Energy Dissipation in RC Beams Under Cyclic Load

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(a) Nmal and Darwin (12); and (b) Scribner and Wight (16) o, - aa(V;) ..... Draper, N. R., and Smith, H., Applied Regression Analysis, 2nd ed., John Wiley. & Sons ...
ENERGY DISSIPATION IN EC BEAMS UNDER CYCLIC LOAD

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By David D a r w i n , 1 M. ASCE and Charles K. N m a i 2 ABSTRACT: The development and application of an "energy dissipation index" to characterize reinforced concrete beams under cyclic load is described. The index is designed to provide an objective measure of the response of beams with different geometries and reinforcement ratios subjected to severe cyclic loading. The effects of flexural reinforcement, transverse reinforcement, concrete strength, and load history are considered. The index is used to evaluate the results of five major experimental investigations. The energy dissipation index may be applied to both research and design. The analysis of the experimental work shows that to improve the cyclic performance of reinforced concrete beams, it is more efficient to decrease the shear stress, by increasing the beam width, than to increase the transverse reinforcement. The analysis also indicates that increasing the ratio Of positive to negative reinforcement may not markedly improve the cyclic performance of a beam. Recommendations are made for the design of reinforced concrete beams that may be subjected to severe cyclic loading. INTRODUCTION

The response of a reinforced concrete member to cyclic loading depends upon the member geometry, the reinforcement details, and the loading history. In an attempt to determine the parameters that improve member behavior, researchers have used test specimens with differing geometries and reinforcement details. However, because of differences in test specimens or loading history, it is often difficult to compare the results obtained in these tests. A good example of the difficulties involved concerns the use of energy dissipation capacity as a measure of member performance. For a beam subjected to cyclic loading of the type expected in an earthquake, the use of increased flexural reinforcement (positive or negative) may increase the energy dissipation capacity of the member. The use of increased positive reinforcement has been specifically advocated for this purpose (11). However, the additional reinforcement will also increase the severity of the loading in the form of higher induced m o m e n t s and shears. The overall result may be n o net gain in performance a n d in some cases the member may exhibit poorer performance, in spite of its higher energy dissipation capacity. Clearly, a comparison of member performance should account for the induced loading as well as the member strength and energy dissipation capacity. In an attempt to make comparisons between members with different geometries subjected to widely varying load histories, Gosain, Brown, 'Prof, of Civ. Engrg. and Dir., Struc. Engrg. and Materials Lab., Univ. of Kansas, Lawrence, KS 66045. 2 Grad. Student, Purdue Univ., West Lafayette, IN; formerly, Grad. Student, Univ. of Kansas, Lawrence, KS 66045. Note.—Discussion open until January 1, 1987. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on August 5, 1985. This paper is part of the Journal of Structural Engineering, Vol. 112, No. 8, August, 1986. ©ASCE, ISSN 0733-9445/86/0008-1829/$01.00. Paper No. 20838. 1829

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and Jirsa (8) developed the "modified work index," l'w, as a measure of the severity of loading for a member. Their procedure takes into account the member geometry and strength, but does not use the actual load history. Instead, I'w is obtained from the product of the number of cycles and the ratio of maximum deflection to yield deflection or deflection ductility, \x,, coupled with modifications for the shear span-to-depth ratio and axial load. Even with the modifications, l'w does not produce a strong correlation with the test data, indicating that a better measure of member response to cyclic load is desirable. This paper discusses the development and application of the "energy dissipation index," D ; , which is designed to provide an objective measure of member response. The new index is used to study the response of beams of varying geometries and reinforcement ratios subjected to severe cyclic loading. The effects of flexural reinforcement, transverse reinforcement, concrete strength, and load history are considered. The energy dissipation index is used to evaluate the results of five major experimental investigations. Recommendations are made for the design of reinforced concrete beams that may be subject to severe cyclic loading. The full details of this work are presented in Ref. 12. ENERGY DISSIPATION INDEX

