Advances in Steel Structures - part 47 pptx

440 Z. Wang and Y. Zhen Figure 1. The comparison between theoretical calculation and the test result Research on the Hysteretic Behavior of High Strength Concrete 441 Figure 2. The comparison between theoretical calculation and the test result in reference 4 Figure 3. The comparison between theoretical calculation and the test result in reference 5 THE FORCE-DISPLACEMENT HYSTERETIC LOOP CHARACTERISTIC We can find some characteristics of force-displacement hysteretic loop of high strength concrete filled steel tubular members from the theoretical analysis and experiment research result. a) The shape of hysteretic loop is closed to that of steel member under the condition of without local buckle. And it is also analogous to the loop of general concrete filled steel tube member. 442 Z. Wang and Y. Zhen b) No matter how the parameters change, the hysteretic loop has great plumpness and no pinched or reduced phenomenon appear. 4.CONCLUSION The calculation method in this paper has its new characteristics on how to select constitutive relationship and construct the model of finite element. On the basis of theoretical analysis and experiment study, the characteristic of force-displacement hysteretic loop of HCFST under compression and bending are discussed. From above, I think the following should be further studied: a) The basic property of polygon HCFST should be studied by making use of programme in this paper. b) The property of the member of eccentric compression should be studied by making use of calculating method in this paper. c) The lateral force resisting property of short column of HCFST should be studied considering of shear deflection. REFERENCE l.Shantong Zhong.(1994). Concrete filled steel tubular structures. Heilongjiang science and technology press,. 2.Linhai Han.(1996) Mechanics of concrete filled steel tubular. Dalian science and engineering university press. 3.Yonghui Zhen.(1998). The hysteretic behavior studies of high strength concrete filled steel tububular members subjected to compression and bending. Master thesis of HUAE. 4.Weibo Yan.(1998). Theoretical analysis and experimental research for the hysteretic behaviors of high strength concrete filled steel tubular beam-columns. Master thesis of HUAE. 5.Yongqing Tu.(1994). The hystersis behavior studies of concrete filled steel tubular membersw subjected to compression and bending. Doctor thesis of HUAE. DESIGN OF COMPOSITE COLUMNS OF ARBITRARY CROSS- SECTION SUBJECT TO BIAXIAL BENDING S. F. Chen 1, j. G. Teng 2 and S. L. Chan 2 1 Department of Civil Engineering, Zhejiang University, Hangzhou 310027, China 2 Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China ABSTRACT In this paper, an iterative Quasi-Newton procedure based on the Regula-Falsi numerical scheme is proposed for the rapid design of short concrete-encased composite columns of arbitrary cross-section subjected to biaxial bending. The stress resultants of the concrete are evaluated by integrating the concrete stress-strain curve over the compression zone, while the stress resultants of the encased structural steel and the steel reinforcing bars are obtained using the fiber element method. A particularly important feature of the present method is the use of the plastic centroidal axes of the cross-section as the reference axes of loading in the iterative solution process. This ensures the convergence of the solution process for all cross-sectional conditions. Numerical examples are presented to demonstrate the validity, accuracy and capacity of the proposed method. KEYWORDS Composite Columns, Arbitrary Cross-Sections, Irregular Cross-Sections, Structural Design, Biaxial Bending INTRODUCTION Composite steel-concrete construction has been widely used in many structures such as buildings and bridges. The concrete-encased composite column is one of the common composite structural elements. Many studies have examined the behaviour and strength of biaxially loaded composite columns of doubly-symmetric cross-sections. Several researchers, including Johnson and Smith (1980), Lachance (1982) and Roik and Bergmann (1984) proposed simple methods for the analysis and design of rectangular