Daniels, J. H.

Lu, L. W.

English

Lehigh University: Lehigh Preserve

Lehigh University
Lehigh Preserve
Fritz Laboratory Reports Civil and Environmental Engineering
1971
Applications of the sway subassemblage methods,
(73-10) original plans were scraped when AISI
indicated that they intended to publish the
information in their Bulletin so an APPENDIX
DESIGN EXAMPLE was written and given 338.4
to agree with item 10.8 in the ASCE Manual No.
41, 1971.
J. H. Daniels
L. W. Lu
Follow this and additional works at: http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-
reports
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Recommended Citation
Daniels, J. H. and Lu, L. W., "Applications of the sway subassemblage methods, (73-10) original plans were scraped when AISI
indicated that they intended to publish the information in their Bulletin so an APPENDIX DESIGN EXAMPLE was written and given
338.4 to agree with item 10.8 in the ASCE Manual No. 41, 1971." (1971). Fritz Laboratory Reports. Paper 1937.
http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1937
APPLICATIONS OF THE SWAY SUBASSEMBLAGE METHOD,
original plans for 338.4 were scraped when AISI
indicated that they intended to publish the infor-
mation in a Bulletin so this APPENDIX was written
APPENDIX.
DESIGN EXAMPLE 17 5-'Rrrz ENGINEERING
!!.A/]ORATORY LIBRARY
The load-deflection behavior of the one-story assemblage at Level 8 of
the frame shown in Fig. 1 will be determined by the subassemblage method.
The uniformly distributed factored gravity loads on the beams (0.321 kips per
inch) and the axial loads in the columns are maintained constant. These loads are
determined in accordance with the working loads shown in Fig. 1, using a load
~.~ .._~_._factor of L3 ..and .. the live load reduction factors suggested by ASA AS8.l. The load-
drift behavior is determined for the wind from left condition only. Although the
analysis is more easily and qUickly accomplished by computer, step-by-step manual
calculations are presented in Plates I to VI to illustrate the procedure.
The first step is to isolate the one-story assemblage at Level 8 from
the frame. The resulting one-story assemblage with known member sizes is shown
in Plate I. Also shown are the distribution of bending moments under gravity loads
alone (6/h = 0), column and beam properties and the initial restraint coefficients.
The analysis of the one-story: Cls'Semblage initially involves the calcula-
tion of the non-dimensional rotational restraint stiffnesses M at each joint
r
before and after the formation of each plastic htnge. In addition the non-dimen-
sional restraining moments M at each joint are calcula ted under the gravity loads
. r
alone and under the combined loads at the formation of each plastic hinge.
The comments which follow are intended to clarify the correspondingly
lettered items in Plates I and III. Comments concerning calculations in Plate III
will also be relevant to corresponding calculations in Plates II and IV.
Plate 1.-
(a) The distribution of bending moments is determined by elastic
~···--·-·~·---ana--rysls·;a-ssuming-each-column -is laterally restrained at both ends at mid-height.
-2
(b) The column axial forces are computed on the basis of a mechanism
condition occurring in each story of the frame under the combined loads, assuming
wind from the left.
(c) The reduced plastic moment capacity M of each column was com-pc
puted from Eq. 5 of Part 1.
(d) The minimum plastic moment required to resist 1.3 times the working
gravity loads is defined as Mpm
M =pm
21.3wL
16
where L = beam span center-to-center of adjacent columns and
w = uniformly distributed working gravity load per unit of span length.
It is convenient to use this moment as a non-dimensionalizing "factor when deter-
mining the total sway resistance of a beam.
(e) The initial restraint coefficients are computed from Eq. 15 of
Part 1.
Plate III.-
(a) "The analysis of interior subassemblage A-C begins by deter-
mining the total
from the initial
change in moment in
dri!Szero~ condition
,...
the columns at joint B as sway ~/h increases
to the occurrence of the first plastic hinge
in the _ subassemblage.
(b) The total change in moment in the columns at joint B is now re-
quired as the drift is further increased up to the formation of the second plastic
hinge in the subassemblage.
(c) With the first two plastic hinges found to occur at the leeward
ends of the two beams, the third or last plastic hinge can only occur somewhere
in the windward half span of beam BC or in the columns at joint B.
I " -- 0-'· o __
-3
(d) The initial moment in the columns under the gravity loads alone
and zeto drift is equal to the net moment from the beams or M = 3514 - 2335 =
e
1179 k-in.
