19980505

Applied Mathematics and Mechanics (English Edition, Vol. 19, No. 5, May 1998)

Published by SU, Shanghai, China

THE APPROXIMATE ANALYTICAL SOLUTION FOR THE

BUCKLING LOADS OF A THIN-WALLED BOX COLUMN

WITH VARIABLE CROSS-SECTION

Xie Yongjiu ( i~ Jg ) ~ Ning Qinghai (~r Chen Minglun (I~,~)~1'~) 2

(Received Jan. 25, 1997; Communicated by Liu Renhuai)

Abstract For a thin-walled box column with variable cross-section, the three governing

equations for" torsional-flexural buckling are ordinary differential equations of the

second or fourth order with variable coefficients, so it is very diJ'ficult to soh,e them by

means of an analytic method.'In this paper, polynomials are ,tsed to approximate the

geometric properties of cross-section and certain coefficients of the differential

equations. Based on the energy principh, and the Galerkin's method, the approxhnate

formulas for calcutathTg the flexural and torsional buckling loads of this kind of

columns are developed respectively., and numerical examples are used to veriJ)' the

correctness of the sohuions obtained. The resuhs calculated in this paper provide the

basis for demonstrating the stability of thin-walled box cohonns with variable cross-

section. This paper is of practical value.

Key words thin-walled box column with variable cross-section, torsional- flext.r:d buckling, approximate solutions for buckling loads

I. In t roduct ion

Thin-walled box columns with variable cross-section are extensively used as compression

members for high bridge piers, water and television towers and other similar structures. At

present, however, few papers dealing with the critical loads of torsional-flexural buckling for

tills kind of structures can be seen. In this paper, by means of the energy principle and the

Galerkin's method, the approximate expressions for calculating the critical loads of flexural

and torsional buckling are developed respectively. An applicable computer program is worked

out, and some concrete numerical examples are given. And a uniform straight bar with an

exact solution can be regardedas a special case of the problems discussed here. By comparison

with the exact solution, it can be seen that the results calculated in this paper are considerably accurate. This demonstrates that the expressions given and the program compiled in this paper are correct.

The Highway Planning, Survey and Design Institute under the Sichuan Provincial

Department of Bridge and Structure Engineering, Southwest Jiaotong University, Chendu 610031, P. R. China

-" Chongqing Jiaotong Institute, Chongqing 400016, P. R. China

445

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduction

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

446 Xie Yongjiu, Ning Qinghai and Chen Minglun

Department of Communications has asked the authors to write this paper. The expressions

developed and the program prepared have been used for demonstrating the stability of the

spandrel columns on the Wanxian Yangtze River Arch Bridge, which has the longest span of

the reinforced concrete arch bridges in the world. Of course, they can also be used to

demonstrate the stability of other similar structures. In consideration of the demands of the

practical engineering structure, the boundary conditions involved in this paper are only that

the Column is fixed at the bottom, but supporte d by a spring for flexural buckling and free for

torsional buckling at the top. However, it is not difficult to extend the above boundary

conditions to any case, in the way as discussed in this paper.

I I . Govern ing Equat ions

Consider a thin-walled box column with a doubly symmetric uniform cross-section. Let

the x and y axes coincide with the principal coordinate axes of the cross-section, the - axis bc

the vertical one, and the origin be at the centre of the top cross-section. The three governing equations are (~' -~1

Ej , , ' ( , ) z , , ( , ) = o ]

E J , + = 0

E J j , arc(z) _ ( GJa - pr2)~'(z) = 0 j _j,

(2.1)

where ~ and I1 are respectively the displacements of tt~e centre at the section z in the x-and y-

direction; 0 is the section angle of twist about the z axis; P is the axial force acting on section

z; J,, and Jr are the second moments of area; Jp is the polar moment of inertia.; Ja is the torsional constant, and J,~ the warping one; E is the Young's modulus, G the shear modulus; and r is the radius of gyration. By definition

r 2 =. J~ + Jr (2.2) F

where F is the area of the cross-section.

It should be noted that for concrete, the modulus E should be replaced with the converted

Young's modulus E1 which is equal to E/ (1 - ~2), where ~t is a Poisson's ratio; and the shear modulus G equals to 0.43E (See Reference 3).

In Eq. (2.t) the first two are the Euler equations, the last is the governing equation for

torsional buckling. When the centroid and the shear centre of the cross-section coincide, as

they do in the case considered here, the three "simultaneous equations in Eq. (2:1) are

uncoupled, and thus they can be solved respectively. And only the smallest of the three buckling loads is of practical significance.

