Seismic Performance and Prediction Equations of Sandwich Beam-Column Joints Subjected to Skew Cyclic Loads

Liqun Hou1, Xinzheng Lu2,*, Hong Guan3, Weiming Yan4, Shicai Chen4

Materials and Structures, Accepted on Nov, 21, 2018, DOI: 10.1617/s11527-018-1288-7

1 Beijing Engineering Research Center of Steel and Concrete Composite Structures, Tsinghua University, Beijing, P.R. China, 100084

2,* Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing, P.R. China, 100084; E-mail: luxz@tsinghua.edu.cn

3 Griffith School of Engineering, Griffith University Gold Coast Campus, Queensland 4222, Australia

4 Department of Civil Engineering, Beijing University of Technology, Beijing, P.R. China, 100124

To study the seismic performance of sandwich beam-column joints constructed with high strength concrete (HSC) column and normal strength concrete (NSC) floor system and joint regions, four specimens with different column to beam concrete strength ratios (a) were tested under skew cyclic loads. The performance indices of the specimens including the failure mode, ductility, energy dissipation were compared and analyzed. The results show that the failure modes of the sandwich joints are in form of joint shear failure after yielding of the beam, while the ductility coefficient is found to be greater than 3.0. Compared to the joints with low concrete strength ratios a, the specimen with a high concrete strength ratio a features larger deformation at the joint. Based on the softened strut-and-tie model, a set of shear strength prediction equations for the sandwich beam-column joint, taking into account the effects of the concrete strength ratio a, floor slabs and plastic region of beams, is proposed. Comparison of the present tests against the published literature confirms that the shear strength of the sandwich joints can be well predicted by the proposed model.

Key words: sandwich beam-column joints; cyclic loads; failure mode; shear strength

1 Introduction

In recent years, high-strength concrete (HSC) columns have been widely used for high-rise building constructions [1]. The use of HSC can reduce the column sections, leading to a larger usable space. However, there is limited benefit of HSC to the flexural strength of a floor system (i.e., beams and slabs). In consequence, normal-strength concrete (NSC) is commonly adopted for beams and slabs. In practice, for simplicity of construction, the NSC beams and slabs are cast continuously over the beam-column joint. Such continuous casting results in a layer of NSC in the intersection between the upper and lower HSC columns, referred to as the sandwich beam-column joint. Concerns are raised that the lower-strength concrete within the joint core may limit the capacity of the sandwich joints.

Such sandwich joints have been studied since 1960s. Bianchini et al. [2] implemented a thorough study on different types of sandwich joints (including corner, edge and interior joints). The influence of the lower-strength concrete in the joint region is represented by an effective compressive strength . Bianchini et al.s work has subsequently been adopted as the design guidelines of the American Concrete Institute [3]. In addition to the studies on the gravity resistance [2, 4-7], significant amount of experimental and analytical studies have also been performed to investigate the seismic performance of various sandwich joints [8-14]. These studies revealed that the concrete strength reduction of the joint had a certain degree of negative impact on the seismic performance of the sandwich joints. However, sufficient seismic performance could be achieved if the design parameters (such as the shear-compression ratio [14], axial load ratio [14], ratio of column to beam concrete strength a, transverse reinforcement in the joint) are well controlled. For some sandwich joints, strengthening measures such as installing vertical or diagonal steel reinforcements may be required [12, 14]. It should be noted however that these measures may inescapably lead to complications in an already congested location in the joint.

Most of the previous experiments examining the seismic performance were based on planar joint specimens [13]. Limited studies have been performed on the seismic performance of spatial beam-column interior joints subjected to skew seismic loads. Note that the existence of floor slabs further complicates the performance and failure modes of the joints, which is rarely considered in previous work. In addition, no computational models to date can accurately predict the shear strength of spatial sandwich beam-column joints. Therefore, it is necessary to investigate the performance and design method of this type of joints with weaker beams and slabs, subjected to skew seismic loads.

This study covers four spatial sandwich interior joint specimens with two-way beams and slabs under skew cyclic loading. Their seismic performance including the failure mode, ductility, energy dissipation, and deformation capacities were analyzed. The influence of different concrete strength between the beams and columns on the failure mechanism of the joints was discussed. Subsequently, a set of prediction equations based on the softened strut-and-tie model was proposed to predict the shear strength of the sandwich joints. Finally, the prediction was compared to the present experimental results and those available in the literature to demonstrate its accuracy.

