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Torsion of beams whose sections are bounded by certain quartic curves
Y.Me&. Phys.solids, 1959, w. 7,pp.272to281.Pergamon
TORSION OF BEAMS WHOSE
Press
Ltd., London.Printed inGreatBritain
SECTIONS ARE BOUNDED
BY CERTAIN QUARTIC CURVES By W. A. BASSALI Faculty
of Science, I’niversity
of Alexandria
CAUCHY integral methods are applied to the torsion of a solid isotropic cylinder whose section is bounded by a quartic curve which is the inverse of an ellipse with respect to any point on its major or minor axis. The complex torsion function, the torsional rigidity and the shearing stresses are explicitly determined in the general case, and particular values are found to agree with those already known for Booth’s lemniscate and the elliptic limacon. The distribution of shearing stress on the boundary is computed in two particular examples.
1. INTRODUCTION MANY solutions have been given to the classical Saint-Venant’s torsion problem for various forms of boundary including polygons, angles, cardioids, limacons, sectors, lemniscates and others ; references are to be found in standard textbooks by LOVE (1944), TIMOSHENKO and GOODIER (1951), SOKOLNEOFF (1956) and MUSKHELISHVILI (1949). Special techniques have been developed by many investigators to treat specific problems. A complete bibliography dealing with the theory of elastic cylinders in torsion and including conventional as well as complex variable methods is contained in a recent Chinese book* (1956) which presents a resumb of almost all existing material relating to this rather special field of elasticity theory. The Schwarz-Christoffel transformation has been used by TREFFTZ (1921) and SETH (1934) to discuss the torsion of beams whose section is a regular rectilinear polygon of n sides. The Cauchy integral methods described by ~USKHEL~SHVI~ (1949, p. 561) wereusedby 1.S.andE.S. SOKOLNIKO~(~~~~)~ discuss a section bounded by two circular arcs of equal radius and intersecting at right angles. A section with the shape of Booth’s lemniscatet has been considered by MUSKEIELISHVILI (1949, p. 581) and in a different but equivalent manner by HIGGINS (1942). STEVENSON (1939),and HOLL and ROCK (1939) treated an ellipticlimacont section. The corresponding problem for a hyperbolic limacon was solved by LIN and WHITEHEAD (1951) and by LIN and YANG (1951). Recently MORRIS and HAWLEY (1958) have developed new complex variable methods to deal with the torsion and flexure of solid cylinders with cross-sections transformable to a ring-space. The particular cases of the elliptic section and a section having a circular arc as internal cut are treated. *See the review of this book in Rppl. Meeh. R#ieus 1958. II, 110. +Booth’s lemniscate ix the inverse of an ellipse with respect to its centre,whiie from
inversion
~~~KHELr6x~lLr
of an eILipse with 1949,
respect
to its focus.
The
p. 175).
272
latter
the elliptic limaeon is the curve resulting
is sometimes
referred
to as Pascal’s
Iimncon
Torsion of beams whose sections are bounded by certGn quartic curves
273
STEVENSON(1943) has considered a grooved column of a special type. The boundary curve, taken in the z-plane, can be mapped on the unit circle y in the b-plane by the mapping function z = cc (1 + m <“),
(1.1)
where c and m are taken to be real and positive constants subject to the restriction 0 < m (n + 1) < 1. Sections bounded by the elliptic limacon or the cardioid are included as particular cases. Recently the author (1959a) has obtained solutions for sections bounded by regular curvilinear polygons that can be mapped on y by the transformation c > 0, 2 = cc/(1 + ?n 5% (1.2) where n is a positive integer and - 1 < m (,n - 1) < 1. This mapping function has been thoroughly studied by the author (195913) where varieties of figures corresponding to different values of m, n and having several axes of symmetry are drawn. For 91,= 2 the section has the shape of Booth’s femniscate, while for 7n = - 2/{m@ - 1)} it was shown that the cross-section is bounded by ?z approximately circular and equal arcs. The present paper is concerned with sections bounded by quartic curves which are mapped on y by the function x = ci/(l
+ n5 + m 5%
c > 0,
(1.3)
where m, n are real and Im] < 1, frz] < 2. It will be shown in Section 3 that the bounding curve is always the inverse of an ellipse with respect to a point on its major or minor axis. For n = 0 we have Booth’s lemniscate while if m > 0 and n2 = &n the section is an elliptic limacon. 2.
FUNDAMENTALEQUATIONS
Consider the Saint-Venant torsion of a homogeneous isotropic cylinder of uniform section S bounded by a simple closed curve r in the x-plane (z = x + iy = r eiB) which is chosen to coincide with the plane Z = 0 perpendicular to the generators of the cylinder. It is known (STEVENSON1943) that the displacements U, ZI,and rp! and the non-vanis~~ing stresses ~2, 2
are given by
U + iv = iTZ2, ze,= 7 f/ (a, y) = 4 7 [D(x)
iii3- iy’-i= p-r
p
(z) -
ii],
+ Q(Z)],
(2.1)
(2.2)
where 7 is the constant twist per unit length, p is the rigidity of the material of the cylinder and Q (z) = 46(z, y) + i # (m,y) is th e complex torsion function which satisfies the boundary condition Q(z) -
D (5) = izi along 1”;
(2.3)
bars are used to denote conjugate complex quantities and accents designate differentiation with respect to the stated argument. Let z = x(t), 5 =peie (2.4) be the function which maps the region S on the circle 161< 1. With the aid of this mapping function curvilinear coordinates (p, 6) are introduced into the x-plane.
W. A. BASSAM
274
The function L?(z) can be written in terms of <, so that $ _t i # = Q (x) = w (<),
(2.5)
where o (6) is regular inside y. It is found (MUSKHELISINILI f.o((,-)
s
x(u) z;(u-1) =$T ~-da + Y
o-5
1949, p. 577) that
constant.
