advanced analysis of pre-tensioned bowstring structures

Steel Structures 6 (2006) 153-162 www.kssc.or.kr

Advanced Analysis of Pre-tensioned Bowstring Structures

J Y Richard Liew* and Jin-Jun Li

Department of Civil Engineering, National University of Singapore, Blk E1A, #05-13, 1 Engineering Drive 2, Singapore, 117576

Abstract

Bowstring steel structure is a novel cable-tensioned space structure in which out-of-plane stability is ensured by pre-tensionedcables acting as bows. In this paper, an advanced analysis method considering both geometrical and material nonlinearities hasbeen developed to predict the collapse behavior of pre-tensioned steel structures. Special emphasis is placed on the modelingof pre-tensioned cable and the nonlinear beam-column connections. Novel connectors connecting rectangular hollow sectionbeam and column have been proposed and tested to obtain their moment-rotation relationship. Numerical work on three-dimensional bowstring structures shows that their load-displacement behavior is dependent not only on structuralconfigurations, but also on the applied load sequence and the magnitude of pretension forces.

Keywords: Advanced analysis, Bowstring structure, Buckling, Pre-tensioned cable, Semi-rigid beam-column connection

1. Introduction

Pre-tensioning technique has been widely and successfully

used in the construction of long-span suspension or cable-

stayed bridges and high-rise steel towers. The adoption of

such technique in building structures provides aesthetically

pleasing and light-weight solutions. Pre-tensioned steel

structures such as pre-tensioned truss (Kosaka, et al.,

1988), arch (Barnes and Dickson, 2000), dome (Wang,

2004) and grid space structures (Barnes and Dickson,

2000; Bradshaw, et al., 2002) have found many engineering

applications in wide-span or large-space buildings. Dong

and Zhao (2004) presented several recent applications of

pre-tensioned steel space structures in China.

Previous works by Liew et al. (2001) and Chan et al.

(2002) studied the behavior of pre-tensioned steel columns.

Very little work has been done on advanced analysis of

pre-tensioned steel structures. Bowstring column and

space frame are novel cable-tensioned space steel

structures in which out-of-plane stability is ensured by

pre-tensioned cables acting as bows. Bowstring structures

have potential applications as supporting members of

building entrance canopy or glass façade wall to reduce

the need of artificial lighting. However, application of

bowstring structures posts a new challenge requiring

more complex nonlinear solution techniques to capture

the behavior of such structures. This paper presents the

modeling techniques using advanced analysis to study the

nonlinear response of bowstring structures. Effects of

cable profile configurations, cable pretension forces, length

and spacing of horizontal struts on behavior of pre-

tensioned steel columns are examined. For the bowstring

frame structure, structural layout, effects of loading

sequence and pretension forces are investigated. A novel

kind of joint connecting rectangular hollow section beam

and column is proposed and tested to obtain the moment-

rotation relationship.

2. Advanced Analysis of Bowstring Structures

2.1. Inelastic beam-column element

Structural steel frameworks can be efficiently modeled

by 3D beam-column plastic hinge element, as shown in

Fig. 1. The element formulation is based on the updated

Lagrangian approach where all physical quantities in

current configuration are referred to the last calculated

configuration. Transverse displacements of the beam-

column element are calculated using the stability interpolation

functions satisfying the equilibrium equation of beam-

column subject to end forces. The beam-column formulations

can capture accurately the member bowing effect and

initial out-of-straightness by modeling each physical

member using only one beam-column element (Liew, et

al., 2000).

A plasticity model that accounts for partial yielding and

hardening is formulated according to the bounding

surface concept proposed by Hilmy and Abel (1985). The

plasticity model employs two interaction surfaces, one

initial yielding surface and the other full yielding

bounding surface, as shown in Fig. 2. The initial yield

surface bounds the region of elastic cross-sectional

behavior, while the bounding surface defines the state of

full plastification of the cross-section. The initial yield

*Corresponding authorTel: +65-6874-2154, Fax: +65-6779-1635E-mail: [email protected]

154 J Y Richard Liew and Jin-Jun Li

surface is assumed to be a scaled down version of the

bounding surface that is fixed in size and translates

without rotation in stress-resultant space. Although the

plasticity model has several parameters for modeling the

behavior of plastic hinges, the one to the most influence

structural inelastic behavior is possibly the size of initial

yielding surface, i.e., surface extension parameter. Surface

extension parameter is actually to be the reciprocal of the

cross-section’s shape factor. But to catch the effects of

initial imperfection of welded members, smaller values

can be selected to reflect pre-emerging initial yielding in

the cross-section due to residual stresses. A discussion

about effects of the surface extension parameter on load-

carrying capacity of CHS columns was provided in

previous work (Liew and Tang, 2000). Once yielding is

initiated, the yield surface will translate so that the state

of sectional resultants remains on the yield surface during

subsequent plastic loading.