_(e) The initial value of non-dimensional restraining moment M
rl is
now determined.
(f) The non-dimensional restraining moment at joint B when the first
plastic hinge occurs is the Sum of the initial restraining moment, 0.388 MpcB
(g) For increased drift beyond the first plastic hinge, beams AB can
no longer contribute to the rotational restraint stiffness at joint B. Thus the
restraint stiffness in the interval between the first and second plastic hinges
decreases to that provided by beam BC alone. The corresponding joint rotation
increment 68 B is 0.00238.
(h) The non-dimensional restraining moment at joint B when the second
plastic hinge forms is again equal to the Sum of the restraining moment at joint
B when the first plastic hinge forms, 0.744 M plus the increase in restrainingpcB
moment up to the second plastic hinge.
(i) Since the second plastic hinge occurred at the leeward end of
beams BC, KBC reduces from 5.949 to 3.0 when calculating the restraint stiffness
M
r3 between the second and third plastic hinges.
(j) Since the third and last plastic hinge forms in the columns at Joint
B the total moment resisted by the two columns M~3 must be equal to twice the
reduced plastic moment capacity M of the restrained column.pcB
Load-Drift Behavior of the Four Subassemb lage s.-
The construction of the non-dimensional load-drift curve for , sub-
assemblage B-D, is shown in Fig. 2. The set of M values calculated in Plate IV
r
-4
will determine the three complete restrained column curves o-a'-c, o-a"-c and
o-b"'-d shown in the figure. These curves are given by Eq. 8 in Part 1. Similarly
the set of M' values will define the four sloping straight lines shown in the
r
figure. The initial segment of the load-drift curve is i-a. This segment is
parallel to o-a' of the load-drift curve corresponding to M
rl • The first plastic
- ---hinge -occurs at point a, which lies on the intersection of curve i-a with the
straight line corresponding to Mri' Similarly, the second segment, a-b, is parallel
to segment a"-b", and the third segment, b-c, is parallel to segment b'''-c'''.
----- ---The last--plastic hinge occurs in the columns at point c on the load-deflection
curve. The final segment, c-d is the second-order plastic mechanism curve for the
subassemblage.
The non-dimensional load-drift relationships of the four subassem-
blages at Level 8 are shown in Fig. 3. In each case, the solid curves indicate
the behavior determined in this analysis. The dashed curves were obtained using
the computer analysis described in Part 1.
Load-Drift Behavior of the One-Story Assemblage
Transforming the ordinates to- the curves in Fig. 3 from Qh/2 M to Qpc
and summing, results in the load-drift curves for the one-story assemblage at Level
8 as shown in Fig. 4. Also shown are the corresponding curves for the one-story
assemblages at Levels 6 and 10 as computed manually (solid) and by computer (dashed).
The sequence of formation of the plastic hinges in the one-story assemblages are
also shown in Fig. 4.
-5
II
-to
-(\J
-co
o
-
o
I
1
I16826
C
I
16826 I
B
I
I
16 YF40 I
tf) 0 ~ 0~ 18VF45 ~ 14YF34 14\'F34 ~~ ~ t\I ~M:.¢ N ~
- do - do en do
rt>
-
en rt>
10 do <0 do ~ do ¢;k ~ ~ ~~
".::t q- 16W40
¢
- 18VF50 - 16YF40 -
-
~ ~ ~ --~-
-do en do en do <0l!. ~ ~ ~>-~ do ~ do ¢ do ~
-
-
m m v18\~F55 18VF55 18VP55 coIt. ~ ~ .!.!.>- :...~ ~ ~ st
- 21VF55 21YF55 - 21VF55
-
([) N C\I <D
ff) do ¢ st do r<>
-~ ~ ~ ~
-~ v g: ~
"7/T 77T /fr 71T11--
2--
3--
4--
5--
.;----g~-=
··7--.
8--
9--
10--
Column A
I
I
Levell-- I
L~30'-L 24' _ L24' -I
Bent Spacing =24'
.Working Loads
Levell
Levels 2-10
Exterior '110 II
Wind
Wo= 60psf
Wo =80pst
Wo =45psf
Ww =20pst
ASTM A36 Steel
WL =30 pst
wL =80 pst
Percent Live Loa d Reduction by ASA A 58.1
Fig. 1 Preliminary Frame Design
_____ - - • 0 - • • • _
.' I
. - - •.---.~-'---. __ ,~.-'" ~.".-- .o.___...,..; ,..... ,.' . .••.__.. _._.;..
