AtA i
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM NO. 1182
THE PROBLEM OF TORSION IN PRISMATIC 1ES1.BRS OF
CIRCULAR SEGIMENTAL CROSS SECTION*
By A. Weigand
SUMMARY
The problem is solved by approximation; byt setting up a
function complying with the differential equation of the stress
function, and determining the coefficient appearing in it in such
a way that the boundary condition is fulfilled as nearly as possible.
For the semicircle, for which the solution is known. the
method yields very accurate values; the approximated stress
distribution is in good agreement with the accurately computed
distribution. Stress and strain measurements indicate that the
approximate solution is in sufficiently exact agreement with reality
for segmental cross sections.
I. FUTDAMENTTAL EQUATION' OF TORSION PAND ITS APPROXMAFTE
SOLUTION BY TEE METHOD OF LEAST SQUARES
The torsion problem for the prismatic member stressed by
twisting moments at the ends is formulated as follows. Find a
function f(y, z) which in the crosssoctional plane satisfies
the partial differential equation
f off
2+ 2 1 (1)
and at the boundary of the cross section the condition
S= 0 (2)
b'"as Torsionsproblem fur StLbe von !ceisabschnittfirmigem
Querschnitt." LuftfahrtForschung, Band 20, Lfg. 12, Feb. 8, 1944,
pp. 333340.
2 NACA TM No. 1182
This function f(y, z) then gives the torsion constant Jd of the
member according to
Jd = f(y, z)dy dz (3)
tho double integral to be extended over the cross section. The
angle of twist 1 of a length 2 Is
MdZ
GJd
Md the applied torque, G the modulus of rigidity of the material.
The components of the shearing stress follow from
21 2M2
TXY d (5)
T T (f)
xy J a i' Iz Jd by
Owing to the equations(5) which satisfy identically the equilibrium
condition
9 + aTx 0
by bz
f(y, z) is called the stress function of the torsion problem.
The differential equation (1) with the boundary condition
equation (2) follows from the consideration of the state of strain
and the relation between stress and strain, which is given by
Hooke's law.
Occasionally, it is appropriate to introduce the polar
coordinates r,Q instead of the rectangular coordinates (y, z)
(fig. 1). The differential equation of the stress function together
with the boundary condition then reads
a f 1 of 1 f
 + +  1 (la)
+ r2 r2r +
f = 0 (2a)
NACA TM No. 1182
while the torsion constant follows from
Jd f(r, cp)r dr drd (3a)
and the shearing stress components from
T 
> (5a)
2M
3d rf
Rigorous methods for solving the potential problem posed by
equations (1) and (2) will not be discussed.
The approximate solution can be effected in three ways. A
function can be assumed that satisfies equation (1) but not
equation (2). If the differential equation is replaced by a
variation problem, it results in the conventional Eitz method; or
a function satisfying the differential equation can be assumed and
the boundary condition met in individual points cr "on the average;"
an exact explanation of what is meant by "on the average" will be
given later. Lastly the differential equation can be replaced by a
difference equation and the linear equation system ensuing from the
boundary condition solved by iteration irith the aid of the Liebmann
Wolf method. Only the second method is discussed in the present
report, after having been pointed out: among others, by Trefftz
(reference 1) and St. BerCmann (reference 2).
Since the torsion problem of the segment is to be treated, we
proceed from the differentia?. equation (la). It has the particular
solutions
r2
to ,f = rk cos kp, k = r sin k? (6)
from which the general solution
r2 n
f + > k ai + bk (7)
4 0 c+
NACA TM No. 1182
can be built up. Now the determination of the coefficients ak
and bk is involved. The next thing is to so determine them
that equation (2a) is complied within individual points. Among
others, problems relating to plate bending have already been
solved by this method.
Another way is the following: Rather than specifying strict
compliance with the boundary condition at originally established
points it is required that by choice of the coefficients the integral
of the squares of the boundary values is least. In this instance
the boundary condition is said to be fulfilled "on the average."