2 Experimental program

2.1 Specimen details

One of the most critical issues associated with the sandwich joints is the difference in concrete strength between the column and the beam. Such a difference can be represented by a concrete strength ratio, a, as defined in Equation (1)

                                                                   (1)

Where  and  are the compressive strengths of concrete of the column and the beam, respectively.

Four half-scale spatial sandwich interior joints (SJ1, SJ2, SJ3 and SJ4, Table 1) were designed and constructed in accordance with the design standards of China [15]. The columns and beams were isolated from the prototype frame structure at the mid-heights or mid-spans of the columns and beams, respectively [16]. Li & Liu and Yang et al.s study [12, 14] identified that the concrete strength ratio and shear-compression ratio ( ) are the key parameters influencing the seismic performance of beam-column joints. Specifically, the concrete strength ratio determines the reduction of the shear strength of the joint, whereas the shear-compression ratio is the parameter governing the failure mode of the specimens. For the shear-compression ratio , bj and hj are the width and depth of the joint core area, respectively;  is the shear resistance of the joints, where fytop and fybottom are the yield strengths of the top and bottom reinforcement of the beam, respectively; As and  are the sectional areas of the top and bottom reinforcement of the beam, respectively; hbo is the effective depth of the beam;  is the distance between the centroid of the resultant force of the longitudinal compressive steel reinforcements and the extreme compression face of the beam; Hc is the column height and hb is the beam depth. In China, the concrete strengths commonly used for beams and slabs are 30 MPa to 40 MPa, and the most popular one is 30 MPa, while the mostly used concrete strengths for columns are 40 MPa to 60 MPa [15]. Furthermore, specimens of similar concrete strength ratios exhibit similar behaviors [17]. For these reasons, all the specimens in this study were designed following the widely used concrete strength ratios (1.3, 1.4, 1.5 and 1.8, for SJ1, SJ2, SJ3 and SJ4, respectively), i.e., the concrete strength of the beams and slabs is 30 MPa, and those of the columns are 40 MPa, 45 MPa, 55 MPa and 60 MPa. Due to the slight deviations of the measured concrete strengths, the actual concrete strength ratios are 1.3, 1.4, 1.5 and 1.8 (SJ1 to SJ4) given in Table 1, based on the measured average cubic compressive strengths of concrete. The tensile strength of concrete is calculated using the expression specified in Chinese design standard [15]: , in which fcu is the average measured cubic compressive strengths of concrete (MPa). Given that the maximum limit of the shear-compression ratio of the joint core area is 0.3 [15], all the specimens were thus designed following the moderate shear-compression ratio (0.2) widely used in engineering practice, which can be employed to determine the beam reinforcement. The dimensions of the specimens, cross sections of the members and the reinforcement details are detailed in Figure 1. The material properties of reinforcements are summarized in Table 1. The concrete cover is 25 mm for both beams and columns, and that for the slab is 10 mm. The design axial load ratio n of the columns is 0.5 ( where N is the design axial force; Ac is the gross-sectional area of the column), which satisfies the requirement specified in the Chinese Code (the maximum limit of the design axial compression ratio for the prototype frame is 0.85) [15].

Figure 1 Dimensions and reinforcement details of the specimens (unit: mm)

Table 1 Material properties and concrete strength ratios of different specimens

Specimen

Concrete strength of column (MPa)

Concrete strength of beam (MPa)

Concrete strength ratio a

Yield strength of Steel (MPa)

Compressive

Tensile

Compressive

Tensile

D6

D10

D12

D14

SJ1

41.6

3.1

32.2

2.6

1.3

292

331

414

/

SJ2

45.3

3.3

32.6

2.7

1.4

292

331

414

440

SJ3

54.3

3.7

36.2

2.8

1.5

388

377

436

432

SJ4

58

3.9

32.2

2.6

1.8

292

331

414

440

The Chinese Design Code for Concrete Structures [15] requires that the shear strength of a reinforced concrete (RC) frame joint, Vjd, should satisfy Equation (2) if the building is located on the site with a seismic design intensity lower than 9-degree (i.e., the design peak ground acceleration (PGA) with a 10% probability of exceedance in 50 years equals 400 cm/s2).

