM2. Proc. Paper 10460, 221-234 (April 1974). 3. Y-M. Kuo and H. D. Conway, The torsion of composite tubes and cylinders. Inr. J. Solids Srrucfures 9, 1553-1566 (1973). 4. Y-M. Kuo and H. D. Conway. Torsion of reinforced square cylinder. J. Engn# Me&. Div. ASCE 166, EM6. 1341-1347 (December 1980). 5. R. C. J. Howland, Potential functions with periodicity in one coordinate. Proc. Comb. Phi/. Sm. 30, 315-326 (1935). 6. A. K. Naghdi, Torsion of a rectangular bar with two circular holes. J. Engn~ Me&. Dk. ASCE 109, 643-648 (April 1983). 7. S. Sokolnikoff, U~tll~~~~li~fl~ Themy ofElust~~ily. McGraw-Hill, New York (1956). 8. I. S. Gradshteyn and I. M. Rythik, Tut& ofifl~e~ru1.s Series und Prodrrtts. New York and London (l%5). 9. L. V. Kantorovich and V. 1. Krylov, Approxinzote Methods of Higher Analysis. Interscience, New York (1964). IO. A. K. Naghdi and J. M. Gerstung, Jr., The effect of a transverse shear acting on the edge of a circular cutout in a simply supported circular cylindrical shell. Ing.-Arch. 42, 141-150 (1973). Il. H. Bateman. Partiul Differential Eqmriotrs of Mothcmutical Physics. Dover, New York (1944). 12. F. B. Hildebrand. Inrrodrrc/ion IO N/rmericul Anul\*sis. McGraw-Hill. New York (1956).
1. Solids Strurfurrs
Printed
Vol. 21. No. 2, pp. 135-139, VSS
0020-7683iss $3.M)+ .oo 0 1985 PergamonRess Ltd.
in the U.S.A.
TORSION OF SOLID AND PERFORATED SEMICIRCULAR CYLINDERS? A. K. NAGHDI Department of Ae~~utic~-As~~~uti~ E~n~~g and Ma#ema~i~ Sciences, Purdue Unjve~ity SchooI of Engineering and Technology and School of Science at Indianapolis, IN 46223, U.S.A. (Received 2 I
A ugw 1984)
INTRODUCTION
First, the solution of the problem of torsion of a solid semicircular cylinder is obtained in closed form. Next, employing a special set of multibipolar coordinate systems the problem of torsion of a perforated semicircular cylinder is formulated. In particular, numerical results for the cases of semicircular cylinders with one circular cavity are presented for various geometrical configurations. METHOD OF SOLUTION
In recent years torsion of multihole circular cylinders as well as torsion of prismatic bars with reinforced cavities have been investigated by Ling [ 11 and Kuo and Conway [Z-4]. These authors employed Howtand functions [S] in order to obtain the solutions of the aforementioned problems. Using another technique, the problem of torsion of a rectangular bar with two symmetrical circular cavities was recently solved by the author [6]. The technique employed in this investigation is quite different from those mentioned previously. Consider a prismatic bar whose cross-section is either a solid or a perforated semicircle as shown in Figs. I(a) and (b). The nondimensional polar coordinates p = r/i?, 0 are chosen for the first step of the analysis. According to the St, Venant’s theory for torsion of prismatic bars 171the equation p+
= -2,
a* 1 p=_+_.._+__ at+
(1)
a
I a*
P dP
P2 ae2
(for polar coordinates)
must be satisfied and the condition 5=0
(2)
on the outer boundary
has to be fulfilled. For the case of the perforated region, the following additional conditions also must be met: +
on the boundary of each inner cutout,
= K,,,
-2 x (nondimensional area of each cavity). Here in relations eqns (3) and (4) K,,, are constants, dS is the dimensionless element of arc length on the inner boundary C, and ii is the direction normal to that boundary. The closed form solution for u solid semicircular section
First, the right-hand side of eqn ( 1) is expanded in Fourier sine series to obtain a*V + I aF + --= 1 a*Q
ap* Pap
P2 de*
- 2 B_+3,5 2 (
>
sin ne.
(5)
t The author wishes to thank the Department of Compulins Services of WPlJf for providing computer time for this probject.