This method is hereafter called the method of least squares.
In the formula the requirement on the factors reads
S= f da = Min. ds = Boundary clement (8)
The integral is to be extended over the entire boundary; this is
indicca ,=' .7 the sign f. Putting equation (7) in equation (8) gives
f i + % + (pi + bkf = Min. (9)
S+ o + atk kM (9)
Ths coc .f: ....nta follcw from the requirement
. 0, o, 0, k=l. .n (10)
From equation (10) follow a linear equation system for the 2n + 1
unknown ao, ak, and bk.
The practical use of the method depends upon whether sufficiently
exact results consistent with a moderate amount of paper work are
obtainable, especially for the stresses, or in other words without
having to solve a great number of linear equations.
NACA TM No. 1182
II. APPLICATION TO TEE SMICIRCLE AND TEE SEGMENT
1. Semicircle; Strict and Approximate Solution
by the Method of Least Squares
The strict solution of the torsion problem for the sector was
given by St. Venant (Handb. d. Physik Bd VI, pp 153154). The special
case of the semicircle is easily treated as will be shown.
To remain in agreement with the notation for the segment
(fig. 6) the coordinate system of figure 2 is shown for the semicircle.
On the straight boundary AB, p = and 31; in the cross
section, = = .
'2 2
The stress function is expressed by
f(r, p) = Xk (r) cos k' (11)
1,3.**
It already fulfills the boundary condition on the straight boundary,
since k is an odd number. The constant 1 in the internal
S< < 3i
= = ~ is expanded in a Fourier series.
k+l
4 cos k
1 = l)
1k (1) (12)
1,3...
Introducing equations (11) and (12) in equation (la), the comparison"
of the coefficients of cos kp on both sides of the equation gives
k+l
Xk +, k2 k 4 (1)2 (13)
S+ k k (13)
The solution of this differential equation, finite for r = 0, reads
k+l
k = Ck (1) 2 (4)
X kCJ m 4 k2
NACA TM No. 1182
Since Xk(R) must be = 0
4.
Ck =
k+1
(1) 2 2k
iT 
(15)
and, hence,
f(r, qp) =
12 3
X 1,3,5.
k+1
() 1 T
k(4 2)
 (r] cos kqg
\E/_
is the solution of the torsion problem for the semicircle. The
torsion constant Jd and tho shearing stress distribution are
computed from equation (16). From equation (3a) follows on three
places exactly
J, = 0.297 4 = 4R
and from (5a)
= *rr3 3 1
)
1 Ap2
sin ip 
.32 4 '
1
52

~ Ki
4r4 r\
( sin 5
1 ) cos
E3 ,3\
1
3(32 4)
/ r4
5(52 4) J
 2) cos 39
 )
(16)
(17)
cos 5a + .
. (18)
" sin 3qp
NACA TM No. 1182
maximum shearing stress occurs at A (fig. 2), that is,
r =0 and =. Here
2
8 Ma
7ax= 
maz 3KC R3
Md
S2.85 
13
(19)
The shearing stress at
C (fig. 2) is
7 C = R.44le
P R3
Following the rigorous solution for the semicircle an approximate
solution r the ac L of least eqsares shall be derived.
Sirncn t e c.ecss scct.on is symmetrical about 9 = i, and
following g equation (7) we wrlto:
r2 \ k
f= / a cos kq (20)
4o o
Instead of coefficient ak the quantity Ik is introduced by
(21)
2 Xk
ak =
4 Rk
So with X = r, formula (20) reads
R
R24 2
4
+ k
0
cos kP)
(20a)
The boundary values are
On AB (fig. 2)
On BC (fig. 2)
 2 (2
f 
AB 4\
BR2
FBC 4= (
+o k
cos !
