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This paper aims to develop an improved understanding of the critical response of structures to multicomponent seismic motion characterized by three uncorrelated components that are defined along its principal axes: two horizontal and the vertical component. An explicit formula, convenient for code applications, has been derived to calculate the cri...
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... excitation is deÿned in terms of design spectra associated with the principal directions of the translational components of ground motion, which are oriented along the two horizontal axes 1 and 2 and the vertical axis z , as shown in Figure 1. The pseudo-acceleration spectra are denoted as A ( T n ) for the major principal axis, A ( T n ) for the intermediate principal axis, and A z ( T n ) for the minor principal axis; T n is the natural vibration period of a single-degree-of-freedom system. Note that the design spectra in the two horizontal directions have the same shape and di er by the ratio of spectrum intensities where 0 6 6 1. The ground acceleration components along the principal axes (1, 2, and z ) are assumed to be uncorrelated [10 ; 11]. They do correlate, however, if deÿned along any other set of axes, for example, along x , y , and z (the reference axes of the structure). As shown in Figure 1, Â denotes the orientation of the earthquake’s major principal axis relative to structural axis x . Deÿned as the incident angle of the ground motion, Â in the counter-clockwise direction is taken to be positive. The peak response of a structure to a single component of ground motion applied along one of the structure axes is commonly evaluated using the response spectrum method. Accounting ...
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... excitation is deÿned in terms of design spectra associated with the principal directions of the translational components of ground motion, which are oriented along the two horizontal axes 1 and 2 and the vertical axis z , as shown in Figure 1. The pseudo-acceleration spectra are denoted as A ( T n ) for the major principal axis, A ( T n ) for the intermediate principal axis, and A z ( T n ) for the minor principal axis; T n is the natural vibration period of a single-degree-of-freedom system. Note that the design spectra in the two horizontal directions have the same shape and di er by the ratio of spectrum intensities where 0 6 6 1. The ground acceleration components along the principal axes (1, 2, and z ) are assumed to be uncorrelated [10 ; 11]. They do correlate, however, if deÿned along any other set of axes, for example, along x , y , and z (the reference axes of the structure). As shown in Figure 1, Â denotes the orientation of the earthquake’s major principal axis relative to structural axis x . Deÿned as the incident angle of the ground motion, Â in the counter-clockwise direction is taken to be positive. The peak response of a structure to a single component of ground motion applied along one of the structure axes is commonly evaluated using the response spectrum method. Accounting ...
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... determine r max and r min , the maximum and the minimum values of r (  ) in Equation (1), usually the two numerical values of  cr determined from Equation (9) are substituted for  . We can, however, derive explicit equations for r max and r min by recognizing that they represent the combined response to three components of ground motion acting in directions 1, 2 and z , with  =  cr , as shown in Figure 1. If the responses to these uncorrelated individual components of ground motion are denoted by r 1 ; r 2 and r z , respectively, the combined response is given by the SRSS ...
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... excitation is deÿned in terms of design spectra associated with the principal directions of the translational components of ground motion, which are oriented along the two horizontal axes 1 and 2 and the vertical axis z, as shown in Figure 1. The pseudo-acceleration spectra are denoted as A(T n ) for the major principal axis, A(T n ) for the intermediate principal axis, and A z (T n ) for the minor principal axis; T n is the natural vibration period of a single-degree-of-freedom system. ...
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... do correlate, however, if deÿned along any other set of axes, for example, along x, y, and z (the reference axes of the structure). As shown in Figure 1, Â denotes the orientation of the earthquake's major principal axis relative to structural axis x. Deÿned as the incident angle of the ground motion, Â in the counter-clockwise direction is taken to be positive. ...
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... determine r max and r min , the maximum and the minimum values of r(Â) in Equation (1), usually the two numerical values of  cr determined from Equation (9) are substituted for Â. We can, however, derive explicit equations for r max and r min by recognizing that they represent the combined response to three components of ground motion acting in directions 1, 2 and z, with  =  cr , as shown in Figure 1. If the responses to these uncorrelated individual components of ground motion are denoted by r 1 ; ;r 2 and r z , respectively, the combined response is given by the SRSS rule r max = {r 2 1 + (r 2 ) 2 + r 2 z } 1=2 ; r min = {(r 1 ) 2 + r 2 2 + r 2 z } 1=2 (12) ...
