Lyse, I.

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Lehigh University
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Fritz Laboratory Reports Civil and Environmental Engineering
1937
A study of structural sections subjected to torsion,
Journal of Faculty Sciences, University of Istanbul,
Jan. 1937
I. Lyse
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Recommended Citation
Lyse, I., "A study of structural sections subjected to torsion, Journal of Faculty Sciences, University of Istanbul, Jan. 1937" (1937). Fritz
Laboratory Reports. Paper 1156.
http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1156
. -
FRiTZ ENGi~1E£j{n~G LABORATORY,
'<._, . LEHIGH UNIVERSITY
, '-," BETHLEHEM, PENNSYLVAI'iIA
A STUDY OF STRUCTURAL SECTIONS SUBJECTED TO TORSTON*;'
by Inge Lyse**
I*"l~
Ilof·7
The general relationship between angle of twist and
tor~ional moment in, a circular section is-given by: T = J.G.e,
where T is tor~ional moment, J is polar moment of inertia,
G fs shearing modulus of elasticity, and e is angle of twist.
For non,:"cir:cular sections we may similarly set: T = K.G.e,
where K represents the resistance of the section to tors'ional
deformation. TheK-value therefore, becomes the important item
for the determination of shearing stresses in structural shapes.
For a rectangular section Saint Venant developed the following
equation for the torsional rigidity:
. 3
K = b.!L _ 2v.n4
3
where: b = length of rectangular section
(1)
n = thickness or breadth of rectangular section
V = a constant depending on the ratio b
n
For the ratio bin greater than 4 the value for'V = 0.10504. Con-
sidering the soap fibrr analogy, the expression -2Vn4 represents
nend lOSS". For any differential length, dx, along the rectangu-
lar' section the rigidity is:
*
**
- - - - - - _.- - - - - - - - - - - - - - - - - - - - - - - -
A detailed description of an extensive investigation of this
problem has been pUblished in a paper: STRUCTURAL BEAMS
IN TORSION by lnge Lyse and Bruce G. Johnston, Proceedings,
of the American Society of Civil Engineers, April 1935,'
pp. 369... 508.
Research Assoc~ate Professor of ~ngineeringMaterials
Lehigh University, Bethlehem, Pennsylvania
(2)
For a trapezoidal section in which the thickness 'at any point
is £ (Fig. la) the ,rigidity for diff~rential length is:
K ~ ~1:3 dxYo-,
2
,Evaluating r in terms of ~ and .!h and integrating:
, ~. ~ JL(~+n') (m2~n2)
12 ,
(3 )
The "end loss" must be deducted from the above value, or:
K = !!....(m+n)(m2 +n2 ) ... Vm~n4 - Vn .n4 c (4)12 .
where Vm and' Vn represeht the constants for· the two'ends.
Cons~dering a section eonsisting of a part of 8 sector such
as shown in Fig. Ib, the fol~owingresultsareobtained:
Vm = 0.10504-0.10000.8+0.08480.82..0.06746.83+0.05153.84 (5)
and
Vn =: 0.10504+0.10000.8+0.08480.82'+0.06746.'83+ 0.05153.84 (6)
where 8 is the slope of the section; that is m-n-.
b
A structural beam with sloping flanges may be cons~d- ,
ered made up partially of trapezoids and rectangles. The
rigidity of the portion shown in
K =' b-W('m+n) (m2 +n 2 )
f. 12
Fig. lc is:
+ ~ wm3 - 2Vnn4 (7)
The beam shown in Fig. 2 consists o,f two such flange sect.ions
plus the web section pl!ls the value contributed at the junc-
tures of these sections. The web contributes:
- <3
(8)
-,
The rigid,ity contributed by the juncture has generally been
considered proportional tb the fdurth power of the largest -
inscribed-circle; that is:
(9)
where: D = diameter of inscribed 'circle, and a. = a factor
depending upon, the ratios .!! and r. The valUes of a. were de-
m m
, ,
termined experimentally by soap -film tests which-gave the
following results for parallel flange sides:
a. = 0.094 + 0.070!,"
m
(10)
For flanges ,with sloping sides the following results-were
obtained. For slope ofl on 20: , a. = 0.066+0.02~: +0.072i
for slope of 1 on 50: a. ~ 0.084 + 0.007w + 0.071r
m m
SUinmari'zing the various elements we obtain-for. par--
allel-sided flanges:
. -2 ' I
' K = _bn3 +-(d-2n) .w3+ ,2a.D4 .. 0.42016n4 (11)3 ,3 -- ,
and for sloping flange sections:
K = b-W(m+n) (m2 +n 2 )+I(d-2m) .w3+2a.D-4Vn .n4 (12)6 3- - ,
The diameter of the inscribed circle is found by the following
formula for the parallel-sided flange sections~
D =--------
2r"", n
and ,for sloping flange sections:,
(13)
(14)
(r.8(~1 +1' -1- !L)+z\ 2+w(r+ !!)D =82 . 2r j 4.