composite columns under biaxial loading. E1-Tawil et al. (1995) developed an iterative computer method for biaxial bending of encased composite columns using the fiber element method and generated numerical results to evaluate the uni- and biaxial bending strengths of composite columns predicted by ACI-318 (1992) and AISC-LRFD (1993) provisions. Munoz and Hsu (1997) proposed a generalized interaction equation for the analysis and design of biaxially loaded square and 443 444 S.F. Chen et al. rectangular columns. They compared their results with test results of many columns and predictions of the current ACI (1992) and AISC (Manual 1986) design methods. In the design of building comer columns, comer walls, and core walls, irregular cross-sections or regular cross-sections with asymmetrically placed structural steel and/or steel reinforcement are often used to suit irregular plan layouts and/or eccentric load requirements. Little work has been carried out on composite columns of such irregular sections. Design rules for these columns also do not exist in design codes such as ACI-318 (1992) and EuroCode 4 (1994). Roik and Bergmann (1990) appears to have presented the only study on the design of rectangular composite section with an asymmetrically placed steel section. They proposed an approximate design method, which is a simple modification of the design approach for composite columns with a doubly symmetrical cross-section given in Eurocode 4 (1984). Only sections with the structural steel mono-symmetrically placed were considered. Rotter (1985) presented a moment-curvature analysis of arbitrary sections subject to axial load and biaxial bending using Green's theorem in integration which can be used to analyze composite steel-concrete columns. The emphasis of his study is on predicting the moment-curvature relationship rather than the design of such sections. This paper provides a brief description of a general iterative computer method for the rapid design and analysis of arbitrarily shaped concrete-encased composite columns with arbitrarily distributed structural steel and steel reinforcement subjected to biaxial bending. A detailed presentation of the method is given in Chen et al. (1999). The method employs the iterative Quasi-Newton procedure within the Regula-Falsi numerical scheme. The stress resultants of the concrete are evaluated by integrating the concrete stress-strain curve over the compression zone, while those of structural steel and reinforcement are obtained using the fiber element method, in which the steel sections are discretized into small areas (fibers). Numerical examples are presented to demonstrate the validity, accuracy and capability of the proposed method. REFERENCE LOADING AXES For any cross-section under biaxial loading, the exact location of the neutral axis is determined by two parameters: the orientation On and the depth dn (Figure 1). The Quasi-Newton method has been adopted for the solution of 0 n and dn and found to be effective in many studies (e.g. Brondum-Nielsen, 1985; Yen, 1991). When dealing with irregular cross-sections, especially those with the arrangement of structural steel and reinforcement being strongly eccentric, convergence of the iterative process cannot be guaranteed. Indeed, when such a column is subjected to an axial load with magnitude approaching the axial load capacity under pure compression, the origin of the loading axes may fall outside the iso- load contour if this origin is located at the geometric centroid of the cross-section as usual (Yau et al., 1993). As a result, the inclination of the resultant bending moment resistance O~m may change in a range less than 2re when On varies from 0 to 2~r, resulting in non-uniqueness or non-existence of the solution of On (Yau et al., 1993). In this paper, this difficulty is overcome by using the plastic centroid as the origin of the reference loading axes. By taking the plastic centroidal axes as the reference loading axes, the existence and uniqueness of the solution of On are always ensured and the convergence of the iterative solution process is guaranteed. For an arbitrary composite cross-section, the plastic centroid may be determined as follows (Roik and