The Equivalent Non-Linear Single Degree of Freedom System of Asymmetric Multi-Storey Buildings in Seismic Static Pushover Analysis

This book deals with earthquake-resistant structures, such as, buildings, bridges and liquid storage tanks. It contains twenty chapters covering several interesting research topics written by researchers and experts in the field of earthquake engineering. The book covers seismic-resistance design of masonry and reinforced concrete structures to be constructed as well as safety assessment, strengthening and rehabilitation of existing structures against earthquake loads. It also includes three chapters on electromagnetic sensing techniques for health assessment of structures, post earthquake assessment of steel buildings in fire environment and response of underground pipes to blast loads. The book provides the state-of-the-art on recent progress in earthquake-resistant structures. It should be useful to graduate students, researchers and practicing structural engineers.


Introduction
In order to estimate seismic demands at low seismic performance levels (such as life safety or collapse prevention), the application of Non-Linear Response History Analysis (NLRHA) is recommended for reasons related to its accuracy. However, for the purposes of simplification, the application of Pushover Analysis is also often recommended. Here, in order to obtain the seismic demands of asymmetric multi-storey reinforced concrete (r/c) buildings, a new seismic non-linear static (pushover) procedure that uses inelastic response acceleration spectra, is presented. The latter makes use of the optimum equivalent Non-Linear Single Degree of Freedom (NLSDF) system, which is used to represent a randomlyselected, asymmetric multi-storey r/c building. As is proven below, for each asymmetric multi-storey building, a total of twelve suitable non-linear static analyses are required according to this procedure, while at least two hundred and twenty-four suitable non-linear dynamic analyses are needed in the case of NLRHAnalysis, respectively, while if accidental ecentricity is ignored (or external floor moments loads around the vertical axis are used) then a total number of fifty six NLRH Analyses are required. The seismic non-linear static procedure that is presented here is a natural extension of the documented seismic equivalent static linear (simplified spectral) method that is recommended by the established contemporary Seismic Codes, with reference to torsional provisions. From the numerical parametric documentation of this proposed seismic non-linear static procedure, we can reliably evaluate the extreme values of floor inelastic displacements, with reference to results provided by the Non-Linear Response History Analysis. More specifically, it is well-known from past research that, as regards the pushover procedure (i.e. pushover analyses of buildings with inelastic response acceleration spectra), its suitability for use on asymmetric multi-storey r/c buildings is frequently questioned because of the following five points: 1. For a known earthquake, an ideal equivalent NLSDF system, which represents the real asymmetric multi-storey r/c building, must first be defined, in order to calculate the seismic target-displacement. In other words, the above-mentioned r/c NLSDF system has an equivalent viscous damping ratio 0.05  = and is combined with "inelastic acceleration spectra with an equivalent viscous damping ratio 0.05 displacement is transformed into a demand extreme seismic displacement of the monitoring point of the real Multi-Degree of Freedom (MDoF) system, namely the real multi-storey building. After that, the remaining demand seismic displacements (and seismic stress on the members) at the other points of the building are easily calculated using the pushover analysis image. Therefore, successfully defining the ideal equivalent NLSDF system is always the most important part of this procedure, and for this reason, it also constitutes a major concern. Many attempts have been made in the past to resolve this problem in the planar frames (Saiidi & Sozen, 1981;Fajfar & Fischinger, 1987a,b;Uang & Bertero, 1990;Qi & Moehle, 1991;Rodriquez, 1994;Fajfar & Gaspersic, 1996;Hart & Wong, 1999;Makarios, 2005;). On the other hand, in the case of asymmetric multi-storey buildings, a mathematically documented optimum equivalent Non-Linear Single Degree of Freedom System can be defined (Makarios, 2009). According to the latter proposal, three coupled degrees of freedom (two horizontal displacements and a rotation around a vertical axis that passes through the floor mass centre) are taken into account for each floor mass centre. The above-mentioned definition of the optimum equivalent NLSDF system was mathematically derived by observing suitable dynamic loadings on the floor mass centres of the multi-storey building, using simplified assumptions. The use of this optimum equivalent NLSDF system, in combination with the inelastic design spectra, provides an acceptable evaluation of the extreme (maximum/minimum) seismic floor displacements required for a known design earthquake, and for this reason constitutes the core issue of the present chapter. 