\dS.
OF TECHNOL
DEC
2 1960
LIBRAR
BUCKLING OF CHANNEL FLANGES DURING
BENDING IN THE WEAK DIRECTION
.by
RICHARD C' HASKELL
BE-CE
University of Southern California
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August 1960
.........
,
..
...
Signature of Author........
Department of Civil and Sanitary
Fngineexing, August 22, 1960
Certified by. ..
/Thesis
Supervisor
Accepted by.....
t7
i............
Chai7 an,
Graduate
Qmmittee on
s
~2
ACKNOWLEDGEMENTS
The author would like to express his
sincere thanks to Professor J. M. Biggs, whose guidance
and patient advice made this thesis possible.
Appreciation is
also extended to Don Gunn for his pre-
paration of test specimens and Saul Nuccitelli for his
advice on strain gage techniques.
ABSTRACT
BUCKLING OF CHANNEL FLANGES DURING
BENDING IN THE WEAK DIRECTION
by
RICHARD C. HASKELL
Submitted to the Department of Civil and Sanitary Engineering on August 22, 1960 in partial fulfillment of the
requirements for the degree of Master of Science.
The problem of channel flange crippling during
bending about an axis parallel to the web had been given
very little treatment since it did not occur too often
in practice. One rigorous analysis and three approximate
analyses of determining critical flange stresses were
discussed, and experiments were performed to spot check
these theories.
A method of determining the ultimate failure
moment for channels was investigated from a semi-empirical approach.
Thesis Supervisor.:
Title:
J. M. Biggs
Associate Professor of Structural Engineering
TABLE OF CONTENTS
Page
Summ
...
ry....
...
...
...
...
1
..
Work...................
2
1.0
Previous
2.0
Theoretical Approaches.............
3
2.1
Method (a)........ ...............
Assuming an Infinitely Long Uniformly Stressed Hinged Flange
7
2.2
Method (b)...................
.
.
Assuming an Infinitely Long Uniformly Stressed Partially Restrained
Flange
7
2.3
Method (c).........
Bijlaard's Theory Assuming a
Uniform Stress Coefficient of
Restraint
8
2.4
Method (d)........................
Bell Aircraft Solution of Channel
in Bending about Axis Parallel to
Web
12
2.5
Ultimate Strength Considerations...
14
3.0
Procedure............
.
15
3.1
Method ofAttack...................
15
3.2
Description of Apparatus...........
17
3.3
Description of Procedure...........
17
3.3.1
Tests to Determine Material Propert ies............................
17
3.3.2
Tests of Channel Sections..........
17
3.4
Methods of Making Computations and
of Plotting Curves.................
l8
3.4.1
Determination of Material Prop ert ie s............................0 0 0 0 0
18
3.4.2
Plotting Values of k vs. bw/bf.....
19
3.4.3
Design of Channel Sections.........
19
.
......
Page
Calculation of Section Pro
perties using Actual Channel
Dimension.................
21
Predicted Buckling Stresse s
using Actual Dimensions...
21
3.4.6
Reduction of Channel Test Data. .
24
3.4 .7
Determination of k at
cr ......
27
3.4 .8
Ultimate Moment.................
28
3.5
Sources of Error................
28
4.0
Results...........
33
5.0
Discussion of Results...........
63
5.1
Critical Moment.................
63
5.2
Ultimate Moment.................
63
6.0
Conclusions.....................
67
6.1
Cbitical Moment.................
67
6.2
Ultimate Moment.................
67
7.0
References......................
69
8.0
Appendices......................
70
3.4.4
3.4.5
...........
List of Figures
Figure
No.
1
Title
Page
Tensile Testing Apparatus............ 71
2
Tensile Testing Specimen in Machine ......
71
3
Channel Test Sections Showing Strain
Gages...........
.. .
..............
72
4
Channel Section in Testing Machine ....... .72
5
Channel Section and Strain Indicator ..... .73
6
Balancing Strain Indicator ............... 73
7
Channel Testing Apparatus................ 74
8
Tensile Specimen ............
9
Channel Section.....
..
75
............. 76
10
Table of Calculated Buckling Stresses
from Actual Channel Dimensions ........... .24
11
Table of Percent Error between Plotted
Elastic Stresses and Calculated Elastic
Stresses ................
29
Table of Percent Error between Calculated
Stress Diagram Moment and Actual Applied
Moments..................................
30
Table of Percent Error between Calculated
Critical Stresses and Experimental
Critical Stresses........................
32
Channel Dimensions and Section
Properties...............................
34
Tensile Test Log Sheets
Specimens 1, 2, and 3...........------
35,36
Tensile Test Log Sheets
Specimens, 4, 5 and 6...................
36,37
Tensile Test Log Sheets
Specimens 7, 8, 9, and 10................
38,39
12
13
14
15
16
17
18
Tensile Stress-Strain Curve
Specimen 4.............................. o
19
20
Page
Title
Figure
No.
Tensile Stress-Strain Curve
Specimen 5............................
.41
Tensile Stress-Strain Curve
Specimen 6............................
42
21
Tensile Stress-Strain Curve
Specimen 7............................... 43
22
Tensile Stress-Strain Curve
Specimen 8.............................
44
Tensile Stress-Strain Curve
Specimen9...........................
45
Tensile Stress-Strain Curve
Specimen 10..............................
46
Channel Buckling Test Log Sheets,
Channel #1...............................
47
Channel Buckling Test Log; Sheets,
..............
Channel #2........ ...
48
Channel Buckling Test Log Sheets
Channel #3..............................
49
23
24
25
26
27
28
Channel Buckling Test Moment-Strain
Curves: Channel #1,
29
0D ,3
,and(.
Channel Buckling Test Moment-Strain
Curves: Channel #1,
30
Gages
Gages
O
3
.--.
Channel Buckling Test Moment-Strain
Curve : Channel #2, Gages
,5,
and.
............................
..-
31
Channel Buckling Test Mome t-Strain
Curves: Channel #2, GagesQ(
}, and
32
Channel Buckling Stress Moment-Strain
Curves: Channel #3, Gages Q,
}
...
0........ 0.
6
33
34
35
36
50
5
52
,
54
Channel Buckling Stress Mo ent-Strain
Curves: Channel #3, GagesQ9 and 0 )....
55
Table to Plot Stress Distribution Curves
from Moment-Strain Curves................
56
Channel Buckling Test Stress Distribution
Curves, Channel #1-.......................
57
Channel Buckling Test Stress Distribution
Curves, Channel #2......................
58
Figure
No.
Page
Title
37
Channel Buckling Test Stress Distribution
Curves, Channel #3 ....................... 59
38
Check of Stress Distribution Curves by
Area and Moment Balance..................
60
39
Values of k vs. bW/bf from Test Results
and Theoretical Methods (a), (b), (c),
and (d).................................. 61
40
Table of Percent Error between Actual
Ultimate Moment and the Predicted
Ultimate Moment..........................