-J:";
ANALYSIS OF LEVEL 8 - FRAME 8 PLATE I
ONE -STORY ASSElvf8LAGE AT LEVEL 8
-
c
A 8 C 0
~I 18VF55 ~I lSI¥"" 55 ~I 18W:- 55 I~]~I ~ ~~ ~ ~I~ 360" ~~ 288" -I~ 288" ~
__ 8ENDING MOMENTS - Gravity Loads Only and L1/h =0 (a)
~/~ ------JP!Y, -2335 21731-2352 1920-_._- i'-... --.t---.. .-f
1607\ - V-589 --89~89 1/-9601607 -589 -960
RESTRAINED COLUMN PROPERTIES - Wind From Left Only
Restrained
'\ Wind From LeftColumn Section 'Y p(b~ F}f? (c)l.fpcy
Units Kips lOps /( -in.
A8 - A9 14 VF //1 //75 23.1 505 0.430 4740
88 - 89 14 vV=-//9 1260 23.0 833 0.661 3040
C8 - C9 14 VF //9 1260 23.0 745 0.591 3660
D8- 09 14tFB4. 889.6 23.5 597 0.671 2040
.-
BEAM PROPERTIES AT LEVEL 8
Girder" Section lr1p M,om
(d) I L
Units K-in. K -in. in.4 in.4
A8 - 88 18 l¥=" 55 4010 2600 889.9 360
88 - C8 IBYF55 4010 1670 889.9 2BB
C8 -08 /81-F55 4010 1670 B89.9 288
INITIAL RESTRAINT COEFFICIENTS (e)
Noles
Joint Krighf
_IVeft
A8 !<AB =5. 750
I. E =2~000 ksi in the
onalysis1ond
88 KBC 5.949 KBA =6.286 2. LeIters in parentheses refer
C8 J( 6.154
-/feB =6.053 to items which are
- -----
_._.-
-'CD-. ----- ---
--------disclJ5sed In the texf~------08 Koc=5.857
-6
PLATE IIr------------------------------, -7
WINDWARD SWAY SUBASSEA4BLAGE A-B
Flrsf Plastic Hinge .. .- A B
I M =4010'
8MBA =496=6.286 x E~:g9.9 x 5.7~0-4 8~ ~~97 :0~
=13.6E8&.4 i :. E8~ =36.5 134813~
.
__ .BMAB~5.750X a::09x 36.5:: 5/8 k-in. KAB = 3.0
Second Plastic Hinge' Mp :40!
_________. J} = o.':Z:X~~~2 =1.54; 9::~~-_n=-LJ
Mmin
"0 ~ = 0.38,· M,nin =0.38 x 2600 ::987k-in. Mp =.4010rpm
Check" 987 < 2 MpCA (OK) 8MAB =987-(-2697) = 3684k-in.
Determine Mr: and M/ Values"
··Initial (iJ/n =0) .. M/=-::~; A-PCA = -0.678 ~CA
M - 29,000:<889.9 .rt - 5.75. .~. 360-j~'"4"40 ~ MpcA =87.0 ~ "1f~A
8A!4= 518 =87.0 x 4740 8~ :. 8~ = 4/;ttOO =0.00126 rad.
J
Mr/:: (87x 0.00126- a678) MpCA = -Q568 t1~cA
_ 29,000 x 889.9 . _ LJ u·
Mr2 - 3.0 x 360 x 4740 ~ "1oCA - 45.4 0J1 I~'RCA
3684 .
BMA =3684 =45.4 x 4740 8~ .0. 8~ = 2/5.500 =0.0171 rad.
1 .
A!r2'=(45.4 xO.017/- 0.568) MpCA =0.209/;'RcA
LEEWARD Sl·YAY SUBASSEMBLAGE C-D C 0
I M'P= 4010 I
Flrsf Plastic Hinge" 8 Moe =4010 -/920:: 2090 k-ini . ~
Determine ~ and /4:' Values" Clo.,,==-~-_-_-::::::=:---"-2005 ~
Initial ( iJ1J =0)" Mr ' =2~:~ MpCD:: 0.941 U;.zco -2005
~I -5.857x 29,000X889.9 L? U -257 LJ ~I
1Ylr/- • 288 x2040 °0 mpcD - °0 mpCD
2090 .
