Dynamic Buckling of Guyed Stacks, Masts and Columns with Constant Inertia and Algebraic Polynomials Generated by use of Operator Tn Attached to Bessel Functions

Dynamic Buckling of Guyed Stacks, Masts and Columns with Constant Inertia and Algebraic Polynomials Generated by use of Operator T, Attached to Bessel Functions by JEAN DANIEL

Civil Engineering Department, Quebec H3G 1 M8, Canada

Concordia

University,

Montreal,

ABSTRACT :

A new method of analysis of the dynamic critical load of columns, guyed stacks and masts with constant inertia, under the combined action of horizontal loads, axial load and unlyormly distributed loads along the vertical axis is presented. The integro-d$erential formulation of the problem leads to fourth- and fifth-order partial dlerential equations. In addition to the solution of the equation of motion in expansion series, the new method of solution proposed to solve thejifth-order partial dierential equation has led to the differential equation of the plate on elastic foundations in the old Winkler hypothesis. Moreover, the operator T,, attached to the Besselfunctions J,, has been used to generate a new set of algebraic polynomials.

I. Zntroduction The ever-increasing dimensions of power-plants and the demands of radio and television have led, for some time now, to the erection of chimneys, towers and masts, many in the height range of more than 300 m. As a result, action due to winds and earthquakes tends to generate, in these structures, disturbing forces. Dangerous vibrations occur and, according to Vandeghen and Alexandre (l), it was only in 1952 that important vibrations for steel stacks were mentioned in the literature of technology. During the last three decades this phenomenon has been extensively studied, and many researchers have contributed to a better understanding of it. For example, Hirsch et al. (2) reported the reasons for the failure of a steel stack 140 m tall. In a previous publication, the author (3) analysed the static buckling load of columns, guyed stacks and masts with constant inertia. A formula was proposed, using the Bessel functions J, and an operator T, attached to these functions. In the present investigation, dynamic buckling is considered, again taking into account the influence of dead weight uniformly distributed along the vertical axis. The undamped motion of the system is formulated, as a fourth-order partial differential equation, and the solution is presented in the form of a power series. In the damped motion, a fifth-order partial differential equation is obtained. The proposed method of solution to that equation has led to the rediscovery of the

0

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95

Jean Daniel

equation of beam on an elastic foundation in the old Winkler hypothesis. Moreover, it is shown that the operator T, can be extended to a solution in seriesid order to generate new sets of algebraic polynomials. II. Formdation

of the Problem

Under the set of applied loads, an element of a stack between two sets of guys is schematized in Fig. 1. Let [F] be the resultant of the external forces. The equilibrium equation of the element, in its displacement around the vertical axis, is expressed by the integral equation (1) :

&I-

(

a2.h 4 + ax2 X

c

EI

IS 14o x s

)=-

wx,4

s ~ax2at (Y(Y,

4 -

~6,

32

a2

t))

&

-

;

Y(X,

t) +

v,x

_

+

c

M2

-Ax,

aY(x,

t)

t)

q.

a2Y(x,

t) (1)

at-gatz

In (l), the second term on the left-hand side represents the influence of the damping

P

I

/

p (x,t)

I

z -)

x

Q(X)

/

-

-

I

P+"A 4(X)

FIG. 1. Set of

acting loads. Jovial

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of the Franklin Institute ~er~amon Press Ltd.

Buckling of Masts, Stacks and Columns are defined below :

stresses. The other parameters EI : g: c,: C: 40: dx, t) :

flexural rigidity gravity strain velocity (internal) damping extenal damping coefficient unit weight of the element moment due to the external distributed

III. Undamped

load p(x, t).

Vibration

Below, the undamped

motion

of the system is considered.

1. Partial diTerentia1 equation of motion In (l), it is sufficient to set C, and C identical

to zero. If,

K = EI then (1) is easily expressed

by (2) :

a4Y(x, t) 1 --+~Cp+~o(~-X)I,,z--ax4

a2Yh

t)

t) + 40

40 dY(X,

a2Y(% Kg at2

ax

Kg

t)

= - ;

2 w, 9).

The homogeneous part of (2) is considered, and the d’Alembert method of separation of variables is applied. That is,

method

or the

(3)

Y(& t) = Y(x)W)* Hence, (4) and (5) are obtained, y(4)(x)+ f

(2)

i.e.