For a bar with variable cross-section, all the geometric constants in Eq. (2.1) are functions

of the vertical coordinate z. If the self-weight of the structure is considered, the axial lbrce P is

also a function of z. In this case, Eq. (2.1) becomes three ordinary differential equations with

variable coefficients, rather difficult of solution by using the analytic method. In this paper,

some approximations are introduced so that more .accurate solutions can be easily Obtained for

the governing equations of the thin-walled box column with variable cross-section.

Abstract


1. Introduction




au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293


Buckling Loads of a Thin-Walled Box Column 447

III. Approx imate Ana lys i s fo r the Eu ler Buck l ing Loads o f a Co lumn

Rest ra ined hy a Spr ing

Consider a thin-walled bo.,~ t;olumn with variable cross-section, as shown in Fig. I (a). The

column has a height of L and the same wall thickness. It is fixed at the bottom and restrained by a spring at the too under the action of a compressive axial force P~. The spring stiffiaess is k; in the range of [0; + co]. The computation model of the structure is shown in Fig. I (b), and the cross-section of the column in Fig. 1 (c).

~ 2

b~ t' l

(a) (b) (r Fig. 1

Let the cross-section number be 1 at the top of the colmnn, and 2 at the bottom. In the following expressions, the notations with a subscript i(i = 1, 2) are used to denote the values of the geometric properties of cross-section and the relevant physical quantities at the corresponding end of the column. Now polynomials of z are formulated to approximate the values of timse properties and quantities at section z. These polynomials are exact at the two ends and approximate within the bar. They ~rre

the area of cross-section F (z ) = F I ( I+ L~--z ) (3.1a)

the self-weight per unit lenth q (z )= q l ( l+~z) (3.1b)

where fl = F2/Fx - 1 (3.1c)

the axial force P(z ) = P,, + o qlr 1+ d~ = P~. + qlz 1 + (3.2)

= ( al ~3 (3.3a) the second moment of area J ( z ) J1 1 + -~z)

where al = ~-)'-2/Ji - 1 (3.3b)

It should be noted that Eq. (3.3) is applicable for both J , , (z ) and Jr(z), but the former is firstly considered here.

Consider the idealized model of the structure, as shown in Fig. 1 (b). For brevity, let

3~ a = ~-~, b = 3a = ~ (3.4)

Choose a trial function as

Abstract


1. Introduction




au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293



n(z ) = dtEsinaz + (1 - Ck)sinbz] (3.5)

which represents an assumed fiexural shape fo the column. Based on the equation above, the

real horizontal displacement of the column top can be expressed as

= ,~(L) = ~C ~

where C is an undetermined parameter to be determined by the stat ionary condition of the

critical load, but its changing range can be discussed as Ibllows. On the one hand, since the

displacement 6 is a real physical quantity t'or any k >I 0, there must be C>0. On the other

hand, if the top end of the column is simply supported, i. e. when k = 0r then ~ = 0. And

there must also be C< 1. Hence the undetermined parameter C must satisfy 0


3re 2 - 12)

9 1 = fl__ 3 1 Dz3 4 ~--2+ 2(~-7 ~)

1 (11) ( 1 1 1 1 = y -, + + )Ol +-2-(

54 13_____552 _ 27(4-52 - ~)a~ D32 = - -~a 1 - 2rc2al

1 (9 .~2 ) (3 ~2) D33 = 40-~- +27 - a1+27 ~-+ a~+

previously stated; the spring constant k t> 0 and

3( 27 + 97r2-4)a: ~4

(3.10)

As the undeternained pa,'ameter 0 < C < 1, so that 0 < C k < 1. From Eq. (3.7) it is concluded that 0 < u< 1. Of all thc possible values taken for C ~. or u which satisfy the above conditions, the correct value is that

dP~, which makes the critical load Pv a minimum. Therefore, we have du = O, i. e.

t t

P'v = { DI[ PoD3 - GoD2 - 2K0(1 - u)] - D'I[ PoD 3 -GoD2

+ Ko(1 - u)2]} /D~ = 0 (3.11)

Eq. (3.9) gives rise to s

D1 = 9u ] t

D2 = Dz~ + 2Dz3 u t

D3 D32 + 2D33 u J

(3.12)

Substituting Eqs. (3.9) and (3.12) into Eq. (3.11), we obtain

QI~ 2+Q2u+Q3=O where

(3.13)

13 Q1 = - 2(PoD32 - GoD~) + 9Ko /

Q2 = P0(D33 - 9D31) - Go(Dz3 - 9D21) - 8KoI (3.14)

Q3 = l (PoD32- GoO~) -Ko

Solving Eq. (3,13) yields

- Q2 + ~ Q~ - 4Q103 (3.15) u = 2Qt

Just as stated above, only when 0 < u < 1, the solution u given by Eq. (3.15) is of practical significance.