The twisting couple N is furnished by N = rD, L’ = P (I + J),
(2.7)
where I) is the torsional rigidity of the cylinder, 1 is the polar moment of inertia I
XZdS for the cross-section, or I = -
Hi
and J=$
z (I?) x’ (u) zs {u-l) d G, s Y
(2.3)
z (0) B (61) w’ (u) d (T. J‘ Y
It was shown by MORRIS and HAWLEY is (of course) real.
(1958,
(2.9)
p. 469) that the expression
I +
J
The shearing stresses ~2 and 8^z at any point of the cross-section are given by (2.10) and substituting from (2.2) in (2.10) we get the convenient formula P^z-
i E
= [IL7 UPjZ (z;,]] [w’ (5) -
3.
MAPPING
iz’ (5) 2 (%,].
(2.11)
FUNCTION
STEVENSON (1943) and the author (1959a) have solved the torsion problem for sections which are regular curvilinear polygons having n sides and ri. rounded vertices (which may become cusps). The boundaries of the sections, taken in the z-plane, are mapped on y by the conformal transformations
z = CJ(1 + m {“f, e > 0, 0 < m (I&+ 1) < 1,
(3.1)
z = cc/(1 + m Se), c > 0, -
(3.2)
and 1 ,( m (n -
1) < I,
where m, n, are two real constants subject to the mentioned restrictions. consider the section which is mapped on ]{I < 1 by z = cl/Cl + n5 + mC2), c > 0,
We here
(3.3)
where n, m are taken to be real parameters, and, in order that the transformation shall be eonformal at all points within the boundary F, z’ (5) must not vanish or become infhrite at points within r, and we therefore have the condition -l
(3.4)
Torsion of beams whose sections are bounded by certain quartic curves
275
If P’ is the inverse of r with respect to the circle 1x1= d and x’ = x’ + iy’, x = x + &Jare the points on P, r, respectively, that correspond to 5 = e@ on y, then
(34 and inserting (3.5) in (3.3) leads to XI -f
= OY(do+ rne-y/c,
where f =
n d2,k.
(34 (3.7)
From (3.3) it follows immediately that 2” is the ellipse (Lr:’-
f)"/a" + p//%2= 1,
(3.8)
where a = (1 + nt) @/c,
Fig. 1.
b =
(l
-
m) rP/c.
(3.9)
inverses of eltipse with respect to points on major axis.
The contour P mapped on y by means of (3.3) may thus be identified with the inverse of the ellipse I” whose axes are Za, 26 with respect to a point on the first axis distant -f from the centre of the ellipse, where, using (3.7) and (3.9),
W. A.
276 n =
2fl(a
+
m
b),
=
(a -
BASSALI b)/(a
d being the radius of the circle of inversion.
+
It is sufficient to consider The parametric
2 (
either positive
equations
2fP/(a
+ b),
(3.10)
If (3.3) is to map the area inside
on to the area inside y then the centre of inversion equation of (3.10) gives the additional condition -
c =
b),
‘I1 <
must lie inside r’
r
and the first
(3.11)
2.
or negative
values of n.
of I’ are
cx/r2 = n + (1 + m) cos 8,
cy/‘r2 = (I -
m) sin 0,
(3.12a)
where c2/r2 = 1 + TL~+ m2 + 2tt (1 + m) cos B + 2m cos 28. The Cartesian and polar equations by
(nx2 + ,ny2 -
of the quartic
curve
(3.12b)
r are therefore
furnished
~a)~/(1 + m)” + c2y2/( 1 ~ m)” = (x2 + y2)2,
(3.13)
and (92~-
c cos 0)2/(1
+
?a)2 + c2 sin2 O/(1
-
m)” 2 r2,
(3.14)
respectively. The two curves corresponding to m = 0.2, n = - 0.2 and m = 0.2, n = - O-6 are illustrated in Fig. 1. They are the inverses of the two ellipses (x + 1)2/36
Fig. 2.
+ y2/16 = 1 and (x + 3)2/36
+ y2/16 = 1,
Inverses of ellipse with respect to points on minor axis.
respectively, with respect to the origin. For negative values of m we have a < b and the centre of inversion lies on the minor axis of the ellipse. Fig. 2 shows the two cross-sections for which m = - 0.2, n = - 0.2 and m = - 0.2, n = -0.4. These are bounded by the inverses of the ellipses (x + 1)2/16 with respect to the origin.
+ y2/36 = 1 and (x + 2)2/16
+ y2/36 = 1
Torsion of beams whose sections are bounded by certain quartic curves
277
For m = 0 the contour r becomes Booth’s lemniscate while if m > 0 (a > b) and f = & d(a2 - ba), so that the centre of inversion coincides with one or other of the two foci of the ellipse r’, then equations (3.10) yield 12= -J2#m.
(3.15)
In this case the mapping function (3.3) takes the form (3.16)
25= c5/(1 + mV, and substituting from (3.15) in (3.14) we obtain, after simplification, elliptic limacons r = A f B cos 0, where A = c (1 + m)/(l - m)2, B = 2c m*/(l - m)“. 4.