Bergmann, 1990) XcAcf~c/rc +XsA~L/r, +XrArfy/7"r YcAcfc~/rc +Y~A~L/r~ +YrArfy/7"r = , r.c= Acfcc/rc+A~L/r~+Arfy/rr (1) Ypc AcLc/rc+AsL/r~+A~fy/rr Design of Composite Columns of Arbitrary Cross-Section 445 where Ac, Ar and As are the total areas of concrete, reinforcing bars and structural steel, respectively; fcc, fy and fi are the respective specified strengths according to design codes such as Eurocode 4 (1994); Yc, Yr and 7~ are the corresponding partial safety factors, Xc, Yc, Xr, Yr. Xs and Ys are the respective centroid coordinates in the global XCY system. v~ T"_~ \ vmax \dn ~''-1~- \1 (a) (b) (c) Figure 1: Arbitrarily shaped cross-section: (a) Cross-section consisting of several regions; (b) Strain distribution; (c) Stress block for concrete in compression CROSS-SECTIONAL DESIGN Basic Assumptions The proposed design method is based on the following basic assumptions: (1) Plane sections before deformation remain plane after deformation. Consequently, the strain at any point of the cross-section is proportional to its perpendicular distance from the neutral axis. (2) The cross-section reaches its failure limit state when the strain of the extreme fiber of the concrete in compression attains the maximum strain gcu. (3) The stress-strain relationship of concrete in compression is represented by a parabola and then a horizontal line: (c f~22/(~<~0)and crc=fic(CO<_C <-Ccu) (2) Crc=fcc 2~oo-Co J The structural steel and the steel reinforcing bars are assumed to be elastic-perfectly plastic. (4) Tensile strength of concrete is neglected. Stress Resultants in the Cross-section The cross-section may assume any shape with multiple openings. The entire section may be divided into several regions if necessary (Figure 1 a). For convenience of calculation, each region is treated as the superposition of a solid section occupying the entire area of the region and a number of negative sub-sections representing the openings (Figure 1 a). Consequently, the entire section consists of ns solid subsections (i.e. regions completed filled with structural materials) and no negative subsections (i.e. 446 S.F. Chen et al. openings). All subsections may be arbitrarily polygonal. The xoy coordinate system represents the reference loading axes with its orgin as determined by Eqn. 1. The uov system is related to the xoy system by a rotational transformation with the u-axis being parallel to the neutral axis. For the ith subsection (either solid or negative), the coordinates of the jth vertex in the uov coordinate system are related to those in the xoy system by uy=-xjcOS On+yjsin O., vj=yscOS On-Xjsin O. (3) where On is the orientation of the neutral axis. In order to evaluate the stress resultants of the concrete, the intersection points of a subsection with the neutral axis are first determined. These intersection points together with the vertices of the subsection above the neutral axis form the compression zone of the subsection (Figure 1). The stress resultants of the concrete can then be found by integrating the concrete stress-strain curve (Figures l b and 1 c) over this polygonal compression zone as given in Eqn. 4, assuming that the vertices of the zone are numbered sequentially (either clockwisely or anti-clockwisely) It Nzci =]Pzcil = ICrc dud~ IJ=l uj o n c Uj+l v(u) n c Uj+l v(u) , Muci = PiZ I I [-ere (~ + vn)]dud~' Mvc i = PiE f f~ (4) j=l uj 0 j=l uj 0 where nc is the total number of vertices of the compression zone; P (u)=v(u)-v. is the linear equation of the boundary line, with v. being the v-coordinate of the neutral axis; pF1 when P~ci >0, and p~ -1 when Pzci<O; Unc+l Ul and Vnc+l V 1. The total stress resultants contributed by the concrete of the whole cross-section are then given by a summation over all subsections n s +n o n s +n o n s +n o Nzc = ZciNzci, Muc= ZciMuci, Mvc= ZciMvci (5) i=1 i=1 i=1 in which c~l for a solid subsection and cu -1 for a negative subsection. The bending moments of the concrete about the x- and y-axes can be easily obtained by coordinate transformation Mxc=MuccOS O Mvcsin 0~, Myc=Mucsin O.