Flexural Behavior and Design of High-Strength Concrete Members

Transcription

1 Flexural Behavior and Design of High-Strength Concrete Members Zhenhua Wu, Wonchang Choi, Amir Mirmiran, Sami Rizkalla and Paul Zia Synopsis: Development of high-strength concrete (HSC) dates back to the early 1930 s, when concrete compressive strengths above 97 MPa (14 ksi) were achieved using autoclave curing. With the advent of super-plasticizers, today HSC has become an economical solution for bridge construction. This paper presents an overview of the past research on flexural behavior of HSC members. The study is based on an extensive search of literature, a national survey of bridge owners and bridge producers, and an international survey of bridge design codes. The study has resulted in identification of factors that affect the flexural behavior and design of reinforced concrete members made of HSC. The paper also identifies relevant design issues to extend the current concrete compressive strength limit of 69 MPa (10 ksi) to 124 MPa (18 ksi) in the AASHTO- LRFD Bridge Design Specifications. Keywords: beam; bridge; flexural members; high-strength concrete

2 Zhenhua Wu received his MS in 2003 and is continuing his research as a Ph.D. student in the Department of Civil, Construction and Environmental Engineering at North Carolina State University. He received his BS in 2000 from the Department of Civil Engineering at Tsinghua University, Beijing, China. His main research interest is the structural behavior and design of high-strength concrete members. Wonchang Choi is a graduate student in the Department of Civil, Construction and Environmental Engineering at North Carolina State University. He received his BS (1999) and MS (2002) degrees from the Department of Civil and Environmental Engineering at Hongik University, Seoul, Korea. His main research interests include the structural use of high-strength concrete and design of high-strength prestressed girders. ACI member Amir Mirmiran is Professor and Chair of the Department of Civil and Environmental Engineering at the Florida International University in Miami, FL. He has served on the faculty of North Carolina State University, University of Cincinnati and the University of Central Florida. His research interests include high-performance concrete and composite materials. ACI member Sami Rizkalla is Distinguished Professor of Civil and Construction Engineering in the Department of Civil, Construction and Environmental Engineering, North Carolina State University. He is the Director of the Constructed Facilities Laboratory and NSF I/UCRC in Repair of Structures and Bridges at North Carolina State University. He is a fellow of ACI, ASCE, CSCE, EIC and IIFC. ACI member Paul Zia is Distinguished University Professor Emeritus at North Carolina State University. He served as ACI president in 1989, and is a member of ACI Committee 363, High-Strength Concrete; the Concrete Research Council; and the Joint ACI-ASCE Committee 423, Prestressed Concrete; and ACI Committee 445, Shear and Torsion. INTRODUCTION In the early 1990 s, the Federal Highway Administration (FHWA) sponsored the use of High Performance Concrete (HPC) in several demonstration projects. Since 1993, a number of HPC bridges have been constructed across the country. The FHWA compilation project 1 reports on 19 such bridges in 14 states. While the highest design concrete strength in these bridges was reported as 97 MPa (14 ksi) in Texas, the achieved strength at the design age reached as high as 110 MPa (15.9 ksi) in South Dakota. The AASHTO LRFD Bridge Design Specifications, first published in 1994, includes an article ( ) limiting its applicability to a maximum concrete strength of 69 MPa (10 ksi), unless physical tests are made to establish the relationship between concrete strength and its other properties. These limitations reflected the lack of research data at the time, rather than the inability of the material to perform its intended function.