C2
2rr
I
v(l
(3.17a) (3.17b)
COMPLEX TORSION FUNCTION
Substituting from (3.3) in (2.6) we find that the torsion function a section mapped on ([I < 1 with the aid of (3.3) is furnished by
w(l) =-
the two
which solves the problem for
a2 do + ncr + m02) (02 + 12(I + fn) (0 -
(4.1)
1) + constantP
where 5 is any point inside y. It is easily shown that, under the restrictions (3.4) and (3.11) on m and n, respectively, the integrand of (4.1), considered as a function of (I, has two simple poles I&, 5s that lie outside y, where 1r + f;s = a;::2
n@,
1r 1s = I/m,
(4.2)
.+ 4m. For large values of D the integrand in (4.1) is of the order 1/1013, and we therefore w (1) = -
c2 i (Ri + R,) + constant,
where R, and R, are the residues of the integrand
w(1)= -
c2 i (&
-
t2)
at o = & and o = [s, respectively.
h2 m [ (Cl - 5) (t12+
c22 12 C1+
m) - (t2 - I) (Cz2+ n C2+ m)1 +
Hence
constant’ (4’3)
and, making use of (4.2), it is found that (4.3) simplifies to
Cd(4.1 =
c2 i (1
_
m)
k
1
fm
1
+
+mn5
nJ
mp
+
+
constant*
(4.4)
where k = (1 _t m)2 -
n2.
(4.5)
In the special case of an elliptic-limacon section n 2 = 4m and the integrand in (4.1) has a double pole outside y at [,, = F m-f. If R is the residue of the integrand at D = &, in this case then OJ(1) = -
c2i R + constant,
and it can be shown that the result agrees with (4.4) after substitution from (3.15). Taking for the constant in (4.4) the value - c2 (1 + m) i/2 (1 - m) k, one finds for the complex torsion function :
w(I) =-
c2iq-nc-mq[2
2k 1 +nl+m[2’
(4.6)
where
q = (1 + m)/(l
-
m) = a/b.
(4.7)
1%'. A.
278
BASSALI
The imaginary part of (4.6) is +“z
p (1 - m2 p4) - n2 p2 + n (q -
1) p
(1
-
p2) cos
e
(4.8)
2k 1 + n2 p2 -C_m2 p4 + 2n p (1 + ‘m p2) cos 0 + 2m p2 cos 20’
Setting p = 1 in (4.8), substituting for q from (4.7), and using (:J.l2b), me see that, on the boundary r, IJ reduces to &z, as it should. TORSIONAL
5.
RIGIDITY
The polar moment of inertia I for the section is obtained by substitution from (3.3) in (2.8) 1 =
_
CT3(1
lic4
4
f
- ml?) do
v(02 + n D + .m)2 (1 + n O +
:
(5.1)
mo3)3’
If X, and X2 are the two roots of the equation u2 + n m + m = 0, so that h, h, = ‘111, jm < 1
; A, + A, = - n,
11q< 2,
(5.2)
then X, and h, lie inside y and 1 = ?j 7rc4 (PI
-1 P2),
(5.3)
where P, and P, are the residues of the integrand in (5.1) at CJ= X, and (J = h,, respectively. It can be proved that P,
=
3x,2 (1 -
(X, -
m X,2)2
2 (1 -
h,)2 (1 + n h, -t m h,2)4 -
(h, -
m2) x,3
h2)3 (1 + n h, +
(5.4)
m A,2)3’
while P, is derived from (5.4) by interchanging h, and X2. Applying (3.2) we find after considerable reduction that* (1 - n2m + ffl.2 +
th3)3
ms(2 - 6
+
2 (I -
+ m + mP)2 +
m)4
m (1
+
m)2
m3)
I
(1 + m) (1 i(1 - m)2 where k is given by (4.5). Introducing (3.3) and the torsion function expression (1 -m)k
We therefore have
s
defined by (4.6) and (4.7) in (2.9) leads to the
m
(1 + m) o3 + 4% o2 (1 + m2 CJ~)
Jo2
+ n 0 + m) (I + 110 + m 02)s
c4 i
Jz--
(5.5)
do.
(3.6)
J = 271~~(Q1 + Q,)/U - m)k,
(6.7)
where Q,, the residue of the integrand in (5.6) at CJ= X,, is given by m2
)-a+
Q
1
(A,
-
A,)
and Q2 is deduced from Q1 by interchanging algebraic manipulation that
X12)Al2 + (1
+
m (1 + nk)
n A,
+
XI3
(5.8)
m A,2j3
(3.2) it is found after some
h, and X2. Applying
.
(5.9)
I When (5.5) and (5.9) are inserted in (2.7), it is seen that the torsional rigidity is? D _ prc4 k4
3 (1 - n2 m + m2 + 2?r~~)~+ 3m2 (2 - n2 + m + ms)2 - 4m2k2 2(1 - m)4
[
+ (1 + m)2 3m + 1 . *It can be shown that the I =
7x4 [(l + m)4 (I
t.411equivalent
expression
for I may
t 4m2 + m4) + 2 (1 + ?Q
and simpler
D = &L”C4[(I
expression
-
(1
+
m)
(1 + m3)
(1 - m,)2
*
(5.10)
be put in the form (1 -
6m + 4m= -
for the torsional
+ m4) (1 + m”)
n2 rn, -
rigidity
6rn3 ml m*) n2
+ en2n4]/!2(1 - my A4
is
4m (1 + m) (1 + m3) 7&Q+ 2mQ n4]/“
(I -
m)4 k4.
Torsion of beams whose sections are bounded by certain qua&c
curves
279
Setting n. = 0 in (5.10) and observing that (1 + m2 + 2rr~~)~+ m2 (2 + m + m3)2 = Yj(1 -
,m2)2 (I -
?n)* + 4 (1 + m2)2 (1 + m)4,
we find that n simplifies to r) = $ II” c4(f + .m4}/(1 - my*
(5.11)
which agrees with the result of MUSKHELISHVILI (1949, p. 582) for Booth’s lemniscate. Substituting from (3.15) in (5.10) yields the following expression for the torsional rigidity of either of the elliptic limaeons (3.17) :
D = gp7r4 (1 + t-w +- 1Gd + 8m3 + mQ)/(1 - m)s, which can be shown to be equivalent
(5.12)
to
D = &,,@A4
“r 8A2B2 + B4) I
(5.13)
on using the values of A and S given by (3.17b). The formula (5.13) was obtained by STEVENSON (1943, equation (3.4)). As a further check on the expression (5.10) we notice that for m = 0 it reduces to D = jpv d4, where c’ = c/(X - n2). This result is to be expected since in this ease it is easily seen that the curve I’ becomes a circle of radius c’. Putting m = n = 0 in (5.10) yields D = &we4 which is the torsional rigidity for a circular cylinder of radius c.
