E

byJa Hartley DanielsLe- II Lu

ELIla TO YfE

Plastic esign ~F Multi-Story Frames

E

FRITZ E~~GINEEFHf\IG

Lt\80HI\T()HY L:BliARY

u

Frif:z Engineering Laboratory Report No. 273&54

273.54

SYNOPSIS

This report presents 78 design charts which are to be used in con­

junction with an earlier report describing the Sway Subassemblage method

of designing unbraced multi-story frames. l These charts relate the

lateral load versus sway deflection behavior of restrained columns per- '

mitted to sway. They have been prepared to cover the range of axial

load ratios pip from 0.30 to 0.90, and the range of slenderness. yratios hlr from 20 to 30.

The composition of the design charts is briefly discussed. The

scope of application of the charts is also given along with two worked

examples to illustrate their use in solving restrained column problems

similar to those encountered in the design of multi-story frames.

i

273.54

TABLE OF CONTENTS

SYNOPSIS

1. INTRODUCTION

2 • SCOPE OF APPLICATION OF THE CHARTS AND EXA:MPLES

3. ACKNOWLEDGMENTS

4. NOTATION

5. REFERENCES

6. CHARTS

ii

i

1

3

7

8

9

10

273.54

1. INTRODUCTION

A new semi-graphical method of designing columns and girders

in rigidly jointed, unbraced multi-story frames was presented in

Reference 1. The design method discussed there, starts with columns

and girders which have been selected initially from a previous pre-

liminary design of the story under consideration. Then, through

application of an analysis procedure which makes use of the charts

presented in this report, the lateral load versus sway deflection~

behavior of that story can be determined.

The design charts give the lateral load versus sway deflection

relationships for restrained columns which are permitted to sway, and

which are loaded by a constant axial load P and a gradually increasing

lateral load Q. In addition to the above loads, the columns are also

subjected to a bending moment M defined byn

Mn

(1)

in which h/2 represents the column height and A/2 the sway deflection.

The reasons for choosing such a moment is discussed in Reference 1.

Each chart is prepared for a given axial load ratio pip andy

a given slenderness ratio h/r for the restrained column. The lateral

load Q is nondimensionalized as Qh/2M with M being the plasticpc pc

-2-

moment of the column corresponding to the given pip ratio. M valuesy pc

for commonly used rolled wide-flange column sections in both A36 and

A44t steels are listed in Part II of Reference 3. Each chart contains

two sets of curves which define the relationship between Qh/2M andpc

A/h for certain values of the restraining function. The derivation

of these curves and the procedure for preparing the charts, is presented

in Appendix I of Reference I. Use is also made of the moment-rotation

curves given in Part III of Reference 3 for the preparation of the charts.

The application of these charts to the design of columns in unbraced

frames, is explained in Reference I and also in lectures 18 and 19 of

Reference 2.

273.54

2. SCOPE OF APPLICATION OF THE CHARTS AND EXAMPLES

-3-

The charti presented in this report give a precise in-plane load­

deflection curve for any initially straight steel wide-flange column'

which is pinned at one end and elastically restrained at the other end,

while free to sway. In addition, the column must be adequately braced

against out of plane movement, and subjected to the loading condition

outlined in Section I of this report. The load-deflection curves take

into account residual stresses in the column, plastification of the

column, reduction of plastic moment due to axial force in the column

and the effect of column axial force, P, acting through the sidesway

displacement, ~, commonly referred to as the P - 6 moment. Although

the columns in an unbraced multi-story frame do not comply exactly

with the above requirements, they can be reduced to this case by making

certain simplifying assumptions which are stated in Reference I. Thus

the charts given in this report can be used to give reasonably accurate

designs of such frames.

The following two examples will illustrate the use of these charts

in solving restrained column problems similar to those encountered in

the design of multi-story frames.

-4-

webs is permitted. The column is subjected ,to a constant axial force

P = 1960 kips and a varying lateral force Q. In addition, a moment Mn

is applied at the top of the column which is related to the forces P

and Q according to Eq. 1. Both the column and the beam are made of A36

"ksteel. Determine the maximum lateral force that can be resisted by

the column, and the corresponding sway and joint rotation.

