Elastica stability analysis of a simple frame under a follower force

B.S.I. ES310

Elastica stability analysis of a simple frame under a follower force T. Avraam and A. Kounadis Department oj" Civil Engineering, National Technical University o[" Athens, 42 Patission Str, Athens, 10682, Greece (Received April 1992; revised version accepted March 1993)

An elastica stability analysis is applied to a geometrically perfect and elastically supported two-bar frame, which is subjected to a follower compressive load at its joint, eccentrically applied to its column centreline. The individual and coupling effect of the loading eccentricity and the support elasticity on the stability and postbuckling response is discussed. It is found that for a slight change in the amount of support elastic stiffness, the static bifurcational mechanism disappears and the frame exhibits a monotonically rising equilibrium path. K e y w o r d $ : Elastica stability analysis, follower loading

The stability conditions of linear mechanical systems under the action of a set of nonconservative follower-type compressive forces, have received considerable attention in the last 20 years. The majority of pertinent studies refer to elastic systems with a finite number of degrees-offreedom or simple continuous systems such as columns and plates under various boundary conditions. An extension of these analyses to simple elastica frames by including geometrical nonlinearities of moderate magnitude has been presented x-6. On the basis of either discrete or continuous models it was established that nonconservative systems due to the action of follower-type loads may lose their stability either through flutter or divergence depending on the boundary conditions or joint stiffness in case of nonconservative frames. Flutter-type nonconservative systems may be analysed by employing only the dynamic method of analysis, whereas divergence-type nonconservative systems may be analysed by employing either the static or the dynamic method. Conditions for the existence of regions of divergence instability have been presented 7,8 In this paper the large displacement response of a simple two-bar frame with sway under the action of a follower compressive force at its joint, eccentrically applied to the column centre-line, will be considered. To this end an elastica analysis is employed. The effects on the critical nonlinear divergence buckling load of the loading eccentricity as well as of the amount of partial fixity of its support are thoroughly discussed. Existing results as well as various phenomena regarding the degeneration of branching points into limit points and of equilibrium paths into physically unacceptable paths are also clarified. The results presented, were obtained by employing the Runge-Kutta and Newton-Raphson numerical schemes. These results, in certain cases, were confirmed by other existing analyses. 0141-0296/94/040238~94 {) 1994 Butterworth-Heinemann Ltd

238

Engng Struct. 1994, Volume 16, Number 4

Mathematical analysis The geometrically perfect frame shown in Figure I is made of homogeneous, isotropic and linearly elastic material with a modulus of elasticity E. Each bar has length Ii, constant cross-sectional area Ai and constant moment of inertia li (i = 1, 2). The frame is subjected at its joint to an eccentrically applied follower compressive force of constant magnitude P, which remains tangential to the centre line of the vertical member during the

x 1 ,u 1

x2'u2 I P

w1

Figure 1 Geometry and sign convention of two-bar frame subjected to follower compressive load

Elastica stability analysis of a simple frame under a follower force: T. Avraam and A. Kounadis deformation. The vertical bar is elastically restrained with a rotational stiffness c* on an immovable support, whereas the horizontal bar is hinged on a simple support. Let w* and u* be the transverse and axial displacement components, respectively, of the centre line of the ith bar. According to one-dimensional elastica theory we can write

dw* dx* - (1 + ei)sin 01

N*

f12 cos 0B -- k 2

g2 - E A 2 -

~22

p sin 0 2

(8)

where 22 = 12(A2/12) 1/2 is the slenderness ratio of the horizontal bar and N* is the axial force, tangential to its deformed axis.

(1)

Method of solution

du* d x * - ( 1 + ~ i ) c o s 0 1 - 1 i = 1,2 where Oi = Oi(x*) and ei = el(x*) are the rotation angle of the tangent to axis x* and the axial strain of the centre line of the ith bar, respectively. The bending moment M l ( x * ) at the point x* is given by M l ( x * ) = S w * ( x * ) - P [ x * + u*(x*)]sin 0B - C*OA

(2) where S is the compressive force of the column and 0 A, 0 B are the rotation angles at the support A and the joint B, respectively. Given that d01

(3)

m i ( x * ) = - Eli d x *

and using equations (1) and (2) the deformed state of member A B is described by the following differential equations 0' 1 :

port C. The axial strain of member B C is given by

-- k 2 w 1 "~ fl2(X 1 "t-

u0sin On +

cO A

(4a)

Equations (4) and (7) with the respective boundary conditions define a boundary-value problem which is solved by the Runge-Kutta-Verner numerical scheme. We begin by assuming enough initial conditions to obtain a starting point (step 1), then integrate along the length of the frame members (step 2) and finally check the boundary conditions at the terminal point (step 3). For the frame A B C and for given values of the parameters #, p, 21, 2 2, c and with starting point the support A, the starting conditions are 0

(9a)

ul(0) = 0

(9b)

01(0) = Oa (arbitrary)

(9c)

WI(0 ) :

