13 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 4 Paper No. 79 REVI AND EVALUATION OF COMBINATION RULES FOR STRUCTURES UNDER BI-DIRECTIONAL EARTHQUAKE EXCITATIO Haluk Sesigur 1, Oguz C. Celik, Feridun Cili 3 SUMMARY Seismic regulations and guidelines for buildings and bridges prescribe simplified combination rules to obtain the maximum structural response under multi-directional earthquake effects. An unfavorable internal force usually develops under the combined effects of an earthquake motion. The spectrum intensity concept can be used to investigate the bi-directional effects of earthquakes on structures. For this purpose, a set of recent and past thirteen earthquakes (M>6) are selected to predict bi-directional effects. Elastic velocity response spectra of these earthquakes are numerically obtained and plotted for damping ratios of ξ =.,.5,., and.5, representing a wide range of undamped and heavily damped structures. Spectrum intensities for both orthogonal directions and for the resultant direction are calculated using a computer program developed for this purpose. Unfavorable response is then calculated by equating the resultant spectrum intensity to principle direction s intensity plus the other direction s contribution as a percentage of the principle component, or equating the resultant spectrum intensity to principle direction s intensity plus a percentage of the other direction s contribution, or vice versa. Although the results obtained are strongly earthquakedependent, sensitivity to damping ratio seems relatively less. Based on this analysis, well-known building regulations are reviewed and evaluated by emphasizing the prescribed combination rules. These numerical results support the combination rules given in the current codes. A safe value of..5 seems reasonable for regular structures. ---------------------------------------------------------------------------------------------------------------------------------- 1 Graduate Research Assistant, Istanbul Technical University, Department of Architecture, Division of Theory of Structures, Taskisla, Taksim, 34437, Istanbul, Turkey, haluk@itu.edu.tr Assoc.Prof., Istanbul Technical University, Department of Architecture, Division of Theory of Structures, Taskisla, Taksim, 34437, Istanbul, Turkey, celikoguz@itu.edu.tr 3 Prof., Istanbul Technical University, Department of Architecture, Division of Theory of Structures, Taskisla, Taksim, 34437, Istanbul, Turkey, cilif@itu.edu.tr
INTRODUCTION A well-designed structure should be capable of resisting earthquake motions equally (preferably) from all possible directions. A structural design based on accounting for the orthogonal earthquake effects separately may result in insufficient member dimensions, as an unfavorable internal force distribution in the structural elements would usually develop under the combined effects of an earthquake motion. Greater column moments, for example, caused by the simultaneous yielding of story beams may be expected when a building is subjected to a skew earthquake loading. On the other hand, if a column is elastic under uniaxial bending moment and axial force, it may be inelastic when subjected to biaxial bending moment and axial force. In fact, in-depth overview of damages to structures during the recent earthquakes has revealed that bi-directional effects must be taken into consideration in seismic analysis. Many studies investigated the bi-directional effects using several methods of analysis. As a common approach, the square- root-of-sum-of-squares (SRSS) procedure is based on the assumption that the actions on an element affected by earthquake excitations in two directions are combined [1]. Cili et al. [] proposed a way of analysis for the multi-directional effects of earthquakes using the characteristics of the selected earthquakes. The 3% and 4% rules are simplified approximations to the SRSS method. Menun and Der Kiureghian [3] presented a response spectrum rule for combining the contributions from three orthogonal components of ground motion to the maximum value of a response quantity. This is called the CQC3 (complete quadratic combination) rule which also offers the most critical orientation of the ground motion components. An explicit formula for the critical response in terms of the responses to single components of ground motion applied separately along the three structural axes and the correlation between these responses are developed by Hernandez and Lopez [4]. They examined the SRSS, 3%, 4%, and IBC rules. Underestimations and overestimations of the critical response are identified for each combination rule and each design spectrum. Current design codes [5 to 9] for buildings and bridges require that members should be designed for 1 percent of the seismic forces in one direction plus 3 percent of the seismic forces in the perpendicular