For the 14W264 section, P = 2795 kips and r = 6.74 in. Sincey x

h/2 = 81 in. , h/rx

= 24. Pp

y

1960=--

2795 == 0.7,"k "k

M = 551 kip-ft.pc

Ix

Fur the 30Wl16 section, Z

4919.1 in.4

377.6 in.3

MP

1132 kip-ft, and

Note that the plastic moment of the beam is larger than two times

that of the column. A plastic hinge will therefore form at the top of

the column. The restraining function M , (Ref. 1) is given byr

3EI(3 ) (29500) (4919.1)x

M 8M 8Mr

LM,pc pc

pc (12) (23 .4) (12) (551)(2)

234 e Mpc

where L is the clear girder span plus half the column depth. (Ref. 1)

By entering Chart 51 (P = 0.70' PY

h = 24 r) and locating the curve for

M = 234 eM, the peak value of Qh/ZM is found to be 0.680. Thusr pc pc

the maximum lateral force-Q which can be resisted by the column ismax

(0.680) (551)

6.75

0;'(

For A44l Steel see p. 50 REF •. .1

Part II. REF. 3

55.6 kips (3)

273.54 -5-

The sway deflection index 6/h corresponding to Q is also foundmax

from Chart 51 to be equal to 0.0115. Thus the sway deflection at Q ismax

(0.0115) (81) 0.932 inches. (4)

Referring again to Chart 51, the value of M' corresponding to Qr max

is 1.94 M The joint rotation corresponding to Q is determined bypc max

equating M and M'r r

1.94 Me = pc

234 Mpc

i'(

= 0.0083 radians. (5 )

Example 2: Determine the maximum lateral force that can be re-

sisted by the column of Example I, and the corresponding sway and joint

rotation, if the beam is reduced in size to a 24Vf76.

For the l4\f264 section, P = 2795 kips, r = 6.74 in.y x

and h/4 24.

p0.7, M 551 kip-ft.p pc

y

For 'the 24Vf76 section Z = 200.1 in.3

M 600 kip-ft. an-d I 2096 in. 4p x

Note that two times the plastic moment of the column is larger than

that of the beam. A plastic hinge will therefore form in the beam at

the column face.

i'(

Note that the numerical value of e will always be equal to or less

than the numerical value of 6/h for positive values of M .r

-6-

The restraining function M is calculated as shown in Example I.r

Mr

100 8Mpc

(6 )

Assuming an idealized elastic-plastic moment-rotation relationship

for the beam, the joint rotation e corresponding to M at the columnp

face is given by

M .{.e = --E.­

3ET

(12) (600) (12) (22.75)

(3) (29500) (2096)

= 0.0106 rad. (7)

where t is the clear girder span. The maximum value of the restraining

function M' is obtained by substituting Eq. 7 into Eq. 6.r

M'r

1.06 Mpc

(8)

h = 24 r)Entering Chart 51 (P = 0.70 PY

and locating the curve for M = 100 8M we find that its intersectionr pc

with the curve M' = 1.06 M gives a value of Qh/2M equal to 0.240.r pc pc

Thus the maximum lateral force Q which can be resisted by the column ismax

(0.240) (551)

6.75= 19.6 kips (9)

The away deflection index cO,rresponding to Qmax

is o. 0115. Thus the

sway deflection at Q ismax

(0.0115) (8l) '0.932 inches (10)

273.54

3. ACKNOWLEDGMENTS

-7-

This study is part of a general investigation "Plastic Design of

Multi-Story Frames" currently being carried out at Fritz Engineering

Laboratory, Department of Civil Engineering, Lehigh University.

Professor L. S. Beedle is acting Head of the Civil Engineering Depart­

ment and Director of the Laboratory. This investigation is sponsored

jointly by the Welding Research Council, and the Department of the Navy,

with funds furnished by the American Insittute of Steel Construction,

American Iron and Steel Institute, Naval Ships Systems Command and Naval

Facilities of Engineering Command. Technical guidance is provided by

the Lehigh Project Subcommittee of the Structural Steel Committee of

the Welding Research Council. Dr. T. R. Higgins is Chairman of the

Lehigh Project Subcommittee.

The authors extend their thanks to all those who assisted with the

preparation of the design charts. To Dr. B. P. Parikh, formerly of

Fritz Engineering Laboratory for his assistance with computer programming,

to Mr. J. R. Dawson of Bethlehem Steel Corporation who made available

their computing facilities for the preparation of some of the design

charts, and to Mr. A. Goodman who translated the computer output into the

initial design charts.

Miss S. D. Gubich and Mr. J. M. Gera Jr. prepared the final charts

and Mrs. D. Eversley typed the manuscript.