67
41
Table Calculating k Values for Methods
(b) and (c)............................... 62
42
Illustration of Stress Distribution
Factor, OC................................
77
Illustration of Stress Distribution in
Channel Section..........................
77
Notation for Analysis of Flange by
. .
....
. . ..
Method ()...
78
43
44
45
...
Theoretical Ultimate Moment Stress
Distribution.............................
78
1
Summary
The object of this thesis was to spot check
various theoretical methods of determining critical
buckling stresses, and to formulate an approach to finding the ultimate moment a channel can resist when loaded
in bending in
the weak direction.
Four theoretical buckling analyses were discussed, and tests were run on three different channel
sections.
Critical buckling moments and ultimate
moments were determined by observation of specimens
and analysis of strain gage data.
Of the four methods shown in figure 39, it
was found that method (a) was the best method of predicting critical buckling stresses for materials with
proportional limit below 18.0 ksi.
For material with
a higher proportional limit method (b) was used to be
conservative.
Methods (c) and (d) indicated large
errors on the conservative side, especially at the low
buckling stresses.
A semi-empirical method of predicting ultimate
strength was discussed, and formulas were developed.
How-
ever, the results were inconclusive and more test data
was required to substantiate the theory.
A conservative
formula for predicting ultimate strength was
Mult
'( crb2t
(19a)
2
1.0
Previous Work
A search of the literature showed that very
little work had been done on this subject.
Tne people
most interested in formed sneet metal sections were in
the air frame industry and these people were usually
concerned with minimum weight design.
For this reason
it was found that a channel section in bending about an
axis parallel to the web was rarely employed in practice.
There were two theoretical approaches to finding
the critical buckling stress of these channel flanges
which had been presented to date.
The first approach
was derived by Bell Aircraft Corporation in an unpublished
report with the aid of references 2, 3, and 4.
The results
of this report were shown in a graph in reference 1.
Another study which is analogous to the problem
was done by Bijlaard and presented in reference 5.
Bijlaard derived formulas for critical compressive stresses
of hinged and fixed flanges under a linearly varying
stress.
The critical stress for a partially restrained
channel flange was somewhere in between the cases of
pinned and fixed flanges.
Since the available methods
of arriving at the amount of restraint were approximate
the Bijlaard theory did not seem as accurate as the
graph used by Bell Aircraft.
Another approach to flange buckling problems
which was often used by aircraft designers was based on
ultimate strength theory.
A semi-empirical method that
gives good results for uniformly loaded flanges was given
in
reference 9.
However, for the case under consideration
another ultimate strength approach was developed.
There was no previously available test data
on either critical buckling stress or ultimate strength
which could be found.
2.0
Theoretical Approaches
A firm understanding of plate buckling theory
was necessary to analyze this problem correctly.
The
basic formulato determine the critical crippling stress
in plates or plate elements with various compressive
stresses was
klT2 YV
cr
E
12(1 - V 2)
2
($)
where:
V/=
Poisson's ratio
UCr = critical compressive crippling stress - ksi
t = plate thickness - inches
b = plate width - inches
= plasticity coefficient used in reference 2
for stresses in the inelastic range
7= Et/E where Et = tangent modulus, the slope of
the stress-strain curve at any particular point.
k = constant which is dependent on the restraint
of the plate along its unloaded edges and the
distribution of the stress across the width
of the plate.
4
The basic differential equation for instability
of plates in the elastic range was
Et3
t72
12(l -1)
d 4w
4w
Wa
(. 4 +2 j
dx dy
J
4w
+ )+i
c y
C)2
x
t
2
o
(2)
whereDT
w
* stress in the direction of loading
* plate displacement perpendicular to the plane
of the plate
For the case of stress in the inelastic range,
equation (2) was modified.
different modifications.
Different authors gave
Probably the most widely ac-
cepted plasticity hypothesis was derived by Stowell in
reference 8, but the results were far too complicated
to be used in design.
A more simple approach was given
by Bleich in reference 2, and was sufficiently accurate
for practical purposes.
'Whenthe buckling stress exceeded the proportional limit of the material, Young's modulus, E, no longer held.
Bleich assumed that when
exceeded pro-
portional limit, the tangent modulus, Et was effective
in the direction of loading and Young's modulus, E,
was effective in the direction perpendicular to loading.
In equation (2) the three terms in parenthesis were
noted.
The first term corresponded to bending of strips
parallel to the x axis.
These strips were stressed
by the longitudinal force, rext.
modified to read
<)x(
This term was then
In the same manner the third
5
term corresponded to strips in bending perpendicular
to the x axis which were free of externally applied stresses.
Therefore this term remained unchanged.
The middle term
in parenthesis was associated with the distortion of a
square plate due to twising moments on the element.
This term was effected by plastic action in a complicated
way, and was multiplied by a coefficient having a value
somewhere between 1 and 7.
what arbitrarily.
Et3
2
-
12(1 -V
) (7'
C
Yx
The value V-7 was used some-
This equation (2) became
w
+2 2
M4
)X
+
w
by
-4)+
dy
w
024
Ix
(2a)
Poisson's ratio:, /
,
was effected slightly in
the inelastic range, but since the effect of if on equation (2a) was small, the change due to inelastic behavior
was ignored.
Solution of equation (2a) resulted in the
algebraic plate crippling equation, (1).
It was known
that the plasticity coefficient lay somewhere between
Es/E and Et/E where Es was the secant modulus.
was a conservative value and was con-
value of
sidered as a lower limit for most cases.
7
Bleich's
Bleich defined
as follows(3)
~cr
Et/E _ (Fy ~ cr)
(UOy Up)OC'p
''
material yield stress - ksi
Cr-=
y
=7
material proportional limit
-
ksi
6
It was noted that since 7' was dependent upon
r-'cr,-a trial and error solution of (1) was necessary.
This was avoided by algebraic manipulation of (1) to read
r2
k 7722
12 (1 -1/ 2)
CFyc
Tables were available to determine
values of OrcVr
(tW2
(la)
b
Orcr from corresponding
for various materials.
For the case of a sheet metal channel in
bending in the weak direction the only unknown in (1)
was the constant, k.
Four different methods of deter-
mining k were considered, and were presented in figure 39,
as a function of (web depth/flange width).
These methods
were summarized as follows+- (a) An infinitely long
hinged flange under uniformly distributed stress was
assumed.
(b) An infinitely long partially
flange under uniform stress was assumed.
restrained
(c)
(Bijlaard's
analysis) The flange was assumed infinitely long and
under a linearly varying stress.
A coefficient of
restraint proportional to the coefficient for a uniformly
distributed stress was used.
(d) (Bell Aircraft graph)
The actual case of a channel loaded in bending in
the
weak direction was assumed.
This paper was only concerned with the most
common case where the web thickness equaled the flange
thickness and the unsupported flange length was great
compared to the flange width.
--I1
7
2.1
Method (a):
Assuming an Infinitely Long.Uniformly
Stressed Hinged Flange
Solution of equation (2a). using the boundary
conditions of method (a) for an infinitely long hinged
flange under uniform stress led to the basic algebraic
equation (1) for crippling of plates.
