ELASTIC AND INELASTIC BEHAVIOR IN MODEL STEEL COLUMNS
by
Michael Shuga
A report submitted in fulfillment
of the requirements in Course C.E.
404, "Structural Research"
Fritz Engineering Laboratory
Department of Civil Engineering and Mechanics
Lehigh University
Bethlehem, Penna.
Fritz
L~boratory
Report No. 205A.18
to
September,
1955
Page
~
I\,
I.
INTRODucrr~ON
II.
THEORETICAL ANALYSIS
III.
DESCRIPTION OF EXPERIMENTAL PROGRAM
AND'TE$T RESULTS
IV.
PISCUSSION AND SUMMARY .
V.
ACKNOW~DGMENT
VI.
NOMENCLATURE
10
VII.
REFERENCES
11
VIII.
FIGURES
o .
S
1
1
7
9
-1
I.
I N TROD U C T ION
The problem under consideration is the determination
of the
m~ximum
loading a cqmpression member can carry.
.
study is limited to
.
th~
.
This
investigation of pin-ended square
cplurnns whose centroidal axes are the diagonals of the square.
Model steel yolumns axially loaded in one series of tests,
and eyeentrica+ly loaded in another series of tests were tested
to failure.
These
exper~meptal
results are Gompared with
analytiGal val~es Gomputed by a virtual displacement methodS 5) (7)
II.
THE 0 RET I CAL
A N A L Y SIS
A +arge number of variables enter in the solution of
this problem and to reduce
~hese
the following assumptions
are made for this approximate solution:
1.
The material under study possesses properties which
are· idealized as perfectly elastic up to the yield
point and then perfectly plastic.
2.
Members are
ori~inally
straight, free fr0m
accidental eccentricities, and of uniform crosssection along their lengths.
3.
Sections
d~formonly
in the plane of the applied
\
moments and will deform in a sine Gonfiguration.
4.
•
Sections plane before bending will remain plane
after bending •
The behavior of a column in the elastic range is
de$cribed by the equation, y" :;: ;~ (y + e) .
For a column bent
.......; -.
-2
.
in single curvature the equation of center line c;1eflection
is:
t
~
= eCI-cos
u)
cos u
~
where
lrP
- 2 'EI
If no moment is present and on+y an axial compressive
load is appliec;1 to the member no solution exists for the
above
This is to say that
eq~ation.
~he
axial thrust, P, for
I
an axially loaded member, can be increased from zero to a
critical value, P cr ,
slightest
increas~
qolumn to
d~form
ass~ed mat~rial
wit~
the member remaining straight.
The
in loaq above P cr , however, will cause the
laterally and then c91lapse.
Pcr, for the
Qehavior, is either the yield load,P y , or the
Euler buckling load, Pe .
.'
For th,e eccentrically loaded column a different type
of behavi9r is observed.
In this case the member stqrts to
deflect. on the slightest application of thrust.
As the load is
increased, the member wil+ deform elastically until a load is
reached a.t which yielding takes place.
ca~ges
:.;"'.
Further increase in load
yielding to progress across and along the member reducing
its resistance to
f~rther
loading.
A point is finally reached
at which an increase in load is impossible and the member continues to deflect as the load remains constant.
critical load for the
ecc~ntrically loade~
This is the
member.
The upper limit of column strength for the case where
bending moments and axial thrusts are present is given by the
Simple Plastic Theory.
uSiJ,1g a set of
these values of
Axial load ar;td IJ).oment are computed
assum~d
a~ial
plastic stress patterns.
Figure 1 shows
+oad and moment plotted in the non-
-3
M
dimensional form of ~
vs. -My
'Y
Th~ ultimate strength of an eccentrically loaded
P
L
column may best be represented by an -P - - curve. To obtain
y
this
r
a virtual displacement method of determining the
stability pf beam columns in the• elastic and inelastic ranges
curv~,
as desmribed by R.L. Ketter(5) in a recent paper, was used.
To obtain the desired
curv~
yield penetrations as shown in
Figure 2 were assumed for the column
s~ct~on.
