Elastic Critical Load and Effective Length Factors of Continuous Compression Member by Beam Analogy Method

ARCHITECTURAL RESEARCH, Vol. 2, No.1(2000), pp. 47-54

Elastic Critical Load and Effective Length Factors of Continuous Compression Member by Beam Analogy Method

Soo-Gon Lee

Prof，Chonnam National University, Kwangju,Korea

Soon-Chul Kim

Associate Prof., Dongshin University, Naju,Kore안

Abstract

The critical load of a continuous compression member was detennined by the b就m-analogy method. The proposed method utilizes the stress-analysis results of the 즈nalogo나s continuous beam, where imaginary concentrated lateral Io음d changing its direction is applied 승t。즈ch midspan. The proposed method gives a lower bound error of critical load and can predict the span that buckles first. The effective length fac­

tors for braced frame columns can be easily determined by the present method, but result in the upper bound errors in all cases, which can lead to a conservative structural design

Keywords: beam analogy method, slope-deflection method, Kinney's fixity factor, modified slope-deflection method, effective length factor, finite element method

1. INTRODUCTION

In the structural design of a besmi-colunm, the effect of axial compressive force, P, is included by multiplying the factor 1/ (1-P/Pc) with the beam moment and deflection.

In the case of a single span beam-column, the elastic criti­

cal load, Per is easily determined, whether the sectional property of that member is constant or variole along its axis. For a multi-span beam-column, however, the conven­

tional neutral equilibrium method, or energy principle­

based Rayleigh-Ritz method cannot be efficiently applied to the determination of critical load. In this case, modified slope-deflection method or a numerical method for exam­

ple, the finite difference or tiie finite element method, be­

comes a useful tool for the determination of critical load or tiie stress analysis of a continuous beam-column.

In this paper, the Beam-analogy method is proposed to determine -the approximate critical loads of continuous compression members. The main idea of the beam analogy method is to replace the continuous compression member by a continuous beam, to which concentrated lateral loads are applied at each mid-span of the multi-span beam. The mid-span concentrated loads are made to chsuige their di­

rections in order to sim니ate the buckling mode. The re­

sults of sti*ess dialysis of the beam me used to calculate Kinney's⑶ fixity factors. Finally, the critical load is ex­

pressed by fixity factors. The Beam analogy method may also be applied to the determination of effective length factors ofbraced frame columns.

2. BEAM ANALOGY METHOD

The Beam analogy method can be explained with the following simple example. Fig. 1(a) shows a 2-span con­

tinuous compression member whose critical load is to be

determined by the proposed method. The first step of the proposed method is to rqjlace the given compression member with a continuous beam. Fig. 1(b) shows the analogous beam, where the direction of an imaginary con­

centrated lateral load, 0 at each mid-span is alternating its direction to simulate the buckling mode.

1.5L

(a) Example member(W. F. Chen)

075L . Q75L . Q.5L . 0.5L

Fig. 1. 2-Span continuous memberfunifbrm section)

The second step is the stress analysis of the analogous beam( Fig.l(b)). In the present example, the usual slope­

deflection method may be conveniently applied to obtain end moments and rotation angles at the supports. Fig. 1(c) shows tiie stoess analysis results.

The third step of the proposed method is to find Kinney's fixity factors⑶ by utilizing the following rela­

tionship.

48 Soo-Gon Lee and Soon-Chui Kim

思=1 一刼 (1)

Where | | denotes the absolute value. The above relation­

ship was introduced by the author⑴ to the eigenvalue problems of the tapered bars. He also applied Eq. 1 to the analysis of single span steel beams⑵ with partially fixed ends. In Fig. 1(c), fcB =0.0 by Kinney's definition (simply supported ends). When Eq.l is applied to the first span AB and to the second span BC at the intermediate support BJba and办care determined in the following way.

0.750：= 4成 以 L5Q£?

8 1.5L l-fBA 16E7

0.75 « 0.273 (2-a)

0.75QL 4EI fBC 1.5gZ2

--- =--- --- - --- 8 L \-fBC 16EI

=——=0.2000.75 3.75

(2-b)

The final step is to determine the critical load or effective length factor (K-factor) using the following expressions.