6. Substitution ;”
-
irz
SHEARING
STRESSES
from (3.3), (4.6) and (4.7) in (2.11) leads to = -
(1 _t np2 + mp2 t)2 {n<+ -
~7 ic T
i_ 2m (1 + rn,) + m2
n 4)
(1 --ml@
[
1,
-I- P (1 - m C2)(1 + 12t + m T2) where k is given by (4.5) and T = (1 + mm2 p4 -
(6.1)
2mp2 eos 2f?)t [I + n 2 pz i_ ms p4 + 2np (1 + mp*) eos l? + Zrnj.? cos 201. (6.2)
Putting 5 = p ei@ in (6.1) and separating real and imaginary parts it is found, after some straightforward computations, that the shearing stresses at any point of the cross-section are 2p Tc (1 - p2) pz=-----------(1 - m)kT
h
n {I + m (3m ‘_t k) p2 + ?a4 p*) sin B + m (1 +
nt) (1
i_ m2 p2) p sin 2Q + ma 72p2 sin 3B
I
9
(6.3)
J
,r;.JE: T
__L-[ (1 -ml&
1
2m (1 + m) (2m + ns) p3 f 2rh2p (1 f m3 p4) + ‘n 1 + 4m (1 -t m) p2(1 + mpe) i
$-m2np2(1 +p2)cos36
I
+p(l
-m2p*)
-+-np2(1
1.
-mp2)cosB
(6.4)
Since w (5) is regular throughout the cross-section S and z’ (5) does not vanish at any point within S by the restriction (3.4), it follows from (2.11) that the stresses (6.3) and (6.4) are necessarily &rite and physically admissible across S. Setting p = 1 in (G.3), (6.4) and substituting for T from (6.2) we And that, on the boundary r of the cross-section,
2
vanishes as expected 1*c7
(Q2h==1 = (1
_
m)k
while the peripheral
shear stress simplifies to
(1 + m) (1 + m2) t_ 2mn. cos 6 l/(1
+_ m2 -
2mcos26)
’
(6.51
W.
280
A.
BASSALI
From (6.5) it is seen that the maximum and minimum values of the tangential stress acting on the boundary I’ occur alternately at the three points B = 0, B = t(, and B = T, where dL= cos-1 [ and the magnitudes
11(1 + rrt)/Z (I + ~?G)],
(6.6)
of these vatues are furnis~~ed by (z),=l,o=o
= FLTC [(I -t_ V/i) (1 -t nr2) + 2VOL]/(l -
(z)p=l,@=a
= p7c [(l + m2)2 -
(G&p=1,8_n
= pre[(l
n%+/(l
+ ?I&)(1 + W&2)-
-
V&)212,
(6.7)
?n)li,
2nr?q/(l
-
(6.8) (9.B)
m)2h-.
Su~sti~uti~~ from (3.15) in (6.71, (6.9) and using the vnlucs of A and B defined by (Y.l?b) it can be verified that the resulting expressions agree with those obtained by STIWENSON (19&d, equations (3.7, (3.8)) for the peripheral shears of the elliptic-limacon section at the bottom of the groove and at the point where T becomes a maximum.
B Distribution
Fig. 3.
of shearing stress on ~)oun(~aries of sections.
Fig. 3 illustrates the vari&ion in peripheral shear on the boun[~aries of the two sections in Figs. 1 and 2 which correspond to m. = 0.2, n -7 - 0.6 and ~a, z - 0.2, n = - 0.4. ‘I’be points at which the stresses attain their maximum and minimum values on the two boundaries are also indicated. REFERENCES BASSALI, W. A.
1959a 1959b
J. Math. Pkys. In press. J, Me&. Pkys. Solids 7, 145.
1956
TheomJ of Elastic CyEi?xiers in Torsion J. Appt. Pkys. 13,457. 2. nngew. Math. Mech. 19, 141.
CHIEN, W. Z., Hu, H. c., LIN, R. S. and
YEH,
C.
Y.
HIGGINS, T. J.
HOI&, D. L. and ROCK, II. I%. LIN, T. C. and WHIRLED, LIN,
T.
c. and
1951
L. G. YANCJ,
II.
1942 1939
T.
1951
(Peking).
U~~~ers~~~ of ~,~a~k~~~~tor~, ~~g~~eer~~l~ ~xp~~rn~nt Station Series, Bulletin 118, 108. University of Washington, Engineering Experiment &ztion Series, Bulletin 118, 112.
Torsion of beams whose sections are bounded Love,
A. E. H.
1944
A
Treatise
by certain quartic curves
on the Mathematical
Theory
281
of Elasticity,
4th Ed. (New York). MORRIS, R. M. and HAWLEY, F. J. MUSKHELISHVILI, N. I. SETH,
B. B.
S~KOLNIICOFF. I. S.
1958 1949 1934 1953
Quart. J. Mech. Appl. Math:ll, 462. Som.e Basic Problems of the Mathematical Theory of Elasticity, 3rd Ed. (Moscow). Proc. Can& Phil. Soc,30, 139. Mathernalical Theory of Elasticity, 2nd Ed. (McGrawHill).
SOKOLNIKOFF, I. S. and SOKOLNIKOFF, E. S. STEVENSON, A. C.
1938 1939 1943
Bull. Amer. Math. Sot. 44, 384. Proc. Lond. Math. Sot. 45, 128. Phil. Mug. 34, 115.
TJMOSHENKO, S. and GOODIER, J. N. TREFFTZ, E.