A. K. N.xittm
156
Fig,
t f Semicircular bars with solid and perforated cross-sections,
Next, the sofution of eqn (5) is sought in the form
The expression (6), which obviously satisfies the condition q = 0 along the diameter MAN [see Fig. l(a)], is substituted in eqn (5). The-integration of the resulting ordinary differential equations and the consideration that 9r = 0 at p = 1 finally lead to:
It is known [g-10] that FikJ, 4) =
F2(p,
$1
1 cash A -I- cos d, p” cos n+ = i: ‘;iln cash h - cas b, ’ n= 1.3.5 n
=
+ Arctan (G(h, 4)) + Arctan (GOI, n - 4))
(I+
1,
m
coshh)tan$
h = -In p, G(X, *) = cos g-z n=l.3.5
n$
n-
’
f>p>O,
1 i In cot $ , rt,#O,n (
n
sin n$ 5
sinh h
1
v
1.3.5 -=;i:*n
Empioying method of partial fractions and utilizing the relations given in eqn 18) the closed form solution for the torsion of a solid semicircular bar is derived. In the foi-
I57
Solid and perforatedsemi-ci~ula~cylinders
lowing the explicit expressions for vi1 warping function &, and shear stress ;izt,are given:
+ (P -k)
SinB}
- $p2(1 - ~0~2%
I >p>o,
V1=~{F,[2-(~+p’)c0~28]-~t(~-p’)sin28
+
ire=
(P
+
$)
--=Ml ap
COStI}
-
--
fp2sin28,
1 >p>O,
FAp, 0) cos 29
P sin 8 f
+ Fdp, 0) sin 28
1>
p >
0,
8
= - - sin 8, 3n p*a
;iLe Ip-4 = - -aT,
ap
- f [2 sin 9 - 2F,(l, 9) sin 281 +I
-
cos
28,
8 z 0, T.
It should be mentioned that the expression for - dvrlap ly-r~in eqn (9) has been directly obtained from the series solution eqn (7). The nondimensional torsional rigidity b is obtained from
(IO)
b = fo” 1,’ p2 $$ dp de.
Employing eqn (7) into eqn (10) and using a similar procedure for summing up the resulting series, it is found
(11) The actual value of shear stress Q, is obtained from Q = the appIied torque. In particular 72%
=
i&
. TIR3, in which T is
J-_w - 2) rr2 - 8 ’
R3
p-1 B-d2
T 728
=:T_=-
p-0
e-d2
R3
its 5
p-o
d-42
T = R3 3(m2’16-8) ’
(12)
Differentiating ;ite IPi in relation eqn (9) with respect to 8, and setting the result equal to zero, it is seen that the location of the maximum shear stress along p = 1 is at 0 = n/2. The highest value of shear stress in the semicircular cylinder occurs at p = 0, 8 = 42 as can be seen from relations eqn (12). It is interesting to note that the maximum
A. K.
158
NAGHDI
shear stress in a solid semicircular bar is approximately 4.48 times that of a solid circular bar of the same radius. It is aIso found that the shear stress T,~ becomes zero at p = 0.48022, 0 = nf2. Solution for- a perforated
senzicircular section
Consider a semicircular region with circular holes symmetrically located with respect to the y axis as shown in Fig. l(b). A set of complementary solutions of eqn ( I) in multibipolar coordinate systems are chosen as follows:
+
fe”“2
-
@n(28-rt2’]
c()s&
- f&6 - @*@+I)] cos&
+
.=* +
- [&
[em%
-
@(*@-TV)]
- @3-?;)]
cosfib
(13)
_ .*. _ [&
_
en(2P-?4},
in which 5, and rli are the bipolar coordinates measured with respect to rectangular coordinate system Xi and Y, [see Fig. I(b)] and are given by [ 1l] -5’=Arctan~2+*;2~~F2, 1
+yi=;, I
(14)
1 (xi f Z)2 + yf z - ’ = 5 sinb a , i=l,2,3 9i = - fn -E R 2 (Xi - Cl* + Yi!’
,a..
N.
Here in eqn (14) p is the common value of all r\is on the semicircular outer boundary of the bar. o! and p are obtained from the following relations [I 11:
P = cash - ‘(a cash a + U), ci = cash-’
1 _ if2 - p
(15) ) 5=4f.
2ZP
R’
z=Q
R’
It should be noted that the coefficient of each x,, in the complementary automatically satisfies the homogeneous condition on the semicircular should also be noted that the origins of the prime coordinates such as e;, the reflections of those of St, 31, (2,312. In fact, the combination of each such as
[em% - emc2~-VJ~0s nti - [em%_
&*e-$)I
cos
ny
solution (13) boundary. It qi, [i, -q; are pair of terms
(16)
produces an odd function withrespect to y having a zero value along the diameter of the semicircle. Adding 9, and Y2 in order to obtain the solution for a perforated semicircular bar, and employing the condition (4) it is found: A,, = 0.
