FLEXURAL ANALYSIS AND DESIGN METHODS FOR SRC BEAM SECTIONS WITH COMPLETE COMPOSITE ACTION

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1 Journal of the Chinese Institute of Engineers, Vol. 31, No., pp (8) 15 FLEXURAL ANALYSIS AND DESIGN METHODS FOR SRC BEAM SECTIONS WITH COMPLETE COMPOSITE ACTION Cheng-Cheng Chen* and Chao-Lin Cheng ABSTRACT Section analysis, that complies with complete composite and strain compatibility conditions of 81 SRC beam sections, was carried out to investigate the flexural behavior of SRC beam sections. As well, a correlation between the curvature ductility ratio and the depth of the neutral axis was established with fairly good agreement. A section analysis procedure, that fulfills all composite conditions with plastic stress distribution, is introduced for the calculation of moment capacity and the curvature ductility ratio of SRC beam sections. In addition, a design method, based on complete composite and plastic stress distribution, is proposed for the design of SRC beam sections to meet both the moment and curvature ductility requirements. The analysis and design methods proposed here are able to provide tools for more economical and rational SRC beam design, and are superior to the strength superposition design method. Key Words: SRC construction, composite construction, flexural design, flexural analysis. I. INTRODUCTION A steel reinforced concrete (SRC) beam section is composed of concrete, steel shape, longitudinal steel bars, and transverse steel bars. Fig. 1(a) shows a typical SRC beam section. This type of construction was originally developed in Japan and recently has been used more often in Taiwan. The Japan and Taiwan use the concept of superposition with plastic stress distribution (Superposition-PSD), as shown in Fig. 1(b), to evaluate the moment capacity of SRC beam sections. Japanese specification (AIJ, 1) uses the working stress method while incorporating the concept of superposition to design SRC beam sections for bending. The Taiwanese specification (CPA, 4) uses the concept of Superposition-PSD to design SRC beam sections for bending. The AISC Specification (AISC, 5) adopts the *Corresponding author. (Tel: ; Fax: ; c3@mail.ntust.edu.tw) C. C. Chen is with the Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei 16, Taiwan, R.O.C. C. L. Cheng is with the Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei 16, Taiwan, R.O.C. & Department of Construction Technology, Tungnan University, Taipei, Taiwan, R.O.C. complete composite concept when shear connectors are provided for concrete-encased steel beams, in which longitudinal and transverse steel bars are not mandatory. The stress distribution of the section can be either complete composite with plastic stress distribution (Composite-PSD) as shown in Fig. 1(c), or complete composite with the stress distribution, fulfilling the strain compatibility condition (Composite- SC) as shown in Fig. 1(d). However, in an SRC beam section, the steel shape is enclosed by the concrete that is well confined by both the longitudinal and transverse steel bars. Under this condition, the use of shear connectors is not necessarily restricted. A total of 3 test specimens and their results from six studies (Chen, 199; Ju, 1993; Tsai et al., 1995; Chen, 1996; Weng et al., 1; Chen & Chen, 1) were collected. The details of the specimens are shown in Table 1 and Fig.. Specimens tested by Chen (199), Ju (1993), Tsai et al. (1995), and Chen (1996) were beam-column sub-assemblage types loaded cyclically with a concentrated load applied at the beam ends. Specimens tested by Weng et al. (1) were simply supported beams loaded monotonically with a concentrated load applied at the midspan. Specimens tested by Chen & Chen (1) were simply supported beams loaded monotonically with 4-point load, as shown in Fig. 3(a). The bending