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... above-mentioned results for the axial force in column a are presented in Figure 10 for a one-storey system with ÿxed T y = 0:5 s and T x over a range of values. As the spectrum intensity ratio increases, we expect r srss and r cr to increase, however r cr =r srss decreases. ...
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... critical angle of incidence depends on the period ratio T x =T y , but not on the spectrum intensity ratio. It varies between 45 and 90 • , as shown in Figure 10(b). If T x =T y is much smaller or much larger than one, Â cr is close to 90 • , implying that the response reaches its critical value when the stronger component of horizontal ground motion is applied along the y-axis of the structure. ...
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... an idealized one-storey unsymmetrical-plan building with a rigid slab supported by any number of lateral resisting elements oriented along directions x and y ( Figure 11). The system has three degrees of freedom: translations of the centre of mass (CM) along x-and y-directions and rotation of the slab about a vertical axis passing through the CM. ...
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... system has three degrees of freedom: translations of the centre of mass (CM) along x-and y-directions and rotation of the slab about a vertical axis passing through the CM. The eccentricity of the centre of rigidity CR relative to CM is given by distances e x and e y ( Figure 11); e x =r and e y =r are the normalized eccentricities, where r is the radius of gyration of the eoor about the vertical axis passing through the CM. A 5 per cent damping ratio is assumed for each of the three vibration modes. ...
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... major and the intermediate principal components of ground motion are deÿned by A(T n ) and A(T n ), respectively, applied at incident angle  (Figure 11), where A(T n ) is the design spectrum of Figure 5. No vertical ground motion is considered. ...
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... motion in the y-direction excites only the mode (period T 3 ) that describes uncoupled motion in the y-lateral direction. The periods of the three natural vibration modes are plotted against T x =T y in Figure 12(a). ...
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...  = 0 • , the displacement increases as T x becomes larger. For the cases when T x =T y 1 or 1, the periods T 1 and T 2 are well separated from T 3 ( Figure 12(a)), the correlation coeecient approaches zero (Figure 12(b)), the critical angle  cr approaches 90 • when T x =T y 1 and 0 • when T x =T y 1 (Figure 12(c)), and the critical response r cr approaches r ( = 90 • ) for T x =T y 1 and r ( = 0 • ) when T x =T y 1 (Figure 12(d)). These results can be explained as follows: when T x =T y 1, ÿ¿1 (Figure 12(b)), which implies r y is much larger than r x , therefore the response is the largest when the ground motion is applied in the y-direction. ...
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...  = 0 • , the displacement increases as T x becomes larger. For the cases when T x =T y 1 or 1, the periods T 1 and T 2 are well separated from T 3 ( Figure 12(a)), the correlation coeecient approaches zero (Figure 12(b)), the critical angle  cr approaches 90 • when T x =T y 1 and 0 • when T x =T y 1 (Figure 12(c)), and the critical response r cr approaches r ( = 90 • ) for T x =T y 1 and r ( = 0 • ) when T x =T y 1 (Figure 12(d)). These results can be explained as follows: when T x =T y 1, ÿ¿1 (Figure 12(b)), which implies r y is much larger than r x , therefore the response is the largest when the ground motion is applied in the y-direction. ...
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...  = 0 • , the displacement increases as T x becomes larger. For the cases when T x =T y 1 or 1, the periods T 1 and T 2 are well separated from T 3 ( Figure 12(a)), the correlation coeecient approaches zero (Figure 12(b)), the critical angle  cr approaches 90 • when T x =T y 1 and 0 • when T x =T y 1 (Figure 12(c)), and the critical response r cr approaches r ( = 90 • ) for T x =T y 1 and r ( = 0 • ) when T x =T y 1 (Figure 12(d)). These results can be explained as follows: when T x =T y 1, ÿ¿1 (Figure 12(b)), which implies r y is much larger than r x , therefore the response is the largest when the ground motion is applied in the y-direction. ...
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...  = 0 • , the displacement increases as T x becomes larger. For the cases when T x =T y 1 or 1, the periods T 1 and T 2 are well separated from T 3 ( Figure 12(a)), the correlation coeecient approaches zero (Figure 12(b)), the critical angle  cr approaches 90 • when T x =T y 1 and 0 • when T x =T y 1 (Figure 12(c)), and the critical response r cr approaches r ( = 90 • ) for T x =T y 1 and r ( = 0 • ) when T x =T y 1 (Figure 12(d)). These results can be explained as follows: when T x =T y 1, ÿ¿1 (Figure 12(b)), which implies r y is much larger than r x , therefore the response is the largest when the ground motion is applied in the y-direction. ...
