EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 26, 849 —858 (1997) EVALUATION OF BUILDING SEPARATION DISTANCE REQUIRED TO PREVENT POUNDING DURING STRONG EARTHQUAKES JOSEPH PENZIEN* International Civil Engineering Consultants, Inc., Berkeley, CA 94704, U.S.A. SUMMARY Presented herein is an analytical procedure for evaluating the separation distance required between two buildings to prevent pounding during strong earthquakes. The procedure is based on equivalent linearization of non-linear hysteretic behaviour and application of the well-known CQC method of weighting normal mode responses. The numerical results obtained are compared with corresponding results obtained using the well-known SRSS and ABS methods. ( 1997 by John Wiley & Sons, Ltd. Earthquake Engng. Struct. Dyn., 26, 849 —858 (1997). No. of Figures: 8. KEY WORDS : No. of Tables: 0. No. of References: 5. pounding of buildings INTRODUCTION Major damage to buildings in metropolitan areas is often caused by the pounding action of one against another during strong motion earthquakes. To avoid such damage, new buildings located adjacent to each other should be provided with adequate separation distances. To specify minimum distances for this purpose, a rational analysis procedure is needed. It is this need that stimulated the author to develop the procedure presented herein. Hopefully, it can provide a basis for developing a code provision specifying minimum separation distances. LINEAR RESPONSE Consider two buildings separated by a distance * and having heights H and H as shown in Figure 1. If both 1 2 buildings are subjected to the same ground acceleration l̈ (t) in the y direction, the normal mode equations gy of motion for the ith and jth modes of Buildings 1 and 2 will be ¸ ½® (t)#2u m ½Q (t)#u2 ½ (t)"! 1i l̈ (t) 1i 1i 1i 1i 1i 1i M gy 1i ¸ ½® (t)#2u m ½Q (t)#u2 ½ (t)"! 2j l̈ (t) 2j 2j 2j 2j 2j 2j M gy 2j respectively, in which H1 P0 ¸ , 1i m (x)/ (x) dx, 1 1i H1 P0 M , 1i (1) (2) m (x)/ (x)2 dx, i"1, 2, 2 , N 1 1i 1 (3) * Correspondence to: J. Penzien, International Civil Engineering Consultants, lnc., 1995 University Avenue, Suite 119, Berkeley, CA 94704, U.S.A. CCC 0098—8847/97/080849—10$17·50 ( 1997 by John Wiley & Sons, Ltd. Received 22 August 1996 Revised 27 February 1997 850 J. PENZIEN Figure 1. Two buildings of heights H and H having a separation distance *: H )H 1 2 1 2 H2 P0 ¸ , 2j m (x)/ (x) dx, 2 2j H2 P0 M , 2j m (x)/ (x)2 dx, j"1, 2, 2 , N 2 2j 2 (4) where N and N denote the number of lower modes to be considered for Buildings 1 and 2, respectively; see 1 2 Figure 4(a) In formulating a procedure for setting the separation distance * required to avoid seismic pounding, the inward relative displacement l (t) between the two buildings at the top level of the shorter building (Bldg. 1) R as expressed by N1 N2 l (t)" + / (H )½ (t)! + / (H )½ (t) (5) R 1i 1 1i 2j 1 2j i/1 j/1 is the response quantity of interest. When l (t) reaches the value *, pounding will occur at level x"H . R 1 Equation (5) can be expressed in the equivalent form (N1`N2) l (t)" + A ½ (t) m m R m/1 (6) in which A ,/ (H ), ½ (t),½ (t), m"1, 2, 2 , N (7) m 1m 1 m 1m 1 (H ), ½ (t),½ (t), m"(N #1), (N #2) , 2 , (N #N ) (8) A ,!