3 Many design provisions stipulated in the LRFD Specifications are still based on test results obtained from specimens with compressive strengths up to 41 MPa (6 ksi). Although such a strength limit is not imposed explicitly by other codes such as the ACI except in its provisions for development length whether these codes are applicable to HSC is not fully addressed either. The NCHRP has initiated four separate projects to expand the LRFD Specifications, allow broader use of HSC, and meet the needs of the bridge design community. The objective of NCHRP Project 12-64, which is the subject of this paper, is to recommend revisions to the LRFD Specifications to extend the applicability of its compressive and combined compressive and flexural design provisions for reinforced and prestressed concrete members to concrete strengths up to 124 MPa (18 ksi). RESEARCH SIGNIFICANCE Strength of commercial concrete has been increasing over the years. Concrete with strength up to 124 MPa (18 ksi) have been successfully used in some high-rise buildings. However, most of the current design codes were developed using experiment results based on normal strength concrete. Therefore most codes specified limit to the maximum concrete strength. These limitations reflected the lack of research data at the time, rather than the inability of the material to perform its intended function. The National Cooperative Highway Research Program (NCHRP) has initiated four projects to expand the LRFD design specifications to accommodate the application of HSC. This paper summarizes the effort of many researchers to extend the applicability of flexural and compression design provisions for reinforced concrete members to concrete strengths up to 124 MPa (18 ksi). OVERVIEW OF THE FLEXURAL BEHAVIOR OF THE HSC BEAMS This section provides a review of 16 different projects on HSC beam tests with a total of 174 test data. The variables in the HSC beam tests include compressive strength of concrete f c, yield strength of steel reinforcement f y, span length L and the distance between the load points L c, width of the section b, depth of tension steel d, shear span to depth ratio a/d, reinforcement ratio for tension steel ρ in percentage as well as a fraction of the balanced reinforcement ratio ρ b, the reinforcement ratio for compression steel ρ, and transverse steel reinforcement ratio ρ s in the constant moment region. It is important to note that for flexure-critical failure, the amount of the stirrups in the constant moment region is not expected to significantly affect the capacity of the member, unless a large amount of stirrups is used so that the concrete in the compression zone is well confined. However, it is potentially an important factor that may affect ductility of the member. Table 1 shows the range of test variables in the HSC beam databank.

4 Of the data obtained on HSC beams, a total of 35 specimens were reported with unusually low capacities 3. The low capacities reported for these tests were attributed to the anchorage failure of the reinforcement near the support for some of the specimens and shear failure in others. These tests were excluded from the data synthesis. Flexural failure has been reported for all other test data irrespective of the a/d ratio. It is important to examine the distribution of test variables in the beam test. Figure 1 shows the distribution of the beam tests based on the strength of concrete. The majority of the tests correlate to concrete compressive strengths below 97 MPa (14 ksi), while only 4% of the beams have concrete compressive strength over 110 MPa (16 ksi). Figure 2 shows the distribution of the beam tests based on the amount of reinforcement and the type of failure. While only 18% of the specimens failed in compression as overreinforced beams, the clear majority involves tension failure of under-reinforced beams. Figure 3 shows the distribution of the beam tests based on the shape of the cross section. About 85% of the beams had a rectangular section, while 5% of the beams were circular, 6% were triangular, and only 4% were flanged sections. It is important to note that even in the flanged sections tested, the compression zone remained in the flange. Therefore, there is a lack of test data on the HSC beams with flanged sections. FLEXURAL RESISTANCE OF FLANGED SECTIONS USING RECTANGULAR STRESS DISTRIBUTION The axial stress-strain relationship of concrete varies with its strength. The ascending and descending portions of the curve become steeper with increasing strength. The curves tend to become more linear for higher strength concretes. As a result, the equivalent stress block for high-strength concrete is expected to be different from that of normal-strength concrete. A generalized stress block is defined by three parameters, k 1, k 2 and k 3, as shown in Figure 4. The parameter k 1 is defined as the ratio of the average compressive stress to the maximum compressive stress. The parameter k 2 is the ratio of the depth of the resultant compressive force to the depth of neutral axis. The parameter k 3 is the ratio of the maximum compressive stress to the compressive strength of concrete cylinder f c. The design values of the stress block parameters are determined at the ultimate strain ε cu, which corresponds to the maximum moment of the section. These parameters are depicted in Figure 4. They were originated from the eccentric bracket tests performed by Hognestad et al. 4 in the 1950 s. The k 1 k 3 value and the k 2 value can be obtained from the equilibrium of the external and internal forces, as follows: ( d k c) + A ' f ( d ') M = k (1) n 1k3 f ' c bc 2 s su d