The analysis
was divided into two parts, Case A, where yielding only on the
compressioq
s~de
pf the specimen is considered, and Case B,
where yielding has occurred on the tension side as well as the
cowpression side qf the
'.
wer~
spec~men.
The following relationships
used in this phase of the investigation:
P ,..
The curvature,
section;
~,
rA<r dA
M
=
~
= EI -E2
fA cr ydA
h
is a function of the strain developed at the
axial load, P, and the moment, M, are functions
th~
of str,ess.
Values obtained using the above relationships and the
vlastic stress
patte~ns
shown in Figure 2, are shown in non-
dimensional form in Figures' 3 and 4.
From these curves another
set of auxili~ry curves (Figure 5) were plotted for ~M' vs. ~
y
P
at constant p'
values of ~
~y
patterns.
y
Figure 3 was entered at spm~ value of ~
X/y
y
and
determined for each of the assumed plastic stress
These values of ~
y
were used to enter Figure 4 to
-4
obtain
~
va~u~q
for
M
My
for eaqh of the assumed
pl~stic
stress
patternq.
Next, two sets of curves (Figures
p and
tangent values of the above auxiliary curves were
These
cu~ves r~preqe~t
i~ter~q+ bending
the
P
7) representing
plotte~.
stiffness
~~
0y
• M \ . t
of a seytion at a given P ratio CEq. 10, R~f. 5), and more
Qr le~s cor~espond to theYEI value computed for elastic considerations.
ln
are plotted as functions of the
coordinates M and 0 . ,Values of AM . 0 y (
were selected and
My
0y
M
60
My int
cor~~spondi~g v~lues of ~ and M were read from Figu~es 6 and 7
.
The tangent
Y
•
val~es
y.y
m
I)1o
@b
P
00 (
thus comput;:Lng - o
The equat1.on - = -, ... - . Eq. 14,
'My
My
My
P e 0y
Ref. 5) ~~l8;tes the end moment" ,mo~ with the moment and unit
rotation quantitie$ at
th~
most stressed center line section
(~o anq 0 0 ) for ~ g~ven value
ps>,int a family of curves
8) since the
~uler
load,
-
of
inte~n~l stiffness ~
P
rEllqt~ng P
P~,
y
and
L
r
.
At this
e
may be computed (Figure
is a function of
t~e
slenderness
L
ratio r •
IP
rna
ec
eccentricity ratio), a line
Since -My -P y =-r~ (the
drawn at a slope ec proportional to 'mo/p (see Figure 8) will
;'2"
' .
My P y
inter~ect the rna ,P ,L curves at the critical load ratio P •
,
Py
My
Py
r
In this manner anot~er family of curves is drawn showing critical
I
load ratios for constant eccentricity ratios.
curves
a~e u~ed
These collapse
to determine the critical load ratio P cr for a
given eccentricity and slendElrness ratio.
Py
•
-5
;: . 1: :. 11:: . .:-.-.:::.D:-rE_S:;:;.....;C::....:.R:.,.....:I--=-P.. .;T;;..·...;;I::.....:::O~N--....;O::.....:::F..,...-..;:::E.-;X;,;;,...;P;.,..,...;::E RIM ,E N TAL
PRO G RAM AND T EST RES U L T S
Fig~re
assembly.
a:fte~
of
A shows a view of the end
fixt~res
f'he principal load carrying pieces are
mild steEilI.
The c:irqu.;J.ar insert is designed. to accommodate a
specimep of 3/4 ipch by 3/4 inch Cross.,section
Smaller sections a.re fitted by shims.
circ~lar
mQvement of the rocker with
fixture.
This movement
threaded to the
perpen~icular
th,~
size).
A small pointer attached
rpc~er
i~
Eccentricity;is applied by
~espect
to the base pf the end
con~rolled
screw.
by two kno'bs which are
Tneroc~er
screw axis is
to the axis of end rotation.
of tests was performed with no
0 th
. ,eIl
(max;i~um
insert orients the specimen with respect to the
appli~d' E!ccen,tricity.
plane of
ser~es
mad~
carbon steel while the rectangular bearing block is of
hig~
to the
before and
4
·ser,j-es
it
0f
exper
men·s
In this study one
applie~
eccentricity while
·
.
ec
wa.s m.ade uSlng
an ~
0"f
one
quarter.