                      (2)

In Equation (2), the seismic adjusting factor gRE = 0.85 [15]; hj is the confinement effect factor of the orthogonal beams. In this work, hj = 1.00 according to the Chinese Code for Design of Concrete Structures [15]; ft is the design tensile strength of concrete; hj and bj are the height and effective width of the core zone of the joint, respectively; N is the design axial force of the upper column; bc is the column width; fyv is the design strength of shear reinforcement; Asvj is the total area of transverse reinforcements within the effective width of the core zone in the loading direction; hbo is the effective depth of beam; s is the spacing of the transverse reinforcements along the beam axis;  is the distance between the centroid of resultant forces of longitudinal compressive steel reinforcements and the extreme compression face of the beam.

If ft is determined based on the concrete strength of the columns, all of the specimens can meet the requirement specified by Equation (2). However, if ft is determined according to the concrete strength of the beams, then only Specimen SJ1 can meet the requirement of Equation (2).

2.2 Test setup and measurement

In order to consider the spatial effect, the seismic load was applied on the top of the columns. Due to the symmetry of the specimens, the skew cyclic seismic load was simulated by a 45 uniaxial cyclic load applied to the column top. Note that 45 is the angle between the loading direction and the longitudinal direction of the beam. The bottom of the column was pinned to the laboratory strong floor, whilst the vertical and horizontal loads were applied to the top of the column using two actuators. The four ends of the orthogonal beams were also connected to the strong floor through steel links, which provided restraint against the vertical translation but allowed for the horizontal translation of the beam along the 45 angle. The test setup is shown in Figure 2.

(a) Perspective view

(b) Side view

(c) Plan view (unit: mm)

(d) Typical displacement transducer locations on specimens (unit: mm)

Figure 2 Test setup and measurements

(1: Load cell; 2: 200t actuator; 3: 20t actuator; 4: Displacement transducers; 5: Steel link; 6: Pin)

(a) Force control (unit: kN)

(b) Displacement control (unit: mm)

Figure 3 Loading history diagrams

Loading history diagram is shown in Figure 3, where Fprevious is the amplitude of the previous load level during force control period; (Fy+, dy+), (Fy-, dy-) correspond to the loads and displacements when the beam reinforcement yield in the positive and negative direction, respectively; Dprevious is the amplitude of the previous load level during displacement control period; Hc is the column height. Firstly, each of the specimens was subjected to a constant design axial load ratio n = 0.5. Then, a quasi-static horizontal load was cyclically applied to the top of the column. Before yielding of the beam reinforcement, the horizontal load was force-controlled and each load level was repeated once. The first load level was 10 kN, and the second level was 20 kN. After that, the load was increased by 20 kN for each subsequent level. After yielding of the beam reinforcement, the load was changed to displacement-control and each load level was repeated twice with a 12 mm increment. The specimen was considered to have failed when the resistance was less than 85% of the peak resistance, or the specimen lost its stability.

The following parameters were monitored during the test (see Figure 2): (1) Displacements at the top and bottom of the column. (2) Axial loads and lateral cyclic loads on the top of the column. (3) Shear deformations of the joint region through transducers installed in an X-shaped configuration. (4) Deflections of the beams and the columns. (5) Strains in the beam and column reinforcement within and adjacent to the joint region and the transverse reinforcement strains within the region. Different displacement components can be estimated according to the measured member deflections and the joint shear deformation proposed by Kimreth et al. [18] and Liu [19]. A clear description of the procedure to estimate the different displacement components is given by Liu [19].