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... the cases when T x =T y 1 or 1, the periods T 1 and T 2 are well separated from T 3 ( Figure 12(a)), the correlation coeecient approaches zero (Figure 12(b)), the critical angle  cr approaches 90 • when T x =T y 1 and 0 • when T x =T y 1 (Figure 12(c)), and the critical response r cr approaches r ( = 90 • ) for T x =T y 1 and r ( = 0 • ) when T x =T y 1 (Figure 12(d)). These results can be explained as follows: when T x =T y 1, ÿ¿1 (Figure 12(b)), which implies r y is much larger than r x , therefore the response is the largest when the ground motion is applied in the y-direction. The opposite situation occurs when T x =T y 1, where ÿ¡1 (Figure 12(b)), implying that r x is much larger than r y ; therefore the response is largest when the ground motion is applied in the x-direction. ...
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... results can be explained as follows: when T x =T y 1, ÿ¿1 (Figure 12(b)), which implies r y is much larger than r x , therefore the response is the largest when the ground motion is applied in the y-direction. The opposite situation occurs when T x =T y 1, where ÿ¡1 (Figure 12(b)), implying that r x is much larger than r y ; therefore the response is largest when the ground motion is applied in the x-direction. For systems with T x = T y , = 0 ( Figure 12(b)); thus based on Figure 2, it would be expected that r cr =r srss = 1. ...
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... opposite situation occurs when T x =T y 1, where ÿ¡1 (Figure 12(b)), implying that r x is much larger than r y ; therefore the response is largest when the ground motion is applied in the x-direction. For systems with T x = T y , = 0 ( Figure 12(b)); thus based on Figure 2, it would be expected that r cr =r srss = 1. This is conÿrmed by the results shown later in Figure 13(a). ...
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... systems with T x = T y , = 0 ( Figure 12(b)); thus based on Figure 2, it would be expected that r cr =r srss = 1. This is conÿrmed by the results shown later in Figure 13(a). ...
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... the contrary, observe that when one of the natural vibration periods, T 1 or T 2 , is equal to vibration period T 3 in Figure 12(a), the correlation coeecient is close to −1 or +1, respectively, ÿ tends to 1 (Figure 12(b)), the critical angle is close to 135 or 45 • , respectively (Figure 12(c)), and the critical response r cr (Figure 12(d)) has two peaks that exceed the two responses r (Â = 0 • ) and r (Â = 90 • ). This observation can be explained as follows: When T 1 or T 2 is equal to T 3 , the correlation between the modal responses increases, therefore the cross term r xy and the correlation coeecient increases; T 1 = T 3 implies that T x =T y = 0:85, and T 2 = T 3 occurs at T x =T y = 1:15, which deÿne the two peaks in the critical response values shown in Figure 12(d). ...
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... the contrary, observe that when one of the natural vibration periods, T 1 or T 2 , is equal to vibration period T 3 in Figure 12(a), the correlation coeecient is close to −1 or +1, respectively, ÿ tends to 1 (Figure 12(b)), the critical angle is close to 135 or 45 • , respectively (Figure 12(c)), and the critical response r cr (Figure 12(d)) has two peaks that exceed the two responses r (Â = 0 • ) and r (Â = 90 • ). This observation can be explained as follows: When T 1 or T 2 is equal to T 3 , the correlation between the modal responses increases, therefore the cross term r xy and the correlation coeecient increases; T 1 = T 3 implies that T x =T y = 0:85, and T 2 = T 3 occurs at T x =T y = 1:15, which deÿne the two peaks in the critical response values shown in Figure 12(d). ...
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... the contrary, observe that when one of the natural vibration periods, T 1 or T 2 , is equal to vibration period T 3 in Figure 12(a), the correlation coeecient is close to −1 or +1, respectively, ÿ tends to 1 (Figure 12(b)), the critical angle is close to 135 or 45 • , respectively (Figure 12(c)), and the critical response r cr (Figure 12(d)) has two peaks that exceed the two responses r (Â = 0 • ) and r (Â = 90 • ). This observation can be explained as follows: When T 1 or T 2 is equal to T 3 , the correlation between the modal responses increases, therefore the cross term r xy and the correlation coeecient increases; T 1 = T 3 implies that T x =T y = 0:85, and T 2 = T 3 occurs at T x =T y = 1:15, which deÿne the two peaks in the critical response values shown in Figure 12(d). ...