2.2. Semi-rigid connection

Modeling of the moment-rotation relationship is a

fundamental requirement for any consideration of

interaction of connection and member behavior. A large

number of experimental tests have clearly demonstrated

nonlinearities between moment and rotation for almost all

types of connections. The four-parameter power model

has been adopted to model a connection (Goldberg and

Richard, 1963; Abbott and Richard, 1975; Liew, 2001)

(1)

where Ke is initial stiffness of connection, Kp strain-

hardening stiffness of connection, M0 a reference moment,

and n shape parameter (Fig. 3). The evaluating procedure

of four parameters in Eq. (1) for the best representation of

the moment-rotation curves has been provided by Attiogbe

and Morris (1991). By differentiating Eq. (1) with respect

to θ, the tangent stiffness of the connection can be

obtained. To allow for unloading of the connection

associated with non-proportional loading and inelastic

force redistribution, the unloading stiffness is assumed to

be equal to the initial stiffness as shown in Fig. 3.

In general, there are two different ways to incorporate

connection flexibility into computer-based structural

analysis of steel frameworks. The first is to introduce

additional connection elements, or say spring elements,

that model the beam-to-column connections directly. In

the second approach each beam-column member with

semi-rigid connections comprises a finite-length member

with a zero-length virtual “rotation spring” attached at

each end which will be represented in the stiffness matrix

of beam-column elements with fixity or rigidity factors

(Xu, 2002). For three-dimensional frame analysis, it is

convenient to model semi-rigid connections with physical

rotational spring elements, since it allows the relative

torsional and flexural rotations between the member end

and the connection. The first method is adopted in the

present analysis.

2.3. Pre-tensioned cable element

Cable may be modeled by line element which can only

resist tension force (Li and Chan, 2004). The elongation

of the cable element is the unique natural deformation.

By analyzing the relationship of elongation with tension

force and incorporating the coordinate transformation, the

incremental elastic stiffness matrix for the cable element

can be written as

MK

eK

p–( )θ

1 KeK

p–( )θ M

0⁄

n+[ ]

1 n⁄---------------------------------------------------- K

pθ+=

Figure 1. 3D beam-column element.

Figure 2. Two surface plasticity model.

Figure 3. Four-parameter power model for semi-rigidconnections.

Advanced Analysis of Pre-tensioned Bowstring Structures 155

(2)

where E is the material modulus of elasticity, A and l0are respectively the sectional area and the original length

of the cable, and C is the direction cosine vector which

can be expressed as

(3)

where the index i takes on value 1, 2, 3 corresponding

to the respective global axes X1, X2 and X3 in Fig. 4.

The geometric stiffness furnishes the relationship

between the change in the global nodal force components

and the global nodal displacements when the element

natural tension is held invariant. Hence, by examining

these nodal tension components before and after

imposition of the nodal displacement, the geometric

stiffness can be expressed as,

(4)

where T is the cable tension and I3 is a 3×3 unit matrix.

The nonlinear stress-strain relationship of the cable

material is assumed to be defined by Eq. (5) originally

proposed by Jonatowski and Birnstiel (1970)

(5)

where ε is cable strain, f stress, E initial elastic modulus

and fu ultimate stress which is taken as the yield stress, fy,

for an elastic-perfectly plastic material, n a constant

relating to the shape of the stress-strain curve.

During unloading and reloading, the stress-strain

relationship is assumed to be linear, with the slope equal

to the initial elastic modulus E. A cable element is

considered to be slacken when its axial strain becomes

less than its permanent strain (see Fig. 5). The axial force

of a slacken cable is set equal to zero, and its axial

stiffness is considered to be negligible.

