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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
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UNIVERSITY OF ILLINOIS BULLETIN
ISSUED WmCLyT
Vol. XXVII July 1, 1930 Wo.44
[Entered as second-claa matter December 11, 1912, at the post offile at Urbana, Ilinois, under
the Act of August 24, 1912. Acceptance for mailing at the special rate of postage provided
for in section 1108, Act of October 3, 1917, authorised July 81, 1918.]
THE TORSIONAL EFFECT OF
TRANSVERSE BENDING LOADS
ON CHANNEL BEAMS
BY
FRED B, SEELY
WILLIAM J. PUTNAM
WILLIAM L. SCHWALBE
BULLETIN No. 211
ENGINE RINV EXE1IM ENT 'TATION
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T HE Engineering Experiment Station was established by act
of the Board of Trustees of the University of Illinois on De-
cember 8, 1903. It is the purpose of the Station to conduct
investigations and make studies of importance to the engineering,
manufacturing, railway, mining, and other industrial interests of the
State.
The management of the Engineering Experiment Station is vested
in an Executive Staff composed of the Director and his Assistant, the
Heads of the several Departments in the College of Engineering, and
the Professor of Industrial Chemistry. This Staff is responsible for
the establishment of general policies governing the work of the Station,
including the approval of material for publication. All members of
the teaching staff of the College are encouraged to engage in scientific
research, either directly or in cooperation with the Research Corps
composed of full-time research assistants, research graduate assistants,
and special investigators.
To render the results of its scientific investigations available to
the public, the Engineering Experiment Station publishes and dis-
tributes a series of bulletins. Occasionally it publishes circulars of
timely interest, presenting information of importance, compiled from
various sources which may not readily be accessible to the clientele
of the Station.
The volume and number at the top of the front cover page are
merely arbitrary numbers and refer to the general publications of the
University. Either above the title or below the seal is given the num-
ber of the Engineering Experiment Station bulletin 6r circular which
should be used in referring to these publications.
F' For copies of bulletins or circulars or for other information 'address
THE ENOINEERINGa EXPVEIMEt T STATION,
UNIVERSITY OF ILLINOIS,
" *'U]W;NA, ILLINOIS1
K1^ ,^ ''' ,. '1 *'.' '1'^ *,^!^
UNIVERSITY OF ILLINOIS
ENGINEERING EXPERIMENT STATION
BULLETIN No. 211 JULY, 1930
THE TORSIONAL EFFECT OF TRANSVERSE
BENDING LOADS ON CHANNEL BEAMS
BY
FRED B. SEELY
PROFESSOR OF THEORETICAL AND APPLIED MECHANICS
WILLIAM J. PUTNAM
ASSISTANT PROFESSOR OF THEORETICAL AND APPLIED MECHANICS
WILLIAM L. SCHWALBE
ASSOCIATE IN THEORETICAL AND APPLIED MECHANICS
ENGINEERING EXPERIMENT STATION
PUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA
UNIVERSITY
5500 430 8127 0 PRSS 0
CONTENTS
PAGE
I. INTRODUCTION . . .. 7
1. Purpose of Investigation . . . . . . 7
2. Previous Work . . . . . . . . . . . 8
3. Scope of Tests and General Method of Testing. 9
4. Acknowledgment 9
II. SHEAR CENTER FOR CHANNEL SECTIONS . . . . . 10
5 Approximate Location of Shear Center . . . . 10
6. Experimental Determination of Shear Center . . 14
7. Observed Values of Longitudinal Stress in Channel
When Loaded through Shear Center . . . 23
III. LONGITUDINAL STRESS IN CHANNEL WHEN THE TRANS-
VERSE LOADS PRODUCE TWISTING WITH THE BENDING 25
8. Influence of Twisting on Longitudinal Stresses. 25
9. Observed Values of Longitudinal Stress in Channel
When Load Did Not Pass through Shear Center 30
10. Simplified Approximate Analysis of Longitudinal
Stresses. . . . . . . . 34
11. Additional Longitudinal Stresses . . . . . . 37
12. Significance of Effect of Twisting of Channel Beams 39
IV. CONCLUSION . . . . . . . . . . . . . 40
13. Summary . . . . . . . . . . . 40
APPENDIX. BARS OF NARROW CROSS-SECTION SUBJECTED TO
TORSION WITH RESTRAINT
I. INTRODUCTION . . . . . . . . . . . . . 43
1. Purpose of Appendix . . . . . . . . . 43
2. Types of Sections . . . . . . . . . . 43
3. Loads and Restraints . . . . . . . . . 43