One of the most important aspects of structural performance under seismic loading is the ability of a structure to adequately dissipate energy. However, when comparing the results from different test specimens, the total energy dissipated must somehow be normalized with respect to the prototype structure. In an adequately designed lateral load resisting frame under severe lateral loading, plastic hinges (inelastic zones) will form in the beams rather than in the columns. The moments and shears to which the beams are subjected are a function of the flexural strength of the members; the higher the strength, the greater the imposed loads. The energy dissipation index, D,, is designed as a realistic measure of member performance in an actual structure. The index normalizes the energy dissipation capacity of a beam with respect to the elastic energy stored in the beam at yield, which may be approximated as one-half of the product of the yield load and yield deflection, 0.5 PyAy (Fig. 1). The elastic energy is characteristic of the design capacity of a beam and is dependent upon the beam geometry as well as the amount of flexural reinforcement. Before proceeding, it is useful to address the dichotomy that exists in seismic design procedures for reinforced concrete. Structures are typically proportioned to support equivalent static lateral loads (17). However, they are detailed (shear, anchorage, relative strength of beams and columns) based on forces that result from imposed displacements (1,17). This approach to detailing is aimed at insuring adequate toughness under cycles of load. Of course, the response of a structure to an earthquake is highly complex, and neither the imposed force nor the imposed displacement representation is by itself correct. However, procedures that are slanted toward the "imposed displacement concept" are more likely to insure toughness in individual members. For this reason, D, was de1830

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dices greater than or equal to SO, 81 % had values of D,- in excess of 40, and 100% had values of D, greater than 35. Assuming that vs/v„, = 1.0, these three values of D, imply maximum shear stresses of 3.3V/], 3.9\/J!c, and 4.3V7^ r respectively, based on Eq. 11 (used here as a guide). Twelve of the 16 beams had vs > vm (14 had vs > 0.95um), and 13 of the 16 had avm< 4.3V/I. Of the 19 beams with \x, s 3.9 that did not sustain five complete cycles of an inelastic loading, 15 had either vs < v,„ or v„, > 4.3v7c • Of these 15 beams, six had both vs < vm and vm > 4.3V/c • Four beams that did not sustain five cycles had vs > v„, and vm < 4.3v7^ • However, these beams, from the study by Ma, Bertero, and Popov (11), exhibited severe bond deterioration within the joint, and are not applicable to establishing design criteria for flexural members alone. Based on this analysis, it is recommended that the current procedure (1,17) of disregarding the concrete contribution to shear capacity, Vc, should be continued for flexural members subjected to severe seismic loading. The result of this requirement is to require the design stirrup capacity, $VS, to be greater than or equal to the factored shear, Vu, (or §vs a Vu/bd). In addition, design values of D, > 35 are desirable in terms of member behavior. Df = 35 represents a maximum shear stress of approximately 4.6V7c for §vs/vm - 1 {vs/vm = 1.18). This figure is equivalent to a shear stress of about 7\/J7c on the core of the section, the upper limit recommended by Gosain, Brown, and Jirsa (8). Higher values of applied shear could be sustained with added shear steel. However, Eq. 11 suggests that vs would have to be increased more rapidly than v,„ (or vu) to maintain a desired value of D,. For example, a 30% increase in vm from 4.6v7c to 6.0V/c / using Eq. 11 to maintain a value of D, = 35, requires a 68% increase in vs from 1.18um to 1.98z;,„. The required increases in shear reinforcement to maintain D,, and thus cyclic performance, become unrealistic very rapidly. It is much more efficient to increase the beam width, with no increase in reinforcing steel. This approach maintains the moment capacity, while reducing the induced shear stress. It also reduces congestion of the reinforcement, while having a minimal effect on the total volume of concrete in a building. Although data is limited, an increase in concrete strength is another option to consider, to be used alone or in conjunction with an increase in vs and a decrease in v,„. CONCLUSIONS

