Design recommendations for achieving strong column-weak

Engineering Structures 126 (2016) 343–352

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Engineering Structures

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Design recommendations for achieving ‘‘strong column-weak beam” inRC frames

http://dx.doi.org/10.1016/j.engstruct.2016.07.0530141-0296/� 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: School of Civil Engineering, Qingdao University ofTechnology, 11 Fushun Rd, Qingdao 266033, China.

E-mail address: [email protected] (N. Ning).

Ning Ning a,b,⇑, Wenjun Qu c, Zhongguo John Ma d,e

a School of Civil Engineering, Qingdao University of Technology, 11 Fushun Rd, Qingdao 266033, ChinabCollaborative Innovation Center of Engineering Construction and Safety in Shandong Blue Economic Zone, Qingdao 266033, ChinacDepartment of Building Engineering, College of Civil Engineering, Tongji University, 1239 Siping Rd, Shanghai 200092, ChinadDepartment of Civil and Environmental Engineering, University of Tennessee, Knoxville, TN 37996-2313, USAe Shandong Jianzhu University, Jinan, Shandong 250101, China

a r t i c l e i n f o

Article history:Received 3 December 2015Revised 26 July 2016Accepted 27 July 2016Available online 9 August 2016

Keywords:RC frames‘‘Strong-column-weak-beam”Cast in-situ slabsEffective slab widthRequired ratio of column-to-beam strength

a b s t r a c t

This paper deals with the realization of ductile failure of RC frames with cast in-situ slabs. The cyclic load-ing experiments of two spatial reinforced concrete (RC) frames, a control specimen and a RC frame withcast in-situ slabs, are carried out. The failure pattern and the role of slabs are experimentally studied.Tests results indicate that slabs can change the failure pattern of RC frames from a typical ‘‘strong-column-weak-beam” failure to the ‘‘strong-beam-weak-column” failure. The slab reinforcement enhancesthe resistance moment of the beams, which is the main reason to form ‘‘strong-beam-weak-column”.The parameters such as axial compression ratio, thickness of slab, concrete strength, reinforcement ratioof slab and stiffness of transverse beam are investigated by Finite Element Analysis (FEA). Strut and tiemodel of slab is proposed to calculate the effective slab width. Simple equations are derived by a 95%guarantee rate based on 200 models. The required ratio of column-to-beam strength is studied basedon the tests and FEA data. A reasonable required ratio is proposed to avoid the brittle failure of the RCframes for seismic design.

� 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Reinforced Concrete (RC) frames are the most popular struc-tural system for multistory buildings in many parts of the world.However, these buildings have shown poor performance duringstrong earthquakes in last few decades. For example, on October8, 2005 an earthquake of 7.6 (Mw) struck the Kashmir of Pakistan,where the main damages of RC frames were the beam-column fail-ure and the story failure (Fig. 1) [1]. Another earthquake of 6.2(Mw) struck the Abbruzzo region of Italy on April 6, 2009. Seismicdamage investigation showed that columns seem to have failed incompression before the yielding of beams (Fig. 2) [2]. WenchuanChina suffered a magnitude 8.0 earthquake on 12 May 2008. Themain failure of RC frames was caused by ‘‘strong beam-weakcolumn” (Fig. 3) [3,4]. Marmara earthquake of August 17, 1999and Van earthquake on October 23, 2011 in Turkey showed thatthe slab affection was one of the reason why RC frames were dam-aged (Fig. 4) [5]. The similar failure modes were observed in the

magnitude 6.6 earthquake in Bam on December 26, 2003 in Iran(Fig. 5) [6]; and in the magnitude of 7.6 earthquake in Chi-Chi Tai-wan in 1999. According to reference [7], the ‘‘strong column-weakbeam” concept was not implemented in the design of those schoolbuildings. Thus, plastic hinges appeared in columns earlier than inbeams (Fig. 6) [7]. The failure of those RC frames was against theductility design concept of the existing Code [8–11]. Can the con-cept of ‘‘strong column-weak beam” be realized according to theexisting Codes?

