S_d=∆/(Γϕ_n )
Γ=({ϕ}^T [M]{1})/({ϕ}^T [M]{ϕ})
α=〖[∑_(j=1)^N▒〖m_i ϕ_i 〗]〗^2/(M∑_(j=1)^N▒〖m_i 〖ϕ_i〗^2 〗)
Response Spectra: Obtain or calculate response spectra converted into the ADRS format as the capacity spectrum in in Sa-Sd format.
Graphical Solution: Plot capacity spectrum and family of damped response spectra on an ADRS format (i.e. S vs S coordinates with period T lines radiating from origin). The a d intersection of the capacity spectrum with the appropriately damped response spectrum represents the estimated demands of the earthquake on the structure.
When the displacement at the intersection of the demand spectrum and the capacity spectrum is within 5 percent of the displacement of the trial performance point, this point becomes the performance point. If the intersection of the demand spectrum and the capacity spectrum is not within the acceptable tolerance, then a new api, dpi point is selected and the process is repeated. The performance point represents the maximum structural displacement expected for the demand earthquake ground motion.
Fig 04
The capacity curve is determined by statically loading the structure with realistic gravity loads combined with a set of lateral forces to calculate the roof displacement ∆ and base shear V , that defines first significant yielding of structural elements. The yielding elements are then relaxed to form plastic hinges and incremental lateral loading is applied until a nonlinear static capacity curve is created. The curve is created by superposition of each increment of displacement and includes tracking displacements at each story (ATC 1982). This procedure is sometimes referred to as the pushover analysis.
There are several levels of sophistication that may be used for the pushover analysis, ranging from applying lateral forces to each story in proportion to the standard code procedure to applying lateral story forces as masses times’ acceleration in proportion to the first mode shape of the elastic model of the structure. For added sophistication, at each increment beyond yielding, the forces may be adjusted to be consistent with the changing deflected shape. For tall buildings the effects of the higher modes of vibration may be considered (12).
Freeman s,a 1998 (10), The CSM is applicable to a variety of uses such as a rapid evaluation technique for a large inventory of buildings, a design verification procedure for new construction of individual buildings, an evaluation procedure for an existing structure to identify damage states, and a procedure to correlate damage states of buildings to amplitudes of ground motion. The procedure has been successfully used to correlate recorded motion and observed performance for buildings that have been subjected to various earthquake ground motions, such as those from the San Fernando (1971), Loma Prieta (1989), or Northridge (1994) earthquakes. The CSM stands up well when compared to other procedures such as the equal displacement method and the secant methods and has the added advantage of giving the engineer the opportunity to visualize the relationship between demand and capacity.
Chopra A.K and Geol 1999 (12), Presented next are two improved procedures that eliminate the errors (or discrepancies) in the ATC-40 CSM procedures, The improved procedures use the well-known constant-ductility design spectrum for the demand diagram, instead of the elastic design spectrum for equivalent linear systems in ATC-40 procedures. the improved procedures differ from ATC-40 procedures in one important sense. The demand diagram used is different: the constant ductility demand diagram for inelastic systems in the improved procedure versus the elastic demand diagram in ATC-40 for equivalent linear systems. The improved method can be conveniently implemented numerically if its graphical features are not important to the user. Such a procedure, based on equations relating the yield strength reduction factor, Ry , and ductility factor, ~, for different period, Tn’ ranges, has been presented, and illustrated by examples using three different R, – ~ – Tn relations.
Fajfar, P., 1999. (14), the idea of using inelastic demand spectra within the capacity spectrum method has been elaborated and is presented in an easy to use format. The approach represents the so-called N2 method formulated in the format of the capacity spectrum method. By reversing the procedure, a direct displacement-based design can be performed. The application of the modified capacity spectrum method is illustrated. The seismic demand in the capacity spectrum method can be represented by inelastic spectra. In principle, any realistic inelastic spectra can be used. However, they should be compatible with the basic elastic spectrum. The specific demand spectra applied in this paper are simple and reasonably accurate for a broad range of design situations. It has been shown that the performance evaluation procedure, called the N2 method, can be formulated in the format of the capacity spectrum method. Furthermore, by reversing the procedure, a direct deformation based design can be performed.