5 The three-parameter generalized stress block can be reduced to a two-parameter equivalent rectangular stress block, by keeping the resultant of the compression force at the mid-depth of the assumed rectangular stress block. The two parameters of α 1 and β 1 can be defined as: k k 1 3 α 1 = (2) 2k2 β = 2k 2 (3) The nominal axial and flexural resistance of the section provided by AASHTO LRFD as well as ACI code can then be shown as: M LRFD β1c = α 1β1 f ' c bc d + As ' fsu ( d d') (4) 2 It is generally agreed that flexural resistance of under-reinforced beams is not significantly affected by the exact shape of the stress block, so long as the internal lever arm for moment resistance is chosen properly. This is mainly due to the fact that strength of under-reinforced beams is controlled by the yield strength of the reinforcement. Compression failure occurs more often in compression members with small eccentricity. In those cases, a more accurate representation of the stress block is crucial to accurately predict the strain profile and the lever arm. In the case of over-reinforced concrete beams, where failure is controlled by compression, the equations can be affected more significantly by the shape of the stress block. Test data on over-reinforced HSC beams are rare. However, some of the data related to compression failure of beam-columns may be used to verify these equations. A cursory review of over 100 test data on HSC beams in the databank shows that the use of rectangular stress block in the LRFD Specifications is not detrimental to the design of flexural members. Table 2 shows the statistical summary of the available HSC beam tests in the databank as a ratio of the experimental flexural resistance M Exp to that predicted using the LRFD Specifications M LRFD. The latter is calculated based on Equation ( ) of the LRFD Specifications, using β 1 of 0.65, because the concrete strengths exceed 69 MPa (10 ksi). Note that ACI and LRFD Specifications give the same nominal strength of the member on this issue. All calculations are based on the actual yield strengths of the flexural reinforcement, if measured. Otherwise, nominal yield strengths were used instead. No strain hardening was considered in the analysis. Plain concrete beams and reinforced concrete beams with reported premature shear failure 3 were excluded from this comparison. In the case of plain concrete beams, the nominal flexural capacity is the cracking moment of the section, which is significantly smaller than that observed in the experiments. In general, test data for over-reinforced concrete beams is scarcer than the under-reinforced beams. A M Exp /M LRFD ratio of less than 1 in the table indicates an un-conservative design.

6 The data is also shown as a function of concrete compressive strength f c in Figure 5. The 125 data points in the figure are broken down into three categories: 22 over-reinforced rectangular sections, 84 under-reinforced rectangular sections, and 19 other shapes. No general trend can be observed in relation to the concrete compressive strength. It can be seen, however, that test data above 103 MPa (15 ksi) are quite rare to make a valid judgment as to the applicability of the design equations. It should be noted that only one research publication 5 has addressed flexural specimens with a triangular or circular compression zone. These members were designed as over-reinforced sections. The flexural resistance of other sections was calculated similar to the procedure for rectangular sections. Even though the assumption of the rectangular stress block is not valid for other shapes, it seems that the flexural resistance of the member is not greatly affected. Figure 6 shows the M Exp /M LRFD ratio with respect to the shear span a/d ratio, where a is the shear span and d is the depth of tension reinforcement. All data points reflect flexural mode of failure, as the specimens were heavily reinforced to avoid shear failure. The figure shows no clear trend, indicating same applicability of the current equation for the majority of the flexure-critical beams. Even though the role of compression reinforcement in flexural members is well established, their behavior and contribution to the resistance of the HSC beams need to be justified. The amount of compression reinforcement is an important factor in calculating the flexural resistance of the section. The presence of compression steel further increases the reinforcement limit for tension steel 6. It should also enhance the rotational capacity of a section, especially if adequate transverse reinforcement is provided. However, no physical test data is available from the literature on doubly reinforced HSC beams. The amount of transverse reinforcement may have an effect on the flexural resistance of HSC beams. However, only eight (8) beam tests were found in the literature with stirrups in the constant moment region. Of these, the amount of stirrups in three (3) of the beams were not reported. Mansur et al. 7 reported that ultimate strength and ductility are enhanced by using stirrups for over-reinforced members. However, further research is needed on this subject. Flexural resistance of flanged sections is also affected by how the stress block is determined and whether the neutral axis or the stress block falls outside the flange area 8. The LRFD Specifications calls for a T-section behavior if c > h f, in contrast with the definition in the ACI and the AASHTO Standard Specifications, where a section is considered a T-section if a > h f. This issue has caused confusion in design, in cases when a < h f < c, and needs to be addressed analytically. Since the use of concrete with compressive strengths in excess of 69 MPa (10 ksi) in bridge decks is rare, the most likely application of flanged sections with HSC would be in continuous beams over interior support with the compression region in the bottom flange of the girders. There is no test data on this type of application. Even though a few tests with T-shaped specimens are found in the literature review, the