Based on the existing design codes, it is required that the totalcolumn moment strength be larger than the total beam momentstrength at a joint in order to guarantee that beams yield beforecolumns do in strong earthquakes, as shown below:

(1) ACI 318-14 [8]

XMnc P

65

XMnb ð1Þ

whereP

Mnc = sum of nominal flexural strengths of columns fram-ing into the joint, evaluated at the faces of the joint;

PMnb = sum of

nominal flexural strengths of the beams framing into the joint, eval-uated at the faces of the joint. Where the slab is in tension under

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Fig. 1. Story failure of RC frame (Kashmir).

Fig. 2. Collapse of hotel (Italy).

Fig. 3. Column damage (Wenchuan).

Fig. 4. Strong beam weak column failure (Turkey).

Fig. 5. The plastic hinge in a weak column (Iran).

Fig. 6. Collapse of school (Taiwan).

344 N. Ning et al. / Engineering Structures 126 (2016) 343–352

moments at the face of the joint, slab reinforcement within an effec-tive slab width as defined in article 8.12 and 8.13 shall be assumedto contribute to

PMnb if the slab reinforcement is developed at the

critical section for flexure.

(2) EC 8 [9]The following condition should be satisfied at all joints of pri-

mary or secondary seismic beams with primary seismic columns:XMRc P 1:3

XMRb ð2Þ

N. Ning et al. / Engineering Structures 126 (2016) 343–352 345

whereP

Mnc = sum of the design values of moments of resistance ofthe columns framing into the joint;

PMRb = sum of the design val-

ues of moments of resistance of the beams framing into the joint.When partial strength connections are used, the moments of resis-tance of these connections are taken into account in the calculationof

PMRb.

(3) Chinese code [10]

XMc P 1:2

XMbua ðwhen the seismic intensity is 9 while frames of class IÞ ð3Þ

XMc ¼ gc

XMb ðother situationsÞ ð4Þ

whereP

Mc = sum of the design values of moments of resistance ofthe columns framing the joint;

PMbua = sum of flexural strengths of

the beams framing into the joint;P

Mb = sum of the design valuesof moments of resistance of the beams framing the joint; The valuesof gc are 1.7 for class I, 1.5 for class II, 1.3 for class III, and 1.2 forclass IV, respectively.

(4) TEC-2007 [11]The TEC-2007 Code requires that the sum of ultimate moment

resistance of columns framing into a beam-column joint shall beat least 20% more than the sum of ultimate moment resistance ofbeams framing into the same joint.

Mra þMr€u P65ðMri þMrjÞ ð5Þ

Based on Eqs. (1), (5), the sum of ultimate moment resistance ofcolumnsat a joint shall be at least 20%more than the sumof ultimatemoment resistance of beams. However, only ACI 318-14 [8] consid-ers the slab effect in ‘‘strong column weak column” design by usingeffective slab width in the calculation of beam flexural strengths,while other Codes underestimate the slab effect. ACI Code 8.12and 8.13 on the T beam flange width are determined on the basisof the shear lag and tomake the stress of effective slab flange equiv-alent to 0.85fc. However, when the slab is in tension, the slab cracksand slab rebars play the leading role. The effective slab width in ACICode may not be useful in the seismic design. The existing researchindicated that the cast in-situ slab can enhance the moment resis-tance of beams, and thus the effective slab width should be consid-eredwhen calculating themoment resistance [12–17]. Although theslab effect on RC frames has been improved greatly, the existingunderstanding is still insufficient. To the best knowledge of theauthors, there are very few studies about the slab effect on the con-tinuous spatial RC frames. Those results cited above are not unifiedand difficult in application. If the effectivewidth is under estimated,the amplified factor in column (6/5 forACI)will be still not enough tomake ‘‘strong column” form. The required ratios of column-to-beamstrength also need further research attention.

The presented paper here first discusses the experiment of RCframes with cast in-situ slabs with the objective of investigatingthe participation of slabs during strong earthquakes. Then theFEA models considering different influence factors are analyzed.And a strut-and-tie model of slabs is proposed to calculate theeffective slab width. The required ratio of column-to-beamstrength is also proposed.

