Huber, A. W.

English

Lehigh University: Lehigh Preserve

Lehigh University
Lehigh Preserve
Fritz Laboratory Reports Civil and Environmental Engineering
1951
Connections for welded continuous portal frames,
theoretical analysisiof haunched knees, 1951
A. W. Huber
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Recommended Citation
Huber, A. W., "Connections for welded continuous portal frames, theoretical analysisiof haunched knees, 1951" (1951). Fritz
Laboratory Reports. Paper 1383.
http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1383
"'......
'llHEORETICAL AEALYSTS OF' HATTNCI-IED KNEES
by
.A1fons W.Huber
A Report Submitted to Mr.Lynn S.Beed1e for
Credit Requirements in Course C.E. 209.
Lehigh UniversIty
August 20,1951.
r.,
TABLE OF' COnTENTS
Introduction 1
Stresses at the Haunch 1
Rotation Analysis 5
Shear and Bending Moment Rotation 5
Equivalent Length Method 8
Deflection Analysis 9
Newmark's Numerical Method 9
Graphical Method 11
Direct Integration 12
Conclusions 13
Bibliography 14
1INTRODUCTION:
Tests have been completed on varj,ous connections for welded
continuous portal' frames in our laboratory*(l).This report presents
analytical methods to predict the stiffness of a haunched knee as
characterized by moment-rotation and moment~leflection relat~,ons
in the elastic range.
STRESSeS AT THE HAu~CH:
It is of importance to know the stresses at the haunch in or-
der to analyse the knee for shear and bending moment rotations.It
has been shollvn that the stress distl'ibutibn in a wedge is no longer
linBar for plane cross-sections(2).However the given solution is
only exact 'when the 'wedge is held by radially directed forces at
the supported end which are distributed in the same manner given
<by the ,solution.
We are an.·'dous to know the stress distribution just at the
supported end where the above l1isntloned solution is not correct.
Therefore the stress distr~bution DOS made on'a more rational ba-
I
s1 sunder :the following s Implifyj.ng assumpti ons:
loThe stres~ di~tribution varies liriearily over the web and is con-
stant over the whole flange areas.
20The shecri~g stresse~ are uniformly, distributed over the whole
"web area.' ,';
, ,I
We will consider the stresse:3 due to V and N separately.
F1Mdes 1gnQ. t e s the force in the upper flanGe due to Vo
F211l designates the force in the lower flange due to No
I
Compression ne£;ati ve and tension po s:i t :i. ve 0
*Figures, in parenthesis refer to the bibliography at the end of
the report o
2~ II N
~h\ ::--~~--"-
r) 1===;==~===='~v
h1.
,
<iU4": FAN2, C,OSa..
Considering the equilibrium we can write the following equat:tons:
w'ne.re..
then rIM::' M eLlM·cx.'Mcx.~M + 01. 1"" CX~t'\
F2. ... -M O£.,t-\
M O!. \M 01.. 'ttol + C\lto\ 0(31'\
too _ ~ I\~ 0
r3M- "M A, a
r.- I ( . .
L'I =Aw V+ f 1."'1 Ilr'\I\1\ IX )
Fa.... ~C5'2t1 ::O ---=->- ~d... ~..... '"AL yn A~
()~:= F.tt~ (shess \n diree..t\on 0\ paY\ge)
It ~e. le.t 0<.=0, A,=A2. 1 A3 =O / Q=b= ~ (ordinory 'pea~) I we ob\ain',
3The small difference in stre s~: :i. s caused by our assumpt :i. on that
the stress is constant in the flan~eso
The equations contain the unl-:noYJns a and b oThose have to ~.Je
guessed for the f~rst approxlmation and the stress computation has
to be repeated if necessaryo
2 0 St res s e:3 d'l e toN 0
Again cons:~derlng the e~nJlibpium:
0= F"2. N t.#tvQl.- 1:'N AV-i
N =-F;NOL'H - t='2.N ClC'2.~
N hi:' - F;M o..~N - Fz.t.l CI.. \tN
oY\d
<:J.'.I = £..uL
I", A. \
(). ... = F~N ~ol
1." Al.