In this case the
value for k was .425 as shown by the straight line in
figure 394,
This well known salution was presented by
Timoshenko in reference 7.
2.2
Method (b):
Assuming an Infinitely Long Uniformly
Stressed Partially Restrained Flange
The case of method (b)
for a restrained flange
was solved by Bleich in reference 2.
Bleich introduced
the concept of the coefficient of restraint, ef,
determine the value of k.
to
The coefficient of restraint
expressed the amount of fixity provided to a plate
element, by the adjoining plate elements.
It depended
upon the cross section dimensions and the stress distribution.
For the case of a channel under uniform compres-
sive stress, the coefficient of restraint for the flanges
was
tf3
bw
tw 3
bf
1
1
.106 (tf/tw)2 (bw/bf)2
-
where:
t
=
flange thickness
-
inches
tw = web thickness -inches
b
= flange width -inches
bw = web width - inches
8
This equation was valid for 9.4 (tw/t9) 2 (bf/bwfel.0.
When
the above value was less than 1.0 the web plate crippled
at a lower load than the flange,
so the flanges then pro-
vided restraint for the web.
To determine the value of k the following
equation was used:
kr=
2
+ .65)2
(5)
where:
kr = k for the restrained case under uniform load.
For the case being considered where t
= t wk
was found as a function of bw/bf in
Figure 41 and plotted
in figure 39.
2.3
Method (c):
Bijlaard's Theory Assuming a Uniform
Stress Coefficient of Restraint.
Mothdd (c) 6d finding 0-cr of a channel was
taken from Bijlaard's paper, reference 5. BiJlaard
analyzed hinged and fixed flanges of infinite length
subjected to a stress that varied linearly with flange
width.
An energy approach was used and the equations
were solved by a method of finite differences.
The flange
was assumed to buckle in the shape of a sine curve in the
longitudinal direction for both the fixed aid pinned
cases.
In the lateral direction, for the fixed case the
deflection was expressed in terms of the normal mode of
vibration of a cantilever beam.
For the hinged case the
9
flange deflection was assumed to increase linearly
with the distance from the hinge.
For application to
the case under consideration, two modifications were
made.
To arrive at a plasticity factor,
',
Bijlaard
published graphs that expressed a plasticity constant
in terms of Es/E at the edge of highest strain.
This
constant was then plugged into an equation to determine -,
and this factor,
in place of
r
i[, was
in turn used in equation (1)
to find rcr.
The advantage of
Bijlaard's plasticity factor was that the graphs were
applicable to all materials, but values of Es/E
were not readily available for various materials.
Also
the method involved a trial and error solution because
was dependent on c'cr.
Since values of Es/E vs.
stress
for mild steel sheet could not be easily plotted, and
since Bijlaard's method involved a great deal of arithmetic, it was decided to use Bleich's
coefficient,
f,rin place of 7.
plasticity
The error involved
was slight.
The second modification to Bijlaard's theory
was the determination of a proper coefficient of restraint.
The procedure was to find values of k for
Bijlaard's hinged and fixed cases, and then determine
an intermediate value proportional to the intermediate
value for the case of a uniformly stressed channel given
10
in reference 2.
This gave results which were generally
conservative as expected, since a web in tension during
bending supplied more restraint than a web in compression
during axial loading.
Values of k vs. bw/bf were plotted in figure 39
(b), and
and compared to values obtained by methods (a),
(d).
These values were determined by the following al-
gebraic manipulation of Bijlaard's equations, and appeared
in figure 41.
In reference 1, k for the fixed and pinned cases
was expressed as a function of
M, where O is an ex-
pression for the linear stress distribution as shown in
Figure 42.
ah (compressive stress
oC = 1 - w
=
+
)
(6)
where:
=
stress at hinged or restrained edge
-
ksi
= stress at free edge -'ksi
To express
OC
in terms of bbf, it was
necessary to locate the elastic neutral axis, y, of the
channel.
A
A
From figure 43,
bf(bf/2) + (b12)bf
+b/2
bf
(b
(bf + bw)
2+b
1
/bf
11
= -(b
- Y)
.. [b
b + b
- 2 +bV /b
b
+ b/br
+ bw
2 + b /b
2 + b/bf
bb
2b
=
-
+ b
bf
h 1
=
+
bf + b
b
w
+ bw
(6a)
It was then possiole to solve for oC knowing
o/or.
In reference 5, the value of (kg)h for ninged flanges
was
16.8
(ks3
(8)
77 2 (4 -OL)
h
where
(k
in
was Bijlaard's constant, k, for hinged flanges.
For fixed flanges (kB f vs. o( is plotted
gure 9 or reference 5, where (kB4
constant,
was Bijlaard's
k, for fixed flanges.
Using equations (4)
and (5)
values of kr were
determined for restrained channel flanges when the channel
was under uniform stress.
The proportioned k value for
moment loading was found by the following equation:
12
k
-k
kf
kh
Bh
B'h
where.
kr
=
k for uniformly loaded restrained flange
kh
=
k for uniformly loaded hinged flange
k
= k for uniformly loaded fixed flange
=
=
.425
1.277
(kB)h =k for Bijlaard's hinged flange under linear
stress distribution (reference 5)
(kB)
=
k for Bijlaard's fixed flange under linear
f
stress distribution (reference 5)
2.4
Method (d):
Bell Aircraft Solltion of Channel
in Bending about Axis Parallel to the Web
Method (d) of determining the plate buckling
factor, k, was a method derived by Bell Aircraft
Corporation in an unpublished report with the aid of
references 2,
3,
and 4.
The results appeared in
a
graph in reference 1 which was reproduced in Figure 39.).
An energy approach was used in conjunction with an
application of the moment distribution method explained
in reference 4.
Figure 44 was considered.
The stability condition was
T = V
and V 2
(10)
where.,T
= work done by external compressive forces
V
= strain energy in the plate
V2
= strain energy in the elastic restraining medium
13
(J )
to-
T =/
dx dy
42 w
b 2
Et3
V1
2
2 x 12(1 - 1/2)
+ 2 (1 -V)
(11)
+
2w
2
7of-2
(
22
j~
)
2
Idx dy
(12)
dx
2~~
2
V2
(13)
So = stiffness per unit length of elastic medium or moment
for 1/4 radian rotation
The proper boundary conditions from Figure 44
were
(14a)
(w) Y= = 0
Et3
32"
12 (1 -V
Ey2
-
=
4x2y=0
(4
12 (1 - V
12 (1
)
4s (
)O
y=0
(14b)
2W
Et3
Et
+1
2
'V2)
)
2
7
+ (2
y =b
- y(
Y7
(14c)
0
=
-
W
y
laxy) y=b.
(14d)
where (14b) and (14c-) expressed the moment condition
at points y=O and y=b respectively, and (14d) expressed
the shear condition at y=b.