The N2 Method
Peter FAJFAR and Matej FISCHINGER 1988 (15), proposed the N2 method where two mathematical models and three steps of analysis are used. In the first step, stiffness, strength and supplied ductility are determined by the non-linear static analysis of a MDOF system under a monotonically increasing lateral load (pushover curve). Then, in the second step, an equivalent single-degree-of-freedom (SDOF) system is defined. Here, it must be assumed that the deflected shape does not change during an earthquake. The non-linear characteristics of the equivalent system are based on the base shear top displacement relationship, obtained by the non-linear static analysis in the first step. In the third step of N2, maximum displacements (and the corresponding ductility demand) are determined by carrying out non-linear dynamic analysis of the equivalent SDOF system. Dynamic analysis, in its simplest form, can also be performed by using inelastic response spectra.
Peter Fajfar 2000 (16), purposed a version of the N2 method that combines the advantages of the visual representation of the capacity spectrum method, developed by Freeman (Freeman et al. 1975, Freeman 1998) (10), with the sound physical basis of inelastic demand spectra. The inelastic spectra have been used in such a context also by Goel and Chopra (1999) (12). The N2 method, in its new format, is in fact a variant of the capacity spectrum method based on inelastic spectra. Inelastic demand spectra are determined from a typical smooth elastic design spectrum. The reduction factors, which relate inelastic spectra to the basic elastic spectrum, are consistent with the elastic spectrum. The lateral load pattern in pushover analysis is related to the assumed displacement shape. This feature leads to a transparent transformation from a multi degree-of-freedom (MDOF) to an equivalent single-degree-of-freedom (SDOF) system.
The steps of the simple version of the N2 method are described as mentioned below. A simple version of the spectrum for the reduction factor is applied and the influence of cumulative damage is not taken into account.
DATA: A planar MDOF structural model is used. In addition to the data needed for the usual elastic analysis, the nonlinear force – deformation relationships for structural elements under monotonic loading are also required. The most common element model is the beam element with concentrated plasticity at both ends. A bilinear or trilinear moment – rotation relationship is usually used. Seismic demand is traditionally defined in the form of an elastic (pseudo)-acceleration spectrum Sae (“pseudo” will be omitted in the following text), in which spectral accelerations are given as a function of the natural period of the structure T. The specified damping coefficient is taken into account in the spectrum
SEISMIC DEMAND IN AD FORMAT, Starting from the acceleration spectrum, we will determine the inelastic spectra in acceleration – displacement (AD) format. For an elastic SDOF system, the following relation applies.
S_de=T^2/(4π^2 ) S_ae
Where Sae and Sde are the values in the elastic acceleration and displacement spectrum, respectively, corresponding to the period T and a fixed viscous damping ratio. A typical smooth elastic acceleration spectrum for 5% damping, normalized to a peak ground acceleration of 1.0 g, and the corresponding elastic displacement spectrum, is shown in Figure 1a. Both spectra can be plotted in the AD format.
For an inelastic SDOF system with a bilinear force – deformation relationship, theacceleration spectrum (Sa) and the displacement spectrum (Sd):
S_a=S_ae/R_μ
S_d=μ/R_μ S_de=μ/R_μ T^2/(4π^2 ) S_ae 〖=μ T^2/(4π^2 ) S〗_a
where μ is the ductility factor defined as the ratio between the maximum displacement and the yield displacement, and R μ is the reduction factor due to ductility, i.e., due to the hysteretic energy dissipation of ductile structures.
Several proposals have been made for the reduction factor Rμ. An excellent overview has been presented by Miranda and Bertero (17). In the simple version of the N2 method, we will make use of a bilinear spectrum for the reduction factor R μ
R_μ=(μ-1) T/T_C +1 T<TC
R_μ=(μ) T≥TC
where TC is the characteristic period of the ground motion. It is typically defined as the transition period where the constant acceleration segment of the response spectrum (the short-period range) passes to the constant velocity segment of the spectrum (the medium period range).