* .
02.K = . ~~ ( .st'(~s'S. \n ~itec.\"\OI'\ o~ \h~ ~l.an9~)
4StreSSGS at the Haunch of Connent~0n B:
1'2." '2.0.'2.5" '2.6.125"
-:-'I' --. - ,.~ - .. ---- ....
I
p
/
M:=0'2..~?
Gb':lov= ·995'
~0l:.OS87
1 "t~== 'T
A.\ -= \ inL.
1 . 'l..p...1.=--" \.00511'\
~c:(.
,.. I
Fa... "" . 0 5 S9 M
F'l.M=-·OSf>\ 1'0'\
p
r y -= -S(.707 -.\8:!> ) :: ·17L+<O P
FIt~::-·2\5P
r'l.K:- ·O~59P
o..l~:= (\+ -t,)-:: '2.·5
Cl ~N= ~ = 1. '2.\
0-.'1.1-4:: (I .. l..~o\) Cffio.. 'C 2,.4-1
C1..-\\oN"" (12. +.s~;2.5) ~o{ ::: \ q.O
C5\t'! -:: --. '2.\5 P
()lK -= - .obS\?
cH.sl,)~ed "
c.: ~ hS
, b -= .4~S h, (o.~~umed 'io11Je. was ·5 h ')J
c..",.\.i\h (\I " .\1\,)
i' the. QS&\lW\e.<:\ "o.\ues O'fe. too ,a(" oTt I
the. c.omputQt\on 'na.s '\0 be. repeated \'(Wn
tne. \YV\'P'fo'ie.d va.lue.s,
cr; ::: 61"" ..... G)N = \,b\S?
cr2.. .. (J2.M 1- ()i~ =-\.t&7P
Si VV\p\e. be.o.m ~'n e.OfY "
(I, .. \,2..&'2..?
6"2. =-\,51ZP
Ble\c.h "
cr, '" \, 'l.bO P
""(j"2, ::: -\.b8~P
Wedge. tneory ( OSC3ood )
6"','= \,468P
lJt()2.. =-\,S9Z.?
5RO'}'A'fl T C)",r t\~" liT -,rc:; T "') 0";:;' 'fTrJl;' K1\Fi'P (' 01'.~Yfi'(~I]'J·OT,f p
__'_~~ _ ~ '." ... :..J..J.L ~. " l: _ j .. l .... _\ .L• .~, .J_~:~,--!-~ '.' _. l.J •
The analysIs will be made unde;.' the following assuJnDt:'Lons:
10The mOll1'8nt at the section AD (Mr ) contr:1.butes to the stIffener
: l-:' .
shorterHng (Mra ) and to the":' shear' oei'c:rma t 1 on (Mrb) 0
Mr : Mfo ~ ""rb
20Bendin~ and normal stresses over the web are neglected.
3.The normE'tl forcesFUIf and F2 N contribute to the shear de:{ormation.
4.The stresses In the flanges (BC,DC) Qnd the diagonal (AC) vary
linearly and are zero at Co
I=,~ ... Q IM Mr
F'2.",,:.-Q2,)r\ Mr
, Flt~ = - Q'lJI N
F2.I'~ =- Q2.N N
The stress at point A is eqUal to the force in the direction of
the dia60naldivided by the stiffener area.
"
6b)Rotat\r:m due to Shear:
"C___ I 1:; -= ~w ( 0 11,\ N\rb - o. N ~) = r G N
"t'l::' ~w (Q2.."" Mrb+ Q2.~N)(~cx-~O<.) - A""
r7 rY ,...,..,. rv l"C''''''{" is only equal ·v2. for, 0(.:= 0 • (,., increase s wi th ~ over 2. .1. ile snear
rotation will be calculated under th~ assumption of pure shear of
the magnitude '[;, which will give a larger rotation than the actual
rotation iso
M _ AwGk'"+OlN N
rb - On-\
and
From tha last equation ~ may be determined.
c)Rotat:i.on rJ::l0_ to Bending:..
