saito 2008 engineering-structures

8/14/2019 Saito 2008 Engineering-Structures

1/16

Engineering Structures 30 (2008) 12241239www.elsevier.com/locate/engstruct

Post-buckling behaviour of prestressed steel stayed columns

Daisuke Saito , M. Ahmer Wadee Department of Civil and Environmental Engineering, Imperial College of Science, Technology & Medicine, London SW7 2AZ, UK

Received 14 May 2007; received in revised form 2 July 2007; accepted 3 July 2007Available online 31 August 2007

Abstract

A steel column that is reinforced by prestressed stays generally has an increased strength in axial compression. A geometrically nonlinearmodel accounting for the post-buckling behaviour of the stayed column is formulated using the RayleighRitz method and then validated usingthe nite element method. It is found that the post-buckling behaviour is strongly linked to the level of the initial prestress. As the prestress isincreased, the following different levels of the responses can be observed in sequence: initial Euler buckling that subsequently restabilizes strongly,the critical load increasing with a post-buckling path that is either stable or unstable, an upper limit for the critical load where the post-bucklingis unstable after an initially rather at response. These ndings are important for designers aiming to achieve safer and more efcient designs forthis structural component.c 2007 Elsevier Ltd. All rights reserved.

Keywords: Cable-supported structure; Nonlinear buckling; RayleighRitz method; Analytical modelling

1. Introduction

A prestressed steel stayed column ( Fig. 1) is a structuralcomponent that is reinforced by either cable stays or rods suchthat its strength is increased in axial compression. Ordinarycolumns have a propensity to buckle under axial compressionprimarily due to their characteristic of being slender. To counterthis, prestressed steel stayed columns are equipped with pre-tensioned stay systems; these restrain the column bucklingdisplacement through the horizontal crossarm placed at someintermediate distance from the column ends. Consequently,this additional system prevents the principal movement duringconventional buckling and potentially provides a considerableincrease in axial strength.

An application of this column type can be found whereslender supports or towers are required, for example, it wasused as a temporary support during the erection phase of themain stage of the Rock in Rio III stadium in Rio de Janeiro,Brazil [1,2], (see Fig. 2).

In this project, it was required to support the large roof structures as high as 36 m above ground level so that

Corresponding author. E-mail address: [email protected] (D. Saito).

completion could be achieved within a limited time constraint.Conventionally, supporting the large roof structure at thisheight would have required a massive and complicated shoringsystem with a commensurate time penalty. To counter this theengineers decided to adopt the stayed column as the shoringsystem. Owing to its structural simplicity and superiority inresisting axial loads, this choice allowed the engineers to savesignicant time in the construction process.

In addition to these practical uses, a number of researchworks on stayed columns have existed since the 1960s,such as those evaluating critical buckling loads [ 38],imperfection sensitivity studies [9,10], and examining thecolumns maximum axial strength [ 11,12].

Despite this, as far as we are aware, the post-buckling

response has not been investigated satisfactorily. Thisinformation is crucial to make the design safer and moreefcient; stability in the post-buckling range implies that thedesign load could potentially be set higher than the criticalbuckling load; conversely instability in the post-buckling rangemeans the design load should be reduced in order to ensuresafety and the potentiality of the structure being sensitiveto imperfections [ 13]. In the current study, the post-bucklingresponse was investigated by developing analytical modelsusing energy methods, the results of which were validated bythe nite element method (FEM).

0141-0296/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2007.07.012
http://www.elsevier.com/locate/engstructmailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2007.07.012http://dx.doi.org/10.1016/j.engstruct.2007.07.012mailto:[email protected]://www.elsevier.com/locate/engstruct


2/16

D. Saito, M.A. Wadee / Engineering Structures 30 (2008) 12241239 1225

Fig. 1. Principle of the prestress steel stayed column: stays are pre-tensionedto provide lateral restraint against overall buckling.

1.1. Methodology

In the current work, a single-crossarm stayed column, whichis the simplest type, shown in Fig. 3, was modelled. It is knownfrom previous work that investigating the stayed column withan analytical procedure inevitably involves mathematically

sophisticated formulations, therefore modelling the simpleststructure is denitely a suitable rst step to revealing its post-buckling response. Moreover, the majority of the literaturedeals with this single-crossarm type; hence validation andcomparisons with previous research is possible.

In order to formulate the model, the total potential energyprinciple was applied in conjunction with the RayleighRitzmethod [ 13]. The total potential energy V for the prestressedstayed column was developed as a multiple-degree-of-freedom

(MDOF) system. A set of algebraic equilibrium equations wasderived from minimizing V using the symbolic computationsoftware M APLE [14]. The structural response was revealed bythis process, and was subsequently validated by the FEM usingthe well-established code ABAQUS [15]. For the analyticalmodelling, the following assumptions were made.

(1) The column is simply supported.(2) The connections between the stays and the column, and

between the stays and the crossarms, are ideal hinges. Theconnections between the crossarm and the column are rigid.

(3) The column is centrally loaded and perfectly straight,i.e. imperfections are not taken into account in the analysis.

(4) The axial deformation of the crossarm and the bendingdeformation of the stays are both ignored.

(5) The stay goes slack the instant it goes into compression;hence it does not carry any stresses in compression.

(6) The analysis is purely elastic; hence, the stressstrainrelationship is completely linear apart from the stayslackening.

(7) Changes in geometries from applying the prestress areignored, i.e. the initial conguration is kept after theintroduction of the initial prestress.