2. I t i s k n o w n t h a t i n t h e c a s e o f R e s p onse History Analysis, three uncorrelated accelerograms (Penzien & Watabe, 1975;) are used simultaneously along the three axes x, y & z. For this reason, in the case of Response Spectrum Analysis (RSA) or Simplified Spectral Method (SSM) or the equivalent static method, the three response acceleration spectra are "statistically independent" (sect.3.2.2.1(3)P of Eurocode EN-1998.01). This practically means that the response spectra (in the case of RSA), or the lateral floor static forces (in the case of SSM), must be inserted separately for each main direction of the building, and then a suitable superposition (i.e. rule of Square Root of Sum of Squares -SRSS) must be applied to the results of each independent analysis, always in the linear area. However, in the non-linear area, as in the example of static pushover analysis, where superposition is generally forbidden due to its non-linearity, a basic question arises; which is the most suitable way of taking the "spatial seismic action" of the three seismic components into account? In order to answer this question, the rule of Eq.(1) that has resulted from a parametric numerical analysis, provides an approximate evaluation of the spatial seismic action during the static pushover procedure (Makarios, 2011): where =0.75 κ is a mean value in the case of displacements/deformations, and I E, is the extreme seismic demand inelastic displacement/deformation due to static pushover analysis using the first seismic component only, along the horizontal real (or fictitious) principal I-axis of the building. Similarly, II E, is the extreme seismic demand inelastic displacement/deformation due to pushover analysis using the second seismic component only, the other horizontal real (or fictitious) principal II-axis of the building.  -1998.01; g is the acceleration of gravity). 3. What is the most suitable "monitoring point" and its suitable characteristic degree of freedom (control displacement) that is related to the degree of freedom of the equivalent NLSDF system, in the case of asymmetric buildings? As has been proven by the mathematical analysis provided below, concerning the definition of the optimum NLSDF system, the centre of mass of the last floor at the top of the building can play the role of the "monitoring point", whilst its "control displacement" must be parallel to the lateral floor static forces. 4. What is the most suitable distribution of lateral floor static forces in elevation? There are various opinions on this matter, since the distribution can be "triangular", "uniform" or in accordance with the pure translational "fundamental mode-shape". Moreover, the distribution can be adapted to the fundamental mode-shape of the building in each step of the pushover analysis, an issue related to the action of higher order mode-shapes in tall buildings. According to sect.4.3.3.4.2.2(1)/EN-1998.01, at least two different distributions ("uniform" and "first mode-shape") in elevation should always be taken into account. On the other hand, fundamental mode-shape distribution is often applied to common asymmetric buildings, while in the case of very tall buildings, Non-Linear Response History Analysis is mainly reccommended. 5. At which point in the plan should the lateral floor static forces be applied during the pushover analysis of asymmetric buildings? According to sect.4.3.3.4.2.2(2)P/EN-1998.01, the lateral static loads must be applied at the location of the mass centres, while simultaneously, accidental eccentricities should also be taken into account. Moreover, it is known that in the case of dynamic methods, such as "Response Spectrum Analysis" or "Response History Analysis", accidental eccentricities are also considered, with a suitable movement of the floor mass centres by a 0.05 ii eL  or 0.10 i L  , where i L is the dimension of the building's floor that is perpendicular to the direction of the seismic component (sect.4.3.2(1)P/EN-1998.01). However, in the case of static seismic methods, such as the "equivalent static method", "simplified spectrum method", "lateral force method" and "static pushover analysis", the accidental eccentricities are examined with  Anastassiadis et al (1998). As we can observe, a calculation of (a) the location of the real/fictitious stiffness centre of the asymmetric building in the plan, (b) the orientation of its horizontal principal directions I & II and (c) the magnitude of its torsional-stiffness radii, must be carried out before the seismic static pushover analysis begins. For this triple purpose, the fictitious elastic Reference Cartesian System o P (, , ) I II III of the asymmetric multistorey building is the most rational, documented method for use (Makarios & Anastassiadis 1998a,b;Makarios, et al 2006;Makarios, 2008;). On the contrary, it is worth noting that, various other floor centres, such as centres of rigidity/twist/shear, are not suitable for seismic design purposes, because they are dependent on external floor lateral static loads (Cheung & Tso, 1986;Hejal & Chopra, 1987;). At a next stage, the static