2. Experimental investigation

2.1. Model description

The experimental work consists of two 1:2.5 scaled spatial RCframes: a control specimen (RC-1) and a RC frame with cast in-

situ slabs (RC-2). Both frames are made of beams with a cross sec-tion of 100 mm � 200 mm and columns with a cross section of160 mm � 160 mm. The thickness of RC-2 slabs is 50 mm. Theframes were designed according to the Chinese concrete structurecode (GB50010-2008). Fig. 7 shows the geometry of the frameswith two bays in the longitudinal (Y) direction and one bay inthe transversal (X) direction. The span of the frame was selected

to consider the length to width ratio of slabs. The material proper-ties were introduced in Ref. [18].

2.2. Instrumentation and loading program

The overall view of the test setup is shown in Fig. 8. The top oftwo frames was connected to multi-function electro-hydraulicservo structure tester, which transferred static lateral cyclic loadsto the specimens. Load cells mounted in the actuators were usedto record the lateral loads in the loading system. Linear VariableDifferential Transducers (LVDTs) were mounted on first and secondfloor of the frames to monitor the lateral displacements of speci-mens and the rotations of beams and columns. Additionally, oneLVDT was used to monitor the displacement perpendicular to thedirection of lateral loading. The strains of the longitudinal barswere measured by strain gauges attached to the steel bars. Loadingprogram was introduced in Ref. [18]. The displacement routine isshown in Fig. 9. Limited by the experimental conditions, the axialload was not applied to specimens.

2.3. Test results

The plastic hinges of the two frames are shown in Fig. 10. Thefailure pattern of RC-1 is the typical ‘‘strong-column-weak-beam”.For the RC-2, the failure pattern is the ‘‘strong-beam-weak-column” type which is different from the design objective of ‘‘strong-column-weak-beam”. That can be attributed to the participation ofcast in-situ slabs which causes the performance of the strongerbeam and weaker column.

Fig. 11 shows the strain variation of the bottom and top longi-tudinal reinforcement in the first floor slab. The strain is largerwhen the slab rebars are closer to the column. So the slab affectioncan be considered as ‘‘slab effective width”. The value of strainincreases with the increase of the lateral displacement. Especially,it increases when the story drift varies from 0.4% to 2% and theincrease rate slows down after 2% story drift. It is illustrated thatthe slab affection can be considered at 2% story drift in design. Itcan also be found that the strains of the top reinforcement are lar-ger than those of the bottom reinforcement. The constructionalslab reinforcement on the top layer should be taken intoconsideration.

3. FEM analysis

Three dimensional spatial frame models have been developedusing ABAQUS. Damaged plasticity model is used to simulate crackof the concrete. Constitutive equations of concrete and steel referto Ref. [19]. Concrete and reinforcement element types areC3D8R and T3D2, respectively. Reinforcement is embedded in con-crete in the model. The main element size is 50 mm. The loadingprogram is the same as the test one. Full Newton solution tech-nique is used in analysis. The comparison of the skeleton curves

3280

400

1440

1440

4760380380 4000

30016002400300 8080

1 1

22

(a) Specimen details 6@

150

6@150

6@150 6@150

6@150

6@15

06@

150

6@15

06@

150

6@15

0

24004000

1600

1600

400

400

40040

040

0

400

400400

Y

X

west east

exterior joint interior joint exterior joint

(b) Details of slabs

2 8

6@70(150)

2-21-1

160 100

200

160

2 10

2 82 10

6@70(150)

(c) Cross sections of column and beam

Fig. 7. Design of specimens.


is shown in Fig. 12. The comparison of cracks from experimentswith ‘‘simulation of damages” by FEA is shown in Figs. 13 and14. The analysis results show that the FEA data fit the experimentalresults well and can be used to conduct parametric analyses.

The axial compression ratio, concrete strength, reinforcementratio of slabs, thickness of the slabs and the stiffness of the trans-verse beams are considered as variables in the FEA, as shown inTable 1. The actual axial load is calculated by scaled model inFEA models. The axial load ratios of RC-3 to RC-9 varies accordingto the proportions of each column in order to analyze the effectiveslab width and the failure patterns of RC frames while the axialratio changes progressively. The stresses of the slab longitudinalreinforcement, beam longitudinal reinforcement and columnlongitudinal reinforcement are obtained from FEA results. The

participation of slabs and the required ratio of column-to-beamstrength can be calculated by FEA results.