“扁=(】+&)(i+r如)/(?) ₍₃₎

"爲L

where X-factor is defined by

K = ((1 + 々)・(1 + /伽)严 (4) The validity of Eq. 3 can be easily demonstrated ; That is, when a single prismatic member is simply supported at its both ends, then f 邱 =fpa~ 0.0, by definition. With f 해 = ffia= 0.0, Eq. 3 yields P^EI/L2 (K^l.0).

In the same way, PCf=2^EI/l/ is obtained for the member with one end simple supported。" 皿 더椅) and the other fixed (f ^=1.0). Finally, the critical load of the member fixed at both its ends = 1.0) is expressed by 4^EI/L2 (K=0.5).

With the fixity factors of Fig. 1(c), Eq. 3 and Eq. 4 give the following res니ts :

(Span4B)

2 El (%应=(1硕1，273)/京0

(1.5匕)

= 5.584—El r

Kab =1/7(1x1.273) a 0.886

(5-a)

(6-a) (Spai BC)

(以•屁=(L2)(L00)/2 El謙 0)

= 11.844—>(^)^ (5-b)

±J

Kbc = 1 시(1XL273) » 0.913 (6-b) As mentioned earlier, Fig. 1(a) is the ex皿pie member chosen by Chen"). He 叫)plied the Neutral equilibrium method to obtain, R尸5.89EI/I?, which is approximately 5.2% la-ger them the result given by Eq. 5(a).

Now above results are to be compared with those by two conventional methods.

(Modified slope-deflection method)

Firstly, the Modified slope-deflection method (M.S.D.M)(5) will be briefly described. When the axial force, P, is considered, the end moments and rotation an­

gles of a beam-column are related by the following :

Fig. 2. Deformation of beam-column

El 、

M邱=(—)ap - ^n°a (7-a)

L EI

M际=(-厂)砂-(afea +a“如) _(7-b) k-t

First, these equations mb applied successively to Fig.

1(a). Next, the moment equilibrium conditions at the ex­

ternal and intermediate supports allows one to obtain the following matrix equation:

anI / 1.5 ap/1.5 o" 6

a^/1.5 (%"L5 + %) «f ♦ « -=・ _{0 • (8)}

0 a{ 0

Where % and % are so-called Merchant's stability fimc- tions defined by

a 虹

a -如

(9-a,b)

n I'

f宀；

with

如"(也1 )2 (1-kLcotkL) (10-a)

虹~ （成，）2 (kL esc kL-Y) (10-b)

kL = y/pL2/EI (10-c)

Oni and % are Eq. 9 with k]L=1.5kL

The characteristic equation for critical load is obtained from Eq. 8. That is :

/1.5 Oyj/1.5 0

det(Zyj/1.5 (awl/1.5 + an) af = 0 (11-a)

_ 0 af %

Elastic Critical Load and Effective Length Factors of Continuous Compression Member by Beam Analogy Method 49

when expanded

知 으 丄

1.5 1.5

5”끙%“一끔心。(H-b)

The least root satisfying Eq. 11(b) is found by a trial and error procedure, which yields kL=2.426S and PcK.888EI/必 by Eq. 10(c). This critical load coincides with that of above mentioned result of Chen.

(Finite element method)

Frequently used finite element gf/rod(F.E.M)⑴⑹⑺ will also be briefly described. Fig. 3 shows an element of a beam-column subjected to constant axial fbrcej* and a set

(a) Element forces

Y

(b) Element displacements

Fig. 3. Beam-column element

nodal forces {«}. In Fig. 3, {切 denotes nodal displace­

ment vector corresponding to {q}. The element stiffiiess matrix, [A] combines {q} and {切 in the following form.

{아그回

㈣

因*广‘俱 (⑵

in which

]2 Symm.

