IR. F. K. LIGTENBERG
STABILITY AND PLASTIC DESIGN
(2)
o
IntroductionV.D.C. 624.042
In consequence of the slenderness of the structure, the collapse load of a portal frame may be lower than is indicated by the elementary methods of plastic design. In this paper it has been endeavoured to assemble the now available data in such a form as to enable this effect to be estimated. To this end, a hypothetical buckling load P e has been deduced from the actual collapse load Pcr and the ultimate load Pp , according to plastic design. This load Pe
is determined from IIPer = IIPp
+
IIPe.From P e it is possible to deduce an effective length le which depends upon the ratio of the direct force in the bottom column to the bearing reaction of the bottom beam, upon the boundary conditions of the bottom column, and upon the shear force to be transmitted by these columns. Although there is no "exact" theoretical basis for the computation rule obtained, the collapse loads of portal frames can nevertheless be satisfactorily predicted. For the designer who bases himself on the methods of plastic design it is important to have some warning of the circumstances in which he must provide diagonal bracings or other means of ensuring the stability of a portal frame structure.
In consequence of the slenderness of the structure, the collapse load of a portal frame may be lower than is indicated by the elementary methods of plastic design. This problem of frame instability has also received a good deal of attention from investigators in many other research laboratories. In this connection the name of BAKER (Cambridge), BEEDLE (Lehigh University) and WOOD (Building Research Station) call for mention. Despite all the investiga-tions that have been carried out, not much progress has been made beyond the point where the behaviour of a single column with known boundary conditions can be predicted. With the aid of such data it is possible to per-form a complex calculation - involving the use of electronic computers - for investigating the behaviour of a simple portal frame under increasing load.
Simple procedures for approximately taking account of the effect of the occurring deformations upon the distribution of forces have, inter alia, been developed by LOOF and BERKELDER in the Stevin Laboratory, Delft. However, this research also has not yet made progress to the extent that practical prob-lems can be solved in a reasonably short time.
In the practical application of plastic design it is, however, of great impor-tance that the designer should be able to judge whether the elementary methods of plastic design provides an acceptable approximation of the actual collapse load. To meet this desire, an attempt was made to obtain, on the basis of the available data, at least some insight into the principal factors which are responsible for the fact that the collapse load calculated by means of the ele-mentary analysis is not entirely attained.
The starting point adopted was the only simple formula that can be consid-ered suitable for the purpose, namely:
I I 1
= +
-Per Pp Pe
where: Per = the actual collapse load
Pp = the ultimate load according to the elementary plastic analysis Pe
=
the buckling load, according to EULER, for a column with alength to be further determined
Various investigators (including W. MERCHANT 1) have used this formula. In
most cases a predetermined value was assigned to Pe (e.g., the elastic buckling load of the portal frame as a whole). In the preceding paper a method of determining Pe experimentally by means of caricature models was indicated. The theory underlying this method is somewhat disputable, however, so that it is better to achieve, as far as possible, realistic stiffness ratios in the "caricature" models.
1 Prelitninary investigation
In order to find out whether something could be attained with a method of this kind, a large number of data relating to tests with portal frames having one or more storeys (constructed to scales ranging from full size to 1 : 20) have been collected from the literature. Very important information was ob-tained more particularly from a series of tests conducted by Low, in which numerous small models of portal frames comprising 3, 5 or 7 storeys were tested to failure. Now that Per (the observed collapse load )and Pp (the cal-culated ultimate load) were known, it was possible to deduce
P
e from the formula:Pe was converted and expressed in an effective length le of the column, accord-ing to the formula:
n2
EI
Pe=
-le2
Low's fairly large series of tests (34 portal frames) was most suitable for obtain-ing a preliminary idea of the important influencobtain-ing factors involved.
In Fig. 1 three lines have been drawn which give a somewhat too un-favourable estimate of the effective length for structures 3, 5 and 7 storeys in height respectively. The relevant formula is:
1) W. MERCHANT, "The failure load of rigid jointed frameworks as influenced by stability". Structural Engineering, 32, No.7, 185 (July 1954).