[P + qo(l - x)]y’2’(x) - & ?+o.?T=

y(l)(x) - j& dy(x)

= 0

0

(4) (5)

where o is a constant. 2. Solution to the equation of motion A solution as an expansion in series is sought in the form of an analytic the whole plane. Consider the change of variable :

function

in

X X-f1

and, define :

P+qol

al =12gK

40 a,=--

40 rn2=-12gK

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February

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1986

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Jean Daniel

Hence, (4) is written as y(4)+ (CIi+ a,x)y’2’ + cX,y(l’+ a,y = 0.

(6)

Consider (6); the ordinary method used in the search for a solution to a secondorder differential equation is applied. The solution sought will be in the form : y(x) = X’ + f C,x’ +n. n=l

(7)

In this case, the indicial equation F(r) is written as : r(r-l)(r-2)r-3)

= 0.

(8)

This indicial equation has four different roots in Z ; and, whatever the root under consideration, the recurrence relation is expressed by (9) :

. Cn+4 = _ (r+n+1)C(r+n+2)~,C,+2+{~2(r+n)+~3}C,+11+~4C, (n+r+4)(n+r+3)(n+r+2)(n+r+l)

i

(9)

1

Solution (10) corresponds to root r = 3 yl(x)

=

X3_$X5_?2$3X6_

f n=O

(r+n+1)((r+n+2)~,C,+2+C(r+n)~2+a31C,+,}+a4C,

x

i

(n+r+4)(n+r+3)(n+r+2)(n+r+l)

I

X7+n

(10)

X5+n

(12)

with : co=1

c,=o

Q-g

c,=---.--

2a, + cl3 120 .

Solution (11) corresponds to r = 2 y2(x) = X2+C,x3-~x4-~(3a,C,+a,+a3)x5-

f n=O

with :

co = 1

C1 : indeterminate

c, = -$

c, =&3n,C,+a,+a,).

Solution (12) corresponds to r = 1

2%+%

y3(x) = x + c,x2 + c2x3 - ----x-E 24

n=O

(r+n+1){(r+n+2)~~C,+2+C(r+n)~2+~31C,+l)+~4C,

x i

98

4

(n+r+4)(n+r+3)(n+r+2)(n+r+l)

I Journal of the Franklin Institute Pergamon Press Ltd.

Buckling of Masts, Stacks and Columns with : C1

Solution

(13) corresponds

y&x) = 1+ ClX + c&

C, : indeterminate

and

to Y = 0 + c3x3 - 2 n=O

x

(n+l){(n+2)a,C,+2+Cna2+a31C,+,}+a,C,

X4+”

(n+4)(n+3)(n+2)(n+l)

i

(I31

I

with : Co = 1 Then, the solution

C,, C2

C, : indeterminate.

and

to the homogeneous

equation

is given by (14) :

Y(X,Q = C~Y,($ + A,Y,($ + A3y3(4 + A~~(41 IX, sin wt + J% ~0s d.

(14)

That is to say y(x, t) is expressed as a linear combination of the preceding solutions. If the initial values at x = 0 and t = 0 are considered, the general solution to (2) can always be written as :

Yb, t) = w, GYO +

x

f

ss0

0

Y(x - v, t - z)f(v, 7) dv dz.

(15)

In (15), Y(x, t) is a 4 x 4 fundamental matrix, while y(0) and f(v, z) represent, respectively, the initial conditions and the right-hand side of (2). 3. Remark 1 Note that, in the search for a solution to (6), one obtains for each solution that corresponds to the roots of F(r) a number of arbitrary parameters identical to the integer which is the difference between the greatest root and the root under consideration. In addition to these parameters, there exist six other constants of integration. The number of relations that one can write corresponds to the number of parameters. However, the problem becomes more and more complicated. The present investigation is restricted to the case where the six parameters obtained in (ll), (12) and (13) are imposed equal to zero. 4. Remark 2 The magnification factor is obtained from the integral part of (15). Since the objective is the determination of the dynamic buckling load, this integral part is ignored. Hence, the linear system associated with the equilibrium equations leads to the natural frequency or to the dynamic buckling load, when free vibration or forced vibration are considered. Vol. 321, No. 2, pp. 95-107, February Printed in Great Britain

1986

99

Jean Daniel 5. Convergence of the power series solution Consider (7) and let :

Assuming

as follows :

CI2 1 then the different terms of (7) can be estimated

ICZIG; LI

G& 2&!