3. Results and discussion

3.1 Test observations and failure modes

Figure 4 Cracking patterns of Specimen SJ4 at the end of test

The cracking patterns of the Specimen SJ4 at the end of the test are displayed in Figure 4. The other specimens showed similar crack patterns. Flexural cracks were mainly located on the beams whereas limited cracks were found on the columns. The majorities of the joint cracks were diagonal cracks. Concrete crushing was found at the corner of the joint. The first set of flexural cracks of all specimens appeared in the beams. Subsequently, the flexural cracks were also observed in the slabs and joints. Shear cracks of the joints occurred after the flexural yielding of the beams. The longitudinal reinforcement in the beams and the stirrups in the joints yielded when approaching the peak strength. After that, the flexural cracks in the beams developed rapidly, and finally propagated into the joint regions. Finally, the concrete in the joint region crushed, particularly in the corner of the joints. In Specimens SJ2, SJ3 and SJ4, more than 60% of the stirrups in the joints yielded. In Specimens SJ1, 60% of the stirrups in the joints yielded at the ultimate displacement. Specifically, three middle stirrups out of the five stirrups in the joint were found to have yielded. Yielding of the stirrups in the joint demonstrated that the shear failure is the major failure mode of the specimen. Furthermore, the specimens with larger concrete strength ratios a underwent more deformation in the joint region as will be discussed in Section 3.3.

3.2 Hysteretic curves, backbone curves and ductility

The measured hysteretic responses of all specimens are illustrated in Figure 5, in which the vertical axis is the shear strength while the horizontal axis is the corresponding drift ratio. All specimens exceeded the ultimate drift ratio criterion (2%) of RC frames specified in Chinese Code [20] and ASCE Code (ASCE/SEI 7-16) [21]. The hysteretic curves of different specimens are similar to each other under the drift ratio of 1.0%~4.0%. Pinching effects were noticed in all specimens. According to the definition of the degradation coefficient (i.e., the ratio of the area in the degraded envelope to that in an idealized full bilinear envelope) of Steelman & Hajjar [22], the degradation coefficients of the last envelops of the hysteretic curves were found to be 0.58, 0.47, 0.58 and 0.52, respectively, for SJ1, SJ2, SJ3 and SJ4.

The skeleton curves [23-24] of all specimens are also indicated in Figure 5(e). The displacement ductility coefficients of all specimens are defined as the ultimate displacement (when the strength decreases to 85% of the peak strength) [14, 23-24] over the yield displacement. All of the specimens have their displacement ductility coefficients greater than 3.0 (4.7, 3.4, 3.1, 4.1 for SJ1~SJ4), which satisfy the requirement in the Chinese Code [15, 20] and ASCE/SEI 41-13 [27] for moderate ductility. Specimen SJ1 demonstrates the largest displacement ductility coefficient, owing to its smallest concrete strength ratio a. Specimen SJ4 with a = 1.8 underwent a relatively small yield displacement. In consequence, the ductility coefficient of Specimen SJ4 is larger than that of Specimens SJ2 and SJ3.

(a) Hysteretic curve for SJ1                                                       (b) Hysteretic curve for SJ2

(c) Hysteretic curve for SJ3                                              (d) Hysteretic curve for SJ4

 

(e) Skeleton curves of four specimens

Figure 5 Hysteretic curves and skeleton curves of different specimens

3.3 Deformation performance

Figure 6 Definition of different components of the total deformation in each specimen

(a) SJ1

(b) SJ2

(c) SJ3

(d) SJ4

Figure 7 Different components of the total deformation in each specimen

The total displacements of the specimens consist of three components: the beam deformation, the column deformation and the joint region deformation. Definition of different components in each specimen [18, 19] is shown in Figure 6. A shown in Figure 2(d), the beam and column deformations are calculated by the measured values of the displacement transducers on the beam and column, respectively. The joint region deformation is measured through transducers placed on the joint core area. Different component deformation ratios are shown in Figure 7. During the initial stage of the test, the beam deformation contributed largely, followed by the column deformation and the joint deformation. With the increase of the overall deformation, the proportion of the joint region deformation increased gradually, that of the column deformation remained relatively unchanged, and the proportion of the beam deformation was found to decrease. The deformation patterns for all specimens were fundamentally similar during the test. The proportion of the joint shear deformation of Specimen SJ1 was obviously lower than that of the other specimens because of its concrete strength ratio a being the smallest amongst all. Although the proportion of the beam deformation of Specimen SJ1 was evidently higher, when the concrete strength ratio a is greater than or equal to 1.39, its influence on the proportions of the beam and column deformations did not show a remarkable difference.

3.4 Equivalent hysteretic damping

The energy dissipation capacity of different specimens can be represented by the equivalent hysteretic damping ratio xh, which is defined in Equation (3) [28].