4. Assumptions. . .. . . . . . . 43
5. Method of Analysis . . . . . . . . . 43
II. TRANSVERSE BENDING WITHOUT TORSION . . . . . 45
6. The Shear Center in General . . . . . . . 45
7. The Shear Center for Channel Sections . . . . 46
4 CONTENTS (Concluded)
III. TORSION WITHOUT RESTRAINT . . . . . . . . 49
8. Torsion Formula . . . . . . . . . . 49
IV. TORSION WITH RESTRAINT . . . . . . . . . 50
9. Torsion with Restraint Equivalent to Bending Plus
Torsion without Restraint . . . . . . 50
10. Center of Rotation for an Angle Section . . . 51
11. Center of Rotation for a Channel Section . . . 52
12. Longitudinal Stress Formula . . . . . . . 53
13. Bending Shear Couple . . . . . . . . . 55
14. Differential Equation for Torsion with Restraint . 56
15. Variation of Torsional Couple and Bending Shear
Couple along Length of Bar. . . . . . 58
16. Bending Moment M' in the Half Section . . . 59
17. Computation of Longitudinal Stresses in a 6-in.
15.3-lb. Ship Channel. . . . . . . . 60
LIST OF FIGURES
NO. PAGE
1. Stress in Channel Loaded Through Shear Center . . . . . . . . 13
2. Sketch Showing Arrangement of Beam for Testing . . . . . . . 14
3. View of Channel with Load Applied through Shear Center at End of
Cantilever . . . . . . . . . . . . . . . . . . 15
4. View of Channel with Load Applied Approximately through the Centroid
of End Section . . . . . . . . . . . . . . . 16
5. Curves Showing Relation between Angle of Twist and Distance from
Fixed End of Cantilever. . . . . . . . . . . . . . 18
6. Location of Shear Center Determined from Test Data . . . . . . 20
7. Observed Longitudinal Stress in Channel Loaded through Shear Center 24
8. General Torsional Effect of Transverse Load on Channel Beam . . . 26
9. Observed Longitudinal Stress in Channel Loaded Approximately through
Centroid .. . . . . . . . . . . . . . . . . . 28
10. Observed Longitudinal Stress in Channel Loaded Back of Shear Center 31
11. Approximate Analysis of Longitudinal Stress in Channel When Load Is
Not Applied through Shear Center . . . . . . . .. . 35
12. Types of Sections . . . . . . . . . . . . . . . . 44
13. Channel Beam Subjected to Torsion with Restraint . . . . . . . 44
14. Section of Channel Loaded through Shear Center . . . . . . . 46
15. Equilibrium of Element of Channel. . . . . . . . . . . . 47
16. Rotation of Channel Subjected to Torsion. . . . . . . . . . 51
17. Rotation of Angle Subjected to Torsion . . . . . . . .. . 52
18. Center of Rotation for Channel Section . . . . . . . .. . 53
19. Section of Beam Subjected to Unsymmetrical Bending . .. . . 54
20. Circle of Inertia . . . . . . . . . . . . . . . . 55
21. Circle of Inertia Applied to Unsymmetrical Bending. . . . . . 55
22. Bending Shear Couple .... . . . . . . . . . . 56
23. Channel Beam with Load Not through Shear Center . .. . . . 59
24. Graphical Determination of Section Factors for a Ship Channel . . . 61
LIST OF TABLES
1. Experimental and Calculated Values of the Distance of Shear Center from
Center Line of Web . . . . . . . . . . . . . . 22
2. Values of a ... . . . . . . . . . . . . . . . . . 37
3. Values of Additional Longitudinal Stress at B Due to Load Applied
through Centroid of Channel Cross-section . . . . . . . . 38
4. Values of Additional Longitudinal Stress at A Due to Load Applied Back
of Shear Center of Channel Cross-section. . . . . . . .. . 39
5. Values of J for Various Channel Sections . . . . . . . . . . 50
6. Longitudinal Stresses in 6-in. 15.3-1b. Ship Channel . . . . . . . 63
THE TORSIONAL EFFECT OF TRANSVERSE BENDING
LOADS ON CHANNEL BEAMS
I. INTRODUCTION
1. Purpose of Investigation.-The shapes of steel rolled I beams
and channels have been designed with the main purpose of giving
large resistance to bending in accordance with the simple flexure
theory for beams; the torsional strength and stiffness of such sections
are known to be relatively small. In order to develop the bending
resistance of I beams and channels without permitting them to twist,
it has been very generally assumed that transverse bending loads on
these sections should be applied through the centroids of the trans-
verse cross-sections and parallel to the webs. In the case of I beams
this assumption is correct, but when a channel is so loaded it twists
appreciably as it bends.