r.S. (VI +1 ..;. 1 - -!!). + r + z
. S2. .' 2r
where z= the maximum flange depth shown in Fig. 2. The shear-
4
'. (15) .
ing stress is a function of the thickn'essof mater'ial ~nd the.:
following',equations were found to give fairly accurate maximum .
.. ,
shearing stress in ,the flange and in the ,web of structure:lbeam: '
For parallel-sided flange:
.... _T(n+O.3r}
:!'f - - K - .
For sloping flange sections:
= !Jm+O.3r)
K
and for the web:
. _. 'r.{w+O.3rt .
.'t"w - ,. K
(16)
(17)
For structural sections·restrained at the ends we, have
the following conditions. Referring' to Fig. 3 the torsional
moment is readily seen to be:
in which
T . = KG ~ = 2K.G.~
11 dx. h dx
(18)
(19) .
where: .K is the torsional constant for the section, and G is
the modulus of shear•. If !l. denotes the larger m~ment of in-
,2
ertia of one flange, we have:
Ely .d3 yQ. = _.
2 dx3
(20)
Thus:
5
(21)
Setting
a = h ,jElY.
2 KG
and reducing t we have:
'd3 y dv .hT' "
a 2 _.:=Jl..+-·-=.O
'dx3 dx, 2K·G (22)
The diff'erential equation (22:) has the gener~l solution:
" ,
y =A ,sinh 2£ +B cosh 2£,+ C + Yl (23)
a . 'a '
where: ' A, 'E, and C are determined' by t~e border conditions
'and Yl is the solution which satisfies equation' (22).
For a section fixed at both ends equation (~3) be-'
comes:
or for x = IJ.·. --t:...
Y = T·h·a (~- 2 tanh .L)2 K·Q a" .' 2a (25) ,
The bending moment in each flange is:
Ely d2 y ,T.a sinh (L _X)
-, 2a a
M = '2' .' dx2 = h cosh' L (26)
2a
the shear:
, (27)
5(28)
I h·ly
Mb· T•a •b sinh' (~ -!')
___ . a a
cosh·1·
. 2a
-=
I
=~=
and the +ongftudinal stresses at the outer fibres of· the
flanges:
The angular twist at the center of the' section is
given by:
. (29)
. d\lr __ 2. dv
ec = ai"h ~. By use of e-quation (24) we obtatn:
cosh ~ - 1.
T 2a
= KG cosh :::e:
2a
Due to the shortening of 'the beam during twisting. a correct-
ion factor must be applied when cros~-sections are restrained
from warping•. Professor Timoshenko* has shown' that a correct-
ion of: 1 + 2.95 1.: should be used, that is:
cosh ~ 1 .. b 2Q. :: L .2a (1 2 95) (30)
. c KG J)" + .• .~2 ".
cosh ~"
or the torsion con~tant for this case is:
t cosh ~t .i ~= T -K . .2a _..;...._--.-e•.G- cosh -1 1+2.95ba" 2a .' ~ (31) .
The theoretical study was substantiated by experiments on full
sized structural sections. T~e loading rig used for these test~"
is shown inFig.4• .An illustration of the observed stress dis-
tribution in "free-ended sections is sho"wn in Fig.5, 'and in fUlly
restrained s~ctions in Fig.5. Fig.? shows the~relationship ~e­
tween computed arid observed stiffnessesas well as Yl~eld points
of fixed-ended beams, and Fig.8 indicates the agreement between
the computed and observed shearing stresses in" the beams.
- - - - - - - - - ~ - - ~.- - - - - - - - - - - - - -
* S. Timoshenko STRENGTH OF MATERIALs, :Vo1. 1
:cr JJ
ro- b--J
lb) PART OF A SECTOR
1101 .