Pushover curve is determined using the distribution of lateral loads assuming that lateral force in the i-th level is proportional to the component of the assumed displacement mode shape weighted by the story mass mi and the displacement shape was exact and constant during ground shaking, then the distribution of lateral forces would be equal to the distribution of effective earthquake forces.
equivalent SDOF model is derived from the capacity diagram as follows:
m^*=∑_(i=1)^n▒〖m_i ∅_i 〗
where m* is the equivalent mass of the SDOF system
D^*=∆/Γ
F^*=V/Γ
Where and D* and F* are the displacement and force of the equivalent SDOF system
Γ=({ϕ}^T [M]{1})/({ϕ}^T [M]{ϕ})=(∑_(i=1)^n▒〖m_i ∅_i 〗)/(∑_(i=1)^n▒〖m_i 〖∅_i〗^2 〗)=m^*/(∑_(i=1)^n▒〖m_i 〖∅_i〗^2 〗)
Assumed displacement shape is normalized – the value at the top is equal to 1. Note also that any reasonable shape can be used for ϕ a special case; the elastic first mode shape can be assumed. Γ is equivalent (but, in general, not equal) to PF1 in capacity spectrum method, and to C0 in the displacement coefficient method (ATC 40 and FEMA 273).
Finally, the capacity diagram in AD format is obtained by dividing the forces in the
force – deformation (F* – D*) diagram by the equivalent mass m*
S_(a,in)=F^*/m^*
S_(d,in)=D^*
seismic demand for the equivalent SDOF system, The seismic demand for the equivalent SDOF system can be determined by using the graphical procedure illustrated in Figure 5 (for medium- and long-period structures; for short-period structures 1). Both the demand spectra and the capacity diagram have been plotted in the same graph. The intersection of the radial line corresponding to the elastic period of the idealized bilinear system T*
T^*=2π√((m^* D_y^*)/f_y )
R_µ=(S_ae (T^*))/S_ay
Note that Rµ is not the same as the reduction (behaviour, response modification) factor R used in seismic codes. The code reduction factor R takes into account both energy dissipation and the so-called over strength. The design acceleration Sad is typically smaller than the yield acceleration Say.
where the elastic demand spectrum Sae defines the acceleration demand (strength) required for elastic behavior and the corresponding elastic displacement demand. The yield acceleration Say represents both the acceleration demand and the capacity of the inelastic system. The reduction factor Rμ can be determined as the ratio between the accelerations corresponding to the elastic and inelastic systems
Fig 05.
Based on the equal displacement rule, the the inelastic displacement demand Sd is equal to the elastic displacement demand Sde for the elastic period T* is larger than or equal to TC.
For T≥ TC
µ=S_d/(D_y^* )
R_µ=µ
S_d=S_de (T^*)
For T< TC
S_d=µ/R_µ S_de
S_d=D_y^*µ=S_de/R_µ [(R_µ-1)*T_C/T^* ]
Steps 6 and 7: global and local seismic demand for the mdof model ,The displacement demand for the SDOF model Sd is transformed into the maximum top displacement Dt of the MDOF system (target displacement) by using the following equation
D^*=∆/Γ
F^*=V/Γ
In general, the results obtained using the N2 method are reasonably accurate, provided that the structure oscillates predominantly in the first mode. Applications of the method are, for the time being, restricted to the planar analysis of structures. The inelastic demand spectra, used in the proposed simple version, are not appropriate for near-fault ground motions, for soft soil sites, for hysteretic loops with significant pinching or significant stiffness and/or strength deterioration, and for systems with low strength.
Pushover analysis is based on a very restrictive assumption, i.e. a time-independent displacement shape. Thus, it is in principle inaccurate for structures where higher mode effects are significant, and it may not detect the structural weaknesses which may be generated when the structure’s dynamic characteristics change after the formation of the first local plastic mechanism. A detailed discussion of pushover analysis can be found in the paper by Krawinkler and Seneviratna (1998) (6) where the pro and cons of the standard pushover methods were discussed Limitations are imposed also by the load pattern choices. Whatever load pattern is chosen, it is likely to favour certain deformation modes that are triggered by the load pattern and miss others that are initiated and propagated by the ground motion and inelastic dynamic response characteristics of the structure. The simplest example is a structure with a weak top story. Any invariant load pattern will lead to a concentration of inelastic deformations in the top story and may never initiate inelastic deformations in any of the other stories. Thus, good judgment needs to be employed in selecting load patterns and in interpreting the results obtained from selected load patterns.. One practical possibility to partly overcome the limitations imposed by pushover analysis is to assume two different displacement shapes (load patterns), and to envelope the results. Also, Modal pushover analysis methods are used in order to capture the higher modes effect.
