Buckling Nonuniform Member

ELSEVIER

J. Construct. Steel Res. Vol. 42, No. 2, pp. 141-158, 1997 © 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain 0143-974X/97 $17.00 + 0.00

P n : S0143-974X(97)00010-2

Equivalent Buckling Length of Non-uniform Members

John Ch. E r m o p o u l o s

National Technical University of Athens, Department of Structural Engineering, Laboratory of Steel Structures, 42 Patission Street, 10682 Athens, Greece

(Received 7 February 1996; revised version received 26 November 1996; accepted 17 December 1996)

ABSTRACT

The non-linear equilibrium equations of framed non-uniform members under compression are established for non-sway and sway mode. The models considered are similar to those proposed by EC3 Annex E for uniform members. Using an iteration procedure the critical loads and the corresponding equivalent buckling lengths are calculated, while results are presented in tabular and graphical form, to make direct use from practising engineers easy. © 1997 Elsevier Science Ltd.

NOTATION

E I/ Ic, Ic'

G

Ix g m = lm/L c g i : li/L i k

Lc Li

M , V

rh, rl2

Modulus of elasticity Moment of inertia of the beams (i = 1...4) Moments of inertia of the tapered bar AB at A and B, respectively Moments of inertia at the middle of the tapered bar AB (i.e. for x = 0.5Lc) Moment of inertia of the tapered bar AB at distance x Reference stiffness coefficient of the tapered member Reference stiffness coefficient of beams 1-4 Equivalent buckling length ratio Length of the tapered bar Lengths of the beams (i = 1...4)

Bending moment and shear force of a bar Distribution factors

141

142 J. Ch. Ermopoulos

P External axial compressive force [D m = pL2/EIm Dimensionless external load /Dcr = PcrLZ/EIm Dimensionless critical load a Distance from origin to A

ai, fi Oln~ Olf

,gAG tx ~)n, ~ f

Dimensionless parameters of the slope-deflection method for the tapered bar AB Dimensionless parameters (i = 1...4) Dimensionless parameters of the slope-deflection method Transverse deflection of node B, regarding node A Angles of rotation at A and B, respectively {(pot2/EIc) - 0.25 Dimensionless parameters

1 INTRODUCTION

The determination of the exact buckling length of axially compressed bars which belong to a framed structure presupposes the stability analysis of the entire structure. Through this analysis the critical loading system and the corresponding axial forces in the various members can be calculated and consequently their equivalent buckling length coefficients can be obtained. In order to avoid this cumbersome procedure, some approximate methods have been proposed in the past, which enable practising engineers to estimate in a simple and accurate way these coefficients [ 1-3], in the case of frames consisting of uniform members (i.e. with constant moment of inertia along their axis).

Based on these methods, various countries have established specifications (e.g. AISC/ASD [4], AISC/LRFD [5], BS5950 [6], DIN 18800 [7], EC3 [8], etc.) according to which simple ways of estimating the buckling length of such members are suggested or required. In addition, the influence of the semirigidity of the connections has also been faced [9] using the same model as in Annex E of EC3.

In the case of non-uniform members, various researchers have proposed approximate procedures to obtain the equivalent buckling lengths [10-15], and some of the recent codes have used these procedures in a graphical form (e.g. AISC/ASD, AISC/LRFD, etc). Regarding in particular Eurocode 3, it has to be noted that the following is stated in Annex E: 'An equivalent buckling length may also be used to relate the buckling resistance of a non-uniform member to that of a uniform member under similar conditions of loading and restraint', without any suggestion of a specific methodology.

In this paper, on the basis of the slope-deflection method as it is applied for tapered bars under compression [ 16,17], the non-linear equilibrium equations for the systems of Fig. 1 (which correspond to EC3, Annex E models) are

Equivalent buckling length of non-uniform members 143

I:1.

x ~,

• , . ~ . ~ . ' ~ _ . ~ - - - ~ - ~

u_ o

,li ~ /11 I

W

x -~ E

¢.. v~

r'~ , . 0

V

0

E

Q

E

e . o

Z

V ,--~

S ,a - -J r


derived (both for non-sway and sway mode). The solution of these equilibrium equations gives the critical loads [18,19] of the models and consequently the corresponding equivalent buckling length ratios of the tapered members.

The results of the study are presented both in a graphical form similar to that of EC3, Annex E for uniform members, and in a tabular form.

2 ANALYSIS

2.1 General

The model that is considered in this study is similar to that used in Eurocode 3 [8], Part 1.1 (Annex E, Figure E.2.3), where the uniform column is replaced by a non-uniform (tapered) one, as shown in Fig. 1. This column is acted upon by an axial compression force P, while the axial force of the beams is assumed to be zero. The geometric characteristics of the tapered column are shown in Fig. 1 (c). The law governing the variation of the moment of inertia I, along the axis of the bar is given by [15]:

where I x is the moment of inertia of the cross-section at a distance x from the origin and Ic is the moment of inertia at a distance t~ from the origin. This law corresponds sufficiently to cross-sections having constant or approximately constant area (simple or built-up tapered members), thus describing most cases of ordinary steel structures. Both non-sway and sway modes are examined and the critical loads and the equivalent buckling length ratios are obtained in a closed form, for the model considered.