Changes in geometries from the prestress do not yieldsignicant effects unless the initial prestress has the same levelas the Euler load of the column. As this level of prestress leadsto a considerable amount of compressive force in the column,which signicantly diminishes the axial buckling resistance,this situation is considered to be impractical.

2. Model formulation

In this section, the MDOF system is developed byconsidering, in turn, the displacements of each component andthe geometrical changes after applying the prestress. This leadsto the total potential energy function.

2.1. Displacement functions for the column

Two different buckling mode shapes for the column areconsidered: a symmetric shape (Mode 1, W 1( x)) and an

(a) Stayed column as a temporary shoring. (b) Crossarm in detail .

Fig. 2. Construction phase of Rock in Rio III main stage.


3/16

1226 D. Saito, M.A. Wadee / Engineering Structures 30 (2008) 12241239

(a) Prole. (b) Structural model.

Fig. 3. Structural model of the stayed column: column length L , crossarm length a , axial load P , angle between the stay and the vertical and stay length L s . Thequantity i X represents the end-shortening of the column, where subscripts i and X represent a buckling mode number (1 or 2), and a buckling type ( A, B or C )respectively. Subscripts 1, 2, 3 and 4 after X represent the number of the individual stays. The quantities A, As and Aa are dened as the cross-sectional areas with E , E s and E a being the Youngs moduli of the column, the stays and the crossarm respectively. The quantities I and I a refer to the cross-sectional second momentsof area of the column and the crossarm respectively.

Fig. 4. Buckling Modes 1 (symmetric) and 2 (antisymmetric).

antisymmetric shape (Mode 2, W 2( x)) about the column mid-

span, as shown in Fig. 4; these are the basic possible deection

shapes for buckling in the single-crossarm stayed column. InMode 1, the maximum curvature can be found at the columnmid-span and zero curvature at both ends; in Mode 2, zerocurvature can be found at the column mid-span and both ends.Each mode can be expressed as a summation of sinusoidalwaves. Dening the column length as L (see Fig. 3) andthe generalized coordinates as qm , where the subscript m isan integer representing a degree of freedom for a sinusoidalwave that has a wavelength of 2 L/ m , the displacementfunctions for the column W 1 and W 2 can be assumed to be asfollows:

W 1( x) = q1 L sin x

L+ q3 L sin

3 x

L+

=n

m= 1q2m 1 L sin

(2m 1) x L

, (1)

W 2( x) = q2 L sin2 x L

+ q4 L sin4 x L

+

=n

m= 1q2m L sin

2m x L

, (2)

where n represents the number of degrees of freedom in themodel. As the individual components of the stayed column tendto be long and thin, EulerBernoulli bending theory can beapplied; the angles of the members to the vertical 1( x) and

2( x) are therefore approximated as the rst derivative of the


4/16


Fig. 5. Buckling types in Mode 2.

displacement with respect to x:

1( x) = q1 cos x L

+ 3 q3 cos3 x L

+

=n

m= 1

(2m 1)q2m 1 cos(2m 1) x

L

, (3)

2( x) = 2q2 cos2 x L

+ 4 q4 cos4 x L

+

=n

m= 12mq 2m cos

2m x L

. (4)

2.2. Displacement functions for the crossarm

2.2.1. Buckling type distinctionThe deected shape of the crossarm and the function for

the end-shortening of the column depend on the stress state of the stays (see Fig. 5). To take these effects into account in thecurrent model, the following four states are considered:

(1) Type A: all of the stays are slack.(2) Type B: all of the stays are active.(3) Type C: two stays are active.

Note that Type A occurs with a small value of the initialprestress; Type B occurs with a sufcient amount of theinitial prestress which allows the stays not to slacken untilbuckling; Type C buckling can occur either after Type A, Bor the fundamental (pre-buckling) state. Shape functions forthe crossarm for each type can be obtained by solving thedifferential equations reecting each type of stress state in the

stays and the reaction forces developed in the crossarm.

Fig. 6. Free body diagram to determine the bending moment at an arbitrarycross section ( y 0) of the crossarm. Note that the subscript X represents thebuckling type classication which can be either B or C .

2.2.2. Shape functionsFirst, the bending moment for y 0 in the crossarm (see

Fig. 6) M a B is given by

M a X = Rh X [h X w2 X ( y)] + Rv X (a y), (5)

where Rh X and Rv X are horizontal and vertical reaction forcesrespectively at the tip of the crossarm; h X is the displacement atthe tip of the crossarm; y is the horizontal axis; and w2 X ( y) isthe deection of the crossarm perpendicular to the coordinate.

Ignoring higher-order terms and the effect of the end-shortening of the crossarm, the basic differential equation forthe bending of the crossarm takes the form:

M a X = E a I a w2 X ( y), (6)

where primes represent differentiation with respect to theindicated independent variable, in this case y, and E a and I a are

the Youngs modulus and the cross-sectional second moment of


5/16


area of the crossarm respectively. Substituting Eq. (6) into (5)leads to

w2 X ( y) + k 2 X w2 X ( y) =

Rv X E a I a

(a y) + hk 2 X , (7)

where

k X =Rh X

E a I a. (8)

The general solution of Eq. (7) is

w2 X ( y) = H X sin k X y + K X cos k X y

Rv X

k 2 X E a I a(a y) + h X , (9)

where H X and K X are constants of integration that aredetermined from the boundary conditions, thus:

w2 X (0) = 0, w 2 X (0) = , w 2 X (a ) = h X , (10)

where is the angle between the horizontal and the crossarmat the mid-point, dened as

= 2( L/ 2) = 2q2 4q4 +

=n

m= 1( 1)m 12mq 2m . (11)