1951 1921
Theory of Elasticity, 2nd Ed. (McGraw-Hill). Math. Ann, 82, 306.
TORSION OF BEAMS WHOSE
Press
Ltd., London.Printed inGreatBritain
SECTIONS ARE BOUNDED
BY CERTAIN QUARTIC CURVES By W. A. BASSALI Faculty
of Science, I’niversity
of Alexandria
CAUCHY integral methods are applied to the torsion of a solid isotropic cylinder whose section is bounded by a quartic curve which is the inverse of an ellipse with respect to any point on its major or minor axis. The complex torsion function, the torsional rigidity and the shearing stresses are explicitly determined in the general case, and particular values are found to agree with those already known for Booth’s lemniscate and the elliptic limacon. The distribution of shearing stress on the boundary is computed in two particular examples.
1. INTRODUCTION MANY solutions have been given to the classical Saint-Venant’s torsion problem for various forms of boundary including polygons, angles, cardioids, limacons, sectors, lemniscates and others ; references are to be found in standard textbooks by LOVE (1944), TIMOSHENKO and GOODIER (1951), SOKOLNEOFF (1956) and MUSKHELISHVILI (1949). Special techniques have been developed by many investigators to treat specific problems. A complete bibliography dealing with the theory of elastic cylinders in torsion and including conventional as well as complex variable methods is contained in a recent Chinese book* (1956) which presents a resumb of almost all existing material relating to this rather special field of elasticity theory. The Schwarz-Christoffel transformation has been used by TREFFTZ (1921) and SETH (1934) to discuss the torsion of beams whose section is a regular rectilinear polygon of n sides. The Cauchy integral methods described by ~USKHEL~SHVI~ (1949, p. 561) wereusedby 1.S.andE.S. SOKOLNIKO~(~~~~)~ discuss a section bounded by two circular arcs of equal radius and intersecting at right angles. A section with the shape of Booth’s lemniscatet has been considered by MUSKEIELISHVILI (1949, p. 581) and in a different but equivalent manner by HIGGINS (1942). STEVENSON (1939),and HOLL and ROCK (1939) treated an ellipticlimacont section. The corresponding problem for a hyperbolic limacon was solved by LIN and WHITEHEAD (1951) and by LIN and YANG (1951). Recently MORRIS and HAWLEY (1958) have developed new complex variable methods to deal with the torsion and flexure of solid cylinders with cross-sections transformable to a ring-space. The particular cases of the elliptic section and a section having a circular arc as internal cut are treated. *See the review of this book in Rppl. Meeh. R#ieus 1958. II, 110. +Booth’s lemniscate ix the inverse of an ellipse with respect to its centre,whiie from
inversion
~~~KHELr6x~lLr
of an eILipse with 1949,
respect
to its focus.
The
p. 175).
272
latter
the elliptic limaeon is the curve resulting
is sometimes
referred
to as Pascal’s
Iimncon
Torsion of beams whose sections are bounded by certGn quartic curves
273
STEVENSON(1943) has considered a grooved column of a special type. The boundary curve, taken in the z-plane, can be mapped on the unit circle y in the b-plane by the mapping function z = cc (1 + m <“),
(1.1)
where c and m are taken to be real and positive constants subject to the restriction 0 < m (n + 1) < 1. Sections bounded by the elliptic limacon or the cardioid are included as particular cases. Recently the author (1959a) has obtained solutions for sections bounded by regular curvilinear polygons that can be mapped on y by the transformation c > 0, 2 = cc/(1 + ?n 5% (1.2) where n is a positive integer and - 1 < m (,n - 1) < 1. This mapping function has been thoroughly studied by the author (195913) where varieties of figures corresponding to different values of m, n and having several axes of symmetry are drawn. For 91,= 2 the section has the shape of Booth’s femniscate, while for 7n = - 2/{m@ - 1)} it was shown that the cross-section is bounded by ?z approximately circular and equal arcs. The present paper is concerned with sections bounded by quartic curves which are mapped on y by the function x = ci/(l
+ n5 + m 5%
c > 0,
(1.3)
where m, n are real and Im] < 1, frz] < 2. It will be shown in Section 3 that the bounding curve is always the inverse of an ellipse with respect to a point on its major or minor axis. For n = 0 we have Booth’s lemniscate while if m > 0 and n2 = &n the section is an elliptic limacon. 2.
FUNDAMENTALEQUATIONS
Consider the Saint-Venant torsion of a homogeneous isotropic cylinder of uniform section S bounded by a simple closed curve r in the x-plane (z = x + iy = r eiB) which is chosen to coincide with the plane Z = 0 perpendicular to the generators of the cylinder. It is known (STEVENSON1943) that the displacements U, ZI,and rp! and the non-vanis~~ing stresses ~2, 2
are given by
U + iv = iTZ2, ze,= 7 f/ (a, y) = 4 7 [D(x)
iii3- iy’-i= p-r
p
(z) -
ii],
+ Q(Z)],
(2.1)
(2.2)
where 7 is the constant twist per unit length, p is the rigidity of the material of the cylinder and Q (z) = 46(z, y) + i # (m,y) is th e complex torsion function which satisfies the boundary condition Q(z) -
D (5) = izi along 1”;
(2.3)
bars are used to denote conjugate complex quantities and accents designate differentiation with respect to the stated argument. Let z = x(t), 5 =peie (2.4) be the function which maps the region S on the circle 161< 1. With the aid of this mapping function curvilinear coordinates (p, 6) are introduced into the x-plane.
W. A. BASSAM
274
The function L?(z) can be written in terms of <, so that $ _t i # = Q (x) = w (<),
(2.5)
where o (6) is regular inside y. It is found (MUSKHELISINILI f.o((,-)
s
x(u) z;(u-1) =$T ~-da + Y
o-5
1949, p. 577) that
constant.
