Determination Method on Initial Pretension of Cables in Steel Mega Frame and Pre-stressed Composite Bracing Structure

TANG Bai-jian(),WANG Fei( ),GU Sheng( )

School of Architecture and Civil Engineering,Jiangsu University of Science and Technology,Zhenjiang 212003,China

Introduction

The concept of mega structures appeared from the 1960’s.Nowadays,mega frame structures and mega truss structures have been widely studied and applied since the Chicago Hancock Center was built.The Tokyo City Hall,the NEC Building,and the Taipei 101 Building[1-2]are typical mega frame structures.Mega frame structures have the advantage of simple member arrangement.Their main disadvantage is that their total height is limited due to their weak lateral rigidity,less than that of mega truss structures.However,joints in a mega truss structure are complicated and difficultly constructed.In order to improve the lateral rigidity of mega frame structures,kinds of bracing systems were investigated and applied[3-5].Tang and Ruan[6]presented a steel mega frame and pre-stressed composite bracing structure (SMFPCBS) with composite bracing systems.Each composite bracing is composed of two rigid Λ shaped steel mega bracings and two flexible V shaped pre-stressed cables.

The initial prestress value is a key parameter for designing a pre-stressed steel structure.Chanetal.[7]researched the initial prestress value of pre-stressed cable-stayed columns with initial imperfection by using the robust point-wise equilibrium polynomial element satisfying equilibrium in moment and shear at mid-span in association with the cable clement.Tang and Zhu[8]obtained optimum initial prestress in a stayed steel column through a general mechanical model and static equilibrium equations.De Araujoetal.[9]studied the structural behavior of double-stayed steel columns by means of an experimental program followed by finite element (FE) simulations aiming to determine the most efficient structural geometries and the corresponding steel ties prestress force magnitudes.Dongetal.[10-11]developed an improved singular value decomposition method,referred to as the DSVD,for the determination of the integral prestress modes for various cable domes.By using this method,the prestress design process of Geiger domes was presented.Cao and Zhang[12]proposed a simplified computational strategy for the determination of the self-internal-force mode based on the nodal equilibrium for the tensegrity system in a suspendome which is grounded on the local analysis method.Tang and Cai[13]obtained prestress values for designing the reticulated dome structure of Wuhan Sports Center Gymnasium based on the principle that pre-stressed cable can control the deformation of the reticulated dome structure.Xue and Liu[14]took the beam string structure (BSS) in Shanghai Yuanshen Arena as a sample.The prestress force of the BSS was optimized through a parametric analysis using ANSYS.

To determine reasonable initial cable pretension values in a steel mega frame and pre-stressed composite bracing structure,the influencing coefficient leveling method is developed based on the equilibrium of internal forces in composite bracings.Numerical examples are given to verify this method.

1 Mechanical Analysis

1.1 Internal force disequilibrium in composite bracings under vertical loads

Fig.1 Single-story mega frame structure

Fig.2 Mechanical analysis at a joint under vertical uniform load q

According to the horizontal force equilibrium of the joint,we have

(1)

That is

So that

Nq>Pq.

(2)

It can be concluded that shear force in a mega column causes internal force disequilibrium in the composite bracing under vertical loads.

1.2 Leveling internal forces in composite bracings

The unbalanced internal force acting on the column increases shear force and bending moment in the column,which can easily lead to the column buckling.To eliminate this disequilibrium,leveling internal force in the composite bracing is necessary.

For a symmetrical structure under symmetrical loads,leveling internal force can only be realized through symmetrical means.In a steel mega frame and pre-stressed composite bracing structure,two pretensions in V shaped cables are symmetrical about the structural symmetric axis if each cable in the same large story is stretched at the same degree.Therefore,leveling internal force can be achieved conveniently through tensile cables.The free-body diagram of a single-story mega story frame under two symmetrical pretensions is shown in Fig.3.

Fig.3 Mechanical analysis under two symmetrical pretensions T

Using the principle of force equilibrium at the horizontal direction,we have

(3)

That is

So

PT>NT.

(4)

Combining Eq.(4) with Eq.(2),if we choose a suitable magnitude ofT,we can establish the following equation

Nq+NT=Pq+PT.

(5)

It means that internal force disequilibrium in composite bracings can be removed by adjusting initial cable pretension values.

Under asymmetric horizontal loads,internal force disequilibrium in composite bracings can not disappear by adjusting cable pretension values for the reason that the axial force distribution in a symmetrical structure is asymmetrical under asymmetrical horizontal loads[15].As a consequence,pretensions are decided only by symmetrical vertical loads in a symmetrical steel mega frame and pre-stressed composite bracing structure.Thus,the criterion for determining initial cable pretensions is the internal force equilibrium in composite bracings when vertical loads and pretensions are applied at the same time.

2 Influencing Coefficient Leveling Method

For a multi-story mega-braced frame with composite bracings，it is still true that the axial force of the mega bracings is larger than that of the cables in the same story when interactions between story and story are considered[16].

As shown above,internal force disequilibrium caused by symmetrical loads imposed on a symmetrical structure can be erased by symmetrical initial cable pretensions.