direction (the 3% rule). ATC-3 [1] requires 4 percent to be used rather than 3 percent. In this study, the spectrum intensity concept is used to investigate the bi-directional effects of earthquakes and the effectiveness of this method is examined. The challenge of this approach is to cover both characteristics of an earthquake and to take into account a wide range of periods of structures. This study excludes the effect of vertical component of the ground motion, assuming that horizontal components are the governing components. However, vertical component can be included in a similar way followed here, when the principle component is in the vertical direction. METHOD OF ANALYSIS The peak ground acceleration is commonly used for evaluating the intensity of an earthquake. Maximum amount of elastic deformation in a structure during an earthquake is a very significant indication of ground motion intensity. Integral of a velocity response spectrum over an appropriate period range can be used as an effective measure of ground motion intensity. Housner introduced a measure of ground motion intensity which defines the integral of the velocity response spectrum over the period range T 1 to T sec as the spectrum intensity (1). T SI = S ( ξ, T, t) dt (1) T 1 v
where SI, ξ, T and S v are intensity, damping ratio, fundamental period of the structure and the ordinate of the velocity spectrum respectively. T 1 and T are proposed as.1~.5sec and.5~5.sec. Velocity response spectrum is obtained following the standard procedure given in [1]. This integral can be evaluated for any desired damping ratio (note that Housner recommended using ξ=.); however, to clarify the impact of damping ratio on the evaluation of this integral, damping ratios of ξ=.,.5,., and.5 are chosen. For near-fault motions (L 15 km, T (fundamental period of soil).sec, near the epicenter, stiff soil conditions and minimum focal depth is H=3km), one component of the spectrum intensity might be negligible, and the other could be significant, (e.g. the 1.18.1989 Loma Prieta and the 1.17.1994 Northridge earthquakes). For the far-field motions (L=4~5km, stiff soil conditions, T.5~.5sec and.5~6.sec) if the horizontal components of the intensities are close to each other then both components must be taken into account, (e.g. the 5.18.194 El-Centro earthquake). For the earthquakes with a long period, the unfavorable conditions occur in the resultant direction, (e.g. the 7.6.1964 Mexico-City earthquake). Table 1 presents the data of thirteen recorded accelerograms used in the present study. Three records from Turkey are chosen with great effects on populated areas. A set of recent and past earthquakes is selected namely the 194 El Centro, the 195 Taft, the 1954 Eureka, the 1966 Parkfield, the 1968 Tokachi-Oki, the 1978 Miyagi Ken Oki, the 1989 Loma Prieta, the 199 Erzincan, the 1994 Northridge, the 1995 Kobe, the 1999 Kocaeli, the 1999 Duzce, and the 1999 Chi-Chi earthquakes. Characteristics of the selected earthquakes are given in Table 1. Elastic velocity response spectra of the above mentioned earthquakes are plotted for damping ratios of ξ =.,.5,., and.5, representing a wide range of structures from undamped systems to significantly damped systems (for example seismically isolated systems or structures with energy dissipation members). Table 1. Characteristics of the selected earthquakes a max a max a max Event Date Station M (T) (L) (R) Chi-Chi 9..1999 CHY8-N Ms=7.6 85.4 64.6 87.3 Duzce 11.1.1999 Ambarli Mw=7. 37.4 64.6 375.7 El Centro 5.19.194 El Centro Ms=7. 341.7 1.1 348.84 Erzincan 3.13.199 Erzincan M=6.9 564.95 46.57 65.7 Eureka 1.1.1954 Ferndale City Hall M=6.6 169.5 71 8.9 Tokachi-Oki 5.16.1968 Tohoku University M=7.9 311.7 6. 31.8 Kocaeli 8.17.1999 Yarimca Mw=7. 3. 3. 34.66 Kobe 1.17.1995 Kobe JMA M=6.9 8.6 619. 85.5 Loma Prieta 1.18.1989 16 LGPC Ms=7.1 559 595.7 651.5 Miyagi Ken Oki 1.6.1978 Tohoku University M=7.4 77.5 51.7 35.9 Northridge 1.17.1994 Sylmar-Olive View Ms=6.7 59.5 87 87.35 Parkfield 7.6.1966 Cholame ML=6.1 347.8 45.7 47 Taft 1.7.195 Lincoln School Tun. M=7. 175.9 15.7 15.1 Under a bi-directional earthquake excitation, the unfavorable response of a structure could be calculated in two alternative ways: One way could be to equate the resultant spectrum intensity to principle direction s intensity plus the other direction s contribution as a percentage (α) of the principle component. The second way could be to equate the resultant spectrum intensity to principle direction s intensity plus a percentage (λ 1 ) of the other direction s contribution, or to the weak direction s intensity plus a percentage (λ ) of the principle direction s intensity. α, λ 1, and λ can be numerically calculated for each of a set of selected earthquakes. These coefficients can then be implemented in the combination rules to obtain the maximum response parameters.