After _obtaining u from Eq. (3.15), by use of Eqs. (3:8), (3.9) and (3.10) the Euler buckling load Pr = Pv in the y-direction can be obtained. In like manner, the Euler buckling load Px in the x-direction can be obtained too.

IV. Approx imate Ana lys is for Tors iona l Buck l ing Load

Consider a thin-walled box column with variable cross-section and set the coordinate system as shown in Fig. 2. The cross-section of the column is the same as shown in Fig. 1 (c)

Abstract


1. Introduction




au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293


450 Xie Yongjiu. Ning Qinghai and Chen Minglun

As stated in Section II, the accurate expression of the go~,erning equation for torsional buckling of the column is

Ef~(z) Jp(z) O~ [ GJd(z)0" + [r(z)]2P(z)O" = 0 -

or written in brief as

a4(z)O. Iv - a2(z)0" + b(z)O" = 0 (4.1b)

In order to solve the ordinary differential equation with variable coefficients, polynomials of z are introduced again to approximate the coefficients in the equation above. Let

( o#)' a4(z) = a4o 1 + (4.2a)

(4. la)

where Fig. 2

a,o = e jo , / ( - Ja,),a4 ='~ J~,4t Jvz Ja= - 1 (4.2b.e)

The coefficient a2(z)is approximately expressed as

( Tz) a2(z) = a2o 1 + a2 (4;3a) where

a2o = GJ~,, a2 = ~ Jd,/J~, -1

For the radius of gyration r(z), the following interpolation function is introduced:

(4.3b,e)

(4.4a)

where

f12 = r2/rt - 1 (4.4b)

On taking note of Eq. (3.2), the expression for b(z) in Eq. (4.1) becomes

For simplicity of writing, in the following, b(z) remains to be used to denote the polynomial of z given in the equation above. In Eq. (4.2) through (4.5), as stated in Section III, the gcomctric constants with an additional subscript i(i = 1, 2) denote the values of the corresponding geometric properties of cross-secti0n at each end of the colu.nn.

After introducing these interpolation functions, the approximate expression of the governing equation becomes

( a4 ~5017 (1+ a_..2 )30,+ b(z)0" =0 (4.6) a4o 1 + --~z ] - a2o L - -

Abstract


1. Introduction




au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293


Buckling Loads of ~:'Thin-Walled Box Column 451

Since all the coefficients of the derivatives of each order for 0 are polynomials of z, based on

Oalerkin's method, the above equation can be approximately solved. For torsional buckling of a column, the main train of thought of Galerkin's method m is as

follows. Suppose that at the moment stability is lacking for the column, its angle of twist is

O(z) = 2 Co~i(z) (4.7) i= l

where Ci is an undetermined coefficient, and Xi(z) is a displacement function which satisfies the boundary conditions of displacement. Substituting Eq. (4.7) into Eq. (4.6) will lead to .a result which does not. in general, equal zero, because the assumed :O(z) is not necessarily the accurate solution. The result is denoted by 8,,,, known as a conversion IoadHL In this case. the internal force in the bar and the conversion load ~,,, are in a state of neutral equilibrium. The work done by the conversion load 6',~ on any arbitrary small virtual displacement 0j equals zero. Assume that the virtual displacements chosen are

Oj = XJ(z) (J = 1 ,2 , " ' , ,0 According to the principle of virtual displacement, we have

f28,,Z/(z)dz = 0 ( j = 1 ,2 , ' " ,n ) (4.8)

From the Viewpoint of mathematics, the equation above is a system of homogeneous linear equations with respect to the undetermined coefficient Ci(i = 1 ,2 , ' " ,n ) . Eq. (4.8) has a nonzero solution if and only if the determinant of the coefficient matrix of Ci equals to zero. Based on this condition, the approximate value of the torsional buckling load can be obtained.