2 16 Journal of the Chinese Institute of Engineers, Vol. 31, No. (8) b h d d b f t w t f d s Steel shape N.A. (steel shape).85f c.85f c.85f c f y f s F y N.A. Fy F f y (RC) s N.A. c PSD N.A. c SC +F y +F y +F y +f s +f y +f y Trans. steel bar Steel shape RC Steel shape RC Steel shape RC Comp. steel bar Ten. steel bar (a) SRC beam section (b) Superposition-PSD (c) Composite-PSD (d) Composite-SC Fig. 1 Stress distribution of SRC beam sections strength of each specimen M t is listed in Table. For specimens loaded with cyclic load, each specimen provides two bending strengths: one with top flange under compression, and another with bottom flange under compression. Among the 3 specimens, the S1 and S1-SS from Chen & Chen (1) are to be discussed first. Then, the bending strengths of all the collected specimens will be predicted and discussed. Figure 3(a) shows the test setup for Specimens S1 and S1-SS. Fig. 3(b) shows the section in the constant moment region of Specimen S1 and S1-SS. S1- SS had shear studs placed at the top flange of the section while S1 did not. Fig. 4 shows the momentcurvature curves measured in the constant moment region of the two specimens. Due to the existence of shear studs, stress concentration was induced, surrounding the shear studs, which resulted in an earlier crush of the concrete. Therefore, the bending strength of S1-SS (655 kn-m) is somewhat lower than that of S1 (691 kn-m). However, S1 showed a more significant sudden load drop than S1-SS did. That was because the concrete in the constant moment region of S1 crushed simultaneously, while the concrete of S1- SS crushed gradually due to stress concentration. Specimen S1-SS showed somewhat better ductility behavior than S1. However, the strength and ductility differences between the two specimens were very limited. Fig. 5 shows the measured strain distribution of the two specimens. The strain readings were obtained from strain gages placed on steel shapes and steel bars. Fig. 5 reveals that both Specimen S1 and Specimen S1-SS remained fully composite up to the point where moment capacity was developed. The bending strength of the 3 test specimens were predicted by Composite-SC, Superposition-PSD and Composite-PSD stress distributions, and, the corresponding bending strengths were designated respectively as (M a ) SC, (M a ) SUP and (M a ) PSD. The predicted bending strengths and the ratio of M t over the predicted bending strengths are listed in Table. In Table, the italic numbers are from the specimens that had shear studs placed on the compression flange of the H shape. The others are from the specimens without shear studs on the compression flange of the H shape. Table 3 lists the statistical results from Table. It can be found that (M a ) SC and (M a ) PSD are consistent with the test results for sections with or without shear studs placed on the compression flange. On the other hand, (M a ) SUP underestimated bending strength of the specimens by 5% on average with a coefficient of variation (COV) of 16.1% which is much higher than that of (M a ) SC and (M a ) PSD. The experimental results revealed that the bending strength of a wide range of SRC beam sections, with or without shear studs placed on the compression flange of H shape, can be predicted with (M a ) SC and (M a ) PSD. Composite-PSD and Composite-SC stress distributions are more accurate, more rational, and more economical than Superposition-PSD stress distribution. In addition, the definition of curvature and curvature ductility in Superposition-PSD is ambiguous since there are two neutral axes, one for steel shape and another for RC part, in the same section (Fig. 1(b)). Consequently, the control of curvature ductility of the designed section is unable to be accurate. On the other hand, the definition of curvature according to Composite-PSD and Composite-SC is very clear, and the control of curvature ductility of the designed section can be carried out rationally. Despite the drawbacks of Superposition-PSD, it is quite easy to apply and is used extensively in design. On the other hand, currently, the use of Composite-PSD or Composite-SC is much more tedious and is not

3 C. C. Chen and C. L. Cheng: Flexural Analysis and Design Methods for Composite SRC Beam Sections 17 Table 1 Details of collected test specimens Authors Loading Specimen Beam Concrete Steel bar Steel shape Shear connector type designation section Comp. Arrangement Yield stress Section Yield stress strength Top Bottom d d f y F y (mm mm) f c (MPa) bar bar (mm) (mm) (MPa) (mm) (MPa) No #1 (1) 66 H H : 363 Top flange No #1 -# H H : 363 Top flange Chen Cyclic No #1 (1) 66 #1 : 611 H H : 363 Top flange (199) load No #1 (1) 66 #8 : 788 H H : 363 Top flange No #1 -# H H : 363 Top flange No #1 -# H H : 363 Top flange No #8 -#8 6 6 H H : 357 Top flange H H : 357 Ju Cyclic No #8 (1) 6 #8 : (PL ) Cover plate : (1993) load 374 Top flange No #8 (1) 6 H H : (PL ) Cover plate : 374 Top flange SRC #1 -#5 7 6 H H : 353 Top flange #1 : 478 Tsai et al. Cyclic SRC #7 -#5 7 6 H H : 353 Top flange #7 : 313 (1995) load SRC #1 -#5 7 6 H H : 37 Top flange #5 : 41 SRC (1) -#5 6 H H : 37 Top flange Top & bottom SP #8 -#8 7 7 H H : 41 Chen Cyclic #8 : 314 flanges (1996) load SP #8 -#8 7 7 #1 : 5 H H : 41 None SP #1 -#1 7 7 H H : 41 None Monotonic/ Weng B-SRC-S #6 -# H H : 367 Top flange Central et al. B4-SRC-N #6 -# #6 : 598 H H : 367 None concentrated (1) B5-SRC-N #6 -# H H : 367 None load S #8 -#1+-#8 7 7 H H : 33 None S1-SS #8 -#1+-#8 7 7 H H : 33 Top flange S1-T #8 -#1+-#8 7 7 H H : 67 None S1-T #8 -#1+-#8 7 7 H H : 67 None S #7 -#8+-#8 7 7 H H : 33 None S #5 3-#7 7 7 #11 : 468 H H : 33 None S #8 -#8+-#7 7 7 #1 : 435 H H : 33 None Chen Monotonic/ #8 : 46 + (PL16 15) Cover plate : 364 & Chen #7 : 44 H H : 33 4 point load S #5 -#8 7 7 None (1) #6 : 46 + (PL 1) Cover plate : 368 #5 : 411 H H : 33 S #5 -#1+-#1 7 7 None #3 : (PL 1) Cover plate : 368 S #7 -#1 7 7 H H : 7 None D #6 -#1 6 7 H H : 33 None D1-T #6 -#1 6 7 H H : 33 None D #3 -# H H : 33 + (PL 9) Cover plate : 368 None Note : (1) Steel bars were not anchored into the column, so, they are neglected.