Since the combination rules in regulations are usually based on the factor (α) the upper and lower limits of this coefficient would be of special interest to review. Apparently, numerical values are expected to be earthquakedependent. A single degree of freedom system subjected to bi-directional strong ground motions and the evaluation of velocity spectrum intensities are illustrated in Figure 1. Figure 1. Single degree of freedom system under bi-directional earthquake excitation and spectrum intensities for each component and resultant x indicates the earthquake direction that produces maximum internal forces. Spectrum intensities for both orthogonal directions and for the resultant direction are defined as SI x, SI y, and SI xy. Depending on these numerically obtained values, the above mentioned coefficients can be calculated using the following equations: α=si xy /SI x 1 () λ 1 =(SI xy SI x )/SI y (3) λ = (SI xy SI y )/SI x (4) For a given ground motion, the resultant velocity response ( (t) ) of a SDOF system can be calculated using the square root of the squares of each orthogonal component s contribution ( x t), x ( ) ), (5). x r 1( t x r x1 ( t) + x ( ) ( t) = t (5) The maximum values of the above equation for a given damping ratio and natural vibration period give the ordinates of the resultant velocity spectra. This is shown in (6). Note that all computations are performed in linearly elastic conditions. ( Sv ) r = x r ( t) (6) max
COMPUTER PROGRAM To carry out the numerical computations, a special computer program that accepts strong ground motion data for the selected earthquakes, damping ratios and natural vibration periods as input, was coded (Figure ). The spectrum intensities are calculated for the two orthogonal and resultant directions. S ξ, T, t, x g, y g S v x SI, SI y x r x1 ( t) + x ( ) ( t) = t ( Sv ) r = xr ( t) max λ 1 =(SI xy SI x )/SI y λ = (SI xy SI y )/SI x α=si xy /SI x 1 E Figure. Flow-chart of the computer program Maximum internal forces (E xy ) in a structural member under bi-directional effects can be evaluated by the internal forces developed in the principal direction (E x ) plus the other direction s contribution (E y ). For this purpose, α, λ 1, and λ obtained through the spectrum intensity concept are used: Exy Ex + λ1 = E (7a) xy = x y E λ E + E (7b) On the other hand, for design purposes, the principle component can be taken into account only, which leads the resultant response as given in (7c). This is actually the fundamental concept in building y
regulations to take into account the bi-directional earthquake effects on structures. Therefore, α coefficient is of special interest to evaluate the combination rules in building codes. E xy = E + αe (7c) x x For the selected earthquake ground motions (a total of thirteen), velocity response spectra for each of the orthogonal directions and for the resultant direction are shown in Figure 3 following the procedure summarized above. α, λ 1, and λ are numerically calculated for each of the selected earthquakes and the results are also summarized in