eccentricities I o, e and II o, e of an asymmetric multi-storey building are defined as the distance between the floor mass centre (CM) and the real/fictitious stiffness centre o P of the building, along the two horizontal principal I & II -axes. In relation to my previous point, and in order to account for uncertainties as regards the real location of floor mass centres, as well as the spatial variation of the seismic motion, accidental eccentricities are also examined (second part). However, it is well-known that according to documented Seismic Codes with reference to torsional provisions, the critical horizontal directions for static lateral loading (in linear equivalent static seismic methods) are the real/fictitious principal I & II-axes of a building. Therefore, the documented static pushover procedure must define the same point; the lateral floor static loads that will be used in the static pushover analysis must be oriented along the real/fictitious principal I & II-axes of the building. In this case, and since the final design eccentricities ( max e & min e along the two principal I & II-axes) have also been used, we can observe that this non-linear static pushover analysis is, undoubtedly, a natural extension of the linear equivalent static seismic method. For illustration purposes, we can see the numerical example of an asymmetric four-storey r/c building with a typical plan, whose real/fictitious principal axes are initially unknown ( Fig.1). The location in the plan and the orientation of the o P (, , ) II II I I system have been calculated according to the above-mentioned relative references, using a 50% reduction of the stiffness properties of all structural elements, due to cracks (sect.4.3.1(7)/ EN-1998.01 To conclude, the real/fictitious principal elastic Cartesian system Ρ ο (Ι,ΙΙ,ΙΙΙ) of the asymmetric multi-storey building is initially calculated. Ιn order to calculate the real/fictitious principal elastic Cartesian system Ρ ο (Ι,ΙΙ,ΙΙΙ) of the asymmetric multi-storey r/c building, all structural elements (columns, walls, beams, cores) of the building model must have effective flexural and shear stiffness, constant along their whole length, corresponding to 50% of their geometric section values. This effective stiffness is more rational, since it leads to a realistic "total lateral stiffness" of the building in the post-elastic, non-linear area. However, if we use much lower values of effective stiffness (i.e. equal to 0.20EI or less), then an artificial increase of the eigen-periods occurs. In such a case, one possible result is that the building model is inadequately seismically loaded, because the state of co-ordination between the fundamental eigen-period of the building and the fundamental period of the actual seismic ground strong motion is removed. Existing views about the use of very small effective flexural stiffness values stem from the following assumption, that "…plastic hinges appeared simultaneously at both ends of each structural element of the multi-storey building." However, this is rather incorrect, because many structural elements do not yield, even in the "near collapse state" of the building, as we can observe from the seismic NLRHA. On the other hand, cracks appeared on the plastic hinges of the r/c elements with yielding steel bars, while at the next moment, when the bending of the elements has the opposite sign, some cracks on the plastic hinges cannot fully close, because the yielded steel bars obstruct their closure. Additionally, in the frame of the static pushover analysis of irregular in the plan, asymmetric multi-storey buildings, a Cartesian system CΜ(xyz) that has the same orientation with the known fictitious elastic system Ρ ο (Ι,ΙΙ,ΙΙΙ) of the building must be adopted as a global reference system. Thus, the lateral floor static loads, according to the static pushover analyses, must be oriented along the two horizontal principal I & II-axes of the building. Otherwise, it is well-known from the linear area that it is not possible to calculate the envelope of the floor displacements/stress of an asymmetric building. In other words, the loading along the principal axes of a building creates the most unfavourable state for static methods. Moreover, as a general conclusion, final design eccentricities ( min e ) must always be taken into account in pushover analysis, and thus the present non-linear static pushover procedure is a natural extension of the equivalent linear static method, with mathematical consistency. With reference to the simulation for the non-linear analysis, we consider that each structural column/beam consists of two equivalent sub-cantilevers ( Fig.2a,b). Each sub-cantilever is represented by an elastic beam element with a non-linear spring at its base. This inelastic spring represents the inelastic deformations that are lumped at the end of the element. In order to identify the characteristics of a plastic hinge, it is assumed that each member (beam/column) deforms in antisymmetric bending. The point of contraflexure is approximately located at the middle of the element and for this reason, two sub-cantilevers appear for each member.