/ m 2,m~N1 1 1 1 2 m 2,m~N1 1 Using the CQC method,1,2 the expected maximum absolute value of l (t) can be expressed in the form R ¸ ¸ (N1`N2) (N1`N2) 1@2 m n o S (m , u )S (m , u ) + + A A (9) Dl (t)D " m n M M mn d m m d n n R .!9 m n m/1 n/1 where C D ¸ ,¸ , M ,M , m"1, 2, 2 , N m 1m m 1m 1 ¸ ,¸ , M ,M , m"(N #1), (N #2), 2 , (N #N ) m 2,m~N1 m 2,m~N1 1 1 1 2 (10) (11) and 8Jm m (m #rm )r3@2 m n m n o "o " mn nm (1!r2)2#4m m r(1#r2)#4(m2 #m2)r2 n m m n Earthquake Engng. Struct. Dyn., 26, 849 —858 (1997) (12) ( 1997 by John Wiley & Sons, Ltd. EVALUATION OF BUILDING SEPARATION DISTANCE Figure 2. Plots of o mn 851 versus r for discrete values of m as given by equation (16) in which r,u /u , u *u (13) m n n m u ,u , m ,m , m"1, 2, 2 , N (14) m 1m m 1m 1 u ,u , m "m , m"(N #1), (N #2), 2 , (N #N ) (15) m 2,m~N1 m 2,m~N1 1 1 1 2 The quantity S (m , u ) in equation (9) representing the earthquake ground acceleration l̈ (t) is the elastic $ m m gy displacement response spectral value for damping ratio m and frequency u . If all damping ratios m m m (m"1, 2, 2 , N #N ) equal the same value m, equation (12) reduces to the simplified form m 1 2 8m2(1#r)r3@2 o "o " mn nm (1!r2)2#4m2r(1#r)2 (16) which is plotted in Figure 2 as a function of r for discrete values of m. Since the value of Dl (t)D is usually dominated by the first-mode responses of Buildings 1 and 2, the R .!9 higher-mode responses can be ignored, in which case equation (6) simplifies to in which 2 l (t)"/ (H )½ (t)!/ (H )½ (t)" + B ½ (t) R 11 1 11 21 1 21 s s s/1 The CQC expression for Dl (t) D R .!9 B ,/ (H ), B ,!/ (H ) 1 11 1 2 21 1 ½ (t),½ (t), ½ (t),½ (t) 1 11 2 21 , equation (9), then reduces to C (18) D 2 2 ¸¸ 1@2 Dl (t) D " + + B B s t o S (m , u )S (m , u ) s t M M st $ s s $ t t R .!9 s t s/1 t/1 ( 1997 by John Wiley & Sons, Ltd. (17) (19) Earthquake Engng. Struct. Dyn., 26, 849 —858 (1997) 852 J. PENZIEN If the first-mode frequencies of Buildings 1 and 2 are widely separated and the corresponding damping ratios are small, the cross terms in equation (19) become negligible. In this case Dl (t) D can be approximated by R .!9 the more conservative SRSS expression C D ¸2 2 1@2 Dl (t) D " + B2 s S (m , u )2 R .!9 s M2 $ s s s s/1 An even more conservative relation is given by the ABS (absolute sum) expression (20) 2 ¸ Dl (t) D " + B s S (m , u ) (21) R .!9 sM $ s s s s/1 The above procedure, based on linear response, has been developed in a previous paper3 using random vibration theory; however, the same results can be obtained by a deterministic approach2 which has been adopted in the above development. NON-LINEAR HYSTERETIC RESPONSE The above procedure for evaluating Dl (t) D is based on linear elastic response of both Bldgs. 1 and 2; R .!9 however, since they normally will respond inelastically during strong earthquakes, one must carefully examine the applicability of this procedure in such cases. Similar to an elastic case, Dl (t) D will be R .!9 controlled primarily by first-mode type of responses, even though inelastic deformations take place. Because of the commonly used weak-girder/strong-column principle of design, one can assume the shape of this type of response to be linear as shown in Figure 3, so that /(x)"x/H. The corresponding generalized SDOF system representing such response is shown in Figure 4(b). The generalized coordinate ½ (t) corresponds to % the displacement at the top of the building relative to the moving ground. The generalized mass M will be given by the second of equations (3), whether or not the response remains elastic, so long as the linear mode-shape function /(x) does not change significantly with respect to time. When responding in the linear elastic range as represented in Figure 4(a), the generalized first-mode spring constant K will equal u2M and its generalized dashpot coefficient C will equal 2Mum, where m is the first-mode damping ratio. When responding in the inelastic range, the equivalent linearized SDOF system, as represented in Figure 4(b), can be used to predict maximum response, provided its generalized spring constant K and % generalized damping coefficient C are selected properly. Assuming a bilinear force-displacement hysteretic % relation for the non-linear generalized spring as shown in Figure 5, the equivalent