Strength of Beam-column Joint in Soft First Story of RC

Strength of Beam-column Joint in Soft First Story of RCBuildings Part 2: Design Equations

, SusumuTakahashi Toshikatsu Ichinode, , Masaomi Teshigawara

Journal of Advanced Concrete Technology, volume ( ), pp.12 2014 146-157

Takashi Kamiya,,Sefatullah HalimHiroshi Fukuyama

Cross-Sectional Damage Index for RC Beam-Column Members Subjected to Multi-Axial FlexureSatoshi Tsuchiya, Koichi MaekawaJournal of Advanced Concrete Technology, volume ( ), pp.4 2006 179-192

Nonlinear FEM analysis of RC beam-column joint strengthened by cast in-situ joint expansionAmorn Pimanmas, Preeda ChaimahawaJournal of Advanced Concrete Technology, volume ( ), pp.7 2009 307-326

Strength of Beam-column Joint in Soft First Story of RC Buildings Part 1: Experiment, Sefatullah Halim Toshikatsu Ichinose, , Go Kotani, Masaomi Teshigawara,

Journal of Advanced Concrete Technology, volume ( ), pp.12 2014 138-145

Susumu TakahashiTakashi Kamiya, Hiroshi Fukuyama

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Journal of Advanced Concrete Technology Vol. 12, 146-157, May 2014 / Copyright © 2014 Japan Concrete Institute 146

Scientific paper

Strength of Beam-column Joint in Soft First Story of RC Buildings Part 2: Design Equations Sefatullah Halim1, Susumu Takahashi2, Toshikatsu Ichinose3, Masaomi Teshigawara4, Takashi Kamiya5 and Hiroshi Fukuyama6

Received 26 June 2013, accepted 19 February 2014 doi:10.3151/jact.12. 146

Abstract Part-1 of this study presents the experimental results of beam-column joints in soft-first story buildings, in which the first-story column depth is twice that of the upper stories. In Part-2, strut-and-tie models (STMs) are developed for the specimens. The flow of internal forces shown by the developed strut-and-tie models agreed with the observed cracks. The developed models for the joints are different from those for the usual joint: the main difference is that, in the specimens large struts were extended into the wall panel. STMs are also developed for the total frame to understand the overall equilibrium. Based on the STMs, new design equations are proposed, where the joint strength is evaluated as the sum of the flexural strengths of the beam and the second-story column. In I-type joint, the effect of the wall panel and beam stirrups is also considered. In O-type joint, the effect of the hoops in the joint is also considered.

1. Introduction

The strength of the beam-column joint is one of the more discussed issues in structural engineering (Shio-hara 2012). Many researchers proposed models to por-tray the failure mechanism of the usual beam-column joints (e.g., Paulay et al. 1978; Shiohara 2001, 2004). However, the application of these models to the joints in soft-first story building is difficult because as discussed in Part-1, the bending moments acting on the joints are opposite to those on usual joints.

This Part-2 paper includes three topics. The first is the development of strut-and-tie models (STMs), (Schlaich et al. 1987) for the specimens described in Part-1 to understand the flow of the forces in the speci-mens. The STMs are compared with those for usual knee joints and exterior tee joints, which are discussed by Liang (2005) and Sritharan (2005). The second topic is the development of STMs for the total frame to un-derstand the overall equilibrium. The third topic focuses on the development of new design equations to compute the strength of such joints considering the effects of the

boundary beam and column.

2. Overview of test results of O-1 and I-1 specimens

Two specimens are reviewed here: Specimens O-1 and I-1 with first-story column extended toward outside and inside of the frame, respectively. The black and red lines in Fig. 1 show the envelopes of the observed load-displacement relationship of Specimen O-1 and Speci-men I-1, respectively. The purple lines show the analyti-cal strength assuming that the beam and the joint panel have infinite rigidity. Figure 1 shows that the observed strengths of the specimens in the joint opening direction are less than two-third of the analytical strengths. The analytical strength is based on the flexural capacity of first-story column as discussed in Part-1. In the closing direction, the observed strength of Specimen O-1 is al-most equal to the analytical strength, whereas the ob-served strength of Specimen I-1 is smaller. In the fol-lowing sections STMs are developed to understand the behavior and failure mechanisms of the specimens.

1Ph.D Student, Dept. of Architectural Engineering, Nagoya Institute of Technology, Nagoya, Japan. E-mail:[email protected] 2Assistant Professor, Dept. of Architectural Engineering, Nagoya Institute of Technology, Nogoya, Japan. 3Professor, Dept. of Architectural Engineering, Nagoya Institute of Technology, Nogoya, Japan. 4Professor, Department of Architecture, Nagoya University, Nogoya, Japan. 5Chief Research Engineer, Yahagi Construction Co., Ltd., Nogoya, Japan. 6Director, Dept. of Structural Engineering, Building Research Institute, Tsukuba, Japan.

-1200

-800

-400

0

400

800

-40 -30 -20 -10 0 10 20 30 40

Late

ral L

oad

(kN

)

Defrmation (mm)

436 kN436 kN

Opening

-950 kN

-1065 kN

Analytical Strength

Specimen O-1Specimen I-1

Maximum Strength

Closing Specimen O-1

Opening directionClosing direction

Specimen I-1

Fig. 1 Load-Deformation relationship.