_<>1"::':: 2.0 1.8 ---1.6 1.2 0.2
-.--.. -+--.--1---
----+ , 1·10·' ! 2 ·10·' o ___~
•
I column I beam 0.5 0.65 1.0 o +3-storey portal frame • 5-storey portal frame 0 7-storey portal frame ...
EI
..
,.
xFig. I. Low's series of tests.
le
=(n+
1+
100H) k
6 6 V
where: n
=
the ratio of the direct force (normal force) in the bottom columns to the vertical loads on the bottom beam (in Low's tests all the loads per storey were equal, so that n in the above formula is equal to the number of storeys)HI
V = the ratio of the total horizontal load to the total vertical loadk
=
the length of the bottom columnsThis formula is very simple, but cannot be applied to all cases. Some further important factors emerge in the other portal frames investigated.
Low's columns were fixed at the base. Obviously, a pin-jointed base will produce a more unfavourable situation. Further information with regard to 15
this is provided by the test series A and B conducted by Lv and GALAMBOS. Some of the portal frames investigated by BAKER were subjected to very large horizontal loads. In that case 1 00HI6 V would become far too large. Also, it is clear that HI V - especially if the collapse mechanism calculated in the elementary analysis is a beam mechanism (Fig. 2) - can easily cause
devia-tions from the collapse load. However, if H is
,. so large that a combined mechanism or panel
mechanism occurs, then there is no reason to anticipate a considerable influence of this kind. The same is true of portal frames such as north-light frames (saw-tooth roof frames) in which the inclined beams exert considerable horizon-tal forces on the columns anyway.
Because of these various factors the formula has been modified to:
QL---~====~~:HH n~5 100 H Fig. 2. le = ~·k
+
l+
-~·k 6 6 V Fig. 4. o 50 100 Per calculated in %H
seriesA n = 8.5 seriesB 11 = 4.3$OJ
(shape factor 1.27).0.." Ia..<'>. .
Ii
t~~t _s.;~ies A' lUtttt~nd Galambos (With partially restrained base)r Lu and Galambos
n
I "',.jr
1.0 L'_OS,'i f ' "
'_""e_, B,-*",*-T-I' ~i ----1f-I---H-++ :! I I I! \ : ~~ ...
u
t calculated for A'I
17,,1\[
'''''~Ij
I 1'1 ~I I LH~lculated for B
0.5 I--J---.J---.I-H~
..-:t,
'Jr-t+-i-t
I I -~J I ,e" ",;e, A L, oed G,I,mbo,
I I
Itt).:
I---+-+-+---~-+-I-I f--I 1/_ .j calculated for A
I I I I I
I I I I I
o 20 40 60 BO 100
~ 0 12C ~
~I
ill > u ~ .~ 110 B ..::! o 0. Q. Q.1 '""
90 80 A cP + o ® OJ B2 Rl 814 )( vlB3 B5 Bl 00 oV~R'2
0 o ~~~
[Z
70 60--~--.--AVID
A I • A 0 A • 50 40r·---r
30 20 10 10 +~
l.JL
-+--'l/l-/1
20 30 40 50 Fig. 3. 60 70 80 90 100 Pdesign rule in % ~plastic+
test series A of Lu and Galambos • test series A' of Lu and Galamboswith base partially retrained • test series B of Lu and Galambos
o Low's tests for 3 storeys
o
Low's tests for 5 storeys lJ.. Low's tests for 7 storeyso
B1 FSF 1 (full scale frame) of Baker with pin-jointed base<>
82 FSF 2 of Baker with fixed base.. 83 Baker's portal frame 4a
X 84 Baker's northlight frame F 5 • B5 Baker's north light frame F 6
T R1 frame of Ruzek, Knudsen,
Johnston and Beedle
I-R2 frame of Ruzek, Knudsen, Johnston and Beedle
V' Driscoll, Lynn and Beedle
; lY," I
11
(C]~t
50"t
I--.~~93 .~t
in mm ~~ in emP~~I
~
18':~~ l~I
>"y 16' :0]/ I-~Ai
rr4"16/j"1
1
~
'";:'S' I 8/~;~!