4 18 Journal of the Chinese Institute of Engineers, Vol. 31, No. (8) d H shape (at the center of the section) d 1 Trans. steel bar Top steel bar Bottom steel bar Fig. Steel bar arrangement of the collected test specimens P/ P/ A B C D LVDT Strong Floor Unit: mm (a) Arrangement of test specimens 55 -#8 -#8 -#1 35 (b) Cross section S1-SS only ASTM-A18 7 #3@ stirrup. H Fig. 3 Test set-up and cross-sections for S1 and S1-SS (Chen & Chen, 1) favored by engineers. In this paper, a design method which complies with the assumption of Composite-PSD and is fairly easy to apply is established for the design of moment capacity and curvature ductility of SRC beam sections. This designed method is designated as the S&D method, where S stands for strength and D for ductility. The curvature ductility here is related to the curvature of the section at the start of crushing the covering concrete. The S&D Method can be used to design an SRC beam section for a required moment capacity and a required curvature ductility ratio when a complete composite assumption is considered appropriate. The required curvature ductility ratio can be selected based on limiting the flexural failure mode to tension failure, or by controlling the crushing of the covering concrete in an earthquake-resistant structural design. It is noted that a strength reduction factor φ b of.85 is specified by the AISC Specification when Composite-PSD and Composite-SC stress distributions are used. Since Composite-PSD is used in developing the analysis and design methods of SRC beam sections, a φ b of.85 is recommended. II. CURVATURE DUCTILITY OF SRC BEAM SECTIONS 1. Model Sections A total of 81 SRC beam sections were arranged and analyzed to investigate the flexural behavior and curvature ductility characteristics of SRC beam sections. All of the sections were rectangular sections, as shown schematically in Fig. 1(a), with a width b equal to 4 mm and a depth h equal to 7 mm. The centroid of the steel shape coincides with the centroid of the section. The most commonly used materials in Taiwan were selected in this study. CNS SD4W (CNS, 5) with a yield strength f y = 414 MPa (4 kgf/cm ) was selected for the steel bars. It is noted that the mechanical properties of CNS SD4W are similar to those of ASTM A76 (ASTM, 199) steel bars. ASTM A57 (ASTM, 1) with a yield strength F y = 345 MPa (35 kgf/cm ) was selected for the steel shapes. Normal weight concrete with compressive strength f c of 7.6 MPa (8 kgf/ cm ), 34.5 MPa (35 kgf/cm ), and 41.4 MPa (4 kgf/cm ) were selected. An equivalent tension steel

5 C. C. Chen and C. L. Cheng: Flexural Analysis and Design Methods for Composite SRC Beam Sections 19 Table Bending strength predicted by Composite-SC, Superposition-PSD and Composite-PSD Authors Specimen Test Composite-SC Superposition-PSD Composite-PSD designation strength M M t (M a ) t SC (M a ) M t M SUP (M a ) t PSD (kn-m) (kn-m) (M a ) SC (kn-m) (M a ) SUP (kn-m) (M a ) PSD No No No Chen (199) No No No No Ju (1993) No No SRC SRC Tsai et al (1995) SRC SRC SP Chen (1996) SP SP B-SRC-S Weng et al. B4-SRC-N (1) B5-SRC-N S S1-SS S1-T S1-T S S Chen & Chen S (1) S S S D D1-T D

6 Journal of the Chinese Institute of Engineers, Vol. 31, No. (8) Table 3 Statistical results for predicted bending strengths Section groups M t /(M a ) SC M t /(M a ) SUP M t /(M a ) PSD Average COV Average COV Average COV All sections Sections with shear studs (italic numbers in Table ) Sections without shear stud Moment (kn-m) 4 S1 S1-SS Curvature (1/m) FIg. 4 Moment-curvature curve of test specimens (Chen & Chen, 1) 55 Strains on rebars Strains on steel shape P =.43 P max P =.65 P max P =.87 P max P = P max Height (mm) Strain ( 1 6 ) Strain ( 1 6 ) Strain ( 1 6 ) Strain ( 1 6 ) (a) S1 specimen 55 P =.46 P max P =.69 P max P =.9 P max P = P max Height (mm) Strain gauge (c) Arrangement of strain gages Strain ( 1 6 ) Strain ( 1 6 ) Strain ( 1 6 ) Strain ( 1 6 ) (b) S1-SS specimen Fig. 5 Strain distribution along beam height (Chen & Chen, 1) bar ratio ρ t and an equivalent compression steel bar ratio ρ c, defined respectively by Eqs. (1) and (), were used for the convenience of section arrangement: ρ c = ρ r + A f bd F y f y d s h () ρ t = ρ r + [A f( d s h ) +.5A w] bd F y f y (1) ρ r, ρ r, A f and A w defined respectively by Eqs. (3) to (6).