+ tk X
o
The
for
Cos kp
8 NACA TM No. 1182
The method of least squares yields as conditional equation for xk
1 efo2
Therefore
VXk cos
dX + 1
h'
+ x cos k)2 d
= Min. (22)
1 (
0
+
2
xkX cos l/ 3 cos dX
+ x0
. Xk
0
cos, cos Iqp dip = 0
For xk the linear equation system with syametrical matrix
2 Akl xk = B = 0,1, .
0
is applicable, with
A00 = 1 +
co!"'.
 = 1, n
2k + 1
(23)
(24a)
(241b)
cos 0 cos 2
k++l
 c+ Z+l1
Bo0=+
B3
1 sa (k )+
2 +
k k3
sin (k + l)+ 
 j2 k
k + I
cos s, sin 2
B = 1 n
1+3 1
f 1 (24c)
(24d)
The numerical calculation was effected for n = 1, 2 6.
For n = 6 the equation system reads
I* +
Akk= 
NACA TM No. 1182
II
!
+
I
!
H
M14
+
II
CU
I
H
!
(0
e
I
Him
M
I I
14 \ r4r
li II
HO H
+ I
+
N
. e0
ilF
ri ,L\
cO
+i0
I
u
\r
w^s
+lo
>rY
mi
14t
ent
1
+
II
I0
a
+
M1
r(^
m( ON
it~
Id
4
+
+
I C
A4
+
eln
Im
+
r
Hi
HjO~
II
I'
14
uPg
alJ
a1~7
14
HJl
en
Nii
HjJ
+P
HIa
+!crj
IN
I
NACA TM No. 1182
The coefficients were changed to decimal fractions and considered only
up to the fifth place after the decimal point. Six approximations
were computed; for the first approximation x2 = x, = = x6 = 0
was used. The result is presented in table I. Insertion of
equation (20a) in equation (3a), gives the torsion constant Jd as
Jd = 2R 4 1 1 x
d k ? X + X.
8 3 1 3x5
and the following approximations for
places:
 1 c +..t. = ~xR (25)
5x7 /5
K computed exact to three
S(1)= 0.414
(1)
K =0.298
(I'.
K = 0.326
(2) 300
k(5) = 0.300
K(3) = 0300
S= 0.298
(6)
The third approximation computed from four linear equations already
gives a torsion constant value that differs by no more than
2/3 percent from the rigorously computed value.
For the stress calculation, equation (20a) is inserted in
equation (5a), so that
r 2KR3
Md .
aR3 1
The shearing stresses
The shearing stresses T '
(D
n k1
1
sin kg
(26)
21 + xckl cos lp) (27)
and Tr ~2 at the straight boundary
 3x3X + 5x 5X + .)
Ir R
(28)
NACA TM No. 1182
T i =2 + 413 6x15 + X) (29)
4P 2kR3 2+x
For X = 1 the shearing stresses are
SX=1 M d
Tr S3 sin + 2 sin 2+. (30)
(T x= (2e + x1 CO q + 22 xCos 2p + (31)
The two maxiSam shearing streeses are:
r=n x1 Md
T IT r 2 T x (32)
r max 2r R3
T T1 2 ( + 2x 3x3+ (33)
Of these expressions T and 7T must at least
approximately disappear.
Now for a check of the extent to which these conditions are
met for the different approximations and also of the oxtent of the
differences between the approximated and the exact values of T
and iC TIhe results are represented in table II and figures 3
to 5. The fourth approximation already gives a serviceable result,
which is somewhat further improved by the fifth and sixth affproxi
nations.
Figures 3 and '4 show the approximations for T '~
and T=l which really should disappear. It indicates gob~
reeme in the fifth sixth approximations. Figure 5 sho
agreement in the fifth and sixth approximations. F1iue 5 shows
NACA TM No. 1182
the shearing stress T9= at the straight boundary plotted
r
r
against X =. The fourth, fifth, and sixth approximations differ
R
little from each other and from the accurate stress distribution
designated by g. A marked departure occurs in the immediate
vicinity of the corner (point B in fig. 2).