The calculatIon will be made very simple under the assumption that
the rotation is determined by the elongation of the outside ~langes.
The total rotation then is
For connect~on B the rotatton of the knee area is:(see also page
4 ~or the flange forces)
Mf=~'l·S?
A..,.,-= :?> \Y'\1.
As -= \ \n1..
~(45+-o<.):: .63\+2-
G 0 £/12..&
Q \I'~" . '2. \5
Q\loI\ = .05SS
02.~ "" .056\
. 2. \5 P
·0559
7\r::=. ZB.95 -6() \04oQ.. = .0'2.77P \0
-'-¢r': r + (b =. 0 fd 5 P \0
In order to compare the theoretical analysis with test results
t'!o have to consider the rotation of the tapered members to the points
where the rotation measurements had been made.This rotation will be
determined later when we make the deflection analysis.~he result of
this analysis is then used to determine the total rotation which
is comparable to the test results.
The rotation of connection Bas r000rted in reference 1 is
,/.. -b
'fJ-r ~ . b ~ 5 P 10
I
There is however evidence that this test rotation is in error.lf
we take the ~:'otation a t the point of measurement and nm.l-c:tply wi th
the distance to the applied force and add the defl~ction of the ,can-
tilever beam from the point of rotation measurement to tho applied
force we should get a value which is a little l&~s than the measu~ed
deflection.Less because we negiected the deflection of the point of
rotation meaDurG~ent.Doing so we obtain ~ value which is biG3er
than the measured deflectlon.The deflection 1s believed to be' correct
since difficulties had been encountered in the rotat 1.on measurements
and the devicG for the measurements had been improved on other con-
nections o
OT= \1+.'29P\Cb <PTL" .b35P\Ob X '25 = 15.BoP\O-ft,
de.ft- aT c:.antile.ve,r = 6· 2.1 . "
-ft, '2'2..0\tP10-ft,
0" '2.2.04 x. .707 =15.(Q P \0 > &T
8Rotation Analysis for 1~~nivalent I,aneth:
• - I
Another approach to calculate the deflection is the analysis of
the knee talcJ.ng the e01)i valent length of the connection Denter
lLnes from one rotation measurement point to the otheroTh:i.s analy-
sis neglects the influence of shear and takes only into account the
bending moment.
Mo~e.Y\t 0\ IY\eftiQ(IY\")
----- ')(. )( \
-
'1.
-5-
5'2.'"
Be.\'\di\"l9 Moment
-
i--~-.~" 'j]I:~9.S\YI't
I ~ i (..---L~ l:t= '1.&0 i~'
1\:: '2,·98 X. ~ ~S. 5
I.1."" 35. b )(.l+ \O~
2~V
M\= \.O~V)(.+ ~~y
M2.= \·O~ Vy..,+ 48·~Y
;"0" 52," ,.J
rt.-;:. ?JtoheMI, ds = 2.
E
YtY'08~ +- ~b dx. ~ r
't' J '2.·9&)t.~~9.S J
o 0
,+. ::. 2-" t 7 ~ or '3.92"'1"" 62·5 +- .\Db + \·2..75 log 118.2.} =
't' E L' "'"":.) ~C3·5 \ 0"-
-~~: .4-\7 P \0
9The deflection will be comp\J.tod for the canto: lev~r beam fixed
, at the haunch AD. The deflection due to:the rotation of the knee
(computed in the forego1ng chapter) will be added.
Vario\.J.s methods are to our disposal to compute. the deflect;:!.on
of a cantilever beam.Three will be considered in this report in
the order of their rational application.
a)Newmark's numerical method (6)0
b)Graphical method (Any textbook on the sUbject)
c)Direct integration (In some cases)
II MAll three methods are based on the basic flexure equation 'I =--'0EI
However the way of the solution is different.
The metbod is based on the following two equations:
.. "It. 0 IV~ ~i:--I ':' -Iyo" ~y~ . yR =- Yo + eRn.).
- " - "..:--
In order to obtain the deflections properly we have to reverse the
bearnoThe fixed end moment of the M/EI load will then be already
the desired deflection.We let ~ start from the fixed end.
