q j mechanics appl math-2004-pi-551-69 oxx

ELASTIC FLEXURALTORSIONAL BUCKLING OF FIXEDARCHES

by YONG-LIN PI and MARK ANDREW BRADFORD

(School of Civil and Environmental Engineering, University of New South Wales, Sydney,NSW 2052, Australia)

[Received 6 May 2003. Revise 22 June 2004]

Summary

This paper is concerned with the elastic flexuraltorsional buckling of laterally fixed circulararches that are subjected to uniform axial compression, and to uniform bending. The finitestrains and the energy equation for the flexuraltorsional buckling of arches have been derivedbased on an accurate orthogonal rotation matrix. Closed form solutions for the elastic flexuraltorsional buckling resistance of laterally fixed arches in uniform compression and in uniformbending have been obtained, which are quite different from those of pin-ended and simplysupported arches. The results demonstrate that the routine effective length approach, which isoften useful for determining the buckling response of straight members, is not suitable for theelastic flexuraltorsional buckling response of fixed arches.

1. Introduction

Arches that are loaded in-plane may suddenly displace laterally and twist out of their plane ofloading and fail in a flexuraltorsional buckling mode as shown in Fig. 1. The elastic flexuraltorsional buckling loads of arches that are subjected to uniform axial compression and to uniformbending play an important role in the investigation of the mechanics of the buckling response ofarches under general loading. They are also often used as the reference load (or moment) in thedesign of steel arches (1 to 3). This paper is concerned with the elastic flexuraltorsional bucklingof laterally fixed circular arches with a doubly symmetric cross-section that are subjected to uniformcompression and to uniform bending.

Elastic flexuraltorsional buckling of arches that are subjected to uniform compression or touniform bending has been investigated by a number of researchers (4 to 14). Closed form solutionsfor the elastic flexuraltorsional buckling load of pin-ended arches in uniform compression andfor the elastic flexuraltorsional buckling moment of simply supported arches in uniform bendinghave been obtained (5,8,10 to 12); however, the elastic flexuraltorsional buckling of laterally fixedarches that are subjected to uniform compression and to uniform bending does not appear to havebeen reported. It might be thought that the solutions for the elastic flexuraltorsional buckling loadof pin-ended arches in uniform compression and for the elastic flexuraltorsional buckling momentof simply supported arches in uniform bending can be extended routinely to fixed arches by usingthe effective length concept (15). The method that adopts the effective length approach has beenused in the elastic buckling analysis of columns and beams (4, 5, 15). For example, the closed formsolution for the flexural buckling load Ny of a column about its minor principal axis is given by

Ny = 2 E Iy/2,Q. Jl Mech. Appl. Math. (2004) 57 (4), 551569 c Oxford University Press 2004; all rights reserved.

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552 Y.-L. PI AND M. A. BRADFORD

Fig. 1 Geometry and loading. (a) Arch in uniform compression; (b) arch in uniform bending; (c)flexuraltorsional buckling; (d) lateral restraints (plan)

where E is Youngs modulus, Iy is the second moment of area of the cross-section, and is theeffective length of the column, related to the actual length S of the column by = kS in which k isthe effective length factor and whose value depends on the end support conditions. For a laterallypin-ended column, it is well known that k = 1 and = S; for laterally fixed column, k = 05and = S/2. Thus, the first mode flexural buckling load of a laterally fixed column in uniformcompression is equal to the second mode flexural buckling load of a laterally pin-end column inuniform compression. This method can also be used for the elastic flexuraltorsional buckling ofbeams, for which the first mode flexuraltorsional buckling moment of a laterally fixed beam inuniform bending is equal to the second mode flexural buckling moment of a laterally pin-endedbeam in uniform bending. However, because the lateral deformations primarily couple with thetwist rotational deformations during the flexuraltorsional buckling of arches, the method of usingthe concept of an effective length may be unsuitable for the elastic flexuraltorsional buckling ofarches.

An analysis of the cross-section of an arch has usually been undertaken in order to derive thestrains for determining its flexuraltorsional buckling response (6, 8 to 10). Most of these types ofanalysis can be considered to be equivalent to using a rotation matrix R that does not satisfy thedesired proper orthogonality conditions that RRT = RT R = I and that det R = +1 in the strainderivations, and so some terms in the strains that are significant for modelling the flexuraltorsionalbuckling of arches may be lost. Therefore, an accurate rotation matrix that satisfies the orthogonalityconditions needs to be sought for the strain derivations that are to be used in the buckling analysis,in order to model properly the mechanics of the buckling response.