Therefore, each sub-cantilever has a "shear length" s,i L , where its chord slope rotation   Fig.(2b). On the other hand, other simulations of plastic hinges, such as the one based on the "dual component model", where each member is replaced by an elastic element (central length of the element) and an elasto-plastic element (plastic hinge) in parallel, are not suitable for analyzing reinforced concrete members, because then the "stiffness degradation" for repeated dynamic seismic loading cycles is ignored. Also, in many cases, and especially for existing r/c structures, a "shear failure" on structural members can occur because the mechanism of the shear forces in the plastic hinges is destroyed, and therefore doubts arise concerning the validity of the well-known modified "Morsch's truss-model" after the yielding. Therefore, the available plastic rotation p θ of a plastic hinge is reduced (by up to 50% in several cases); (see Eq.(A.12) of EN-1998.03). Also, in the case where plastic hinges appeared simultaneously at both ends of an structural element, the flexural effective stiffness of the section is calculated by where Μ is the flexural yielding moment and V is the respective shear force on a plastic hinge. In the case of tall-walls, the "shear length" s L can be calculated by the distance of the zero-moment point to the base in elevation, from a temporary lateral static floor force.

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The "shear span ratio" is calculated by ss a= MV h = L h , where h is the dimension of the section that is perpendicular to the axis of the flexural moment Μ. Moreover, the yielding moment M of the diagram M   cannot be greater than the value s max R M =V L , where R V is calculated by Eq.(A.12) of Eurocode EN/1998.03, with the axial force N of the column at zero (i.e. due to the action of the vertical seismic building vibration of vertical and horizontal seismic components) and the plastic ductility of the element is equal to pl py Δ 5    (conservative value), because then the "brittle shear failure" appears prematurely, due to the doubts of the validity of the modified "Morsch's truss-model" after the yielding. On the other hand, in the case of seismic Non-Linear Response History Analysis (NLRHA), we have to use suitable pairs (or triads) of uncorrelated accelerograms as seismic action. However, the seismic demands of asymmetric multi-storey buildings are often unreliable for the following reasons: i. It is possible for the accelerograms to be deemed unsuitable (they often do not have the necessary frequency content or are inadequate as regards the number of strong cycles of the dynamic loading or the strong motion duration). In order to cope with this problem, according to contemporary seismic codes, at least seven triads (or pairs, in the case where the vertical seismic component is ignored) of accelerograms must be taken into consideration and then, the average of the response quantities from all these analyses should be used as the "final design values" of the seismic action effect. Note that, for each pair (or triad) of accelerograms, the latter must act simultaneously (sect.3.2.3.1.1(2)P/EN-1998.01) and be "statistically independent". This means that the "correlation factor" among these accelerograms must be zero (uncorrelated accelerograms). However, each accelerogram of each pair must be represented by the same response acceleration spectrum (sect.3.2.2.1(3)P/EN-1998.01). The acceleration response spectra (with 0.05 equivalent viscous damping ratio) of artificial (or recorded) accelerograms should match the elastic response spectra for 0.05 equivalent viscous damping ratio. This should occur over a wide range of periods (or at least between 0.2T and 2T, where T is the fundamental period of the structure), as is defined by sect.3.2.3.1.2(4c)/EN-1998.01. Also, no value of the "mean elastic spectrum", which is calculated based on all the used accelerograms, should be less than 90% (or greater than 110%) of the corresponding value of the tagret elastic response spectrum. The peak ground acceleration of each accelerogram must always be equal to g aS  , where g a is the design ground acceleration of the local area and www.intechopen.com S is the soil factor according to EN-1998.01. Simultaneously, for strong earthquakes, the minimum duration of the strong motion of each accelerogram must be sufficient (about 15s in common cases), whilst, from personal observations, I consider that there should be a minimum number of approximately ten "large" and thirty-five "significant" loading loops, due to the dynamic seismic cyclic loading (i.e. a "large" cycle has an extreme ground acceleration of over 0.75PGA and a "significant" cycle has an extreme ground