7 C. C. Chen and C. L. Cheng: Flexural Analysis and Design Methods for Composite SRC Beam Sections 1 ε < εcu : f c = ε c ε cc x = ; r = E sec = f c f c ε cc f c xr r 1 + x r E c E c E sec εc = Concrete strain f c = Concrte stress f c = 8 Day Comp. strength εcc = Strain at peak stress =. εcu = Ultimate concrete strain E c = Elastic modulus E sec = Secant modulus ε s < ε y : f s = E s ε ε s < ε sh : f s = f y f u ( ) s < ε su : f s = f u (f u f y ) εsu ε εs εsu εsh εs = Steel strain f s = Steel stress f y = Yield stress f u = Fracture stress εy = Steel strain εsh = Strain at strain hardening εsu = Failure strain E s = Elastic modulus Stress f c Stress f s f y ε cc Strain εc ε cu ε y εsh Strain εs εsu Fig. 6 Stress-Strain curve for unconfined concrete in compression (Mander et al., 1988) Fig. 7 Stress-Strain curve for steel shapes and steel bars ρ r = A rt /bd, (3) ρ r = A rb /bd, (4) M SC M y A Crushing of concrete B A f = b f t f, (5) A w = t w (d s t f ), (6) Moment where, A rt is the area of tension steel bars, A rb is the area of compression steel bars. And d, d s, b f, t w, t f are referred to Fig. 1(a). In Eq. (1), ρ t includes the contribution from tension steel bars and the contribution from the part of the H shape. And, in Eq. (), ρ c includes the contribution from compression steel bars and the contribution from the part of the H shape. Based on the considerations of steel shapes available, constructability, and practical application, the sections were arranged within the ranges described as follows: (1) ρ t =.5,.1,.15,. and.5; () ρ c /ρ t =.55 and.7; (3) ρ r /ρ t =.5,.35 and.5; (4) A w /A s =.3,.4 and.5; (5) d s /h =.5,.6 and.7, where A s is the cross-sectional area of steel shapes. In addition, all of the steel shapes have a b f equal to 175 mm. It is assumed that an SRC beam section reaches the concrete crushing limit state when the maximum concrete compressive strain reaches.3. The corresponding depth of the neutral axis, calculated based on Composite-SC, is designated as c SC, as indicated in Fig. 1(d). The corresponding depth of the neutral axis based on Composite-PSD is designated as c PSD, as indicated in Fig. 1(c). The relationship between curvature ductility ratio and c SC was first established, as described in the following Section. However, Composite-PSD, instead of Composite-SC, was O adopted in the S&D Method for simplicity. Therefore, a correlation between c SC and c PSD was established, as described in the subsequent Section.. Composite-SC φ y Curvature A commercial program named XTRACT () was used to analyze the moment-curvature relationship of the sections up to the concrete crushing limit state. The concrete stress-strain relationship proposed by Mander and Priestley (1988) was used, as shown in Fig. 6. The tensile strength of the concrete was neglected. The elastic-yield plateau-parabolic strain-hardening model, as shown in Fig. 7, was used for the stress-strain relationship of steel bars and steel shapes. The accuracy of XTRACT was verified by comparing the results from XTRACT and BIAX (Wallace, 199). The solid curve in Fig. 8 is a typical moment-curvature curve φcr Fig. 8 Typical moment-curvature curve of SRC beam sections

8 Journal of the Chinese Institute of Engineers, Vol. 31, No. (8) of an analyzed section, and the concrete crushing limit state of the section is indicated as B. The curvature corresponding to Point B is designated as φ cr, and the depth of the neutral axis corresponding to Point B is designated as c SC. The moment-curvature curve is approximated by straight lines OA and AB. OA was obtained by linear regression analysis of the data points before the first yielding of steel, either steel shape or steel bars with the constraint of passing through Point O. AB was obtained by means of linear regression of the data points in the inelastic range with the constraint of passing through Point B. The curvature corresponding to Point A is defined as yield curvature and is designated as φ y. The curvature ductility ratio µ cr is defined by Eq. (7): µ cr = φ cr /φ y. (7) Fig. 9 shows the µ cr versus d/c SC relationships of the 81 sections that were analyzed. It can be found that µ cr versus d/c SC relationships are close to linear. A linear regression analysis was carried out, and Eq. (8) was obtained with a coefficient of determination.987: µ cr = 1.31(d/c SC ) 1.6 (8) 3. Composite-PSD The assumption of Composite-PSD was adopted here for the section analysis. The property of flanges of steel shape is assumed to lump at the center line of the flanges. The distance from the top edge of the section to the thickness center of the top flange is designated as d s, as shown in Fig. 1(a). There are three possibilities for the location of the neutral axis: (1) below the top flange; () above the top flange; and (3) right on the top flange. The method proposed by Chen & Cheng (3) was adopted for the determination of moment capacity M PSD and depth of neutral axis c PSD, and is stated as follows. At first, the neutral axis is assumed to be below the top flange, as shown in Figs. 1(b) and 1(c), and, the corresponding depth of the neutral axis is designated as c PSD1. It can be found that the net tension of the steel shape T net can be expressed by Eq. (9): T net = t w (h c PSD1 )F y. (9) With the condition that the axial force of the beam section is zero, an equilibrium equation for the section can be established, and an expression for c PSD1 can then be established, as shown in Eqs. (1) and (11), respectively: t w (h c PSD1 )F y + A rt f y =.85f c β 1 c PSD1 + A rb f y, (1) µ cr d/c SC Fig. 9 µ cr versus d/c SC relationship c PSD1 = (A rt A rb ) f y + ht w F y t w F y +.85β 1 f c b, (11) where β 1 is the same β 1 defined by ACI Code (ACI, 5). If the resulting c PSD1 is greater than d s, the neutral axis is below the top flange as was assumed, and the depth of the neutral axis equals c PSD1. In addition, the corresponding moment capacity of section M PSD, can then be obtained according to Eq. (1): M PSD = F y Z t w ( h c PSD1) +(A rt + A rb ) f y d d +.85β 1 f c c PSD1 b( h β 1c PSD1 ), (1) where Z is the plastic modulus of the steel shape and d is referred to Fig. 1(a). If the resulting c PSD1 is not greater than d s, the assumption is incorrect. Then, it is assumed that the neutral axis is above the top flange, as shown in Fig. 1(d). The entire cross section of the steel shape is under tension. And, according to the condition that the axial force of the beam section is zero, the depth of the neutral axis c PSD can be obtained by Eq. (13): c PSD = (A rt A rb ) f y + A s F y. (13).85β 1 f c b If the resulting c PSD is less than d s, the neutral axis is above the top flange as was assumed, and the depth of the neutral axis equals c PSD. In addition, M PSD can then be obtained according to Eq. (14): M PSD = A rt f y (d c PSD )+A s F y ( h c PSD) +.85β 1 f c (c PSD ) b(1 β 1 ) µ cr = 1.31(d/c SC ) A rb f y (c PSD d ) (14)