The second condition comes from the assumption that W 2( x)intersects the crossarm at right angles. Applying this conditionyields the following expressions:

H X =

1

k X

Rv X

k 2 X E a I a + , K X =

Rv X a

k 2 X E a I a h X ,

h X =( E a I a k 2 X Rv X ) sin k X a + k X Rv X a cos k X a

k 3 X E a I a cos k X a. (12)

In order to nd the actual shape of the crossarm with Eq. (9),it is also necessary to establish equations for Rv X and Rh X .With reference to Fig. 7 and then by taking the leading terms of 2 X and h X , the changes in the axial force in Stay 3 and Stay4, F X 3 and F X 4 respectively, resulting from the structuraldisplacement can be expressed as follows:

F X 3

= E s A

s

Ls2 X 3 Ls

Ls

2 X +

2h X

Lcos2 , (13)

F X 4 = E s As Ls2 X 4 Ls

Ls 2 X +

2h X L

cos2 . (14)

From the expression for F X 3 and F X 4 , the vertical and thehorizontal reaction forces for Type B, Rv B and Rh B , can beobtained thus:

Rv B = (T + F B3) cos 2 B3 (T + F B4) cos 2 B4

=4h B L

(T sin2 + E s As cos2 ) cos , (15)

Rh B = (T + F B3) sin 2 B3 + (T + F B4) sin 2 B4

= 2[T + (T E s As ) 2 B cos2

] sin . (16)

Fig. 7. Elongation of the stays and reaction forces at the tip of the crossarm.

As only one stay is active on each side in Type C, RvC and RhC , can thus be obtained from F X 3. However, including thehC term in the RhC equation causes a computation problemthat leaves the governing equation intractable. To rectify this weapply the approximation hC = 0 in RhC , which applies whenthe stays rst slacken, thereby enabling us to obtain RhC :

RvC = (T + F C 3) cos 2C 3 = 1 2C 2hC L

sin2

+ E s As

T cos2

T cos , (17)

RhC = (T + F C 3) sin 2C 3

= 1 + 2C 2h C L

1 E s As

T cos2 T sin

1 + 2C cos2 T E s As 2C cos2 sin . (18)

2.3. Stress and geometrical changes in the structure

Stress and geometrical changes in the structure areinvestigated prior to the energy formulation presented in thefollowing section. The investigation includes such items as thestress changes by the prestress, the elongation of the stays andthe end-shortening of the column.

2.3.1. Initial stress of the column with prestressWith reference to Fig. 8, the initial prestresses that are

introduced to the column T c and the crossarm T a are

T c = 2T cos , T a = 2T sin . (19)

Therefore, the strains in the stay st , the column ct , and thecrossarm at are respectively:

st =T

E s As, ct =

2T cos

E A, at =

2T sin

E a Aa. (20)


6/16


Fig. 8. Effect of the initial prestress.

2.3.2. Tip displacement coefcient The tip displacement of the crossarm is necessary to nd the

elongation of the stays. The tip displacement for Type B canalso be obtained from h B, which is given in Eq. (10). Expandingh B to the fth order with respect to R B and then taking theleading order with respect to 2 B and yield

h B = c Ba , (21)

where c B is a factor expressing the magnitude of the tipdisplacement of the crossarm for Type B:

c B = 1 + 2 E s As3 E a I aa 2 sin cos2

1

. (22)

and R B is

R B = a2 1 + 2 B cos2 T E s As 2 B cos2 sin

E a I a.

(23)

The same expression for the tip displacement can be obtainedusing the work of Smith et al. [5]. The tip displacement for TypeC can also be obtained from hC , which is given in Eq. (10).However, as the direct expression that can be obtained from h C

is too complicated for the analytical model, this is simplied byusing the Taylor expansion to the fth order with respect to RC .Then taking the leading order with respect to 2C and giveshC as

hC = cC a + cC a 2C + cC 0a , (24)

where cC , cC and cC 0 are factors expressing the magnitude of the tip displacement of the crossarm in Type C:

cC =15 E a I a E s As a 2 sin a 2T sin cos2 + 15 E a I a

5 2 (25)cC

=3 E a I a + 2 E s As a 2 sin cos2 T sin2 + E s As cos2 a 2 cos

2 (26)

cC 0 = a 2T cos

, (27)

where

= (3 E a I a + E s As a 2 sin cos2 ), (28)

and R is

RC = a1 + 1 E s AsT 2C cos

2 T sin

E a I a. (29)

Note that this simplication becomes less accurate when theinitial prestress T is large.

2.3.3. Elongation of the staysThe post-buckling shapes are sketched in Fig. 9; these

geometries allow the new stay length L s i X j , where the subscript j refers to the stay number as indicated in Fig. 8, to be evaluatedthrough Pythagorass theorem, which leads to the strain inthe stays purely arising from the applied load P in the stays.Subsequently, this equation is expanded as a Taylor series up tosecond order with respect to qm and i X . In this process, thecross and quadratic terms of i X such as i X qm and 2i X aredropped, as these terms are considered to be small.

By combining the expanded strain i X j with the initialprestress T , the total strains in the stays s i X j can be obtained,giving the following:

s i X j = i X j + st . (30)

2.3.4. End-shortening of the columnIn order to nd the end-shortening expression of the column

i X , equilibrium is considered at the end of the column wherethe external load P is applied with the free body diagramapproach shown in Fig. 10.