As shown in Fig. 2, the support condition of the column is free at the top and fixed at the bottom. Suppose that the coordinate system set twists along with the cross-section of the column top. In this ca~e, the boundary conditions of displacement are 0(0) = O'(L) = O. Assume at the moment the stability of the column is lost, the angle of twist is

O(z) = Clsinaz + C2sinbz (4.9) where the constants a and b are identical With the ones given in Eq. (3.4). It is obvious that

each of the displacement functions (both sinaz and sinbz) satisfies the boundary conditions of

displacement. Substituting Eq, (4.9) into the governing Eq. (4.6) yields the conversion 10ad

6,, = a40 1 +.--~z] (Cla4sinaz + C2b4sinbz)

Ct 4 ~3 + a~ 1 + --'~z) (Cta2sinaz + C2b2sinbz)

- b(z)(Cla2sinaz + C2b2sinbz)

Multiplying the above expression by the virtual displacements sinaz and sinbz respectively, integrating over the interval [O ,L ] and letting the results vanish, we can get a system of homogeneous linear equations

T21 + Q21P~ T= + Oz~P~ J c2

where

Abstract


1. Introduction




au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293



r~a2L[[(1+/32)3 1] 6(/32)2(2 /32)}, Qn =- 6-"--~2 ' - + ,"~, +

1 r12f12 ( R 021 = ~ Q12 = 1-g~'- + 5/32)

r~b2Z[F (1+ 6( _fl--Z2 / 2 + Qzz = 6/32 I. L" f12) 3 -1 ] + ~3~/-- (2 /32)}

(4 .11)

T# is the sum of three terms:

T~ = Z~/+ Bi/+ W 0. ( i = 1,2; j = 1,2) (4.12)

where Ai/ and BO. are related to the results of integration of the first and second terms. respectively, while W i/ is a modifier terrn to take account of the self-weight of the structure. In order to simplify the expressions in the following, let

;tr2 = 7t ' /~2 ' 7t'4 = ;R ' /~4 l d2 = I+ a2, d4 = I +a4 I (4.13)

fl'z = fl + fl=, f12-= fl+4/32J

Then the constants in Eq. (4.12) can be expressed as

a4oa4L[ 1 6 I ) 6 } "i'~(d4 1)+ 5(-~--~4 [ 4 4 1) - 12rc4Z(d42 + +

- - - re4 ( d4 + 1) 48] An a4

1 5a4oa4L[ 1) 6 A:I = Am = - 8a, k~-J4 [rr~(5d~ + 3) - 3;r42(17d42 + 15) + 192]

f l 5(1~ ~ A99- a4~ -~(d6-a4 1)+ "2"k~4' [(37t'4)4(dl + 1)-12(3f f4)2(dl + I )+48]}

and (4.14)

Bn - 2a2 {~- (d2 - 1) + - - ~ - L 2k 2 + 1) -- 4}

1 3a~a2L( 1 I 4 B=, = ~-B,= =- ~ ~-~2! [Tr22(5d~ +3) - 16]

a2ob2L[ 1 3 1 4 B, - ~ l.-~-(d 4 - 1 )+ (~'~2)[(3zr2)2(d~+ 1) -4 ]}

(4.15)

and

: 10( 6 1)] _ + ; 111

: 2)} +s + 5-+ 1

Abstract


1. Introduction




au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293



10( ~-~2 1)] 3 (~2- 1)] _-- - - _ - - + /--42/-42[ ~ --

~ 1) {9zr2 + 2)} +~,z~-g-+ +~ 2

Let the determinant of the coefficient of CI and C._, in the system of honloge,lous lineal" Eqs. (4.10) vanish. Then we obtain

AP 2, + BP2~ + C = 0 (4.17)

which is a quadratic equation with respect to the concentratedload Pv acting at the top of the column. The approximate value for the torsional buckling load P,, is the minimum root ol" the above equation. Thus,

9

2A

In the two equations above,

A = QnQ22- Q12Q21 ]

B = (QnTz~ + Q22TI1) - (Q12T21 + QzlT12)I (4.19)

C = Ttt T22 - T12 T2x

For a thin-walled box column with uniform cross-section, there exists an exact solution of the torsional buckling load. In order to make comparison with the exact solution, the approximate formula of the torsional buckling load for the column with uniform cross-section will be derived as follows, using the method given in this paper.