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BUCKLING OF FIXED ARCHES 553

Fig. 2 Axes system, basis and position vectors

The purpose of this paper is to investigate analytically the elastic flexuraltorsional buckling oflaterally fixed circular arches with a doubly symmetric open thin-walled cross-section using anenergy approach that is based on an accurate rotation matrix, and to obtain the analytical solutionsfor the elastic flexuraltorsional buckling load of fixed arches in uniform compression and for theelastic flexuraltorsional buckling moment of laterally fixed arches in uniform bending.

2. Rotation and curvatures

In general, the centroidal axis os of a circular arch has an initial curvature x0 about the majorprincipal axis ox (that is, in the direction of the minor principal axis oy of the cross-section) asshown in Fig. 1. To describe the deformation of the arch, a body-attached curvilinear orthogonalaxis system oxys is defined as follows. The axis os passes through the locus of the centroids of thecross-section of the undeformed arch and the axes ox and oy coincide with the principal axes of thecross-section, as shown in Figs 1 and 2. After the deformation, the origin o displaces u, v, w to oand the cross-sections (that are assumed to remain rigid in their plane and so do not distort) rotatethrough an angle , and so the body attached curvilinear orthogonal axis system oxys moves androtates to a new position oxys as shown in Fig. 2.

A unit vector ps in the tangent direction of os, and unit vectors px and py in the direction of oxand oy form a right-handed orthonormal basis as shown in Fig. 2. The unit vectors px , py, ps areused as the fixed reference basis. They do not change with the deformation, but their directionschange from point to point along the arch axis os. In the deformed configuration, a unit vector qs isdefined along the tangent direction of the axis os of the axis system oxys, and unit vectors qxand qy are defined along the principal axes ox, oy of the rotated cross-section at o as shown

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in Fig. 2. The unit vectors qx , qy , qs also form an orthonormal basis. They attach to the arch andmove with the arch during the deformation with the vector qs being normal to the cross-section atall times.

The rotation matrix that describes the rotation from the basis vectors px , py , ps in the undeformedconfiguration to the basis vectors qx , qy , qs in the deformed configuration can be obtained in matrixform as (see the Appendix)

[qx , qy, qs] = [px , py, ps]R, (2.1)

where the rotation matrix R is given by

R = Rxx Rxy RxsRyx Ryy Rys

Rsx Rsy Rss

(2.2)

with

Rxx = (1 u2)C uvS, Rxy = (1 u2)S uvC, Rxs = u, (2.3)Ryx = (1 v2)S uvC, Ryy = (1 v2)C + uvS, Rys = v, (2.4)Rsx = uC vS, Rsy = uS vC, Rss = w, (2.5)

where C cos , S sin , u = u/(1+), v = v/(1+), w = (1+w)/(1+), v = vwx0,w = w +vx0, ( ) d( )/ds, (1+ ) =

(u)2 + (v)2 + (1 + w)2, is the longitudinal normal

strain at the centroid and = 1/(1 + w).The rotation matrix R in (2.2) belongs to a special orthogonal rotation group denoted by SO(3)

because it satisfies the proper orthogonality and unimodular conditions (16)

RRT = RT R = I and det R = +1. (2.6)

For SO(3), the curvatures and strains are invariant during rigid-body rotations.A fixed (space) right-handed rectangular coordinate system O XY Z is defined in space as shown

also in Fig. 2. The position of the undeformed and deformed arch can be defined in the axissystem O XY Z . Unit vectors PX , PY , PZ in directions O X , OY and O Z also form a right-handedorthonormal basis.

In the undeformed configuration, the position vector of the centroid o in the fixed axes O XY Z isr0 (Fig. 2), and so the unit vector ps tangential to the centroidal axis os can be expressed in termsof the position vector r0 as

ps = dr0/ds. (2.7)

The FrenetSerret formulae in terms of basis vectors px , py , ps in the undeformed configurationcan then be written as

dpids

= K0pi (i = x, y, s), (2.8)

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Mx Mx Mx Mx

(d)(c)

s

sy

yx

x

(b)(a)

Fig. 3 Curvature and bending moment conventions. (a) Positive initial curvature; (b) negative initialcurvature; (c) positive bending; (d) negative bending

where K0 is the skew-symmetric matrix for the initial curvature x0 and is given by

K0 = 0 0 00 0 x0

0 x0 0

(2.9)

and the positive and negative curvatures are defined as shown in Fig. 3.In the deformed configuration, the position vector of the centroid o in the fixed axis system

O XY Z is r as shown in Fig. 2, and so the vector qs tangential to the deformed centroidal axis oscan be expressed in terms of the position vector r of the centroid o as

qs = drds =1

1 + drds

, (2.10)

where the relationship ds = (1 + )ds is used. Because the differentiation of the position vectoris taken with respect to the deformed length s, qs is a unit vector.