(17)
The remaining condition to be_satisfied by3 = T, f q2 is (3). The constants Kt , Kz, * * * are evaluated along with A I, AZ) . . . A,, by satisfying the mentioned condition@) on the boundaries of the inner circular holes. In order to achieve this goal, p terms in the series solution (13) are retained and the boundary condition(s) are satisfied at q points (q > p) of the boundary (or boundaries) of the inner circular cutouts. This procedure leads to a set of q x p linear algebraic equations which are normalized and solved approximately by the technique of least square error [12]. For all the numerical results presented here q and p are chosen as 35 and 24 respectively. The obtained results are remarkably accurate. For example, for a case of a semicircular bar with one hole along the y axis the maximum value of relative error in satisfaction of
Solid and perforated semi-circular cylinders
159
Table 1. The values of dimensionless shear stress sze = T,s/E and dimensionless torsional rigidities E for
various F and ii
z
z
0.35 0.35
0.15
:: 0:so 0.50 0.50 :::
0.20 0.25 0.20 0.25 0.30 0.35 0.25 0.30
r:e at A
& at B
T:~ at C
$8 at E
Torsional rigidity D
- 2.8855 -3.021 I -2.9416 -2.7212 - 2.6988 - 2‘7293 - 2.8700 - 2.8266 - 2.88194
- I .a879
0.2287
- 2.3750 - 2.2184 -0.7551 - 1so876 - i .4586
0.4963 I.1620 1.6387 I .%32 2.3526 2.8781 2.8%7 3.7002
2.4150 2.4130 2.4795 2.5921 2.7245 2.9429 3.31IS 3.2488 3.9021
0.30057 0.30034 0.30091 0.29774 0.29430 0.28655 0.27213 0.27266 0.25395
- 1.9190 - 1.6311
- 0.4363
the inner boundag condition is of the order of IO”‘*. The values of dimensionless torsional rigidity D for a hollow bar is numerically determined by a highly accurate eight order polynomial approximation for numerical integration [ 121. In Table 1 the values of dimensionless shear stresses S;* = S&is at points A, B, C, E [see Fig. I(c)] as welt as the nondimensional torsional rigidities E are presented for various Z and i?. It is seen that for lower values of Z and B the maximum shear stress occurs at point A. However, for higher values of T and Z the maximum shear stress is shifted to point E. REFERENCES 1. C. 8. Ling, Torsion of a circular tube with logitudinal circular holes. Q. uppl. Muth. 5, 168-181 (1947). 2. Y-M. Kuo and H. D. Conway, Torsion of cylinders with multiple reinforcement. J. En~n# Meclr. Dip. ASCE 100, EM2. Proc. Paper 10460, 221-234 (April 1974). 3. Y-M. Kuo and H. D. Conway, The torsion of composite tubes and cylinders. Inr. J. Solids Srrucfures 9, 1553-1566 (1973). 4. Y-M. Kuo and H. D. Conway. Torsion of reinforced square cylinder. J. Engn# Me&. Div. ASCE 166, EM6. 1341-1347 (December 1980). 5. R. C. J. Howland, Potential functions with periodicity in one coordinate. Proc. Comb. Phi/. Sm. 30, 315-326 (1935). 6. A. K. Naghdi, Torsion of a rectangular bar with two circular holes. J. Engn~ Me&. Dk. ASCE 109, 643-648 (April 1983). 7. S. Sokolnikoff, U~tll~~~~li~fl~ Themy ofElust~~ily. McGraw-Hill, New York (1956). 8. I. S. Gradshteyn and I. M. Rythik, Tut& ofifl~e~ru1.s Series und Prodrrtts. New York and London (l%5). 9. L. V. Kantorovich and V. 1. Krylov, Approxinzote Methods of Higher Analysis. Interscience, New York (1964). IO. A. K. Naghdi and J. M. Gerstung, Jr., The effect of a transverse shear acting on the edge of a circular cutout in a simply supported circular cylindrical shell. Ing.-Arch. 42, 141-150 (1973). Il. H. Bateman. Partiul Differential Eqmriotrs of Mothcmutical Physics. Dover, New York (1944). 12. F. B. Hildebrand. Inrrodrrc/ion IO N/rmericul Anul\*sis. McGraw-Hill. New York (1956).