15"3"11/~i
{L!:i.= '/,
W ;;',J;n
p P T -SWF40:r
1--8B13 1 ~0~ __ ~~'i l
P P P i P/4~t
P/4 ~, ~w,j .:l 40 ' i----"'---iwhere I denotes the effective length of the bottom column, if the bottom beam is regarded as infinitely stiff (but able to undergo displacement, if any). Hence for a fixed base we have I = k, for a pin-jointed base we have I = 2k, and for a fixed base and a beam restrained by rigid walls we have I = 0.5k.
100 H
The term
-6V·k
must not exceed the value2/3k,
which is therefore the case forHI
V = 4%. There are not enough data available to give a better or more accurate rule for this.The result is indicated in Fig. 3.
The collapse load represented in this fashion satisfactorily reflects the in-fluence of various factors, even when considered in detail, e.g., compared with the test series of Lv and GALAMBOS, in which one and the same type of
portal frame with progressively increasing column lengths was tested (see Fig. 4), or with their calculations for a particular case (see Fig. 5).
---=-:::----",-, ~~m~t~~a~~a~s _ _ _ _ _ _ _ _ _ ,
-,,-'1,,-°1.0 " '_ '\
r---'---'-'
'~"~-:::'-='-"--.--,~:;::::::==:::::::=_I..c=-.:::re~du~ce~d..:.c:O~'~lap~.'e~load
due to buckling of columns0.8
\.
~ elastic buckling of portal frame
0.6 '
'"
"-I
1'-'",--,
~1 1 1~ ---.
collapse load predicted with formula p = p +
p-Cr p e w 0.4 P = wL 0.2 20 40 60 80 100 120 ~ slenderness ratio }.
Fig. 5. Comparison with Lu's theoretical results.
2 Preliminary conclusions
The method of determining the collapse load of a portal frame, as described in the foregoing, can be expected to be suitably serviceable for practical pur-poses. The object is not so much to obtain very great accuracy as to provide a
warning to the designer if a structure which he has designed should happen to have an excessively low collapse load in comparison with the collapse load determined from the elementary analysis.
In such a case the stability will have to be ensured by fairly simple means, such as, for example, the provision of local wind bracings or rigid diaphragms. The rule for calculation will have to be so extended as to enable the effect of such improvements to be ascertained and their adequacy to be judged. Un-fortunately, the number of available data is still very limited. Not only are no experimental data concerning the effect of wind
bracings available, but also the number of portal frame shapes investigated is very small. The most com-monly employed multiple portal frame types (Fig. 6) have not yet been the subject of any investigation. Far too few data are available on portal frames in which the columns are pin-jointed at the base or in which one of the columns is stiffened in some degree. Only a limited proportion of all the portal frames investigated consisted of I-section members. Most of the smaller models were constructed of members hav-ing a rectangular or closed box-type cross-sectional shape.
Because of these circumstances, research into the structural strength of a number of portal frames has been undertaken in collaboration with the Research
~~ >;;''7'" ~ ~>;;' ~ ~b7:; Fig. 6. ~ ~ 7'
m
Committee on Steel Structures of the Vereniging van Constructiewerkplaatsen (Netherlands Structural Steelwork Fabricators' Association). Some of these tests are reported in the following.
