A study of structural sections subjected to torsion ...

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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. 1937I. Lyse

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Recommended CitationLyse, I., "A study of structural sections subjected to torsion, Journal of Faculty Sciences, University of Istanbul, Jan. 1937" (1937). FritzLaboratory Reports. Paper 1156.http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1156

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. -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:. 3K = 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 bn

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 thisproblem has been pUblished in a paper: STRUCTURAL BEAMSIN 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 ~ngineering MaterialsLehigh 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 + 0.071rm 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. dvec = 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 JJro- b--Jlb) PART OF A SECTOR

1101 .+-~,,-~ b .. I

(e) SLOPING FLANGE SECTION~- q-'-~~

Fig. 1Sections 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 onof 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

MaterialYield Point

Stress Distributionat Beam Yield Point

(b) SHEARINGSTRESS

DISTRIBUTIONON SURFACE

OF FLANGE ANDWEB

0.002 0.004 0.006Unit Twist in Radians

0.008 0.010

7

Fig. 5 Free-Ended Tqrsion Test

~5"1I I

I

I

Shear Tensometers

(el TOP VIEW OF FLANGE

Stiffeners

Tensometersfor~DirectStress

Idl DIRECT STRESS IN OUTER FIBERS OF FLANGE ATYIELD POINT OF THE BEAM .

5"

40f----f:c-----

rI

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 0.003Unit 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,

.,~ 60c

cE!1

"g "40'@0=

100

oo

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\('"'' ! I2 __1~_'~11-1--+----14 __ _ __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.-.ms,-.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 OnTorsional Rigidity of Beams

,J

.. ~.

10 r----r--j---,--,-----,-----,----,------.------,

10

/

"c:::;

3·IN. I·BEAM 3B.96 IN. LONG

1----I~-I-___=_____",¥---+-----1

o 2Distance 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

---~~ --- .6

V/.....~ ~~ _..;:;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

:10o...

'"• c, Ir

;;

". c,o

Q.

Fig. 8 _ Relation Between Computed and ObservedShearing stresses in Fixed-Ended Beams