The twisting couple N is furnished by N = rD, L’ = P (I + J),
(2.7)
where I) is the torsional rigidity of the cylinder, 1 is the polar moment of inertia I
XZdS for the cross-section, or I = -
Hi
and J=$
z (I?) x’ (u) zs {u-l) d G, s Y
(2.3)
z (0) B (61) w’ (u) d (T. J‘ Y
It was shown by MORRIS and HAWLEY is (of course) real.
(1958,
(2.9)
p. 469) that the expression
I +
J
The shearing stresses ~2 and 8^z at any point of the cross-section are given by (2.10) and substituting from (2.2) in (2.10) we get the convenient formula P^z-
i E
= [IL7 UPjZ (z;,]] [w’ (5) -
3.
MAPPING
iz’ (5) 2 (%,].
(2.11)
FUNCTION
STEVENSON (1943) and the author (1959a) have solved the torsion problem for sections which are regular curvilinear polygons having n sides and ri. rounded vertices (which may become cusps). The boundaries of the sections, taken in the z-plane, are mapped on y by the conformal transformations
z = CJ(1 + m {“f, e > 0, 0 < m (I&+ 1) < 1,
(3.1)
z = cc/(1 + m Se), c > 0, -
(3.2)
and 1 ,( m (n -
1) < I,
where m, n, are two real constants subject to the mentioned restrictions. consider the section which is mapped on ]{I < 1 by z = cl/Cl + n5 + mC2), c > 0,
We here
(3.3)
where n, m are taken to be real parameters, and, in order that the transformation shall be eonformal at all points within the boundary F, z’ (5) must not vanish or become infhrite at points within r, and we therefore have the condition -l
(3.4)
Torsion of beams whose sections are bounded by certain quartic curves
275
If P’ is the inverse of r with respect to the circle 1x1= d and x’ = x’ + iy’, x = x + &Jare the points on P, r, respectively, that correspond to 5 = e@ on y, then
(34 and inserting (3.5) in (3.3) leads to XI -f
= OY(do+ rne-y/c,
where f =
n d2,k.
(34 (3.7)
From (3.3) it follows immediately that 2” is the ellipse (Lr:’-
f)"/a" + p//%2= 1,
(3.8)
where a = (1 + nt) @/c,
Fig. 1.
b =
(l
-
m) rP/c.
(3.9)
inverses of eltipse with respect to points on major axis.
The contour P mapped on y by means of (3.3) may thus be identified with the inverse of the ellipse I” whose axes are Za, 26 with respect to a point on the first axis distant -f from the centre of the ellipse, where, using (3.7) and (3.9),
W. A.
276 n =
2fl(a
+
m
b),
=
(a -
BASSALI b)/(a
d being the radius of the circle of inversion.
+
It is sufficient to consider The parametric
2 (
either positive
equations
2fP/(a
+ b),
(3.10)
If (3.3) is to map the area inside
on to the area inside y then the centre of inversion equation of (3.10) gives the additional condition -
c =
b),
‘I1 <
must lie inside r’
r
and the first
(3.11)
2.
or negative
values of n.
of I’ are
cx/r2 = n + (1 + m) cos 8,
cy/‘r2 = (I -
m) sin 0,
(3.12a)
where c2/r2 = 1 + TL~+ m2 + 2tt (1 + m) cos B + 2m cos 28. The Cartesian and polar equations by
(nx2 + ,ny2 -
of the quartic
curve
(3.12b)
r are therefore
furnished
~a)~/(1 + m)” + c2y2/( 1 ~ m)” = (x2 + y2)2,
(3.13)
and (92~-
c cos 0)2/(1
+
?a)2 + c2 sin2 O/(1
-
m)” 2 r2,
(3.14)
respectively. The two curves corresponding to m = 0.2, n = - 0.2 and m = 0.2, n = - O-6 are illustrated in Fig. 1. They are the inverses of the two ellipses (x + 1)2/36
Fig. 2.
+ y2/16 = 1 and (x + 3)2/36
+ y2/16 = 1,
Inverses of ellipse with respect to points on minor axis.
respectively, with respect to the origin. For negative values of m we have a < b and the centre of inversion lies on the minor axis of the ellipse. Fig. 2 shows the two cross-sections for which m = - 0.2, n = - 0.2 and m = - 0.2, n = -0.4. These are bounded by the inverses of the ellipses (x + 1)2/16 with respect to the origin.
+ y2/36 = 1 and (x + 2)2/16
+ y2/36 = 1
Torsion of beams whose sections are bounded by certain quartic curves
277
For m = 0 the contour r becomes Booth’s lemniscate while if m > 0 (a > b) and f = & d(a2 - ba), so that the centre of inversion coincides with one or other of the two foci of the ellipse r’, then equations (3.10) yield 12= -J2#m.
(3.15)
In this case the mapping function (3.3) takes the form (3.16)
25= c5/(1 + mV, and substituting from (3.15) in (3.14) we obtain, after simplification, elliptic limacons r = A f B cos 0, where A = c (1 + m)/(l - m)2, B = 2c m*/(l - m)“. 4.