I”’ ’ (4+r)!

Hence, (16) is an upper bound

for (7)

a, (1 + n)4a”r! x’ +n Y(X) = c (n+r)! ’ n=O

(16)

That is, for a fixed x :

The series defined by (16) is absolutely

IV. Damped

Vibration

The damped c

d5Y(X>t)

s ax4 at

convergent.

vibration

is considered.

+ a4Y(x, 0 -+~rP+eo(l-x)l~--ax4

In this case, (1) leads to (17) d2Yk

t)

40 aY(x,

Kg

c ay(x, t) + 40 a~Y(x~t) -= Kg at2 +K at

t)

ax I a2 - E Q Mx, t)).

(17)

The operation defined by (3) is applied to the homogeneous part of (17), and the new equation is differentiated with respect to the time-variable, as has been proposed by the author (4). That is, .. d + Y(d)(x) Y(x); C,Y@n(x) + ; Y(x) f + $ z >’ + ~((‘+q,(l-x))~~2~(x)--q,Y”‘(x)}

= 0.

(18)

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Journal of the Franklin Institute Pergamon Press Ltd.

The two equations (19) and (20) are obtained, i.e. 1 -3

c,Y(4qx}-l- g

(

T-+am-nT

Y(x) = 0 ) L=T

0.

(1%

cm

ln (20), /z is another constant of integration. It is interesting to observe that, if ~2~~/~C < 1, (19) is of the same form as the di~~rentia~ equation of the beam on elastic support, formulated in the old Winkler hypothesis* So, according to the value of o, the quantity (I- o’g,/gC) can be either positive, negative or zero. Therefore, the method has led to the formulation of ~ffere~tia~ equations with constant coefficients. From the boundary values, one obtains the ~~~en~e of the axial load, whether or not il is considered. The following elements are defined :

(23) %I. 321, No. 2, PP. 95-107, February 1986 Printed in Great Britain

102

Jean Daniel (c) l-

$

= 0

Yij(X) =

i > 1: Then, the general solution

x0 (j-l)!

Yi,(X) = & (Y[,- ljj(X)) j = 1,. . . ,4. to (17) is expressed

(24)

by (15).

V. Dynamic Buckling Load Equation (15) leads to the dynamic buckling load of the element when one considers the combined action of the horizontal loads, the uniform vertical load, the axial load and the support conditions. It suffices to use components of [9] at extremities 1 and 2 of the element under consideration, that is V, and M,, V, and M, or shear force and bending moment at 1 and 2 (see Fig. 1).

VI. Bar on Fixed

OP

Elastic Supports

Consider (15), the slope deflection method can be used to analyse the bar under multiple fixed or uneven supports. This method is used conveniently to analyse the bar on elastic supports such as guyed stacks or masts. With the above definition of an element, the system is constituted by IZelements. By using [F], the equilibrium conditions at the collar of each set of guys are expressed by the mapping _Y such that : 9 : LP + W”.

(25)

The operator A expressing (25) is a 2n x 2n matrix. The homogeneous system of linear equations associated with A has a solution if the determinant of A is equal to zero. The equation f(o) obtained being transcendental, different values will be assigned to o. If, det A = 0

(26)

then, the least value of o which satisfies (26) leads to the first fundamental frequency. If ooj and Go) represent the first fundamental frequencies when the axial load P is equal to or different from zero, the dynamic critical load P,, can be obtained by using the approximate formula proposed by Kolousek (5) or Massonnet (6). O(i) = WC1) l(

l’?t

&

(27)

>

In (27), xcl) is the safety factor in the first mode of lateral buckling.

deflection

induced

by

TThis formula is not applicable when CL+,)and Us(,) are very close to each other. A new formula, the object of another paper, is proposed by the author (7).

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Buckling of Masts, Stacks and Columns VII. Application

(4)

Equation (15) has been used to determine the first circular frequencies coo) and &j(i) of a chimney of diameter 2.7 m and height 200 m. The stack is considered to be hinged at the bottom and stayed by five sets of four guys. The vertical distance between two sets of guys is 38 m and an overhang of 10 m terminates the stack. The flexural rigidity is taken as 2,434,796.94 Tm’. The matrix A being a 10 x 10, (26) is zero for the following values.