The position vector r of the centroid o can be expressed as (Fig. 2)r = r0 + upx + vpy + wps . (2.11)

Substituting (2.11) into (2.10) and considering (2.7) produces

qs = 11 + drds

= 11 + [u

px + vpy + (1 + w)ps] = upx + vpy + wps . (2.12)In the deformed configuration, according to the FrenetSerret formulae, the relationship between

the derivatives of the basis vectors and the curvatures and twist can be written asdqids

= 11 +

dqids

,= Kqi (i = x, y, s), (2.13)

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where the matrix K for the curvatures and twist in the deformed configuration is given by

K = 0 s ys 0 x

y x 0

(2.14)

in which x and y are the curvatures about the unit vectors qx and qy (that is, about ox and oy),respectively, and s is the twist about the unit vector qs (that is, about os) after the deformation.

Differentiating (2.1) with respect to s yields[dqxds

dqyds

dqsds

]= [px py ps] dRds +

[dpxds

dpyds

dpsds

]R (2.15)

and substituting (2.8) and (2.13) into (2.15) and considering ds = (1+)ds leads to the curvaturesin the deformed configuration as

(1 + )K = RT dRds

+ RT K0R. (2.16)

Substituting (2.2) and (2.9) into (2.16) leads to the curvatures x and y and the twist s in thedeformed configuration expressed as

x = {uS vC w(uS vC) + [(1 u2 w2)C uvS + wC]x0}/(1 + ),y = {uC + vS w(uC + vS) [(1 u2 w2)S + uvC + wS]x0}/(1 + ),s = [ + (uv uv) + ux0]/(1 + ),

where S sin and C cos .

3. Position vectors and strains

The position vector a0 of an arbitrary point P(x, y) on the cross-section of the arch in theundeformed configuration can be expressed as (Fig. 2)

a0 = r0 + xpx + ypy, (3.1)where x and y are the coordinates of the point P in the principal axes ox and oy (Fig. 1(c)).

The position of the point P(x, y) in the deformed configuration is determined based on thefollowing two assumptions. First, the cross-sectional plane remains both plane and perpendicularto the member axis during the deformation (the EulerBernoulli hypothesis). Secondly, the totaldeformation of the point P results from two successive motions: translation and finite rotationof the cross-section, and a superimposed warping displacement along the unit vector qs in thedeformed configuration. Under these two assumptions, the position vector a of the point P1, whichis the position of the point P after the deformation, can be expressed in terms of the basis vectorsqx , qy, qs as (Fig. 2)

a = r + xqx + yqy (x, y)sqs (3.2)in which (x, y) is the normalized section warping function.

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The warping function (x, y) in (3.2) can be obtained by considering the Saint-Venant uniformtorsion problem for a prismatic bar, which results in the Laplace equation

2 = 2

x2+

2

y2= 0 (3.3)

and which may be solved by considering that the shear stresses sx and sy conjugate to the shearstrains sy and sx satisfy a traction-free boundary condition on the lateral surface. The solutionfor the warping function (x, y) can be uniquely specified by using the following three additionalorthogonality conditions:

A(x, y) d A =

A

x(x, y) d A =

Ay(x, y) d A = 0. (3.4)

In order to obtain the expressions for the finite strains, the following simplifications are made. Itis assumed that the effects of third and higher order terms of the strains are very small and can beignored and that w + u2/2 + v2/2 + w2/2. The longitudinal normal strain ss at the pointP(x, y) on the cross-section can then be obtained as

ss = 12(

a

s

a

s a0

s

a0s

)

w + 12 u2 + 12 v2 + 12 w2 x{uC + vS x0S}+ y{uS vC + x0C 12 u2x0 x0} { + ux0} + 12 (x2 + y2){ + ux0}2, (3.5)

where the approximation = 1/(1 + w) = 1/(2 + w) 1/2 is used. Similarly,

sx = 12(

a

s

a

x a0

s

a0x

)

(y +

x

)s (3.6)

and

sy = 12(

a

s

a

y a0

s

a0y

)