The internal force equilibrium is satisfied if

(6)

In a steel mega frame and pre-stressed composite bracing structure,cables are short enough to neglect their nonlinear effects.Therefore,when all the cables are jacked at the same time,internal force change of composite bracings can be seen approximately as superposition of internal forces caused by each cable pretension at each large story.So for a steel mega frame withnlarge stories,

(7)

(8)

Substituting Eqs.(7) and (8) into Eq.(6),we have

(9)

3 Numerical Examples

Using the software SAP2000,the numerical 2-D FE models of the mega frame and pre-stressed composite bracing structures are developed to validate the suggested method (Fig.4).This 54-story frame structure,diagonally symmetrical,has the same 4 m floor height.Its total height is 216 m and its square plan dimensions are 36 m×36 m.There are four same 6 m×6 m square lattice mega columns at each corner.Each mega column,consisting of four box-sectioned columns,connects to each other through a Λ shaped bracing.The mega beams connecting to the mega columns are set up at Levels 9,18,27,36,45,and 54.Each mega beam with a height which is the same as the floor height is composed of four chords,several vertical and oblique web members.At the upper part of each large story,there are two rigid Λ shaped mega bracings,and at the lower part there are two flexible V shaped pre-stressed cables (Fig.4 (a)).Every rigid mega bracing is a steel truss.The main member sizes are listed in Table 1.Four different models are considered for different cable diameters or top sub-frame layer numbers (Table 2).

(a) Elevation of the main structure (b) Elevation of the whole structure

Table 1 Member sizes in numerical models

Table 2 Computation models

All the beams and columns are modeled using the 2-D beam element.All the trusses in beams,columns,and bracings are modeled using the 2-D truss element.All the slabs are modeled using the 4-node shell element,and the cables are modeled using the cable element.FE models incorporate nonlinear deformation characteristics.

On each large story,the dead load is 6.0 kN/m2and the live load is 2.5 kN/m2.On each floor,the dead load of 4.5 kN/m2and the live load of 2.5 kN/m2are imposed.

Symbols c1-c6 stand for the cables,and b1-b6 are the mega bracings from the ground to top large story.

3.1 Influencing coefficients

Using SAP2000,the influencing coefficients are obtained when the pretensionsT1-T6from the ground to sixth large story applied to the mega frame in turn (Fig.5).

(a) T1

(b) T2

(d) T4

(e) T5

(f) T6

From Fig.5,we find as follows.

(1) The influencing coefficients at adjacent stories are opposite in sign.For example,underT2the influencing coefficients of b2 and c2 in Story 2 is positive but those in Stories 1 and 3 are negative.The absolute values of influencing coefficients reflect the influence of the pretension on the axial forces of composite bracings.

(2) The pretension influence fades away due to force diversion at junctions.The more joints the pretension flows through,and the smaller the influencing coefficient is.For example,in Model 1 the influence ofT1on b6 is so small that it can be neglected (Fig.5 (a)).

(3) The influencing coefficient decays quickly upwards but slowly downwards for the reason that the pretension flow upwards through a K shaped node.For example,underT4the influencing coefficient of b3 is -0.71 but -0.32 for c5.Namely,the pretension affects the substructure in greater extent than the superstructure.This conclusion can be applied to planning a jacking scheme.

(4) Compared Model 1 with Model 2,when the cable diameter increases,influencing coefficients decay at a slower rate.

3.2 Leveling internal forces in composite bracings

The internal force under vertical loads has been shown in Fig.6.It can be seen that the cable tension is larger than the compression of the mega bracing at the same large story no matter in Model 1 or in Model 2.Their difference is largest at the ground large story and reaches 1794.14 in Model 1 as two cables at the first story are anchored to the ground.Compared Model 1 with Model 2,Fig.6 exhibits that internal forces in composite bracings are influenced by cable diameters.

Fig.6 Internal forces in composite bracings only under vertical loads

The results of leveling internal force by using the influencing coefficient method proposed above are shown in Fig.7.The coincidence of internal forces in Models 1 and 2 explains cable diameters have no influence on internal force equilibrium.

Fig.7 Internal forces of composite bracings under vertical loads and prestresses

3.3 Fluctuation of internal force in composite bracings

Compare the balanced internal forces among Models 1,3,and 4 (Fig.8).Models 1,3,and 4 have different sub-structural floor numbers at the top large story.

Fig.8 Internal force fluctuation of composite bracings

In Model 3,there is a great difference of internal forces in different composite bracings.The internal force at the sixth large story is 8.67 times of that at the fifth large story.The internal force at the fourth large story is 4.48 times of that at the third large story.And the internal force at the second large story is 3.3 times of that at the first large story.The internal force difference values decrease from the top to bottom story.In each odd story,the axial force in the mega bracing located on the windward side is small and tends to change from compression to tension under wind loading.At the same time,some cables step out of working.This is not a good situation for the mega column.