For a bar with uniform cross-section, based on the expressions (3.1c), (4.2c), (4.3c) and (4.4b),

f l = a4 = a2 = f12 = 0 (4.20a) If the self-weight is neglected,

Let

ql = 0 (4.20b)

r = r 1,ct2a = GJ d ]

a4o = EJ, oJp/(J~ - Ja) I (4.21)

Substituting the above values into Eqs. (4.11) and (4.14) through (4.16), with their lilnit values calculated if necessary, and considering Eq. (4.13), we obtain

1 1 2 2L ~1 1 4 Qn = -~-Q= =-~' r a An = A22 = , "~a40a L 1 1 2 ,Bn = --c-B= = ~'a~ L

(4.22a)

(4.22b) Qq = Aq = B~ = 0 ( i # j )

and

Wq = 0 ( i = 1 ,2 ; j = 1,2) (4.22e)

On substituting Eqs. (4.22b, c) into Eq. (4.12) and considering the three equalities on the leR

Abstract


1. Introduction




au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293



side of Eq. (4.22a), Eq. (4.19) is simplified as

a = l B = 18Qn(5An + Bit)

C = 9(A~ + Bn)(9An + Bn)

(4.23)

The critical load P,~ is obtained again by using Eq. (4.18). In fact, a bar with uniform cross-

section is only a special case of the problem discussed in this paper.

V. Example and Er ror Ana lys i s

5.1 Example Consider a concrete column, whose boundary condit ions for Euler buckl ing are as shown

in Fig. I (a) and for torsional buckling as shown in Fig. 2, and whose cross-sections at the two

ends are as shown in Fig. I (c). Take ht = h2 = 2.4m, bt = 1.3m, b2= 2 .5m, t=0.1m,

L = 60.1m, E = 3 .0x 107kPa, /z = 1/6 , the weight pet" unit volume 7 = 26kN/m 3, and

the stiffness of the support spring k = lkN/cm. The Euler buckling loads obtained by Eq. (3.8)

in this paper are Pr = 1.8021 x 104kN and P~, = 2.19318 x 104kN, and the torsional buckling

load by Eq. (4.18) is Pd = 7..36959 x 107kN. Only the smallest of the three buckling loads is

the real buckl ing load. The critical load is therefore P,r = Pr = 1.8021 x 104kN. A straight uniform bar with an exact solution can be regarded as a Special case of the

problems discussed in" this paper. As few expressions of the exact solution have been seen for

Table i Compar i son of the Resu l ts of This Paper with Exact So lu t ions for an Un i form

Bar

No. Case Conditions* for calculation Exact solution Rcsult of this paper Error

|

|

|

t P,

L [

I P,

/,, I

- L 7 . . . . ~ .

P,

IP,

Free at the top

k=0, F=0

Simply supported at the top

k= **,Y = 0

Free at the top

k -- 0, y = 26kN/m },

b = 1.3m, h = 2.4ra,

t =0.1m

Spring-supported at the top k = IkN/em,7 = 0,

The others are the same as i,1 case |

~2E] P,, = 4L 2

20.19 g] P,, = L 2

P,, = 4704.5kN

P,. = 9856.4kN

~2 EJ P , ,= 4L 2

20 .~E J P~, = L 2

P~, = 4723.3kN

P~, = 9867.6kN

0.0%

0.2%

0.44%

0.11%

*(1) The column is fixed at the bottom, but the conditions restrained at the top are given out in the

table.

(2) Apart from case | self-weight is neglected.

Abstract


1. Introduction




au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293



such problems, in the following case, a uniform bar will be mainly used for comparison so as

to determine the errors of the solutions given in this paper.

5.2 Compar i son of the so lut ions for Eu le r buck l ing load

A uniform bar has the same cross-sectional dimensions at its two ends. Use the same

symbols with the subscript i(i = I, 2) removed to denote the corresponding dimensions and the

geometric properties of cross-section and the relevant pl!ysical quantities for the bar. But now

the lesser o f EJ,, and EJ r should be taken as the flexural stiffness E J, the lesser of the Eulcr

buckling loads could be obtained. For the straight uniform bars with exact solutions, the

comparison of the results calculated is shown in Table 1. From the table, it can be seen that

the results obtained in this paper are considerably accurate.