(x

y

)s . (3.7)

4. Buckling analysis

4.1 Stress resultants

In classical flexuraltorsional buckling analysis, the following assumptions are usually made. First,there are no lateral and twist deformations before buckling. Secondly, the conservative loads that actas well as the in-plane stress resultants are constant during the flexuraltorsional buckling. Hence,during flexuraltorsional buckling, there are no changes of in- plane deformations. Thirdly, theprebuckling strains are small so that linearized prebuckling strains can be used in the bucklinganalysis, while fourthly, the effects of prebuckling deformations on the flexuraltorsional bucklingcan be ignored. According these assumtions, prior to buckling, u = = 0 and the prebuckling

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shear strains sx , sy and shear stresses sx sy are equal to zero. Hence, only the in-plane stressresultants prior to buckling exist and they are given by

N =

Ass d A and M =

A

ss y d A,

where A is the area of the cross-section, N is the axial force, M is the bending moment aboutthe axis ox and ss is the longitudinal normal stress given by ss = Ess with E being Youngsmodulus. During the buckling, N and M remain constant.

4.2 Arches in uniform compressionA laterally fixed circular arch (u = = u = = 0 at s = 0 and s = S) with a radial loadq uniformly distributed around its centroidal axis (Fig. 1(a)) can be considered to be primarilysubjected to the uniform compression Q = N = q R, because the bending moments in the archare very small and can be ignored (M = 0). Under the uniform compression, an arch may bifurcatein a flexuraltorsional buckling mode.

Because there are no changes of in-plane displacements during flexuraltorsional buckling, thepotential energy of in-plane uniformly distributed conservative loads q due to the lateral andtorsional buckling deformations is equal to zero. The potential energy of the arch due to lateraland torsional buckling deformations can then be written as

= 12

V(E2ss + G 2sx + G 2sy) dV, (4.1)

where V indicates the volume of the arch and G is the shear modulus of elasticity.Substituting (3.5) to (3.7) and M = 0 into (4.1), and considering the potential energy given by

(4.1) is produced by the lateral and torsional buckling deformations only, leads to

= 12

S0

{E Iy

(u +

R

)2+ E Iw

( u

R

)2+ G J

( u

R

)2

+ N[(u)2 + r20

( u

R

)2]}ds, (4.2)

where R 1/x0 is the radius of the circular arch because the curvature x0 of an arch is in thenegative direction of the axis oy, S is the length of the arch, Iy is the second moment of area of thecross-section about its minor principal axis, Iw is the warping constant of the cross-section, J is theSaint-Venant torsional constant of the cross-section, and Iy , Iw and J are defined by

Iy =

Ax2 d A, Iw =

A

2 d A and J =

A

[(x

y

)2+

(y +

x

)2]d A.

From symmetry, the first mode possible buckled shapes of a laterally fixed arch can be defined by

u

uC=

C= 1

2

(1 cos s

S/2

), (4.3)

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where uC is the central lateral displacement and C is the central angle of twist during flexuraltorsional buckling. The mode shapes of (4.3) satisfy the kinematic boundary conditions u = =u = = 0 at s = 0 and s = S and the symmetric conditions u = = 0 at s = S/2.

By substituting (4.3) and N = Q into (4.2) and integrating (4.2), the potential energy due tolateral and torsional buckling deformations becomes

= 18

S2

(

S/2

)2 {E Iy

[u2C

(

S/2

)2 2uCC

R+ 3

2C

R2

(S/2

)2]

+[

G J + E Iw(

S/2

)2] [u2CR2

2uCCR

+ 2C]

u2C Q(

1 + r20

R2

)+ 2uCC Q r

20R

2Cr20 Q}

. (4.4)

According to the principle of stationary potential energy (17), the buckling equilibriumconfiguration is defined by the equations

uC= 0 and

C= 0

which leads to [k11 k12k21 k22

] {uCC

}=

{00

}, (4.5)

where

k11 =[

1 + a2f b2f Q

Ny f

(1 + r

20

R2

)]Ny f ,

k12 = k21 =(a f

b f a f b f + QNy f

r20R

Ny fMys f

)Mys f ,

k22 =(

1 + 3a2f

b2f Q

Ny fNy fNs f

)r20 Ns f ,

in which Ny f and Ns f are the first mode flexural buckling load and the torsional buckling load of afixed column of length S that is subjected to uniform axial compression respectively, and are givenby