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... the contrary, observe that when one of the natural vibration periods, T 1 or T 2 , is equal to vibration period T 3 in Figure 12(a), the correlation coeecient is close to −1 or +1, respectively, ÿ tends to 1 (Figure 12(b)), the critical angle is close to 135 or 45 • , respectively (Figure 12(c)), and the critical response r cr (Figure 12(d)) has two peaks that exceed the two responses r (Â = 0 • ) and r (Â = 90 • ). This observation can be explained as follows: When T 1 or T 2 is equal to T 3 , the correlation between the modal responses increases, therefore the cross term r xy and the correlation coeecient increases; T 1 = T 3 implies that T x =T y = 0:85, and T 2 = T 3 occurs at T x =T y = 1:15, which deÿne the two peaks in the critical response values shown in Figure 12(d). ...
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... the contrary, observe that when one of the natural vibration periods, T 1 or T 2 , is equal to vibration period T 3 in Figure 12(a), the correlation coeecient is close to −1 or +1, respectively, ÿ tends to 1 (Figure 12(b)), the critical angle is close to 135 or 45 • , respectively (Figure 12(c)), and the critical response r cr (Figure 12(d)) has two peaks that exceed the two responses r (Â = 0 • ) and r (Â = 90 • ). This observation can be explained as follows: When T 1 or T 2 is equal to T 3 , the correlation between the modal responses increases, therefore the cross term r xy and the correlation coeecient increases; T 1 = T 3 implies that T x =T y = 0:85, and T 2 = T 3 occurs at T x =T y = 1:15, which deÿne the two peaks in the critical response values shown in Figure 12(d). In addition, the parameter ÿ is close to 1 in the same period range. ...
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... addition, the parameter ÿ is close to 1 in the same period range. Thus, as pointed out previously in Figure 2, the simultaneity of ÿ approaching 1 and approaching −1 or +1 leads to an increase in the critical response with respect to the responses r (Â = 0 • ) and r (Â = 90 • ), as conÿrmed by the results shown in Figure 12(d). ...
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... ratio of the critical value of the response to its SRSS value, r cr =r srss , is computed from Equation (15) and plotted against the period ratio T x =T y in Figure 13. This ÿgure is organized in three parts to show (a) the eeect of the spectrum intensity ratio , for e x =r = 0, e y =r = 0:3 and T x =T Â = 1; (b) the eeect of T x =T Â , for e x =r = 0, e y =r = 0:3 and = 0; and (c) the eeect of the eccentricity e x =r, for T x =T Â = 1, e y =r = 0:3 and = 0. ...
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... ÿgure is organized in three parts to show (a) the eeect of the spectrum intensity ratio , for e x =r = 0, e y =r = 0:3 and T x =T Â = 1; (b) the eeect of T x =T Â , for e x =r = 0, e y =r = 0:3 and = 0; and (c) the eeect of the eccentricity e x =r, for T x =T Â = 1, e y =r = 0:3 and = 0. The system presented in Figure 13(a) is the same oneway, unsymmetrical system subjected to a single ground motion component that was presented previously in Figure 12. The largest value of r cr =r srss is 1.24 when = 0 and T x =T y = 1:15. ...
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... ÿgure is organized in three parts to show (a) the eeect of the spectrum intensity ratio , for e x =r = 0, e y =r = 0:3 and T x =T Â = 1; (b) the eeect of T x =T Â , for e x =r = 0, e y =r = 0:3 and = 0; and (c) the eeect of the eccentricity e x =r, for T x =T Â = 1, e y =r = 0:3 and = 0. The system presented in Figure 13(a) is the same oneway, unsymmetrical system subjected to a single ground motion component that was presented previously in Figure 12. The largest value of r cr =r srss is 1.24 when = 0 and T x =T y = 1:15. ...