, El -61 4l2 wb=—

I5 -12 61 12 _-6Z 2Z2 61 8Z2

(13-a)

18 Symm.

, 1 -1.5Z 2l2

wg=—

g 15/ -18 1.5Z 18 -1.5Z 0.5 户 1.5Z 2Z2

(13-b)

In the above two equations, [A]b denotes ttie flexural( or bending ) stiffness matrix and (瓦 is called geometric( or initial stress) stiffness matrix. As one co니d see in Eq. 12, flexural stiffness is decreased, due to the axial compressive force, P.

The structural stiffness matrices for the entire member

are obtained by transforming the individual element matri­

ces from element to structural coordinates and then by assembling the resulting matrices. Finally, boundary condi­

tions should be applied to the assembled matrices. The procedures are easily found from textbooks on the finite element method or structural stability⑸⑹⑺.hence, a de­

tailed explanation will be omitted.

The external force vector, {Q\ and corresponding dis­

placement vector, {A} are related by

{Q} = [K] {시 (14)

where [K], the structure stiffness matrix obtained after the application of boundary conditions, takes the form

[K] = [K] -P[K] (15)

b g

With external force vector, {Q}={0}, Eq.(14) and (15) constitute a typical eigenvalue problem.

([K] -P[K] ){A} = {0} (16) b g

To obtain the least eigenvalue (here, the elastic critical load corresponding to the first mode of buckling) by com­

puter-aided iteration method, Eq.(16) should be trans­

formed into the following form

([K]：[K] - &邛{A} 그 {0} (17) b g P

where [7] is the unit(or identity) matrix.

When the finite element method is applied to Fig. 1(a) one can obtain nearly the same critical load as that by the modified slope-deflection method. Here, it is observed that the representative two methods, (the one, an analytical method or modified slope-deflection method and the other, a numerical or finite element method) can result in the smne value of the elastic critical load.

The procedures necessary to obtain the final result by above mentioned methods involve complicated calcu­

lation when one relies on hmd calculation. Furthermore,

(a) Given continuous member

I 3Q |Q

" t 5 多一 M i D

L o. 6L-」一0.5，」 L-Q5Z」

(b) Analogotis beam

0 儿그。.，砧2 払=0 心 fK=0.0

I

2£l 〜_二二~、-一 El 广'一一＞스 —바 戶、_ ——一/泸 4 日心22花 C“mi

Kv=。66 Km=083 ^{Kd 92}

(c) FEfy factors and K-foctors

Fig- 4. 3-Span continuous member

50 Soo-Gon Lee and Soon-Chul Kim

these methods can neither predict the span that buckles first, nor determine the effective length factor, K, of each span.

For a clearer explanation of the proposed method, an­

other example of 3-span continuous compression member is adopted. Fig.4 (b) shows the first step of proposed method. The second step, that is, the stress analysis of the jmalogous beam shown in Fig.4(b) can proceed in the following ways(the slope-deflection method is adopted).

(End moment equations)

2EI 、 3.6QL QL

mab =?77(2°8)--- — = -5.7792(—)

1Q丄 o o

,, 2EI 2QL QL、

M bc - --- --- = +0.7585(—)

2EI 2QL QL、

Mcb= —(6>s + 20c)- -亍=-0.5604(%-)

2EI QL QL、

Mcd = — (20c + 盼 _ %- = +0.5604(%-)

2EI QL

mdc = 一厂伊 c +2^d) + —= 0.0

(18-a)

(18-b)

(18-c)

(18-d)

(18-e)

yr 3OZ L

= 0， %+诺c +eD=^■— (19-b)

8 2EI

■Di QL L

£虬=0, E"으布

The solution of the simultaneous equations yields Oie following rotation an옹les.

(19-c)

(20-a) QI?、

0c = 72.8(——) 84857

(20-b) QU、

勺 =-69.2( -쓰一) 848£7

(20-c) (End moment)

Above rotation uigles are substituted into slope­

deflection equations to obtain the results shown at the right-hand sides of the same equation.

The third step is the determination of fixity factors by using Eq.(l). In the present problem

0,758501 8£7 /以 69.30?