7 compression zone has remained in the flange. Therefore, there is a lack of data on the HSC beams with flanged sections. REINFORCEMENT AND STRAIN LIMITS FOR FLEXURAL MEMBERS block where d The LRFD Specifications specifies a limit of c on the depth of stress A f d + A f d ps ps p s y s e =. Over-reinforced sections may be used in Aps f ps + As f y prestressed and partially prestressed members, only if it is shown by the analysis and experimentation that sufficient ductility of the structure can be achieved. Over-reinforced reinforced concrete sections shall not be permitted. The maximum reinforcement limit in flexural members is established based on ductility requirements. It is important to discuss the definition of ductility to better understand the basis of the Specifications in this regard. The term ductility is defined as the ability of the material or member to sustain deformation beyond the elastic limit, while maintaining a reasonable load-carrying capacity before total collapse. Depending on the type of material or member, the deformation measure for ductility may be strain, curvature, displacement or rotation. In reinforced concrete beams, curvature ductility is a method preferred by many, while displacement ductility is perhaps easier to measure. The two measures of ductility are given by φ u µ φ = (5) φ y u µ = (6) y d e where µ φ and µ are the curvature and displacement ductility, respectively, φ u and φ y are the ultimate and yield curvature, respectively, and u and y are the ultimate and yield displacements, respectively. The curvature and displacement at failure are often defined as those corresponding to a strength loss of about 20% for the member 9. Test data on ductility of HSC beams is quite limited. A total of 55 HSC beam tests have been reported in six different research publications, three of which used curvature ductility 10, 11, 12, two used displacement ductility 6, 13, and only one reported rotation ductility 14. All beams were of rectangular section, and were tested in four-point bending. Only the specimens by Pam et al. 6 were equipped with stirrups. Test results indicate that in the absence of confinement in the constant moment region, the dominant

8 factor in determining the load-deflection curve of the member is the c/d e ratio. Ductility of the member is greatly reduced, as this ratio increases. For all available test data, Figures 7 and 8 show the measured curvature and displacement ductility as functions of c/d e ratio, respectively. Weiss et al. 10 considered failure of the specimen at 25% strength reduction, whereas Sarkar et al. 11 used 30% strength reduction to signify failure. Suzuki et al. 12 considered the actual collapse of the beam. Although not shown in the graphs, rotation ductility of beams tested by Alca et al 14 ranged between 1.5 and 2.8 for the c/d e ratios between 0.38 and LRFD Specifications limits the c/d e ratio to 0.42 to distinguish between underreinforced and over-reinforced beams. Figure 9 shows the ratio of the measured flexural resistance of HSC beams in the databank to that predicted using the equations in the LRFD Specifications, as a function of the c/d e ratio. The line drawn at c/d e of 0.42 identifies the LRFD definition of over-reinforced and under-reinforced beams. Test data marked as other shapes refer to beams with triangular or circular compression zone 5. It is noteworthy that no value below 0.8 was observed in the literature for either overreinforced or under-reinforced HSC beams that were carefully detailed to avoid shear or bond failure. Tests by Ahmad and Lue 3 were not included since they reported premature shear failure. Figure 9 also shows that the use of rectangular stress block parameters in the current LRFD Specifications could be improved to avoid over-estimating the flexural resistance of the section in HSC beams. Figure 10 shows the same data in terms of the reinforcement ratio as a fraction of the so-called balanced reinforcement ratio that is defined as the amount of reinforcement for which steel yields (ε y of 0.002) at the same time that concrete reaches its ultimate strain ε cu of Note that a balanced reinforcement ratio corresponds to a c/d e ratio to 0.60, whereas the limit of 0.42 for the c/d e ratio corresponds to 70% of the balanced reinforcement ratio. While the majority of the tests have been carried out on under-reinforced beams, no clear trend can be observed in the figure regarding the accuracy of the LRFD Specifications for either over-reinforced or under-reinforced beams. The ultimate strain of unconfined concrete has a profound effect on the ductility requirements and the reinforcement limits of flexural members. ACI 441R indicates that the strain limit of is quite satisfactory for both HSC and NSC, although it may not be as conservative for HSC 7. Figure 11 shows the ultimate strains in HSC beams as the average over the constant moment region. The figure confirms the findings of the ACI 441R Except for one data point, which is from an over-reinforced beam without any stirrups 7, all others show an ultimate strain greater than Tests by Van Mier 16 on plain concrete prisms of 2, 4 and 8 in (51, 102, and 203 mm) heights showed the scale effect and size dependency of the ultimate strain of concrete due primarily to the damage localization phenomenon. Alca et al. 14 tested 12 simply-supported beams with concrete strengths ranging from 50 to 90 MPa (7.3 to 13 ksi), and found that the plastic rotation capacity exceeded the values predicted by using the ultimate strain value of This can be explained by simple statics. The neutral axis in an HSC beam is