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... T x =T Â ( Figure 13(b)) leads to a decrease or increase in r cr =r srss , depending on the period ratio T x =T y ; r cr =r srss is largest for systems when T x =T Â = 1 and T x =T y = 1:15. Similarly, varying the eccentricity e x =r ( Figure 13(c)) may lead to an increase or a decrease in the values of r cr =r srss , depending on T x =T y ; r cr =r srss is largest for systems when e x =r = 0 and T x =T y = 1:15. ...
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... T x =T Â ( Figure 13(b)) leads to a decrease or increase in r cr =r srss , depending on the period ratio T x =T y ; r cr =r srss is largest for systems when T x =T Â = 1 and T x =T y = 1:15. Similarly, varying the eccentricity e x =r ( Figure 13(c)) may lead to an increase or a decrease in the values of r cr =r srss , depending on T x =T y ; r cr =r srss is largest for systems when e x =r = 0 and T x =T y = 1:15. For all systems considered herein, the largest value of r cr =r srss is 1.24, corresponding to e x =r = 0, e y =r = 0:3, T x =T Â = 1, and T x =T y = 1:15, when the excitation is a single seismic component ( = 0). ...
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... structural response to two principal components of horizontal ground motion acting along any incident angle Â, relative to the structural axes. The critical response r cr , deÿned as the maximum response considering all possible incident angles Â, is given by Equation (14). Also, the SRSS response r srss was deÿned as the larger of the two responses for  = 0 and 90 • . ...
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Citations
... Menun and Der Kiureghian [33] estimated the critical direction of seismic excitation and the corresponding response quantities based on CQC3 rule. Similarly, Lopez et al. [34] & López et al. [35] investigated that the maximum seismic demand at critical seismic direction for a response quantity can be 20% higher than the seismic demand observed when the structure is excited along the reference axes. In addition, the seismic demand corresponding to the critical angle of seismic excitation for the most unfavourable combinations of various engineering demand parameters was determined by Menun and Kiureghian [36] & Menun and Kiureghian [37]. ...
This research presents the influence of varying seismic orientations on the statistical distribution of nonlinear seismic response considering a numerically validated stiffness-eccentric structural model. Firstly, the experimental results were utilized for the validation of the numerical model. The experimental results were obtained through shaking table test where stiffness-eccentric reinforced concrete (RC) structure was exposed to progressive seismic excitations. The asymmetric structure experienced inelastic deformations under high seismic excitations, which eventually caused the transformation of the structure from a state of elastic deformation to a state of highly inelastic deformation. The experimental observations were compared with numerical illustrations for each stage of the seismic excitation from elastic to in-elastic state of the structure. Then, based on the validated numerical models, the numerical investigation was extended for nonlinear seismic response under varying seismic orientations. Seismic responses of the flexible and stiff edges of the stiffness-eccentric structure were compared under varying seismic excitation to present the influence of asymmetry, response variability and their statistical distribution. The results were further investigated from seismic uncertainty viewpoint to evaluate the response variability and design shortcomings. It has been concluded that the stiffness-eccentric structures are more sensitive towards the rotational seismic response as compared to the translational seismic response. Similarly, the relative rotational response at the stiff edge of the structure proved to have more response variability than the flexible edge of the structure under varying seismic orientations.
... A series of bidirectional ground motions (20) are synthesized by using 18 design ground motion accelerograms provided by Specification for Highway Bridges (JRA) 1 for Earthquake Types I and II of Level-2 earthquakes with three soil conditions (I, II, II), and 2 simulated ground motions for Level-1 and Level-2 Earthquakes in the design guideline by the Building Center of Japan (BCJ) 40 as the original components ( ) of Equation (11). The target spectra of these design F I G U R E 1 0 Difference of directionality characteristic before and after spectral matching of GM2 ground motions are shown in Figure 12(A)-(C). ...
In this study, the effects of the degree of directionality on critical seismic performance assessment are investigated from two aspects: the bidirectional seismic demand represented by maximum‐direction spectrum and the performance indices obtained from nonlinear time‐history analysis. Firstly, typical representations of the bidirectional demand of ground motions are reviewed and compared, in seismic design practices (e.g., ASCE 7‐16, Eurocode8, and JRA). Statistical analysis using 83 ground motions underlines the implication of the spectrum‐compatible condition that has not been fully considered in past studies investigating the bidirectional effects. In the second part, a series of spectrum‐compatible bidirectional ground motions with various degrees of directionality are generated as inputs. The corresponding seismic performance assessment results of a base isolation building with the friction pendulum system are investigated. According to the simulation results, the current design practices could introduce unconservative assessment on the maximum bearing displacement due to the omission of the directionality effect included in the design ground motions, particularly, when the directionality parameter is at a large value (less directionality). The cases with nondirectionality always tend to result in a nearly constant envelope of the maximum interstory drift and the maximum floor acceleration assessment for the full directionality case, over all incident directions. For the critical seismic performance assessment, the two extreme cases, namely the full directionality and nondirectionality cases, are recommended to be considered.