For the model shown in Fig. l(a), the following relations of the slope- deflection method are valid:

4EII MaE- L~ Oa (la)

4EI2 MBF = L20B (lb)

4EI3 MAC-- ~3"0A (lc)


4E14 MAD -- L-4 0A (1 d)

2EIc[_Lc ~cc] MBA = [ot30a + ~40B + (a3 + a4) ( le)

2Elc I-_ Lc ~ ] MAB = [C~IOA + ~2'9B + (~1 + ~2) ( I f )

2EI¢[_ ~ - P ~ ] ( lg)

For the analysis, the following dimensionless parameters are introduced:

pL2c Pc,L2c /Pot 2 1 e LC, ,Por eT-m,p` = = - - ! ~>0. ( 2 )

VEI~

The coefficients in eqns (le) and (If) which refer to the non-uniform member AB are the following [16,17]:

(fz - f4)(p` 2 + 0.25)£2 (3a)

2fo

~2 =f4(p`2 + 0"25)/?2 (3b) 2fo

{(/"2 - f4) cos[p, ln(1 + e)] + (f3 - fl) sin[~ ln(1 + e)]}(t, 2 + 0.25)e2X/1 + e ~3 = 2fo (3C)

{f4 cos[p` ln(1 + e)] - f3 sin[p, ln(1 + e)]}(p` z + 0.25)e2Xfi + e

~ 4 ~"~- 2fo (3d)

where

f 0 ~-~ f d 4 - - f2.f3

f~ - ~ cos[/* ln(1 + e)] - ~ sin[p` ln(1 + e)] - o.5e

(4a)

(4b)


f 2 - /x cos[tx ln(1 + f)] + ~ sin[tx ln(l + f)] - txf (4c)

f 3 = xfi + f cos[Ix ln(1 + f)] - 0.51~ - 1 (4d)

f 4 = " ~ "1- ~ sin[ix ln(1 + f)] -- lXf. (4e)

Introducing eqns (4a-e) into eqns (3b) and (3c), it can easily be proved that a2 = a3.

2.2 Non-sway mode

The equil ibr ium of the moments at A (EMA = 0) and B (~MB = 0), taking into account that ~ = 0 in the case of non-sway mode, lead to the fol lowing equi l ibr ium equation:

1 - (5)

where T~l and T h are the same as the EC3, Annex E distribution factors, given here by:

KB ~h = KB + K1 + 1(2 (6a)

KA 772 = KA + K3 + K4 (6b)

and

~' /c K A = L ~ , K B - Lc - Lc(1 + e ) 2 (6c)

Ii Ki = ~ (i = 1...4). (6d)

In the particular case of a uniform column, which corresponds to ~ = Lc/a = 0, the equi l ibr ium equat ion takes the form:

Equivalent buckling length of non-uniform members 1 4 7

where

(7)

~n ~Of an - 2(q~z _ q~2), ogf- 2(q~nZ _ q~fZ) (8a)

1-P~m/tan(P~m) ~mm/Sin( P ~ m ) - - 1

~On~--- iPm ' ~ f = Pm (8b)

em = PLy (Im = constant for uniform member). Elm

(8c)

2.3 Sway mode

For the model shown in Fig. l(b) where ~ : / :0 , the relations ( l a ) - ( lg ) are valid, and the equilibrium system consists of the following equations:

ZMA = 0 (9a)

ZMB = 0 (9b)

V. = O. (9c)

Equations (9a-c), after some manipulation and taking into account that ~z = ~3, lead finally to the following equation, which corresponds to sway mode:

[ 1 0/2 + . . . . . . . . .

[ ~ - ~ , + ~ +~+ ~4~] [2(1_~, ,/~,+,~2+

O~ 4 -1-

( ~ 3 + a a ) Z _ z . 2 1 - 1 + o q +

Pm (~+~2+~3+~4 - - - - 2 2


In the particular case of uniform column (i.e. /? = 0), the equilibrium equation (10) takes the form:

O/f+P2 2(o~ + O/r)J g +O/n+ oo+of,2 il

I( on+ f,2 ]] 2 1 _ l ) + a . + . . . . . . O. (11) ~22 [P2_2(a.+ae)

2.4 Design procedure

In order to determine the equivalent buckling length ratios k of the non-uniform member AB [Fig. l(a) and (b)], the critical load Per of this member has to be established first.

Thus, using the equilibrium equations obtained in the previous paragraphs [eqns (5) and (7) for the non-sway mode and eqns (10) and (11) for the sway mode], the critical loads ecr can be calculated for non-uniform and for uniform members by means of an iteration procedure.

The following Euler 's formula is valid:

7r2EIm ecr - (kLc)2 (12)

and using eqn (2), the equivalent buckling length ratio k is finally obtained from:

'B-

k - # c r " (13)

This ratio corresponds to that given by EC3, Annex E for uniform members, but also covers the case of tapered columns, as stated in paragraph E. 1.4 of Annex E.

3 NUMERICAL R E S U L T S - - E X A M P L E S

On the basis of the above analysis a computer program has been constructed which enables the calculation of the buckling length ratio k for the case of tapered columns in non-sway and sway modes.


By utilizing this program, the dimensionless critical loads /Ocr and the ratio k have been calculated and the results are presented in Tables 1 and 2, for various values of the ratio e = Ldo~, and the distribution factors r/l, 772.