8 1.2Z i-fBA 848成

(Joint equation) ... = 0.1482 (21-a)

'^匚 M b = 0, —0g + .. --- --- (19-a) O.75850Z 4EI fBC 69.3QZ?

1-2 8 2EI 8 L 1 - /bC 848A/

nble. I. Critic이 Load coefHcient, C(Pc产CEI/I?)- Reference values a?衫 the M.S.D.Mresults.

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用 g雌5螂

R 幻^阳弋 .： 治命樂織？설*'工 ■' . ' ；■.：" 天 ； ■: ” 空 - ■-•.：. „ X. 資 .. ■•: 、'.疽 ，.电

匚可漏

心 此이『

思■■”曜

f脚

冬 B C D 투

Pl & 釜 月 急 四 給 &Tp

fA=O.0

fE.o 9.8696 9.8696 9.8689 (ALL) 0.0 10.6221 10.6223 10.6221

(DE) 5.6 fA=0.0

ff=0.0 12.7800 12.7801 12.7801 (BC,CD) 4.9

P ■* 1 P ■*—j P - " 1

A 广 倉 沮 最 단 带 E! 1

fA=0.0

ZQ 5.6468 5.6469 53889 (AB) 4,6 fA=0.0

fiO 5.7209 5.7210 5.4226 (AB) 5.2 fA=0.0

zo 7.1344 7.1344 &6068 (CD) 7.4 f神.Q

fE=0.0 7.5874 7.5877 7.2198 (CD) 4.8

. 戶—r 户―1

妍 ！ 决 普 理 * 2EI 盐 EI 1 "

L a L 山 _L.... j

fA=0.0

ZO 9.8696 9.8689 9.8689 (ALL) 0.0 g。

ZQ 10.2130 10.2131 10.0142 (AB) 1.9 fA-0.0

fL이) 11.243& 11.2439 10.1393 (DE) 9.8 fA=0.0

12.6131 12.6132 1L7S64 (CD) 6.6

, 1 「 尸

4P a 陌 令 处/ 结 w 金 0 %『

, M § £ 耳 £ R L ,

fA=0.0

ff=0.0 5.8586 5.8583 5.4897 (AB) 6.3 fA=0.0

zo 5.S748 5.8746 5.5110 (AB) 6,2 fA-0.0

9.4846 9.4843 9.2461 (AB) 2.5 go

fr~o.9 9.7375 9.7373 93270 (AB) 4.2

Elastic Critical Load and Effective Length Factors of Continuous Compression Member by Beam Analogy Method 51

fBC = 0.2248 (21-b) 0.56040, _ 4EI fCB 62.90?

8 L \-Jcb 848E7

•- /cb 그 /cd =°.191O (21-c) The fixity factors are given in Fig. 4(c). The last step is to find the A-factor and die critical local of each span by applying Eq.(3) and (4).

Span AB:

Kab = {(1 +1)(1 + 0.1482)}t” a 0.66 수 QEI) (3%)如 =2x1.1482—느七 (1.2L)2

、 El

•••아诲』广1。・49玲 (22-a) L

Sp 取! BC:

Kbc ={(L2248)(1.191)}~°s ® 0.83

, 、 수 El

(2Pcr)BC = 1.2248x1.191-^-

^7.198? (22-b)

Span CD :

Kcd ={(l.^lXl)}-0-5 저 0.92

z 、 /El El

(PQcd = L191^ 저 11.754万 (22-c)

Lj L

Eq. (22) is the final step, where one can easily see that the member of span BC may buckle under the magnitude of load, 7.198EI/I}.