-8-

A

E

I

L =

M =n

M =P

Mpc

Mr

M' =r

p =

p =y

P-b, =

Q =

z

h

h/r =

~ =

r =

~/h =

e . =

4. NOTATION

area of member

elastic modulus

strong axis moment of inertia

center to center span length of girder

moment at joint or end of member

plastic moment

reduced plastic moment

restraining moment or restraining function

maximum restraining moment

applied axial force

axial yield force = A x ay

secondary overturning moment

shear resistance of column

plastic modulus

story height

slenderness ratio·

clear span length of girder

radius of gyrarion

static yield stress of material

joint translation' or sway

deflection index or cho~d rotation

joint rotation

273.54

5. REFERENCES

-9-

1. Daniels, J. H. and Lu, L. W.THE SUBASSEMBLAGE METHOD OF DESIGNING UNBRACED MULTI­~TORY FRAMES, Fritz Engineering Laboratory ReportNo·. 273.37, Lehigh University, March 1966.

2. Driscoll, G. C. Jr., Beedle, L. S., Galambos, T. V.,Lu, Le-Wu, Fisher, J. W., Ostapenko, A., and Daniels, J. H.

PLASTIC DESIGN OF MULTI-STORY FRAMES ---LECTURE NOTES,Fritz Engineering Laboratory Report No. 273.20, LehighUniveristy, Summer 1965.

3. Parikh, B. P., Daniels, J. H., and Lu, L. W.

PLASTIC DESIGN OF MULTI-STORY FRAMES ---DESIGNAIDS BOOKLET, Fritz Engineering Laboratory ReportNo. 273.24, Lehigh University, Summer 1965.

273.54

6. CHARTS

10

i.O

0.8

--.QL2Mpc

0.2

o

-0.20.03

CHART

to

DB

0.6

'0.4

0.2

o

-0.2

1.0

0.8

....QJL2Mpc

Oa4

0.2

o

-0.2

CHART 2

1.0

OB

Oa6

0.4

Oa2

o

-0.2

0.2

0.6

o

0.4

08

-0.20.03

CHART :3

0.03 1.0

P =0.30 FYh = 24r

l.O

0.8

0,.2

o

0.4

-0.2

--.WL2Mpc

1.0

0~8

--.Q1L2Mpc

0.2

o

-02

CHART 4

1.0

08

0.6

0.4

0.2

o

-0.2

1.0 .-~~ ....

0.8

--L2Mpc

0.2

o

0.03

CHART 5

1.0

os

0.6

0.4

0.2

o

-0.2

1.0

0.8

.JilL2Mpc

0.2

o

-0.20.03

CHART 6

1.0

08

0.6

0.4

02

o

-0.2

0.8

o

0.2

CHART 7

0.03 • 1.0P =0.35 P

h =20r

1 -02- ,,~

1.0

o

0.2

0.4

-0.2

0.8

0.6

--.Q.L2Mpc

0.6

0.4

~-o.,J

~~r;·" +

~ ~~. t

""'.>~::l' ~~ ,"

--~~

c.---

1.0

O~8

....Q!L2Mpc

0.6

0.4

O~2

o

r;,~ -0.2r:~~

....;

OJ CHART 8

1.0

08

0.6

0.4

0.2

o

-0.2

O~03

P =o. 3 5 F? • 1.0

h =24r yLO 08

o

0.4

02

0.03

CHART 9

L..J=:E~::::t:4:-L~~~R::::::t-=+-+:::±::=*=-+-+=AH4::t===F~r-r-r-cr t::t:::±=::bJ -0.2

o

0.2

0.6

-0.2

1.0

0.2

OB

0.6

0.4

1.0

0.8

0.2

o

0.6

-.Q.!L2Mpc

-O.z

to

h =28r

[[[:

0.03

P :: 0.35 FY I0.01 0.02

-- ----- J --...-,~"l--'-"--""-""-""""-'-"""'---""-""

0.6

0.4

02

o ~ o

1.0

o

0.6

0.4

02

OB

1.0

0.8

o

"

0.03 -0.2

CHART 12

0.6

-0.2

~2Mpc

1.0

0.6

0.4

08

02

i.O

0.8

o _----................ -_- --- 0..........- ----u ~

I ------..~ r ~__ 'to ....,--___~- :~~- -~~ - ~-

r----r--- - r-------- ~"""'--- ......... 0~~~ ~~~ .2~~---- -----... ~~------~ __..___ ...........---- ... 0.1~___-------- ~~-

------ - 0 ___-- ~~r---.L.........