... This common engineering practice can result in significant underestimation of seismic demands (see, e.g., Athanatopoulou [2]; Rigato and Medina [3]; Kostinakis et al. [4][5][6][7]; Fontara et al. [8]; Pavel and Nica [9]; Cavdar and Ozdemir [10]; Skoulidou et al. [14]; Lucchini et al. [43]; Nguyen and Kim [44]; and Roy et al. [45]). Finally, special mention must be made of the significant research of Smeby and Der Kiureghian [46] and Menun and Der Kiureghian [47], who developed the CQC3 modal combination rule, which has been successfully used for the investigation of the influence of the seismic excitations' angle on structures' responses within the context of the linear response spectrum analysis (see, e.g., Lopez et al. [48,49]; Sessa et al. [50,51]). ...
The angle of seismic excitation is a significant factor of the seismic response of RC buildings. The procedure required for the calculation of the angle for which the potential seismic damage is maximized (critical angle) contains multiple nonlinear time history analyses using in each one of them different angles of incidence. Moreover, the seismic codes recommend the application of more than one accelerograms for the evaluation of seismic response. Thus, the whole procedure becomes time consuming. Herein, a method to reduce the time required for the estimation of the critical angle based on Multilayered Feedforward Perceptron Neural Networks is proposed. The basic idea is the detection of cases in which the critical angle increases the class of seismic damage compared to the class which arises from the application of the seismic motion along the buildings’ structural axes. To this end, the problem is expressed and solved as Pattern Recognition problem. As inputs of networks the ratios of seismic parameters’ values along the two horizontal seismic records' components, as well as appropriately chosen structural parameters, were used. The results of analyses show that the neural networks can reliably detect the cases in which the calculation of the critical angle is essential.
... The influence of this angle on the structural response has been widely studied. [7][8][9][10][11] In general, they have found that the seismic response of a structure significantly varies depending on the incidence angle. ...
A simplified procedure is developed to consider the azimuthal orientation of buildings when estimating seismic risk. Two square-plan reinforced concrete building models are considered as a testbed, one with similar and one with dissimilar properties along the two principal horizontal axes. The fragility of both structures is analysed using a set of ground motion records rotated to multiple incidence angles to develop orientation-dependent fragility functions. It has been observed that, reorienting all records so that these structures have the same azimuth vis-à-vis the corresponding epicentre leads to significant differences compared to assuming random orientations. Additional results stemming from single-degree-of-freedom oscillators further confirm such findings, showing a dependence to the proximity to the faults and the level of dissimilarity in the principal horizontal axes of the structure. The end results point to a non-negligible bias in assessment studies when a structure's orientation with respect to governing rupture scenarios is not taken into account. It is shown that the median of fragility curves calculated for un-rotated incidence angles can be bias-corrected through shifted by an amount that depends on the azimuthal orientation and level of axes-dissimilarity of structures. K E Y W O R D S directionality effect, cloud analysis, consistent rotations, seismic risk assessment
... To explore these issues, in-depth research has been conducted on the principal axes of ground motion [5][6][7] and critical angle of seismic incidence (ASI) [8][9][10][11][12][13][14][15][16] (note that, all abbreviations used in this paper are speci¯ed in Table A.1) to ensure the maximum responses of structures. For a ground motion with two horizontal acceleration components, such a method is needed to combine the directionality-varying singlecomponent spectral acceleration values into a single numerical value, which is representative of the two-dimensional horizontal ground motion. ...
... Furthermore, the MEAC has a normal-JT distribution according to Fig. 3(b), following the unique Johnson transformation function, expressed by Eq. (12). The AI ÀJT probability density function is very close to that of ' aÀJT , implying that there may be a strong correlation between the PCoA and the MEAC. ...