Furthermore, in order to have the results in the same form as in Annex E of EC3, the diagrams of Figs 2 and 3 have been plotted. From these figures, for a given ratio f = Lc/O~ and for a certain pair of the factors r/1 and r/2 [expressed by eqns (6a) and (6b)] the corresponding buckling length ratio k is obtained.

For example, let us consider that the beams have an equal length Li = 5 m with moments of inertia Ie = 50,000 cm 4, while the tapered column has a length Lc = 7.5 m, moment of inertia Ic = 10,000 cm 4 and a distance o~ = 1.5 m.

The moments of inertia Im in the middle of the column and I~' at end B are:

( x ) 2 (°t +~-tc/2) 2 (1 -5+3-75 /2 Im = Ic = lc = 10,000 1.5 j = 122,500 cm 4

( _ - + +__7.5? lc' = tc c\ a ~ / = 10,000 1.5 j = 360,000 cm 4.

Moreover,

Lc 7.5 e - - - 5

t~ 1.5

IB 360,000

Lc 750

I1 12 360,000 250,000 r/~ IB+ + - - +

L2 750 500

= 0.706

IA 10,000

Lc 750 r/2 - IA 13 14 - 10,000 250,000

Lc + L3 + L4 750 + 500

= 0.06

and the ratio k is found to be:

non-sway mode (Fig. 2, e = 5): k = 0.84

sway mode (Fig. 3, e = 5): k = 1.68

while the corresponding ratio for the uniform column with Im = 122,500 cm 4 and


TABLE 1 Critical L o a d s / 3 and Equivalent Buckling Length Ratios k for a Tapered Member in a Non-

sway Mode

"0/ "02 /~ = 0 /? = 0.5 f = 1.0 e = 5.0 P,, ~ P,r ~ Per k i:,,,~

0.0 39.43 0.500 38.34 0.507 36.37 0.521 24.60 0.633 0.1 37.35 0.514 35.87 0.525 33.66 0.541 21.39 0.679 0.2 35.10 0.530 33.32 0.544 30.98 0.564 19.00 0.721 0.3 32.79 0.549 30.85 0.566 28.53 0.588 17.32 0.755 0.4 30.52 0.569 28.58 0.588 26.38 0.612 16.13 0.782

0.0 0.5 28.3;/ 0.590 26.56 0.610 24.54 0.634 15.26 0.804 0.6 26.38 0.612 24.78 0.631 22.98 0.655 14.59 0.822 0.7 24.58 0.634 23.22 0.652 21.65 0.675 14.07 0.837 0.8 22.95 0.656 21.87 0.672 20.52 0.694 13.65 0.850 0.9 21.48 0.678 20.68 0.691 19.54 0.711 13.30 0.861 1.0 20.17 0.700 19.64 0.709 18.70 0.726 13.02 0.871 0.0 37.35 0.514 36.65 0.519 34.95 0.531 23.96 0.642 0.1 35.43 0.528 34.34 0.536 32.40 0.552 20.87 0.688 0.2 33.34 0.544 31.94 0.556 29.86 0.575 18.55 0.729 0.3 31.18 0.563 29.61 0.577 27.51 0.599 16.92 0.764 0.4 29.05 0.583 27.44 0.600 25.45 0.623 15.76 0.791

0.1 0.5 27.01 0.604 25.50 0.622 23.67 0.646 14.90 0.814 0.6 25.12 0.627 23.79 0.644 22.16 0.667 14.25 0.832 0.7 23.38 0.650 22.28 0.665 20.87 0.688 13.73 0.848 0.8 21.82 0.673 20.96 0.686 19.76 0.707 13.32 0.861 0.9 20.40 0.696 19.81 0.706 18.81 0.724 12.98 0.872 1.0 19.13 0.718 18.79 0.725 17.99 0.741 12.69 0.882 0.0 35.10 0.530 34.77 0.533 33.36 0.544 23.23 0.652 0.1 33.34 0.544 32.63 0.550 30.97 0.564 20.27 0.698 0.2 31.42 0.560 30.40 0.570 28.58 0.588 18.04 0.740 0.3 29.43 0.579 28.21 0.591 26.37 0.612 16.45 0.774 0.4 27.45 0.600 26.17 0.614 24.40 0.636 15.32 0.802

0.2 0.5 25.54 0.622 24.33 0.637 22.70 0.659 14.49 0.825 0.6 23.75 0.645 22.69 0.660 21.24 0.682 13.85 0.844 0.7 22.11 0.668 21.24 0.682 20.00 0.702 13.34 0.860 0.8 20.61 0.692 19.97 0.703 18.93 0.722 12.94 0.873 0.9 19.25 0.716 18.86 0.723 18.00 0.740 12.61 0.885 1.0 18.03 0.740 17.87 0.743 17.20 0.757 12.33 0.895 0.0 32.79 0.549 32.76 0.549 31.60 0.559 22.37 0.664 0.1 31.18 0.563 30.78 0.566 29.38 0.580 19.56 0.710 0.2 29.43 0.579 28.72 0.586 27.16 0.603 17.43 0.752 0.3 27.61 0.598 26.69 0.608 25.09 0.627 15.91 0.788 0.4 25.78 0.619 24.79 0.631 23.23 0.652 14.82 0.816

0.3 0.5 24.00 0.641 23.05 0.654 21.62 0.676 14.01 0.839 0.6 22.33 0.665 21.50 0.678 20.23 0.698 13.38 0.859 0.7 20.78 0.689 20.12 0.700 19.04 0.720 12.89 0.875 0.8 19.36 0.714 18.91 0.722 18.01 0.740 12.50 0.888