2. The Segment
(a) Approximate solution by the method of least squares. Since
the cross section is symmetrical to q = 0 (fig. 6) the
formula (20a) is applied to the stress function f. The boundary
values are given by
AB 4 2
n xcosk cosk
0 x  coskV
0 cook
(34a)
0 = = a .
+ <
BC= 1 + I=cos k e
O"
(34b)
If dsI is an element of the straight
an element of the arc BC,
boundary AB
des = R cos C
s Rcos2q
ds2 = R dp
The expression that is to
coefficients xk reads
cos2a_
cos2q
be made a minimum by the choice of the
+
0
kcosk
xk o
cosk q
cos 2
cos tp j
cos a
d(
cos2cp
+ f 1 + 0 co ~) gq 3
0
and ds2
(35a)
(35b)
 =i
(36)
NACA TM No. 1182
J
From = 0 follows a linear equation system with symmetrical
matrix for xk, which is to be written in the form of equation (23).
The coefficients Ak and the right sides are given by
A00 = I a + sin a (37a)
2
2k+ a. cos kp C
Ak = cos k o 2k2 d( + 
J cos k 2
sin 2ka
in 2 k = 1, 2, .n (37b)
1;k
k+l+l I cos kp cos 1) i ein (k l)p
Ak1 = cos a. +2 d.p 2 k I
S k+1+2 2 ki
sin (k + l)p k = 1, 2, n
(370)
k + 1 J = 0, 1. n
Bo= a + sin a (cs + sin2a (37d)
1 0+3 cos 2et sin 2a
B = cos a+m  d p p = 1, 2, ... n (37e)
cos) "0.49 1
The integrals appearing in equation (37) are of the form
f Cos9; they can be defined by expressing cos pcp by cosP),
0 cos pe
cos'2 9, etc. A reproduction of the somewhat elaborate formulas is
omitted.
The matrix Ak: including the righthand sides B, of
equation (23) were computed to five places with the calculating
NACA TM No. 1182
machine as functions of a. To keep the paper work within tolerable
limits the process was carried to k, I = 6. The result is shown in
figure., The unknowns Xo x xl .. x6 were computed by
equation (23), .by the Gauss method. The result is given in table IV.
After x0, xl 6 have been determined the torsion
constant Jd and the shearing stresses can be computed.
By equation (3a) the torsion constant is
J = f (r, 9p)r dr d(p
The double teal is to be extended over the area
The double integral is to be extended over the area ABC
(fig. 6)
cos a
r = R OS
cos q
= I+j f (r, p)r dr
ABC AOC OBC 40
+ d(p f(r, cp)r dr
IM" to
Insertion of the expression for the shearing function
equation (23) in this formula gives
f from
sin 2a4 2
4 s
4 
JT = R
+ x0
1 sn
+. sin 2
3
sin 2)\
a + 2
sin 4
r)^
kT k+ 2 \
(38)
NACA TM No. 1102
with
Jk k+2a f( cos kd
k = cos 0a k+c
cos k p
By equations (5) and (5a) the hearing stresses are
Nd
Txz 
=  ^
2% cos
[2 cos
n
( +

1
k x, k cos (]t 1)
k xk kl sin (k l)(
xkk1 cos k.\)
I.T n ]
T  k x1 )
r
sin 1:p
(43)
Particularly important .e t7 c; :'cnrrulas for the stresses at the
boudr lri,. T: 
on A i
T 2 cos
X ':? y
n
X
1
k1
:: k ^~1
o0^:J
cos (k 1) i
sin (k 1) i
(39)
(40)
(41)
(42)
(40a)
n
+ k
1
os 9)
Xk K008
(41a)
2=  2 cc a ten a
2iq<1> _
NACA TM No. 1182
on BC
BC Md (2
9 3~ S3
2ivB
S X cos k 1
1 /
(42a)
(43a)
BC M n
T =  k k sin k9
r 2KR23 1
Of these equations (41a) and (43a) must disappear
approximately).