In both serieq of experiments the centroidql axis of
the three
~u~rter
45
degr~es
1
occ~rred
by three quarter
inc~
columns was oriented at
to the plane of applied eccentricity such that
about, or
Tension
pa~allel
co~pons
be~ding
to, the diagonal axis of the specimen.
and column specimenp were annealed at
11400 F fOF one and one half hours at temperature and then
cooled.
qua~ter
Tension specimens were milled to one half such by three
inch
s;iz~
tor two lpch gCi-ge lenths.
fA.
Huggenberger
straip gage set at Qne :riaIf inch gage length wa;s used to determine
~lastic
load
d~flections.
A Peters strain gage was used to
record the larger strains encountered beyond the yield point of
-6
the material.
A
straip curves for
Ten
and
fre~dom
'rabIes
th~
of the tension test results and the stress
tension specimens is shown on Figure F.
col~ spec~mens
from
of the
su~ary
summ~ry
i~itial
were chosep
fo~
their straightness
twist and were milled to length.
me~surements
for these
sp~cimens
A
is shown in
and 2.
~
In performing the column tests, the column
speci~ens
were first aligned uslng the alignment apparatus shown in Figure
B.
The
he~~
circul~r
wedge plates shown resting
hydraul~c
of the 60,000 lb.
spheric~ll~ seat~d
movable cros~head.
fixed cross-
testing machine were
and then the specimen was geometrically
thrqugh a
onth~
al~gned.
l~veled
LQad was applied
bloqk which was attached to the
Finfll alignment f9:r a g~ven eccentrici,ty was
made by checking the computeq elastic eurves and also by checking
symmetry of gendipg.
expe~imental
def~ec~ion
;f
Computed. elastic load
points qre plotted in Figu!e Q.
curve for ppeci,men
E~~,
is 0.16, is shown in Figure H.
refe~red
to the
ecce~trici,ty.
~ages
defl~ction
colu~
curves and
The elastic load
where the ecceptricity ratio
Deflection readi1;1gs are
axis and are independent of the end
The adjustable frame used to support the Ames dial
is shown in Figu:re C.
A,fte.r fj,nal alignment was
was applied
i~ e~en
for a specimen, load
obtaine~
increments s9 as to
ge~
about eight load
deflection readings withi,n the elastic range.
of loading was
was reaQhed.
qont~nued
This procedure
until the critical" load
~or
the column
After the column had buckled, the load was
permitted to reach a state of
equ~librium.
The load was then
held constant until the deflection reached equilibrium.
This
-7
process of un19ading was continued until sufficient data was
accum~~ated
for
points
obtained where possible.
w~re
anun~oading cu~ve
collapse occurred
the critical
~t
for the specimen.
Sudden and complete
~oad
for specimens having
of 100 and 120,and an unloading curve was not
r~ti9s
these specimen$.
Load
specimens appear in
curv~s
def1~ction
Fig~re
J.
Unloading
L
r
ob~ained
for
curves for the axi q 11y loaded
Figure K shows load deflection
for the eQcentrica1J,.y'loaded columns.
A summary of
predicted coJ,.lapse loads c;J.nd actuc;J.1 experimental collapse loads
is contained in Tables 1A and 2A.
~o det~rmi~e
t~e
on
end fixture and the bearing block, a
lo~ding
and unloading
w~thin
the elastic range was performed for one
sp~cimen~
complr~ed
elastic' qurves, as well as the unloading
9urv~
cycle
The
the effect of friction between the rocker
this $pecimen is shqwn
~~
Figure H.
DIS C U S S ION
~v.
for.
AND
SUM M AR Y
The ten qo1umn tests, although far from conclUsive,
$eem tq indicate that the virtual displacement method for
d~te~mining co11~p$e
-
1arg~st
loads for
co1u~s
is slightly conservative. The
deviation (5.8%) from a predicted collapse load was
~oted for the axi q11y 1q?ded specimen with an ~ ratio of 120.