The table on the next page contains all the value requl~ed ~or
the numerical procedure.
--'2.6.\'2.5"· -. . '2.0.2.5 "._-
\51'+ l?l It 1\ 10 9 i 7 b ~ \+ ~ t I 0
5' _I(.= .~-~:~-=---L_ r ~- i\ ~ 12"
---...1
.--
)(-.,
I: -:: t w (n - ~)~+ btw('n+t _ 2i)L.
,x. \"- 5 '2. 5
I 1/ '
till'" 'it h '" \\.S"
I....., _. ti~e.d.. e.nd ~OWl tnt Oit ~. \OQQ\Ylg
will be. the desife.d defLe.ct\OY\.
~.sulls: £0 b8.89 x.707 Plci"- It·S8P,c)i>
&8· 1 "'-~'21& PlO" :: .30\ P \(5~
o• .00 I
1..75
'aO.7S
1't.iS
II 8.15
II
II
-.1'2.5
2·875
S.B15
'2.M Ie..
VI
50.15 ;~:; -= 50.15
5 '· 75 3Cj·S - 5'" "'-
'0. 4\.9 - V. J
"
"
.
rII
39.5
1t\.9
,-
M
V
71
n
1 ch'
l 01
10-
; &0.75 ;3q.!? ':: !.+Pt -
• 1100 I
!
I
!86.9
t
ii 77.0
""V
----~-----------+--;------4--'I '29.&75 .
6 1I
t
38·815
12. 41.875
i
I
~0
I
1'+ 35.&15
: 3'2..87 S
I· IIS,
: !
: i Z9.S15
t>e.~led:\on
1\.5 a~.15 41.0 13.15 \Ob.5 \45.15 \90.0 ~1t015 'l.CJ\t.QS 34lt.'2.S 39\."35 4:%.15 "'17.55 51b.95
20.15 ~El.75 32.75 38.15 't'tJ5 50.15 5~.3 50:2.. '+7.\ 44-.8 4-\.4- 39."t
e
IOe~~9nQt ..
1
II
. roe-toro'2..s678
---_.- .._---------------------
910\ \1'2.1 15 lit 13
--------
I
i 2..75 8.15 \4.75
I
r----~-"---- --
______.__••__• __ . ••• __• "_0' .. __.00_
81.50 \bl.'S 'ZJo7.75 \4-1'?>.0
.._---_._-----~-----_.- --_.•..-
i
"'t----..- ...
V).1- I
'2.EI: c 1 y
. - _.. -. ~ .
•
_& I
\0 P Ij7.515.050.\01.6'2.
~
G>03.0 S4-:5.15 \\'37.8 I't8,.o5 \~1~·4 Z:lI09 .55 ~787.\ 330'+.~
1
I
8.&9 i
!
1.\05.71B.2.\0 f .4~2.
1'+:25 'to.50
.O':;&~ .10&
'l..15
.OO7\t
11
. b) Grapl1i'cal I,Tethod:
e I
The m.oment diagram of theM!EI loading which give the deflec-
tions is obtained graphically.Care has to be taken in choosing the
scales for length and force o
le.Y\g\:h I" .. 10" (LS)
. F'ovc.fit. I":. '2.00t~ - ,,,1.1
It'
EI.c. P
I IS.SOY
2. ·4-:!>.ioV
3 68~10V
"t. 9,3·20Y
5 118·2,0'1
" 12.5·50'11
7 1l4-.50Y
a \O~.oov
9 .99·'2.OV
&./ /
'/
. ,/9 I
. 10 . -~~. =~.50Y (,n) 6 --- ~.50" ':, 12·50 V \0Wl8QS. . , '2.~ \0· .
10 I -E>a/":\·50\1-- : . \S,+ P ,0
76· 2. ~q06 '3\.\.5
I.
\
The .results are very close to the ones obtained by Newmark's.
method.
12
c)Direct Int0£ration:
In som,e ca~es it is possible to integrate y"=M/EI diI'ectly ..