BMD =2090 =257x 2040 880 .·.88D = 525.000 ::0.00398rad.1
_....------.-.- '--",'-;-=-(257[0."00398 + 0.941) tyPCD :: /.963 ItfpcD- --- - -, --- - ---_...... -- --
./
PLATE 11/
INTERIOR SWAY SUBASSEMBLAGE A -C
-8
First Plastic Hinge: (a)
EX889.98M8A =496=6.286x 360 8lle=15.5E8&ei ... E8fle=31.9
8M8C =5.949x e::09 x 31.9=586k-in. i 8A{.o =6.053x e:::x31.9x 5.9;9-4=581k-in
:. z;: =2.0 . Mmin =2.0x/670=3340k-in.
Second Plastic Hinge:
,~I - - 0 Ex889.9 -8mco-1256-6. 53x 288 x8tlc-18.7E8fJci
a;"8c=5.949x ~e:a9x67.IX 6.0J3 - 4 = 1265 k-in.
Third Plastic Hinge:
Fi = 16X40/0 =240
~ Q32/ x2882 .
:. E80c =67.1
(b)
(e)
Check: 4010 +3340=7350> 211t!pcB (NG) :. Mmin=2x3040-4010=2070k-in.
8Afoc=2070 - (-484)=2554 k-in.
Calculate Mr. and M/ Values:
Initial (A17 =0) Mr ' = 3~;; MpcB =0.388 A"P..CB (d)
M =(6.286X 29,OOox889.9 + 5.949 x 29,000 x 889.9 ) c9, M. (e)·
rI' 360x3040 . 288 x3040 8 pcB
= (148+175) °0 "'PCB =32308 11o..cB
8/.18=1082 =(148+175)3040861;
1082
:. 86'8= 983
,
000 =0.00110 rod.
Mr / =(323 x 0.001/0 +0.388) MpCB = 0.744 MRCB
Mr2 =175 00 ~CB
(t)
(g)
/2658Mo= 1265 =175 x 3040890 j :.8°0 = 533 000 =0.00238rad.
1
Mr ; = (/75 x 0.00238 +0.744) A{ocB =/./61 "RCB (h)
M - 3.0 X 29,000 x 889.9 LJ U =88.5 fJ.8 Mp..CB . (0r3 -. 288 x3040 °8 lr"'pcB
25548M8=2554 =88.5 x 3040 86'8" ... 86'0 =269,000 =0.0095 rod.
Mr ;= (88.5 x 0.0095 + 1.161) MpCB :: 2.000 I¥fRCB (Checks) (j)
J ..~
.0. £800= 9.50
-9
PLATE IV
INTERIOR SWAY SUBASSEMBLAGE B-D
First Plastic Hinge ..
BMco=1836=6.053x E;::9.9 80c =18.7£88c 0°0 E8tlc=98.3
BA-fco =6.154x ~~~9 x98.3=/870k-in.i 8J,,'oc =5.857x 8::: x98.3X 6.1S;-4=191{3
• /(-m.
.- -. -- Second Plastic Hinge"
Ex 889.98Moc=172=5.857x 288 880 = 18./£8&0
8Afco=6.154x 889.9x9.50x 5.857-4 =168 k-in.1-__ - - - --- _...._-288_2
Third Plastic Hinge ..
. .. ":,-~'
-- . -c._.. --J= ai~;x4~~~2 =2.40 .eo :~~ =2.0 Mmin =2.0 x1670 =3340k-in.
Check: 4010 +3340= 7350 > 21vfpcc (NG) .'. Mmin=2x3660-4010=33/0k-in.
BMco =3310 -(-311) =3621 k-in.
C · I' OJ d·· I l.l Ialcu,are {tiro on -Mr. va ues ..
Initial (LJ/n =0) '~'=- 3::0 !'.1pcc =-0.049/t1pcc
M. - /~ 053 29,000x889.9 6 154x 29,000x889.9) LJ AIrJ - 10. x 288x3660 +. 288x 3660 °c IVtpCC
=(/48'"150.5) 6;; J.Jpcc = 298.50c A~cc
BMc=3707=(148+150.5)3660 80c :. 80c = ~O:£~% 0.00339 rod.