14
The assumed deflection was
w
A
=
+ B
(Y)5 + ai(Y)4 +
a2( ) + a() ]os
(15)
where A and B were arbitrary deflection amplitudes.
When A = 0 the edge was clamped, and when B = 0 the
edge was pinned.
The assumed deflection inaequation (15)
was a sine curve in the direction of length, and the sum
of a straight line rotation deflection and cantilever
beam deflection in the direction of the width.
Method (d) above was a rigorous solution of
the problem of a channel in bending in the weak direction.
Methods (a), (b), and (c), demonstrated conservative
approximations which might be used by designers if more
accurate data were not available.
2.5
Ultimate Strength Considerations
Determining ultimate strength of compression
flanges was a very complicated mathematical problem,
and empirical or semi-empirical methods were usually
used.
Gerard in reference 9 had developed a method that
seemed to work for plates and flanges under uniform load.
His method assumed that after tha flange buckled at the
free edge, the member continued to take load until the
yield point was reached at ithe flange-web connection.
the present case this theory did not hold up, since the
connection area between flange and web carried small
stresses at ultimate load.
In
15
A theory of predicting ultimate moment was
discussed in paragraph 5.2 of this thesis.
It assumed
a particular stress distribution at failure, which
varied with cross section dimensions.
The method was
semi-empirical in nature, and much more test data was
necessary before it could be considered accurate.
3.0
Procedure
3.1
Method of Attack
The purpose of these tests was to spot check
the theoretical methods of determining critical buckling
stresses, and to formulate an approach of finding the
ultimate moment a channel could resist when loaded in the
weak direction.
The first step in testing was to determine
the material to be used and its properties.
Cold
rolled annealed mild steel strip was selected because
of its thickness tolerances (+.002"),
its freedom, from
residual stresses, and its linear stress-strain curve.
Tensile tests were run on specimens cut from the same
strip as the channel sections,-and stress-strain diagrams were plotted.
From the stress-strain diagrams.'
average values of proportional limit, yield point, and
Young's modulus were found.
Using the Bell Aircraft curve in figure 39,
three channel test sections were designed and constructed.
16
The material used was the same thickness throughout,
and the width of web was held constant.
The flange
width was varied on the three sections to give
critical crippling stresses (1) below the proportional
limit, (2) at the porportional limit, and (3)
inelastic range.
in the
The channels were all the same
length and were all loaded in pure bending at the
same loading and support points.
Sheared edges were
ground off to eliminate strain hardening.
The 20 inch span between load application was
ground down to assure the highest stresses occurred
in
this area and not at the point of load application
where shear stresses were present.
Care was taken to
avoid any pounding or straightening which might work
harden the material in critical areas.
A mathematical check was made of channel #3
to determine if the channel would fail by lateral buckling
below the ultimate load.
It was found that later.all
bickling was not critical.
Strain gages were located on the channel at
midspan to determine the stress distribution at various
applied moments.
Gages were located as shown in
figures 28, 30, and 32.
The double gages were placed
on both sides of one flange and connected in series to
eliminate any effects from the flange bending out of its
plane.
Since no eccentric load occurred in the web single
gages were adequate there.
The single gage on the opposite
flange was to indicate if the channel was loading
17
concentrically.
The double gage nearest the web was
located at the elastic neutral axis to note at what
moment the neutral axis started shifting.
3.2
Description of Apparatus
Photographs of the tensile testing and channel
testing apparatus were shown in figures 1 through 6.
A sketch of the channel testing apparatus was shown in
figure 7, and sketches of tensile specimens and
channel sections were shown in
3.3
figures 8 and 9.
Description of Procedure
3.3.1
Tests to Determine Material Properties
Tensile specimens were made from the strip
used to form the channel sections.
These specimens con-
formed to the standard ASTM specifications for tensile
testing sheet metal as described in reference 10 and
figure 8.
The specimens were measured with micrometers
and tested in a 5,000 lb. capacity Baldwin tensile
testing machine, using a Metzger extensometer with a 2 inch
grip reading to .0001".
Incremental load and deflection
readings were taken as shown in figures 15, 16, and 17.
3.3.2
Tests of Channel Sections
-
Three sheet metal channel sections were formed
as shown in figure 9.
Baldwin A-7 120 ohm, 1.96 gage factor
strain gages were attached as shown in figures 28, 30,
and 32.
Double gages were connected
18
in series.
The resistance changes were read with a
Baldwin SR-4 strain indicator which read strain directly
to the nearest 0.1 microinch.
The channel sections were
measured using micrometers and placed in a 10,000 lb.
capacity hand operated beam testing machine as shown
in figure 7.
Loads were applied incrementally as
shown in figures 25,
26, and 27.
The loading machine
was accurate to the nearest two pounds and strain
gage readings were recorded at each incremental load.
On channel #1 loads were increased on up to failure, but on channels #2. and #3 the load was applied,
released and reapplied alternately.
The loads causing
crippling in the extreme fibres and ultimate failure
were noted.
3.4
Methods of Making Computations and of Plotting Curves
3.4.1
Determining Material Properties
The data from the tensile specimens was reduced
in the usual manner and plotted as stress vs. strain
in figures 18 through 24.
The yield point was determined
by the .2% offset method as shown on the graphs.
The E
value was taken as the initial straight slope ofthe curve.
The porportional limit, which was difficult to obtain
consistantly, was taken as the stress at that point of the
curve which first deviated from a straight line.
Due to
initial unrecorded stresses which were unavoidable,
each curve had a small offset stress from the zero point
which was compensated for in the calculations.
Material
19
properties for each specimen were calculated on the
stress-strain curve, and the sum was averaged to determine the yield point, proportional limit, and Young's
modulus of the material
Ultimate strengths were re-
corded.
3.4.2
Plotting Values of k vs. bv/bf
Four curves of k vs. bl/bf were computed in
figure, 39 and plotted in accordance with methods (a),
(b),
(c), and (d) of paragraph 2.0.
The computations
for these plots were as follows.
Method (a) k = .425
3.4.3
(straight line)
Method (b)
k from equation (5)
(see figure 41)
Method (c)
k from equation (9)
(see figure 41)
Method (d)
k from reference 1.
Design of Channel Sections
Channel sections were designed using equation (1)
and the Bell Aircraft curve reproduced in figure 39.
Material properties as found in paragraph 3.4.1 were
used in the calculations.
For each channel the follow-
ing dimensions were constant:
Web thickness = flange thickness = .0625"
Web depth = 4.00" (outside dimension)
Channel span = 40.0"
Distance between load points = 20.0"
Bend radius = .062"
The flange widths were varied to give crippling stresses (1) in the elastic range, (2) at the
20
proportional limit, and (3) in the inelastic range.