4. Effective width of cast in-situ slabs

4.1. Strut and tie model

The strut and tie model for slabs is shown in Fig. 15 according tothe experimental observations as well as FEM results. Take the lon-gitudinal slab reinforcement and the transverse beam reinforce-ment as tie. Take the slab concrete between cracks as strut. Thestress of transverse slab reinforcement can be ignored because itsvalue observed in test and FEM results is very small.

(a) RC-1 (b) RC-2

Fig. 8. The loading setup.

story drift

cycle0.1%0.2% 0.3% 0.4% 0.6%

3.5%4.2%

Fig. 9. The displacement routine.

(a) RC-1 (b) RC-2

Fig. 10. The location of plastic hinges.


Taking the static moments of the forces about point A:

f y�trAs�trðlp þ btr � asÞ þ f y�trAs�trðlp þ asÞ ¼XAsbif ybi

sbxx2

ð6Þ

where x is the effective slab flange width and can be calculated byEq. (6), the expression of x is shown in Eq. (7).

x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisbf y�trAs�trð2lp þ btrÞ

Xf ybiAsbi

.rð7Þ

According to the experimental results, the effective slab widthwas influenced by the length to width ratio. The span of main beamto transverse beam p is introduced to reflect the span of the mainbeam. According to the test results of Refs. [13,14,16,17], x isadjusted by fitting analysis introducing the coefficient pi.

x ¼ pi

1:5

� �0:25ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisbf y�trAs�trð2lp þ btrÞ

Xf ybiAsbi

.rð8Þ

The effective slab width can be calculated by

bef ¼ bþ x ¼ bþ pi

1:5


Xf ybiAsbi

.rð9Þ

where sb = spacing of slab longitudinal reinforcement, f y�tr=yieldstrength of longitudinal reinforcement of transverse beam,As�tr = area of longitudinal reinforcement of transverse beam,hb = height of the main beam, btr = width of the transverse beam,f ybi = yield stress of the slab reinforcement, Asbi = area of the ithlayer(top or bottom) slab reinforcement, pi = span of main beamand transverse beam the ith span slab, b = width of the main beam,as = distance between center of rebar and outer edge of transversebeam, and lp = length of the plastic hinge.

The influencing factors such as the height of transverse beam,longitudinal reinforcement of transverse beam, area and spacingof the slab reinforcement are considered in Eq. (9). The experimen-tal data and calculation results of RC-2 to RC-22 at 2% story driftare shown in Table 2. RC-9 is destroyed at 2% story drift due tothe high axial compression ratio, and thus the data is not available.Based on the existing study [12] and our earlier study [18], theexperimental and FEM values were calculated by Eq. (10),

bef ¼Pn

i¼1rbisrbmax

þ b ð10Þ

(a) West exterior joint (b) East exterior joint

(c) West interior joint (d) East interior joint

Fig. 11. Strain of longitudinal slab reinforcement (bottom).

(a) RC-1 (b) RC-2

Fig. 12. The comparison of skeleton curve of frames.


where rbmax is the maximum stress of slab reinforcement, rbi is theith slab rebar’s stress, b is the width of beam, s is the spacing of theslab reinforcement.

The average absolute error is 10.01%, and the coefficient of vari-ation is 0.57. It can be concluded from Table 2 that the calculationresults agree with the experiment results. However, the ones influ-enced by axial compression ratio and the height of the transversebeam differ considerably. The results indicate that the strut andtie model can be used to calculate the effective slab width. Coeffi-cient m and k are introduced to consider the influence of axialcompression ratio and the height of the transverse beam, respec-tively. Thus, Eq. (9) is modified as follows:

bef ¼ mk bþ ð pi

1:5Þ0:25

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisbf y�trAs�trð2lp þ btrÞ

Xf ybiAsbi

.r� �ð11Þ

where m and k were derived by fitting calculations asm ¼ �0:1799n2 þ 0:3891nþ 0:9847, k ¼ 0:618þ 0:0019htr .