TABLE 1 Continued

~h "02 e = 0 f = 0 . 5 e = l . 0 f = 5 . 0 Pcr k ~'cr k ]'cr k ~'cr k

0.9 18.07 0.739 17.84 0.744 17.12 0.759 12.17 0.900 1.0 16.90 0.764 16.89 0.764 16.34 0.777 11.90 0.910 0.0 30.52 0.569 30.67 0.567 29.71 0.576 21.38 0.679 0.1 29.05 0.583 28.86 0.585 27.67 0.597 18.74 0.726 0.2 27.45 0.600 26.96 0.605 25.62 0.621 16.72 0.768 0.3 25.78 0.619 25.09 0.627 23.70 0.645 15.28 0.804 0.4 24.09 0.640 23.32 0.650 21.96 0.670 14.23 0.833

0.4 0.5 22.45 0.663 21.70 0.674 20.44 0.695 13.44 0.857 0.6 20.89 0.687 20.24 0.698 19.13 0.718 12.85 0.876 0.7 19.44 0.713 18.94 0.722 18.00 0.740 12.37 0.893 0.8 18.10 0.738 17.78 0.745 17.01 0.762 11.99 0.907 0.9 16.88 0.765 16.76 0.767 16.16 0.781 11.68 0.919 1.0 15.76 0.791 15.86 0.789 15.41 0.800 11.41 0.930 0.0 28.37 0.590 28.58 0.588 27.77 0.596 20.25 0.698 0.1 27.01 0.604 26.91 0.606 25.89 0.617 17.79 0.745 0.2 25.54 0.622 25.17 0.626 24.00 0.641 15.90 0.788 0.3 24.00 0.641 23.45 0.649 22.22 0.666 14.54 0.824 0.4 22.42 0.663 21.81 0.673 20.61 0.692 13.55 0.853

0.5 0.5 20.93 0.687 20.30 0.697 19.19 0.717 12.80 0.878 0.6 19.48 0.712 18.94 0.722 17.96 0.741 12.23 0.898 0.7 18.12 0.738 17.71 0.746 16.89 0.764 11.78 0.915 0.8 16.86 0.765 16.62 0.770 15.95 0.786 11.41 0.930 0.9 15.70 0.793 15.65 0.794 15.14 0.807 11.11 0.942 1.0 14.64 0.821 14.79 0.817 14.43 0.827 10.85 0.953 0.0 26.38 0.612 26.55 0.610 25.81 0.618 18.98 0.721 0.1 25.12 0.627 25.01 0.628 24.08 0.640 16.71 0.768 0.2 23.75 0.645 23.40 0.649 22.34 0.665 14.97 0.812 0.3 22.33 0.665 21.81 0.673 20.70 0.690 13.69 0.849 0.4 20.89 0.687 20.30 0.697 19.21 0.717 12.76 0.879

0.6 0.5 19.48 0.712 18.90 0.723 17.89 0.743 12.06 0.904 0.6 18.13 0.738 17.62 0.748 16.74 0.768 11.52 0.926 0.7 16.85 0.765 16.48 0.774 15.73 0.792 11.09 0.943 0.8 15.67 0.794 15.45 0.799 14.85 0.815 10.74 0.959 0.9 14.57 0.823 14.53 0.824 14.08 0.837 10.45 0.972 1.0 13.57 0.853 13.72 0.848 13.41 0.858 10.21 0.983 0.0 24.58 0.634 24.63 0.633 23.90 0.643 17.59 0.749 0.1 23.38 0.650 23.18 0.652 22.28 0.665 15.52 0.797 0.2 22.11 0.668 21.70 0.674 20.68 0.691 13.91 0.842 0.3 20.78 0.689 20.22 0.698 19.17 0.717 12.74 0.880 0.4 19.44 0.713 18.82 0.724 17.79 0.745 11.87 0.912

0.7 0.5 18.12 0.738 17.52 0.751 16.57 0.772 11.22 0.938 0.6 16.85 0.765 16.33 0.777 15.49 0.798 10.71 0.960 0.7 15.65 0.794 15.25 0.804 14.54 0.824 10.31 0.978 0.8 14.54 0.824 14.29 0.831 13.72 0.848 9.98 0.994