Assume flange width = 4.00" (outside dimension)
Channel #1:
ff
e
=
Cr-
k17 2E
E- )
12(1 -1/ )
t f2
f
(1)
t
= .0625"
b
= 4.00 -
.06/2 = 3.97"
b
= 4.00 -
.06 = 3.94"
v
= .30
E = 28.6 x 103 ksi
P.L.= 17.7 ksi
From figure 39, method (d):
b/bf
3.94/3.97
. . k = 1.30
.994
=
0625)2 =
1-30T 2 x 28.6 x 10
12(l -. 350'
c,:I
8
.35ksi< 1 7 7
= 8.35 ksi
r
Assume flange width = 2.50" (outside dimension)
Channel #2:
b
=
bf
4.0
-
.O
=
1.60
.*.
1.157 2 x 28.6 x 103
12(1 - .302)
.S-r
18.0 ksi
.k= 1.15
-
o65 2 = 19.0 ksi =l17.7
21
Channel #3:
bw/bf
3.4.4
4.00 _ .06
=
=
Ei29.0 ksi
1.00" (outside dimension)
.
4.10
= .96r2 x 28.60 )10
rl
*
Assume flange width
k = .906
0625 2 =97.1 ksi>l7.7
(slightly less than yield point)
Calculation of Section Properties Using Actual
Channel Dimensions
The following section properties of each channel
were calculated using the measured dimensions of each
section and material properties found in paragraph 3.4.1.
When the two flange dimensions varied the least of the
two was used.
Refer to figure 14.
y = neutral axis ibacation - inches
S = section modulus - in. 3
I = moment of inertia - in.4
3.4.5
Predicted Buckling Stresses Using Actual Dimensions
The extreme fibre buckling stresses were
calculated using the actual measured cross-section
dimensions.
When the two flange dimensions varied, the
least of the two was used.
22
Method (a)
Channel #1:
4.00 x 4.01 x .o61o
-= 1.0
tf
LFcr
12 (1 -I
')
k772 (1) x 28.6 x 103
12 (1 - .302)
(tf
bf
2
= 25.9 x 103 k
b/bf= 3.94/3.98 = .990, .'.
Ucr
=
25.9 x 103 x 1.300('
Channel #2:
b
b
k = 1.300
8)
= 7-91 ksi
4.00 x'2.49 x .0618
= 3.94/2.46 = 1.60
k = 1.152
25.9 x 1.152 ('0 6)8=
(try ~0cr)g-r
(30.0
18.8 ksi > 17.7
'r) C~cr
(30.0 - 177.)
Solve by trial and error
_
17.7
I
I
30.(Cq
218
- Ur
2
23
1
U'r
2
Ucr 2
5
4
3
3o04
-
3
2
6
7
4/218
YT.
1813
335
549
214
. 982
.991
18.~6
18.5
342
555
213
.978
.988
18.6
I Ucr
Channel #3:
ksiI
3.99 x 1.00 x .0628
b/b
= 18.5
= 3_93= 4.05
.97
1
.'.
r=25.9 x 10(068
cr/
..
1
2
O~cr Ocr
4
3
2
97
5-
30Tcr
2
> 17.7
= 98.6 ksi
5
7'
6
4/218
k = .910
7
r VI-qr/
29.5
870
885
15
.0689
.263
26.0
29.3
859
880
21
.0964
.310
30.5
I
-'r
= 29.4 ksi
24
In a similar manner Ccr was solved using
methods (b), (c), and (d).
The results were summarized
below in figure 10.
Method
Channel
k
(a)
(1)
1.300
(2)
1.152
18.8
18.5
(3)
0.910
98.6
29.4
(1)
1.260
(2)
1.026
16.7
16.7
(3)
.702
76.3
28.6
(1).
.854
5.19
5.19
(2)
.702
10.44
10.44
(3)
.425
46.1
27.2
(1)
.425
2.58
2.58
(2)
.425
6.94
6.94
(3).
.425
(b)
(d)
(d)
Figure 10-.
CFc
ksi
7.91
7.65
46.1
76r (ksi)
7.91
7.65
27.2
Table of Calcnlated Buckling Stresses from
Actual Channel Dimensions
3.4.6
Reduction of Channel Test Data
Graphs were plotted of applied moment vs. strain
for each strain gage of each channel,
and were shown in
figures 28 and 29, 30 and 31, and 32 and 33 for channels
#1, #2, and #3 respectively.
Strain gage locations for
each channel were shown in figures 28, 30 and 32.
25
By studying the moment-strain curves and
the log sheets, the critical crippling moment and ultimate
moment were established for each channel section.
Stress
distribution curves were plotted to scale for the Mcr, Mult,
and two other significant moments in figures 35, 36,
and 37.
Stress was determined from the strain readings
by use of Young's modulus
C~ (ksi)
=
E(ksi)
x
E(microinches)
It was noted that gage
©
= 28.6 x strain
and gage
©
plotted different moment-strain curves which indicated
that the channel was being loaded eccentrically.
error was particularly pronounced in channels #(l)
#(2).
At lower loads gage @1
This
and
read a greater amount
When a certain irtermediate load was
of strain.
reached the channel had deformed sufficiently to fit
the loading appartatus, and from that point on each flange
took the same amount of strain.
from gage 0
At higher loads readings
became insignificant due to high bending
strains as the flange assumed its buckled shape.
This eccentric loading was compensated for
in the following manner as shown in the table of figure 34.
In the range of lower loads, before the flanges were
accepting an equal amount of applied moment, the actual
strain was taken as an average of gages Q
and 0
.
This averaging method was used up to the point where the
26
channel had warped sufficiently to no longer load
Further incremental loading applied equal
eccentrically.
At this point gage
stress to each flange.
a higher strain in
that flange than gage
(Q) showed
(j.
Half the
difference of these two strains was subtracted from the
strain readings of gage
intermediate moment.
0
(D
at all moments above the
-Strains from gages
(Q} ,
(}
, and
were all adjusted in a proportional manner, depending
on the distance from the neutral axis.
The intermediate moment for the different
channels was:
Channel #1
2,000 inch lbs.
Channel #2
2,000 inch lbs.
Channel #3
1,000 inch lbs.
The following checks were made of the stress
distributions:
1.
Using the formula C'=
4,
the extreme fibre
stresses were calculated for moments in the elastic range
and compared to the measured stresses.
2.
(see figure 11)
The area of compressive stress equalled the
area of tensile stress since thickness was a constant.
(see figure 38)
3.
The area of compressive stress x flange
thickness x moment arm between compressive and tensile
27
area centroids equalled the applied moment.
(see
figures 38 and 12)
In the case of channel #3 where the flange
strain exceeded the yield strain, the points of stress
were plotted in figure 37 as if the material was
infinitely elastic.
These points were connected by the
dotted lines, but the actual stress distribution was
shown by the cutoff at (Ur.For the 1,000 inch lb.
unloading moment the stress diagram was found by subtracting half of
1,000 inch lb. loading moment dia-
gram from the 1,500 inch lb. diagram.
The 1,500
inch lb. diagram was shown dotted.
3.4.7 Determining k at Ucr
The values of Mcr were established by
studying the test data and moment-strain curves.