(at least
Lastly there are the formulas for the shearing stresses in
A and C (p = n).
A
T =T
xy max
(2 cos a 
nk x kl \
1 k xk cos a
1
2R
2rR3
(1) klk
(42b)
The numerical values for the torsion constant and the
particularly interesting shearing stresses T A and r C follow
xyp
from equations (38), (4ob), and (42b). These are also included in
table IV and in figure 7 plotted against a.
(b) Solution formula by Fourier series.
segmental cross section can also be solved by
series. The method is briefly explained.
The torsion problem for
means of the Fourier
To transform equation (la) we put
(44)
r2
f=  + (r, P)
4
0 must be a potential function which assumes the values
4a
at the section boundary.
(45)
(40b)
NACA TM No. 1182
Therefore
cos a
cos2P
This even
interval
function of q) 1s developed in a Fourier series in the
 = +
= /_
0
0 ( C
R2 2 2
 = cos ,
4 L
an cos nqp
sin 2)
+ae
Scos ng sin na
/ ~cap 
Scos rp 31
The potential function $(r, T) is built up with the aid of yet
to be detenrin ed :ooffiiele. from per'.cular solutions.
n
 = "n cos nq
0
(48)
At the boundary 0 as : ts fo olrowing values
0 n cosnt
0 9 a n = osna cos no
0 cosn
a= = = it = b n cos nc
0
1 (49)
'f
a = T= I
(46)
vith
(47)
(47a)
f 1
I ;
C4
_L2
NACA TM No. 1182
This even function of ( is also developed in interval
 i = C = + i in a Fourier series
Bo=2
+ ir
1 A
Bn = 2 bk
+ ( C
0 b\
= / Bn cos np
0
bkk coska cos ko d
COS CO /
to k cos k) dP
k coska
R k cos kp cos n dp
cos /
Scos k cos
bo k Cos kc)) cos n> dI
(50)
(50a)
The .Fourier series (equations (47) and (50)) obtained for the
boundary values of D must be identical, that is
Bn = an
n = 0, 1, .
This is an infinite linear equation system for the lookedfor
coefficients bn. With
n 4
(51)
(52)
L
F
NACA TM No. 1182
19
the system reads
+sin 2a
rxo+ amo = R a + 2
1 2
&i cos kg sin k,
a n = Cos2 cos kq sn (53)
1 cos2) k
The coefficients aon and ak are given by
n a cos n) sin na
a = cos cosn d  n = 1, 2, .
on mf cosp2n n ea sin 2na
Ca cc d + 2 4n
en co ko cos rqp sin(n k)a sin(n + k)a
aim cos a dq
cos n 2(n k) 2(n + k)
Obviously a J an, that is, the matrix (al) is not symmetrical.
To solve for given a the torsion problem by this process the
Fourier series must be limited to finitely many terms; in other
words, the system (53) must be approximated by the section method.
For example, going as far as x7 inclusive means that 72 = 49 factors
aan have to be computed. The numerical calculation thus becomes
very tedious and is therefore omitted.
III. CHECK OF THEORETICAL RESULT BY TEST
With the setup described in reference 3 the torsion constant Jd
of a member of segmental section was optically determined; while the
maximum shearing stress T (point A in fig. 6) was determined by
means of stress measurements. The shaft sketched in reference 3,
means of stress measurements. The shaft sketched in reference 3,
NACA TM No. 1182
figure 1 tas machined to d = 70 millimeters ahd a flat surface
milled out which gave the desired section. The milled surfaces
corresponded to the angles a = 200, *0, 600, and 80. The
comparison is illustrated in figure 7. The agreemefzt is plainly
sufficient.
Translated by J. Vanier
National Advisory Committee
for Aeronautics
REEE~NCES
1. Trefftz, E.: Ein Gegenstick zum Ritzschen Verfahren.
Verhandlungen des 2. internationalen Kongresses fir
technische Mechanik, Zirich 1926, p. 131.