T A B L E 1

Continued

T~I '~2 g = 0 f = 0 .5 f = 1.0 ~ = 5.0 P,,r k P,,. k Pcr k Pcr k

0.8

0.9

1.0

0.9 13.50 0.855 13.42 0.857 12.99 0.871 9.70 1.008 1.0 12.55 0.887 12.65 0.883 12.36 0.894 9.48 1.020 0.0 22.95 0.656 22.83 0.657 22.06 0.669 16.11 0.783 0.1 21.82 0.673 21.47 0.678 20.55 0.693 14.21 0.833 0.2 20.61 0.692 20.08 0.701 19.06 0.719 12.76 0.879 0.3 19.36 0.714 18.71 0.726 17.66 0.747 11.68 0.912 0.4 18.10 0.738 17.40 0.753 16.38 0.776 10.88 0.952 0.5 16.86 0.765 16.18 0.781 15.25 0.804 10.28 0.980 0.6 15.67 0.794 15.07 0.809 14.24 0.832 9.81 1.003 0.7 14.54 0.824 14.06 0.838 13.36 0.859 9.43 1.023 0.8 13.48 0.856 13.15 0.866 12.58 0.885 9.12 1.040 0.9 12.49 0.889 12.34 0.894 11.90 0.910 8.87 1.055 1.0 11.58 0.923 11.60 0.922 11.30 0.934 8.65 1.068 0.0 21.48 0.678 21.16 0.683 20.31 0.697 14.57 0.823 0.1 20.40 0.696 19.87 0.705 18.89 0.723 12.83 0.877 0.2 19.25 0.716 18.56 0.729 17.51 0.751 11.51 0.926 0.3 18.07 0.739 17.28 0.756 16.20 0.780 10.53 0.968 0.4 16.88 0.765 16.05 0.784 15.01 0.811 9.80 1.003 0.5 15.70 0.793 14.91 0.813 13.95 0.841 9.25 1.033 0.6 14.57 0.823 13.87 0.844 13.01 0.871 8.81 1.058 0.7 13.50 0.855 12.92 0.874 12.19 0.900 8.46 1.079 0.8 12.49 0.889 12.06 0.904 11.46 0.928 8.18 1.098 0.9 11.56 0.924 11.29 0.935 10.82 0.955 7.95 1.114 1.0 10.69 0.961 10.59 0.965 10.25 0.981 7.75 1.128 0.0 20.17 0.700 19.63 0.709 18.68 0.727 12.99 0.871 0.1 19.13 0.718 18.40 0.732 17.33 0.754 11.39 0.931 0.2 18.03 0.740 17.16 0.758 16.03 0.785 10.19 0.984 0.3 16.90 0.764 15.94 0.787 14.81 0.816 9.31 1.030 0.4 15.76 0.791 14.79 0.817 13.69 0.849 8.65 1.068 0.5 14.64 0.821 13.71 0.848 12.70 0.881 8.14 1.101 0.6 13.57 0.853 12.73 0.880 ! 1.82 0.913 7.75 1.128 0.7 12.55 0.887 11.83 0.913 11.05 0.945 7.43 1.152 0.8 11.58 0.923 11.02 0.946 10.36 0.976 7.17 1.173 0.9 10.69 0.961 10.29 0.979 9.76 1.005 6.95 1.191 1.0 9.86 1.000 9.62 1.012 9.22 1.034 6.77 1.207

Im 122 ,500

Lc 750

71 72 im 122,500 I1 ~ 2 5 0 , 0 0 0

L ~ + ~ + L 2 750 + - 5 0 0

= 0.45


TABLE 2 Critical Loads ~bcr and Equivalent Buckling Length Ratios k for a Tapered Member in a Sway

Mode

"01 '12 e = 0 ~ = 0.5 e = 1.0 e = 5.0 Pcr k ecr k Pcr k ecr k

0.0 0.0 9.86 1.000 9.63 1.012 9.23 1.034 6.77 1.207 0.1 9.34 1.028 9.01 1.046 8.55 1.074 5.96 1.286 0.2 8.75 1.062 8.34 1.087 7.84 1.121 5.30 1.364 0.3 8.09 1.105 7.63 1.137 7.13 1.176 4.78 1.437 0.4 7.37 1.157 6.90 1.195 6.43 1.239 4.36 1.504 0.5 6.60 1.223 6.17 1.265 5.75 1.310 4.02 1.566 0.6 5.79 1.306 5.44 1.347 5.11 1.389 3.75 1.622 0.7 4.95 1.412 4.73 1.444 4.51 1.479 3.52 1.673 0.8 4.11 1.550 4.05 1.560 3.96 1.578 3.33 1.721 0.9 3.28 1.735 3.41 1.700 3.45 1.691 3.17 1.763 1.0 2.46 2.003 2.81 1.873 2.98 1.818 3.03 1.803

0.1 0.0 9.34 1.028 9.21 1.035 8.87 1.055 6.61 1.221 0.1 8.85 1.056 8.63 1.069 8.23 1.095 5.83 1.300 0.2 8.31 1.090 8.00 1.111 7.56 1.142 5.19 1.378 0.3 7.70 1.132 7.33 1.160 6.88 1.197 4.68 1.451 0.4 7.03 1.185 6.64 1.219 6.21 1.260 4.27 1.519 0.5 6.30 1.252 5.94 1.289 5.56 1.331 3.94 1.581 0.6 5.53 1.336 5.24 1.372 4.95 1.412 3.67 1.638 0.7 4.74 1.443 4.56 1.471 4.37 1.502 3.45 1.689 0.8 3.93 1.585 3.90 1.590 3.83 1.604 3.27 1.736 0.9 3.12 1.779 3.28 1.734 3.34 1.719 3.11 1.779 1.0 2.33 2.058 2.69 1.913 2.88 1.850 2.98 1.820