First frequency (radians per second) Undamped motion Damped motion Undamped motion (using Kolousek (5) functions)

W(l)

W(l)

2.9805 2.9798

2.4707 2.9637

2.9783

2.9732

VIII. Operator T, and Generated Polynomials 1. Sudden disappearance of the compressive force Consider (lo), (ll), (12) (13) and (15) as equations of motion of a system. It is assumed that the first term a, suddenly disappears. This can be the case of a column sustaining a load with a mechanism capable of moving the load. Oscillation continues and the system tends to recover its equilibrium position. The following parameters are defined : a, = 0 c12= a;(j - T,d) CQ= aj(j-

T,d)

t14 = a:( j - T&I.

(28)

In (28), al,, a;, a> are real numbers, j and d are integers, and T, is the linear operator attached to the Bessel functions J,, subscript n is the integer appearing in (7). T,=l The properties

Vn.

of T, have been defined in (3). The substitution

of (28) in (7) + :

(r+n+1)[(v+n)a2+cr,]C,+l+a,C,

Cn+4

=

-

i (n+r+4)(n+r+3)(n+r+2)(n+r+l)

(29)

I

CI~= 0 in (lo), (ll), (12) and (13) =zYl(X) = x3 -

m2

+

%I

p

120

Vol. 321, No. 2, pp. 955107, February Printed in Great Britain

_

f

(r+n+l)[a,(r+n)+a,]C,+,+a,C,

.=,(n+r+4)(n+r+3)(n+r+2)(n+r+l)

x7+n

(30)

1986

103

Jean Daniel yz(x)=

x2 +

c,x3-&2+a3)x5 - .=,(n+r+4)(n+r+3)(n+r+2)(n+r+l) 1m tr+n+1)C(r+n)a2+a31C,+,+a4C,

X6+n

(31) y3(x)=

x+c,x2+c2x3-$x4- cm (r+n+1)Ctr+n)a2+a31C,+,+a4C,

X5+n

.=,(n+r+4)(n+r+3)(n+r+2)(n+r+l)

(32) y4(x) = 1 + c,x + c,x2 + c,x3 Case 1 Consider

m(n+1)Cncc2+a31C.+1+cc4C,X4+n . (33)

c n=O (n+4)@+3)(n+2)(n+l)

(28), and impose as a condition

that for a given n :

jT,,+, = d.

(34a)

This implies coefficients of C, +k = 0, k = 1,2,. . . co, which means that oscillation stops. A possible occurrence of this case would be the motion of a satellite in space, suddenly caught by the mechanical arm of a Space-Lab. Hence : n=O n=2 The following

polynomials

*

T,j=d

=z- T3j=d ..........................

are defined :

r=3*: D,(x) = D,(x) = D,(x) = x3 n=3:D,(x)=x3--x

2%+%

6

120

n > 3 : D,(x) = D,-1(x)-

n[(n-l)a,+a,]C,-,+a,C,_,

x3 +“.

n(n + l)(n + 2)(n + 3)

(34b)

r=2*: D,(x) = x2

D,(x) = D,(x) = x2 + C1x3

n = 3 : D3(x) = x2 + C,x3 - &(Lx~ +a,)~~

n > 3 : D,(x) = D,-1(x)-

(n-l1)C(n-22)a2+a3lC,-3+a4C,-4

(n_l)(n)(n+l)(n+2) Journal of

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Buckling of Masts, Stacks and Columns

D,(x) = x + c,x2

D,(x) = x

D,(x) = x +

c,x2 + &x3

n = 3: D3(x) = x+CIx2+C2x3-$x4

(n-2)C(n-3)a2+a31C,-3+a4C,-4

n > 3: D,(x) = D,-1(x)-

(n-2)(n-l)n(n+l)

Xl+n

I

.