2. Berngann, St.: Ein Maherungsverfahren zur Ldsung gewisser
partieller, linearer Differentialgeeichungen. Z. angew.
Math. u. Mech. Bd. 11 (1931), p. 323.
3. Weigand, A.: Ermittlung der Formziffer der auf Verdrehung
beanspruchten abgesetzten Welle mit Hilfe von Feindehnungs
messungen. LuftfForschg. Bd. 20 (1943), Lfg. 0, p. 217.
(Also available as NACA TM No. 1179.)
!4ACA TM No. 1182
TABIE I
THE APPROXIMATIONS FOR THE UNKNOWN IN THE EQUATION
SYSTEM (23) APPLICABLE TO TEE SEMICIRCLE
Xo x1 x2 3 4 x5 6
First approximation
0.486 0.654     
Second approximation
0.1135 1.399 0.638    
Third approximation
0.0136 1.6553 0.9412 0.3004   
Fourth approximation
0.0086 1.7364 1.0859 0.4566 0.10095  
Fifth approximation
o0.o68 1.7310 1.0711 0.43465 0.0835 0.00855 
Sixth approximation
0.00022 1.6978 0.9967 0.3251 0.0308 0.0900 0.0359
NACA TM No. 1182
m id .
SI OU %O
rI o I rl(
h S r m mo r6 mo n o 0 r o ao
\ 0 r 4 r I 0C Cu
t'K t ..:t P. trfC(n O bO
rI 1 ,.4 r,
o30 t, mH l o m0o m
H OO(O O 0 O IOOO
0 Oni i 0rCJ m
S 000000000
00 I I I I I
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S' , d u o i' r\ 1
fn 'i 10* Cl
K ooo
MM 0F0 1%ic y ^ o
0
oI
IJACA TM ,To. 1182
H. U o UNo o ng. t. o o* .
o p' l
En *8. rI % 1, 9. *oo, o Mo ID O .T
i 0i < io
8 o0 / .r4 1 0000 HoI 0 00 0O r 0i 0
T S o, to F
Add" o 4,'' r9 S .l o m .
S9111111 0 91 i i i i o 9111
o. o t. ., 0 rQ W \oo : .
A i A o A iA A 1 i i 1i 1 i ii
9 v1 N r. D. lp a. n .
0 0 I i Ig 0 ro i i i i I 8 i B A' gI=' A A I8
H01H o H o 111 0 p 1 Q 1Q
0 o.c o. 0 H. R9 00 8 t Q c? ..i '. ?.
UN 19 n 9 ng 0 M En
&145 8i r0n Q0 cm oofc 01 a rm ococo
911 9 H 0911 0;1110
00 H. 0 H p "R
0 90 I I I I I r4I. r4 A A oA
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V\ r o. ir S S2>?8 ag Cr4 a s R ? g
0 Ao 9 A A AOIm cm HH go I H1 0H
d 0 p a I a MR m I6 0 9'" a I I0 I0
NACA TM No. 1182
Figure 1.
The coordinates of the section points and the shearing
stress components.
C
A B
"*
Figure 2. Semicircular section with coordinate system.
NACA TM 10o. 1182 25
0)
S__ O O Or
S N0 0 00
0_ 0_ 0
/ r/ \ 0 '.
[1
      ~wT \" 7 r e )*
.. .. / 0 Q .
0 C 0
. , I
jh 0 00
*sG CY s) 5 a
CH
^ ^^ < \ l iooo
0.
ali r ar
< o
Fi 00
4 0hD
0 43 0 0
FL, COd, c
0 0 0
ILI s' A
a 3 ~ % za EbD
NACA TM No. 1182
3
Figure 5.
The approximations for T at the boundary AB of the
r
1:1.
2:2.
3:3.
semicircular section.
Approximation 4:4. Approximation
Approximation 5:5. Approximation
Approximation 6:6. Approximation
g: exact solution
Figure 6. Notation at segment.
NACA TM No. 1182
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UNIVERSITY OF FLORIDA
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