Test
values of k were determined as follows, and plotted on
figure 39.
Channel #1:
Dc
Mcr = 3,750 in-lbs
M - 3.75 = 7.78 ksi
k=r12(1 -V2)
72E
b 2
tf
2-=71 28
25.9 x 10
0610
28
Channel #2:
-
Mcr = 3,750 in-lbs
=
9
r
19.05 ksi
try -crh)
~
k
(y
p
-
cr~
Up
(2 46
19.4
_
-
.1(30
-
19.05) 19.05
1(30 - 17.7)
17.7
2
25.9 x 10 (.9 8 )b-
Channel #3:
U'cr
Mcr
=
1000
(30.0
30.0
1000 in-lbs
28.3 ksi
-
28.3) 28.3
1-7.7) 17.7
28.3
k
25.9 X103 x (.470)
.470
.969
(.0628
2
) =.554
3.488 Ultimate Moment
A semi-empirical method of predicting ultimate
moment was presented in paragraph 5.2 of this report.
3.5
Sources of Error
The chief source of error in the results was
the variation of strain gage readings due to eccentric-.
loading.
Other sources of error were variations in
material thickness, initial eccentricities in flange
straightness, inaccuracies in load application at low
loads, differences from critical moment in an infinitely
=
-
29
long channel and a finite length channel,
variations
in channel material properties from the tensile test
results, inaccuracies in gage and instrument readings,
and inability to detect the exact critical load.
It
was extremely difficult to calculate the exact percent
error due to each of
these factors.
However, it was
possible to check the final results in several ways to
find the overall percent error.
The.rmain error due to eccentric loading was
compensated for by the method explained in paragraph
3.4.6,
By comparing the stresses on the plotted
elastic stress distributions (figures 35,
the calculated stresses a
36, and 37) to
percent error was
determined in figure 11.
Channel
Applied Mom.
(in. - lbs.)
Calc. Max.
Plotted
M
Max.Stress(ksi)
Stress = g(ksi) (Figs.35,36,
and 37)
% Error
#1
2,000
4.15
3.7
-1o.8
#2
1,000
5.08
5.0
-
#2
2,000
10.16
9.3
-'8.
#2
3,750
19.0
18.3
-
#3
1,000
28.3
30.0
+ 6.0
Figure ll-.
1.6
3.7
Table of Percent Error Between Plotted
Elastic Stresses and Calculated Elastic Stresses
This error indicated the difference from theoretical elastic stress and strain gage readings.
Positive error
30
indicated the plotted stress was higher.
To determine the percent error in the inelastic
stress regions, the applied moment was compared to the
moment of the stress distribution diagrams.
This was
done in figure 38 and summarized in Tigure 12.
Positive
error indicated the stress diagram moment was higher.
Channel
Applied Mom.
(in.
- lbs.)
Stress Diagram
Moment, (in9lbs)
% Error
#1
3,750
3,390
-.9.6
#1
5,000
4,650
- 7.0
#1
7,000
7,260
+ 3.7
#2
4,750
3,770
-20.6
#3
1,500
1,630
+ 8.7
#3
1,000 (unload)
1,000
0
#3
1,550 (reload)
1,620
+ 4.5
Figure 12-.
Table of Percent Error Between Calculated
Stress Diagram Moments and Actual Applied
Moments
The 20.6% error in channel #2 was probably
cde to an erroneous strain gage reading at the extreme
fibre,
since this
recorded strain was actually lower than
the strain at lower moments.
The above errors repremnted differences from
applied moments and strain gage readings.
To detect the
error in the moment at which buckling occurred was more
difficult, and the value could vary as much ae 10%.
This
was especially true in channel #3 which buckled in the
inelastic range.
Due to dimensional differences and
31
eccentric loading one flange always buckled at a lower
load, so it was necessary to interpolate between to get
the actual critical moment.
The percent error between the calculated critical
(b), (c) and (d); and the
stresses of figure 10 using (a),
test results were summarized in figure 13.
In the inelastic
range it was noted that a large variation in the value of k
had a small effect on the critical stress.
Positive error
indicated the test results were higher.
Error in the critical stress due to the flange
not being infinitely long was approximated from the case
of a uniformly loaded hinged flange.
Reference 6 gave for
this case the following formula for k;
k = .456 + (b)
(16)
where:
a = distance between simply supported loaded
edges of flange
b = flange width
For the worst case of channel #1 assume
b = 4.oo" and a = 24.0"
k = .456 + (4)2
=
.485
% Error = .485 - .456
6.4%
.456
This estimate was high since the linear varying
stress and flange restraint tended to reduce the effective
value of b.
The effect of the flange length not being in-
finite was neglected.
Channel
Critical Crippling Stresses (ksi)
% Error
Experimental
Mcr
Met.
Met.
Met.
Met.
Met.
Met.
Met.
Met.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
#1
7.78
7.91
7.65
5.19
2.58
-1.7
+1.7 +33.3 +67.9
#2
19.05
18.5
16.7
10.44
6.94
+2.9 +12.3 +45.2 +63.6
#3
28.3
29.4
28.6
27.2
Figure 13
27.2
-3.9 -11.1 + 3.9 + 3.9
Table of Percent Error Between Calculated Critical Stresses
and Experimental Critical Stresses
\AJ
I')
335
4.0
Results
The results of this experiment were presented
in the following log sheets and graphs1. Channel dimensions and section properties (figure 14).
2. Tensile test log sheets (figures 15 through 17).
3.
Tensile stress-strain curves (figures 18 through 24).
4.
Channel buckling test log sheets (figures 25 through 27).
5. Channel buckling test moment-strain curves (figures 28
through 33).
6. Table to plot stress distribution curves from momentstrain curves (figure 34).
7.
Channel buckling test stress distribution curves
(figures 35 through 37).
8.
Check of stress distribution curves by area and
moment balance (figure 38).
9. Values of k vs. blbff from test results and theoretical methods (a), (b), (d), and (d) (figures 39
and 41).
34
Channel #1
Channel #2
Channel #3
4.o1
2.49
1.00
4.01
2.49
1.01
tL
.0622
.0620
.0628
tR
.0610
.o618
.0628
b
4.00
3.99
2.65
1.762
.810
3.99
A
.734
.555
.376
I
1.278
.348
.0287
.197
,0354
S
Figure 14.:
Channel Dimensions and Section Properties
35
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Applied
(in.-lbs)
. 2.70x3.7/2)2
A Compr.
A Tension
Moment
2.80x6.9/2)2
4.ol+l.14)3.8
19.3x.0 6 1x2.88x1013
5000
A Compr.
A Tension
Moment
(3.0o8x8.6/2) 2
(4.01+.86) 5.5
26.5x.o61x28 8 xlo30
7000
A Compr.
A Tension
Moment
[13.00(10.2+5.0)/21
4.ox1.4
45.7x.061x2.55xio3
[1000
A Compr.
A Tension
Moment
2000
A Compr.