acceleration of between 0.30PGA and 0.75PGA). However, the subject of the exact definition of the Design Basis Earthquake and the Maximum Capable Earthquake of a seismic hazard area is open to question, while recently, a new framework for the simulation and definition of the seismic action of Design Earthquake Basis for the inelastic single degree of freedom system, using the Park & Ang damage index on the structures, has been developed (Moustafa, 2011;). ii. It is possible for the numerical models from the building simulation to be inadequate, because many false assumptions about the non-linear dynamic properties of plastic hinges can be inserted into the building model. For example, each non-linear spring must possess a suitable dynamic model M   for cyclic dynamic seismic loading (Dowell et al, 1998;Takeda et al, 1970;Otani, 1981;Akiyama, 1985;), where after each loading cycle, a suitable "stiffness degradation" must be taken into account (Fig.3). iii. It is possible for the numerical integration method to be inadequate, as regards accuracy & stability. iv. The critical dynamic loading orientation of the pair of horizontal seismic components is unknown or does not exist and leads to the examination of various other orientations (at least one more orientation with a rotation of 45 degrees relative to the initial principal horizontal orientation must be examined). Alternative, a special methodology about the examination of various orientations usimg pairs of accelerograms, gives the envelope of the most unfavourable results (in the linear area only) was presented by Athanatopoulou (2005). Note that, for each pair of accelerograms, all possible sign combinations of seismic components must be taken into account, resulting in four combinations for each pair. Moreover, seven (7) pairs must be taken into account. In addition, due to the accidental eccentricities, four different positions of the floor mass centres must be considered. Therefore, (2 orientations)x(4 combinations of signs per pair)x(7 pairs)x(4 positions of the mass centres)=224 NLRH Analyses. Alternatively, in order to reduce the number of said analyses, the action of accidental eccentricities can be taken into account using equivalent external floor moments M m,i around the vertical axis, which are calculated by the floor lateral static forces, . In this case, we arrive at a total number of fifty six (56) NLRH Analyses. Consequently, the most important issues that we come across during the static pushover analysis are the following: (1) Why is the use of a single degree of freedom system that represents the real structure required? (2) How can the spatial action of the two horizontal seismic components during the static pushover analysis be taken into account? (3) What is the most suitable monitoring point in the case of asymmetric buildings? (4) Which distribution of lateral floor static forces is most suitable? and (5) At which point, in the plan of an asymmetric building, should the lateral floor static forces be applied? The design dynamic eccentricities and accidental eccentricities must also be taken into account, which means that the real or fictitious elastic axis of the building with the real or fictitious horizontal main elastic axes, where the lateral static forces must be oriented during the static pushover analysis, must be calculated. On the other hand, the Non-Linear Response History Procedure has a high computational cost and, simultaneously, many of its results present a high sensitivity. For these reasons, in practice, the mathematically documented static non-linear (pushover) analysis, presented below in this chapter, is a reliable alternative approximate method, which can envelope the accurate seismic demand floor inelastic displacements by the above Non-L i n e a r R e s p o n s e H i s t o r y P r o c e d u r e , i n a r a t i o n a l w a y . H o w e v e r , i t i s k n o w n t h a t t h e remaining seismic demands, such as the inter-storey drift ratios, real distribution of the plastic hinges on the building, yielding/failure mechanism of the building, etc, are only predicted correctly by using Non-Linear Response History Analysis. Jm l  ii due to its diaphragm (where s,i l is the floor inertia radius). Every diaphragm is supported on massless, axially loaded columns and structural walls, in reference to the global Cartesian mass system CM(x,y,z) that is parallel to the known fictitious principal Cartesian reference system   o P I,II,III , as presented in the previous paragraph. Next, the original point CM has three degrees of freedom for each floor; two horizontal displacements x,i u and y,i u relative to the ground, along axes x and y respectively, and a rotation z,i θ around the vertical z-axis (Fig.4).