The internal force difference in different composite bracings can be explained as follows: the vertical load on the top large story is undertaken only by two mega bracings.However,in other stories it is carried by both the Λ shaped mega bracings and the V shaped pre-stressed cable bracings.As a result,the axial force in each mega bracing at the top story is larger than those at other stories.In order to reach internal force equilibrium,the cable tension at the top story is larger than those at other stories.A large cable tension at the top story causes a little mega bracing compression and a little cable tension at the next lower story.

In Model 1,there is a smaller difference of internal forces in different composite bracings compared with Model 3.The internal force at the sixth large story is 1.51 times of that at the fifth large story.The internal force at the fourth large story is 1.34 times of that at the third large story.And the internal force at the second large story is 1.25 times of that at the first large story.

In Model 4,there is almost no difference of internal forces in different composite bracings.The internal force at the sixth large story is 0.99 times of that at the fifth large story.The internal force at the fourth large story is 0.96 times of that at the third large story.And the internal force at the second large story is 0.94 times of that at the first large story.

4 Conclusions

Based on the force flow mechanism of composite bracings,we have found the decisive factors for determining cable pretension values.Then,according to the criterion that the axial force values in a composite bracing must be equal,the influencing coefficient leveling method is proposed.The main conclusions are as follows.

(1) Under vertical loading,internal forces in a composite bracing are unbalanced due to the existence of the column shear rigidity,and the cable tension is less than the axial force in the mega bracing.

(2) Internal force disequilibrium in composite bracings can be eliminated only by symmetrical loads.For leveling internal forces,a pair of symmetrical initial cable pretensions is a good choice.

(3) The initial pretension is decided only by the vertical loading.The determination criterion for initial cable pretensions is the internal force equilibrium of composite bracings when vertical loads and initial cable pretensions are imposed at the same time.

(4) The influencing coefficient leveling method is feasible,reliable,and accurate.

[1] Zhang X A,Wang D,Jiang J S.The Controlling Mechanism and the Controlling Effectiveness of Passive Mega-Sub-controlled Frame Subjected to Random Wind Loads [J].JournalofSoundandVibration,2005,283(3/4/5): 543-560.

[2] Fan H,Li Q S,Tuan A Y,etal.Seismic Analysis of the World’s Tallest Building [J].JournalofConstructionalSteelResearch,2009,65(5): 1206-1215.

[3] Sarno L D,Elnashai A Y.Bracing Systems for Seismic Retrofitting of Steel Frames [J].JournalofConstructionalSteelResearch,2009,65(2): 452- 465.

[4] Moon K S.Stiffnessed-Based Design Methodology for Steel Braced Tube Structures: a Sustainable Approach [J].EngineeringStructures,2010,32(10): 3163-3170.

[5] Kari A,Ghassemieh M,Abolmaali S A.A New Dual Bracing System for Improving the Seismic Behavior of Steel Structures [J].SmartMaterialsandStructures,2011,20(12): 120-125.

[6] Tang B J,Ruan H T.Analysis on Performance of Mega Steel Frame and Prestressed Composite Brace Structure [J].JournalofGuangxiUniversity:NaturalScienceEdition,2010,35(4): 574-581.(in Chinese)

[7] Chan S L,Shu G P,Lü Z T.Stability Analysis and Parametric Study of Pre-stressed Stayed Columns [J].EngineeringStructures,2002,24(1): 115-124.

[8] Tang B J,Zhu J J.Perfect Theoretical Equation of Optimum Initial Prestress of Stayed Steel Column [J].EngineeringMechanics,2011,28(9): 143-148.(in Chinese)

[9] de Araujo R R,de Andrade S A L,da S Vellasco P C G,etal.Experimental and Numerical Assessment of Stayed Steel Columns [J].JournalofConstructionalSteelResearch,2008,64(9): 1020-1029.

[10] Yuan X F,Chen L M,Dong S L.Prestress Design of Cable Domes with New Forms [J].InternationalJournalofSolidsandStructures,2007,44(9): 2773-2782.

[11] Wang Z H,Yuan X F,Dong S L.Simple Approach for Force Finding Analysis of Circular Geiger Domes with Consideration of Self-weight [J].JournalofConstructionalSteelResearch,2010,66(2): 317-322.

[12] Cao Q S,Zhang Z H.A Simplified Strategy for Force Finding Analysis of Suspendomes [J].EngineeringStructures,2010,32(1): 306-318.

[13] Tang H,Cai Y Q.Application Study on Cable Prestress Design for Large-Scale Reticulated Shell Structure [J].EngineeringJournalofWuhanUniversity,2006,39(2): 119-122.(in Chinese)

[14] Xue W C,Liu S.Design Optimization and Experimental Study on Beam String Structures [J].JournalofConstructionalSteelResearch,2009,65(1): 70- 80.

[15] Tang B J,Gu S,Liu H P.Determination Method on Diameter of Cables in Mega Steel Frame and Composite Brace Structure [J].JournalofCivil,Architectural&EnvironmentalEngineering,2012,34(6): 39- 45.(in Chinese)

[16] Gu S.Determination Theory on Cable’s Sectional Area and Prestress in New Vertical Pre-stressed Steel Structure [D].Zhenjiang: Jiangsu University of Science and Technology,2012.(in Chinese)