A Tension
Moment
(1.78x9.3/2)2
(3.99+.65 3.4
16.6x.061x1.80x103
16.6
15.8
1840. *
(1.78x18.3/2)2
4.64x7.0
32.6x. 0618x1.80x10 5
32.6
3750
A Compr.
A Tension
Moment
L4750
A Compr.
A Tension
Moment
(2.08x16.3/2)2
2.42x16.0
33.9x1.80x.0618x103
38.7
3,770.
1000
A Compr.
A Tension
Moment
(.83x30.0/2)2
6.3x2.05x2
24.9x.0628x.65x103
24.9
25.8
1,010
1500
A Compr.
A Tension
Moment
2[(.87+.51)/2]130.0
12.0x2.03x2
41.4x.0628x63x103
A Compr.
A Tension
Moment
8.8x2x2.03
3750
1
2
3
10.0
A Compr.
A Tension
Moment
[2000
1000
(unload)
1550
Figure 38:
A Compr.
A Tension
Moment
4.0121. 24)1.8
0.0 x .061(1.80+1.21)103
1.78x5.O/2)2
3.99+. 65)1.9
.9lx1.80x.0618x103
9.8
1,840
19.3
19.6
3,390.
26.5
24.8
4,650.
45.7
45.9
7,260.
8.91
8.81
990.
32.5
3,630
33.9
41.4
48.8
1,630
[.45(23.3+15.0)/2+(.40x23.3/2)]2
26.6x.60x.0628x130
[130.0(.89+.62)/2]2
14.0x2x2.03
57x. 0628x10 5
45.3 -x.
Check of Stress Distribution Curves by
Area and Moment Balance
35.7
1,000.
45.3
56.9
11620
26.6
61
Figure 39
4-4- 114+4
-4
at
&
Ac
-
-1
----
a
-
--
-
-
-
-
-
-
-
--
-
-
/
-
-
-
71
~~~~
.~
.
T
.j
- - --
----
---
L7:4
I
1_
qB~
1--
44
-
4-'
-4-
L1-
f.
T_
(3ia:
tE
(J
/
r-
7
r
-
- -
_T
-L
T4
T, -
-7
l7+
-FL
r4
M
t
y-
-
--
4---
.f
-
L
-4
-A
-
-
1-
-T
-
-1f4
-
-
-
- --
-
-
-4+-
.567
.54
0O
a
.0
2.0
3.0
bw/4jb
4.0o
5.O
62
I
2+bib f
bp/br
4
2
1±bf/
6
5
b
16.8
bib
2
7
9
a
- -106(
7?(4-ad)
11
r
fb
2
""-B f
10
+ 4
3,f
)
.65
'2
kr
5:3 + 4
kr- .425'
.852
12
13
25
-hkr
(k -- (
(_ kB
( kB
r-
[ k )
0
t500
1.150
1.322
1 .051
1.300
2.218
.0250
.075
.. 491
1.141
1.302
.998
1.291
2.128
.oo166
.1252
.3756
. .457
1.107
1.225
.939
1.283
2.012
2.03
.00662
.252
.756
.420
1.070
1.144
.844
1.256
1.833
.730
1.96
.0265
.514
1.542
.362
1.012
1.024
.703
1.230
1.595
1.571
.703
1.89
.0596
.798
2.394
.314
.964
.929
.591
1.187
1.405
1.000
1.500
.680
1.86
.1060
1.119
3.36
.
272
.922
.850
.499
1.180
1.269
1.500
1.400
.654
1.81
.238
1.970
5.91
.202
.852
.726
.353
1.156
1.062
2.000
1.333
.637
1.77
.424
3.47
10.41
.139
.789
.622
.231
1.133
.899
2.500
1.286
.627
1.75
.662
7.40
22.2
.. 076
-726
.527
.1199
1.123
.762
3.00
1.25
.619
1.73
.954
195.6
.. 010
.660
.435
.0117
1.1111.
.632
3.07
1.247
.619
1.73
.0
.650
.425
0
.619
4.00
1.20
.608
1.65
o0
0
.650
.425
0
.6o8
5.00
1.167
.601
1.69
00
0
.650
.425
0
.601
1.000
.567
1.61
*c:
0
.650
.425
0
.567
2.00
.850
2.15
0
.0250
1.973
.839
2.13
0
.125
1.889
.807
2.09
.250
1.800
.774
.500
1.667
.750
0
1.00
0
65.2
**o
Figure 41: Table
0
00
Calculating
k
Values for Methods (b) and (c).
-(kBh
63
5.0
Discussion of Results
5.1
Critical Moment
The test results of critical stresses showed
very good agreement with theoretical methods (a) and (b).
They indicated that methods (c) and (d) were too conservative.
The large variations of k in the different
theoretical methods did not appreciaoly effect the
crippling stress in the inelastic range.
However, in
Iae elastic range the critical stress was directly proportional to k.
For this reason for the materials used
methods (a) and (b) predicted critical stresses which
were almost identical.
However, for a material with a
proportional limit above 20 ksi, method (b)
would pre-
dict conservative stresses when the ration of bjbf was
greater than 1.6.
Channel #2, which buckled at 19 ksi, seemed to
indicate better agreement with method (a) than method (b).
This was the area on figure 39 where curves (a) and (b)
began to separate and indicated that method (a) gave
better results in the high ratios of bw/bf.
However,
the results were not conclusive on this point.
5.2
Ultimate Moment
For each channel section an ultimate moment
was recorded which was somewhat greater than the critical
buckling moment.
A semi-empirical method was developed
for predicting this moment.
64
Assume at ultimate moment the web was completely in tension and the flange in compression.
Assume
the stress in the compression flange was equal to the
extreme fibre buckling stress and the tension web was at
some stress not greater than the material yield point.
Consider figure 45, if the flange stress was
-cr, and since the tension area equalled the compression area, then,,by proportioning, the web stress was
approximately (for one flange)
b ft
O'cr
2
0
bw
-T
(17)
cr bf
w
t
For most practical channel dimensions and most
materials it was found that the web stress was less than
a-Y*
Therefore, figure 45 seemed like a reasonable
assumption for a first approximation of the stress distribution at ultimate moment.
Consider the stress distributions at ultimate
load for the various channel (figures 35, 36 and 37).
Channel #1,
with a deep flange first
extreme fibre stress.
buckled at a low
However, after buckling the ex-
treme fibre still maintained the critical stress.
As
the moment kept increasing the stress in the fibres
closer tbathe web increased to their critical stress, and
the neutral axis shifted down.
The fibres inside the
extreme fibre all buckled at higher critical stresses
65
than the extreme fibre, so the actual stress distribution
looked like the 7,000 in-lb moment condition in
figure 35.
Finally the moment got so large that the
flange buckled completely.
The fibres immediately ad-
jacent to the web took small stresses since the propagation of the buckle created local stresses which
failed these fibres.
The actual distribution in figure 35
may be approximated by the theoretical distribution in
figure 45.