Vertical force equilibrium and moment equilibrium aroundthe point O give the following equations:

T i X 1 cos i X 1 + T i X 4 cos i X 4 + P C i X cos i Si X cos i = 0, (31)

M i X d x Si X cos i X W i (d x) Si X sin i W i (d x)C i X cos i + d xC i X sin i = 0, (32)

where T i X j is the axial force in stay j ; with C i X , Si X and M i being an axial force, a shear force and a bending momentrespectively in the column at a point which is a small distanced x away from O ; i is the angle between the column and thevertical; i X 1 and i X 4 are the angles between each stay andthe vertical. These angles, internal forces and moments needto be dened in order to solve the equilibrium equations andto obtain an expression for i X . Firstly, i can be obtained bysubstituting x = 0 into i ( x) dened in Eqs. (3) and (4):

1 = 1(0) = q1 + 3q1 +

=n

m= 1(2m 1)q2m 1, (33)


7/16


(a) Mode 1. (b) Mode 2 Type A. (c) Mode 2 Type BC.

Fig. 9. Geometry of the stayed column in buckling Modes 1 and 2.

Fig. 10. Equilibrium free body diagram for the column. Note that R HiX is thehorizontal reaction force at the end of the column.

2 = 2(0) = 2q2 + 4q1 + =

n

m= 12mq 2m . (34)

With reference to Fig. 9(a)(c), cos i X 1 and cos i X 4are obtained through trigonometry; subsequently, thoserelationships are expressed to the leading order with respect toqm and i X . For example, cos 1 X 1 is given as

cos 1 X 1

=12 L(1 1 X )

12 L(1 1 X )

2+

n

m= 1 L( 1)m 1q2m 1 + a

2

(1 1 X sin2

) cos . (35)

As all of the required angles are dened, the forces andmoments T i X , C i X , Si X , and M i X in the free body diagramneed to be investigated. Firstly, with the strain expressions of the stays shown in the previous section and the assumption thatthe stays do not resist compression, the axial forces in the staysT i X 1 and T i X 4 are dened as follows:

T i A1 = T i A4 = T iC 4 = 0,T i B1 = i B1 E s As , T i B4 = i B4 E s As ,

T iC 1 = iC 1 E s As .

(36)

The axial strain in the column c1 X is expressed as a summationof the components i X and ct minus the effect of therelaxation from the buckling displacement. Therefore, the axialstrain for each mode is expressed as follows:

c1 X = 1 X + ct 1

L L

0

12

W 21 ( x)d x

= 1 X +2T cos

E A

n

m= 1

(2m 1)2 2q 22m 14 , (37)

c2 X = 2 X + ct 1

L L

0

12

W 22 ( x)d x

= 2 X +2T cos

E A

n

m= 1m2 2q 22m . (38)

Thus, the axial force C i X is expressed as

C i X = E Aci X . (39)

With linear bending theory, the bending moments M i are

expressed as the following equations:


8/16


M 1 = EIW 1 (d x)

=n

m= 1

(2m 1)2 2 EIq 2m 1 L

sin(2m 1) d x

L, (40)

M 2 = EIW 2 (d x) =n

m= 1

(2m)2 2 EIq 2m L

sin2m d x

L. (41)

The shear force Si X can be dened by substituting Eq. (39) andeither Eq. (40) for Mode 1 or Eq. (41) for Mode 2 into Eq. (32)and then by taking the limit d x 0.

By substituting Eqs. (36), (39) and an expression for theshear force, either (40) for Mode 1 or (41) for Mode 2, intoEq. (31), expressions for i X can be obtained. Subsequently,the solution is expressed as a Taylor series with respect to T , Pand qm up to second order, which gives the following simpliedexpressions:

1 X = b pX P + bt X T + b1 X q1 + b3 X q3 + + b11 X q 21 + b13 X q1q3 + b

233 X q

23 +

= b pX P + bt X T +n

m= 1b2m 1 X q2m 1

+n

m= 1, l= 1,m lb2m 12l 1 X q2m 12l 1q2m 12l 1, (42)

2 X = b pX P + bt X T + b2 X q2 + b4 X q4 + + b22 X q 22 + b24 X q2q4 + b

244 X q

24 +

= b pX P + bt X T +n

m= 1b2m X q2m

+n

m= 1, l= 1,m l

b2m2l X q2m2l q2m2l (43)

where b pX , bt X , bm X and bml X are coefcients for P , T , qm ,and qm ql respectively.

2.4. Energy formulation

The total potential energy V i X comprises components of thestrain energy and the work done by the load. In a general stateof deection, there are four components of strain energy: frombending in the column ( U cbi ) and the crossarm ( U abiX ) withaxial strains in the column ( U caiX ) and stays (U si X ). Note that

the bending energy in the crossarm ( U abiX ) only exists in Mode2 in buckling Types B and C as the crossarm does not bend inthe other cases.

2.4.1. Bending energyThe bending energy components in the column arise

from a linear curvature expression; thus, W i give followingexpressions for U cbi :

U cb 1 =12

E I L

0W 21 ( x)d x U cb 0

=n

m= 1

(2m 1)4 EIq 22m 14

4 L U cb 0, (44)

U cb 2 =12

E I L

0W 22 ( x)d x U cb 0

=n

m= 1

(2m)4 EIq 22m 4

4 L U cb 0 , (45)

where U cb 0 is the existing column bending energy at the

beginning of each buckling type.In a similar way, the bending energy in the crossarm forMode 2 Types B and C can be obtained. Note that crossarmsymmetry accounts for the doubling of the standard bendingenergy expression:

U ab 2 X = E a I a a

0w 22 X ( y)d y U ab 0

= E a I a k 3 B{2 H X K X H 2

X cos k X a sin k X a

+ H 2 X k X a 2 H X K X cos2 k X a

+ K 2 X cos k X a sin k X a + K 2 X k X a }/ 2 U ab 0 , (46)

where U ab 0

is the existing crossarm bending energy at thebeginning of each buckling type. Note that U cb 0 and U ab 0 haveindependent values from qm , therefore they do not affect thecritical load nor the post-buckling path as they simply vanishon differentiation.