                                                                     (3)

where Eh is the energy dissipated of the lateral load-displacement hysteretic curves for each cycle at all load levels; Ee is the energy dissipation of an equivalent elastic cycle of the lateral load-displacement hysteretic curves for each cycle at all load levels. Liao et al. [26] give a detailed description for the calculating of Eh and Ee.

Figure 8 Energy dissipation capacity curves

The variation of  for all specimens is shown in Figure 8. It is clear that xh increased with a larger story drift, which was attributed to the yielding of the beams. When the drift ratio was smaller than 2%, the value of xh generally increased smoothly. At the peak load, the equivalent hysteretic damping ratio xh of Specimens SJ1, SJ2 and SJ3 SJ4 are 0.16, 0.13, 0.16 and 0.17, respectively. The viscous damping ratio for an elastic seismic design specified for RC frames is 0.05 according to the Chinese Code [15]. Therefore, the equivalent hysteretic damping ratio differs from the viscous damping ratio. After the peak load, xh exhibited an obvious increase toward the failure of the specimens, and xh of the first cycle is less than that of the second cycle at each load level after the peak load, which is exactly opposite to that before the peak load. The reason for such a phenomenon is that the energy was mostly dissipated by the beams before the peak load, while the joint also participated in the energy dissipation after the peak load, leading to a larger energy dissipation in each load cycle. Our test results do not disclose a clear relation between the strength ratios and the equivalent hysteretic damping ratios.

3.5 Strain analysis

At the peak load, the strains of the top longitudinal reinforcement in the beams are illustrated in Figure 9(a), measured at five key points. The first and last points were located in the beams whereas the middle three were located within the beam-column joint. A larger strain adjoining the column was found with a smaller concrete strength ratio a in the tension zone of the beam. The strain measurements also indicated that all the longitudinal reinforcement of the beam yielded within the joint.

At the peak load, the strains measured in the column longitudinal reinforcement of the tested specimens are shown in Figure 9(b). For all specimens, the strains in the longitudinal reinforcement in the column were smaller than those in the beam, nevertheless the measured strains in the column longitudinal reinforcement (approximately 2000 me) still exceeded the yield strain. Note that different specimens exhibited similar maximum strains in their columns.

Figure 9(c) shows the strains measured in the transverse reinforcement within the joints for all specimens at the peak load. It can be found that the measured strains in the transverse reinforcement within the joints exceeded the yield strain. A larger concrete strength ratio a led to a larger transverse reinforcement strain.

(a) Longitudinal reinforcement strains in beams going through the joints

(b) Longitudinal reinforcement strains in columns going through the joints

(c) Transverse reinforcement strains within the joints

Figure 9 Strains of developed in reinforcement

4 Prediction model for predicting shear strength

Remarkable differences exist in the shear strength prediction models for a beam-column joint in different design codes and literature [29]. A widely used softened strut-and-tie model [29-32] has been proposed by Hwang & Lee [33], but it does not consider all of the key variables that influence the response of sandwich joint (e.g., floor slab). Further to this model, a modified calculation model for sandwich joints is proposed in this study taking into account the influences of the concrete strength ratio a, floor slabs and plastic region of beams.

4.1 Softened strut-and-tie model

Hwang & Lee [33] proposed the following equation for determining the shear strength based on the softened strut-and-tie model (see Figure 10).

                                                                   (4)

where f is the strength reduction ratio with a value of 0.85; K is the strut-and-tie index taken as K = Kh+Kv-1;  is the cylinder strength of concrete (MPa); n is the softening coefficient of the concrete calculated using the expression , in which er is the average principal tensile strains in the normal direction of the diagonal strut, and n can be approximated by  [33-35], bs is the width of the diagonal strut, which is taken as the depth of the column section; as is the depth of the diagonal strut estimated by , cb is the depth of the compression area of the beam obtained by taking , hb is the beam depth, cc is the depth of the compression area in the column obtained by taking , Nc is the axial force acting on the column, hc is the depth of the column in the loading direction; q is the angle between the diagonal strut and horizontal axis assuming ,  and  are the distances between the extreme longitudinal reinforcement in the beams and columns, respectively.