Table. MULTI-COMPONENT COMBINATION RULES IN CURRENT CODES In seismic design of structures under multi-component earthquake loads, the current codes widely propose either the use of the SRSS rule or an assumed percentage rule. The 3% and 4% rules are linear approximations of the combined response. While the former percentage (3%) is taken in most structures (including buildings), the latter (4%) is recommended in ATC-3 [5] for the analysis of nuclear structures and bridges. The SRSS and 3% rules are prescribed in Eurocode 8 [6] where the unfavorable earthquake loads are obtained by the following combinations: E x, λe y, µe z λe x, E y, µe z λe x, λe y, E z (8a) (8b) (8c) where E x, E y and E z are the internal forces produced by the design earthquake forces in x and y (horizontal components), and z (vertical component) directions respectively. For the design purposes, λ and µ coefficients are chosen as.3 and.. The UBC97 [7] requires the use of either the SRSS rule or the 3% rule, but only for structures having certain types of irregularities. There are two alternative rules prescribed in IBC3 [8]. One of them combines the responses with two horizontal seismic components using the SRSS rule; the result is multiplied by the redundancy coefficient and added to the effect of the vertical component, which is written as a linear term in the design load combination. The other alternative rule combines the responses with two horizontal components using the 3% rule, and adds the effect of the vertical component in the same way. In Turkish earthquake code Specification for structures to be built in disaster areas the 3% rule is recommended [9]. To compare the numerical results obtained from this study and the percentage rules given in most codes, the variation of α, λ 1, and λ coefficients with respect to selected damping ratios is further illustrated in Figures 6a,b,c. For the selected earthquake ground motions, it is observed that the maximum value of the coefficient α is always smaller than.3. The lower and upper bounds of α for ξ=.,.5,., and.5 are.14~.7,.3~.7,.3~.3 and.~.19 respectively. For practical purposes the 3% rule seems to be on the conservative side. Nevertheless some of the cases show that α could be taken as approximately.~.5 in regular structures. λ 1 and λ take values always greater than or in some cases equal to α. From Table and Figure 6b, λ 1 is in the range of.6~.41. Similarly for λ a wider range of.4~.63 is obtained. All of the coefficients are typically earthquake-dependent. In other words, there is no a specific value for the coefficients. Figure 6a,b,c also provide an information about the sensitivity to damping ratio. It is seen that, for most of the earthquake data, this sensitivity is less especially for the damping ratios of 5% or greater.