Table 2 shows the calculation result using Eq. (11). The data ofRC-3 � RC-8, RC-19 and RC-22 are shown in brackets. Table 2shows that the adjusted results are on the safe side, with an aver-age absolute error of 6.61%, and the coefficient of variation of 0.74.

For interior joints, the calculation results at 2% story drift usingEq.(9) are shown in Table 3.

The average absolute error is 11.1% and the coefficient of varia-tion is 0.82 by Eq. (9). In order to take axial compression ratio andheight of transverse beam into consideration, m0 and k0 are intro-duced to modify Eq. (9) as follows:

bef ¼ m0k0 bþ pi

1:5


Xf ybiAsbi

.r� �ð12Þ

(a) test result (crack) (b) FEM result (damage)

Fig. 13. The torsion crack and damage of transverse beam.

(a) test result (crack) (b) FEM result (damage)

Fig. 14. The crack and damage of slab.

Table 1Variables considered in FEA (The variables in different models are in bold type).

Models Parameters

Axial compression ratio Concrete strength/MPa

Reinforcement ratio of slabs Thickness of slabs/mm

Dimension of transversebeam/mm

Eastcolumn

Middlecolumn

Westcolumn

Reinforcement Reinforcementratio (%)

RC-3 0.10 0.18 0.13 30 U6@150 0.377 50 100 � 200RC-4 0.20 0.36 0.26 30 U6@150 0.377 50 100 � 200RC-5 0.25 0.45 0.33 30 U6@150 0.377 50 100 � 200RC-6 0.30 0.54 0.39 30 U6@150 0.377 50 100 � 200RC-7 0.35 0.63 0.46 30 U6@150 0.377 50 100 � 200RC-8 0.40 0.72 0.52 30 U6@150 0.377 50 100 � 200RC-9 0.50 0.90 0.65 30 U6@150 0.377 50 100 � 200RC-10 0.10 0.18 0.13 20 U6@150 0.377 50 100 � 200RC-11 0.10 0.18 0.13 25 U6@150 0.377 50 100 � 200RC-12 0.10 0.18 0.13 30 U6@200 0.283 50 100 � 200RC-13 0.10 0.18 0.13 30 U6@100 0.566 50 100 � 200RC-14 0.10 0.18 0.13 30 U8@150 0.670 50 100 � 200RC-15 0.10 0.18 0.13 30 U8@100 1.005 50 100 � 200RC-16 0.10 0.18 0.13 30 U6@150 0.377 30 100 � 200RC-17 0.10 0.18 0.13 30 U6@150 0.377 40 100 � 200RC-18 0.10 0.18 0.13 30 U6@150 0.377 60 100 � 200RC-19 0.10 0.18 0.13 30 U6@150 0.377 50 100 � 100RC-20 0.10 0.18 0.13 30 U6@150 0.377 50 100 � 150RC-21 0.10 0.18 0.13 30 U6@150 0.377 50 100 � 250RC-22 0.10 0.18 0.13 30 U6@150 0.377 50 100 � 300


A

f y-trAs-tr lp

f yAs

sbsb

sbsb

x

xf ybAsb/sb

x/2

Fig. 15. The strut and tie model.

Table 3The comparison of bef in interior joints (2% story drift).

Modelnumber

Experimentalvalue

Calculatedvalue

Absolute error% (using Eq.(12))

RC-2 446 428 4.3RC-3 440 428(448) 2.7(1.8)RC-4 458 428(466) 6.6(1.8)RC-5 472 428(471) 9.3(0.2)RC-6 492 428(473) 13.0(3.9)RC-7 463 428(471) 7.6(1.7)RC-8 447 464(506) 3.8(13.1)RC-10 372 428 15.1RC-11 387 464 19.8RC-12 440 478 8.6RC-13 294 367 24.8RC-14 320 341 6.6RC-15 391 297 24.0RC-16 402 428 6.5RC-17 429 428 0.2RC-18 439 428 2.4RC-19 322 428(330) 32.9(2.4)RC-20 342 428(355) 25.1(3.9)RC-21 391 428(407) 9.4(4.0)RC-22 427 428(432) 0.2(1.3)


where m0 ¼ �0:8542n02 þ 0:665n0 þ 0:9747 and k0 ¼ 0:6505þ0:0012htr based on fitting calculations.