9 closer to the extreme compression fiber than in an NSC beam with the same reinforcement ratio, simply because a shallower compressive stress block is needed in an HSC beam. The shallower neutral axis depth is expected to result in higher plastic strains in the tension reinforcement, leading to a ductile behavior. This effect counteracts the fact that HSC exhibits a lower ultimate strain in the extreme compression fiber. The average measurements of the ultimate strain in beams ranged from to at the peak strength, and from to at the time of failure. Some local strain values reached as high as Weiss et al. 10 also tested 8 beams with f c of approximately 100 MPa (14.5 ksi) and two reinforcement ratio ρ/ρ b of 0.3 and 0.8. They reported strains ranging between and Local strains for a beam with shortest constant moment region reached as high as CONCLUSIONS There is adequate test data to establish the necessary parameters for the flexural resistance of rectangular section as described above. However, due to large scatter of the available test data and the scarcity of data for concrete compressive strengths above 97 MPa (14 ksi), validation tests such as eccentric bracket tests are needed. Unless the experiments show the existing relationship to be unreliable, only the stress block parameters need to be calibrated. To the extent possible, a seamless transition would be suggested between the normal strength concrete and high-strength concrete after further research. ACKNOWLEDGEMENTS The authors would like to acknowledge the support of the NCHRP through the project and the Senior Program Officer, David Beal. The authors also thank the contributions of Henry Russell of Henry Russell, Inc. and Robert Mast of Berger/ABAM Engineers, Inc. who serve as consultants on the project. The findings and the conclusions reported here are of a preliminary nature and are those of the authors alone, and do not reflect the views of the supporting agency. REFERENCES 1. Russell, H.G., Miller, R.A., Ozyildirim, H.C. and Tadros, M.K., Compilation and Evaluation of Results from High Performance Concrete Bridge Projects, Volumes 1 & 2, Federal Highway Administration, Washington, D. C., 2003.

10 2. ACI Committee 318, "Building Code Requirements for Structural Concrete (ACI ) and Commentary (ACI 318R-02), American Concrete Institute, Farmington Hills, MI, 2002, 443 pp. 3. Ahmad, S. H. and Lue, D. M., Flexure-Shear Interaction of Reinforced High- Strength Concrete Beams, ACI Structural Journal, Vol. 84, No. 4, 1987, pp Hognestad, E., Hanson, N.W. and McHenry, D., Concrete Stress Distribution in Ultimate Strength Design, ACI Journal, Vol.52, No.4, 1955, pp Kahn, L. F. and Meyer, K. F., Rectangular Stress Block for Nonrectangular Compression Zone, ACI Structural Journal, Vol. 92, No.3, 1995, pp Pam, H. J., Kwan, A. K. H. and Islam, M. S., Flexural Strength and Ductility of Reinforced Normal- and High-Strength Concrete Beams, Proceedings of the Institution of Civil Engineers: Structures and Buildings, Vol. 146, No. 4, 2001, pp Mansur, M. A., Chin, M. S. and Wee, T.H., Flexural Behavior of High-Strength Concrete Beams, ACI Structural Journal, Vol. 94, No. 6, 1997, pp Naaman, A. E. Rectangular Stress Block and T-section Behavior, PCI Journal, Vol. 47, No.5, 2002, pp Park, R. and Paulay, T., Reinforced Concrete Structures, John Wiley and Sons, New York, N.Y Weiss, W. J., Guler, K. and Shah, S. P. Localization and Size-Dependent Response of Reinforced Concrete Beams, ACI Structural Journal, Vol. 98, No.5, 2001, pp Sarkar, S., Adwan, O. and Munday, J.G. L., High Strength Concrete: An Investigation of the Flexural Behavior of High Strength RC Beams, Structural Engineer, Vol. 75, No.7, 1997, pp Suzuki, M., Suzuki. K., Abe. K. and Ozaka. Y., Mechanical Properties of Ultra High Strength Concrete Beams Subjected to Pure Bending Moment, Fourth International Symposium on the Utilization of High Strength/High Performance Concrete, May 1996, Paris, France, pp Bernhardt, C. J. and Fynboe, C. C., High Strength Concrete Beams, Nordic Concrete Research, No.5, 1986, pp