The following conclusions are based on the analyses presented in this paper: 1. The energy dissipation index, D,, is a useful conceptual tool. It appears to provide a consistent measure of beam performance under cyclic loading, and should prove useful in applications to structural design. 2. D, is primarily controlled by the maximum shear stress, the concrete strength, and the nominal capacity of the shear reinforcing. It is also sensitive to modifications in detailing, anchorage slip, and beam geometry. 1843

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3. The performance of reinforced concrete beams subjected to cyclic loading will improve with a decrease in the maximum shear stress a n d an increase in the transverse steel capacity. Although results are limited, performance also appears to improve with an increase in concrete strength. To improve performance, it is more efficient to reduce the maximum shear stress through the use of wider beams with n o increase in the longitudinal reinforcement t h a n to increase the transverse reinforcement. 4. A sizeable portion of the increased energy dissipation capacity obtained with an increase in the ratio of positive to negative reinforcement at the face of the support m a y be required by a structure to satisfy the added energy dissipation d e m a n d caused by the additional positive reinforcement. 5. Special reinforcement schemes can improve beam performance significantly, while b o n d slip within a joint will markedly reduce beam performance. 6. The shape a n d size of a beam may affect its performance u n d e r cyclic loading. Beams with low width-to-effective d e p t h ratios, b/d, a n d large effective depths appear to be less efficient w h e n compared to beams with higher b/d values and smaller effective depths. 7. A value of 35 for D, appears to provide adequate performance under cyclic loading. This corresponds to a maximum shear stress, v„ = Vu/bd, of approximately 4.6V/J for VS = Vu. ACKNOWLEDGMENTS

This research was supported by the National Science Foundation u n der NSF Grant PFR 79-24696. Numerical calculations w e r e performed on the Honeywell 66/60 computer system of the University of Kansas Academic Computer Service. APPENDIX I.—REFERENCES

1. ACI Committee 318, "Building Code Requirements for Reinforced Concrete (ACI 318-83)," American Concrete Institute, Detroit, Mich., 1983, 111 pp. 2. ACI-ASCE Committee 426, "The Shear Strength of Reinforced Concrete Members," Journal of the Structural Division, ASCE, No. ST6, June, 1973, pp. 1091-1176. 3. Bertero, V. V., Popov, E. P., and Wang, T. Y., "Hysteretic Behavior of Reinforced Concrete Flexural Members with High Shear," Report No. EERC 74-9, Earthquake Engineering Research Center, Univ. of California, Berkeley, Calif., Aug., 1972, 126 pp. 4. Bertero, V. V., and Popov, E. P., "Seismic Behavior of Ductile Moment-Resisting Reinforced Concrete Frames," Reinforced Concrete Structures in Seismic Zones, Publ. SP-53, American Concrete Institute, Detroit, Mich., 1977, pp. 247-291. 5. Bresler, B., and Scordelis, A. C , "Shear Strength of Reinforced Concrete Beams," Journal of the American Concrete Institute, Vol. 60, No. 1, Jan., 1963, pp. 51-74. 6. Brown, R. H., and Jirsa, J. O., "Reinforced Concrete Beams under Load Reversals," Journal of the American Concrete Institute, Vol. 68, No. 5, May, 1971, pp. 380-390. 1844