This can be done when I is a function of x which can be inteL-rated.
\
If I is to C0TI19lex an approximation can be made by a sim~oller fnnc-
ti 0l1 o The latter has been made in the following calculation fol"'
connectLon B.
M,o::. .,C)1 P x. M I = ('2.~·\'2.5 +X 1) ·101 P
M': ·l()"P M: : (?~H1.S +~t)· 707
1:)11 = ~8(7.5+~)"b ... ~(1.'5 +- ~,)1..
-"S = \~.~ ~ \0 '
The deflect10n computed by 9irect integration is somewhat bigger
than the preViously obtained deflection.Thi~ is due to the repl~ce-
ment of Ix by a linear relation in order to facilitate the inte-
, grationo
. The deflection 6, has been computed using as the canti.lever
beam length the distance from the junction of the centerlines of
the knee legs to the point of applied force Po
13
CONCLUS 1.£;""'1S':"
Using the deflection curve as obtained by Newmarkb nUTIlerical.
method we obtain the following final r~sults:
a)Rotation:
.
. -b . -b¢ 0; cPr +8g c(30 \ + .08\5) P \0 =.~825 P \0
<PT= .6~5~::no6 (tes," 1tSult)
On page 7 it has been explained why the test result is believed
to be erroneouso
b)Deflection:
-b ( )-b&=(8.&'Q-x'2. + .08l5X4b.~15).707P\O = \'l·S8+1.'1 Ptt>
6 = \5.2.5 P \~fo .
<Pr '=.08\5 (5ee PQ9~ 7)
- ~
bT:: \4.2.'H> \0 ('\.e~t Ye~\)H:)
8.g<t P \Ofo '4eft\c.~'l ot.~ltc\'\o~
0\ \net ,c.o."\:'l tve'f.
The analysts gave a deflection 607?~ higher than the deflection
measured during the testoIt should be noted th::,t the normal force
was neglected in the deflection analysis o .
It is customary in ordinary frame analysis to assume the connec-
tions to be fully rigid at the points of junction of the centerlines
. -
of the 'meeting members and the length of the latter to l)e determined
by the distances between those junctionso
Therefore the deflection ~, has been computed (page l2)"which
gives a mean of comparison bet1;veen analysis assumption and actual
behavior.
It j.s evident that connection B is actually more rigid than the
analysis would yield under above stated assumptionsoThe ratio of
deflections beeing ~~~~~ ='o~4:
14
This report dealt only with one type of haunched
connection, connection Bo The procedure for the analysis
of other haunched connections will be identical as far as
the deflections of the cantilever beams are concernedo In
computing the knee rotations adjustments will have to be
made, for the different stiffener portions in those
connections. The equivalent length method will be
identical again.
LIST OF RSFE~EECES:
Topractsoglou, A. A.
Timoshenko, So
Osgood, Wo Ro
Topractsoglou, A. Ao
Beedle, L. S.
Johns t on , BoG 0
Topractsoglou, A. A.
Bee dIe, L. S.
Johns ton, B. Go
Newma rk, N. M.
15
I'Connecti ons for Welded Con-
. IItinuous Portal Frames
Fritz Engineering Laboratory,
Leh:~gh Uni versi ty, September,
1950
IITheory of Elasticity"
McGraw Hill Book Coo, 1934
"A Theory of Plexural fOl~
Beams with NonRarallel
Extreme Fibers
Journal of Applied Mechanics,
Vol. 6, No.3, September 1939
"Connections for Welded
Continuous Portal Frames,
Part r"
'I'he WeI ding Journal, July ,
1951
"Cormecti ons for Welded
Continuous Portal Fr8mes,
Part rr ll
Mimeograph, Fritz Engineering
Laboratory, March 1951
"Numerical Procedure for
Computing Deflection, Moments,
and Buckling Loads II
Transactions AoSoC.~., Vol. 108,
1943


Connections for welded continuous portal frames, theoretical analysisiof haunched knees, 1951

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Connections for welded continuous portal frames, theoretical analysisiof haunched knees, 1951

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