In order to cause a channel beam to bend without twisting, and to
develop stresses in accordance with the simple flexure formula for
beams, the loads must be applied in a plane parallel to the web but at
a considerable distance back of the back of the channel. The inter-
section of this plane of loading with the neutral surface is called the
"axis of bending"; and the intersection of this axis with a transverse
cross-section of the beam is called the "shear center" for the section.
When I beams and channels are free from lateral restraint, and
hence are free to twist, and are loaded so that the transverse bending
loads do not pass through the axis of bending, then these members
will twist as they bend. This twisting of the beam will produce large
additional longitudinal stresses, provided that the beam has a trans-
verse cross-section that remains plane and hence does not warp as the
beam twists, such as the center section of a simple beam subjected to
symmetrical loading, or the fixed end of a cantilever beam.
Only channel beams were tested in the investigation herein re-
ported. In some of the channels tested the additional longitudinal
stress due to the twisting caused by a load applied through the cen-
troid of the section was more than 50 per cent of the stress that the
same load developed when applied through the shear center. The
maximum additional stress occurred at the section that was com-
pletely restrained from warping.
It is evident, therefore, that in the use of steel rolled channels as
beams it may be important to determine where the loads should be
ENGINEERING EXPERIMENT STATION
applied to produce bending without twisting, and also to determine
the longitudinal stress in a channel beam if the channel is loaded so
that twisting accompanies the bending. It is the purpose of this
bulletin to present the results of tests and of analysis that will help
toward a solution of these problems.
2. Previous Work.-In 1909-10 Prof. C. Bach* published the re-
sults of bending tests of a 30 cm. (11.8 in.) steel rolled channel used as
a simple beam loaded with two equal concentrated loads, at the third
points. The loads were applied through the centroids of the cross-
sections. He measured the strains at the center of the beam along
each of the four edges, and found that the strain along one edge of the
top was much greater than that along the other edge of the top, and
that a similar condition existed on the bottom edges. The experi-
mental results thus indicated that the usual flexure formula gave
values of stress in a channel beam largely in error when the channel
was loaded according to the conditions that had been assumed to
make the flexure formula applicable. Professor Bach also applied the
loads through the center of the web and found that the channel acted
more nearly in accordance with the flexure formula, but nevertheless
the discrepencies between the stresses found from the measured
strains and the values given by the flexure formula were appreciable.
Professor Bach did not offer an explanation of the inconsistency be-
tween the test results and the assumed theory.
These tests, however, served to direct attention in a practical way
to the difference in the behavior of a channel in bending under trans-
verse loads acting through the centroids of the cross-sections from
that of an I beam similarly loaded, but no satisfactory analysis of the
problem appears to have been given until about ten years later.
In 1920 H. Schwyzer presented a thesis at the University of Zurich
in which the shear center was determined by mathematical analysis.
In 1920-21 R. Maillart,t A. Eggenschwyler,f and H. Zimmerman§ by
somewhat different analyses brought to the attention of engineers the
location and significance of the shear center for channels and some
other thin sections. These investigators also obtained mathematical
expressions for the additional longitudinal stress caused by the twist-
ing of a channel when the transverse loads on the channel do not pass
through the axis of bending, and they found that the experimental
*Zeit. d. Vereins deutscher Ingenieure, 1909, p. 1790, 1910, p. 382.
tSchweiz, Bauz., Vol. 77, 1921, p. 195.