0.2 0.0 8.75 1.062 8.71 1.064 8.45 1.081 6.41 1.240 0.1 8.31 1.090 8.18 1.098 7.85 1.121 5.67 1.318 0.2 7.81 1.124 7.60 1.139 7.23 1.168 5.06 1.396 0.3 7.25 1.167 6.98 1.189 6.59 1.223 4.57 1.469 0.4 6.63 1.220 6.33 1.248 5.96 1.286 4.17 1.538 0.5 5.96 1.287 5.67 1.319 5.34 1.359 3.85 1.600 0.6 5.24 1.372 5.00 1.404 4.75 1.440 3.59 1.656 0.7 4.49 1.483 4.35 1.505 4.20 1.533 3.38 1.708 0.8 3.72 1.629 3.73 1.627 3.68 1.637 3.19 1.756 0.9 2.94 1.832 3.13 1.776 3.20 1.755 3.04 1.800 1.0 2.18 2.128 2.56 1.962 2.76 1.890 2.91 1.840

0.3 0.0 8.09 1.105 8.14 1.101 7.95 1.114 6.17 1.264 0.1 7.70 1.132 7.66 1.135 7.41 1.154 5.47 1.343 0.2 7.25 1.167 7.13 1.176 6.83 1.201 4.89 1.420 0.3 6.75 1.209 6.56 1.226 6.24 1.257 4.42 1.494 0.4 6.18 1.264 5.96 1.286 5.65 1.321 4.04 1.562 0.5 5.57 1.331 5.35 1.358 5.08 1.394 3.74 1.624 0.6 4.90 1.419 4.73 1.444 4.52 1.477 3.49 1.682 0.7 4.20 1.533 4.12 1.548 3.99 1.571 3.28 1.734 0.8 3.48 1.684 3.52 1.674 3.50 1.678 3.10 1.782


TABLE 2 Continued

"0/ "02 v = 0 e = 0.5 e = 1.0 v = 5.0 Pcr k Pcr k Pcr k Pcr k

0.9 2.74 1.898 2.94 1.829 3.04 1.801 2.95 1.826 1.0 2.01 2.216 2.40 2.026 2.61 1.942 2.83 1.867

0.4 0.0 7.37 1.157 7.49 1.148 7.37 1.157 5.86 1.297 0.1 7.03 1.185 7.06 1.182 6.88 1.198 5.21 1.375 0.2 6.63 1.220 6.58 1.224 6.36 1.245 4.67 1.452 0.3 6.18 1.264 6.07 1.274 5.83 1.301 4.24 1.526 0.4 5.68 1.318 5.53 1.335 5.29 1.366 3.88 1.594 0.5 5.13 1.387 4.97 1.408 4.76 1.440 3.59 1.658 0.6 4.52 1.478 4.40 1.497 4.24 1.525 3.35 1.716 0.7 3.87 1.597 3.83 1.604 3.74 1.623 3.15 1.769 0.8 3.20 1.756 3.27 1.736 3.28 1.734 2.98 1.818 0.9 2.51 1.983 2.73 1.900 2.84 1.862 2.84 1.863 1.0 1.82 2.329 2.21 2.111 2.44 2.011 2.71 1.905

0.5 0.0 6.60 1.223 6.75 1.209 6.68 1.215 5.46 1.344 O. 1 6.30 1.252 6.37 1.244 6.25 1.256 4.88 1.422 0.2 5.96 1.287 5.96 1.286 5.80 1.304 4.39 1.499 0.3 5.57 1.331 5.51 1.338 5.32 1.361 3.99 1.573 0.4 5.13 1.387 5.03 1.400 4.84 1.427 3.66 1.641 0.5 4.63 1.460 4.53 1.475 4.36 1.503 3.39 1.705 0.6 4.09 1.553 4.01 1.567 3.89 1.591 3.16 1.765 0.7 3.51 1.677 3.49 1.680 3.44 1.693 2.98 1.818 0.8 2.89 1.848 2.98 1.820 3.01 1.810 2.82 1.868 0.9 2.25 2.094 2.47 1.996 2.60 1.947 2.68 1.916 1.0 1.60 2.484 1.98 2.227 2.22 2.107 2.57 1.957

0.6 0.0 5.79 1.306 5.92 1.291 5.88 1.295 4.93 1.414 0.1 5.53 1.336 5.60 1.327 5.51 1.337 4.43 1.491 0.2 5.24 1.372 5.25 1.371 5.13 1.387 4.01 1.569 0.3 4.90 1.419 4.86 1.424 4.72 1.445 3.65 1.642 0.4 4.52 1.478 4.45 1.489 4.30 1.513 3.36 1.713 0.5 4.09 1.553 4.01 1.568 3.88 1.593 3.12 1.778 0.6 3.61 1.653 3.55 1.665 3.47 1.686 2.91 1.839 0.7 3.09 1.787 3.09 1.786 3.06 1.795 2.75 1.894 0.8 2.53 1.975 2.62 1.938 2.67 1.921 2.60 1.945 0.9 1.95 2.250 2.16 2.134 2.30 2.070 2.48 1.995 1.0 1.34 2.714 1.71 2.398 1.95 2.250 2.37 2.039

0.7 0.0 4.95 1.412 5.01 1.402 4.97 1.409 4.24 1.526 0.1 4.74 1.443 4.75 1.441 4.67 1.454 3.83 1.604 0.2 4.49 1.483 4.45 1.488 4.35 1.506 3.48 1.683 0.3 4.20 1.533 4.13 1.545 4.01 1.568 3.18 1.759 0.4 3.87 1.597 3.78 1.614 3.66 1.641 2.94 1.831 0.5 3.51 1.677 3.41 1.700 3.30 1.728 2.73 1.900 0.6 3.09 1.787 3.02 1.807 2.95 1.829 2.56 1.963 0.7 2.63 1.937 2.6l 1.941 2.59 1.949 2.41 2.022 0.8 2.14 2.148 2.20 2.114 2.25 2.092 2.28 2.078 0.9 1.61 2.476 1.79 2.344 1.92 2.265 2.17 2.128