It should be
~ot~d
crookedness in
that, because of some unavoidable initial
a true axial load condition.
showed a
Elastic
it was very difficult to approach
th~ ~pecimens,
devi~tion
of about
de~ormations
The eccentrically loaded specimens
4%
above predicted collapse load.
for these specimens were symmetriG and
-8
ch~cked oomp~ted d~flections
reasonably well.
Friction between
the end fixture and the bearing plate is considered negligible.
Ecpentr~cities
were set wit;lin the accuracy of the end fixtures.
When the variation of material properties from specimen to
specimen, qS eompared to the average
v~lue
used for tnese
material properties is considereq, the experimental results can
be said to
che~k
the theoretical results with reasonable
acc~racyo
-9
V.
~he
A C K NOW LED G MEN T S
author
~
wisp~s
tq
expres~
his sincere appreciation
to R.L. Ketter for his guidance and counsel throughout the
prog!['ess of this study.
the
de~~~n
Mr. Ketter is. also responsible for
of the end fixtures.
Tpis project has been carried out at the Fritz
Engineering Laboratory
~s
a course in structural researqh with
funds provideq by tpe Ipstitute of Research.
under the directiop of R.L.
Ket~er.
The program was
-10
VI •
NOM i E N C L A T U R i E
I
•
I'
A
Area of cross-section.
c
~xtr~~~
e
End eccentricity of axial load app+ication.
E
Young's modulus of elasticity.
I
Moment of
L
Lepgth of te$t
M
Bending m0rp.ent.
fiber diptance from the ceptroid of a section.
~nertia
Of total cross-section about the
prin~1p~1 axis of tne section ~der copsideration.
speci~en.
Bending moment at qepter line section of member.
Th~t
part of center liI}e moment whi'ph is independent of
q~flect,ion.
moment at which yield is first
Ben~ing
~eached ~n
fixture.
Axial lqam.
Maximum load a member can carry.
Euler
buc~ling
load.
That lQad qeyond which a member deforms inelastically.
load correspond~ng to yield stress level over
entire section.
~xial
' 1
.
I'
r
Radius of gyration of a section.
y
Deflection
Yo
Center line deflection.
yfl
Second derivative of deflection with respect to
longitudinal column axis. '
,
I
~n
the
p~ane
.
of moment application.
"I
Center line deflection.
I
Unit normal stress.
Lower yield point stress.
Curvature
Curvature at initial yield.
-11
VII.
REF ERE N C E S
:1-.
Beedle,Lynn S., "Plastic Strength of Steel Frames",
Fr1tz Labor~tory Report 205.~6, Progress Report No. 14,
February, 1955.
2.
Bleich, F., "Buckling Strength of Metal Structures",
McGraw-~ill Book Company~ New York, 1st. Ed., (1952).
British C,onstructional Stee~work AssO,C,ia1:;ion, "The Collapse
Method of De~ign", B.C.S.A. Publication No.5, (Br.),
1952.,
. '
5.
7,
Ket~er,~Qbert~
L., Ka~insky, E~mund~L., and B~edle, L~ S.,
. "Plastic Deformation of Wide-:F1a~ge Beam-Columns",
A.S.C.E. Pro~., Vo+. 79, Separate No. 330, October, 1953.
8~
Thl1rJ,imapn,B., Drucker, D.C., Report 2toCo1UlD.l?-Resl?arch
C01.p1.ci1 find the Rhpde I~land Dep~rtment of Public- ,
Roads, "Investigation of the AASHO apd'.A.R'~·~~ef!tcat1ons
Co1wnns Un;der Combined Thrust and Bending Moment",
Brown University, May, 1952.
9.
Timoshenko, S., "Theory of Elastic Stability", ;McGraw-Hill
Book Company, N~w ~ork, 1936.
10.