(36)

r=O*: D,(x) = 1+ c,x

D,(x) = 1

D,(x) = 1+ c,x + c,x2

D3(x) = 1+ ClX + c,x2 +

c3x3

~~~~~~~~~~~~~~~~~~~~~~~~~~~~ n(n-3)(n-2)(n-1)

n > 3: D,(x) = D,_1(x)-

Case 2 The following new conditions are imposed. In (31), (32) and (33), the six parameters are taken as zero. Moreover LX1= 0 cl3 = -(r+n)a,

Vn

c14= ak(j- T&y). The set of conditions

(38) leads to

C4n

(-l)“-

=

air! (r+4n)!

C 4n+1 --C4nf2=C4n+3=0 Hence, (9) is reduced

(38)

n=O,l,...,

co.

to (39) : y(x) = z.

(- ly$+‘“.

If

for a given n, then D,(x) = x’ (-l)nr!an4 D4”W

Vol. 321, No. 2, pp. 95-107, Printed in Great Britain

February

=

D4,n

- l)(X)

+

(r+4n)!

Xr+4n

’

(40)

1986

10.5

Jean Daniel That is, (-1)“3!a;x3+4”

r = 3 =+-:

D4,&c) = D4~n-1~(~)+

r = 2+:

D4n(~) = D~~,~li(x)+(-1~~~~~+4" n.

(3+ 4n)!

(- l)“czi;xl+‘+n r = 1= : D&4

= D+

dx) +

(1 + 4nl! (- l)“c@

r = 0 * : D4n(x) = D4,,- I)(X) +

(4nj!

.

IX. Conclusion The preceding investigation shows the complexity of the analysis of the dynamic buckling of stacks or columns when its own weight is considered as distributed along the vertical axis of the system. One finds that it is impossible to determine a priori either the dynamic buckling load or the first natural frequency of the bar. However, the proposed method of solution has led to a differential equation similar to that of plates on elastic foundations as in the old Winkler hypothesis, when viscous stresses are taken into account. Furthermore, it has been shown that it is possible to generalize the operator T.attached to the Bessel functions J,,in order to generate new sets of algebraic polynomials. References (1) A. Vandeghen and M. Alexandre, “Vibration des grandes cheminbes en acier sous l’action

du vent”, Publications Z.A.B.S.E., Vol. 29-1, pp. 95-132, 1969. (2) G. Hirsch, H. Ruscheweyh and H. Zutt, “Failure of a 140 m High Steel Chimney Caused by Wind-excited Oscillations Transverse to the Wind Direction”, pp. 8099837, Steel Plated Structures, Dowling Harding & Frieze, Cox & Wyman, London, 1977. (3) J. Daniel, “Bessel functions and buckling of the uniform column under combined actions due to its own weight and an axial load”, J. Franklin Inst., Vol. 317, No. 3, pp. 159-169, March 1984. (4) J. Daniel, “Static and dynamic buckling analysis of guyed stacks and masts with variable inertia”, Ph.D. thesis, Concordia University, Montreal, 1982. (5) Vladimir Kolouseck, “Dynamics in Engineering Structures”, p. 385, Butterworth, London, 1973. (6) Ch. Massonnet, Bull. Cows et Lab. Constr. GCnie Civil, LiZge, Vol. 1, p. 353 (1940). (7) J. Daniel, “Bessel functions and undamped dynamic buckling of guyed stacks and masts with constant inertia under combined action due to an axial load and to their own weight uniformly distributed”, to be submitted.

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Buckling of Masts, Stacks and Columns 4ppendix

Proof of Equations (19) and (20)

Consider equation (18) : d -[{(C,Y”‘(x)+;Y(x) dt

> {(P+q,,(l-x))Y’2’(x)-q0Y(1)(x)]

+ 6

1

= 0.

(18)

Let Yl(x) and Y,(x) be defined as follows: Yl(X) = c,Y’4’(x)+ ;

Y&c) = g

Since Y(x) is independent

Y(x)

Y(x).

oft, then (18) is written as : .

=0. $[ we +Y,(x)g 1

(184

. (>

The last equation is equivalent to :

Yl(X)$ g =-Y&

(’ >

g ,

Hence,

_\y,o_w2

VW

Y,(x)

with o : angular frequency. From (18b), (19) and (20) are easily obtained: c,Y’4’(x)+

; (

1- s

~tco=~--1T

Vol. 321, No. 2, pp. 95-107, February 1986 Printed in Great Britain

>

Y(x) = 0

= 0.

(19) (20)

107