linearized spring constant K and damping ratio m can be obtained using the following procedure4: % % (1) The values of the equivalent-linearized stiffness K and damping ratio m depend upon the value of ½ , % % 6 as expressed by which is a weighted value of D½ (t) D % .!9 ½ "aD½ (t) D (22) 6 % .!9 The value of a which yields the optimum linearized system depends upon the characteristics of the ground-motion input v (t); however, for typical seismic inputs, the value 0·65 gives reasonably realistic gy results. The value of K , as shown by the slope of the dashed line 402 in Figure 5, is given by % c#b(k!c) K "K (23) % k C D in which c,1/a"D½ (t) D /½ % .!9 6 b,KM /K Earthquake Engng. Struct. Dyn., 26, 849 —858 (1997) (24) (25) ( 1997 by John Wiley & Sons, Ltd. EVALUATION OF BUILDING SEPARATION DISTANCE 853 Figure 3. Assumed linear displacement shape for both elastic and inelastic response Figure 4. Generalized first-mode SDOF systems representing elastic and inelastic responses of the building in Figure 3 Figure 5. Bilinear force—displacement hysteretic relation for non-linear generalized spring and k is the maximum ductility demand defined by k,D½ (t) D /½ % .!9 : The period ¹ of the equivalent-linearized system can now be obtained using % ¹ "2nJM/K % % ( 1997 by John Wiley & Sons, Ltd. (26) (27) Earthquake Engng. Struct. Dyn., 26, 849 —858 (1997) 854 J. PENZIEN (2) The viscous damping ratio m is evaluated as the sum of that ratio m considered appropriate for % %1 response in the linear elastic range, usually assigned the value 0·05, and that ratio m which will %2 produce an energy loss E during a full cycle of oscillation at frequency u "JK /M and amplitude 7 % % ½ that is equal to the hysteretic energy loss E during the same full cycle of oscillation. It is easily 6 ) shown2 that E is given by 7 (28) E "2nm Mu2½2"2nm K ½2 %2 % 6 % 6 7 %2 in which K ½2 is the sum of the areas under triangles 025 and 046 in Figure 5. The hysteretic energy % 6 loss E is the area under parallelogram 1234 in this same figure, which can be evaluated using ) E "4½ (½ !½ ) (K!KM ) (29) ) 6 6 : Setting the right-hand side of equation (28) equal to the right-hand side of equation (29), making use of equations (22)—(26), and then solving for m yields %2 2 (u!c) (1!b)c m " (30) %2 n k[c#b(k!c)] G H Specifying m equal to 0·05 and adding this value to m as given by equation (30) and using %1 %2 c"1/a"1/0·65"1·54, functions of m vs. k can be generated for discrete values of b as shown in % Figure 6. Having established the equivalent-linearized stiffness K and damping ratio m for each building, equa% % tion (12) can be used to generate the cross-term o , which equals o , by introducing the corresponding %12 %21 equivalent-linearized value of r define by in which ¹ r , %11 , % ¹ %21 ¹ *¹ %21 %11 (30) ¹ "2nJM/K (31) % % The high values of o , obtained for typical values of ductility demand, result in large values for the % cross-terms as compared with the corresponding values obtained for low-damped (o"0·05) linear elastic response. For buildings having fundamental periods in the range 0·5(¹(3·0 s, the maximum values of the generalized displacements ½ (t) and ½ (t) for the non-linear hysteretic model, Figure 4(b), will be 11 21 approximately the same as for the linear elastic models, Figure 4(a); thus, the elastic displacement response spectral values can be used in equation (19) when evaluating Dl (t) D for the non-linear models. Often, as in 3 .!9 the 1994 Uniform Building Code 5 (Figure 16—3, pp. 2—42), these displacement response spectral values, S (m, ¹ ), are specified to be proportional to ¹ over the above-mentioned period range. In this case, for $ m "m "m, the following relation is satisfied: 11 21 ¹ S (m, ¹ )" 21 S (m, ¹ ) (32) $ 21 $ 11 ¹ 11 Assuming a uniform mass distribution over the height of each building, the linear mode shape, /(x)"x/H, yields 3/2 for all values of ¸ /M and ¸ /M in equation (19); and, the values of B and B in this same s s t t 1 2 equation equal 1 and H /H , respectively. The ratio H /H can be expressed in terms of the period ratio 1 2 1 2 ¹ /¹ using available empirical relations. For example, the above-mentioned Uniform Building Code 1 2 specifies that ¹ for all buildings can be determined using the approximate relation ¹"C H3@4 t Earthquake Engng. Struct. Dyn., 26, 849 —858 (1997) (33) ( 1997 by John Wiley & Sons, Ltd. 855 EVALUATION OF BUILDING SEPARATION DISTANCE in which the coefficient C depends upon type of building (C equals 0·035 for steel moment-resisting frames, t t 0·030 for reinforced concrete moment-resisting frames and eccentrically braced frames, and 0·020 for all other buildings) and where H is measured in units of feet. In this case, the period ratio ¹ /¹ is expressed by 11 21 (¹ /¹ )"(C /C ) (H /H )3@4 (34) 11 21 t1 t2 1 2 This equation can be converted to the equivalent form (H /H )"(C /C )`4@3(¹ /¹ )~4@3 (35) 1 2 t2 t1 21 11 Making use of equations (32) and (35) and the above stated values of B , B , ¸ /M , and ¸ /M , Eq. (19) 1 2 s s t t simplifies to the CQC form C A B A B A B A B D C 4@3 ¹ ~1@3 C 8@3 ¹ ~2@3 1@2 Dl (t) D 3 R .!9 " 1!2o t2 21 21 # t2 %12 C S (m, ¹ ) 2 ¹ C ¹ $ 11 t1 11 t1 11 after dividing both sides by S (m, ¹ ). The corresponding SRSS and ABS relations are $ 11 G C A BA B D C A BA B D 3 C 8@3 ¹ ~2@3 1@2 21 1# t2 2 C ¹ t1 11 Dl (t) D R .!9 " S (m, ¹ ) $ 11 3 C 4@3 ¹ ~1@3 21 1# t2 2 C ¹ t1 11 (36) (37) (38) respectively. Equations (36)—(38) apply to that case where the building fundamental period ¹ is proportional to H3@4 as indicated in Equation (33), i.e. buildings of the same height and structural type have the same period. Another case of interest is where both buildings have the same height (H "H ) but they have different fundamental 1 2 periods due to structural differences. As for the previous case, assume that the maximum values of ½ (t) and 11 ½ (t) for the non-linear hysteretic model are approximately the same as for their linear elastic models and 21 that the displacement response spectrum S (m, ¹ ) is proportional to ¹ over the period range 0·5(¹(3·0. $ Again ¸ /M and ¸ /M both equal 3/2 due to the assumed linear mode shape and uniform mass distribution; s s t t however, in this case B "B "1. Letting m "m "m, Equation (19) now simplifies to the CQC form 2 1 1 2 C A B A BD Dl (t) D ¹ ¹ 2 1@2 3 R .!9 " 1!2o 21 # 21 %12 ¹ S (m, ¹ ) 2 ¹ $ 11 11 11 (39) if dividing both sides by S (m, ¹ ). The corresponding SRSS and ABS relations are $ 11 G C A BD C A BD 3 ¹ 2 1@2 1# 21 2 ¹ 11 Dl (t) D R .!9 " S (m, ¹ ) ¹ 3 $ 11 1# 21 ¹ 2 11 (40) (41) NUMERICAL EXAMPLE Case 1: Consider two steel moment-resisting frame buildings subjected to an earthquake excitation corresponding to the 5 per cent-damped S soil-type (deep cohensionless or stiff clay soils) normalized 2 ( 1997 by John Wiley & Sons, Ltd. Earthquake Engng. Struct. Dyn., 26, 849 —858 (1997) 856 J. PENZIEN Figure 6. Equivalent linearized damping ratio as a function of maximum ductibility demand for discrete values of b(b,KM /K) acceleration response spectrum in the 1994 Uniform Building Code5 (Vol. 2, Fig. 16—3, pp. 2—42) with the effective peak ground acceleration specified to be 0·4 g. Over the period range 0·58(¹(3·00, this normalized spectrum is given by S (0·05, ¹ )/0·4g"1·45/¹ ! which corresponds to the displacement response spectrum (42) S (0·05, ¹ )"14·4¹ (43) $ in which units of centimeters and seconds are used. Assuming b"0·05 for both buildings and that they have been designed for