The above-mentioned modified slope-deflection method is to be 叩plied to the member of Fig.4(a). When Eq(7-a) and (7-b) are applied to this member, one obtains

M BC = ~(an2^B + a/2^c) (23-a)

-

e

-

l

-

e

-

e

X)/

、、丿

Ay -b -c -d - e 3

3

3

3

where a차, a^, — are Merchant's stability functions (see Eq. 10) with

kL =加?际 k〔L = J3kL, k2L = 4ikL

To obtain a characteristic equation, one can start with Mac =0 to obtain 6b =-使丿％ Then, the moment equilibrium at the support 瓦(》农=0) and

give tiie following matrix equation

(5anl/3 + an2) afl [이」

af2 an2+(an ~af)/an 门謨

(24) From the above equation, one gets tiie characteristic equation

/ 3 + %)• [a”？ 十«#—#)/%]

-(5 )2=0 (25)

The least root satisfying tiie above equation is found to be kL=2.72787, gives the elastic critical load

Pcr =7.441EI/I)

The proposed beam analogy method (B.A.M.) can be easily applied to multi-span continuous con屮ression members without regard to changes in the sectional prop­

erties or the lengths of each span. Some of the critical loads of four-span continuous compression members m

shown in Table 1. In this table, the boundary condi- tions(B.C) are either a fHctio이ess hinge (fa =0.0) or a fixed end (fa =1.0). When compared with results by F.E.M or modified slope deflection method, the beam analogy method (B.A.M) gives lower boundary errors for critical loads. In the column “AB” in the parenthesis denotes that the AB span buckles first. For example, (DE) means that span (DE) will buckle first under the given conditions.

3. ^FACTORS FOR BRACED COLUMNS

In the practical design of framed columns, A-factor concept (rather thwi elastic critical load) is widely used.

Several authore(11)(12) have proposed diflEerent methods of X-factor determination for friuned columns. Until today, however, no unified method is available. Now, the pro­

posed method can be applied to the determination of effec­

tive length factors for multi-story braced frame columns.

In the braced frames of Fig 5, columns are not permitted to move horizontally and so they can be modeled as a con­

tinuous member ofFig. 5(c).

Actually, Fig. 5 is the braced frame and its modeling ex­

ample was chosen by Hellesland<8) for effective length factors. Helleslan시s A-factors for first and second story columns are K^O.656 and K2=0.928, respectively, if one assumes Lf= L2, Pf= 2P2 in Fig. 5(c). Meanwhile, the

52 Soo-Gon Lee and Soon-Chul Kim

beam analogy method 叩plied to Fig. 5(c), gives Kf^,707(error, +7.8%) andiT f=1.00(error, +7.8%).

Fig. 6 shows a two-story steel frame chosen by McCor- for its effective length 曲tor determination. He adopted AISC'S Alignment chart, which is based on the following equation.

Ga •잉 勿/K

一 + I---Hl---)

2 tan(;r/K)

2tan(0.5 勿/K)

+ ——1---!一 = 1.0 (26) 지 K

4

with

E (으%

G =a L a “El

早砂

_ sum of column stiffness at joint a sum of beam stiffness at joint a

(27-a)

changes in the manner of Fig. 7(a). Fig. 7(b) shows the

»ialogous beam. The stress analysis results from the slope-deflection method are given by Fig. 7(c). Finally, the effective length factor for each column detennined by Eq.

4 are shown in Fig. 7(d).

. a * , P 기 L—'-…

A즈 *…. 丄…，华B / 尊 C 卜——一丄4--- h_______ 4_______ I

(a) Assumed cdumn load

(b) Anak^ous benn

Ma=0.0 M^i2.78QU88 Mc^0.0 9^=7.56QI?/88EI

(c) Absolute end moments & rotation angles

_ sum of column stiffness at joint § sum of beam stiffness at joint p

A3WZ的

A

fHB=0.0

"後 gg78 ―B二二£

如=0336 Jcb=0.0 fK=0297

(d) Fixity factors & K-bctors (27-b)

The G-factors for each joint determined by Eq. 27 are given in the Fig. 6(b) proposed by McCormac.

Finally, the K-factors satisfying Eq. 26 can be found by a trial and error procedure. The least roots of K for the col­

umns are also given in Fig. 6(b).

Fig. 7. Analogous Beam

The A-factor for the first story gives 13% error when compared with the result by the Alignment chart. But for the second-story column, the proposed method yields laige upper bound errors, which can be acceptable as far as the actual structural design is concerned.