Figure 10 Softened strut-and-tie model of interior joint

The strut-and-tie index K can be obtained as:

 for                                                       (5)

 for                                                       (6)

                                                               (7)

                                                               (8)

                                                              (9)

                                                        (10)

                                                          (11)

                                                          (12)

where gh and gv are the fractions of diagonal compression transferred by the horizontal tie in the absence of the vertical ties and the vertical ties in the absence of the horizontal ties, respectively; fyh and fyv are the yield strength of the joint hoop reinforcement and intermediate column bars, respectively;  and  are the balanced forces of the horizontal and vertical ties, respectively; and  are the indexes of the horizontal and vertical ties, respectively.

4.2 Modified softened strut-and-tie model

The existence of the floor slab help to increase the depth of the concrete compression zone at the bottom of the beam, leading to a lager predicted shear strength. However, the same depth proposed by Hwang & Lee [33] is rather small [37, 38]. Therefore, the depth of the concrete compression zone of the beam can be taken as [38-39]:

                                                (13)

where Asb is the reinforcement area in the tension zone of the beam; fyb is the yield strength of the beam reinforcement; Asp is the reinforcement area of the floor slab located within the beam effective flange width bf = 2 6 ts, where ts is the thickness of the floor slab; fyp is the yield strength of the slab reinforcement; bb is the width of beam.

The compressive resistance of concrete for sandwich joints can be modified to the effective compressive strength instead of the cylinder strength of concrete to account for the confinement of the beams and slab, resulting in a larger predicted shear strength using Equation (14). The effective compressive strength [3] of the sandwich joint is

 for                                         (14)

The softening coefficient of concrete compressive strength can be calculated by considering the effect of plastic region of beams [40] as shown in Figure 10, which reduces the predicted shear strength. The calculation process is as follows:

(1) Assume the rotation angle of the column as:

                                                                  (15)

when D is the lateral deformation at the top of the column; Hc is the column height.

(2) The longitudinal strain of the beam reinforcement in the plastic hinge region of the joint is calculated by

                                                    (16)

where ebf is the tensile strain of the beam at yielding; fpmp and fpmn are the positive and negative plastic rotational angles, respectively;  is the distance between the centroids of upper and lower beam reinforcement; lp is the plastic hinge length of the beam.

(3) The length of the plastic hinge region is calculated by

    for                                              (17)

where M /(V h) is the shear span ratio, h and hbo are the overall and effective depths of the section, respectively.

(4) The joint longitudinal strain is calculated as

                                                             (18)

where Vjbf is the joint shear strength when the beam reinforcement yields; Vj1 is the shear strength of Type 1 joints for structures in a non-seismically hazardous area according to ACI recommendations [41], based on the diagonal concrete strut and truss mechanisms.  (MPa), and the constant g=24.

(5) The average principal tensile strain  in the joint is calculated as:

                                                                            (19)

where the definition of angle q is shown in Figure 9. Note that . ed is the average principal compressive strain in the joint which is much smaller than the corresponding principal tensile strain. For this reason, the effect of the principal compressive strain in the joint can be neglected.

(6) The softening coefficient of the concrete in the joint is calculated using Equation (20)

                                          (20)

By substituting Equation (13), Equation (14) and Equation (20) into Equation (4), the shear strength of the sandwich joint in the principal direction can be predicted.

                                                            (21)

where  is the area of the diagonal strut.

Due to symmetry of the specimen, the response of the spatial joint can be effectively separated in the two principal directions. Then the resultant of the 45º directional resistances is  times the uniaxial strength in the principal direction.

                                                    (22)

4.3 Comparison with experiments

The experimental shear force-drift ratios for all the four specimens are compared in Figure 5 with the predicted values using Equation (21). It is worthy to mention that in this comparison, the softening coefficients n  is based on the test results. It can be observed that the proposed method, taking into account the effects of the concrete strength ratio a, the floor slabs and the plastic region of beams, is able to evaluate the shear strength and deformability of the sandwich joints within a reasonable range.