S. Halim, S. Takahashi, T. Ichinose, M. Teshigawara, T. Kamiya and H. Fukuyama / Journal of Advanced Concrete Technology Vol. 12, 146-157, 2014 147

3. Strut-and-tie analysis of specimen O-1 to opening load

Figure 2a shows the STM for O-1 specimen. Solid blue lines indicate the bars in the tensile yielding, while dashed lines indicate below the yield point. Red zones indicate struts in compression.

The STM for the opening direction was determined assuming the following: (1) Three layers of the column main bars (TAD in Fig.

2a and 9-D19 in Fig. 2c) resist the tension1. The beam bottom bars, the vertical bars in the wall panel, and the stirrups also resist the tension. Each tie is located at the centroid of the reinforcement.

(2) The effective compression strength of concrete to determine depth of strut is assumed to be 85% of the concrete strength (Yun and Ramirez 1996). Strut’s width is determined as follows:

0.85 'strut

strutc

CW

f b=

× (1)

where Cstrut is the compression force of a strut, fc' is the concrete strength, and b is the width of each element that the strut goes through. Because the thickness of the wall panel was 100 mm and 1/4 of the width of the beam (400mm), the width of each strut widens by four times at the boundary of the beam and the wall panel. Similar changes also occur at the boundary of the beam and the column. (3) Node B is located at the centroid of the beam bot-

tom bars such that the outer edge of the nodal zone coincides with the edge of the outermost hooked beam bottom bars (Fig. 2d). Node C is located at the re-entrant corner considering the geometric re-strictions. The inclinations of the struts KJ, LI, and EH are simply assumed as 45° because the effects of the inclinations on the strength are negligible.

(4) The compressive force of the column reinforce-ment is neglected because it is located outside node B.

(5) The stresses in the reinforcement are determined within the yield strength so that the lateral load ca-pacity is maximized.

(6) The effect of the hoop reinforcement is neglected in the joint because the hoop reinforcement ratio is very small (0.17%).

As shown in Fig. 2a, the strut distribution agreed with the crack pattern observed in the test. The load carrying capacity based on STM (Q) agreed with the test result with error less than 10% (Fig. 3). This strength is closer to the observed strength than that of the computed capacity based on conventional methods (Fig. 1). There are two reasons for this. The main reason is that in the conventional analysis method, the full

depth of the first-story column was considered effective. In the STMs, node B is located at the outermost point of the hooked beam bottom bars (Fig. 2d) resulting in a smaller distance between the compressive strut and the tensile reinforcement. This is equivalent to the reduced effective depth of the first-story column. The other rea-son is the location of the critical section. In the conven-tional analysis, the critical section is assumed at the level of the beam bottom face. In the STM, node B is located at the level of the beam bottom reinforcement, which is equivalent to a longer shear span (M/Q) than usual.

The anchorage details of the outer corner of the speci-mens are similar to the usual knee joint. The observed cracks in Fig. 2a are also similar to those in knee joint (Wight and MacGregor 2012). To compare O-type joint with knee joints from the view of the force-resistance mechanism, a STM is constructed for a knee joint as shown in Fig. 4. The procedure and assumptions to con-

Fig. 2 Strut and tie model for O-1. (Opening).

Fig. 3 Observed strengths vs. STM.

1 If we assume two or four layers instead of three, we obtain a smaller horizontal force Q.


struct the STM are the same as before and the rein-forcement details are similar to those of the test speci-mens. The biggest difference is that the wall panel and the second-story column are removed.

The strength of the knee joint is approximately three-fourth of that of the test specimen. The capacity of the knee joint is governed by beam flexural capacity. The difference of the capacities in the opening load is attrib-utable to the difference of the location of node C. Note that strut BC in Fig. 2a is less inclined than that in Fig. 4. This leads to a larger compressive force in strut AB and a larger shear force in the column. Moreover, note that the inclination of strut CD in Fig. 2a is opposite to that of strut AB, indicating that the sign of the shear force in the second-story column is opposite to that of the first-story column. The cracks observed in the test also confirm this tendency. The negative shear force is compensated by strut HE in the wall panel (Fig. 2a).

4. Strut-and-tie analysis of specimen I-1 to opening load

Figure 5a shows the STM for Specimen I-1. This model is developed based on the same assumptions made for Specimen O-1, except that the first and third assump-

tions are modified as follows: (1') Ties TAK and TAD in Fig. 5a represent 9-D19 and

2-D19 in Fig. 5d, respectively. (3') Inclination of struts LN and JO is 45°. Strut KM

is steeper (51°) considering the prominent crack ap-peared in the test.

Node K is located at the centroid of the two layers of the main bars of the first-story column such that the upper edge of the nodal zone coincides with the upper edge of the hooked bars (Fig. 5e).

As shown in Fig. 5a, the strut distribution agreed with the crack pattern appeared in the test. The com-puted tensile forces in the ties agreed with the observed values. For example, in the test the beam bottom bars yielded and the observed strain of the first-story column main bars was about 90% of its yield strain as was com-puted in STM (Fig. 5a). The strength based on the STM agreed with the observed strength with an error less than 5% (Fig. 3). Recall that the observed strength is smaller than that obtained by the conventional analysis (Fig. 1). The reason for this decrease is the different failure mode: the beam failure of Specimen I-1 was associated with the yielding of the beam reinforcement (tie BM) and stirrup (tie ML).