+MvccOS O. (6) The fiber element method (Mirza and Skrabek, 1991) is used to calculate the stress resultants carried by the structural steel and the steel reinforcement. The steel section is subdivided into small areas referred to as fiber elements and the reinforcing bars are treated as individual fibers. Both the structural steel and the reinforcing bars are assumed to be elastic-perfectly plastic. The stress resultants of the whole composite cross-section, axial force and bending moments about the x- and y- axes, can then be written as mr ms N z =Nzc+~-'~croAo+ff'o',jAsj j=l j=l mr ms M x = Mxc - ~" (O'rj -Crcj)ArjYrj - ~ (o',j -Crcj)Asjysj (7) j=l j=l mr ms My = My c + Z ( O'rj O'cj ) mrj X rj + Z ( O'sj O'cj ) Asj X sj j=l j=l Design of Composite Columns of Arbitrary Cross-Section 447 where mr and ms are the numbers of reinforcing bars and steel fibers into which the structural steel is discretized, Crrj, Crsj and ere are the stresses of the reinforcement, the structural steel and the concrete at the center of thejth bar or steel fiber. If vj Vn <0, Crcj =0. Iterative Solution Procedure For a given composite cross-section subjected to an axial load Nzd at o and bending moments Mxd and Myd about the x- and y-axes respectively (Figure 1), the depth and orientation of the neutral axis dn and On can be determined by the following iterative procedure: (1) Initial values of On and d, are first specified, for which the axial force Nz is calculated using the first expression of Eqn. 7. (2) The calculated axial force Nz is compared to the design value Nza and iteratively adjusted using the following equation until Nz is equal to Nzd with a given tolerance: dn '-dn dn, k = d n + (Nzd - Nz) (8) N~'-Nz in which Nz' and Nz are the axial force capacities calculated with the neutral axis depths dn' and dn respectively, with Nz' being greater than the design value Nzd and Nz being smaller than Nzd. (3) The bending moments Mx and My are found using the second and third expressions of Eqn. 7, and then the angle Ctm=arctg(My/Mx) is determined. (4) The value of am is compared to the design value ama=arctg(Myd/Mxd) and iteratively adjusted using the following equation: O,'-On On, k = O n + (Ctmd am) (9) a m t a m in which am' and am are the inclination of the resultant bending moment calculated with the neutral axis orientations 0,,' and 0,, respectively, with am' being greater than the design value Ctnd, and am being smaller than Ctmd. Steps (1) to (3) are repeated until O~m and amd are identical within a given tolerance. (5) If the task is to check the adequacy of an existing design, the calculated resultant bending moment Mr is compared to the design value Mra~(Mxd2+My2) 1/2. If Mr > Mrd, the cross-section is adequate, otherwise the structural steel and/or reinforcing bars should be increased and/or the section enlarged. (6) If the task is to design a section, all structural parameters except the reinforcing bars are first specified. Only steel bars of identical properties and size may be used, and the bar diameter can be determined iteratively starting from an initial assumed value. For any value of the bar diameter, the corresponding resultant bending moment capacity can be found and the bar diameter is iteratively adjusted using the following equation: ~bk=~b + r (Mrd_Mr) (10) Mr'-M r where Mr' and Mr are the resultant bending moments resisted by the section with bar diameters ~b' and r respectively, with Mr' being greater than the design value Mrd and Mr being smaller than Mrd. The required bar diameter is found when Mr is equal to Mra within a given tolerance. 448 NUMERICAL EXAMPLES S.F. Chen et al. A computer program has been developed based on the method presented in this paper. Two numerical examples are presented below to demonstrate the validity, accuracy and capability of the proposed method. Further numerical examples can be found in Chen et al. (1999). Tolerances used for the axial load, the inclination of resultant bending moment and the resultant bending moment are 10 -5, 10 -4 and 10 -5 respectively. The larger tolerance for the inclination of resultant moment was used as it was found to converge more slowly than the other two parameters. Rectangular Columns with Asymmetrically Placed Steel H-Section Two rectangular cross-sections with asymmetrically placed structural steel are shown in Figure 