The equivalent non-linear SDF system of asymmetric multi-storey buildings in non-linear static pushover analysis
Moreover, for the needs of this mathematical analysis, we consider that this N-storey building is loaded with the 3N-dimensional loading vector P of external lateral floor static loading, where the floor forces are oriented in parallel to one fictitious principal axis (i.e. along the IIaxis) of the building. The loading vector P relating to the 3N-dimensional vector u of the degrees of freedom of the floor masses of the asymmetric multi-storey building is formed according to the pure translational (along the principal II-axis) mode-shape's distribution of lateral floor static forces in elevation (Eqs.2a,b). For the needs of this analysis, the lateral floor static forces of the loading vector P in the plan, are located at I max e distance from the fictitious stiffness centre o P to the mass centre of the building (Fig.4a). Next, a primary static pushover analysis of the asymmetric N-storey building is performed, using this increased static loading vector P, until the collapse of the building model.    Next, consider the known pure translational mode shape distribution Y of the lateral floor forces in elevation (from the linear modal analysis), where, Eq.(4); In order to define the optimum equivalent Non-Linear SDF system, the following two assumptions are set (Makarios, 2009): of the inelastic displacement distribution of the floor mass centres of the asymmetric multi-storey building is a 'notional mode-shape'. 2 nd Assumption: The lateral force P causes translational and torsional accelerations on the concentrated mass of the asymmetric multi-storey building. Therefore, if we consider that the vector loading P of the external loads is a function of time t, i.e. () t P is a dynamic loading, this function is written as follows in Eq. (5) f( ) a b  t= t + , where a & b are coefficients) and i P, is the value of the floor lateral force. Therefore, the equation of the motion of the masses of the multi-storey building, which are loaded with the dynamic loading P(t), is described by the mathematical system of linear second order differential N-equations, which can be written in matrix form, Eq. (6): where o () t u is the 3N-dimensional vector of displacements from Eq.(3), Κ and Μ are the 3Nx3N-dimensional square symmetric matrices (Eqs.7a,b) of lateral elastic stiffness and masses respectively, while C is the damping 3Nx3N matrix.
where, Jm l  ii is the mass inertia moment of the diaphragm (floor) around the vertical axis passing through the mass centre, and s,i l is the floor inertia radius. The floor lateral stiffness sub-matrix i , k , as well as the global lateral stiffness matrix K of the building, are calculated with reference to the global Cartesian mass system CM(x,y,z) using a suitable numerical technique pertaining to the "stiffness condensation" of the total stiffness matrix. Alternatively, we enforce a displacement (that is equal to one) of a floor mass centre along one degree of freedom and simultaneously all the remaining degrees of freedom of the building are fixed. The generated reactions on the degrees of freedom provide the coefficients of a column of the global lateral stiffness matrix K, and this procedure is repeated for all other mass degrees of freedom of the building. Using Eqs.
Next, we pre-multiply Eq.
Eq.(12) presents an equation of the motion of a single-degree-of-freedom (SDF) system. This equation must be transformed, so that the "optimum equivalent Non-linear SDF system" represents the initial asymmetric multi-storey building. As we can see in Eq.(12), the degree of freedom is the displacement N , y u of the mass centre CM, while loaded with the base shear oy V . Thus, the "effective lateral stiffness" * k of this SDF system must be obtained through an additional, special, secondary, non-linear (pushover) analysis of the initial, asymmetric multi-storey building, where the lateral static force is applied to the mass centre CM along the y-axis (Fig.4b). Hence, the lateral stiffness * k of the first branch and the lateral stiffness  * α k of the second branch of the bilinear diagram o, N yy V-u are the result of this special, secondary pushover analysis (Fig.5a). Therefore, we can consider that the slopes * k and  * α k are already known from the secondary pushover analysis. Thus, for reasons of convergence between the known analytic SDF system of Eq.(12) and the (unknown at present) required optimum equivalent NLSDF system of the asymmetric multi-storey building, Eq.(12) is multiplied with the coefficient where, Therefore, the optimum equivalent NLlinear SDF system is represented by Eq.(13) and is presented in Fig.(6a). In addition, we can assume that the NLSDF system possesses the equivalent viscous damping * c , hence: where, * is the circular frequency (in rad/s) of the equivalent vibrating SDF system in the linear range: and  is the equivalent viscous damping ratio that corresponds to the critical damping of the SDF system (about 5% for reinforced concrete).
In order to transform the diagram o, N yy V-u of Fig.(5a,b) into the capacity curve ** P-δ of Fig.(5b) of the equivalent NLSDF system, factor  is directly provided by:   , i ι Moreover, y ι is the global 'influence vector', that represents the displacements of the masses resulting from the static application of a unit horizontal ground displacement of the building along the seismic loading (i.e. y-direction). Hence, the optimum equivalent NLSDF system is characterized by Eq.(13) and has a bilinear capacity curve ( ** P-δ ) according to Fig.(5b). The maximum elastic base shear * el P of the infinitely elastic SDF system, the yielding base shear * y P of the respective equivalent non-linear SDF system, its ultimate base shear * u P , its yielding displacement * y  and its ultimate displacement * u  , are given ( Fig.5b): The "effective period" * T of the optimum equivalent NLSDF system is given by Eq. Next, in order to evaluate the demand seismic displacement of the asymmetric multi-storey r/c building, we must first calculate the seismic target-displacement t,inel * δ of the optimum equivalent NLSDF system using the known Inelastic Response Spectra, since this is the only credible option; otherwise, a Non-Linear Response History procedure must be performed. Therefore, the inelastic seismic target-displacement t,inel * δ is calculated directly using Eq.(22), Fig.(6b). Secondly, we must transform the inelastic seismic target-displacement t,inel * δ into the target-displacement of the monitoring point (namely, the centre mass at the top level of the real initial asymmetric multi-storey building, along the examined direction), multiplying it with factor  , Eq.(28). Finally, the maximum required seismic displacements of the other positions of the multi-storey building can be obtained from that step in the known primary pushover analysis, where the seismic target-displacement of Eq.(28) of the monitoring point appears.