3 Test series
So far, a series of model tests has been carried out in which the test arrangement is comparable with that adopted by Low. Small portal frames with members of rectangular cross-section were loaded to failure. These frames can be subdivided into three groups:
1. 16 single-bay portal frames comprising 3, 5 or 7 storeys, in which the effect of the horizontal force and of the pin-jointed base was investigated in order to obtain some extension of Low's test series (see Table I). The results have been plotted in graph form in Fig. 7. The test results are in fairly good agreement with the calculation rule indicated on page 16. 2. 16 two- or three-bay portal frames, with the object of including more
currently employed portal frame shapes in the investigation (see Table II). The results are plotted in the graph in Fig. 8. Although the actual collapse load is usually somewhat higher than the value obtained by means of the computation rule, especially in the cases with fixed-base columns, there is nevertheless no very good reason for establishing a different rule for portal frames of this type.
*
100 c~I
'
Jj ~ 90 ~n..c...r
80 70 60 50 40 30 20 10 Fig. 7.*
100 90 70 60 40 30 20 Fig. 8.20
I IJ7
I 10 I IJ/
i I ~---V
010V
19 03 9 +4 I--IV.
I 011e----+--VL
, 014+ 6 ! : I IV
r2
+15 , i-V
16 I , _---~~L. , ! 10 20 10 20 f--30 40 I-o frames with fixed bases
I + frames with pin-jointed bases
i 50 60 70 80 90 100 _______ Pcomputatlon rule in ro Pp1a.>tic +28[18
/ojio
~o: /!
I 2 1 V !o frames with ftxed bases
+ frames with pm-jointed bases - - + - - j - - - j ---- --+,--,--~-~--I 30 40 50 60 i 70 80 90 100 ~Pco'nputatlOnrUlein% Pplamc
3. 4 portal frames in which one of the columns is of double construction (see Table III). The results are plotted in the graph in Fig. 9.
036 a": 100 , , , -so 40
i-~
30 f---+---+--+---+---+---I---+---t----+ I 20 1---+--+---+---t---t-------1-~--f,,-m"
wich;xed~
+ fee,"", wiCh pm-jomcod b"" __
j
10Fig. 9. 10 20
As was to be expected, a stiffening of this kind has a favourable effect upon stability. Such cases do not call for the establishment of a different calculation rule.
The investigations are continuing. In the first place, the effect of the cross-sectional shape ofthe members (I-section) will be studied.
4 SUlUll1.ary
On the basis of available experimental results concerning the collapse loads of portal frames it was possible to establish an empirical design rule which enables the effect of the slenderness and of various other factors on the collapse load to be estimated with a reasonably good approximation.
Some supplementary tests conducted by the author did not provide suffi-cient grounds for modifying the design rule. The effect of the cross-sectional shape of the structural members will have to be further investigated.
The importance of such a design rule to the designer is more particularly that it provides him with a simple warning if he designs a structure to which the elementary plastic design is no longer correctly applicable.
In view of the very considerable effect of the number of storeys it is advisable, in the design of high buildings, to pay a good deal attention to the prevention of a panel mechanism (e.g., by the provision of wind bracings or stiffenings). For the dimensional design of the individual columns the normal procedure should of course be applied. In the present case only the overall stability of the framework has been investigated.
References
1. W. MERCHANT, Structural Engineering 32, No.7.
2. Low, The Institution of Civil Engineers, Proceedings,July 1959, Vol. 13, Paper No. 6347. 3. Lu (thesis Lehigh University, 1960).
4. RUZEK, KNUDSEN, JOHNSTON and BEEDLE, Welding Journal, Vol. 33, September 1954. 5. DRISCOLL and BEEDLE, Welding Journal, Vol. 36,June 1957.
6. BAKER, HORNE and HEYMAN, The Steel Skeleton, II.
7. Lu, A survey of literature on stability of frames. Welding Research Council Bulletin, No. 81, Sept. 1962 (this publication gives 146 literature references).