Ny f = 2 E Iy

(S/2)2and Ns f = 1

r20

[G J +

2 E Iw(S/2)2

], (4.6)

Mys f =

r20 Ny f Ns f is the first mode flexuraltorsional buckling moment of a laterally fixed beamof length S that is subjected to uniform bending, and the parameters a f and b f are defined by

a f = S/2 R

and b f = Mys fNy f S/2 . (4.7)

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Alternatively, the potential energy given by (4.4) is stationary when = 0 (17),

uCuC +

CC = 0 (4.8)

which can be written as {uCC

}T [ k11 k12k21 k22

] {uCC

}= 0 (4.9)

which must hold for any variations {uC , C }T and leads to (4.5).Equation (4.5) has a non-trivial solution for uC and C when

k11k22 k12k21 = 0 (4.10)which can be written as the quadratic equation

A1(Qays f /Ny f )2 + A2 (Qays f /Ny f ) + A3 = 0 (4.11)

in which A1 = 1,A2 =

(2a2f 1 3a4f

)

(1 + 3a2f /b2f

)Ns f /Ny f ,

and

A3 =(

1 2a2f + 3a4f + 2a2f /b2f)

Ns f /Ny f .

The first mode flexuraltorsional buckling load of a fixed arch in uniform compression is thenobtained by solving (4.11) as

Qays fNy f

= 12

Ns fNy f

{(1 + 3a

2f

b2f

)+

(1 2a2f + 3a4f

) Ny fNs f

[1 (1 2a2f + 3a4f ) Ny fNs f

]2

(1 + 6a2f 9a4f

) 2a2fb2f

Ny fNs f

+ 6a2f

b2f+ 9a

4f

b4f

. (4.12)

By using the I-section shown in Fig. 4 with typical material properties for steel (E = 2 105MPa and G = 8 104 MPa; these values are also used throughout the paper), the first modeflexuraltorsional buckling load Qays f of fixed arches in uniform compression given by (4.12) withn = 1 are compared in Fig. 5 with the first and second mode flexuraltorsional buckling loads Qaysn(n = 1 and 2) of pin-ended arches in uniform compression given by (13)

QaysnNyn

= 12

NsnNyn

[(1 + a

2n

b2n

)+

(1 a2n

)2 NynNsn

(

1 + a2n

b2n

)2+ 2

(a2nb2n

1) (

1 a2n)2 Nyn

Nsn+ (1 a2n)4

(NynNsn

)2 , (4.13)

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Fig. 4 (a) Cross-section, and (b) dimensions

Fig. 5 Flexuraltorsional buckling of fixed arches in uniform compression

where the parameters an and bn are defined by

an = Sn R

and bn = n MysnNyn S (4.14)

in which Mysn =

r20 Nyn Nsn is the nth mode flexuraltorsional buckling moment of a simply

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supported beam of length S that is subjected to uniform bending. Here Nyn and Nsn are the nthmode flexural buckling load and the torsional buckling load of a pin-ended column of length S thatis subjected to uniform axial compression respectively, and are given by

Nyn = (n)2 E Iy

S2and Nsn = 1

r20

[G J + (n)

2 E IwS2

]. (4.15)

It is noted in Fig. 5 that the reference second mode flexural buckling load Nys2 of a pin-endedcolumn is equal to the reference first mode flexural buckling load Nys f of a fixed column (4, 5, 15).It can be seen from Fig. 5 that when the included angle is small, the difference between the firstmode buckling load Qays f of a fixed arch and the second mode buckling load Qays2 of a pin-endedarch is very small. The difference increases rapidly with an increase of the included angle .The first mode buckling load ratio Qays f /Nys f of a fixed arch decreases slowly with an increaseof the included angle and still has a substantial value when = 180, whereas the first andsecond mode buckling load ratios Qays/Nys and Qays2/Nys2 of the corresponding pin-ended archdecrease rapidly with an increase of the included angle and the first mode buckling load becomeszero and the second mode buckling load becomes very small when = 180. Hence, using thesecond mode buckling load of a pin-ended arch as the first mode buckling load of a fixed archwill significantly underestimate the flexuraltorsional buckling resistance of the fixed arch. This isquite different from columns that are subjected to uniform axial compression. The second modeflexural or torsional buckling load of a pin-ended column is equal to the first mode counterpart ofthe corresponding fixed column (that is, when = 0 in Fig. 5) (15).