2.4.2. Axial energyThe axial energy U caiX in the column accounts for the

energy gained through the axial compression from the loadP together with the effect of the relaxation from the bucklingdisplacement; using Eqs. (37) and (38) as the ending points of integration for each mode, the axial energy is obtained as

U caiX = ci X

cX 0 EAL d =

12 EAL (

2ci X

2cX 0), (47)

where cX 0 is the existing strain at the beginning of each type.The axial energy in the stays is obtained by integrating the

stressstrain relationship over the stay volumewritten as theproduct of the cross-sectional area As and the length L s :

U si X =4

j= 1U s i X j =

4

j= 1 s iX j

s X 0 As Ls ( s i X j )d, (48)

where U s i X j is the strain energy stored in stay j for Mode iType X ; s X 0 is the existing strain at the commencement of each type. The stressstrain curve of the stays is assumed to bepiecewise linear thus:

s ( s i X j ) = E s s i X j for s i X j > 0,0 for s i X j 0.

(49)

From Eqs. (49) and (48), the total stay energy for Mode i Type X in stay j is described as follows:

U s i X j =12

E s As Ls ( 2s i X j 2s X 0) for s i X j 0,

0 for s i X j 0.(50)

Note that cX 0 and s X 0 affect neither the critical load nor thepost-buckling path, because they are independent of qm and

vanish on differentiation.


9/16


2.4.3. Work done by the load The work done by the load P E i X is dened as the external

axial load P multiplied by the corresponding end-shortening i X L:

P E i X = P i X L P E 0 X , (51)

where P E 0 X is the work done by the load before thecommencement of each buckling type. Note that, again, thisvalue affects neither the critical load nor the post-buckling pathfor the same reason as stated in the previous section.

2.4.4. Total potential energy functionThe total potential energy is a summation of U cbi , U abiX ,

U caiX , U si X minus P E i X :

V i X = U cbi + U abiX + U caiX + U si X P E i X . (52)

In the Mode 2 Type C analysis, higher terms of P are thentruncated as they are not dominant terms in the function and

leave the governing equation intractable. For equilibrium, thetotal potential energy V i X must be stationary with respect to thegeneralized coordinates qm . Therefore, the equilibrium pathscan be computed from the condition:

V i X qm

= 0. (53)

3. Critical buckling

Having formulated the total potential energy, the criticalbuckling load of the stayed column is investigated using linear

eigenvalue analysis. From the earlier work of Hafez et al. [8],it is known that the critical load is divided into three zones inrelation to the magnitude of the initial pretension in the stays.

Zone 1 The tension in the stays disappears completely beforethe external load reaches the buckling load. Therefore,the critical load is exactly the Euler load (Type Abuckling).

Zone 2 The strain in the stays becomes zero when the appliedload reaches the critical load, i.e. the structure resistsbuckling until the tension in the stays becomes zero.Thus, all the stays remain effective until buckling,which sends the critical load potentially to a level that

is signicantly higher than the Euler load (Type Cbuckling). Zone 3 The tension in the stays is nonzero at the instant of

buckling. As a large amount of the pretension has beenintroduced, all the stays remain effective for somewhile after buckling. The value of the critical load fallssomewhat as the initial prestress increases because theinitial compressive stress in the column diminishes itsaxial load capacity (Type B buckling).

As the formulation of the model ensures that the proleof the structure maintains perfect symmetry during thefundamental state, a bifurcation point can be observed when

qm = 0. For Type B buckling conventional linear eigenvalue

analysis, i.e. nding when the Hessian matrix for V i X becomessingular, yields the critical load for Zone 3 P CiZone3 directly. Thedetails on the process and equations obtained can be seen inAppendix .

For Zones 1 and 2, it is necessary to consider geometricallynonlinear effects in order to nd the critical load, because in

these zones, the end-shortening of the column releases the axialenergy in the stays during the fundamental stage, which doesnot allow linear eigenvalue analysis to yield the critical load.Moreover, linear buckling analysis in the FEM does not detectthe critical load for Zones 1 and 2 either, the analytical methodbeing therefore essential to nd the critical load in this rangeof T .

In Zone 1, where the axial energy in the stays is alreadylost before buckling, this problem can be simply resolved byadopting the Type A buckling energy formulation and thenfollowing the same process as for Zone 3. Because in Zone 1(Type A buckling) all of the stays are slack at the instant of

buckling, no substantial changes in the way of determining thecritical load are necessary. This analysis yields the critical loadsfor Zone 1 for Mode i being:

P CiZone1 =i 2 2 E I

L2. (54)

As can be seen from this equation, the critical load for Zone 1is the exactly same as the Euler buckling load PE.

For Zone 2, the critical load can be found from utilizingthe condition that the strain in the stays becomes zero at theinstant of buckling. As all of the stays are active during the pre-buckling stage, substituting qm = 0 and h B = 0 into Eq. (30)with the adoption of subscript B and solving the equation for Pgives the following critical load for Zone 2 for Mode i :

P CZone2 =T

b pB E s As cos2 , (55)

where

b pB = [ 2 E s As cos3 + E A] 1. (56)

Note that Modes 1 and 2 have the same expression for theZone 2 critical load. In fact, the instability behaviour in Zone 2is not a classic bifurcation response: at the point of buckling

there is a sudden release of the axial energy of the column,forcing the column to buckle, which is immediately followed bythe reactivation of the convex side stays as the column displaceslaterally.