Figure 5c shows the detail of node B. the depth of strut AC, wAC, is narrower than the gap between the column edge and the anchorage point; instead, it is de-termined by the equilibrium of the axial and shear forces and the tensile forces in the ties. If the axial force of the first story column is in tension and extremely large, wAC may disappear.

If we carefully examine Fig. 5a, we find that the flex-ural cracks in the second story disappear around node D, which contradicts the STM showing the constant tensile stress in tie AD. This contradiction is attributable to the negligence of the tensile stress of concrete, which would gradually bend strut BC to the left absorbing the tensile stress of tie AD.

Figure 6 shows an STM for an exterior joint. Proce-dure, assumptions, and reinforcement details are similar to those for Specimen I-1. The beam axial force is as-sumed to be zero.

The strength of the exterior joint based on STM (230 kN) is approximately 55% of the specimen’s strength (425 kN). This difference is attributable to the location and orientation of the strut and tie in the second-story column. In Fig. 5a, tie AD is located to the left of strut BC, producing a bending moment opposite to that of strut ED and tie BC in the usual exterior joint (Fig. 6).

5. Strut-and-tie analysis of specimen O-1 to closing load

Figure 7a shows the skeleton of the STM of Specimen O-1 for the closing direction. Among the six assump-tions for the opening direction, the first, third, and fourth are changed as follows:

(1') The three layers of the column main bars (TAC

Fig. 4 Strut and tie model of knee joint. (Opening).

Fig. 5 Strut and tie model for I-1. (Opening).


and TAD in Fig. 7d) resist the tension. The beam top bars (TCH) also resist the tension.

(3') The locations of nodes B, I, F, and H are deter-mined so that the strength is maximized.

(4') The first layer of the first-story column main bars resists the compression (CAE in Fig. 7d) because it is continuous throughout the first- and second-story col-umns.

Figure 7b shows the STM with the compression zones, which agrees with the cracks observed in the test. The computed tensile forces in the ties also agree with the observed strains. The beam top bar (tie CH) yielded as it was observed in the test. The stresses in the first-story column bars (tie BC) were 95% of the yield strength, as was observed (Fig. 7e). The strength based on STM is 88% of the observed strength (Fig. 3). The STM and the cracks appeared in the test show that a large amount of forces is transferred into the wall panel.

To investigate the effect of the wall panel, an STM is developed for a knee joint, as shown in Fig. 8. Virtually, no axial force is applied both in the opening and closing directions, except those required for the equilibrium. The axial force of 2,250 kN (see Part-1), which was applied in the closing direction, cannot be applied to the knee joint because the beam cannot resist a shear force of 2,250 kN. The orientation of struts IH and IF in Fig. 7 is different from that of IH and ID in Fig. 8; this dif-ference can be attributed to the presence of the second-story column and the wall panel in the test specimen. In Fig. 7, most of the shear force is transferred to the shear wall panel, which leads to a larger capacity of the test specimen than the usual knee joint.

On the other hand, there are important similarities be-tween the test specimens and the knee joint. In the open-ing load, as discussed above, node B (Fig. 2d) is located inside the hook of the beam bars, limiting the capacity of the first-story column, as will be discussed later. In the closing load, node C (Fig. 7b) is resisted by the ten-sile forces of the column and beam bars, which indicates that beam top bars in O-type joint should be anchored, according to the provisions for the knee joint making the tail length long enough (O-1t, Fig. 5 of Part-1).

6. Strut-and-tie analysis of specimen I-1 to closing load

Figure 9a shows the skeleton of an STM of I-1 speci-men for the closing load. The STM is developed similar to that of Specimen O-1, except that the first layer of the

Fig. 6 Strut and tie model for exterior joint. (Opening).

Fig. 8 Strut and tie model for knee joint. (Closing).

Fig.7 Strut and tie model for O-1. (Closing).


first-story column main bars does not resist the com-pression force because it terminates in the joint panel. Figure 9b shows the STM with compression zones. The strut distribution agrees with the crack patterns observed in the test. The computed forces in the ties also agree with the observed strains (e.g., the beam top bars as well as the first-story column main bars yielded as was ob-served). The strength based on STM is 80% of the ob-served strength (Fig. 3). The STM as well as the ob-served cracks show that a large shear force is transferred to the wall panel in the second story. As concluded in paper-1 the additional beam top bars provided in I-1t prevented the compressive failure of the wall panel. This phenomenon can also be discussed in term of the strut and tie model (Fig. 9b). Increasing tensile force in tie TCH would result in larger forces in strut IC and IH, and smaller compressive force in strut IE. Similarly, considerable amount of vertical force can be resisted by the beam stirrups making a truss model along the beam length which is similar to Fig. 5a but upside down. The truss is not shown in Fig. 9b because the effect in speci-men I-1 is much smaller than those of the other struts. Figure 10 shows the STM for the closing load of the exterior joint. The procedure and assumptions are simi-lar to those for Specimen I-1. The strength of the exte-rior joint (110kN) is approximately 15% of the speci-men’s strength (755 kN). The strength in Fig. 10 is gov-erned by the flexural capacity of the beam. This differ-ence is attributable to the following reasons: (a) Struts IH and IE in Fig. 9 are much larger than

struts BE and BF in Fig. 10 because they extend the forces into the wall panel.