The variations of the dimensionless first mode flexuraltorsional buckling load ratio Qays f /Nys fof fixed arches ( = 90) given by (4.12) with the out-of-plane slenderness ratio S/ry are shownin Fig. 6, where ry =

Iy/A is the radius of gyration of the cross-section about its minor

principal axis. The variations of dimensionless first and second mode flexuraltorsional bucklingload ratios Qays/Nys and Qays2/Nys2 of the corresponding pin-ended arches given by (4.13) withthe slenderness ratio S/ry are also shown in Fig. 6. It can be seen that the difference between thefirst mode buckling load of fixed arches and the second mode buckling load of pin-ended archesincreases with an increase of S/ry .

4.3 Arches in uniform bendingWhen an in-plane simply supported and out-of-plane fixed arch is subjected to equal and oppositemoments M , the arch is under uniform bending and the axial force N = 0. Under the uniformbending, an arch may also bifurcate in a flexuraltorsional mode. Because there are no changesof in-plane displacements during flexuraltorsional buckling, the potential energy of external in-plane end moments M due to the lateral and torsional buckling deformations is equal to zero. Thepotential energy of the arch due to flexuraltorsional buckling deformations can also be given by(4.1).

Substituting N = 0 and (3.5) to (3.7) into (4.1) and considering the potential energy given in(4.1) is produced by the lateral and torsional buckling deformations only leads to the followingexpression for the potential energy due to these deformations:

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Fig. 6 Effects of the slenderness ratio S/ry on the buckling of fixed arches in uniform compression

= S

0

12

{E Iy

(u +

R

)2+ E Iw

( u

R

)2+ G J

( u

R

)2

+M(

2u + 2

R+ u

2

R

)}ds. (4.16)

Substituting (4.3) into (4.16) and then integrating gives

= 18

S2

(

S/2

)2 {[u2C

E Iy2

(S/2)2 2uCC E IyR + 3

2C

E IyR2

(S/2

)2]

+[

G J + E Iw2

(S/2)2

] [u2CR2

2uCCR

+ 2C]

2uCC M + 32CMR

(S/2

)2+ u2C

MR

}.

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As in section 4.2, we obtain (4.10), where

k11 =(

1 + a2f b2f + a f b fM

Mys f

)Ny f ,

k12 = k21 =(a f

b f a f b f MMys f

)Mys f ,

k22 =(

1 + 3a2f

b2f+ 3a f

b fM

Mys f

)r20 Ns f .

This leads to the quadratic equation for the buckling moment Mays f

B1(Mays f /Mys f

)2 + B2 (Mays f /Mys f ) + B3 = 0 (4.17)in which B1 = 1 3a2f ,

B2 = a f b f 3a3f b f a fb f

3a3f

b fand B3 = 2a2f 3a4f

2a2fb2f

1.

Equation (4.17) has two solutions, corresponding to positive bending (Fig. 3(c)) and negativebending (Fig. 3(d)).

Typical relationships between the first mode flexuraltorsional buckling moment Mays f of alaterally fixed arch given by (4.17) and the included angle are shown in Fig. 7. The first andsecond mode flexuraltorsional buckling moment Maysn (n = 1 and 2) of a simply supported archgiven by (5)

MaysnMysn

= anbn2

an2bn

(

anbn2

an2bn

)2+ (1 a2n)2 (4.18)

are also shown in Fig. 7. It is noted in Fig. 7 that the reference second mode flexuraltorsionalbuckling moment Mys2 of a simply supported beam is equal to the reference first mode flexuraltorsional buckling moment Mys f of a laterally fixed beam (4, 5, 15). It can be seen fromFig. 7 that under positive uniform bending (Fig. 3(c)), the dimensionless buckling moment ratioMays f /Mys f of laterally fixed arches increases with an increase of the included angle , whereasthe dimensionless first and second mode buckling moment ratios Mays/Mys and Mays2/Mys2 ofsimply supported arches decreases with an increase of the included angle (the first mode bucklingmoment of simply supported arches becomes zero when = 180). The differences betweenthe buckling moments of laterally fixed arches and the second mode buckling moments of simplysupported arches are significant. This is quite different from beams that are subjected to uniformbending. Under negative uniform bending (Fig. 3(d)), however, the absolute values of the bucklingmoments of both laterally fixed arches and simply supported arches increase with an increase of theincluded angle . The buckling moments of laterally fixed arches in negative uniform bending alsodiffer from the second mode buckling moments of simply supported arches.