C2
2rr
I
v(l
(3.17a) (3.17b)
COMPLEX TORSION FUNCTION
Substituting from (3.3) in (2.6) we find that the torsion function a section mapped on ([I < 1 with the aid of (3.3) is furnished by
w(l) =-
the two
which solves the problem for
a2 do + ncr + m02) (02 + 12(I + fn) (0 -
(4.1)
1) + constantP
where 5 is any point inside y. It is easily shown that, under the restrictions (3.4) and (3.11) on m and n, respectively, the integrand of (4.1), considered as a function of (I, has two simple poles I&, 5s that lie outside y, where 1r + f;s = a;::2
n@,
1r 1s = I/m,
(4.2)
.+ 4m. For large values of D the integrand in (4.1) is of the order 1/1013, and we therefore w (1) = -
c2 i (Ri + R,) + constant,
where R, and R, are the residues of the integrand
w(1)= -
c2 i (&
-
t2)
at o = & and o = [s, respectively.
h2 m [ (Cl - 5) (t12+
c22 12 C1+
m) - (t2 - I) (Cz2+ n C2+ m)1 +
Hence
constant’ (4’3)
and, making use of (4.2), it is found that (4.3) simplifies to
Cd(4.1 =
c2 i (1
_
m)
k
1
fm
1
+
+mn5
nJ
mp
+
+
constant*
(4.4)
where k = (1 _t m)2 -
n2.
(4.5)
In the special case of an elliptic-limacon section n 2 = 4m and the integrand in (4.1) has a double pole outside y at [,, = F m-f. If R is the residue of the integrand at D = &, in this case then OJ(1) = -
c2i R + constant,
and it can be shown that the result agrees with (4.4) after substitution from (3.15). Taking for the constant in (4.4) the value - c2 (1 + m) i/2 (1 - m) k, one finds for the complex torsion function :
w(I) =-
c2iq-nc-mq[2
2k 1 +nl+m[2’
(4.6)
where
q = (1 + m)/(l
-
m) = a/b.
(4.7)
1%'. A.
278
BASSALI
The imaginary part of (4.6) is +“z
p (1 - m2 p4) - n2 p2 + n (q -
1) p
(1
-
p2) cos
e
(4.8)
2k 1 + n2 p2 -C_m2 p4 + 2n p (1 + ‘m p2) cos 0 + 2m p2 cos 20’
Setting p = 1 in (4.8), substituting for q from (4.7), and using (:J.l2b), me see that, on the boundary r, IJ reduces to &z, as it should. TORSIONAL
5.
RIGIDITY
The polar moment of inertia I for the section is obtained by substitution from (3.3) in (2.8) 1 =
_
CT3(1
lic4
4
f
- ml?) do
v(02 + n D + .m)2 (1 + n O +
:
(5.1)
mo3)3’
If X, and X2 are the two roots of the equation u2 + n m + m = 0, so that h, h, = ‘111, jm < 1
; A, + A, = - n,
11q< 2,
(5.2)
then X, and h, lie inside y and 1 = ?j 7rc4 (PI
-1 P2),
(5.3)
where P, and P, are the residues of the integrand in (5.1) at CJ= X, and (J = h,, respectively. It can be proved that P,
=
3x,2 (1 -
(X, -
m X,2)2
2 (1 -
h,)2 (1 + n h, -t m h,2)4 -
(h, -
m2) x,3
h2)3 (1 + n h, +
(5.4)
m A,2)3’
while P, is derived from (5.4) by interchanging h, and X2. Applying (3.2) we find after considerable reduction that* (1 - n2m + ffl.2 +
th3)3
ms(2 - 6
+
2 (I -
+ m + mP)2 +
m)4
m (1
+
m)2
m3)
I
(1 + m) (1 i(1 - m)2 where k is given by (4.5). Introducing (3.3) and the torsion function expression (1 -m)k
We therefore have
s
defined by (4.6) and (4.7) in (2.9) leads to the
m
(1 + m) o3 + 4% o2 (1 + m2 CJ~)
Jo2
+ n 0 + m) (I + 110 + m 02)s
c4 i
Jz--
(5.5)
do.
(3.6)
J = 271~~(Q1 + Q,)/U - m)k,
(6.7)
where Q,, the residue of the integrand in (5.6) at CJ= X,, is given by m2
)-a+
Q
1
(A,
-
A,)
and Q2 is deduced from Q1 by interchanging algebraic manipulation that
X12)Al2 + (1
+
m (1 + nk)
n A,
+
XI3
(5.8)
m A,2j3
(3.2) it is found after some
h, and X2. Applying
.
(5.9)
I When (5.5) and (5.9) are inserted in (2.7), it is seen that the torsional rigidity is? D _ prc4 k4
3 (1 - n2 m + m2 + 2?r~~)~+ 3m2 (2 - n2 + m + ms)2 - 4m2k2 2(1 - m)4
[
+ (1 + m)2 3m + 1 . *It can be shown that the I =
7x4 [(l + m)4 (I
t.411equivalent
expression
for I may
t 4m2 + m4) + 2 (1 + ?Q
and simpler
D = &L”C4[(I
expression
-
(1
+
m)
(1 + m3)
(1 - m,)2
*
(5.10)
be put in the form (1 -
6m + 4m= -
for the torsional
+ m4) (1 + m”)
n2 rn, -
rigidity
6rn3 ml m*) n2
+ en2n4]/!2(1 - my A4
is
4m (1 + m) (1 + m3) 7&Q+ 2mQ n4]/“
(I -
m)4 k4.
Torsion of beams whose sections are bounded by certain qua&c
curves
279
Setting n. = 0 in (5.10) and observing that (1 + m2 + 2rr~~)~+ m2 (2 + m + m3)2 = Yj(1 -
,m2)2 (I -
?n)* + 4 (1 + m2)2 (1 + m)4,
we find that n simplifies to r) = $ II” c4(f + .m4}/(1 - my*
(5.11)
which agrees with the result of MUSKHELISHVILI (1949, p. 582) for Booth’s lemniscate. Substituting from (3.15) in (5.10) yields the following expression for the torsional rigidity of either of the elliptic limaeons (3.17) :
D = gp7r4 (1 + t-w +- 1Gd + 8m3 + mQ)/(1 - m)s, which can be shown to be equivalent
(5.12)
to
D = &,,@A4
“r 8A2B2 + B4) I
(5.13)
on using the values of A and S given by (3.17b). The formula (5.13) was obtained by STEVENSON (1943, equation (3.4)). As a further check on the expression (5.10) we notice that for m = 0 it reduces to D = jpv d4, where c’ = c/(X - n2). This result is to be expected since in this ease it is easily seen that the curve I’ becomes a circle of radius c’. Putting m = n = 0 in (5.10) yields D = &we4 which is the torsional rigidity for a circular cylinder of radius c.