The pretension force T0 (only positive in tension) in

the cable can be modeled by imposing a temperature

change ∆t (positive as increase) (Liew et al., 2001),

∆t = −T0/αEA (6)

where α is coefficient of thermal expansion, EA axial

stiffness.

3. Bowstring Column

Bowstring steel column was first investigated by Liew

et al. (2001), where effects of geometric imperfection and

cable pre-tension forces were considered. Chan et al.

(2002) presented a stability analysis and parameter study

of pre-stressed cable-stay columns. In this section, the

nonlinear behavior of pre-tensioned bowstring columns is

examined with advanced analysis. Different structural

configurations, cable pretension forces, and length and

spacing of horizontal struts are examined.

3.1. Main concept

Pre-tensioned steel column generally comprises main

column, horizontal strut and cables, as shown in Fig. 6.

The column is the main component which resists gravity

load, while horizontal struts and pre-tensioned cables

work together as the intermediate lateral support to the

main column. The study in this paper is carried out using

only circular hollow sections for the main column and

horizontal struts, due to the aesthetic reasons and the fact

that circular hollow section offers better resistance to

lateral torsional buckling compared to H or I sections. A

summary of the material properties and section size used

for analysis is shown in Table 1. Pre-tensioned steel

columns with length of 10 m, 15 m and 20m are selected

in that their slenderness ratios LV/r (LV is the spacing of

horizontal struts and r is the radius of gyration), 83, 125

Ke[ ]Cable

EA

l0

-------CC

TCC–

T

CC–TCC

T=

Ci

1

l0

--- XJi XIi–( )=

Kg[ ]Cable

1

l0

---I3CC

T– I

3CC

T–( )–

I3CC

T–( )– I

3CC

T–

=

fEε

1Eε

fu------

n+

1 n⁄---------------------------=

Figure 4. Cable element in global coordinate.

Figure 5. Stress-strain curve of cable material.


and 166 respectively, are within the general range of

engineering application.

The bottom end of the pre-tensioned steel columns is

assumed to be restrained in three translational directions

and torsion while free to rotate. However, the support at

the top of the column is free to translate in the vertical

direction and rotate but it is restrained in the other two

translational directions and torsion.

Initial geometric imperfection of L/500 magnitude is

introduced to the main column (L is the column length).

The imperfection pattern is assumed to be the same as the

first buckling mode of the main column.

3.2. Structural configurations

There are many types of cable shapes, or restraining

configurations, for the pre-tensioned steel columns, such

as triangular, rectangular and bowstring shapes as shown

in Fig. 6. The bowstring column configuration is defined

by a second-order polynomial equation which describes a

parabolic curve. Configurations defined with other

higher-order polynomial equations will not be included in

present analysis. The numerical examples are designed

for triangular, rectangular and bowstring columns with

three lengths of 10 m, 15 m and 20 m and with 3

horizontal struts uniformly spaced. The maximum length

of horizontal struts at mid-span is assumed as L/16 (L is

the column length) for three configurations. The same

cable forces of 11 kN are pre-tensioned for different

column lengths and different column configurations. All

pre-tensioned columns are subjected to initial geometric

imperfection and axial compression force. The results of

axial load capacities are listed in Table 2 and the

corresponding failure modes are shown in Fig. 7.

Figure 6. Three types of pre-tensioned steel column: (a)triangular; (b) rectangular; (c) bowstring.

Table 1. Summary of the material properties used foranalysis of bowstring columns

Elements Material properties Section size

Vertical strutfy = 275 MPaE = 205 GPa

CHS 88.9×4.0 mm

Horizontal strutfy = 355 MPaE = 205 GPa

CHS 60.3×4.0 mm

Pre-tensioned cablefy = 460 MPaE = 150 GPa

Diameter 10 mm

fy is the design strength and E is the Young’s modulus

Table 2. Axial load capacities of different columnconfigurations with different lengths (Unit: kN)

Length

Configuration10 m 15 m 20 m

Triangular 122 94 59

Rectangular 165 115 69

Bowstring 183 121 74

Figure 7. Failure modes of pre-tensioned steel columns with lengths of 10 m, 15 m and 20 m: (a) triangular; (b) rectangular;(c) bowstring.