TABLE I
frame frame shape collapse shape actual beams in mm H/Vin
%
No. with according to collapse columns in mm Pp in kg
dimensions elementary shape av! beam in kg/cm2 P w in kg
In mm theory a,,! col. in kg/cm2 Pcr in kg
M p beam in kgcm P,v/Ppin
%
I M p col. in kgcm Pcr/Pp in%
I 6.5 X 6.5 2~r
~ ~
6.5 X 6.5 255 2900 231 ~i x, 2900 212 011 200 91 380 f---i 200 83 -6.5 X 6.5 6 2~:l
~ ~
6.5 X 6.5 255 2900 207 2900 202 380 200 81 ]--.----1 200 79-~~j
$-~
\~
6.5 X 6.5 10 3 6.5 X 6.5 255 2900 177 2900 202 380 200 69 ~--; 200 79 -6.5 X 6.5 2 4f1~:
~,
ft\
6.5 X 6.5 246 2900 153 ':;1 2900 156 200 62 f-1BQ"" 200 64-*,
~
6.5 X 6.5 6 5~~!
6.5 X 6.5 231 2900 117 011 2900 133 ~ 200 51 200 58-~g!
~ ~
6.5 X 6.5 10 6 ~I 6.5 X 6.5 219 ~1 2900 102 380 2900 129 f.oo----1 200 46 200 59-~~l
~
~
7 6.5 X 6.5 0 6.5 X 6.5 435 3000 360 3000 347 206 83 i-~BQ., 206 80 ,-\\
8~'
+~
6.lx6.1 2 a:: , 6.0 X 8.0 230 2000 210 xl 2710 204 ",I I III 91 .1 262 89 r-l§~,-~!
,~
~
9 6.5 X 6.5 6 6.5x 6.5 385 2900 240 2900 238 200 62 380 200 62 I--~---ITABLE I (continued)
frame frame shape collapse shape actual beams in mm H/V in
%
No. with according to collapse columns in mm Pp in kg
dimensions elementary shape (Jvl beam in kg/cm2 Pw in kg
Inmm theory (Jvl col. in kg/em 2 Pc,' in kg
M p beam in kgcm Pw/Ppin
%
M p col. in kgcm Pcr/Pp in%
10g!
,~
l
6.lx6.1 6 6.0x8.0 225 2000 160:1
2710 186"'1
III 71 .2~ 262 83 II~1
¥\
~
6.5 X 6.5 10 6.5 X 6.5 345~I
2900 185 2900 221 "': 200 54 J 200 64 380 f----o-1 12~I
~
~
6.5 X 6.5 6 6.5 X 6.5 355'",
~l
2900 2900 130 143 200 37 200 40 >----' 13~~I
~ ~
6.5 X 6.5 10 6.5x 6.5 235 2900 100 2900 119"'1
200 43 f-~ 200 50 14 6.5x6.5 I 6.5 X 6.5 609 3000 287 3000 336 206 47 i 206 551
cE()~ 15 6.5 X 6.5 6 6.5 X 6.5 504 3000 189 3000 231 206 37 206 46 380 I---~ 16 6.5x 6.5 2 6.5 X 6.5 567 3000 161 3000 168 206 28 206 29 380 f---'"TABLE II
frame frame shape collapse shape actual beams in mm M1) batt.
No. with according to collapse top col. in mm in kgcm
dimensions elementary shape batt. col. in mm H/Vin
%
Inmm theory (Jv! beam in kg/cm' P1) in kg
(Jv! top in kg/cm' Pw in kg (Jv! batt. in kg/cm' Per in kg M 1) beam in kgcm PW/P1) in
%
M 1) top in kgcm Per/P1) in%
17 6.5 X 6.5 211 6.5 X 6.5 2m;~!i~
m
6.5 X 6.5 840 3090 846 3090 680 3090 101 l()o 150 30~ 211 81 211 -18 6.5 X 6.5 211ml=thfH
~
6.5 6.5 X X 6.5 6.5 5 840 3090 660 3090 584 300150300 3090 79 ~-.~ 211 70 211 -19 6.5 X 6.5 211mf~W=1
m
6.5 6.5 X X 6.5 6.5 8 840 3090 570 3090 584 3090 68 JQoJ..5QjClCL 211 70 211 -20 6.5 X 6.5 262 6.5 X 6.5 8~~*~~jl
~~~
6.0x 8.0 840 3090 624 3090 670 2710 74 300150300 211 79 I I 211 -21 6.5 X 6.5 211HM[¥~Wl
m
6.5 6.5 X X 6.5 6.5 2 840 3090 435 3090 422 3090 52 1QO~<lCL 211 50 211 -22 6.5 X 6.5 262m~i~t1
m
6.5 X 6.5 2 6.0x 8.0 840 3090 510 3090 535 300150300 2710 61 ; - - - "'j---, 211 64 21125
TABEL II (continued)
frame frame shape collapse shape actual beams in mm Mp bott.