a. b. Fig. 6. a. The optimum equivalent NLSDF system, b. The seismic target-displacement of the equivalent NLSDF system.
More specifically, with reference to the optimum equivalent NLSDF system, its elastic spectral acceleration * a S is calculated by the elastic response acceleration spectrum, for the known period * T . Consequently, the maximum required inelastic seismic displacement (target-displacement) t δ of the new equivalent NLSDF system arises from the inelastic spectrum of a known earthquake, according to the following equations: , then the response is non-linear and post-elastic and thus the targetdisplacement is inelastic, because this means that the yielding base shear * y P of the NLSDF system is less than its elastic seismic base shear; therefore this system yields:   where, * aa () * S= S T and * dd () * S= S T are the "elastic spectral acceleration" and the "elastic spectral displacement" from the elastic response acceleration and displacement spectrum respectively, for equivalent viscous damping ratio 0.05 ξ = for r/c structures, ** ye l y R =P P is the reduction factor of the system (Fig.5b) * o L =k k the "convergence factor" between the initial Multi Degree of Freedom system and the NLSDF d μ is the demand ductility of this non-linear SDF system (Eq.23): c is a coefficient due to the second branch slope from Eqs. (24,25,26), using a suitable slope ratio = 0%, 2%, 10% α of the equivalent NLSDF system (Fig.5b), according to Krawinkler & Nassar (1992) Note that the demand ductility d μ by Eq.(23) simultaneously satisfies both the rule of "equal energies" and the rule of "equal displacements" according to Veletsos & Newmark (1960) and Newmark & Hall (1982).
, then the response is linear elastic and thus the target-displacement * t,el δ is elastic, because this means that the yielding base shear * y P of the NLSDF system is greater than the elastic seismic base shear and therefore this system remains in the linear elastic area, It is worth noting that, in order to achieve adequate seismic stability of the building, the demand ductility d μ must be less than the available ductility ** uy μ = δδ. Next, in the first www.intechopen.com approach (index "1"), the maximum required seismic displacement ,N,1 y u of the mass at the top level of the asymmetric multi-storey building, along y-direction, is directly given by: To sum up, the above-mentioned optimum equivalent Non-Linear Single Degree of Freedom System allows us to provide the definition of the behavior factor of the multi-storey building. Indeed, it is well known that the global behavior factor q of a system is mathematically defined by the single degree of freedom system only. Therefore, in order to define the behavior factor of an asymmetric multi-storey building, an equivalent SDF system (such as the above optimum equivalent NLSDF system) must be estimated at first, and then its behavior factor, which refers to the asymmetric multi-storey building, is easy to determine. According to recent research (Makarios 2010), the "available behavior factor" av q of the optimum equivalent NLSDF system indirectly represents the "global available behavior factor" of the asymmetric multi-storey building, while av q is given by Eq.(29). Note that, in order for a building to cope with seismic action, the total "available behavior factor" av q has to be higher than the "design behavior factor" q, that was used for the initial design of the building.
where o q is the "overstrength partial behavior factor" that is part of the av q and is related to the "extent of static-indefiniteness / superstaticity" of the asymmetric multi-storey building, as well as the constructional provisions of the Reinforced Concrete Code used, and can be Furthermore, m q is the "ductility partial behavior factor" that is another part of av q that is related to the ductility of the equivalent elasto-plastic SDF system, and is calculated by Eq.(31), which is an inversion of Eq.(23): www.intechopen.com where uy   ** μ = is now the "available ductility" of the NLSDF system.