9 C. C. Chen and C. L. Cheng: Flexural Analysis and Design Methods for Composite SRC Beam Sections 3 b d b f A rb d s ε cr ε rb φ cr 1c PSD1 β.85f c f y c PSD1 β 1 c PSD.85f c F y f y N.A. c PSD h d d s t f N.A. N.A. ε s A rt ε rt +F y +F y +f y +f y RC Steel shape RC Steel shape (a) Simplified section (b) Simplified distribution (c) Stress distribution with c PSD > d s (d) Stress distribution with c PSD < d s Fig. 1 Strain and stess distribution for Composite-PSD If the resulting c PSD is not less than d s, the neutral axis is neither below nor above the top flange. In this case, it is regarded that the neutral axis is right on the top flange, and the depth of the neutral axis equals d s. The resultant force of the top flange is assumed to equal zero, and, accordingly, Eq. (15) can be established for the calculation of M PSD : M PSD = A rt f y (d d s )+A rb f y (d s d ) + A s F y d s t f +.85β 1 f c (d s ) b 1 β 1 (15) Figure 11 shows the distribution of M SC /M PSD of the 81 analyzed sections. The average M SC /M PSD is 1.1 and the corresponding coefficient of variation is.5%. The difference between M SC and M PSD is very small, and the Composite-PSD along with the analysis method used here gives a reasonably good estimation of moment capacity. Fig. 1(a) shows the distribution of the calculated depth of neutral axis c PSD with respect to c SC. Apparently, Composite-PSD can not give a good estimation for the depth of the neutral axis. There are three groups of data points in the figure. Data points indicated with diamonds are sections with their neutral axis located below the top flange. Linear regression analysis of these data points and replacing c SC by c PSD gave Eq. (16) with a coefficient of determination of.9. With Eq. (16), c PSD1 is modified and a better estimation of the depth of the neutral axis c PSD can be obtained: c PSD = c PSD1 + 8 (mm). (16) Data points indicated with triangles are sections with. M SC /M PSD M SC (kn-m) Fig. 11 Distribution of M SC /M PSD their neutral axis located above the top flange. Linear regression analysis of these data points and replacing c SC by c PSD gave Eq. (17) with a coefficient of determination of.9. c PSD = c PSD AVG = 1.1 cov =.5% Max = 1.11 Min = (mm). (17) Data points indicated with marks are the sections with their neutral axis located right on the top flange. The distribution of these data points is in a pattern of three horizontal lines. It was found that c SC varies almost linearly with respect to the average value of c PSD1, c PSD and d s. Regression analysis and replacing c SC by c PSD resulted in Eq. (18) with a coefficient of determination of.96.

10 4 Journal of the Chinese Institute of Engineers, Vol. 31, No. (8) 3 below top flange above top flange at top flange 3 c PSD (mm) c PSD (mm) 1 c PSD = c PSD1 + 8; R =.9 1 c c PSD PSD + 45 = ; R = c SC (mm) (a) c SC versus c PSD 1 3 c SC (mm) (b) c SC versus c PSD Fig. 1 Distribution of c SC versus c PSD and c PSD c PSD = c PSD1 + c PSD + d s (mm). (18) Start The accuracy of Eqs. (16) to (18) was evaluated by plotting c SC with respect to c PSD. Fig. 1(b) shows the distribution of c SC versus c PSD for all the sections analyzed, and the correlation between c SC and c PSD is fairly good. By substituting the value of c PSD into c SC in Eq. (8), a µ cr can be obtained. The analysis procedure introduced here is quite straightforward and is able to provide a reasonable estimation of moment capacity and the curvature ductility ratio of an SRC beam section. III. S&D DESIGN METHOD 1. Limiting Depth of Neutral Axis Provided with a required moment capacity (M n ) req and a required curvature ductility ratio µ req, a beam section can be designed according to the design method proposed here. By replacing µ cr in Eq. (8) with µ req and solving for c SC, an expression for limiting the depth of the neutral axis c L, as shown in Eq. (19), can be obtained: c L =1.31 d µ req +1.6 (19) Limiting the depth of the neutral axis within c L is an important criterion for the designed section to meet the ductility requirement and for determination of the area of compression steel bars. Figure 13 is the flow chart of the S&D method. Stage 4 Stage 3 Stage Stage 1 Base on (M n ) req Select b, h, & H Shape Base on µ req Calculate c L Determine M S, c S and c S c L > c S M S (M n ) req Calculate req. extra moment M RC & req. tension steel bars A rt1 Steel bar congestion Calculate c and c c < c L Calculate req. comp. steel bars A rb modify req. ten. Steel bars A rt End Yes No No No No Yes Yes Yes Add complementary ten. & comp. steel bars End Use ten. Steel bar A rt1 Add complementary comp. steel bars End Fig. 13 Flow chart for S&D design method