By plotting the critical loads against T , the relationshipbetween the buckling load and the initial prestress, whichwas discovered by Hafez et al. [8], can be reproduced. Thisrelationship is shown in Fig. 11 , where T min represents theinitial prestress at the boundary between Zones 1 and 2 theminimum effective pretension required to raise the bucklingload above the Euler load and Pmax represents the theoreticalmaximum buckling load that is observed at the boundary

between Zones 2 and 3.


10/16


Fig. 11. Critical buckling load P C versus initial prestress T .

3.1. Numerical results

In this section the aim is to compare theoretical Pmax values

obtained from the previous section with those from the Hafezmodel as a benchmark for validation. In the Hafez model,Pmax was obtained by the FEM, so that the accuracy of thecurrent model in terms of the critical load can be evaluated.In the Hafez model, Pmax was sought with a variation in threeparameters: crossarm length, stay diameter and stay Youngsmodulus; for the current model, the same parameters are varied.The dimensions of the structure used in the Hafez model wereas follows:

Column Youngs modulus: E = 201 kN / mm2

Crossarm Youngs modulus: E a = 201 kN / mm2

Stay Youngs modulus: E s = 202 kN / mm2

Column length: L = 3.05 mCrossarm length: a = 0.305 mOutside diameter of the column: co = 38.1 mmInside diameter of the column: ci = 25.4 mmOutside diameter of the crossarm: ao = 38.1 mmInside diameter of the crossarm: ai = 25.4 mmStay diameter: s = 3.2 mm or

s = 4.8 mm.

While the crossarm length a is varied from 0.305 to 3.05 m,the stay diameter is xed to s = 3.2 mm; when thestay Youngs modulus E s is varied from 64 .8 kN/ mm2 to204 kN / mm2, the stay diameter is xed to s = 4.8 mm.Fig. 12(a), (b) and (c) respectively show Pmax varying with eachparameter along with that of the Hafez model.

In the case of the single-degree-of-freedom (SDOF) modelfor Mode 1, there is a certain degree of error shown in Fig. 12between the Hafez and the current model. However, with thetwo-degree-of-freedom (2DOF) model, this error between thetwo becomes almost negligible. In Mode 2, however, gapsbetween the Hafez model and the current model can be seen tobe more signicant. With the three-degree-of-freedom (3DOF)model, which is the most sophisticated model presented andtherefore is expected to have the least error, some differencesare still evident. Although these gures show relatively less

good agreement compared with those of Mode 1, the trend is

(a) Crossarm length.

(b) Stay diameter.

(c) Stay Youngs modulus.

Fig. 12. Comparison of Pmax values with those of the Hafez model: (a) varyingcrossarm length, (b) varying stay diameter, (c) varying Youngs modulus.Symbols ( ), ( ) and ( ) represent the cases of n = 1, n = 2 and n = 3respectively.

that increasing the number of freedoms increases the accuracybut with computational expense and analytical complexity.

Considering that the difference between the 2DOF and the

3DOF models is not signicant, and that the solutions from the


11/16


Table 1Selected points for the post-buckling investigation

Point Initial prestress T Criterion expression Mode 1 (kN) Mode 2 (kN)

1 0 0.00 0.002 T min / 2 0.23 0.93

3T

min 0.46 1.864 (T opt T min )/ 3 + T min 1.47 2.505 2(T opt T min )/ 3 + T min 2.48 3.146 T opt 3.48 3.787 2T opt 6.97 7.558 4T opt 13.93 15.10

2DOF model are relatively close to the benchmark solutions,the 2DOF model will be used in order to obtain reasonablyaccurate solutions for the post-buckling behaviour without itbeing excessively demanding computationally.

4. Post-buckling response

Eq. (53) expresses the equilibrium states after buckling,which can be solved using M APLE . In Mode 1, the samedimensions and properties as in Section 3.1 were also appliedfor the post-buckling analysis, with the stay diameter, s =4.8 mm being chosen. The critical buckling loads obtainedwith those dimensions against the initial prestress are shownin Fig. 13.

As illustrated, eight points are picked up from each diagramto investigate changes in the post-buckling response as T changes, the selection criteria being expressed in Table 1 .

4.1. Zones of behaviour

The post-buckling responses for Modes 1 and 2 in each zoneare represented in Figs. 14 and 15. For Mode 1 the relationshipbetween P and q1 q3 is shown. The latter quantity being thenormalized horizontal displacement at the column mid-span,obtained by evaluating W 1( L/ 2)/ L.

For Mode 2 the relationship between P and q2 2q4 isshown. The latter quantity being the normalized rotation at thecolumn mid-span, obtained by evaluating 2( L/ 2)/ 2 .

For both modes the post-buckling path in Zone 1 has twodistinct stages, as shown in (a) and (b) in Figs. 14 and 15respectively; P remains practically at the critical load in Type Abuckling (all stays slack) for a while, then the equilibrium pathstabilizes with Type C buckling (convex side stays reactivated).Note that, as shown in (b) in Figs. 14 and 15, the initial atrange becomes shorter as the prestress T is increased.