0.03 -0.2

CHART 13

0.6

().2 ()

~2Mpc

1.00.01 0.02 0.03

P =0.40 JY1.0

08

0.6 ... 0.6

;;;;;

~ 0.4

;;:;:;;

L....-4--:

0.2 0.2

I 1_

~

o o

-----"o.lLIUW UW -0.20.02 ~ ( d)-0.2 0.01 T Ra . CHART I 4

LO r-,0.03 1.0

-L2Mpc

~:::::f"""C

0~6 ~ O~6

0.4 0~4

----:..-+==

o---

CHART 15

0.03

~

----I I

I ---.. -0.2

1:1 (Rod.)h

0

I I-0.2

0.03 1.0

h = 26rTl

p =0.40 FY~III

0.02

1I

0.01

I1.0~t--t-....l..-

0.8

-ltL2Mpc

0.6

OB

0.6

~

~ ~

0.4

02I ---

~

r-t---4--l..-

I_LLJ-HHH-t-i-ilill-' I---r--t--_..q~ I I I I 0:03 -UolL-----J~---t____r 0.02 ~ J

-0.2 h (Rod. CHART I 6

-=a

---~

o

LO

08

0.6

0.4

0.2 0

o

r-r-r---j--r--r--r--r--r---t---+---=P::...t-~=-+---t~F=F---k:::+-+--+=--+==~~~::::;:::~==-+:::::t:==-J I0 ~~--+---I

0.03 -0.2

CHART 17

1.0

0.8

0.4 0.2

--.QJL2Mpc

1.0

0.6

0.4

08

02

1.0

0.8

0.2 0

--.Q..L2Mpc

-O.l

CHART 18

I 1 1 10.01

1 1 I0.02n 1 J 1 1

0.03 1.0

~

0.6

08

-

~

~

.....~

~

-......-mO.4

--..---~O.2

~

.....

~O

~

~

~,o

~

~

~

~o

0.8

~ II, •

2Mpc I --P*......I ........., -..=-10.6

0.4

0.2

-0.20.03

CHART 19

-0.2

-0.2

1.0

0.8

0.2

o

0.03

CHART 20

1.0

08

0.6

0.4

02

o

1.0

0.6

o

0.4

08

02

-0.2

CHART 21

0.03.A. (Rod.)h

0.020.01

1.0 . 0.01 0.02 0.032 , i , , , , , , , , , 0 , , , , i , , i , iii i J i i

P =0.45 FYh = 24r

0.2

o

0.6

0.8

0.4

-0.2

~2Mpc

1.0

o

08

02

0.6

0.4

h = 26r

0.03

P :: O. 45 ~

-0.2~ --0:02 ~ (Rod.) CHART 22GOI

1.0

0.8

0.6

0.2

o

0.4

-0.2

...QL2,Mpc

1.0

0.6

o

0.4

08

02

h =28r

0.03

P 11: 0.45 'YLO

0.2

0.8

o

0.6

...Q..L2Mpc

-0.2 -0.2

CHART 23

1.0

-0.20.03

CHART 24

h =30r

0.03

P =0.45 FY1.0

-0.2

0.8 I-c-::::± I I~ W--1!::::tl v: ~_ I 7'f ::;Pl'5:::ttac:::r= L 1771_IVI;~ 2.n ~6 I I I I I I I I I I I I I OS

~2Mpc

0:6 0.6

0.4 0.4

0.2 I--4tHHW~~<::;t,e ~i>CL: I:::::=+'~ =+==::J::;:::I:-j---r '4=:::::J I r=-=~ I r-- 1--=±:: I P==-I~ I 0.2

0 Iacc::l: EJ?tI I I I E?::t~ I I =p== +=:::::i t I ===t' ack::: I I l::==:at ac:::L:::: I I ?t= -+:::::::: I I F"'-t cc:±:: I 0