The bowstring column has the highest axial load

capacity as compared to the triangular shape column the

lowest capacity. The rectangular columns can be the

alternative choice if the column has relative large

slenderness ratio (more than 100). For triangular cable

shape, the horizontal struts at quarter-span are not fully

effective to provide lateral restrain forces to the main

column so that bulking occurs earliest. Although

insufficient support also exists at mid-span horizontal

strut due to the equal length for all struts in rectangular

cable shape, this disadvantageous effect will decrease

with the development of lateral deflection at mid-span.

Bowstring cable shape can provide a harmonious lateral

restrain forces to both mid-span and quarter-span

horizontal struts, and give a highest capacity. Since

bowstring column is the most efficient configuration, it

will be used in the subsequent sections to investigate the

effects of cable pre-tension force, length and spacing of

horizontal struts.

3.3. Cable pre-tension force

Cable pre-tension force is an important parameter that

affects the stiffness and strength of the pre-tensioned steel

columns. The effects of cable pre-tensions on the load

capacity of bowstring columns with lengths of 10 m,

15 m and 20 m are investigated. The columns studied

here have 3 horizontal struts uniformly spaced and the

maximum length of horizontal struts at mid-span is

assumed as L/16 (L is the column length). It is assumed

that the yield stress of cable material is 460 Mpa. Since

the diameter of cables is 10 mm, the yielding load of the

cable is about 36 kN. The bowstring columns are

analyzed using advanced analysis by assuming different

pre-tensioning forces in the cables. The results are

presented in Fig. 8 showing the axial capacity of the

column versus the pre-tension force to which is

normalized by the cable yield load, Ty. Cable pre-tensions

equal to 5% of cable yield load can evidently improve the

column capacity when compared to the cases without

cable pre-tensions. The preferred range of cable pre-

tension force is related to the column slenderness and can

be found generally between 30% and 50% of cable yield

strength. When the pre-tension force is too high (more

than 50% of the cable yield strength) the axial capacity of

bowstring columns will reversely decrease, as shown in

Fig. 8. This is because high pre-tensioning force in the

cables will introduce high compression load on the main

column and thus reduce its load-bearing capacity.

3.4. Length of horizontal strut

The length of horizontal struts in bowstring columns is

limited for the sake of aesthetic reasons and restraint

effectiveness. By changing the length of horizontal struts

of bowstring columns, its effects on the column ultimate

capacities can be examined. The numerical results are

listed in Table 3, where the three lengths of horizontal

struts studied are L/8, L/16 and L/24 (L is the column

length). For the 10 m long columns, yielding of

horizontal struts are not observed in numerical analysis

and the longer the horizontal struts, the higher the column

capacity. However, in the case of L/8 for 15 m and 20 m

long columns, yielding of horizontal struts occurs before

column failure and causes the pre-matured collapse of

columns when compared with the case of L/16, as shown

in Table 3. If the section of horizontal struts is

strengthened and replaced by CHS 80.0×4.0, the

capacities are 135 kN and 76 kN respectively for 15 m

and 20 m long bowstring columns with L/8 long

horizontal struts, which are higher than those of L/16

because yielding of horizontal struts are prevented. It can

be drawn from investigation in this section that horizontal

struts should be adequately designed to prevent pre-

matured yielding or buckling and when longer struts are

used more attention should be paid to the possibility of

yielding of the horizontal struts.

Figure 8. Axial load capacity vs. cable pretensions forbowstring columns.

Table 3. Axial load capacities of bowstring columns ofdifferent lengths of horizontal strut and column lengths

(Unit: kN)

Column Length (L)

Horizontal strut length10 m 15 m 20 m

L/8 210 103 63

L/16 183 121 74

L/24 118 104 77

Table 4. Axial load capacities for bowstring columns ofdifferent number of horizontal strut and column lengths

(Unit: kN)

Column Length (L)

Number of horizontal strut10 m 15 m 20 m

3 183 121 74

4 194 154 96

5 196 163 113


3.5. Spacing of horizontal strut

The spacing of horizontal struts LV is varied as L/4, L/

5 and L/6 for 10 m, 15 m and 20 m long bowstring

columns. The length of horizontal struts is kept to be L/

16 and the cable pre-tensioned to be 11 kN as above. The

axial capacities obtained from advanced analysis are

listed in Table 4. A general conclusion can be drawn that

the more number of horizontal struts are provided, the

higher compressive capacity, especially for the columns

with relatively high slenderness ratio. This is because of

the reduction of the effective buckling length of the main

column between two lateral restrained points. However,

increasing the number of restrained points will not further

increase the capacity because the failure will be caused

by the overall buckling of the bowstring column.