8 6 4 8 6 4 8 6 4 CHI CHI 9..1999 ξ=% 1 3 4 5 DUZCE 1.11.1999 ξ=% 1 3 4 5 EL CENTRO 5.18.194 ξ=% 1 3 4 5 15 1 5 15 1 5 15 1 5 CHI CHI 9..1999 ξ=% 1 3 4 5 DUZCE 1.11.1999 ξ=% 1 3 4 5 EL CENTRO 5.18.194 ξ=% 1 3 4 5 1 9 6 3 1 9 6 3 1 9 6 3 CHI CHI 9..1999 ξ=5% 1 3 4 5 DUZCE 1.11.1999 ξ=5% 1 3 4 5 EL CENTRO 5.18.194 ξ=5% 1 3 4 5 8 6 4 ERZINCAN 3.13.199 ξ=% 1 3 4 5 15 1 5 ERZINCAN 3.13.199 ξ=% 1 3 4 5 1 9 6 3 ERZINCAN 3.13.199 ξ=5% 1 3 4 5 8 6 4 EUREKA 1.1.1954 ξ=% 15 1 5 EUREKA 1.1.1954 ξ=% 1 9 6 3 EUREKA 1.1.1954 ξ=5% 1 3 4 5 1 3 4 5 1 3 4 5 8 6 4 8 6 4 HACHINOE ξ=% 1 3 4 5 KOCAELI 8.17.1999 ξ=% 1 3 4 5 15 1 5 15 1 5 HACHINOE ξ=% 1 3 4 5 KOCAELI 8.17.1999 ξ=% 1 3 4 5 1 9 6 3 1 9 6 3 HACHINOE ξ=5% 1 3 4 5 KOCAELI 8.17.1999 ξ=5% 1 3 4 5 Figure 3. Velocity response spectra for each orthogonal and resultant directions
8 6 4 8 6 4 8 6 4 8 6 4 KOBE 1.16.1995 ξ=% 1 3 4 5 LOMA PRIETA 1.17.1989 ξ=% 1 3 4 5 MIYAGI KEN OKI 6.1.1978 ξ=% 1 3 4 5 NORTHRIDGE 1.17.1994 ξ=% 1 3 4 5 15 1 5 15 1 5 15 1 5 15 1 5 KOBE 1.16.1995 ξ= 1 3 4 5 LOMA PRIETA 1.17.1989 ξ=% 1 3 4 5 MIYAGI KEN OKI 6.1.1978 ξ=% 1 3 4 5 NORTHRIDGE 1.17.1994 ξ=% 1 3 4 5 1 9 6 3 1 9 6 3 1 9 6 3 1 9 6 3 KOBE 1.16.1995 ξ=5 1 3 4 5 LOMA PRIETA 1.17.1989 ξ=5% 1 3 4 5 MIYAGI KEN OKI 6.1.1978 ξ=5% 1 3 4 5 NORTHRIDGE 1.17.1994 ξ=5% 1 3 4 5 8 6 4 PARKFIELD 6.8.1966 ξ=% 15 1 5 PARKFIELD 6.8.1966 ξ=% 1 9 6 3 PARKFIELD 6.8.1966 ξ=5% 1 3 4 5 1 3 4 5 1 3 4 5 8 6 4 TAFT 7.1.195 ξ=% 15 1 5 TAFT 7.1.195 ξ=% 1 9 6 3 TAFT 7.1.195 ξ=5% 1 3 4 5 1 3 4 5 1 3 4 5 Figure 3. Velocity response spectra for each orthogonal and resultant directions (continued)
E V E N T Table. Spectrum Intensities and combination coefficients ξ=. ξ=.5 ξ=. ξ=.5 SI SI SI SI α λ (cm/sec) 1 λ α λ (cm/sec) 1 λ α λ (cm/sec) 1 λ (cm/sec) α λ 1 λ Chi-Chi L 744.11.3.4 561.88.3.16 447.77.14.13 343.99.13.13.18 T 89.77.1 613.46.15 448..13 36.5 R 15.59 74.7 58.64 387.67 Duzce L 651.7..3.6 451.66..19 344.49..11 6.4.1.7 T 63.34 468.7.18 381.43.1 36.16.6 R 795.86 553. 419.8 33.34 El Centro L 443.4.34.7 8.56.9.8 7.57.19.1.8 149.78.11.13.8 T 497.33.4 89.31.7 189.1 15.9 R 614.95 367.55 47.55 166.69 Erzincan L 868.5.14.17.31 643.54.11.15.36 479.95.13.17.37 35.38.13.18.39 T 716.14 483.83 36.41 6.3 R 986.85 715.45 54.1 399.5 Eureka L 318.97.3.8 6.6.7.7.8 159.19.3.4.6 17.45.1..7 T 334.16.7 4.83 154.4 1.3 R 44.96 87.91 195.86 19.83 Hachinoe L 455..7.4 48.5.7.1 165.63.6.11 13..16.9 T 475.6.3 31.81.1 199.7.9 141.7.8 R 585.16 331.5 18.38 153.9 Kocaeli L 978.3.3.1 575..1.19 333.4.17.17.17 6.16.14.15.3 T 993.7.1 588.3.19 33.3 187.69 R 1. 698.7 388.49 35.3 Kobe L 873.3..6.37 636.7..4.37 488.99.18.1.31 395.3.19.1.9 T 74.44 53.44 47.51 356.61 R 165.69 766.39 578.41 47.4 Loma Prieta L 1388.38.4.9.63 99.93.3.7.6 69.93.3.6.57 38.6.5.9.51 T 557.88 398.1 85.96 5.3 R 1438.67 117.6 646.91 4.35 Miyagi Ken Oki L 379.33.4.5.8 34.46.19.16 181.11.17.13 15.5.8.9.16 T 364.76 45.66.15 189.9.1 138.84 R 47.8 8.3 1.5 16.74 Northridge L 766.15.41.1 568.68.38.15 411.19.36.5 94.18.4.4 T 13.91.15 776.45.11 68.85.3 468.33. R 119.1 864.9 69.41 479.61 Parkfield L 39.9.9.1.38 159.97.8.1.5 113.4.17.17.18 1.77.18.17 T 168.58 13.75 11.19 11.67.17 R 6.88 173.5 13.91 118.77 Taft L 189.86.34.4 115..3.17 84.56.31.19 65.7.34.1 T 19.89.1 14.3.14 98.9.16 77.71.18 R 65.16 159.4 115.15 91.4