As shown in the graphs in (c) of Figs. 14 and 15, in Zone2, stable paths can be observed in the initial post-bucklingrange with relatively low values of the prestress, such as forPoints 3, 4 and 5, whereas unstable paths can be observed withrelatively high values of the prestress, such as for Point 6. Theinitial prestress at the transition from stability to instability canbe found when T = 2.79 kN for Mode 1. The reason forthis transition in Zone 2 can be considered as follows: with a

relatively large value of the prestress in Zone 2, a large amount

of the axial energy can be stored in the fundamental state dueto the presence of effective axial forces in the stays, whichprevents the release of axial energy from the column. Therefore,this excessive amount of the energy is suddenly released at theinstant of buckling, which is conjectured to cause the unstableresponses. By contrast, with a relatively small value of the

prestress, although the axial energy has been able to be storedin the fundamental state, more than in the case of Zone 1,this additional energy can completely be absorbed into thestays after buckling; therefore, stable paths are seen. Despitethis difference within the zone, for any case in Zone 2 post-buckling, the convex side of the stays are active throughout thepost-buckling range, with the stays on the concave side beingslack, which implies that in Zone 2 the post-buckling responsehas Type C characteristics.

As shown in the graphs in (d) of Figs. 14 and 15, thereis also a discontinuity in the post-buckling response in Zone3. The load P remains nearly at the critical load in Type Bbuckling for a while, and this initial stage is followed by TypeC buckling with a sudden loss of the stability; unstable pathsare then observed when the concave side stays go slack. Thediscontinuity of Zone 3 is basically a mirror image of theresponse in Zone 1, where slackening of stays occurs rather thantheir reactivation.

For all zones, the only difference between Modes 1 and 2is the activating stays in the Type C buckling response: theactivating stays are 1 and 2 for Mode 1, and 1 and 3 for Mode 2.

4.2. Validation

Using the FEM program ABAQUS, a purely numerical

model was developed and the post-buckling response wasrevealed by a nonlinear Riks analysis to validate the resultspresented in the previous section. In this procedure, the columnand the crossarm were modelled as beam elements and thestays were modelled as truss elements. The No compressionoption, which prevents any compression force entering thetruss elements, was also adopted to simulate any slackeningin the stays. Furthermore, it is essential in this type of nonlinear analysis to introduce an imperfection. In the currentstudy, this was achieved through using the Euler bucklingdisplacement generated by eigenvalue analysis. The magnitudeof the imperfection was intended to be deliberately small suchthat the perfect response would be approximated. To triggerMode 1, an out of straightness of L / 10 000 was imposed at themiddle of the column. To trigger Mode 2, an out of straightnessof L/ 14 142 was imposed at the quarter and the three-quarterpoints along the column such that the horizontal displacementat those points would be the same as that in Mode 1.

4.2.1. ComparisonsFigs. 16 and 17 show the post-buckling responses from the

FEM along with those from the analytical models at Points1, 3, 6, 7 and 8. As can be seen in Fig. 16, for Mode 1, thepost-buckling paths of the FEM model almost coincide withthose of the analytical model. However, Fig. 17 shows less

good agreement between the FEM and the analytical models


12/16


(a) Mode 1. (b) Mode 2.

Fig. 13. Critical buckling load P C versus the initial prestress T showing the selected points for the post-buckling investigation.

(a) Zone 1. (b) Initial part of zone 1.

(c) Zone 2. (d) Zone 3.

Fig. 14. Post-buckling responses for Mode 1 represented by axial load P versus mid-span buckling displacement q1 q3 .

in Mode 2. Regardless of the less good agreement in Mode

2, the same trend can still be detected from these two models;

therefore, the analytical models for Mode 2 are still useful for

predicting the qualitative buckling behaviour.


13/16


(a) Zone 1. (b) Initial part of zone 1.

(c) Zone 2. (d) Zone 3.

Fig. 15. Post-buckling responses for Mode 2 represented by axial load P versus mid-span buckling rotation q2 2q4 .

From this comparison, it can be said that the Mode 1buckling of the stayed column can be modelled as the current2DOF analytical model with great accuracy. Also, it can be saidthat with the current analytical 2DOF model for Mode 2, theapproximated post-buckling response can be obtained; howeverit has to be admitted that the analytical model involves a certaindiscrepancy with the numerical model. This inaccuracy can be

reduced by increasing the number of degrees of freedom, butthis process is computationally demanding as discussed earlier.

5. Design implications and further work

Some implications for design can be deduced from theresponses that have been presented. Firstly, when the prestressis relatively low, such as in Zone 1 and in an initial part of Zone 2 in which stable post-buckling paths can be seen, thedesign load can be set on the basis of the maximum strength,which can be larger than the critical load as long as plasticityand excessive deection can be avoided. However, when theprestress is relatively high, such as in Zone 3 and the rest

of Zone 2 in which unstable paths can be seen, the design

load should be much less than the critical load because withimperfections, the maximum strength of the structure usuallybecomes signicantly lower than the critical load and elasticfailure ensues. The margin between the maximum strength andthe design load should be decided by imperfection sensitivitystudies, which the authors are currently undertaking.

Also, with a very large amount of prestressing, much higher

than T opt , the critical load may become relatively reliable for thecollapse load because large amounts of buckling displacementare necessary before unstable post-buckling occurs. Hence,it can be said that, when greater stability is required for astructure, introducing a large value of the prestress is stronglyrecommended. However, it should be noted that a large amountof prestressing would require a signicant increase in thecross-sectional area of the structural components to counteractpotential plasticity effects, which may lead this component tobeing less cost-effective.

The current studies do not account for geometric imperfec-tions nor plasticity of steel. Hence, it does not necessarily reect

the actual response of the stayed column, although the model


14/16


(a) Zone 1.