(b) The vertical tie BD in Fig. 9 is located on the right of the strut CD, producing a bending moment oppo-site to that of the usual joint (Fig. 10).

7. Strut-and-tie model for a prototype frame

ASTM is developed for the frame of a full-scale struc-ture to understand the distribution of the inner forces in a real structure under applied external loads. The build-ing structure is assumed to be a five-story single bay apartment building (Fig. 11) with a span length of 12m in both directions and gravity load of 12.5 kN/m2, which leads to a total gravity load of 12.5 × 5 × 12 × 12 = 9000 kN. For simplicity, the load is applied as concen-trated load at two points to depict its distribution and ease constructing the strut and tie model. The resultant force of the two point loads acts at the center of gravity. The first-story columns are extended toward outside similar to that for Specimen O-1. The dimensions of the beams, columns, and wall panels are twice of those of the specimen. The reinforcement ratio and details are the same as those of the specimen. Recalling that the computed strengths of Specimen O-1 to the opening and closing loads were 400 kN and 940 kN, respectively, the earthquake force is assumed to be (400 + 940) × 4 = 5360 kN and is applied at two-thirds of the building

height. The assumed vertical and horizontal forces cause the axial forces in the right and left columns in the first story to be 0 and 9000 kN, which are four times the ax-

Fig. 9 Strut and tie model for I-1. (Closing).

Fig. 10 Strut and tie model for exterior joint. (Closing).


ial forces applied to Specimen O-1 during the opening (0 kN) and closing load (2250 kN), respectively.

The red zones and the blue lines in Fig. 11 represent the compressive and tensile forces in the frame. There is one difference between Figs. 11 and 2. Strut AH in Fig. 11 is supported by tie AG, which represents the tensile forces in the vertical reinforcement in the wall panel and the stirrups in the beam. Strut HE in Fig. 2 is supported by tie HG, which represents the tensile forces in the lon-gitudinal reinforcement in the vertical stub. Otherwise, the stress distribution near the opening joint in Fig. 11 is identical with that in Fig. 2. Furthermore, there are no differences between Fig. 7 and Fig. 11 for the closing side. These similarities indicate that the vertical stub constructed for the test specimen represents the rest of the frame if the wall panel of the actual frame is pro-vided with appropriate amount of vertical reinforcement, which is indicated by ties AG and BH in Fig. 11. The required quantity is discussed by Shobu et al (2012).If we assume a frame taller than that the in Fig. 11, the location of the horizontal force moves upward causing tensile axial force in the column. This will reduce the amount of the compressive strut in the tensile column but will not change the strut and tie system drastically. Similar strut-and-tie model can be obtained for I-type frame.

The diagrams at the bottom of Fig. 11 show the axial-force, shear-force, and bending moment of the boundary beam. The forces in the struts in the beam are decom-posed into vertical and horizontal components to calcu-late the beam axial and shear forces in each section. The beam moment diagram is calculated as the moment of the forces in the struts and ties about the centerline of the beam. In the axial force diagram, graph above the horizontal axis represents compressive force. In the opening joint (right), the beam is subjected to a tensile axial force, leading to a smaller flexural capacity of the beam, while in the closing joint (left), it resists the com-pressive axial force with lager bending moment capacity. The shear force in the opening joint (right) is resisted by the stirrups, while the shear force in the closing joint (left) is resisted by the concrete strut. The bending mo-ment diagram indicates that the beam bottom rein-forcement needs to be provided all along the span, whereas the top reinforcement needs to be provided only near the end regions. In the proposed design equa-tions, the above mentioned effects of the stirrups and the beam axial force are considered to determine the beam moment capacity.

8. Simplified equations

On the basis of the observed failure modes and the re-sults of the strut-and-tie analysis, simplified equations are proposed. The equations are exclusively utilizing conventional flexural analyses.

The observed failure modes can be classified into two groups: (1) the failure associated with horizontal cracks

at the top of the first-story column (Figs. 10b and 13b of Part-1) and (2) the failure associated with inclined cracks in the beam or the joint (Figs. 7b, 8b, 16b, and 17b of Part-1). The strength of each specimen Qu (or the lateral force which acts on the first story column) is as-sumed to be the smaller value of the first story column strength Qc and the joint strength Qj.

min ,u c jQ Q Q⎡ ⎤= ⎣ ⎦ (2)

(a) First-story column strength Qc As discussed with Fig. 10b of Part-1, the critical cracks of the column failure in the O-type specimen caused by the opening load tend to propagate toward the point of anchorage or node B in Fig. 2a. Therefore, the critical section is assumed as shown in Fig. 12a; where dco is the distance between the bottom of the second story beam and the centroid of the beam bottom bars, and L is the length between the point of contra flexure and the top of the first story (column shear span). The first-story column strength Qc can be obtained considering the equilibrium of the moments around the white circle shown in Fig. 12. For simplicity, however, we assume that Qc can be calculated by the following equation:

1cc

co

MQ

L d=

+ (3)

where Mc1 is the moment capacity of the first-story col-umn

Mc1 must be computed considering that the concrete outside the point of anchorage may not resist the com-pression, as assumed in Fig. 2d. Such a phenomenon