6. Substitution ;”
-
irz
SHEARING
STRESSES
from (3.3), (4.6) and (4.7) in (2.11) leads to = -
(1 _t np2 + mp2 t)2 {n<+ -
~7 ic T
i_ 2m (1 + rn,) + m2
n 4)
(1 --ml@
[
1,
-I- P (1 - m C2)(1 + 12t + m T2) where k is given by (4.5) and T = (1 + mm2 p4 -
(6.1)
2mp2 eos 2f?)t [I + n 2 pz i_ ms p4 + 2np (1 + mp*) eos l? + Zrnj.? cos 201. (6.2)
Putting 5 = p ei@ in (6.1) and separating real and imaginary parts it is found, after some straightforward computations, that the shearing stresses at any point of the cross-section are 2p Tc (1 - p2) pz=-----------(1 - m)kT
h
n {I + m (3m ‘_t k) p2 + ?a4 p*) sin B + m (1 +
nt) (1
i_ m2 p2) p sin 2Q + ma 72p2 sin 3B
I
9
(6.3)
J
,r;.JE: T
__L-[ (1 -ml&
1
2m (1 + m) (2m + ns) p3 f 2rh2p (1 f m3 p4) + ‘n 1 + 4m (1 -t m) p2(1 + mpe) i
$-m2np2(1 +p2)cos36
I
+p(l
-m2p*)
-+-np2(1
1.
-mp2)cosB
(6.4)
Since w (5) is regular throughout the cross-section S and z’ (5) does not vanish at any point within S by the restriction (3.4), it follows from (2.11) that the stresses (6.3) and (6.4) are necessarily &rite and physically admissible across S. Setting p = 1 in (G.3), (6.4) and substituting for T from (6.2) we And that, on the boundary r of the cross-section,
2
vanishes as expected 1*c7
(Q2h==1 = (1
_
m)k
while the peripheral
shear stress simplifies to
(1 + m) (1 + m2) t_ 2mn. cos 6 l/(1
+_ m2 -
2mcos26)
’
(6.51
W.
280
A.
BASSALI
From (6.5) it is seen that the maximum and minimum values of the tangential stress acting on the boundary I’ occur alternately at the three points B = 0, B = t(, and B = T, where dL= cos-1 [ and the magnitudes
11(1 + rrt)/Z (I + ~?G)],
(6.6)
of these vatues are furnis~~ed by (z),=l,o=o
= FLTC [(I -t_ V/i) (1 -t nr2) + 2VOL]/(l -
(z)p=l,@=a
= p7c [(l + m2)2 -
(G&p=1,8_n
= pre[(l
n%+/(l
+ ?I&)(1 + W&2)-
-
V&)212,
(6.7)
?n)li,
2nr?q/(l
-
(6.8) (9.B)
m)2h-.
Su~sti~uti~~ from (3.15) in (6.71, (6.9) and using the vnlucs of A and B defined by (Y.l?b) it can be verified that the resulting expressions agree with those obtained by STIWENSON (19&d, equations (3.7, (3.8)) for the peripheral shears of the elliptic-limacon section at the bottom of the groove and at the point where T becomes a maximum.
B Distribution
Fig. 3.
of shearing stress on ~)oun(~aries of sections.
Fig. 3 illustrates the vari&ion in peripheral shear on the boun[~aries of the two sections in Figs. 1 and 2 which correspond to m. = 0.2, n -7 - 0.6 and ~a, z - 0.2, n = - 0.4. ‘I’be points at which the stresses attain their maximum and minimum values on the two boundaries are also indicated. REFERENCES BASSALI, W. A.
1959a 1959b
J. Math. Pkys. In press. J, Me&. Pkys. Solids 7, 145.
1956
TheomJ of Elastic CyEi?xiers in Torsion J. Appt. Pkys. 13,457. 2. nngew. Math. Mech. 19, 141.
CHIEN, W. Z., Hu, H. c., LIN, R. S. and
YEH,
C.
Y.
HIGGINS, T. J.
HOI&, D. L. and ROCK, II. I%. LIN, T. C. and WHIRLED, LIN,
T.
c. and
1951
L. G. YANCJ,
II.
1942 1939
T.
1951
(Peking).
U~~~ers~~~ of ~,~a~k~~~~tor~, ~~g~~eer~~l~ ~xp~~rn~nt Station Series, Bulletin 118, 108. University of Washington, Engineering Experiment &ztion Series, Bulletin 118, 112.
Torsion of beams whose sections are bounded Love,
A. E. H.
1944
A
Treatise
by certain quartic curves
on the Mathematical
Theory
281
of Elasticity,
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B. B.
S~KOLNIICOFF. I. S.
1958 1949 1934 1953
Quart. J. Mech. Appl. Math:ll, 462. Som.e Basic Problems of the Mathematical Theory of Elasticity, 3rd Ed. (Moscow). Proc. Can& Phil. Soc,30, 139. Mathernalical Theory of Elasticity, 2nd Ed. (McGrawHill).
SOKOLNIKOFF, I. S. and SOKOLNIKOFF, E. S. STEVENSON, A. C.
1938 1939 1943
Bull. Amer. Math. Sot. 44, 384. Proc. Lond. Math. Sot. 45, 128. Phil. Mug. 34, 115.
TJMOSHENKO, S. and GOODIER, J. N. TREFFTZ, E.
1951 1921
Theory of Elasticity, 2nd Ed. (McGraw-Hill). Math. Ann, 82, 306.