Integration of equations (4) for a fixed value of/32 presumes that two starting values are assumed for the axial force k 2 (taken near//2) and the joint rotation 0n (arbitrary). The values of wl, Ul, 01 at the joint B give the starting conditions for integrating equations (7). These conditions are

w'1 = (1 + el)sin 01

(4b)

w2(1) = - u l ( 1 ) / p

(10a)

u'1 = (l + ~l)cos 01 - 1

(4c)

u2(1) = w l ( 1 ) / p

(10b)

0z(1) = 01(1)

(10c)

where ( )'/= d( ) / d x i (x, U, W)i = (X, U, W)*/l i c = c * l l / E I 1 is the nondimensionalized stiffness, f12= PlZ/EI1 and k Z = SI2/EI1 are the nondimensionalized

load and axial force of the column, respectively. The axial strain of member A B is given by N* ~1

-

- -

EA 1

k2 cos01 + flZsinOBsin01 -

22

(5)

where 21 = 11(A1/I1) 1/2 is the slenderness ratio of the column and N* is the axial force tangential to its deformed axis. The bending moment M E ( X ~ ) a t the point x~ is given by M 2 ( x * ) = ( P cos 0B - S ) [ x * + u'~(x*) - u*(0)]

(6)

Using equations (1), (3) and (6) the deformed state of member B C is described by ( k 2 _ f12 c o s OB)

0'2 =

p ( x 2 + u 2 - A)

w~ = (1 + e2)sin 02 U'2 = ( l -t- e2)COS 0 2 --

(7a) (7b)

1

(7c)

where # = I211/Ill2, p = 12/11, and A = u'~(O)/l 2 is the nondimensionalized horizontal displacement of the sup-

As can be seen from equation (7a) for integrating equations (7) we must assume the value of A (taken near w2 (1)). At the terminal point the hypothetical values 0 A, 0 s, k 2 and A must satisfy the following conditions w2(0) = 0

(1 la)

ml(1) + m2(1) + Pe = 0

(llb)

01(1) = 02(1) = 0B

(11c)

A = u2(0 )

(1 ld)

where e = e / l v If these equations are not satisfied then they are corrected using the Newton-Raphson scheme. Subsequently the integration is repeated from A to C, until the hypothetical values take their true values.

Numerical results and discussion For the case p = p = 1, 21 = 2 2 = 40 and c = 0 the frame loses its stability through an asymmetric bifurcational point 1. The critical load is equal to f12r = 2.46758; namely identical with that obtained in ref 1, which is f12 = 2.467. If the effect of the axial shortening of the bars is neglected (standard elastica) the critical load becomes equal to flcz~= 2.46740; namely this effect is practically negligible. Note that for a constant directional (conservative) load

Engng Struct. 1994, Volume 16, Number 4 239

Elastica stability analysis of a simple frame under a follower force: T. Avraam and A. Kounadis it was found that the critical load fl~zr is equal to 1.37283 which coincides with that obtained in ref. 11. From the foregoing results one can compare the load-carrying capacity of the frame under nonconservative follower loading with that of a conservative loading. The

~2

curve 1

/ 2.47 2.468

c=I~-~ curve 2 / c=O

2.465

2.46

0

0.0025

0.005

w 111)

Figure 2

Degeneration of t w o bifurcational paths into t w o independent equilibrium paths w h i c h intersect at A as c varies from c=0toc>0.

load-carrying capacity in the first case is higher than the second case by 80% [ = (2.46758- 1.37283/1.37283]. Moreover, the primary and secondary equilibrium paths when c differs slightly from zero (c = 1.E 10) degenerate into a monotonically rising path and a complementary (not physically acceptable) path, corresponding to curves 1 and 2 of Figure 2. As c increases, curves I and 2 become more distant from each other and no critical state exists. The above phenomenon is clearly shown in the extreme case where c = 1.E7, (c-* ~c~), ll = 16, p = 0.25 and 2 = 80 (Figure 3). This finding contradicts previous results based on linear (static and dynamic) analysis 6 as well as on nonlinear moderately large deformation analysis 9. Such a discrepancy regarding the first case is due to the linearization of the governing equations which force the system to lose its stability through a bifurcational point associated with trivial equilibrium paths. The discrepancy in the second case is due to the numerical solution, obtained with the aid of the hand calculator after making certain approximations 6. In view of the latter the limit point instability results of the Reference 6 should be revised according to the present exact elastica analysis. It should be noted that a nonlinear dynamic analysis is not necessary for verifying the present elastica analysis results, if the effect of the compressibility of the column centre-line is neglected (standard-elastica). On the other hand the discrepancy between the nonlinear static and dynamic critical load when the effect of the compressibility of the column centre-line is included is of minor importance 1o. One should also observe that the critical load based on the theory of moderately large displacements 9"11.12 practically coincides with that of the present elastica analysis. In the case of an eccentrically applied loading the

~2

15

\ ~

physical

\

path

\

unacceptable path

IO

5.0

I

-0.1 Figure 3

240

I

-0.05

i

0

0.05

>

w1 (1)