For a fixed-end column shown in Fig. 2, when the stiffness E J (z ) of the ba," varies

linearly with the vertical coordinate z, an exact solution has been given in Reference 5. In the

example, ht =h2, so E J r (z ) is a linear function of z. If we let the specific weight 7 = 0and the

stiffness of the supporting spring at the top k = 0, then the conditions for calculation turn into

the same as in Reference 5. The result obtained in this paper is P~ = !.75513 x 104kN, while

the exact soluiion is P,, = 1.68557 x 104kN. The error is only 4.1% , meeting the accuracy

requirements in civil engineering.

5.3 Compar i son of the so lut ions for to rs iona l buck l ing load

In the example, the self-weight of the column = 1.3438 x 103kN, which is less than 0.018%

of the torsional buckling load P~,, so its effect is negligible. For a uniform bar with a doubly

.symmetric thin-walled closed cross-section and subjucted to axial compression, if self-weight is

neglected, the torsional buckling load can be written as tal

E J~ 2 + C,.Jd lJ (5.1a) P,~ = r2 t

where

u = 1 - Ja / Jp , t = 1 +

For a column free at one end fixed at the other end,

EJeo. 2 ~-~-px (5.1h, e)

A = ~-~ (5.1d)

In Eq. (5.1), the geometric and physical constants are the same as stated in Section I.

By means of Eq. (5.1), it can be calculated that P,o I = 5.9327 x 106kN when the cross-

section has a height h = 2.4m, and a width b = 1.3m. with the other Constants being the same as

in the example, and P% = 9.4579 x 106kN when h =2.4m and b =2.5m. The former cross-

section is identical to that of the columnat the top in the example, and the latter one to that

at the bottom. P,, of the example lies between P%. and P '2 ' and is a little less than

0.5(P,, 1 + P,02)[ = 7.6953 x 106kN]. This indicates that the result obtained in this paper is reasonable.

If we take h=2.4m and b = 1.3m, to go further and say, the critical load calculated by

Eqs. (4.22), (4.23) and (4.18) is P,, = 5.9357 x 106kN, with an error of 0.05% only, in

comparison with P,q. This indicates that the expressions given in this paper are considerably

accurate.

Abstract


1. Introduction




au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293



Also., the authors have made a "great deal of computation with the geometric dimensions

and the wall thickness of the structure being changed. The results obtained are rather accurate too. This indicates that both the formulas given and the program prepared in this paper are all

correct. Therefore, the results calculated in the way as shown in this paper can taken as one of

the bases for demonstrating the stability of thin.-walled box column, with variable section.

VI- Concluding Remarks

For a thin-walled box column with variable cross-section, the three governing equations for torsional-flexural buckling are ordinary differential equations ot" the second or fourth order with variable coefficients, therefore, it is very difficult to solve them by using all analytic method. In this paper, polynomials are used to approximate the geometric properties of the cross-section, the relevant physical quantities and each coefficient of the governing differential equation for torsional buckling, so that the governing equations are turned i'nto differential equations which can be solved with approximation methods. Based on the energy principle and Galerkin's method, the approximate analytical solutions for tile flexural and torsional buckling loads of this kind of column are developed respectively. An applicable program is prepared and the calculations are run in a computer. Concrete numerical examples are used for verifying the correctness of the solutions given in this paper. The distinguishing features are: the

computer program is short and easy to compile; the time for arithmetic operation is very short: the results obtained is i'ather accurate; but the derivation of formulas is somewhat lengthy and complicated. The results calculated in this paper provide the basis for demonstrating the st,ability of thin-walled box column, with variable cross-section. This paper is of practical value.

Acknowledgement The authors are sincerely grateful to Senior Engineer Xie Bangzhu, chief engineer of the Highway Planning, Survery and Design Institute under the Sichuan Provincial Department of Colnmunications, China, for his generous support to this paper.

References

[1] Bao Shihua, et al., Structural Mechanics of Thin-Walled Bars, China Construction Industry Publishing House, Beijing (1991). (in Chinese)

[2] N. W. Murray, hTtroduction to the Theory of Thin-Walled Structures, Oxford University (1984), 172~177.

[3] Highway Planning and Survey Institute, Ministry of Communications, Highway Bridge and Cuh,ert Design Specifications, People's Communications Publishing House, Beijing (1989), 118. (in Chinese)

[4] A. Gjelsvik, TheTheory of Thin Walled Bars, John Wiley &Sons, Inc. (1981), 185~189. [5] Li Shaomin, The Theory oJ'Stability, People's Communications Publishing House, Beijing

(1989), 26~29, 44~45, 74~77. (in Chifiese )

Abstract


1. Introduction




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19980505

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