α.5.4.3. 4% rule (ATC-3) (a) 3% rule (FEMA,UBC97,CALTRA,IBC3) ξ=. ξ=.5 ξ=. ξ=.5.1. Chi-Chi Duzce El Centro Erzincan Eureka Hachinoe Kocaeli Kobe Loma Prieta Miyagi Ken Oki Northridge Parkfield Taft.7.6.5 ξ=. ξ=.5 ξ=. (b).4 ξ=.5 λ 1.3..1. Chi-Chi Duzce El Centro Erzincan Eureka Hachinoe Kocaeli Kobe Loma Prieta Miyagi Ken Oki Northridge Parkfield Taft.7.6.5 ξ=. ξ=.5 ξ=. (c).4 ξ=.5 λ.3..1. Chi-Chi Duzce El Centro Erzincan Eureka Hachinoe Kocaeli Kobe Loma Prieta Miyagi Ken Oki Northridge Parkfield Taft Figure 6. α, λ 1 and λ coefficients.
CONCLUSIO This study proposes a basis which uses the characteristics of an earthquake motion, for the bi-directional analysis of structures. Among several methods of analyses (e.g. the CQC, 3%, 4%), the Housner s spectrum intensity concept is used to investigate bi-directional earthquake effects. For this purpose, velocity response spectra are plotted for a set of thirteen ground motions for structures having damping ratios of ξ=.,.5,.,.5, and first elastic vibration periods up to 5 sec. This selection enables to evaluate the bidirectional response of structures with undamped systems to significantly damped systems (e.g. for seismically isolated systems or systems with seismic energy dissipation devices). Spectrum intensities for both orthogonal directions and for the resultant direction are obtained. Relevant coefficients commonly used in the percentage rules recommended in codes are numerically calculated using the velocity spectrum intensities. Well known regulations are reviewed and evaluated by emphasizing the proposed combination rules. From the numerical results obtained in this study, it is concluded that the combination rules given in the current codes are conservative (especially for buildings with larger importance) as a safe value of α=. ~.5 seems reasonable for structures having regular load carrying systems. REFERENCES 1. R.W. Clough and J. Penzien, Dynamics of Structures, McGraw-Hill, New York, 1993.. Cili F., Celik O.C., Sesigür H., Behavior.of reinforced concrete buildings subjected to multidirectional earthquakes and related codes, Proceedings of the 3rd National Conference on Earthquake Engineering, İstanbul, 1995: 1-31. 3. Menun C, Der Kiureghian A. A replacement for the 3%, 4% and SRSS rules for multicomponent seismic analysis Earthquake Spectra 1998; 14:153-56. 4. Hernández, J.J., López, OA. Evaluation of combination rules for peak response calculation in three-component seismic analysis Earthquake Engineering and Structural Dynamics 3; 3:1585-16. 5. ATC-3. Improved Seismic Design Criteria for California Bridges: Provisional Recommendations, Applied Technology Council 1996. 6. European Committee for Standardization (ECS). Eurocode 8: Design Provisions for earthquake resistance of structures. Brussels, Belgium. 1998. 7. International Conference of Building Officials (ICBO). 1997 Uniform Building Code. Structural Engineering Design Provisions, Vol., Whittier, CA, 1997. 8. International Code Council, Inc. (ICC). 3 International Building Code (IBC), Birmingham, AL, 3. 9. Ministry of Public Works and Settlement, Specification for Structures to be Built in Disaster Areas, 1998.