Mr ;= (298.5xO.00338- 0.049) Af;cc =0.963 M~cc
Mr2=150.5 0c APcc
8Mc=168 =/50.5 x 3660 86'c " :. BOc = 5;'~0 =o.000305rad,
Mr ; =(/So.S x 0.000305 + 0.963) Irfpcc =1.009 M,qcc
Mr3 =3.0 x 29~~~~~~~.9 ~ Mpcc =73.5 ~ I~cc
362/8Mc =3621 =73.S x 3660 SOc i :0 BOc =269000 =0.0/347rod.,
Mr ;= (73.5 x 0.01347+ 1.009) Mpcc =2.000 Irfpcc (Checks)
.-_.._---._-----_._---.------------------------- --_._._--
J-----------+-----------+----------~~~;:::::_____i0o
0.01 0.02 0.03 I01.0~----..;....----~---------,----------r---O---' .
P =0.6 Py
h = 24 rx
0.8~-----------",.~~~M,.:3::<.OA1.-----+-----------+-----i0.8
'Pee
0.6 0.6
Qh
2Mpc
d
0.4 ~ (ic 0.4
'58C \>
...:10.~t't>'"
0.2 0.2
,
I
....
o
-0.2'------------J.:::III----------'----------"'------'-0.2
0.01 0.02 0.03
I
I,
Fig. 2 Construction of Load-Drift Curve for Sway Subassemblage BD
-11
0.01 0.02
- --- --- --oEFLECTION INDEX Li/h-
0.41------F------t---------t------iQh
2Mpcc 0.3
(c) Interior Sway Subassemblage B- D
"Q.21---+-- ------l
P =0.60 'Y h =24 f x
"0.1
0lf----------J----------J------1
-0.11-- -+- "_"_.-+- -;
0.9
0.8 1----j~----==:lI~""'_d_-------+_---__I
' ........Qh 0.7 ........ "
2MpCD (d) Leeward Sway ................
0.61----;(- Subass emblagee - D ---~~-.l:.------1
0.5 P=0.65Py h=24fx
0.4L..-.- -J...- --.L ---I
o
O,------------r----------r--------,
5
-- ---- . 'r---......:.:-:::.::-::;:t
Qh -0.1 __----------
2M pCA -0.21--------=~~--- (a) Windward SwaySubassemblage A-B
-0.3 P =0.45 Py h =24 rx
-0.4 ~------_f---::;;tI""..:::::.,,-....-..:::.~--_+_---____l0.7
0.6
0.51----~-----+--------t----~
Qh
2M 0.4pcB (b) Interior Sway Subassemblage A-C
-"----"---"--c-""-----·_---"-----O.3-~-- --~
P =0.65 Py h =24 f X
---_·:·_..-0.2
O.II-------_--,..+ ~_t::_-----I
0.7
0.61---------+---~=----+---~~
0.5
Fig. 3 Four Sway Subassemblage Curves
-12
100.0....----------,---~----,--------,
5
2 6
4
(c) Level 6
5
-~ ''-4-
(0) Level 10
·10.01-;:--c~-----_+_---'-----_t_---_;
70.01----=:-H---+--~~---
30.0 ~---I-f-----__t_--__=_~>__"'~t_+_---____j
20.0 I--/f----~:..---
50.0 ~--+--...
80.0I------~(.L-_!_------_+_~--__i
90.0
60.01-----I--~-~-!--------+---~~
40.01----I--I-----_!_-----'-----f----____j
IQ
- ---(KIPS)
o 0.01 0.02
DEFLECTION INDEX t1/h
Fig. 4 One-Story Assemblage Curves for Levels 6, 8 and 10


Applications of the sway subassemblage methods, (73-10) original plans were scraped when AISI indicated that they intended to publish the information in their Bulletin so an APPENDIX DESIGN EXAMPLE was written and given 338.4 to agree with item 10.8 in the ASCE Manual No. 41, 1971.

https://preserve.lehigh.edu/cgi/viewcontent.cgi?article=2936&amp;context=engr-civil-environmental-fritz-lab-reports

Applications of the sway subassemblage methods, (73-10) original plans were scraped when AISI indicated that they intended to publish the information in their Bulletin so an APPENDIX DESIGN EXAMPLE was written and given 338.4 to agree with item 10.8 in the ASCE Manual No. 41, 1971.

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