To investigate the variability of ground motion characteristics (GMC) with the angle of seismic incidence (ASI) and the impact of seismic incident directionality on structural responses, first, a large-scale database of recorded ground motions was used to analyze the causes of GMC variability due to the seismic incident directionality effect (SIDE). Then a single-mass bi-degree-of-freedom system (SM-BDOF-S) with different types of symmetrical sections was selected to explore the influence mechanism of SIDE on the seismic responses. The results illustrated that the GMC has substantial variability with the ASI, which is independent of the earthquake source, propagation distance, and site condition, and exhibits complex random characteristics. Additionally, a classification method for ground motions is proposed based on this GMC variability to establish a criterion for selecting ground motions in seismic analysis considering the SIDE. Moreover, for an SM-BDOF-S, the response spectral plane is proposed to explain the transition behavior of spectral responses that are very similar among different stiffness ratios, but divergent for different types of ground motions. The influence of SIDE on structures is determined by their stiffness and stiffness ratio in the x- and y-directions, as well as the type of ground motion.
... In another research, Menun and Der Kiureghian [17] proposed the CQC3 rule for the computation of the critical incident angle and the corresponding maximum response. Lopez et al. [18,19] proved that the critical value for a single response quantity can be up to 20% larger than the usual response produced by seismic components applied along the structural axes. ...
Power Spectrum Analysis (PSA) consists one of the seismic analysis methods proposed by modern code provisions. In the context of PSA, the computation of seismic behavior is achieved by a stochastic analysis that yields mean square values. A method that can be used for the determination of maximum or mean square response is the response history analysis. In the present study, analytical formulas are developed for the determination of the maximum mean square value and the corresponding critical incident angle of any scalar response quantity under two horizontal seismic components for any time interval of ground motion within the context of linear response history analysis. An R/C building is analyzed in order to demonstrate the variation of the maximum mean square response with the incident angle. The maximum mean square response values over all incident angles are calculated using the developed formulas. The results show that the application of the earthquake components along the structural axes can underestimate the mean square response.
... Menun and Der Kiureghian [6] in 1998 developed a new general combination rule for the multi-component excitation called complete quadratic combination with three components (CQC3) and showed that the use of this rule results in improved predictions. Lopez et al. [7] in 2000 developed a formula, using the CQC3 method, to predict the maximum response of structures in a multi-component response spectrum analysis. Anastassiadis et al. [8] in 2002 proposed a design procedure for predicting the critical direction and maximum responses under bi-directional seismic excitations. ...
... Although the response of the individual records was found to be sensitive to the variations of angle θ, when the mean response was considered the variations were relatively small. (7) Although the use of the 30% rule provides conservative predictions of the radial displacements when compared with the predictions using 3D analyses, when directivity effects are not considered, this conservatism allows for some increases in the radial displacements due to different excitation angles, θ. The probability of exceeding the mean radial displacements of the columns, obtained from the 30% rule, was about 15 to 20%, for the cases studied when directionality effects were considered. ...
Inelastic dynamic analyses were carried out using 3D and 2D models to predict the mean seismic response of four-span reinforced concrete (RC) bridges considering directionality effects. Two averaging methods, including an advanced method considering displacement direction, were used for the prediction of the mean responses to account for different incident angles of ground motion records. A method was developed to predict the variability of the mean displacement predictions due to variability in the incident angles of the records for different averaging methods. When the concepts of averaging in different directions were used, significantly different predictions were obtained for the directionality effects. The accuracy of the results obtained using 2D and 3D analyses with and without the application of the combination rules for the prediction of the mean seismic demands considering the incident angle of the records was investigated. The predictions from different methods to account for the records incident angles were evaluated probabilistically. Recommendations were made for the use of the combination rules to account for the directivity effects of the records and to predict the actual maximum displacement, referred to as the maximum radial displacement.
... 31 Another interpretation of this in the framework of structural design is the critical angle of incidence. 32,33 While the seismologists aim to address the directionality or polarization, the structural designers focus on accounting for the critical orientation. In the process, a consistent definition of spectral acceleration must be adopted both in hazard analysis and structural design. ...