The calculation results using Eq. (12) can be found in Table.3.The average absolute error is 7.32% and the coefficient of variationis 1.04. The average error and the coefficient of variation decreasesafter adjustment. Eqs. (11) and (12) can be used in the calculationof the effective slab width.

4.2. Proposed effective slab width equations

A total of 100 models of exterior joints and 100 models of inte-rior joints based on the experimental model were developed in

Table 2The comparison of bef of exterior joints (2% story drift).

Model number Joint type Experimental and FEM v

RC-2(test) West exterior joint 405East exterior joint 359

RC-3 West exterior joint 503East exterior joint 407



















order to derive simple equations. The influencing factors and thevariation range are shown in Table 4. The effective slab width isexpressed by bþ Khb, where b is the main beam width, hb is themain beam height. The curves of K and the guarantee rate of

alue Calculated value Absolute error% (using Eq. (11))

464 14.6428 18.9464(479) 7.8(4.8)428(437) 5.2(7.5)464(498) 8.3(1.5)428(451) 2.6(8.3)464(507) 10.9(2.6)428(458) 5.9(0.7)464(515) 14.4(5.1)428(464) 14.1(6.7)464(522) 12.3(1.3)428(470) 7.4(1.8)464(528) 11.3(1.0)428(476) 2.1(8.9)464 14.6428 7.5464 1.1428 5.9520 1.9478 2.8397 2.0367 13.3368 16.7341 4.2319 12.8297 12.1464 3.3428 7.4464 10.4428 12.1464 7.0428 12.1464(375) 18.9(3.9)428(346) 23.7(0)464(419) 14.3(3.2)428(386) 18.2(6.8)464(507) 4.5(4.4)428(467) 3.3(5.8)464(551) 17.3(1.7)428(508) 17.2(1.7)

Table 4Influencing factors considered for interior and exterior joints.

Influencing factors Variation range No. of calculation models

Axial compression ratio 0.1–0.85 30Height of transverse beam 150–550 mm 10Diameter of slab reinforcement 6–12 mm 4Spacing of slab longitudinal reinforcement 50–250 mm 17Diameter of the transverse beam longitudinal reinforcement 8–25 mm 10Width of the transverse beam 100–300 mm 8Height of the main beam 100–550 mm 10Length to width ratio of slab 1–1.5 11

(a) Exterior joint (b) Interior joint

Fig. 16. The value K and guarantee rate.

Table 5Ratios of column-to-beam strength at the 2% story drift.

Models East exterior joint Interior joint West exterior jointP

Mc/kNmP

Mb/kNm g (P

Mc=P

Mb)P

Mc/kNmP

Mb/kNm g (P

Mc=P

Mb)P

Mc/kNmP

Mb/kNm g (P

Mc=P

Mb)

RC-2 8.74 13.09 0.67 13.34 14.27 0.91 9.47 13.32 0.71RC-3 11.22 15.55 0.72 11.03 16.43 0.67 8.74 15.48 0.56RC-4 13.25 15.00 0.88 34.30 15.78 2.17 23.90 12.17 1.96RC-5 20.95 15.20 1.38 43.40 16.31 2.66 30.86 12.95 2.38RC-6 32.93 15.92 2.07 56.49 17.56 3.22 43.81 14.93 2.93RC-7 34.85 16.28 2.14 59.04 18.00 3.28 48.70 15.62 3.12RC-10 9.39 13.62 0.67 13.23 15.93 0.83 7.56 13.67 0.55RC-11 9.66 14.74 0.66 13.31 15.54 0.86 10.69 13.74 0.78RC-12 8.56 13.99 0.61 13.31 16.30 0.82 10.31 14.82 0.70RC-13 9.64 17.15 0.56 13.31 14.22 0.94 11.15 17.85 0.60RC-14 10.68 17.76 0.60 13.31 15.56 0.86 9.80 16.51 0.59RC-15 10.73 19.60 0.55 13.31 17.89 0.74 10.03 19.94 0.50RC-16 10.83 15.79 0.69 13.31 16.66 0.80 10.35 15.10 0.69RC-17 11.02 16.33 0.67 13.31 16.18 0.82 8.45 15.59 0.54RC-18 12.53 16.04 0.78 13.31 17.33 0.77 10.54 14.86 0.71RC-19 7.29 13.54 0.54 7.29 11.22 0.65 7.29 12.83 0.57RC-20 7.51 13.99 0.54 9.26 12.98 0.71 7.91 13.99 0.57RC-21 8.12 16.35 0.49 11.65 14.98 0.78 8.64 15.02 0.58RC-22 8.86 17.31 0.51 13.19 17.28 0.76 9.03 16.33 0.59