4. Bowstring Frame

The concept of bowstring column can be extended to

3D frames, if the stability of support columns in the out-

of-plane is ensured by bowstring cables while that in-

plane stability is provided by frame action. Based on the

study of bowstring column, structural behavior of

bowstring frame is analyzed in this section, where the

structural layout, novel semi-rigid connection, structural

nonlinearities under environmental loads and effects of

cable pretensions are addressed.

4.1. Structural layout

A bowstring structure supporting a glass facade is

shown in Fig. 9. It consists of seven dependent frames (3

on left and 4 on right) plus one lateral bracing frame. The

detailed model of a bracing frame is shown in Fig. 10,

which includes the elevation view in X-Y and Y-Z planes.

When the bowstring frames are assembled, pre-tension

force is applied to the bowstring cables and bracing

cables to provide lateral stiffness in the in-plane and out-

of-plane direction of the frame. After roof slab and glass

façade are constructed, the bowstrings are under

compression and a second stage pre-tension force is

applied to the cables so that they can have sufficient

lateral stiffness to resist further loadings such as imposed

roof load and wind load. The load sequences of the

bowstring structures are tabulated in Table 5.

4.2. Testing of mullion-to-transom and transom-to-

hanger connection

Mullion-to-transom and transom-to-hanger connections

of frame in-plane are designed as those of semi-rigidity.

Figure 9. The structural model of bracing bowstringframe and a roof slab.

Figure 10. The structural model of bracing bowstring frame: (a) X-Y plane; (b) Y-Z plane.


The rotational stiffness and moment capacity of these two

connections are tested so that nonlinear analysis models

of bowstring frames can be correctly established and then

the structural behavior can be assessed.

The test setup and specimen instrumentations are

illustrated respectively in Fig. 11. Test specimen 1

represents the hanger (100×100×4 mm RHS) to transom

(150×100×5 mm RHS) connection, which consists of one

M24×60 Grade 8.8 HEX. HD. bolt connecting the 20 mm

plate welded on the inside of the hanger to the 30 mm

Table 5. Loads and loading sequence in analysis

Load sequence Load type Load description

1 1st stage pre-tensiona) Tension inside and outside cables to 52 kN and 50 kN respectively;b) Tension bracing cables to 10 kN

2 Deal load

a) Apply dead loads = 301 kN, at upper column points;b) Apply glass façade loads of 1.75 kN/m on vertical mullions and 1.4 kN/m on transoms;c) Apply self-weight

3 2nd stage pre-tensiona) Tension the inside and outside cables to 78 kN and 82 kN respectively;b) Tension bracing cables to 10 kN

4Lateral bracing force(see Fig. 10b)

a) Apply bracing force at left side = 18.834 kN per node;b) Apply bracing force at right side = 23.082 kN per node

5 Temperature effectTemperature inside glass façade reduces 15oC and that outside increases 25oC simultaneously

6

Wind pressure Apply wind pressure = 2.0 kN/m on mullions and 1.6 kN/m on transoms, up to failure

Wind suction Apply wind suction = 1.4 kN/m on mullions and 1.12 kN/m on transoms, up to failure

Figure 11. The test set-up and specimen instrumentation of semi-rigid connections in bowstring frames: (a) set-up; (b)specimen instrumentation; (c) elevation view of connections; (d) A-A view of specimen 1; (e) A-A view of specimen 2.


plate welded on the surface of the transom (see Fig. 11

(c)). Test specimen 2 represents the mullion (200×100×

10 mm RHS) to transom (150×100×5 mm RHS) connection,

which consists of two M24×60 Grade 8.8 HEX. HD.

bolts connecting a 20 mm plate welded on the inside of

the transom to a 30 mm plate welded on the surface of the

mullion (see Fig. 11(c)). The moment-rotation curves

obtained from the tests are shown in Fig. 12. The test

results are fitted with the four-parameter connection

model in Eq. (1) and then used in the analysis, as shown

in Fig. 12.