a maximum ductility demand, k, equal to 3·0 when subjected to the above specified seismic input, m equals 0·33 as shown in Figure 6. % Because k and b are the same for both buildings, the period ratio ¹ /¹ representing linear elastic 21 11 behaviour equals the corresponding period ratio ¹ /¹ representing the equivalent linearized behaviour. %21 %11 Using the results in Figure 2 for m"m "0·33, values of o can be obtained for discrete values of ¹ /¹ . % %12 21 11 Letting the ratio C /C equal unity, consistent with both buildings being of the same structural type, and 52 51 introducing the values of o into equation (36), the desired response ratio Dl (t) D /S (m, ¹ ) can be %12 R .!9 $ 11 obtained as a function of ¹ /¹ . This function, designated CQC, is shown in Figure 7 along with the 21 11 corresponding SRSS and ABS functions obtained from equations (37) and (38), respectively. The abscissa scale in this figure is also shown in terms of the building-height ratio H /H in conformance with 2 1 equation (34). Case 2: Consider two buildings of the same height (H "H ) but having different periods due to 1 2 structural differences. Subject both buildings to the same seismic excitation as characterized by the displacement response spectrum shown in equation (43). Using the same values of k and b as in the previous Case 1 which resulted in m "0·33, evaluating o for discrete values of ¹ /¹ , the response ratio % %12 21 11 Dv (t)D /S (m, ¹ ) expressed by equation (39) can be obtained as a function of ¹ /¹ . This function, R .!9 $ 11 21 11 designated CQC, is shown in Figure 8 along with the corresponding SRSS and ABS functions obtained from Earthquake Engng. Struct. Dyn., 26, 849 —858 (1997) ( 1997 by John Wiley & Sons, Ltd. EVALUATION OF BUILDING SEPARATION DISTANCE 857 Figure 7. Normalized building separation to prevent pounding: Case 1 Figure 8. Normalized building separation to prevent pounding: Case 2 equations (40) and (41), respectively. Note the results in this case are significantly different from those in Figure 7 for Case 1. CONCLUDING REMARKS One should realize that predicting the fundamental periods of two buildings located adjacent to each other is often subject to considerable error. Therefore, such variations should be taken into account when evaluating separation distance. In addition, one should consider the possibility of the displacement response spectral values being larger than specified in the code. The exceedance probabilities for a given period of time associated with the normalized inward relative displacement values given by the CQC relations in Figures 7 ( 1997 by John Wiley & Sons, Ltd. Earthquake Engng. Struct. Dyn., 26, 849 —858 (1997) 858 J. PENZIEN and 8 would be similar to the exceedance probabilities of the response spectral values used. All uncertainties should be considered when evaluating or specifying separation distances. REFERENCES 1. A. Der Kiureghian, ‘Structural response to stationary excitation’, J. eng. mech. div. ASCE, December 1980. 2. R. W. Clough and J. Penzien, Dynamics of Structures, 2nd edn., McGraw-Hill, New York, 1993. 3. V. Jeng, K. Kasai and B. F. Maison, ‘A Spectral difference method to estimate building separations to avoid pounding’, Earthquake spectra, J. Earthquake Engineering Research Institute (EERI) 8 (1992). 4. H. B. Seed and I. M. Idriss, ‘Soil moduli and damping factors for dynamic response analyses’, Report No. EERC 70-10, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1970. 5. International Conference of Building Officials, Structural Engineering Provisions, Uniform Building Code, Vol. 2, 1994. . Earthquake Engng. Struct. Dyn., 26, 849 —858 (1997) ( 1997 by John Wiley & Sons, Ltd.

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