(b) Zone 2.

(c) Zone 3.

Fig. 16. Equilibrium paths for Mode 1 comparing the FEM and the analyticalmodels.

has revealed the principles behind the post-buckling responsein relation to the initial response in the ideal situation. The im-portance of geometric imperfections was already discussed inearlier works [ 912]. To predict the more realistic response of the structure, it is necessary to incorporate imperfections andthe plastic behaviour of each component into the current model,

and the authors are currently working on this.

(a) Zone 1.

(b) Zone 2.

(c) Zone 3.

Fig. 17. Equilibrium paths for Mode 2 comparing the FEM and the analyticalmodels.

Another aspect that is currently neglected is the effect of stress relaxation that may occur due to creep and changes in theambient temperature with the stays and the column changingtheir lengths, and thereby their internal forces, causing a change

in their stress state. This may manifest itself in stay relaxation;


15/16


in Zone 3, where the optimal prestress is now considered tobe located, relaxation may change the response of the columnby reducing the prestress from where the column has relativelystable initial post-buckling responses (points 7 and 8) topurely unstable post-buckling responses (towards point 6). Thisadverse effect from stress changes would be a key sensitivity to

focus on in future work. If this sensitivity is signicant, it wouldalso be suggested that designers should take into account boththe initial prestress and the effective value of prestress after along time period. Of course, the situation would become morecomplicated if materials of different coefcients of thermalexpansion are used in the column and the stays respectively,or if the temperature changes are non-uniform within the wholecomponent.

Interactive buckling is another issue to be tackled. This isa phenomenon in which different types of buckling modesoccur simultaneously. Fig. 12 shows that there is a boundarybetween Modes 1 and 2; it is one of the places where interactivebuckling possibly occurs [ 13]. From work on other structuralcomponents [16], it has been known that interactive bucklingis also triggered by the interaction between local and globalinstabilities. Catastrophic failure can often be observed withthis type of instability, and therefore, interactive buckling needsfurther investigation.

Experimental studies focusing on the post-buckling responseare also necessary in order to validate the analytical and thenumerical models. This process becomes vital, especially whenthe structure is more complicated than the current structureand therefore it becomes more difcult to formulate analyticalmodels. Moreover, the current modelling is limited to two-dimensional (2D) behaviour, it may become important to

develop three-dimensional (3D) models. Recent work [ 17,18]has used 3D modelling to address 3D collapse responses witha variety of structural congurations and boundary conditions,and with different levels of the prestress, but certain otherstability issues need to be investigated in 3D such as localbuckling mentioned above and torsional buckling if, forexample, open sections are used instead of closed sections forthe main column component.

The work outlined above could be used as a basis to producedesign guidance. Currently, codes of practice, such as theEuropean design code for steel structures [ 19], are lacking inthe design procedures for such potentially efcient and cost-effective structures; consequently, case and sensitivity studiesin conjunction with engineering judgement are necessary todesign stayed columns in practice. Establishing such guidancefor stayed columns would facilitate designers to adopt thisstructural component more effectively.

6. Concluding remarks

The post-buckling behaviour of the prestressed steel stayedcolumn has been investigated using the RayleighRitz method.It has been shown that the post-buckling response is stronglylinked to the zone distinction of the critical loads that was foundby Hafez et al. [ 8] for the rst two buckling modes. In Zone 1,

the response is initially similar to that of Euler buckling, which

is followed by a rather stable path thanks to the reactivationof the stays. In Zone 2, the critical load is increased to morethan the Euler load and either a stable or an unstable pathemerges after buckling, depending on values of the prestressand other structural properties. In Zone 3, the critical loadreaches its theoretical maximum, and the post-buckling path

becomes unstable, after an initially at but slightly stableresponse, due to some of the stays slackening. These resultshave been validated using the FEM. It has been shown that thecurrent analytical model for Mode 1 has excellent agreementwith the FEM model; however it is less accurate for Mode 2when compared to Mode 1, even though the model is still usefulto nd approximate post-buckling responses for that mode.Design implications have also been deduced from those results;it has been stated that any design loads should be carefullydetermined with consideration for the post-buckling responsein order to achieve safe and efcient designs because the resultshave shown that the maximum strength depends primarily onthe prestress rather than the classically evaluated buckling load.

Appendix. Hessian matrix for Zone 3

When m = 2 and in Mode 1, the critical load for Zone 3can be obtained through calculating the following determinantof the Hessian Matrix:

2V 1 Bq 21

2V 1 Bq1q3

2V 1 Bq3q1

2V 1 Bq 23

= 0. (57)

Each component of the matrix can be expressed as

2V 1 Bq 21

= p11 P + t 11 T + 11 , (58)

2V 1 Bq1q3

= 2V 1 Bq3q1

= p13 P + t 13 T + 13 , (59)

2V 1 Bq 23

= p33 P + t 33 T + 33 . (60)

The coefcients of the above equations are as follows:

p11

=a 8b pB E s As b11 B 2cos 2 cos3 + b pB E A 4b11 B 2 4b11 B

tan ,

(61)

t 11 =2a 8cos2 2 cos

tan , (62)

11 =64a 2 E s As cos3 + E I 4 tan

4a(63)

p13

=2a 2b pB E s As 4cos 2 + b13 B cos3 + b13 B(b pB E A 1)

tan , (64)

t 13 = 16a cos3

tan , (65)

13 = 16aE s As sin cos2

, (66)


16/16

saito 2008 engineering-structures

Documents