Typical relationships between the dimensionless first mode flexuraltorsional buckling momentratio Mays f /Mys f of laterally fixed arches ( = 90) and the slenderness ratio S/ry are shown in

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Fig. 7 Flexuraltorsional buckling of laterally fixed arches in uniform bending

Fig. 8. The relationships of the dimensionless first and second mode flexuraltorsional bucklingmoment ratios Mays/Mys and Mays2/Mys2 of the corresponding laterally pin-ended arches withthe slenderness ratio S/ry are also shown in Fig. 8. It can be seen that for positive bending, thedifference between the first mode buckling moment ratio Mays f /Mys f of laterally fixed arches andthe second mode buckling moment ratio Mays2/Mys2 of laterally pin-ended arches is significant andincreases with an increase of the slenderness ratio S/ry . For negative bending, the difference is lesssignificant.

5. Conclusions

This paper has investigated the elastic flexuraltorsional buckling of laterally fixed circulararches in uniform compression and in uniform bending. The nonlinear relationship between thedisplacements and strains during the flexuraltorsional buckling has been obtained based on anaccurate orthogonal rotation matrix. A classical energy approach for investigating the buckling oflaterally fixed arches was then formulated. Closed form solutions for the elastic flexuraltorsionalbuckling resistances of laterally fixed arches in uniform compression and in uniform bending havebeen obtained in the paper.

It was found that for arches that are subjected to uniform axial compression, when the includedangle is small, the difference between the first mode flexuraltorsional buckling load of a fixed

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Fig. 8 Effects of the slenderness ratio S/ry on the buckling of laterally fixed arches in uniform bending

arch and the second mode flexuraltorsional buckling load of a pin-ended arch is very small. Thedifference increases rapidly with an increase of the included angle and becomes very large whenthe included angle exceeds a certain value. The difference also increases with an increase of theslenderness ratio S/ry .

For a laterally fixed arch that is subjected to positive uniform bending, as the included angleincreases, the first mode buckling moment of a laterally fixed arch increases whereas the secondmode buckling moment of a laterally pin-ended arch decreases and the difference between themgrows rapidly with the included angle. The difference also grows with S/ry . For a laterally fixedarch that is subjected to negative uniform bending, as the included angle increases, the absolutevalue of both the first mode buckling moment of a laterally fixed arch and the second mode bucklingmoment of a laterally pin-ended arch increases. The absolute value of the second mode bucklingmoment of a laterally pin-ended arch is somewhat higher that of the first mode buckling moment ofa laterally fixed arch.

Acknowledgement

This work has been supported by an Australian Professorial Fellowship and a Discovery Projectawarded to the second author by the Australian Research Council.

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References

1. Standards Australia, AS4100-1998, Steel structures (SA, Sydney, Australia 1998).2. Y.-L. Pi and N. S. Trahair, Out-of-plane inelastic buckling and strength of steel arches, J. Struct.

Engng 124 (1998) 174183.3. and , Inelastic lateral buckling strength and design of steel arches, Engng Struct. 22

(2000) 9931005.4. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, 2nd edition (McGrawHill, New

York 1961).5. V. Z. Vlasov, Thin-Walled Elastic Beams, 2nd edition (Israel Program for Scientific

Translation, Jerusalem 1961).6. C. H. Yoo, Flexuraltorsional stability of curved beams, J. Engng Mech. 108 (1982) 13511369.7. SSRC, Guide to Stability Design, Criteria for Metal Structures, 4th edition (ed. T. V. Galambos;

Wiley, New York 1984).8. J. P. Papangelis and N. S. Trahair, Flexuraltorsional buckling of arches, J. Struct. Engng 113

(1987) 889906.9. Y.-B. Yang and S.-R. Kuo, Effects of curvature on stability of curved beams, ibid. 113 (1987)

821841.10. S. Rajasekaran and S. Padmanabhan, Equations of curved beams, J. Engng Mech. 115 (1989)

10941111.11. N. S. Trahair, FlexuralTorsional Buckling of Structures (E & FN Spon, London 1993).12. Y.-L. Pi, J. P. Papangelis and N. S. Trahair, Prebuckling deformations and flexuraltorsional

buckling of arches, J. Struct. Engng 121 (1995) 13131322.13. M. A. Bradford and Y.-L. Pi, Elastic flexuraltorsional buckling of discretely restrained arches,

ibid. 128 (2002) 719727.14. Y.-L. Pi and M. A. Bradford, Elastic flexuraltorsional buckling of continuously restrained

arches, Int. J. Solids Struct. 39 (2002) 22992322.15. N. S. Trahair and M. A. Bradford, The Behaviour and Design of Steel Structures to AS4100,

3rd editionAustralian (E & FN Spon, London 1998).16. R. P. Burn, Groups. A Path to Geometry (Cambridge University Press, 2001).17. G. J. Simitses, An Introduction to the Elastic Stability of Structures (PrenticeHall, Englewood

Cliffs 1976).18. J. Argyris, An excursion into large rotations, Comput. Meth. in Appl. Mech. and Engng 32

(1982) 85155.