11 C. C. Chen and C. L. Cheng: Flexural Analysis and Design Methods for Composite SRC Beam Sections 5 b b 1 c S β c S N.A. (for M S ) 1c S β d RC 1 c RC β N.A. (for M RC ) c RC (a) Section for M S (Stage ) (b) Section for A rt1 (Stage 3) Fig. 14 Section at Stages and 3 (c) Stress change in web (Stage 3) The design of the SRC beam sections can be divided into 4 stages, as follow.. Stage 1 Select a set of b, h, and steel shape by referring to the required moment capacity (M n ) req and the required ductility ratio µ req. In addition, calculate c L by substituting the required ductility ratio µ req into Eq. (19). 3. Stage In this Stage, the section is composed only of concrete and steel shape, as shown in Fig. 14(a). Also, Composite-PSD is assumed here. The nominal moment capacity of the section is designated as M S, and the corresponding depth of neutral axis is designated as c S. The procedure described in Section II.3 is adopted for section analysis. With A rt and A rb set equal to zero for the Eqs. in Section II.3, the resulting moment capacity is the M S and the resulting depth of the neutral axis is the c S. c S is modified with one of the equations from Eqs. (16) to (18) to give a more accurate depth of the neutral axis designated as c S. If c S is greater than c L, the section does not satisfy the curvature ductility requirement. The section should be rearranged and the procedure goes back to the beginning of Stage 1. If c S c L and M S (M n ) req, the section satisfies both the strength and ductility requirements. Then, equal amounts of complementary tension and compression steel bars are added to compose an SRC section, and the design can be terminated. If c S c L and M S < (M n ) req, the section requires additional moment capacity M RC, as defined by Eq. (), and the design goes to Stage 3: 4. Stage 3 M RC = (M n ) req M S. () Since the concrete, shown as shaded area in Fig. 14(a), is used to produce M S, it can not provide any extra strength. In this stage, M RC is assumed to be provided by the b d RC rectangular section along with tension steel bars, as shown in Fig. 14(b). By designing the b d RC section as a singly reinforced concrete section to provide M RC, a required tension steel bar area A rt1 can be determined according to Eq. (1): A rt1 =.85f c bd RC f y 1 1 M RC.85f c bd RC. (1) It is noted that the ductility of the section will be checked later; therefore, the limitation of steel ratio, which is.75 of the balanced steel ratio, is ignored here. It is also noted that stress in part of the web of the steel shape would change from tension to compression, as shown in Fig. 14(c). This change is ignored in establishing Eq. (1) since its effect on A rt1 is insignificant. However, the stress change can not be neglected in determining the depth of the neutral axis. As a result, Eq. () was established for calculation of the depth of the neutral axis c RC : c RC = A rt1 f y.85β 1 f c b +t w F y () It is possible that β 1 c S is less than d S and stress change may also occur in the top flange. In this condition, Eq. () may induce error. However, for simplicity, Eq. () is still used, and the error induced is quite acceptable. The total depth of the neutral axis c, is the sum of c S and c RC. The depth of the neutral axis c, which is based on Composite-PSD, is modified to provide a better estimation of the depth of the neutral axis. When c is greater than d s, c is substituted into Eq. (16) as c PSD1, and the resulting c PSD is the modified depth of the neutral axis, and is designated as c. On the other hand, when c is equal to, or less than, d s, Eq. (17) is used to obtain c. When c is no greater

