J. Cent. South Univ. Technol. (2008) 15(s1): 061-066

DOI: 10.1007/s11771-008-315-1

Effects of edge beams on mechanic behavior under lateral load in reinforced concrete hollow slab-column structure

CHENG Jie-yun(成洁筠)¹, YANG Jian-jun(杨建军)², TANG Xiao-di(唐小弟)¹

(1. College of Civil Architectural Engineering and Mechanics, Central South University of Forestry and Technology, Changsha 410004, China;

2. College of Civil and Architectural Engineering, Central South University, Changsha 410075, China)

Abstract: In order to get the formulae for calculating the equivalent frame width coefficient of reinforced concrete hollow slab-column structures with edge beam, the finite element structural program was used in the elastic analysis of reinforced concrete hollow slab-column structure with different dimensions to study internal relationship between effective beam width and the frame dimensions. In addition, the formulas for calculating the increasing coefficient of edge beam were also obtained.

Key words: reinforced concrete; hollow slab-column structure; edge beam; equivalent frame; width coefficient; increasing coefficient

1 Introduction

Large number of theoretical researches and tests proved that reinforced concrete hollow slab-column structure which is composed of cast-in-situ concrete core tube floor and column is similar to that of the slab-column structure on mechanic behavior, so the equivalent frame method is also used to study reinforced concrete hollow slab-column structure under lateral load^[1-2]. The equivalent beam width is affected by many factors, such as geometric dimensions, material properties and loading^[3]. The key factor is geometric dimensions of structures. Large number of documents indicates that the ratio of high-section of the column (c₁) to the span of slab (l₁) is very important to the slab-column structure. The equivalent beam width (b) could be seen as a linear combination of c₁ and l₁, the corresponding linear formulas were deduced in Ref.[4-6]. To the reinforced concrete hollow slab-column structure, the high-section of the column, slab-span, and hide beam width (b_al) were controlling parameters among these geometric dimensions^[1]. Through analysis, these formulae of the beam width of middle equivalent frame and edge equivalent frame were obtained in Ref.[1].

The above analysis did not consider the effects of edge beams on the width coefficient of equivalent frame. In fact, edge beam could effectively reduce the peak moment in the slab, the center deflection of slab, the thickness of slab, and the amount of steel slab. The existence of edge beams can help to improve the seismic performance of structure and anti-punching ability of node, at the same time, the existence of edge beam does not affect the use of functions. So studying the effects of edge beams on the width coefficient of equivalent frame in reinforced concrete hollow slab-column structure under lateral load is very necessary.

2 Equivalent width coefficient

Under lateral load, equivalent frame method is that column is directly equivalent to the calculation model, and slab is equivalent to beam which has the same rotation in width direction^[7]. In this work, the analysis was also based on this model. As shown in Fig.1, in the middle of the floor under lateral load, along symmetry axis and anti-bending line, a representative of the node (node No.1) was selected as a calculation unit. Assuming that anti-bending point of column was at middle of layer height (h/2), h was the representative of layer height. The slab has simple boundary conditions along the vertical direction of the slab-span (l₁). Along the vertical direction of the slab-width (l₂), boundary conditions of slab were free. The boundary conditions of the bottom of column were hinged. 8 nodes of the 3D unit were used. The node model of hollow slab-column structure is shown in Fig.2. The horizontal force F was exerted at the top of the column along the direction of l₁, the apical displacement of column (?x₁) in the node of slab-column could be calculated.

Fig.1 Layout of hollow slab-column structure

Fig.2 Node model of hollow slab-column structure

Node No.1 of slab-column was equivalent to the node of beam-column. Equivalent beam was instead of slab whose width was l₂. The width-section of equivalent beam was b, and the high-section of equivalent beam was thickness of slab, and h_s was the representative of thickness of slab. The geometry of column was not changed. The apical displacement of column (?x₂) in the node of beam-column could be calculated by structural mechanics methods.

                     (1)

Let ?x₁＝?x₂, the equivalent width coefficient () of equivalent beam could be deduced

                    (2)

The equivalent width of equivalent beam was

  　     　　               (3)

where  I_c is the inertial moment of column; I_b is the inertial moment of equivalent beam, I_b=αI_s; I_s is the inertial moment of slab.

3 Effects of edge beams on width coefficient of equivalent beam

There are two cases: the one is effect of torsional stiffness of edge beams perpendicular to the equivalent frame; the other is effect of bending stiffness of edge beams parallel to the equivalent frame.

3.1 Effect of torsional stiffness of edge beams perpendicular to equivalent frame

As shown in Fig.1, in the middle equivalent frame, along symmetry axis and anti-bending line, a representative of the edge node (node No.2) was selected as a calculation unit. Because of the effects of torsional stiffness of edge beams perpendicular to the equivalent frame, the apical displacement of column was reduced, the equivalent width coefficient was changed.

The models of reinforced concrete hollow slab-column structures with edge beam with different dimensions were established by using finite element structural program in order to study the equivalent frame calculation modes for reinforced concrete hollow slab-column structures under lateral load in the elastic analysis. In all examples, c₁/c₂ was 1.0, and diameter of tube filler (D) was 180 mm, and rib width (b_w) was 60 mm, and the thickness of slab was 280 mm, and the layer height h was 4.2 m. Elastic modulus of concrete column and concrete slab, E_c and E_b, were 3.0×10¹⁰ Pa, Poisson ratio (v) was 0, the horizontal force (F) was 100 kN. Other parameters were described in Table 1.

                       (4)

where  a and b are short-length and long-length of torsional component block, respectively^[8].

For reinforced concrete hollow slab-column structures with different parameters, the relationship of α and single parameter is described in Figs.3-6.