No. with according to collapse top col. in mm in kgcm
dimensions elementary shape bott. col. in mm H/Vin
%
lnmm theory av! beam in kg/em 2 Pp in kg
av! top in kg/em2 Pw in kg av! bott. in kg/em2 Pcr in kg M p beam in kgcm M p top in kgcm Pcr/Pp in Pw/Pp in
%
%
23 6.5 X 6.5 211 6.5 X 6.5 4:wq~l
1M
~~
6.5 X 6.5 414 3090 402 3090 322 3090 97 pq 350 I 211 78 211 -24 6.5 X 6.5 211 6.5 X 6.5 10mo'
m
~
6.5 X 6.5 414El
3090 315 3090 322 3090 76 150350 211 78 ~--. 211 -25 6.5 X 6.5 211 6.5 X 6.5 10~I
ffi
~
6.5 3090 X 6.5 414 357 3090 322 3090 86 .2S-0J.59 211 78 211 -26 6.5 x 6.5 262 6.5 x 6.5 10m~l
~
m
6.0x 8.0 414 3090 387 3090 343 2710 93 ~5SL??5?.., 211 83 211 -27 6.5 x 6.5 211 6.5 x 6.5 4{~r~
~
6.5 x 6.5 414r~'
3090 195 xi 3090 220 ~J 3090 47 150 350 1---..., 211 54 211 -28 6.5 x 6.5 262 6.5 x 6.5 4:fM~1
~
m
6.0x 8.0 414 3090 324 3090 273 ~j 2710 78 150 350 211 66 ~~ 211TABLE II (continued)
frame frame shape collapse shape actual beams in mm Mp bott.
No. with according to collapse top col. in mm in kgcm
dimensions elementary shape bott. col. in mm H/Vin
%
Inmm theory (Jvl beam in kg/em 2 Pp in kg
(Jvl top in kg/em2 Pw in kg
(J.vl bott. in kg/em 2 Perin kg M p beam in kgem M p top in kgem Pw/Pp Pcr/Pp in in
%
%
29 6.5 X 6.5 211ffi:~*~
m
6.5 6.5 X X 6.5 6.5 576 2 3080 465 3080 469 3080 81 f--3~? _.J~_o_ 211 81 211 -30 6.5 X 6.5 211~~~
6.5 X 6.5 6f1f~~J
f=fi
6.5 X 6.5 576 3080 408 3080 412 3080 71-
350-+-3~ 211 72 211 -31 6.5 X 6.5 2621~\
6.0x 8.0 2ffi~[
fl'l
6.0x8.0 384 3080 336 2710 286 350 350 2710 88 r- -- -....;---1 211 75 262 -32 6.5 X 6.5 262ffrffl
m
6.0 6.0 X X 8.0 8.0 6 384'1
~t
3080 256 2710 261 _3_S_?--"-3~_Q-i 2710 66 211 68 26227
TABLE III
frame frame shape collapse shape actual beams in mm H/Vin %
No. with according to collapse columns in mm P v in kg
dimensions elementary shape a vl beam in kg/cm2 Pwinkg
lnmm theory avl col. in kg/em' Per in kg
M v beam in kgcm M v col. in kgcm Per/Pv in Pw/Pv in % % 33