2. The maximum load-carrying capacities of the cross-sections under uniaxial loading at different eccentricities were determined in tests by Roik and Bergmann (1990). The cube strength of concrete and yield stresses of structural steel and reinforcing bars are listed in Table 1. Figure 2: Rectangular cross-sections with asymmetrically placed structural steel TABLE 1 MATERIAL PROPERTIES (ROIK AND BERGMANN, 1990) Specimen Cross- Section Vll, V12, V13 V1 V21, V22, V23 V2 Lk fsflange fs.web A,test _ (N/ram 2 ) (N/mm ~) (N/mm 2 ) fN/mm ~ ) 37.4 206 220 420 37.4 255 239 420 TABLE 2 LOAD CARRYING CAPACITIES AND COMPARISON WITH TEST RESULTS Specimen er(mm) Nz.comp(kN) Nz.test(KN) (Nz.comp-Nz.test)/Nz.test VII 0 3608 3617 -0.25% V 12 -40 2654 2825 -6.05% V13 100 1937 1800 7.61% V21 0 2880 2654 8.52% V22 -40 2107 1998 5.46% V23 100 2036 1706 19.34% Design of Composite Columns of Arbitrary Cross-Section 449 In order to compare the present results with the test results, all material partial safety factors were taken to be unity in the analysis. The stress-strain curve from Eurocode 4 (1994) for concrete was used, that is, fcc = 0.85fck, c0 = 0.002 and ecu=0.0035 in Eqn. 2. Table 2 gives the computed load- carrying capacities of the six specimens and their comparison with the test results, where er is the load eccentricity in the Y-direction with reference to the geometric centroid of the cross-section (Figure 2). It is seen that the computed values agree closely with the test results. This demonstrates the accuracy and validity of the proposed numerical method. Column of Asymmetric Polygonal Cross-Section A composite column cross-section, as shown in Figure 3, is subject to the design loads as follows: Nzd = 4320.5 kN, Mxd = 950.15 kNm, Mrd = -577.49 kNm in which X and Y are the geometric centroidal axes of the cross-section. The size and layout of the encased structural steel and the distribution of the steel reinforcing bars are shown in Figure 3. The reinforcing bars are assumed to have the same diameter of 12 mm. The task here is to check if the cross-section has an adequate capacity to carry the given design loads or to determine the required bar diameter if this section is inadequate. The stress-strain curve of Eqn. 2 for concrete (Eurocode 4, 1994) is used in the calculation, with fcc=O.85fck/Yc, c0=0.002 and 6cu=0.0035. The specified strengths and safety factors are taken as follows: fck=30 N/mm2,fs=355N/mm2,fy=460 N/mm2; yc=l.5, ?~=1.1 and yr=l.15 (1) Checking the adequacy of the pre-defined cross- section Coordinates of the plastic centroid o of the pre- defined cross-section (Figure 3): Xpt =-25.962 mm, Ypt =-27.694 mm Design loads with reference to the plastic centroidal axis system xoy (Figure 3): Nza = 4320.5 kN, area = 150.739 ~ Mra-=951.966 kNm Calculated load carrying capacity of the cross- section: Nz = 4320.5 kN, am = 150.739 ~ Mr=774.575 kNm As Mr<Mra, the pre-defined cross-section is inadequate. Figure 3: Cross-section of a column (2) Deisgn of the required bar diameter The program was instructed to find the required bar diameter for the corner column to achieve an adeqaute resistance for the design loads. The required bar diameter was found to be ~eq=20mm, with which the cross-section has a load carrying capacity of Nz=4320.5kN, Ctm=150.739 ~ and Mr = 951.972 kNm. CONCLUSIONS An iterative numerical method for the rapid design of short biaxially loaded composite columns of arbitrary cross-section has been presented in this paper. In the proposed method, the plastic centroidal axes of the cross-section are taken as the reference loading axes, which ensures the uniqueness and . yr=l.15 (1) Checking the adequacy of the pre-defined cross- section Coordinates of the plastic centroid o of the pre- defined cross-section (Figure 3): Xpt =-2 5.962 mm, Ypt =-2 7.694 mm Design. Arbitrary Cross-Sections, Irregular Cross-Sections, Structural Design, Biaxial Bending INTRODUCTION Composite steel- concrete construction has been widely used in many structures such as buildings and. by the structural steel and the steel reinforcement. The steel section is subdivided into small areas referred to as fiber elements and the reinforcing bars are treated as individual fibers.