exterior and interior joints are shown in Fig. 16. According to a 95%guarantee rate, the effective slab width of the exterior joint andinterior joint are bþ 3:2hb and bþ 2:7hb, respectively. In the caseof the main beam with slabs on both sides, the effective slab widthof the exterior joint and interior joint can be bþ 6:4hb andbþ 5:4hb, respectively.

5. The required ratio of column-to-beam strength

The required ratio of column-to-beam strength can be definedas Eq. (13)

g ¼P

McPMb

ð13Þ

In which,P

Mc = sum of moment of columns calculated at the 2%story drift;

PMb = sum of moment of beams calculated at the 2%

story drift. Where the slab is in tension under moments at the faceof the joint, slab reinforcement within an effective slab width can becalculated with the proposed equations above.

Table 5 shows the calculated results of the required ratio ofcolumn-to-beam strength at the 2% story drift based on FEA anal-yses. The damage to columns in RC-8 and RC-9 was very severe at2% story drift due to the high axial compression ratio. Thus, thedata are not available and cannot be used to calculate the moment.So RC-8 and RC-9 are not included in Table 5. It can be concludedthat the required ratio of column-to-beam strength g, is increasedwhen the axial compression ratio is increased. In fact, this ratio


exceeds 2.0 when the axial compression ratio is larger than 0.3.When the axial compression ratio reaches 0.39, g is about 3.0.

In order to ensure a ‘‘strong column weak beam” failure mode,Eq. (14) should be used.X

Mc P gX

Mb ð14Þwhere g = 1.5, 2.5, 3.0 when the axial compression ratio of columnis smaller than 0.25, within the range of 0.25–0.4, and larger than0.4, respectively.

6. Conclusions

Based on the experimental investigation and FEM analyses, thefollowing conclusions can be drawn:

(1) Experimental results demonstrate that slabs can change thefailure pattern of RC frames from a typical ‘‘strong columnweak beam” failure to the ‘‘strong beam weak column”failure.

(2) According to a 95% guarantee rate, the effective slab width ofthe exterior joint and the interior joint are ‘‘bþ 3:2hb” and‘‘bþ 2:7hb”, respectively. In the case of the main beam withslabs at both side, the effective slab width of the exteriorjoint and the interior joint can be ‘‘bþ 6:4hb” and‘‘bþ 5:4hb”, respectively.

(3) The axial compression ratio is the most sensitive factor influ-encing the required ratio of column-to-beam strength. Thisratio increases with the increase of the axial compressionratio. When the axial compression ratio reaches 0.39, thisratio reaches 3.0. The reasonable ratio of column-to-beamstrength is proposed to avoid the brittle failure of the RCframes in seismic design.

(4) The concrete strength, reinforcement ratio of slabs, thick-ness of the slabs and the stiffness of transverse beams areless sensitive influencing the required ratio of column-to-beam strength.

Acknowledgments

The authors wish to acknowledge the financial support by theChina National Natural Science Foundation for Youth (Grant No.

51508289) and the Taishan Scholar Priority Discipline TalentGroup program funded by the Shandong Province. Visit to Univer-sity of Tennessee Knoxville by the first author and visit to TongjiUniversity, China by the third author were made possible byDepartment of Civil and Environmental of University of TennesseeKnoxville, USA.

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