12 6 Journal of the Chinese Institute of Engineers, Vol. 31, No. (8) b b c c L β 1c 1 (c c L ) β c c β L 1 (c ) d c c L c c L (a) Limiting depth of neutral axis (b) Concrete stress block and compression steel bars Fig. 15 Section at Stage 4 (c) Stress change in web than c L, the section is regarded as having sufficient ductility. If A rt1 is considered too big to avoid steel congestion, the section should be rearranged and the design goes back to Stage 1. If A rt1 is not going to cause steel congestion in construction and c c L, complementary compression steel is added, the design is completed. If A rt1 is not going to cause steel congestion in construction and c is greater than c L, the section does not meet the ductility requirement, and the design goes to Stage Stage 4 It is known that the depth of the neutral axis c is greater than c L, as shown in Fig. 15(a). In order to meet the ductility requirement, the depth of the neutral axis has to be pushed upward by the amount of c c L. Consequently, the depth of concrete stress block should be reduced by β 1 (c c L ), as shown in Fig. 15(b). The released concrete stress is compensated for by adding compression steel bars. For cases where c L is greater than, or equal to, d s, stress in part of the web, as the shaded area shown in Fig. 15(c), changes from compression to tension. The stress change in the web should also be compensated for by compression steel bars. Consequently, Eq. (3) is established for the required area of compression steel bar: A rb = (c c L)(.85β 1 f c b +t w F y ) f y (3) Again, stress change may involve the top flange; however, this possibility is ignored in Eq. (3), and, Section VI, the error induced is in the controllable range. For cases where c L is less than d s, there is no stress change in the steel shape; therefore, the required area of compression steel bar can be calculated according to Eq. (4): A rb = (c c L).85β 1 f c b f y (4) Since the compression force from the shaded area shown in Fig. 15(b) is pushed upward to the position of the compression steel bar, the moment capacity of the section increases accordingly. Eq. (5) is the expression for the increased moment capacity M rb. Consequently, the area of tension steel bars can be reduced. By linear interpolation, Eq. (6) is proposed for obtaining the required area of tension steel bars A rt for the section: M rb = A rb f y β 1 c c c L d (5) A rt = A rt1 M RC M rb M RC. (6) Finally, A rt is the required area of tension steel bars, A rb is the required area of compression steel bars, and the design is completed. IV. VERIFICATION OF THE S&D METHOD Three series of SRC beam sections were designed according to the S&D method with a µ req = (or SD- ): S&D method with a µ req = 3 (or SD-3), and Superposition-PSD were carried out. The nominal strength of the materials used were f y = 414 MPa for steel bars, F y = 345 MPa for steel shapes, and f c = 7.6 MPa for concrete. Table 4 lists the sections and required moment capacity selected for design. In each series, d s and b f were kept constant, t f varied every mm, and t f /t w were kept equal to. For each combination of section dimension, steel shape, and (M n ) req, the required area of tension steel bar and the required area of compression steel bar were determined by SD-, SD-3 and Superposition-PSD, respectively. There were a total of 18 possible combinations of section dimension, steel shape, and (M n ) req. However, sections with area of tension steel bar greater than.5bd are abandoned due to the consideration of steel congestion. In addition, steel shape for some sections was too big for the section to meet the curvature ductility requirement, and

13 C. C. Chen and C. L. Cheng: Flexural Analysis and Design Methods for Composite SRC Beam Sections 7 Table 4 Sections and required moment capacity selected in design Series Section Steel shape dimensions No. of different (M n ) b h req d s b f t f t w steel shapes (kn-m) (mm mm) (mm) (mm) (mm) (mm) ~3 4~ , ~4 6~ 15 98, 15, ~5 7~ , 147, M prov /(M n ) req AVG = 1.1; σ =. AVG = 1.1; σ =.5 AVG = 1.1; σ = (M n ) req (kn-m) (a) SD (M n ) req (kn-m) (b) SD (M n ) req (kn-m) (c) Superposition-PSD Fig. 16 Moment capacity strength of designed sections prov µ (M n ) req (kn-m) (a) SD (M n ) req (kn-m) (b) SD-3 Fig. 17 Curvature ductility of designed sections (M n ) req (kn-m) (c) Superposition-PSD these sections were also abandoned. Therefore, only 51, 1 and 91 sections were successfully designed for SD-, SD-3 and Superposition-PSD, respectively. It is noted that the area of compression steel bars is kept at least 5% of the area of tension steel bars for Superposition-PSD for the purpose of providing a certain level of ductility to the section designed. This measure is used by SRC design specification in Taiwan (CPA, 4). After the sections were designed, they were further analyzed by using XTRACT () to obtain the moment capacity M prov and curvature ductility ratio µ prov of the sections. Fig. 16 shows the distribution of the M prov /(M n ) req of the designed sections. The M prov /(M n ) req for SD- varies from.97 to 1.9 with an average value of 1.1 and a standard deviation of.. The M prov /(M n ) req for SD-3 has an average value of 1.1 and a standard deviation of.5. The M prov /(M n ) req for Superposition-PSD has an average value of 1.1 which is much higher than that of SD- and SD-3. Moreover, the standard deviation for Superposition-PSD is.9 which is also much higher than that of SD- and SD-3. It shows that the S&D method results in reasonably good design and is more accurate than Superposition-PSD as far as moment capacity is concerned. Figure 17 shows the distribution of µ prov /µ req. All of the sections designed by SD- have a ductility