Equivalent buckling length of non-uniform members

T A B L E 2

Continued

155

7h r/e e = 0 e = 0.5 e = 1.0 /? = 5.0

['cr k ecr k P c r k i'cr k

1.0 1.06 3.051 1.38 2.665 1.60 2.477 2.08 2.177 0.8 0.0 4.11 1.550 4.05 1.561 3.95 1.580 3.32 1.724

0 . l 3.93 1.585 3.83 1.604 3.71 1.630 3.01 1.809 0.2 3.72 1.629 3.59 1.657 3.45 1.689 2.75 1.893 0.3 3.48 1.684 3.32 1.722 3.18 1.760 2.52 1.977 0.4 3.20 1.756 3.04 1.801 2.90 1.844 2.33 2.055 0.5 2.89 1.848 2.73 1.900 2.61 1.944 2.17 2.131 0.6 2.53 1.975 2.40 2.025 2.31 2.063 2.03 2.202 0.7 2.14 2.148 2.06 2.187 2.02 2.208 1.91 2.268 0.8 1.70 2.409 1.71 2.401 1.73 2.385 1.81 2.331 0.9 1.23 2.833 1.35 2.699 1.45 2.607 1.73 2.388 1.0 0.74 3.652 0.99 3.147 1.18 2.892 1.64 2.447

0.9 0.0 3.28 1.735 3.04 1.801 2.85 1.860 2.14 2.146 o. l 3.12 1.779 2.86 1.855 2.66 1.925 1.93 2.259 0.2 2.94 1.832 2.67 1.921 2.46 2.001 1.75 2.371 0.3 2.74 1.898 2.46 2.003 2.25 2.093 1.60 2.482 0.4 2.51 1.983 2.22 2.104 2.03 2.204 1.46 2.591 0.5 2.25 2.094 1.97 2.233 1.80 2.339 1.35 2.696 0.6 1.95 2.250 1.71 2.400 1.57 2.507 1.26 2.798 0.7 1.61 2.476 1.43 2.626 1.33 2.718 1.17 2.893 0.8 1.23 2.833 1.13 2.944 1.10 2.994 1.10 2.992 0.9 0.82 3.469 0.83 3.433 0.87 3.368 1.04 3.079 1.0 0.38 5.096 0.53 4.309 0.64 3.913 0.98 3.165

1.0 0.0 2.46 2.003 2.01 2.211 1.69 2.415 0.68 3.784 o. l 2.33 2.058 1.87 2.293 1.54 2.526 0.56 4.192 0.2 2.18 2.128 1.72 2.395 1.39 2.661 - - - - 0.3 2.01 2.216 1.55 2.522 1.23 2.3831 - - - - 0.4 1.82 2.329 1.36 2.687 1.06 3.046 - - - - 0.5 1.60 2.484 1.16 2.907 0.89 3.330 - - - - 0.6 1.34 2.714 0.95 3.215 0.71 3.722 - - - - 0.7 1.06 3.051 0.72 3.680 0.53 4.306 - - - - 0.8 0.74 3.652 0.49 4.480 0.35 5.301 - - - - 0.9 0.38 5.096 0.24 6.349 o. 17 7.617 - - - - 1.0 - - ~ - - o o _ o o _ o~

is f o u n d t o b e :

n o n - s w a y m o d e ( F i g . 2 , e = 0 ) : k = 0 . 6 6

s w a y m o d e ( F i g . 3 , e = 0 ) : k = 1 .38 .

F o r t h i s e x a m p l e i t c a n b e s e e n t h a t t h e d i f f e r e n c e b e t w e e n t h e b u c k l i n g


-4 , 0 \ I"h \,, 0.9

o.~ "-~\L "\ \ o.7 ~ \ \\ ",\ \

0.s \ \ - \ \ " \ ' ¢ ~ ~* \~\ ,

\ \ ~ \ ", 0.3 - \ . \

0.2 "° . \ \ \' " 0 . 1 \ ~ r

\ \ \ \ \ o.o~,- \ \ \ \ \

Fixed

\ \ \

\ , , \ \ \

- \ '\,,

\ , , " \ . , \ \

\ \ \ \ " \

Pinned

\ \ " \ \ , \ \ \ ?

\ \ \ "~,~'~, \

\ , \ ",,~o\ \ \ \ "~,~ \,, \ , ",\ ",~o \ \ \ •

"\o_ "N \ \ ' , \ ~ \ \ \ ",

\ ,~ ' \ , \ ,~-.. \ \ \ "\

0.0 " \ \ o~ \ \ 0.8 , \ 0.7 x \ \ 0.6 \ N 0.s , \ , \ 0.4

0.3

0.1 , \ 0.0' ~'-

" \ \ \ ~ .~'<-.

",~ \ \ \

\ \ \ \ \

\ \ \ 0.0 0.1 02. 0.3 0,4 0.5 0.6 0.7 0.8 09 1.0

\ \ \ \ \ ' ~ . ~ .