In Figs.3-6, because of the existence of edge beam, torsional stiffness increases, a is bigger than that of no edge beam.  is the ratio of torsional stiffness of edge beam to bending stiffness of slab.  is used to reflect the relationship with . It can be seen from Fig.6, a and  has linear relationship with .

Above all, for the reinforced concrete hollow slab-column structure which has the torsion edge beams perpendicular to the equivalent frame,  is

Table 1 Frame dimensions of reinforced concrete hollow slab-column structure

Fig.3 Curves of

      (5)

The results of Figs.3-6 can be substituted to Eqn.(5), k₁, k₂, k₃ and k₄ can be calculated, k₁=1.45, k₂=0.2, k₃=0.55, k₄=0.35;  is

Fig.4 Curves of

        (6)

For considering the impact of Poisson ratio, the right side of Eqn.(6) would be multiplied by 1/(1-ν²). Dotted lines in Figs.3-6 were calculated by the Eqn.(6),it reflects the actual changes in a law of the equivalent width coefficient(a).

Fig.5 Curves of

Fig.6 Curves of

3.2 Effects of bending stiffness of edge beams parallel to equivalent frame

As shown in Fig.1, in the middle equivalent frame, along symmetry axis and anti-bending line, a representative of the edge node (node No.3) was selected as a calculation unit. The actual edge beam could be equivalent beam. Taking into account the slab to participate in work, bending stiffness of edge beams (I_b) would be multiplied by the increasing coefficient, β. The bending stiffness of slab (E_bI_b) could be deduced by Eqn.(1):

                     (7)

The bending stiffness of actual edge beam (E_bI_b) is:

                        (8)

The increasing coefficient (β) is:

                              (9)

where  I_b is the inertial moment of slab; I′_b is the inertial moment of actual edge beam.

For different geometric parameters listed in Table 1, the increased coefficient of bending stiffness of edge beams (β) could be obtained. In Figs.7-10, the relationship curves of ,, ,  are shown. From the relationship curves, increasing coefficient (β) and c₁/l₁, b_al2/l₂ are basic linear relationship.  is the ratio of bending stiffness of slab to that of edge beam, reflecting the relationship with β. It can be seen from Fig.10, β and  are not a linear relationship.

Above all, for the reinforced concrete hollow slab-column structure which has the bending edge beams parallel to the equivalent frame, β is:

                             (10)

The results of Figs.7-10 can be substituted to Eqn.(10), k₁, k₂, k₃ and k₄ can be calculated: k₁=7.0, k₂=0.6, k₃=2.5, k₄=-2.75, k₅=1.37; β is:

                         (11)

As shown in Fig.1, in the edge equivalent frame, node No.4 was selected for analysis. The corner node had torsional edge beams perpendicular to the equivalent frame and also had bending edge beams parallel to the equivalent frame. In order to analyze the bending edge

Fig.7 Curves of

Fig.8 Curve of

Fig.9 Curves of

Fig.10 Curves of

beam and torsion edge beam on the impact of increase coefficient, the first step, is to maintain the same size of bending edge beams, only to change the size of torsional edge beam, to analyze the relationship between the increasing coefficient (β) and . The second step is to maintain the same size of torsional edge beam, only to change the size of bending edge beams, to analyze the relationship between the increasing coefficient (β) and . The relationship curve is shown in Figs.11 and 12.

From Fig.11 and Fig.12, for the corner node, the impact of torsional edge beam is very small, and the impact of bending edge beam is very important. Therefore, in fitting formula, the impact of torsional edge beam is not considered. From Figs.7-10, the results of edge node are similar to that of corner node, so Eqn.(11) can be also used in corner node of the edge equivalent frame.

Fig.11 Curve of  

Fig.12 Curve of

4 Conclusions

1) The existence of edge beams in the middle equivalent frame, the torsional stiffness of edge beam and the thickness of slab are the controlling parameters in calculation of the equivalent width coefficient. The equivalent width coefficient can be calculated by formulae.

2) In edge equivalent frame, the actual edge beam can be equivalent beam. Taking into account the slab to participate in work, bending stiffness of edge beams would be multiplied by the increasing coefficient. The increasing coefficient can be calculated by formulas.

References

[1] CHENG Jie-yun. Research on calculation method of reinforced concrete hollow slab-column structure under lateral load [D]. Changsha: School of Civil and Architectural Engineering, Central South University, 2006. (in Chinese)

[2] WANG Jing. Research on force resistance properties of cast-in-situ hollow concrete floor[D]. Changsha: School of Civil and Architectural Engineering, Central South University, 2004. (in Chinese)

[3] HWANG S J, MOEHLE J P. An experimental study of flat-plate and lateral loads[M]. Berkeley: College of Engineering, University of Callfornia, 1993.

[4] LI Jun-lan, L? Xin-lin. Study on the calculation model for flat-plat structures under earthquake action[J]. Journal of Building Structures, 1999, 20(1): 39-45. (in Chinese)

[5] ZHANG Yu-feng, L? Zhi-tao. Effects of edge beams on the width coefficient of equivalent frame in flat-plate structures[J]. Building Structure, 2001, 31(2): 43-46. (in Chinese)

[6] LIU Jin-song, TANG Jin-chun, TAO Qiu. Equivalent beam width affected by slab column-cap in slab-column structure[J]. Building Structure, 2004, 34(2): 64-66. (in Chinese)

[7] Building Code Requirement for Reinforced Concrete[M]. American Concrete Institute, 2005.

[8] NILSON A H. Design of concrete structure[M]. Beijing: Chinese construction Industry Publishing House, 2003. (in Chinese)

(Edited by YANG Hua)

Received date: 2008-06-25; Accepted date: 2008-08-05

Corresponding author: CHENG Jie-yun, Master candidate; Tel: +86-13975800132; E-mail: lucy_cjy@126.com