Next, p q is the "post-elastic slope partial behavior factor" that is the last part of the av q and is related to the slope of the post-elastic (second) branch of the bilinear capacity curve of the equivalent NLSDF system and is calculated by Eq.(32): Note that, if the second (post-elastic) branch has a negative slope (i.e. intense action of second order phenomena), then the "post-elastic slope partial behavior factor" p q is less than one. Next, with reference to the spatial seismic action (signs of seismic components, simultaneous action of the two horizontal seismic components) in the presented non-linear static (pushover) procedure, we can observe the following issues: Being aware, from the above mathematical analysis, that two static pushover analyses are needed for one location of lateral floor static forces, we can conclude that, if we use the final four design eccentricities, then twelve (12) individual static pushover analyses are required, as following: -For the first seismic horizontal component along principal I-axis, six load cases are needed, Fig.(7). Note that from each primary analysis (where one "final design eccentricity" out of the four is used), the demand seismic displacements of the asymmetric multi-storey building are obtained, whilst from the secondary analysis (where the lateral static forces are located on the mass centres) the effective lateral stiffness * k of its NLSDF is defined. Then, the twelve individual pushover analyses are  (Makarios, 2009(Makarios, & 2011, where the correctness of these final results has been verified by the Non-Linear Response History Analysis. In addition, contemporary Sesmic Codes, such as Eurocode EN-1998, do not give details and quidance regarding the static seismic pushover analyis of irregular in plan, asymmetric, multistorey r/c buildings. In reality, Annex B of EN-1998.01 refers to the calculation of the seismic target-displacement of planar frames only, whilst for spatial irregular in plan multistorey buildings there are no relevant provisions. The present work intends to precisely cover this gap. The present non-linear static (pushover) procedure is a natural extension of the established equivalent linear (simplified spectral) static method, recommended by the Seismic Codes (EAK/2003, NBCC/1995, which are adequately documented as regards torsional provisions. The twelve static pushover analyses of the irregular asymmetric building involve an important computational cost, but on the other hand, this is much lower compared to the two hundred and twenty-four (224) non-linear dynamic analyses that the Non-Linear Response History Procedure requires. Thus, the presented non-linear static (pushover) procedure is a very attractive option, despite its twelve non-linear static analyses. In addition, within the framework of the proposed procedure, there is also natural supervision, which constitutes a very important issue.

Conclusions
Consequently, the most important issues that we come across during the static pushover analysis, in order to avoid the exact non-linear response history analysis, have been noted. More specifically, we have explained (1) why it is necessary to use a single degree of freedom system that represents the real structure; (2) how the spatial action of the two horizontal seismic components during the static pushover analysis can be taken into account; (3) what is the most suitable monitoring point in the case of asymmetric buildings; (4) which distribution of the lateral floor static forces is suitable; and (5) at which point in the plan of an asymmetric building, lateral floor static forces should be applied. Design dynamic eccentricities plus accidental eccentricities must be taken into account. This point leads to the calculation of the real or fictitious elastic axis of the building and the real or fictitious horizontal main elastic axes, where the lateral static forces must be oriented during the static pushover analysis (Fig.1). Extensive guidance has been given about the simulation of plastic hinges using Eurocode EN/1998.03. Moreover, various important issues have been discussed concerning seismic non-linear response history analysis. Next, a documented proposal of the optimum equivalent NLSDF system that represents asymmetric multi-storey buildings is presented. The definition of the NLSDF system is mathematically derived, by studying suitable dynamic loadings on the masses of each r/c system and using simplified assumptions. The coupling of the translational and the torsional degrees-of-freedom of the asymmetric building has also been taken into consideration, without dividing the asymmetric building into various, individual (two planar) subsystems. This NLSDF system is used in combination with the inelastic design spectra in order to calculate the seismic demands. The natural meaning of the characteristics of the asymmetric building, such as the fundamental eigenperiod, is not distorted by the optimum equivalent NLSDF system, since both the fundamental period of the multi-storey building and the period of the NLSDF system are very close, due to the fact where the total methodology is derived in a mathematical way. Extended numerical comparisons have verified the correctness of the use of the equivalent NLSDF system (Makarios, 2009(Makarios, & 2011). However, on the other hand, it is a known fact that, the sequence of plastic hinges on a building and the yielding/failure mechanism of multi-storey buildings are not correctly calculated by the various pushover analyses, which can only provide approximate results; nevertheless, this question is beyond the scope of the present chapter.