I

10.0' W16X 36

W8X 24

E

W16 X 57

W8X 40 W8X 24

W18X 50 W18X 97

W8X 24 W8X 40 W8X 24

e D« 1 4

12.d

(a) Floor plan

써--- *

.i.1

---T

1

1 i

wails 22Jem |

J. F3% 曲1

\ *님一& 1

■l1 卜 86 + 'I1 W___』'1'1

Fig. 6 (a). Two story frame

（金匚— _/E哗、 ©5瑚 _/申tAG/wO.328

KebO.653 KgO.625 KsO.626

_ 一 _ 冶^

서).380 fGM.272 *>GM.26O

KM.787 Kq&・0.765 Kg所0.763 gjGi-10.00 @Gd-10.00 @Gg»10.00

Fig. 6 (b). G-foctors & K-foctors

Now, the proposed method is to be applied to this &we

witii tiie assumption that the total load of each story Fig. 8. Design example

Elastic Critical Load and Effective Length Factors of Continuous Con^pression Member by Beam Analogy Method 53

Fig. 8 shows a design exan耳)le ofbraced columns. This design example is adopted from Steel Designer's Man­

ual'^. In the original paper, Jf-factors assume K넉),8 for the lower two-story columns and K^O. 7 for the remaining upper story columns. Fig. 9 shown on the final page of 也is paper shows an analogous beam, the stress analysis, and the K-factors for the design example columns. As can be seen in Fig. 9(c), the proposed method gives slightly larger values for the fi-fectors. For this reason, the proposed method, when applied to braced columns, will result in a conservative design.

4. CONCLUSION

Contrary to conventionally used analytical or numerical methods, beam analogy method requires only easy hand calculations for the determinations of elastic critical load and /【-factors of a continuous compression member. Even in tiie cases of continuous compression members with variable span, and sectional dimensions and also with dif­

ferent boundary conditions, the proposed method can pre­

dict the span that buckles first.

The beam analogy method gives lower-bound errors of critical loads when compared with those by conventional methods. Meimwhile K-factors for the braced frame col­

umns by proposed method are larger tiian those by AISC's dignment chart, and upper-bound errors of effective length factors, which lead to a conservative design of either con­

tinuous compression members or multi-story braced col­

umns.

REFERENCES

(1) S.G. Lee, Eigenvalue Problems of Tapered Bars with Partially Fixed Ends, Graduate School, Seoul Nat'l Univ., 1979

(2) S.G. Lee, "Analysis of Steel Members with Partially Restrained Connections'', The Third International Kerensky Conference, Singapore, 1994

(3) J.S. Kinney, Indeterminate Structural Analysis, Addi­

son Wesley Publishing Co., Inc., 1957

⑷ W.F. Chen and et al, Structural Stability, Elsevier Science Publishing Co., Inc., 1987

(5) A. Chajes, Principles of Structural Stability Theory, Prentice-Hall, Inc., 1974

(6) 金尙植,”構造安定解析"，文運堂，1999

⑺ 李 守 坤, "構造物의 安定理論”, 全南大學校 出 版部，1995

(8) Jostein Hellesland and et al, ''Restraint Demand Factors and Effective Lengths of Braced Columns ", J.

Struct. Engng., ASCE, Vol. 122, No. 10, pp.1216〜1224,

1996

(9) CONSTRADO(U.K), "Steel Designers Manual, William Collins Sons & Co., Ltd., 4th Rev., 1983 (10) J. C. McConnac, ^Structural Steel Design, LRFD

Method”, Harper & Row Publishers. Inc., 1989 (11) P.K. Basu & S.L. Lee, ^Effective Length Factors for

Type-PR Frame Members” Proceedings of ASCE, Structural Div., ppl85-194, May 1989

(12) R.O. Disque, '"Inelastic K-factor for Column Design9', J. Struct. Engng； AISC, Vbl.ll, No.2,2nd Qtr., pp33- 35,1974

54 Soo-Gon Lee, Soon-Chul Kim

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