Continuously rising (stable) equilibrium path for frame with,/1 = 16, p = 0.25, 21 = ~2 = 2 = 80, c = 1 . E l 0

Engng Struct. 1994, V o l u m e 16, N u m b e r 4

Elastica stability analysis o f a simple frame under a follower force: T. Avraam and A. Kounadis f52

f52

y

2.436 2.4

2.2

,

12 .L o a d

~ ' " '

--Z,

'

11

e=O

10

e=O. 10

8

6

2.0 5 3 2 1 i

0.1

Limit point instability andp=p=l

Figure 4

0.2 for e = 0.025,

-0.I

wi(I)

c = O, ),1 = )'2 =

J

40

f r a m e ( c o r r e s p o n d i n g to # = p = 1, e = 0.025) with c = 0 e x h i b i t s a s n a p - t h r o u g h b u c k l i n g (Figure 4) w h e r e a s for c > 0 a n d e > 0 it experiences a m o n o t o n i c a l l y rising e q u i l i b r i u m p a t h (Figure 5). As is k n o w n , the limit p o i n t l o a d is a n u p p e r b o u n d of the d y n a m i c critical l o a d w h i c h is a s s o c i a t e d with a critical p o i n t lying i n the v i c i n i t y of the u n s t a b l e e q u i l i b r i u m p a t h 13. F r o m t h e a b o v e results it is clear t h a t the c h a n g e of b u c k l i n g b i f u r c a t i o n a l m e c h a n i s m to a m o n o t o n i c a l l y rising e q u i l i b r i u m p a t h i m p l i e s t h a t the f o r e g o i n g f r a m e is n o t i m p e r f e c t i o n sensitive.

Conclusions The most important findings of this investigation based on a simple two-bar frame with sway are as follows. For the first time an exact large displacement response using elastica has been performed on a partially fixed supported nonconservative two-bar frame. The derived results were obtained by employing the Runge-Kutta-Verner numerical scheme and subsequently they were confirmed in certain cases by other analyses. When the partially fixed support becomes a hinge the foregoing frame loses its stability through an unstable (asymmetric) bifurcation point. Such a point degenerates into a limit point when the loading is applied eccentrically by e > 0 to the column centre-line. Contrary to conservative bifurcational systems, the nonconservative frame under consideration is not imperfection sensitive since the effect of loading eccentricity does not appreciably reduce its load carrying capacity. The perfect frame with a partially fixed support does not buckle at all, contrary to the corresponding results obtained by using a linear dynamic or linear static stability analysis.

-0.05

)

0

w I (I)

Figure 5 Two monotically rising (stable) equilibrium paths for

frame corresponding to e= O, e= 0.10 and c = 1.E10, # = 16, )-1 = ),2 = 80 and p = 0.25

References 1 Kounadis, A. N., Giri, J. and Simitses, J. G. 'Divergence buckling of a simple frame, subject to follower force', J. Appl. Mech. Trans. ASME 1987, 45, 426-428 2 Kounadis, A. N. 'The effect of some parameters on nonlinear buckling of a nonconservative simple frame', J. de M~can. Appl. 1979, 3, 173-185 3 Panayotounakos, D. and Kounadis, A. N. 'Elastic stability of a simple frame subjected to a circulatory load ~, J. Sound Vibr. 1979, 64 (2), 179-186 4 Avraam, T. P., Pantis, M. A. and Kounadis, A. N. 'Snap-through buckling of a simple frame with tangential load', Acta Mechanica 1980, 36, 119-127 5 Kounadis, A. N. and Economou, A. 'The effects of joint stiffness and of the constraints on the type of instability of a frame under a follower force', Acta Mechanica 1980, 36, 157-168 6 Kounadis, A. N. and Avraam, T. P. 'Linear and nonlinear analysis of a nonconservative frame of divergence instability', AIAA J. 1981, 19 (6), 761-765 7 Kounadis, A. N. 'Divergence and flutter instability of elastically restrained structures under follower forces', Int. J. Eng. Sci. 1981 19, 553-562 8 Kounadis, A. N. 'The existence of regions of divergence instability for nonconservative systems under follower forces', Int. J. Solids Struct. 1983, 19 (8), 725-733 9 Brush, D. O. and Aimroth, B. O. Buckling of bars, plates and shells, McGraw-Hill, New York, 1975 10 Sotiropoulos, S. and Kounadis, A. N. 'The effects of nonlinearities and compressibility on the static and dynamic critical load of nonconservative discrete systems', Ing.-Archive 1990, 60, 399-409 11 Koiter, W. T. 'Postbuckling analysis of a simple two-bar frame', in Recent progress in applied mechanics, Almquist and Wiksell, Stockholm, 1966 12 Roorda, J. 'Stability of structures with small imperfections', J. Eng. Mech. 1965, 91 (EM1), 87-106 13 Kounadis, A. N. 'Nonlinear dynamic buckling of discrete dissipative or nondissipative systems under step loading', AIAA J. 1991, 29 (2), 280-289

E n g n g Struct. 1994, V o l u m e

16, N u m b e r 4

241