A similar stress distribution occurred in
channel #3 where the critical stress was very close to
the yield.
In this case the critical stress occurred
on down to the fibres quite
close to the neutral axia,
and failure was analogous to that of
a cross section
not critical in local crippling.
However, in the case of channel #2 where the
critical extreme fibre stress was close to the prop.ortional limit, the critical stress did not increase in
the fibnes: closer to the web.
The buckle propagation
occurred earlier and the ultimate moment was only
slightly greater than the critical moment.
distribution at ultimate is
This stress
shown by the 4,750 in-lb
moment condition in figure 36, and was close to a
triangular distribution.
In consideration of these observations it was
decided to predict ultimate moment by the distribution
of figure 45, and reduce it
of the individual channels.
by a factor to fit the cases
The ultimate moment by
66
the distribution of figure 45 Was:
Mult = 2 C'er bt
' = Ucr
(18)
2
where:
Mu
= ultimate moment
0 -cr
= critical buckling stress at the extreme
fibre calculated by method (a).
This moment was reduced by some function of
the two ratios (U/0cr) and (t/bf).
A factor that fitted
the test results was
cr br 2 t (E)
Mult
(19)
where:
=16.0 (
- ) (t
3
U'cr
b
Equation (19)
had as its
lower limit the
case of a triangular stress distribution and as its
upper limit a rectangular stress distribution.
Therefore equation (19)
was written as follows:
uit = .667 -er bf2 t
uit ~
lt
yft
2
(C)
cr bf
t
where 16 .0
Cucr
<.667
where 1.0>,16.oC- >.667 (19b.)
WCr
f
where 16.0
c
t
V cr B
> 1.0
Applying these equations to the channels
tested and comparing to the ultimate moments gave the
results of figure 40.
results were higher.
(19a)
f(
Positive error indicated the test
(19c)
67
Test Results
Mult, Eqns.
Channel
Mult(in.-lbs.)
(19c)&(19d)
#1
7,250
7,100
+ 2.1
#2
4,750
4,600
+
#3
1,550
1,735
-11.9
%oError
3.2
Table of Percent Error Between Actual Ultimate
Figure 40:
Moment and Predicted Ultimate Moment
6.0
Conclusions
As a result of studying test results, the
following conclusions were reached.
6.1
Critical Moment
It was found that methods (a) and (b) of
predicting buckling stresses showed good agreement with
test results.
Methods (c) and (d) were too conservative,
especially in the region below the proportional limit.
Method (a)seemed to indicate better agreement
than method (b), but more testing with different materials
and different size channels was necessary to be sure.
From the discussion of paragraph 5.1, it was
recommended to use method (a) for materials with a
proportional limit below 18.0 ksi, and method (b) for
other materials.
6.2
This would assure a conservative design.
Ultimate Moment
From the discussion of paragraph 5.2 a semi-
empirical approach of predicting ultimate moment was
developed.
Formulas that fit the test results were:
68
Muit = .667 T
Mlt
=ybt
b 2t
2(i)
Mlt =Ucr bf t
where 16.0
where
Uy
o-6
bf
<. 667
1.0;>.6.0 O-yU-bct >.667
where 16.0
t
1.0
if>1.0-
where C)cr was the extreme fibre crippling stress as given
by method (a) and V= 16.0 Cy
Ucr
t
bf
More testing was necessary to substantiate
these results.
A temporary method of predicting ultimate
moment which gave conservative results for all cases was
given by equation (19a) .
(19a)
(19b)
(19c)
69
7.0
References
1.
"Bell Aircraft Corporation Structuras
Manual", 1955.
2. Bijlaard, P.P.; "Buckling of Plates Under
Non-homogeneous Stress", ASCE Proceedings,
1957 1293 EM3.
3. Bleich, Friedrich-; "Buckling Strength of
Metal Structures", McGraw-Hill Book Co., 1952.
4. Lundquist, E.E., Stowell, E.A., and
Schuette, E.H.; "Principles of Moment
Distribution Applied to Stability of
Structures Composedcf Bars and Plates".
NACA ARR 3K06, November, 1943.
5. Lundquist, E.E. and Stowell, E.A.;
"Critical Compressive Stress for Outstanding
Flanges",
NACA TR 734,
6. Timoshenko, Stephen;
1942.
"Theory of Plates
and Shells", McGraw-Hill Book Co., 1940.
7.
Timoshenko, Stephen; "Theory of Elastic
Stability", McGraw-Hill Book Co., 1936.
8.
Stowell, E.A.; "A Unified Theory of Plastic
Buckling of Columns and Plates",
NACA Technical Note 1556, 1948.
9.
Gerard, George; "The Crippling Strength
of Compression Elements", Journal of the
Aeronautical Sciences,
10.
January,
1958.
"ASTM Standards", Tensile Test Procedure
E8-54T.
70
8.0
Appendices
The following illustrations were included in
the appendices.
1.
Photos of test apparatus.(figures 1 thro ugh 6)
2.
Sketch of Channel testing apparatus. (figiure 7)
3.
Sketch of tensile specimen,(figure 8)
4.
Sketch of channel section. (figure 9)
5.
Stress distribution factor,
6.
Stress distribution in
7.
Analysis of flange by method (d).
8.
Theoretical ultimate moment stress distr ibution.
(figure 45)
. (figure 42)
channel section.
(figure 43)
(figur e 44)
Figure 1:
Photo - Tensile Testing Apparatus
Figure 2:
Photo - Tensile Testing Specimen in Machine
72
,Figure 3:
Photo - Channel
Test Sections Showing
Strain Gages
Figure 4:
Photo - Channel
Section in Testing Machine
73
F
Figure 5:
Photo
-
Channel Section and Strain Indicator
S
Figure 6:
Photo - Balancing Strain Indicator
If
40.0-a
o
200.
A
I~~
~
f. P/ja
fiis-i.o"
I
__NI
I
I
4I
P/at
PUNCH
FOR
MARKS
EXTENSOMETER
.4262
57TX.
-8.0"W.
GRADUAL
TAPER
FROM
TEST
SECTION
TO
END.
~.1~
* .0/a
.06. RAD
a.
SECT/ON
.O20
#/
4.00
q6.
25.0
#3
/.00
4.00
MATL.-
.062
GRINtD
PREE
Do
NOT
C.R.
ANNEAL.ED
OP AL..
POUND
INTO
STOCK
BURRS
SHAPE
F
CHA NNEL
SEC TION
77
HINGED OR RESTRAINED EDGE~
O-
EE EDGA~
Figure 42:
=
0-,. +
+.-O
oC.
=F/.0
OL. =+/.O
Illustration of Stress Distribution Factor, o<
7
Figure 43:
Illustration of Stress Distribution in
Channel Section
X
ELASTICALLY
RE~STRAINED
.It
EDG-E
Y
Figure 44:
Notation for
Analysis of Flange by
Method d)
EDGE
I
t-
N. A.y
____
Figure 45:
Pig
Theoretical Ultimate Moment Stress Distribution