For specimen 1, the left side of the connection is stiffer

than the right. This could be due to non-symmetry in the

specimens and the fitting of connections. Failure of

connection occurred due to yielding of the transom

member caused by the compression force from the hanger

beam. Weld failure, as shown in Fig. 13, was observed

upon subsequent loading after attaining the maximum

load. The moment capacity is about 8kN-m. For

specimen 2, the transom to mullion connection failed

when the flange of the transom which connects into the

mullion yielded under compression. The moment

capacity of the connection is about 17 kN-m. Failure

modes of two specimens are shown in Fig. 13.

Experimental results provided reliable parameters for

modeling the nonlinear rotational spring elements used in

the structural analysis.

4.3. Nonlinear analysis

By preliminary calculation, two critical load

combinations of the bracing frame are identified:

1) 1.0 (dead load) + lateral bracing load + ambient

temperature difference + wind pressure;

2) 1.4 (dead load) + lateral bracing load + ambient

temperature difference + wind suction.

Lateral bracing loads in the direction of Z axis (in-plane

of frame) represent the required bracing forces provided

by the bracing frame to the other seven dependant frames

Figure 12. Moment-rotation curve of semi-rigid connections:test data vs four-parameter model in Eq (1) for specimen 1(hanger-to-transom) and specimen 2 (mullion-to-transom).

Figure 13. Connection detail and failure modes: (a) specimen1 (hanger-to-transom); (b) specimen 2 (mullion-to-transom).

Figure 14. Nodal displacement vs load factor of wind at ambient temperature: (a) wind pressure; (b) wind suction.


within one load-bearing unit (see Fig. 10 and Table 5).

According to BS5950-1 (2000), where three or more

intermediate lateral restraints are provided each intermediate

lateral restraint should be capable of resisting a force of

not less than 1% of the compression force.

For sake of simplification, only nonlinear analysis

results of these two load cases are presented in this paper.

The loads and loading sequences used in analysis are

listed in Table 5.

The typical load-displacement curves and cable tension

variations are illustrated in Fig. 14 and Fig. 15. Nonlinear

structural response is observed when wind loads are

applied. The nonlinearity is due to second-order effects

associated with the axial force acting on the deflected

geometry of the structure, the change in cable tension and

nonlinear behavior of semi-rigid connections. No plastic

hinge is observed up to the structural failure, while

buckling of vertical mullions occurs at the ultimate states.

One of reinforcing cables to vertical mullions will slack

soon after wind loads applied, as illustrated in Fig. 15, but

this does not trigger the collapse of the system because

the other reinforcing cable is still in tension and the structure

is capable of resisting more loads. Large deflections, in

both X-direction and Z-direction, are observed when

instability occurs. But the deflection in X-direction is

dominant rather than that in Z-direction, which implies

the bracing cables are providing effective lateral stiffness

in the frame plane to other dependant frames.

4.4. Effects of pretension

As mentioned above, two-stage pre-tensioning technique

is needed in the construction of the bowstring frames

because the weight of roof and glass façade will reduce

the cable pre-tension forces and the frame stiffness

against wind loads will be insufficient if only first-stage

pre-tensioning applied. In this section, the effects of

pretension forces on the capacity and stiffness of the

bowstring frame, represented with five pretension cases,

case (a to e) as shown in Table 6, are studied.

Fig. 16 illustrates the curves of displacement in X-

direction of Node A with pressure wind factor under

different magnitudes of pretension forces. Local

instability has been found from the numerical results for

the states of “Pretension a” and “Pretension b”, which

indicates that both the first and second stage pretensions

are both essentially necessary for bowstring structures.

“Pretension e” does not produce an evidently higher

ultimate capacity against the wind pressure, compared to

“Pretension d” and “Pretension c”. The rationality of

“Pretension d”, namely the designed pretension force, is

proved as well from Fig. 16 where a satisfied structural

stiffness at nominal pressure wind load (factor = 1) is

shown.

5. Conclusion

This paper studies the nonlinear behaviors of bowstring

steel structures with advanced analysis, where modeling

techniques used include those for steel beam-columns,

semi-rigid connections and tensioned cables.

It can be found that for pre-tensioned steel columns, the

bowstring cable shape has relatively higher capacity than

triangular and rectangular shapes. The preferred cable

pre-tensioned forces are ranging from 30% to 50%

Figure 15. Cable tension vs. load factor of wind atambient temperature: (a) wind pressure; (b) wind suction.

Figure 16. Effects of cable Pretensions (see Table 6 forcable pre-tension cases).