\ \ \ N \" x \

\ 5 \ ,,

0.0 0.1

\

0.2 0.3 0/, 0.5 0.6 0.7 0.8 0.9 1.0

F i g . 2 . Equivalent buckling length ratio k for uniform ( f = 0 ) and non-uniform columns ( e =

0.5, 1, 5) in a non-sway mode.

length ratios of a non-uniform and the corresponding uniform column is approximately equal to 20%.

4 CONCLUSIONS

In the present paper the non-linear equilibrium equations of framed compression uniform and non-uniform members were established, for both the non-sway and the sway modes.

Solving these equations by an iteration procedure, one can obtain the equiv-


1.0

o~ ~ o.~ ~

0.6

0.5 "~- " " ]

OA

0.3 ~ " \

0.2

0.o. :o \ Fixed

~ .

C \ ."\"k,\ ~\

© × I ~ \, \ , \ ,\~

, . \ ~ \ ! \ \ .,., \ \ \ \ ~\

~. w, k \ \ \ \ \ \ \ \~

, \x ' i \ \ \

\\

Pinned

\

\ > \ \

\ \ \ \1

0.0

03

~410"8 0.7

0.6

o.~ ", \ ~ \ \ o,~ \ \ \ 0.3 ~ " ~ "~ . \ \

o., -~; \ ~ \

0.0 0.I 0.2 0.3 0.~ 0.5 0.6 0.7 0~ 09 1.0

F-c -w, OZ) 0.1

\ \ \ , % "~ \, \ \ \

0.2 03 0Z 05 0.S 0.7 0B 0.9 1.0 .q2=

Fig . 3. Equivalent buckling length ratio k for uniform (¢ = 0) and non-uniform columns (e = 0.5, 1, 5) in a sway mode.

alent buckling length ratios of the members considered, while the corresponding values of the critical loads are also calculated.

The equations for uniform members correspond directly to the diagrams given in EC3, Annex E (figures E.2.1 and E.2.2), while the equations for non- uniform members satisfy paragraph E. 1.4 of the same Annex.

Using the diagrams presented in Figs 2 and 3 it is easy to obtain graphically the equivalent buckling length ratio for each particular combination of the geometric characteristics of the frame under consideration (i.e. non-sway or sway mode, lengths and moments of inertia, tapering ratio of the column, etc.).

In addition, the corresponding critical loads can also be determined from Tables 1 and 2.


It is worth mentioning that the buckling length ratios k for tapered members, depending on the combination of the distribution factors 771 and 7/2, become either bigger or smaller than the uniform ones (with constant moment of inertia equal to Ira), as can be seen from Fig. 2 or Fig. 3.

REFERENCES

1. Kavanagh, T. C., Effective length of framed columns. Part II. Transactions, ASCE, 1962, 127, 81-101.

2. Wood, R. H., Effective lengths of columns in multi-storey buildings. Structural Engineering, 1974, 52, 235-346.

3. Wood, R. H. and Roberts, E. H., A graphical method of preventing sidesway in the design of multi-storey buildings. Proceedings of the Institute of Civil Engin- eers, 1975, 59, 353-372.

4. AISC/ASC, Manual of Steel Construction, Allowable Stress Design, 9th edn. AISC, Chicago, 1989.

5. AISC/LRFD, Manual of Steel Construction, Load & Resistance Factor Design, 1st edn. AISC, Chicago, 1986.

6. British Standards Institution, Structural Use of Steelwork in Building. BS 5950: Part 1. BSI, London, 1990.

7. DIN 18800, Teil 2, Stabilitatsf~ille, Knicken von St~iben und Stabwerken, Berlin, Beuth, 1990.

8. Eurocode 3, Design of Steel Structures, Part 1.1, Commission of the European Communities, DDENV 1993-1-1, 1992.

9. Ermopoulos, J. Ch., Buckling length of framed compression members with semi- rigid connections. Journal of Constructional Steel Research, 1991, 18, 139-154.

10. Fraser, D. J., Design of tapered member portal frames. Journal of Constructional Steel Research, 1983, 3(3), 20-26.

1 I. Galea, Y., Flambement des poteaux a inertie variable. Construction Metallique, No. 1, 1981, 20-46.

12. Lee, G. C., Ketter, R. L. and Hsu, T. L., The Design of Single Story Rigid Frames. Metal Building Manufacturer Association, Cleveland, 1981.

13. Lee, G. C., Morrell, H. L. and Ketter, R. L., Design of Tapered Members, WRC Bulletin, No. 173, June 1972.

14. SSRC, Guide to Stability Design Criteria for Metal Structures, 4th edn, ed. T. V. Galambos. Wiley Interscience, New York, 1988.

15. Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability, 2nd edn. McGraw- Hill, New York, 1961.

16. Ermopoulos, J. Ch., Elastic stability analysis of plane rectangular frames with varying stiffness members. Ph.D. thesis, National Technical University of Athens, 1984.

17. Ermopoulos, J. Ch., Slope-deflection method and bending of tapered bars under stepped loads. Journal of Constructional Steel Research, 1988, 11, 121-141.

18. Ermopoulos, J. Ch., Buckling of tapered bars under stepped axial loads. Journal of Structural Engineering, ASCE, 1986, 112(6), 1346-1354.

19. Ermopoulos, J. Ch. and Kounadis, A. N., Stability of frames with tapered built- up members. Journal of Structural Engineering, ASCE, 1985, 111(9), 1979-1992.

Buckling Nonuniform Member

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