Fig. 11 Strut and tie model for prototype frame.


was prominent in Specimen O-1, where the hoop rein-forcement ratio in the joint was 0.17%; the inclined cracks propagated toward the point of anchorage (Fig. 16b in Part-1). In Specimen O-1t, where the hoop rein-forcement ratio was 0.80%, the hoop reinforcement in the joint reduced the opening of such cracks (Fig. 21b in Part-1). Figure 13a shows the assumed strut created by the contribution of the hoops in the joint, where ∑awσy represents the horizontal force of the hoops in the joint and Cout represents the compressive force acting outside the point of anchorage. This assumption leads to

22( )b

out w yb c

DC a

l Dσ= ∑

− (4)

where Db is depth of the beam, lb is embedment length of beam bottom bars and Dc2is the depths of the second-story column. The depth of the strut outside the anchor-age point is

2' 2 ' ( )out b

w yc c b c

C DD a

f b f b l DΔ σ= = ∑

− (5)

where f'c is the concrete strength and b is the column width. Therefore, considering the contribution of the hoops in the joint panel, the effective depth of the first-story column is computed as follows:

122 ' ( )

beff b b w y c

c b c

DD l D l a D

f b l DΔ σ= + = + ∑ ≤

− (6)

In case Dc1 equals to Dc2, the strut outside the anchor-age point would be straight as shown in Fig. 13b and therefore Deff would be equal to Dc1.

Computing the moment capacity of the first-story col-umn, Mc1, the moment due to eccentricity of the axial force should also be considered because the center of the effective section does not coincide with that of the full section of the first-story column. Therefore,

11 2

c efc ef

D D fM M f N

−= − i (7)

where Deff is the effective depth of the first-story column,

Dc1 is the actual depth of the first-story column, and Meff is the moment capacity of the column computed by con-ventional flexural analysis considering the effective depth.

For I-type joints, the moment capacity of the first story column, Mc1, is approximated to the moment ca-pacity of the full section of the first-story column (Fig. 12b); because the strut AC (Fig. 5a) can resist vertical compressions without any help from the hoop. In case the tensile axial force is extremely large and the strut AC in Fig. 5 disappears, the strength will be slightly overestimated.

In the closing direction, as discussed with Fig. 12b of Part-1, the critical cracks tend to propagate toward node I (Figs. 7a and 9a), which is close to the centroid of the beam bottom reinforcements. Therefore, Eq. 3 is again assumed to apply and the column section is as-sumed to be fully effective (Fig. 12c).

(b) Strength associated with joints, Qj, for O-series specimens As discussed in Part-1, the critical crack of the beam failure caused by the opening load tends to start from the inside corner and propagate toward node C in Fig. 2a. Therefore, the critical section is assumed as shown

J B

TvCv

NQc

Tv

L

dco

dco

Dc1

Cv

NQc

Deff

L

This distance isalmost dco

Dc1

b

L

Tv

Cv

a) Opening load (O-type) b) Opening load (I-type) c) Closing load (O-and-I-types)

Qc N

Fig. 12 Critical sections in column.

Fig.13 Contribution of hoops in the joint.


in Fig. 14a. Ch and Cv in Fig. 14a represent the horizon-tal and vertical components of the strut acting on nodes C, respectively. Th and Tv represent the tensile forces of the beam and column reinforcement, respectively. Thj represents the tensile forces of the hoops in the joint assuming that the crack in the joint penetrates till about 1/3 of the beam depth (height of the joint panel).

3b

hj j y cD

T P bσ= × (8)

where pj is the hoop reinforcing ratio, σy is the yield strength of the hoops, and bc is the width of the joint panel.

The strength associated with the joint, Qj in Eq. 2, can be obtained considering the equilibrium of the moments around the white circle shown in Fig. 14. In other words,

/ 2j

jb

MQ

L D=

+ (9)

where Mj is the moment caused by the inner forces in Figs. 14a and 14b (Ch, Cv, Th, Thj, and Tv).

Referring to Fig. 16a of Part-1, we decompose Mj as follows:

2c bMj M M′= + (10)

where Mc2' is the moment caused by Cv and Tv (regarded as the moment capacity of the second-story column around the center of the first-story column) and Mb is the moment caused by Ch, Th, and Thj (regarded as the moment capacity of the beam).

The critical crack of the beam failure caused by the closing load is similarly assumed, as shown in Fig. 14b.

The following paragraphs are devoted to discuss the calculation of Mc2' and Mb. 1) It is assumed that the magnitude and the location of

Cv and Tv in Fig. 14 are similar to the compressive and tensile forces obtained by conventional flexural analysis of the second-story column using the axial force of the first-story column. Thus, Eq. 9 is ap-proximated as

1 22 2 2

c cc c

D DM M N

−′ = ∓ i (11)

where Mc2 is the moment capacity of the second-story column around its center and the second term is the moment due to the axial force around the center of the first-story column. The sign of the second term is nega-tive and positive for opening and closing loads, respec-tively. 2) The moment capacity of the beam, Mb, caused by

the horizontal inner forces is assumed as

( ) ( )2

bb h b h hj

DM C D d T Tα= × + − × × (12)

where αDb represents the distance of the compressive force Ch from the centerline of the beam (Figs. 14a and 14b). Based on the STMs, α is assumed to be 0.4 and 0.2 for opening and closing loads, respectively (Table 1). For simplicity, the location of Thj is assumed to be the same as that of Th. On the other hand, the equilib-rium of horizontal forces leads to h h hj jC T T Q= + − and h h hj jC T T Q= + + for opening and closing loads, respectively. Thus, we get

( )2

bb b h hj b j

DM D d T T D Qα α⎛ ⎞= + − × + ×⎜ ⎟

⎝ ⎠∓ (13)

The sign of the second term is negative and positive for opening and closing loads, respectively.