However, for actual practical projects, it is impossible to get the measured softening coefficients. So  is used to compute the shear strength in Equation (21) for simplification. To further validate the accuracy of the proposed calculation method, an experimental database on sandwich interior joints failing in shear, after the yielding of the beams, was constructed in Table 2. It covers the measured shear strengths Vt of the present study and those reported in the literature [8, 10-13]. The values predicted by Equation (21) ( , with ) is also compared with the predictions of Hwang & Lee [33] (Vsst) (i.e., Equation (4)), Chinese Code [2] (Vjd) (i.e., Equation (2) using the concrete strength of the beam), ACI Code [3] (VACI) (i.e., Equation (23)) and CSA Standard [42] (VCSA) (i.e., Equation (24)).

The shear strength is based on the ACI 318 [3] is given by Equation (23):

                                                          (23)

where g is equal to 20, 15, or 12, depending on the geometry and the level of confinement exerted by the beams framing into the joint, and Aj is the effective joint area.

The shear strength is based on CSA Standard [42] is calculated by:

                                                            (24)

where g is equal to 2.2, 1.6, or 1.3, depending on the geometry and the level of confinement exerted by the beams framing into the joint, and Aj is the effective joint area.

Table 2 Shear strength of the sandwich joints

Reference

Specimen

Vt /kN

/kN

Vt /

Vt /Vsst

Vt /Vjd

Vt /VACI

Vt /VCSA

Present study

SJ1

857

780

1.10

1.36

1.01

1.09

1.08

SJ2

965

881

1.10

1.45

1.15

1.18

1.02

SJ3

1048

1069

0.98

1.34

1.16

1.18

1.01

SJ4

1059

1072

0.99

1.53

1.24

1.53

1.23

Yu & Li [8]

J-1

287

207

1.39

1.50

1.12

0.92

0.82

Xu et al. [10]

SJ-3

844

1303

0.65

0.83

1.21

0.89

0.73

SJ-4

927

1289

0.72

0.92

1.46

0.97

0.67

Yang [11]

G-1

913

807

1.13

2.47

1.04

1.34

1.29

G-2

952

926

1.03

1.43

1.13

1.31

1.16

G-3

1064

829

1.28

1.56

1.12

1.52

1.35

G-4

1041

941

1.11

1.65

0.86

1.51

1.76

G-5

847

820

1.03

1.28

1.08

1.18

1.09

G-7

936

1049

0.89

1.67

1.09

1.24

1.14

G-8

937

1060

0.88

1.52

0.82

1.30

1.59

G-9

938

895

1.05

1.50

0.86

1.35

1.57

Li & Liu [12]

JX1

964

1265

0.76

1.61

1.38

1.31

0.95

JX2

1174

974

1.21

1.72

1.51

1.54

1.02

JL-1

874

1127

0.78

1.42

0.75

1.18

1.57

JDL

1095

1016

1.08

1.69

0.95

1.48

1.56

Yan et al. [13]

G1

705

686

1.03

1.26

0.92

0.86

0.93

G2

806

807

1.00

1.33

1.04

0.94

0.90

G3

903

892

1.01

1.40

1.15

1.01

0.88

Total

22

AVE

1.01

1.47

1.09

1.22

1.15

SD

0.17

0.31

0.19

0.22

0.30

CoV

0.173

0.208

0.175

0.178

0.259

Vt is the experimental shear strengths;  is the predicted strength by Equation (21) of present study (with ); Vsst is the predicted strength of Hwang & Lee, [33] (i.e., Equation (4)); Vjd, VACI and VCSA are the predictions of Chinese Code [2] (i.e., Equation (2) using the concrete strength of the beam), ACI Code [3] (i.e., Equation (23)) and CSA Standard [42] (i.e., Equation (24)), respectively.


Also presented in Table 2 are the average (Avg), standard deviation (SD) and the coefficients of variation (CoV) of the ratios of the experimental shear strengths and the predicted shear strengths (Vt/ , Vt /Vsst , Vt /Vjd, Vt /VACI and Vt /VCSA). The comparison clearly confirms that the proposed model ( ) agrees better with the experimental shear strength (Vt).

5. Conclusions

Four spatial sandwich beam-column joints with different concrete strength ratios were tested under skew cyclic loads. Based on the experimental and analytical results, the following conclusions can be made.

(1) The failure modes of all the sandwich joints are in form of joint shear failure, subsequent to the flexural yielding of the beams, while the ductility coefficient is greater than 3.0 which satisfies the ductility demands.