+-~,,-~ b .. I
(e) SLOPING FLANGE SECTION
~- q-'-~~
Fig. 1
Sections With
~loping Sides Fig. 2 - structural
Beam Sections
Fig. 3 Illustration or Twisting of Beam
Fig. 4 - Loading Arrangement for Twisting Beams
48
40
8
/'
V
/
/1/ I
Yield Point~ Strain Lines on
of Beam Fenter Line or FlangeI \Drop of Beam and
_§.J Strail Lines along lillet5
~: I _I~ 't'= 12000 Ib per sq in.by Formula~ II ~'t'= 12000 lb per sq in.by Tensometers
''''V (a) i-DIAGRAM
~ ~
~~
~~
fj,-E.I" 10
<n
Material
Yield Point
Stress Distribution
at Beam Yield Point
(b) SHEARING
STRESS
DISTRIBUTION
ON SURFACE
OF FLANGE AND
WEB
0.002 0.004 0.006
Unit Twist in Radians
0.008 0.010
7
Fig. 5 Free-Ended Tqrsion Test
~5"1
I I
I
I
Shear Tensometers
(el TOP VIEW OF FLANGE
Stiffeners
Tensometersfor
~DirectStress
Idl DIRECT STRESS IN OUTER FIBERS OF FLANGE AT
YIELD POINT OF THE BEAM .
5"
40f----f:c-----
r
I
40
__________ 6·0··c.- -.,
I
~ I:Jl 20 f----f---~---':-"...=-~.~ Tension in Top
~ ~ or-__O_"t_er_F_ib_er -=::::,-.."""';:- +~ .
~~V5=~ 0 20
~
0.0050.0040.002 0.003
Unit Twist in Radians
O.OOt
0
,/----I~
IF Yield Pointof Beam
)
.7 00p.
>~. .-
"It" I . (b) SHEARING STRESS
AT SECTION a-a AT 1
o . THE YIELD POINTI) I ~"''' " '"' .".
(a) f-DIAGRAM
I
12
~
c,
If. 80
~
"0
~
c
,
.,~ 60
c
c
E
!1
"g "40
'@
0=
100
o
o
Fig. ·6 Fixed-Ended Torsion Test
o \
o \
\
IO~\-'---'--1-'-,,/---,----I~,-----,
\ I", RELATION BETWEEN LENGTH AND STIFFNESS
\
\
\
-\---- -
\
\
\
o \
" 6 -- \-- ---c----!I-----l------I------j-----i
\
\
\ Theoretical
'r/
~ 4 f-'------'-\~ ----~-I-----~I---~__j
! , "f\('"'' ! I
2 __1~_'~11-1--+----1
4 __ _ __l J~~==±=---~---- _I I Comp",d FT'End'd SI.Hnm I ' '
I I I I.
</., RELATION BETWEEN LENGTH AND YIELD,POINT STRENGTH
I\ I6~\-\ \--I--f---j
~Th'O"'"''To,o" 10' "m•• =38900 I. po< so 'n.
, \ \,,~",.',-nlO-'8.-.
m
s,-.YT--+est---1----+-------j
8 \ ~Theoretical Torque fOf'Tm;u =21 800 lb per sq tn.
~ . I '1" '........... __ F,ee,End.d Y",'d-'~~=2' 8001. POI So ,n,
------~--::::-:::,,--i----,L--t---=--=--=--=.::.r-=.-====t-=:==
4 L-------:'IO=-----2:':O--,----:3'::-0------:40;;-----;';SO=----~60
length 01 Beam Inches
Fig.' " - Effeot ,of End Fixity On
Torsional Rigidity of Beams
,J
.. ~.
10 r----r--j---,--,-----,-----,----,------.------,
10
/
"c:::;
3·IN. I·BEAM 3B.96 IN. LONG
1----I~-I-___=_____",¥---+-----1
o 2
Distance from Center Line in Inches
("i TEST T-IO
-----..- --'--Ic----I-·~.-I----I
4
02--\------------Ii» TEST T-B 3·IN. I-BEAM IB.10 IN~ LONG
~
---
\ Jf\
\
----------
"
/
c k!/'~ :::;" Theoretical" c---~ ~ --- .6V/
.....~ ~~ _..;:;i
0
------------I---
10 8 6 4 2
\
\
\
B' -\~ ----
\
\
\
\
6 - -\--I----I----I----~---I----j----I
\
. 2
~
" 0~-_____t5~-~--{------,;----+----:\:---:;-~-~20 15 . 10 5 0 5 10 - 15 20
Distance from Cenler line in Inches
:10
o
...
'"• c
, I
r
;;
". c
,
o
Q.
Fig. 8 _ Relation Between Computed and Observed
Shearing stresses in Fixed-Ended Beams


A study of structural sections subjected to torsion, Journal of Faculty Sciences, University of Istanbul, Jan. 1937

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A study of structural sections subjected to torsion, Journal of Faculty Sciences, University of Istanbul, Jan. 1937

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