$Schweiz, Bauz., Vol. 76, 1920, p. 266.
§Bauingenieur, Vol. 2, 1921, p. 202.
EFFECT OF TRANSVERSE BENDING LOADS ON CHANNEL BEAMS 9
results of Bach were in substantial agreement with the results of their
analyses.
In 1925 Foppl and Huber* tested a 10.3-in. channel as a simple
beam on a span of 20.8 ft. with a concentrated load of 5500 lb. at the
middle. They also made a test of the same beam with two concen-
trated loads at the third points. They measured angles of twist and
longitudinal strains at various sections for different lateral positions
of the load. Only one beam was tested but the results of the tests
agreed well with the calculated position of the shear center, and with
the values of the stresses given by the mathematical analysis of the
stresses in the channel; the mathematical analysis used was a modi-
fication of Timoshenko's analysis,t made in 1910, of the stresses in
an I beam subjected to transverse loads that cause twisting.
3. Scope of Tests and General Method of Testing.-The channels
tested in the investigation herein reported were the following:
4 in.- 5.3 lb. 6 in.-15.2 lb. (heavy section)
6 in.- 8.2 lb. 10 in.-15.3 lb.
6 in.-15.3 lb. (ship channel) 15 in.-33.9 lb.
These channels offer a rather wide range in values of the ratio of flange
area to web area and in relative dimensions.
Each channel was tested as a horizontal cantilever beam with a
vertical load applied at the end. The load consisted of a weight hung
on the free end of the beam, the other end being restrained so that the
section remained practically plane.
For each channel the angle of twist was measured at six or seven
sections along the length of the beam for each of several lateral posi-
tions of the load, the load being applied at two or three points on each
side of the back of the channel (see Figs. 2, 3, and 4 for general
arrangement).
From the results thus obtained the position of the load that causes
no twisting of the channel is easily found, and the location of the shear
center for each channel section is thereby determined.
Also, for each of several lateral positions of the load the strains
along each of the four edges of the channels were measured at several
sections along the beam, the strains being measured over a 2-in. gage
line. Thus the effect of the twisting of the channel on the longitu-
dinal stresses at different sections along the beam was determined.
4. Acknowledgment.-The main body of the bulletin is the work
*Bauingenieur, Vol. 6, 1925, p. 455.
tZeitschrift fir Math. und Physik., Vol. 58, 1910.
ENGINEERING EXPERIMENT STATION
chiefly of the first two authors, and the analysis and results given in
the Appendix are due to the last-named author.
The tests herein reported were made as a part of the work of the
Engineering Experiment Station, of which DEAN M. S. KETCHUM is
the director, and of the Department of Theoretical and Applied
Mechanics, of which PROF. M. L. ENGER is the head.
II. SHEAR CENTER FOR CHANNEL SECTIONS
5. Approximate Location of Shear Center.-As stated in Section 1
the shear center for a channel section is the point on the axis of sym-
metry through which the plane of the loads must pass, parallel to the
web, to cause the channel beam to bend without twisting, in accord-
ance with the simple flexure theory of beams.
A simple method* of determining the approximate position of the
shear center of a channel section is as follows: Let Fig. la represent a
channel beam (assumed to be a cantilever beam for convenience)
subjected to loads P1, P2, etc., that lie in a plane, parallel to the web,
so located that the loads cause the channel to bend without twisting,
and cause stresses on any transverse section in accordance with the
flexure formula as indicated by the stresses shown acting on section
AB (Fig. ib).