1.0

0.6

o

0.4

08

02

~~;;;;::::a...... -0.20.03

CHART 25

LO 0.01 003

P =0.50 FYh =20r

0.8

0.2

o

0.6

0.4

-0.2

....Q!L2Mpc

1.0

0.8

--.Q.L2Mpc

0.6

0.2

o

1.0

OB

0.6

0.4

02

o

-0.2

CHART 26

to

0.6

08

h = 24r

0.03

P =0.50 JY1.0

0.8

0.4 0.4

0.2 02

0 0

-0.2 -0.2- _.~ -- - --

CHART 27

~2Mpc

1.0

~2Mpc

0.2

o

-0.2

1.0

os

0.6

0.4

02

o

Q:-t--:t:::::±:::::I;~-.L..-'---'-..... -0.2

CHART 28

-0.2

LO

0.8

--WL2Mpc

0.4

o

-0.20.03

CHART 29

1.0

08

0.6

0.4

0.2

o

CHART 30

1.0

o

0.2

0.6

0.4

08

-0.20.03

O~03

P = O.50Pyh =30r

A (Rod.)h0.02

O~02

30

0.01

0.01

O~8

1.0

o

0.2

0.6

0.4

-0.2

--IDL2Mpc

1.0

o

0.2

0.6

0.4

08

h=20r

0.03

P =0.55 Py

-....,....;~---.....~-..- -0.20.03

CHART 31

r<:f::::::L I r==r k:J ~20FM~~4:::t:~-L-.J

0.011.0

0.2

o

0.8

0.4

-0.2

....Q.!L2Mpc

1.0

0.8

....Q!L2Mpc

0.2

o

-0.2

1.0

08

0.6

0.4

0.2

o

-0.2

CHART 32

-O~2

1.0

0.8

....QL2Mpc

0.6

0.4

o

-0.20.03

CHART 33

1.0

08

0.6

0.4

0.2

o

1.0

0.8

....QlL2Mpc

0.2

o

-O~2

1.0

OB

0~6

0.4

0.2

o

-O~2

CHART 34

1.0

0.6

o

0.2

0.4

08

-...........L~ -0.20.03

CHART 35

1.0

0.2

o

~2Mpc

1.0

0.8

--.21L2Mpc

0.6

0.4

0.2

o

-0.2

1.0

OB

0.6

0.4

0.2

o

-0.2

CHART 36

-0.2

i.0

0.8

.JlL2Mpc

0.2

o

-0.20.03

CHART 37

to

08

0.6

0.4

0.2

o

to

0.8

....Q!L2Mpc

0.2

o

-0.2

1.0

os

0.6

0.4

0.2 ,

o

-0.2

CHART 38

1.0

0.6

0.2

o

0.4

OB

0.03

CHART 39

~

-0.2

1.0

0.2

o

0.8

-0.2

--.QL2Mpc

-0.2

1.0

0.8

~2Mpc

0.2

o

-0.20.03

CHART 40

1.0

DB

0.6

0.4

0.2

o

0.6

o

0.4

02

-0.20.03

CHART 41

h = 28 r

0.03 1.0I P = 0.60 Py

0.2

o

-0.2

. _Ul-- 08

...Q!L2Mpc

.:

1.0

0.6

o

0.4

08

02

0.03

CHART 42

"'L ! '::>' ! ~ -0.2

1.0

0.2

0.8

o

-0.2

~2Mpc

0.8

-..WL2Mpc

0.6

0.4

0.2

o

-0.2

CHART 43

1.0

OB

0.6

0.4

02

o

1.0

0.2

o

0.6

0.4

os

-"" J I 7>n' '-0.20.03

CHART 44

1.0

0.2

0.8

o

-.QlL2Mpc

· -to'---_·

0.8

0.4

0.2

o

-0.2

.-", "_. 0.03._. --.-, 1.0

p z O.65Py, h =24 r

I----i

08

0.6

0.4

02

o

-0.2

45

1.0

0.2

o

0.6

0.4

08

I;>' I ,I -0.20.03

CHART 46

0.8

1.0

0.6

0.2

0.4

o

-0.2

.Jlli.­2Mpc

1.0

08

0.6

0.2

o

0.4

0.03

CHART 47

0.03---.,..-r-"'-.,-n,-TTTr=D=r=+++=R=t=t=fIEa:p~=~o. 65 FYh =28r

LO.-

0.8

0.2

o

0.6

-0.2

...QL2Mpc

-0.2

1.0

0.8

--WL2Mpc

0.2

o

-0.20.03

CHART 48

1.0

08

0.6

0.4

02

o

0.6

o

0.4

02

-0.20.03

CHART 49

0.03 1.0

I I I P =0. 70 PyI I I I h= 20 r

o

0.2

0.6

-LO OB

-0.2

~2Mpc

-0.2

1.0

0.8

....Q.L2Mpc

0.2

o

-0.03

CHART 50

1.0

OB

0.6

0.4

0.2

o

o

0.6

0.4

02

-0.20.03

CHART 51

0.03 , 1.0P =0.70 Py

I h=24r

0.2

o

.La 08

-0.2

.JiL2Mpc

1.0

0.8

~2Mpc

o

-02

1.0

DB

0.6

0.4

0.2

o

~-- ......... -0.2

CHART 52

-"'0o

a::

NoI

oNd

CDd9

q CQ (.Q ~ C'4 C\J0 0 0 0 0 0

IroLO

Q?'I-

0'" a::r--CX) <t.N :I:rt') o ",...,

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