APPENDIXRotationsThe rotation matrix R that rotates a vector pi (i = x, y, s) to a new position q j ( j = x, y, s) about a rotationvector can be expressed as (18)

R = I + sin

S() + 12

sin2(/2)(/2)2

S2(), (A.1)

where = p = x px + ypy + sps and the auxiliary matrix S() is skew-symmetric and given by

S() = 0 s ys 0 x

y x 0

. (A.2)

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The scaled rotation vector given by

= 2p tan (/2) (A.3)is used to construct the rotation matrix R.

Substituting the rotation vector of (A.3) for the rotation vector of (A.1) gives

R = I + 11 + /4 S() +

12

1(1 + /4)S

2(). (A.4)

The vector can be expressed in component form by using basis vectors px , py , ps as

= x px + ypy + sps , (A.5)and S() is given by (A.2), where x , y , s replace x , y , s , respectively.

By using (A.2) and (A.5), the components of the matrix R of (A.4) can be written as

Rxx = [1 14 (1 + cos )(2y + 2s )], Rxy = 12 (1 + cos )(s + 12xy), (A.6)Rxs = 12 (1 + cos )(y + 12xs), Ryx = 12 (1 + cos )(s + 12yx ), (A.7)

Ryy = [1 14 (1 + cos )(2s + 2x )], Rys = 12 (1 + cos )(x + 12ys), (A.8)Rsx = 12 (1 + cos )(y + 12sx ), Rsy = 12 (1 + cos )(x + 12ys), (A.9)Rss = [1 14 (1 + cos )(2x + 2y)]. (A.10)

It can be derived from (A.5) that

2x + 2y + 2s = 4 tan2

2= 4(1 cos )

1 + cos (A.11)

and from (2.1) and (A.6) to (A.10), the unit vector qs can then be expressed in terms of the basis vectors px ,py , ps as

qs = Rxspx + Ryspy + Rssps = 12 (1 + cos )(y + 12xs)px+ 12 (1 + cos )(x + 12ys)py + [1 14 (1 + cos )(2x + 2y)]ps . (A.12)

On the other hand, the unit vector qs can also be expressed by (2.12). Comparison of (2.12) and (A.12)leads to

12 (1 + cos )(y + 12xs) = u, (A.13)

12 (1 + cos )(x + 12ys) = v, (A.14)

1 14 (1 + cos )(2x + 2y) = w. (A.15)In order to express the rotation matrix in terms of general displacements, in addition to the displacements

u, v, w, a fourth displacement parameter s associated with the twist rotation of the cross-section isintroduced such that

s = 2 tan (/2). (A.16)Substituting (A.16) into (A.11) and then solving (A.11) and (A.15) simultaneously gives

cos = 1 + (1 + cos )(1 + w)/2. (A.17)

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Substituting (A.16) and (A.17) into (A.13) and (A.14) yields

x = 2{u tan (/2) v} and y = 2 {u + v tan (/2)} , (A.18)

where = 1/(1 + w). Substituting (A.16) to (A.18) into (A.6) to (A.10) leads to the following relationshipbetween basis vectors px , py , ps in the undeformed configuration and basis vectors qx , qy , qs in the deformedconfiguration being stated in tensor form as

qi = R ji p j , i, j = x, y, s, (A.19)or in a matrix form as

[qx , qy , qs ] = [px , py , ps ]R, (A.20)where the matrix R is given by (2.2) with components given by (2.3) to (2.5). These components are expressedin terms of the four conventional displacement parameters u, v, w, and their derivatives, and they becomeinfinite only when 1 + w = 0 and/or 1 + = 0. These situations cannot occur during the deformation of areal structure.

The rotation matrix R of (2.2) belongs to the special orthogonal rotation group SO(3) because it satisfiesthe orthogonality conditions of (2.6). For the SO(3) group, the invariant requirement needed for the rigid bodyrotation is satisfied, namely, the curvatures and strains are invariant during the rigid body rotations.

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