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7. Draper, N. R., and Smith, H., Applied Regression Analysis, 2nd ed., John Wiley & Sons, Inc., New York, N.Y., 1981, pp. 241-249. 8. Gosain, N. K., Brown, R. H., and Jirsa, J. O., "Shear Requirements for Load Reversals on RC Members," Journal of the Structural Division, ASCE, Vol. 103, No. ST7, July, 1977, pp. 1461-1476. 9. Hwang, T. H., and Scribner, C. F., "R/C Member Cyclic Responses during Various Loadings," Journal of Structural Engineering, ASCE, Vol. 110, No. 3, Mar., 1984, pp. 477-489. 10. Lee, D. L. N., Wight, J. K., and Hanson, R. D., "RC Beam-Column Joints under Large Load Reversals," Journal of the Structural Division, ASCE, Vol. 103, No. ST12, D e c , 1977, pp. 2337-2350. 11. Ma, S. M., Bertero, V. V., and Popov, E. P., "Experimental and Analytical Studies on the Hysteretic Behavior of Reinforced Concrete Rectangular and T-Beams," Report No. EERC 76-2, Earthquake Engineering Research Center, Univ. of California, Berkeley, Calif., May, 1976, 254 pp. 12. Nmai, C. K., and Darwin, D., "Cyclic Behavior of Lightly Reinforced Concrete Beams," Structural Engineering and Engineering Materials SM Report No. 12, Univ. of Kansas Center for Research, Inc., Lawrence, Kans., June, 1984, 129 pp. 13. Park, R., Kent, D. C , and Sampson, R. A., "Reinforced Concrete Members with Cyclic Loading," Journal of the Structural Division, ASCE, Vol. 98, No. ST7, July, 1972, pp. 1341-1360. 14. Paulay, T., "Simulated Seismic Loading of Spandrel Beams," Journal of the Structural Division, ASCE, Vol. 97, No. ST9, Sept., 1971, pp. 2407-2419. 15. Popov, E. P., Bertero, V. V., and Krawinkler, H., "Cyclic Behavior of Three Reinforced Concrete Flexural Members with High Shear," Report No. EERC 72-5, Earthquake Engineering Research Center, Univ. of California, Berkeley, Calif., Oct., 1972, 78 pp. 16. Scribner, C. F., and Wight, J. K., "Strength Decay in Reinforced Concrete Beams under Load Reversals," Journal of the Structural Division, ASCE, Vol. 106, No. ST4, Apr., 1980, pp. 861-875. 17. Uniform Building Code, International Conference of Building Officials, Whittier, Calif., 1985, 817 pp. 18. Wight, J. K., and Sozen, M. A., "Strength Decay of RC Columns under Shear Reversals," Journal of the Structural Division, ASCE, Vol. 101, No. ST5, May, 1975, pp. 1053-1065. APPENDIX II.—NOTATION

The following symbols are used in this paper: As A's Av b D, d

= area of top reinforcing steel at face of joint; = area of bottom reinforcing steel at face of joint; = area of shear reinforcement; = width of beam web; = energy dissipation index; = effective depth (distance from bottom of beam to centroid of top reinforcement); di = effective depth (distance from top of beam to centroid of bottom reinforcement); E = total energy dissipated for cycles in which P„, g 0.75Py; f'c = compressive strength of 6 x 12 in. (152 x 305 mm) concrete cylinders; fvy = yield strength of shear reinforcement; L = span between two columns; M + = positive moment capacity at a column face; 1845

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M~ Pm Py P'y r s V Vc Vm Vn Vs Vu v,„ vn vs a (3 Ay Ay

= = = = = = = = = = = = = = = = = = =

negative m o m e n t capacity at a column face; m a x i m u m b e a m load; b e a m load at yielding of t o p flexural reinforcement; b e a m load at yielding of b o t t o m flexural reinforcement; correlation coefficient; stirrup spacing; shear force d u e to lateral deformation = (M^ + M^/L; nominal shear strength provided by concrete; maximum shear force; nominal shear strength; nominal shear strength provided by shear reinforcement; factored shear force; m a x i m u m shear stress = Vm/bd; n o m i n a l shear stress = V„/bd; nominal shear stress provided by shear reinforcement; coefficient of vs used in regression analysis; coefficient of f'c u s e d in r e g r e s s i o n analysis; load-point deflection of yielding of top flexural reinforcement; load-point deflection at yielding of bottom flexural reinforcement; 4> = strength reduction factor = 0.85 for shear; |x = displacement ductility factor; and p = flexural reinforcement ratio.

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