11 14. Alca, N., Alexander, S. D. B. and MacGregor, J. G., Effect of Size On Flexural Behavior of High-Strength Concrete Beams, ACI Structural Journal, Vol. 94, No.1, 1997, pp ACI Committee 441, State-of-the-Art Report on High-Strength Concrete Columns (ACI 441R-96), American Concrete Institute, Farmington Hills, MI, 1996, 13 pp. 16. Van Mier, J.G.M., Multi-axial Strain-Softening of Concrete, Part 1: Fracture, Materials and Structures, No. 111, 1986, pp Kaminska, M. E., High-Strength Concrete and Steel Interaction in RC Members, Cement and Concrete Composites, Vol. 24, No.2, 2002, pp Ozcebe, G., Ersoy, U. and Tankut, T., Minimum Flexural Reinforcement for T- beams Made of Higher Strength Concrete, Canadian Journal of Civil Engineering, Vol. 26, No. 5, 1999, pp Shin, S., Ghosh, S. K. and Moreno, J., Flexural Ductility of Ultra-High-Strength Concrete Members, ACI Structural Journal, Vol. 86, No. 4, 1989, pp Leslie, K. E., Rajagopalan, K. S. and Everard, N. J., Flexural Behavior of High- Strength Concrete Beams, Journal of American Concrete Institute, Vol. 73, No. 9, 1976, pp Swamy, R. N. and Anand, K. L., Structural Behavior of High Strength Concrete Beams, Building Science, Vol. 9, No.2, 1974, pp Lambotte, H. and Taerwe, L. R., Deflection and Cracking of High-Strength Concrete Beams and Slabs, High-Strength Concrete, Second International Symposium, Weston T. Hester (Ed.), SP-121, American Concrete Institute, Detroit, MI., 1990, pp Wiegrink, K., Marikunte, S. and Shah, S. P., Shrinkage Cracking of High-Strength Concrete, ACI Materials Journal, Vol. 93, No.5, 1996, pp LIST OF NOTATIONS a: depth of equivalent rectangular stress block A g : gross cross-sectional area of member A ps : area of prestressing steel A s : area of tension reinforcement b: width of web, which is the same as the width of compression flange in rectangular sections c: distance between the neutral axis and the extreme compression fiber

12 d: depth of tension steel from the extreme compression fiber d : depth of compression steel from the extreme compression fiber d e : effective centroid depth of overall tension steel from the extreme compression fiber E c : modulus of elasticity f c : specified compressive strength of concrete at 28 days, unless another age is specified f ps : actual stress of the longitudinal prestressing tendon f pu : specified tensile strength of prestressing steel f su : ultimate strength of longitudinal steel f y : yield strength of longitudinal steel f yh : specified yield strength of spiral reinforcement, k 1 : ratio of the average compressive stress to the maximum compressive stress k 2 : ratio of the depth of the resultant compressive force to the depth of neutral axis k 3 : ratio of the maximum compressive stress to the compressive strength of concrete cylinder f c M Exp experiment result from the literatures M n : nominal flexural resistance of the section M LRFD : nominal flexural resistance of the section specified by the LRFD code α 1 : reduction factor of concrete strength β 1 : stress block parameter φu: curvature at ultimate stage φy: curvature at equivalent yield of tension reinforcement u: displacement at ultimate stage y: displacement at equivalent yield of tension reinforcement µ φ : curvature ductility µ d : displacement ductility Table 1 Range of test variables in the databank for HSC beams Test f c f y L L c b d Variable ksi (MPa) ksi (MPa) in (mm) in (mm) in (mm) in (mm) Range (365 - (813- (152 - (102 - (102 - (69-124) 862) 7,239) 3,048) 330) 508) Test a/d ρ ρ ρ ρ / ρ s Variable b % % % Range * *Of the 174 beam tests in the HSC databank, 143 beams (82%) did not have any stirrups.