Orientation of a structure in a site is generally known but not the direction of maximum shaking during a future seismic event. Two different types of intensity measures (IMs) are usually used to approximately account for this directionality effect, namely, the rotation dependent such as RotDxx and GMRotDxx and the rotation independent such as RotIxx and GMRotIxx. Rotation dependent IMs are presently constructed by performing time history analysis for all possible orientation (usually @ 1 degree) of the input ground motion set followed by picking the xx‐percentile spectral ordinate. In other words, the construction of RotDxx spectrum requires a set of 180 time history analysis of an oscillator per spectral ordinate. Similarly, the construction of GMRotDxx requires time history analysis of an oscillator against 90 pairs of orthogonal components per spectral ordinate. This paper presents a framework that enables the construction of rotation dependent IMs by performing time history analysis against a pair of as‐recorded components with some nominal supplemental processing. This reduces the computational cost more than 90% when compared with the state of the art. Rotation independent IMs are defined through finding out the rotation that minimizes the error (often termed as the penalty function) with respect to the target spectra of associated rotation dependent IM as the benchmark. Resulting rotation independent IMs show somewhat sensitivity on the maximum time period used in spectral representation. This paper presents an alternate definition (involving scaling and rotation) for rotation independent IMs that nearly eliminates such sensitivity.
... Since the final demand depends on the ASI and the level of conservativeness of the demand is uncertain when only one ASI is considered, a rigorous procedure that determines the ASIcrit and the critical demand was developed (Smeby & Der Kiureghian 1985;Menun & Der Kiureghian 1998;López et al. 2000a). The procedure, which involves the CQC3 combination rule, takes into account the correlation and the relative intensity of three orthogonal components, where the minor component is the vertical one. ...
... Based on the fact that different EDPs reach their maximum values under different ASIs, López et al. (2000a) did not focus on determining the ASIcrit, but on calculating the critical demand as a function of the SRSS response when only one ASI was considered. The developed procedure is based on the independent unidirectional application of the orthogonal response spectra along the structural axes and the SRSS combination of the unidirectional responses. ...
Although earthquake engineering has significantly advanced during the past decades, recent examples of seismic events that resulted in disproportionally large losses,
including, among others, human and economic losses, prove that the discipline still
requires improvements. The weaknesses often met in current practice, usually reflect
shortcomings in up-to-date standard provisions and guidelines, thus rendering the
revision and the improvement of the latter an imperative task.
In light of this, the present thesis addresses the active area of research dealing with the
angle of incidence of the seismic action during structural analysis of buildings.
Particularly, the main objective of the thesis is to determine the effect of the angle of
seismic incidence and to provide methods to account for it in the context of seismic safety assessment of existing buildings. In accordance with commonly used seismic standards, the input seismic action can be represented either by a suite of ground motion records or by a single response spectrum. As such, two different approaches are applied with respect to the determination of the effect of the angle of seismic incidence, depending on the method of representing the input seismic action.
... A spatial combination rule which considers the correlation among ground motion components and also critical response aspect, called CQC3, was introduced as a replacement for the percentage (30% & 40%) [6,7] and SRSS rules by Menun and Der Kiureghian in [8]. An explicit formula was derived for the calculation of critical value of a response, without a need to determine a-priori the critical angle, when the two horizontal principal components were applied at any angle with respect to structural axes along with vertical component of ground motion in [9]. Apart from the previously discussed works one can find other interesting studies related to response spectrum p r e -p r i n t and equivalent lateral static analysis, which were carried out in later years, and confirmed the similar conclusions drawn earlier [10][11][12][13][14]. ...
The general practice in structural assessment or design using 3D seismic analysis of inelastic structures is to obtain the responses (such as storey displacement, inter-storey drift, etc.) by applying the bidirectional components along any arbitrarily chosen structural axes. However, such response quantities may not remain the same when the bidirectional components are rotated with respect to the structural axes. It becomes relevant to identify the angle of seismic loading which maximizes the response quantity in the structure, and that angle would be called as critical angle of incidence (CAI), and the associated response as critical response. Nevertheless, a single CAI which can simultaneously maximize all the responses of the structure does not exist, rather the critical angle varies according to the type of response and earthquake loading. Therefore, any structural assessment or design of buildings based on the responses obtained from applying the earthquake loading along user defined structural axes may not be always conservative. To avoid such scenarios the critical responses should be estimated considering the effect of angle of seismic incidence. To address such an issue, pertaining to inelastic structures, this study presents a new approach which essentially combines bidirectional pushover analysis with an already existing non-linear static procedure called the Extended-N2 method.