(c) Strength associated with joint, Qj, for I-series specimens It is assumed that Eqs. 8 and 9 also apply to I-series specimens. However, Mc2’ in Eq. 10 needs to be evalu-ated differently because of the different failure modes. Figure 15a shows the forces around the joint of I-series specimens subjected to opening load. As shown in Fig. 7b of Part-1, the prominent crack in the opening load starts from the beam and proceeds to the joint above the hook of the first-story column main bars in the extended area. In the STM, the struts BK and MK (Fig. 5a) are anchored inside the hook. Thus, the α value for the opening load is smaller (0.3) than that for the O-Series (0.4), as shown in Fig. 15a and Table 1. The moment capacity of the beam is calculated using Eq. 14.

2b

b b h b jD

M D d T D Qα α⎛ ⎞= + − × ×⎜ ⎟⎝ ⎠

∓ (14)

Figure 16 shows the contribution of the beam stirrups. As discussed with Fig. 7a of Part-1 paper, the prominent

Table 1 Variation of coefficient α.

Load Cases Opening Closing

O-Type 0.4 0.2 I-Type 0.3

NQj

J B

Tv

Cv

α.Db

Th

Thj

Thj

NQj

TvCv

CvCC

ThB

Ch

CvCI

Ch

Db

Dc2

d α.Db

C C2

bDd

Ch

ICh

2bDL+

(a) Opening load (b) Closing load Fig. 14 Critical sections in O-type joint.


crack proceeds to point A in Fig. 16 (the top of the hooks of vertical bars in the extended area of the first story column); and assuming that the crack is inclined at 45°, the tensile force of the beam stirrups Tst1 in the opening case (Fig. 16) is then equal to the tensile force of the beam stirrups in the length of ld−2dc−r (Fig. 16). ld is the embedment length of the vertical bars of the first-story column anchored in the joint, dc is the con-crete cover of the beam and first story column and r is radius of the hook.

1 ( 2 )st w y b d cT p b l d rσ= × × − − (15)

where pw is the stirrup reinforcing ratio, and bb is width of the beam. Though the stirrups were neglected in Fig. 9b for specimen I-1, stirrups in specimen I-1t consid-erably contribute to the strength.

Moreover, the wall vertical reinforcement (Tw) also contributes to the joint strength in the opening case. The force Tw acting at node K is the resultant force of the wall vertical reinforcement above the joint. Because the direction of Tst1and Tw is vertical, their contribution is added to the moment capacity of the second-story col-umn; thus, we get

1 22 ' 2 1 12

c cc c st st w w

D DM M N T l T l

+= + ⋅ + ⋅ + ⋅ (16)

where Mc2 is the moment capacity of the second-story column around its center. lst1 is the distance between Tst1 and Cv in Fig. 15a because Tst1 increases Cv, and is ap-proximated as 1 10.5 0.9st d c cl l d D= − + . lw is the dis-tance between Tw and the Cv, and is approximated as

10.6w cl D= . In Specimen I-1 subjected to the closing load, the in-

clined crack in the beam and the crushing of the wall

panel concrete were prominent (Fig. 8 of Part-1). There-fore, the critical section is assumed as shown in Fig. 15b. Ch and Cv in Fig. 15b represent the horizontal and vertical components of the strut acting on node I, re-spectively. For simplicity, Ch is assumed to act at the beam section and Cv is assumed to act at the T-section composed of the wall panel and the second-story col-umn. The length of the wall panel L is assumed as the sum of the length above the joint (Dc1-Dc2) and 80% of the beam depth.

1 2( ) 0.8c c bL D D D= − + (17)

where 0.8Db represents the projection of the beam depth above the centroid of the beam compression zone. It is further assumed that the moment created by Cv and Tv can be approximated by the flexural strength of the T-

lst1=0.5ld −dc+0.9Dc1

dcdc

Cv

ld−2dcr

−r

A

Tst1

Tw

45

ld

Fig. 16 Contribution of beam stirrups in opening direc-tion. (I-type joint).

Cv

ChL=(Dc1 -Dc2 )+0.8Db

Dbdd

NQj

Dc2

Tv v

Th

N

Tv

Cv

h

ChTh

B

K

J

C

e

45

2bDL+

2bDd

CJ

C

CB

Qj

α.Db

α.Db0.4Db

Tst2Tst1

I

Tw

(a) Opening load (b) Closing load

Fig. 15 Critical sections in I-type joint.


section in Fig. 15b. Thus, Mc2' in Eq. 9 is computed by the following equation:

2' 2 2 2c c st stM M N e T l= + ⋅ + ⋅ (18)

where Mc2 is the moment capacity of the T-section around its centroid and e is the distance between the centroid and the center of the first-story column, and corresponds to:

22 1 2

2

( )2( )

c c c

c

D b D D L te

D b L t⋅ ⋅ − + ⋅

=⋅ + ⋅

(19)

where b is the width of second-story column and t is the thickness of wall panel, as shown in Fig. 15b. The last term in Eq. 18 shows contribution of the beam stirrups in the closing direction. The tensile force of the beam stirrups (Tst2), is different from that in the opening direc-tion (Tst1). Tst2 is assumed to be equal to the tensile force of the beam stirrups in the length equal to 0.8Db and acting at the distance 0.4Db from the face of the column.