14 8 Journal of the Chinese Institute of Engineers, Vol. 31, No. (8) ratio no less than. More than 86% of the sections designed by SD-3 provide a ductility ratio no less than 3. The majority of the sections designed by the S&D method are able to provide the required ductility. Superposition-PSD has less control on the ductility of the sections designed. More than 31% of the sections designed by Superposition-PSD possess a ductility ratio less than. This shows that the S&D method is able to control the curvature ductility ratio of the sections designed reasonably well and is superior to Superposition-PSD. V. SUMMARY AND CONCLUSIONS Section analysis that complies with complete composite and strain compatibility conditions of 81 SRC beam sections, was carried out to investigate the flexural behavior of SRC beam sections. It was found that the curvature ductility ratio at concrete crushing limit state of SRC beam sections is closely related to the ratio of the effective depth of the section to the depth of the neutral axis. Accordingly, a correlation between the two was established with fairly good agreement. A section analysis procedure that fulfills all composite conditions with plastic stress distribution is introduced. This procedure is quite straightforward and easy to apply, and can be used to evaluate both the moment capacity and curvature ductility ratio of SRC beam sections. In addition, a design method, based on complete composite and plastic stress distribution, is proposed for the design of SRC beam sections, to meet both the moment and curvature ductility requirements. The accuracy and applicability of the design method was verified by designing 3 series of sections and comparing the required moment and ductility capacities with the provided moment and ductility capacities of the designed sections. It was found that the proposed design method has good control of moment and curvature ductility capacities of the SRC beams designed. The aforementioned flexural analysis and design methods can be used for cases where complete composite action is considered appropriate. Compared to the strength superposition method, the design methods proposed here are able to provide tools for more economical and rational SRC beam design. However, a strength reduction factor φ b of.85 for bending strength, which is adopted by the AISC code, is recommended when the proposed methods are used. REFERENCES ACI, 5, Building Code Requirement for Structural Concrete ACI , American Concrete Institute, Farmington Hills, MI, USA. AISC, 5, Manual of Steel Construction-Load & Resistance Factor Design, American Institute of Steel Construction, Inc., Chicago, IL, USA. AIJ, 1, AIJ Standards for Structural Calculation of Steel Reinforced Concrete Structures, Architectural Institute of Japan, Tokyo, Japan. (in Japanese). ASTM, 199, Standard Specification for Low-Alloy Steel Deformed Bars for Concrete Reinforcement, ASTM A76/A76M-9, ASTM International, West Conshohocken, RA, USA. ASTM, 1, Standard Specification for High- Strength Low-Alloy Columbium-Vanadium Structural Steel, ASTM A57-1, ASTM International, West Conshohocken, RA, USA. CPA, 4, Design Specification for Steel Reinforced Concrete Structures, Constructions and Planning Agency, Ministry of Interior, Taipei, Taiwan, R. O.C. (in Chinese). Chen, J. R., 199, A Study on the Strength and the Deformability of Pre-Cast Steel Reinforced Concrete Beam-to-Column Connection, Master Thesis, Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C. (in Chinese) Chen, C. J., 1996, Seismic Resistance Characteristics of Steel Reinforced Concrete Beam-to-Column Connection, Master Thesis, Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C. (in Chinese) Chen, C. C., and Chen, C. C., 1, Flexural Behavior of Steel Encased Composite Beams, Journal of the Chinese Institute of Civil and Hydraulic Engineering, Vol. 13, No., pp. 63~75. (in Chinese) Chen, C. C., and Cheng, C. L., 3, A Simplified Flexural Analysis Method and Flexural Design Strength of Steel Reinforced Concrete Beams, Structural Engineering, Chinese Society of Structural Engineering, Vol. 18, No. 3, pp. 3~18 (in Chinese). CNS, 5, Steel Bars for Concrete Reinforcement, Chinese Standard No. 56 A6, Bureau of Standards, Metrology and Inspection, Ministry of Economic Affairs, Taipei, Taiwan, R.O.C. (in Chinese) Ju, J. S., 1993, The New Design and Construction Method of Pre-cast Steel Reinforced Concrete Beam-to-Column Connection, Master Thesis, Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C. (in Chinese) Mander, J. B., and Priestley, M. J. N., 1988, Observed Stress-Strain Behavior of Confined Concrete, Journal of Structural Engineering, ASCE, Vol. 114, No. 8, pp Tsai, K. C., Yu, R. T., Lian, Y., and Shianq, W. B., 1995, Seismic Resistance Characteristics of Half

15 C. C. Chen and C. L. Cheng: Flexural Analysis and Design Methods for Composite SRC Beam Sections 9 Pre-cast Steel Reinforced Concrete Beam-to-Column Connection, Structural Engineering, Chinese Society of Structural Engineering, Vol. 1, No., pp. 35~51. (in Chinese) Wallace, J. W., 199, BIAX: Revision 1, A Computer Program for the Analysis of Reinforced Concrete and Reinforced Masonry Sections, Report No. CU/CEE-9/4, Department of Civil and Environmental Engineering, Clarkson University, Potsdam, NY, USA. Weng, C. C., Jiang, M. H., and Yen, S. I., 1, Experimental Study on Ultimate Flexural Capacity of SRC Beams, Journal of the Chinese Institute of Civil and Hydraulic Engineering, Vol. 13, No., pp. 49~61. (in Chinese) XTRACT,, XTRACT v..6. Release Notes, Imbsen Software Systems, Sacramento, CA, USA. Manuscript Received: Mar. 3, 7 Revision Received: Aug. 13, 7 and Accepted: Sep. 13, 7