13 Table 2 Statistical data for over-reinforced and under-reinforced HSC beams Type of Specimen Over- Reinforced Under- Reinforced Number of Test Specimens in Databank M Exp /M LRFD Minimum Maximum Average Standard Deviation < f' c < 16 ksi 55 Tests 32% 12 < f' c < 14 ksi 31 Tests 18% 16 < f' c < 18 ksi 6 Tests 3.4% 18 ksi < f' c 2 Tests 1% 10 < f' c < 12 ksi 80 Tests 46% Figure 1 Distribution of concrete strengths in 174 beam tests with f c > 10 ksi (69 MPa)

14 ρ>ρ b 32 Tests 18% ρ < ρ b 142 Tests 82% Figure 2 Distribution of reinforcement ratios in 174 beam tests with f c > 10 ksi (69 MPa) T Shape 7 Tests 4% Circular Section 8 Tests 5% Triangular Section 11 Tests 6% Square Section 148 Tests 85% Figure 3 Distribution of section types in 174 beam tests with f c > 10 ksi (69 MPa)

15 b ε cu k 3 f c α 1 f c d A s c k 2 c C = k 1 k 3 f c bc β 1 c β 1 c/2 C = α 1 β 1 f c bc d A s Strain Distribution Generalized Stress Block Parameters Rectangular Stress Block Parameters Section Figure 4 Stress block parameters for rectangular sections (A s f su and A s f su are omitted for clarity) 1.8 Concrete Compressive Strength f'c (MPa) Moment Ratio M Exp /M LRFD Rectangular Overreinf. Rectangular Underreinf. Other Shapes Concrete Compressive Strength f'c (ksi) Figure 5 Comparison of the experimental and predicted values of flexural resistance using the LRFD specifications

16 Moment Ratio M Exp /M LRFD Overreinf. Rectangular Underreinf. Rectangular Other Shapes Shear Span to Depth Ratio a/d Figure 6 Effect of shear span to depth ratio on flexural resistance Curvature Ductility Weiss and Shah ( 2001) Sarkar and Adwan (1997) Suzuki (1996) Note: Weiss and Shah (2001) consider the failure of specimen when the load drops to 75% of the maximum capacity. Sarkar and Adwan (1997) use 70% c/d e Ratio Figure 7 Effect of c/d e ratio on curvature ductility

17 6 Displacement Ductility Pam and Kwan (2001) Bernhardt and Fynboe (1986) Note: Pam and Kwan (2001) consider the failure of specimen when the load drops to 85% of the maximum capacity c/d e Ratio Figure 8 Effect of c/d e ratio on displacement ductility c/d e =0.42 Overreinf. Rectangular Underreinf. Rectangular Other Shapes Moment Ratio M Exp /M LRFD c/d e Ratio Figure 9 Comparison of the experimental and predicted values of flexural resistance using the LRFD specifications as function of c/d e ratio

18 Moment Ratio M Exp / M LRFD ρ/ρ b Ratio Figure 10 Comparison of the experimental and predicted values of flexural resistance using the LRFD specifications as function of ρ/ρ b ratio 8 Concrete Strength f c (MPa) Ultimate Concrete Strain ε ult (x10-3 ) Kaminska (2002) Weiss and Shah (2001) Mansur and Chin (1997) Alca and Macgregor (1997) Concrete Strength f c (ksi) Figure 11 Effect of concrete compressive strength on the ultimate compressive strain measured in flexural members