At any transverse section such as CD the bending stresses and
shearing stresses together hold in equilibrium the external forces P1,
P2, etc., that lie to one side of the section. The vertical shear or
resultant of the loads P1, P2, etc., is denoted by V. By introducing
two equal and opposite forces at the shear center 0, each equal to V,
the vertical shear may be resolved into a force V1 = V at 0 and a
couple, the forces of which are V and V2, constituting a bending
couple whose moment is M. This bending moment M is held in
equilibrium by the bending stresses on the section CD similar to those
shown on section AB (these bending stresses on CD are not shown in
Fig. lb). The vertical force Vi = V causes an equal (but opposite)
resisting shear V,, which will be assumed to be developed only in the
web of the section, and to act along the center line of the web, since
the web is narrow. The forces Vi and Vr form a twisting couple
whose moment is Ve, where e is the distance from the center line of
the web to the shear center 0. For equilibrium this twisting moment
*See A. Ostenfeld, "Teknisk Elasticitetslaere," Fourth Edition (1924) for a similar method.
EFFECT OF TRANSVERSE BENDING LOADS ON CHANNEL BEAMS 11
must be equal (and opposite) to the twisting moment of the lateral
shearing forces H in the flanges of the channel. That is,
Ve = Hh
where h is the distance between the center lines of the flanges, but
may be considered without serious error to be the full depth of the
section.
A value of H may be found as follows: The difference T2 - Ti =
dT of the total bending stress in two sections of a flange the distance
dx apart causes a longitudinal shearing stress on the area connecting
the flange with the web; this area is tdx, where t is the thickness of the
flange (in a rolled steel channel t will be taken as the mean thickness
of flange). If s, denotes the shearing unit stress on this area, then,
for equilibrium,
s8 t dx = dT
But the total bending stress T in one flange at any section is s bit
h
M
where bi is the width of the outstanding flange, and s - is the
bending unit stress in the flange; I is the moment of inertia of the
whole cross-section of the channel with respect to the centroidal axis
parallel to the flanges. Thus
Mh
T = - bit
21
dM h bit
and dT = -
21
where dM is the difference in the bending moments at the two sec-
tions. Therefore
dM h bit
s, t dx -
2I
dM h bi Vh bi
or s - dx 21I 21I
12 ENGINEERING EXPERIMENT STATION
dM
in which the familiar relation d- = V is used. As noted, s, is the
value of the longitudinal shearing unit stress on the area connecting
the web and flange. But an equal shearing unit stress must exist at
the same point on the transverse section of the flange, and this lateral
shearing unit stress decreases to zero at the outer edge of the flange.
It will be assumed that this variation is a linear one, and hence the
Vhbi
average shearing unit stress in each flange is 4 I . The total lateral
shear, H, in each flange then is
Vhb,
H=- bit
and the moment of these H forces is
Hh = Vbilh2t
Hh-
41
Since the moment Ve must balance this moment in order to prevent
the beam twisting we have
Vb dh' t
Ve -= 41 (1)
But I =2 bit + - wh = - bth2 1+ -
in which w is the thickness of the web.
1 1
-bx -bi
2 2
Hence, e =wh = 1a (2)
1+- 1
6 bit 6 as
in which a, is the area of the web and a1 is the area of one outstanding
flange. This expression shows that e is large when the ratio of the
area of the web to the area of the outstanding flange is relatively
small, but that the value of e is always less than one-half the width of
the outstanding flange. In the tests herein reported e was found to
EFFECT OF TRANSVERSE BENDING LOADS ON CHANNEL BEAMS 13
(qv/oroA)
FIG. 1. STRESS IN CHANNEL LOADED THROUGH SHEAR CENTER
vary from approximately 0.20 to 0.36 of the width of the outstanding
flange, or from 0.18 to 0.32 of the full width of the flange; and the
ratio -- varied approximately from 1.7 to 5.7 in the channels tested.
a1
The foregoing approximate analysis, although useful, does not yield
values of e that agree closely with the experimental value of e for the
larger channels; this is discussed further in Section 6.
A slightly more general expression for e is obtained in Section 7 of
the Appendix by a different method of analysis. The expression is
I',, h
e =- (3)
in which I', is the product of inertia of one of the halves of the cross-
section of the channel-shaped beam with respect to rectangular axes
passing through the center of the web, one of the axes being an axis of
symmetry, h is the distance between the intersections of the center
lines of the flanges with the center line of the web, and I, has the same
meaning as has I in Equation (1). Equation (3) can be applied to a
section in which the flanges are not perpendicular to the web; it
reduces to Equation (2) for the section considered in Fig. 1.
ENGINEERING EXPERIMENT STATION
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