Table 6. Five pretension cases for bowstring frame

CaseFirst stage

pretension (kN)Second stage

pretension (kN)

a 0 0

b 50 0

c 50 45

d 50 80

e 50 115


yielding strength of cables. The maximum length of

horizontal struts at mid-span of bowstring columns can

generally be 1/16 of column length and for longer struts

more attention should be paid to preventing strut from

yielding as larger restrain forces are required. The spacing

of horizontal struts depends more on the column capacity

required and when the column is relatively short L/4

spacing, namely three sets of struts along column length,

is sufficient and recommended.

In addition to bowstring columns, a bowstring frame is

presented and analyzed with advanced analysis. The

structural layout and loading sequence of bowstring frame

are described. A novel kind of connectors connecting

rectangular hollow section beam and column, with one or

two bolts, has been tested to obtain the moment-rotation

relationship. Effects of cable pretension forces are

examined and the two-stage pre-tensioning of cables is

necessary for the bowstring structures simultaneously

subjected to vertical and lateral loads. Although the

higher pretensions cannot produce evident increase of

ultimate capacity, it substantially provides better stiffness

to reduce deflection due to wind load.

References

Abbott, B J and Richard, R M (1975), “Versatile elastic-

plastic stress-strain formula”, Journal of Engineering

Mechanics Division, ASCE, V101(4), pp. 511-515.

Attiogbe, G and Morris, G (1991), “Moment-rotation

functions for steel connections”, Journal of Structural

Engineering, ASCE, V117(6), pp. 1703-1718.

Barnes, M. and Dickson, M. (2000), Wide span roof

structures, Thomas Telford.

Bradshaw, R et al. (2002), “Special structures: past, present

and future”, Journal of Structural Engineering, ASCE,

V128(6), pp. 691-709.

BS5950-1 (2000), Structural use for steelwork in building,

Code of practice for design - rolled and welded sections

Chan, S L, Shu, G P and Lü Z T (2002), “Stability analysis

and parameter study of pre-stressed stayed columns”,

Engineering Structures, V24, pp. 115-124.

Dong, S. L. and Zhao, Y. (2004), “Pre-tensioned long-span

steel space structures in China”, International Journal of

Applied Mechanics and Engineering, V9(1), pp. 25-36.

Goldberg, J E and Richard, R M (1963), “Analysis of non-

linear structures”, Journal of Structural Engineering,

ASCE, V89(8).

Hilmy, S L and Abel, J F (1985), “A strain-hardening

concentrated plasticity model for nonlinear analysis of

steel buildings”, Proceedings of NUMETA’85, pp. 301-

314.

Jonatowski, J J and Birnstiel, C (1970), “Inelastic stiffened

suspension space structures”, Journal of Structural

Division, ASCE, V96(6), pp. 1143-1166.

Kosaka, K et. al. (1988), “Development of pre-stressed steel

truss “super wing””, Kawasaki Steel Giho, V20(4), pp.

315-324.

Li, J J and Chan, S L (2004), “An integrated analysis of

membrane structures with flexible supporting frames”,

Finite Elements in Analysis and Design, V40, pp. 529-

540.

Liew, J Y R and Tang, L K (2000), “Advanced plastic hinge

analysis for the design of tubular space frames”,

Engineering Structures, V22(7), pp. 769-783.

Liew, J Y R, Chen, H, Shanmugam, N E and Chen, W F

(2000), “Improved nonlinear plastic hinge analysis of

space frame structures”, Engineering Structures, V22(10),

pp. 1324-1338.

Liew, J Y R (2001), “State-of-the-art of advanced inelastic

analysis of steel and composite structures”, J. Steel and

Composite Structures, Techno Press, V1(3), pp. 341-354.

Liew, J Y R, Punniyakotty, N. M. and Shanmugam, N. E.

(2001), “Limit-state analysis and design of cable-

tensioned structures”, International Journal of Space

Structures , V16(2), pp. 95-110.

Wang, B B (2004), Free standing tension structures: from

tensegrity to cable-strut system, Spon, New York.

Xu, L (2002), “Design and optimization of semi-rigid

framed structures”, in: Recent advances in optimal

structural design, Chapter 7, Edited by Burns, S. A.,

ASCE.

advanced analysis of pre-tensioned bowstring structures

Documents