2 0.8st w y b bT p b Dσ= × × 2 0.8st w y b bT p b Dσ= × × (20)

2stl in Eq. 18 is the distance between Tst2 and Tv in Fig. 15b because Tst2 increases Tv, and is approximated as 2 10.4 0.9st b cl D D= + .

Mb in Eq. 10 is computed using Eq. 14, and α is as-sumed to be 0.3 (Table 1) based on Fig. 15.

(d) Summary and verification The procedure to compute the strength of the joint is summarized here with the help of Table 2. To compute the first-story column strength (Qc), the first-story col-umn moment (Mc1) should be computed. For the open-ing strength of O-type joints, Eqs. 4-7 are used consid-ering the effective depth of the first-story column. For the other cases (closing strength of O-type joints as well as opening and closing strengths of I-type joints) full section of the first-story column should be considered (Table 2).

To compute the joint strength (Qj), the beam bending moment (Mb) and second-story column moment (Mc2') should be calculated. The second-story column moment (Mc2') of O-type joints should be computed using Eq. 11 both for opening and closing loads. Mc2 in Eq. 11 is the

moment capacity of the rectangular section of the sec-ond-story column (Fig. 14). The second story moment (Mc2') of I-type joints for the opening load should be calculated from Eqs. 15-16. Mc2 in Eq. 16 is the moment capacity of the second-story column with rectangular section (Fig. 15a). For the closing load of I-type joints, Eqs. 17-19 should be used to find Mc2'. Mc2 in Eq. 18 should be computed by flexural analysis of the T-section (Fig. 15b).

The beam bending moment Mb is defined by Eq. 13. The α values are given in Table 1 for different joint types and load cases. Because Eq. 13 includes the joint strength (Qj), Qj is computed with an iteration method.

Figure 17 shows the comparison between the ob-served and calculated strengths. Qexp in the vertical axis is the observed strength, while Qj in the horizontal axis is the computed joint strength. The denominator Qc is the computed column strength. Assume that the pro-posed equations are perfect (Qu= Qexp). Then Eq. 2 leads to the following:

If Qj<Qc, Qexp= Qj, which is represented by the in-clined red line connecting (0, 0) and (1, 1) in Fig. 17.

If Qj>Qc, Qexp= Qc, which is represented by the hori-zontal black line connecting (1, 1) and (2, 1) in Fig. 17.

These lines constitute the border between the green and yellow zones. A plot in the green zone indicates that the observed strength is larger than the expected value and vice versa. Because most plots are located around the border, we conclude that the proposed design equa-tions appropriately evaluate the strengths of the speci-

Table. 2 Procedure to compute lateral strength of joint.

Loading Cases Shear Force Moment Joint Type Opening Closing

O-Type Eqs. 4-7 and Flexural Analysis with Deff

Qc (Eq. 3) 1cM

I-Type Flexural Analysis with Full Section O-Type Eq. 11 and Flexural Analysis of Rectangular Section

2cM ′ I-Type Eqs. 15-16 and Flexural Analysis of Rectangular Section

Eqs. 17-19 and Flexural Analysis of T-Section

O-Type Eq. 13 + Table 1

Qj (Eqs. 9-10)

bM I-Type Eq. 14 + Table 1

0

0.5

1.0

1.5

0.5 1.0 1.5 2.0

Column failure is estimatedBeam failure isestimated

O-1 O-1t

O-1O-1t

I-1

I-1I-1t

CaseLoad

Joint ColumnOpeningClosing

Observed failure modes

I-1t

Hanai-1

Hanai-2Hanai-3

Fig. 17 Estimated and observed strengths.


mens. The plotted data include the specimens of Hanai et al. (2009). The vertical broken line at Qj/Qc= 1 is the border of beam or column failure: if Qj/Qc> 1, column failure is expected; if Qj/Qc< 1, beam failure is expected. The observed failure modes, which are indicated by circles and squares, mostly agree with the expectations.

Figure 18 compares the observed and computed strengths in terms of average shear stresses. The ob-served strength (Qexp) and computed strength (Qcomputed) are divided by the gross area of the first-story column (Ac1). For most of the specimens, the observed and com-puted shear stresses have good agreement.

(e) A model for pushover analysis Though Eq.2 is useful to evaluate the test specimens, it cannot be applied to actual structures. Figure 19 shows a model for pushover analysis which may be used for design. The shear wall and boundary columns in the upper stories are modeled according to Kabeyasawa et al. (1983). In this model, infinitely rigid beams are as-sumed at the top and bottom of each floor level. The axial stiffness of the boundary columns is represented by truss elements. The first-story columns are modeled

as One-component model (Giberson 1974). Namely, each column is idealized as a perfectly elastic line ele-ment (Fig. 20a) with two non-linear rotational springs at the two ends. The lower spring is located at a distance dco from the beam bottom face (Fig. 19b). Figure 20b shows the characteristics of the lower spring: the yield strength Mc is computed according to Table 2 and the yield rotation θcy is computed by conventional methods. To consider the beam failure, another non-linear rota-tional spring is located at the center of the joint panel (Fig. 19b). Figure 20c shows the characteristics of the joint spring: the yield strength Mj is computed according to Table 2 and the yield rotation θjy is discussed by Ichi-nose et al (2013).