(2) Compared to the joints with lower concrete strength ratios a, the specimen with the larger ratio features larger deformation at the joint.

(3) Based on the softened strut-and-tie model, a shear strength model of spatial sandwich joints, taking into account the effects of concrete strength ratio, the floor slabs and the plastic region of beams, is proposed herein. The predicted shear strengths are compared with the test results in this work and those in the literature. The comparison indicates that the proposed equations can accurately predict the shear strengths of the sandwich beam-column joints with higher accuracy than other available models. Notwithstanding, further study is necessary to more general situations where the maximum load applied at a different angle or different loading histories in the two orthogonal directions.

Acknowledgement

This work was financially supported by the National Natural Science Foundation of China (No. 51778341) and the National Key R&D Program (No. 2017YFC0702902).

Compliance with Ethical Standards

Funding: This study was funded by the National Natural Science Foundation of China (grant number 51778341) and the National Key R&D Program (grant number 2017YFC0702902).

Conflict of Interest: The authors declare that they have no conflict of interest.

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Notations

Ac

gross-sectional area of the column;

As,

sectional areas of the top and bottom reinforcement of the beam, respectively;

Asb

reinforcement area in the tension zone of the beam;

Asp

reinforcement area of the floor slab located within the beam effective flange width;

area of the diagonal strut

Asvj

total sectional area of transverse reinforcements within the effective width of the joint core zone in the loading direction;

distance between the centroid of resultant forces of longitudinal compressive steel reinforcements and the extreme compression face of the beam.

as, bs

depth and width of the diagonal strut, respectively;

bb, hb

beam width and depth, respectively;

bc, hc

column width and depth, respectively;

bj, hj

effective width and height of the core zone of the joint, respectively;

cb, cc

depth of the compression zone of the beam and column, respectively;

 

distances between the extreme longitudinal reinforcement in the beams and columns, respectively;

hbo

effective depth of beam;

distance between the centroids of upper and lower beam reinforcement;

Hc

height of column;

lp

length of the plastic hinge region of the beam;.

s

spacing of the transverse reinforcements along the beam axis;

ts

thickness of the floor slab;

,

compressive strengths of concrete of the beam and the column, respectively;

cylinder strength of concrete;

effective compressive strength;

ft

design tensile strength of concrete;

fyv

design strength of shear reinforcement;

fyh, fyv

yield strength of the joint hoop reinforcement and intermediate column bars, respectively;

fyb, fyp

yield strength of the beam reinforcement and slab reinforcement, respectively;

N

design axial force;

Nc

axial force acting on the column;

VACI

horizontal joint shear strength predicted by ACI 318-14;

VCSA

horizontal joint shear strength predicted by CSA A23.3-04;

Vjbf

horizontal joint shear strength when the beam reinforcement yields;

Vj1

horizontal shear strength of type 1 joint;

Vjd

horizontal joint shear strength predicted by Chinese Design Code for Concrete Structures;

Vsst

horizontal joint shear strength of the softened strut-and-tie model;

horizontal joint shear strength of the modified softened strut-and-tie model;

Vt

measured shear strength;

Eh

energy dissipated in each load cycle;

Ee

energy dissipation of an equivalent elastic cycle;

K

strut-and-tie index;

M /(V h)

shear span ratio;

a

concrete strength ratio;

q

angle between the diagonal strut and horizontal axis;

gRE

seismic adjusting factor;

gh, gv

fractions of diagonal compression transferred by the horizontal ties in the absence of the vertical ties and the vertical ties in the absence of the horizontal ties, respectively;

,

balanced forces of the horizontal ties and vertical ties, respectively;

,

indexes of the horizontal and vertical ties, respectively;

D

lateral deformation at the top of the column;

eb

longitudinal strain of the beam reinforcement in the plastic hinge region of the joint;

ebf

tensile strain of the beam at yielding;

er, ed

average principal tensile and compressive strains in the joint, respectively;

ej

average concrete strain in the center of the joint;

xh

equivalent hysteretic damping ratio;

hj

confinement effect factor of the orthogonal beams;

n

softening coefficient of the concrete;

f

strength reduction ratio;

fm

rotation angle of the column;

fpmp, fpmn

positive and negative plastic rotational angles, respectively.

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