W~st1und
G., Berg~trom S.G., "Buck:ting of Compressed Stl?el
Members", Trans. of the Royal Inst:i:tl:lte of Tecb+101ogy,
Stockholm, Sweden, 1949.
for
-12
Tdole I - Co/u.mn Spec/mens (AX/Blly Looded)
Specimen
AV~(' Area
e
Lengfh
No.
A-/
A-2
8.7G
13.00
17.16
21.J8
26.00
A-3
A-4A-5
T:Jb/e J-A - 5L1mm;;wy of Predlcfed
Spe'1men
No.
A -I
J1.
tn )
0.565
0.563
0.562
0.562
( Jf7.)
40.5
Go. a
79.3
98.7
120./
0.5GJ
I
Acfu.1/ Column (Ax/CJ/Iy LOdded) CO//;Jpse LOdds
R/C5Jlculc:lfedJ
Pe (CalculafedJ
(165.)
(Jbs)
J8,070
.,fi
/12, /00
45,400
·25,890
Ji3.030 '*
17.980 .t
/6, 770
18,000
A-5
18,030
I 1,340
.. Pf'edlcled FaJiw'e Load
A-Z
A-j
A-4-
Per. (exper.) % DeVIation from
(/65,)
Predlded Per.
17:'500
/8. JOO
'II(
:+;
-3.2
+ 0.4
/8,000
16,700
+ 0.1
/2. 000
-/ 5.c3
- 0.4-
Tdb/e 2 - Column SpecImens (£ccentncd/ly Loaded)
Lengfh
(m,)
Specimen
No.
£ -1
£-2
E-3
E. -4£-5
Yr
Avp,- Area
Jnz)
8.75
0.563
40,4
13.00
0.5640.503
GO.O
79.5
93./
120.0
/7.22
2/.45
25.38
0.564
0.562
Out of 5tralghf Measuremenfs
2
.A
3
+Readlr19s
4
..
Readlllgs referred
fo "X~ number
.sfamp~d on speC/ffJen
as
.spec.
£-1
£-2
£'3
£-4
£-5
A
13
I
2
3
4
5
+0.005
- 0.003
+-0.0005
- 0.002
1"0.0005
1'0.001
- 0.004
+0.001
0
0
-r0.001
- 0.00/
- 0.001
-0.0005
rO.OOI
1'0.003
+0.003 .
+0.00/
.,.0.003
-0.0005
+-0.002
-0.002
-0.00/
A
to.OOI
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.A
fJ
to.0005
-0.005
+ 0. OOZ
+0005
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- 0.004
t 0.00/
A
fJ
A
13
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0
0
0
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0
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-0003
-0.0005
+0.010
0
0
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-O.OOIS
. -0.006
+ 0.006
t 0.008
+0.0005
+0·002
-O.OOZ
-13
T;;;b/e 2-4 - Summdr!J of PredJc fed ~ AcfuJI Co/wrm (£ccenfnca//y LO.Jdfld) Co/lapse Loads
Spec Jn7en Pel. (C~/c)
Pe (eald P:; (Cald PCI' (Calc) f(r (E:.Jc.pr.) /'0 Devlc;ltOn from
(Jbs)
'(165)
No
(/bo5)
(Jbs)
(Jbs
Predlcled !?r
£-1
J3>050
100,070
18,020
/6,200
/5,500
-4.3
£-2
I 1,200
45,390
18,050
/5,120
/5,750
+4.2
£-3
6,650
25,900
/8,020
/3.4/0
14,000
+4.4
£-4-
6,400
/6,670
18,050
/1,250
/ /, 750
f4.4
£-5
6,070
/ /, 330
/8,000
~800
~350
~/.G
F:{
Nofe
7T
2
Average values or 6J ¢ E.from TenSIon Coupons used
£ A
(~y
R.y
In
cal cu ICJhon~.
.32, 000 psi
U; :;
£.,. 2:3, 4=;
R.r (Calc~
- ObfaIned
;:'l" .25
from
F'J' 8 uSing
. ex.cepf for SpectrJ'um
£ -5 where
~:s ./G .
J(
10" psi
-14
Figure A
•
Figure B
-15
FIgure C
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LoC/ding ¢ Unloadm9 Curve fo Show [Ffeet of
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