To compute Mc and Mj, N and Qj are required. A pre-liminary elastic analysis of the structure is required to obtain the approximated values of these variables.

9. Conclusions

(1) The STM for a prototype frame validates the specimens constructed in this study. In other words, the specimens (part of a structural frame) represent the structural behavior of beam-column joint and connecting elements appropriately. The model also implies that the vertical wall reinforcement plays an important role in a wall with an open frame.

(2) The comparison between STMs for test specimens and those for usual exterior and knee joints shows that the flow of the forces of the specimens is dif-

a) Idialized model of frame b) Joint details

dco Column

Hinge

end speing

Jointrotationalspring

Rigid

Elastic

Fig. 19 Model for pushover analysis.

θ (rad)O

M

θjθcy

Mj

θ (rad)

M

O

Mc1

M

φ (rad/m)

Mc1

φy

b) Column end spring c) Joint rotational springa) 1F-Column elastic element

O

Fig. 20 Rotational springs and elastic element properties.

O-1 O-1t

O-1t

O-1

20

2

4

6

4 6

-10%

+10%

I-1Hanai-1

I-1

I-1t

I-1t

Hanai-2Hanai-3

2

1

( / )computed

c

QN mm

A

2

1

(/

)te

st c

QN

mm

A

Fig. 18 Comparison between observed and computed strengths.


ferent from that of the usual joints. All the speci-mens were stronger than the usual joints and a huge amount of shear force was transferred to the wall panel.

(3) In the case of the opening load of O-type joint, the effective depth of the first-story column is defined by the embedment length of the beam bottom bars and shear reinforcement in the joint panel.

(4) The strength related to the beam failure can be ap-proximated as the sum of the beam and second-story column strengths including the effects of the wall panel, hoops in the joint, and stirrups in the beam.

(5) The strength and failure modes predicted by the proposed procedure (Table 2) are in good agree-ment with the test results.

References Giberson, M. F., (1974). “Two nonlinear beams with

definition of ductility.” Journal of the Structural Division, ASCE, 95(7), 137-157.

Hanai, N., Goto, Y. and Ichinose, T., (2009). “Anchorage of column re-bars where column depth greatly reduces and structural wall appears.” AIJ J. Technol. Des., 15(29), 143-146. (in Japanese)

Ichinose, T., Kawai, T., Takahashi, S. and Teshigawara, M., (2013). “Second stiffness of beam-column joint in soft-first story RC building.” Proceedings of the Japan Concrete Institute, 35(2), 265-270. (in Japanese)

Kabeyasawa, T., Shiohara, H., Otani, S. and Aoyama, H., (1983). “Analysis of the full-scale seven-story reinforced concrete test structure.” Journal of the Faculty of Engineering, University of Tokyo, 36(2), 432-478.

Liang, Q., (2005). “Performance-based optimization of

structures: Theory and applications.” 1sted. New York: Spon Press.

Wight, J. K. and MacGregor, J. G., (2012). “Reinforced Concrete Mechanics and Design.” USA: Prentice Hall.

Paulay, T., Park, R. and Priestley, M. J. N., (1978). “Reinforced concrete beam-column joints under seismic actions.” ACI Journal, Proceedings, 75(11), 585-593.

Schlaich, J., Schäfer, K. and Jennewein, M., (1987). “Toward a consistent design of structural concrete.” PCI journal, 32(3), 77-150.

Shiohara, H., (2001). “New model for shear failure of RC interior beam-column connections.” Journal of Structural Engineering, ASCE, 127(2), 152-160.

Shiohara, H., (2004). “Quadruple flexural resistance in R/C beam-column joints.” Proceedings, the 13th World Conference on Earthquake Engineering, Vancouver, B. C., Canada, No. 491.

Shiohara, H., (2012) “Reinforced concrete beam-column joints: An overlooked failure mechanism.” ACI Structural Journal, 109(1), 65-74.

Shobu, H., Teshigawara, M., Nakamura, A., Izumi, N., Matsumoto, K., Ichinose, T., Takahashi, S., Takakoshi, S., Kamiya, T., Fukuyama, H., Suwada, H. and Kabeyasawa, T., (2012). “Effects of vertical reinforcement and boundary beam on the shear strength of RC wall located above an open frame.” Proceedings, 15th World Conference on Earthquake Engineering, Paper ID 4639.

Sritharan, S., (2005). “Strut-and-tie analysis of bridge tee joints subjected to seismic actions.” Journal of Structural Engineering, ASCE, 131(9), 1321-1331.

Yun, Y. M. and Ramirez, J. A., (1996). “Strength of struts and nodes in strut-tie model.” Journal of Structural Engineering, ASCE, 122(1), 20-29.

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Strength of Beam-column Joint in Soft First Story of RC