Advanced nonlinear static analysis techniques:
The nonlinear static analysis procedures considered an appealing method of analysis in order to capture the behavior of the nonlinearity induced from earthquake excitation, There are good reasons the use of the inelastic pushover analysis for demand prediction, since it provide much more accurate information than an the elastic static analysis ,but it must not addresses this method as a solution in all cases. The pushover analysis is a useful, but not effective tool for assessing inelastic strength and deformation demands and for exposing design weaknesses in all cases . Accordingly it’s limitation must identified clearly as mentioned below (4) (19):
Limitations are imposed also by the load pattern choices. Whatever load pattern is chosen,
time-independent displacement shape. and it may not detect the structural weaknesses, which may be generated when the structure’s dynamic characteristics change after the formation of the first local plastic mechanism
proposed adaptive force distributions that attempt to follow the time-variant distributions of inertia forces
Analysis of planar structures. torsional flexible plan-asymmetric buildings
Accordingly in order to overcome this limitation, advanced pushover analysis methods were introduced, each to overcome the above mentioned limitations.
Modal Pushover Analysis
Anil K. chopra (20), Introduce an improved pushover analysis method based on the same theoretical basics of the procedure with invariant force distribution, but provides superior accuracy in estimating seismic demands on buildings. Similar to the standard response spectrum analysis (RSA), as to account for the effects of higher modes on structural response and it was suggested that by considering the contributions of a sufficient number of modes, The MPA was performed on case study frames and a comparative evaluation of MPA and traditional pushover analysis with invariant lateral load patterns in predicting the seismic demands was conducted. the procedure was applied to linearly elastic buildings and it was shown that the procedure is equivalent to the response spectrum analysis. Then, the procedure was extended to estimate the seismic demands of inelastic systems by describing the assumptions and approximations involved. Earthquake induced demands for a 9-story SAC building were determined by MPA, nonlinear dynamic analysis and pushover analysis using uniform, “code” and multi-modal load patterns. The comparison of results indicated that pushover analysis for all load patterns greatly underestimates the story drift demands and lead to large errors in plastic hinge rotations. The MPA was more accurate than all pushover analyses in estimating floor displacements, story drifts, plastic hinge rotations and plastic hinge locations. MPA results were also shown to be weakly dependent on ground motion intensity based on the results obtained from El Centro ground motion scaled by factors varying from 0.25 to 3.0. It was concluded that by including the contributions of a sufficient number of modes (two or three), the height-wise distribution of responses estimated by MPA is generally similar to the ‘exact’ results from nonlinear dynamic analysis.
The seismic demand due to individual terms in the modal expansion of the e<ective earthquake forces is determined by a pushover analysis using the inertia force distribution for each mode. Combining these ‘modal’ demands due to the first two or three terms of the expansion provides an estimate of the total seismic demand on inelastic systems. When applied to elastic systems, the MPA procedure is shown to be equivalent to standard response spectrum analysis (RSA). When the peak inelastic response of a 9-storey steel building determined by the approximate MPA procedure is compared with rigorous non-linear response history analysis, it is demonstrated that MPA estimates the response of buildings responding well into the inelastic range to a similar degree of accuracy as RSA in estimating peak response of elastic systems. Thus, the MPA procedure is accurate enough for practical application in building evaluation and design.
The MPA procedure developed to estimate the seismic demands on inelastic systems is
Organized in two phases: First, a pushover analysis is used to determine the peak response rno of the inelastic MDF system to individual terms, The base shear–roof displacement (Vbn − urn) curve is developed from a pushover analysis for the force distribution s∗n . This pushover curve is idealized as a bilinear force–deformation relation for the nth-‘mode’ inelastic SDF system (with vibration properties in the linear range that are the same as those of the nth mode elastic SDF system), and the peak deformation of this SDF system—determined by nonlinear response history analysis (RHA) or from the inelastic response or design spectrum is used to determine the target value of roof displacement at which the seismic response rno is determined by the pushover analysis. Second, the total demand ro is determined by combining the rno according to an appropriate modal combination rule (e.g. SRSS rule).
The results from chopra analysis lead to the following conclusions, Comparing the peak inelastic response of a 9-storey SAC steel building determined by the approximate MPA procedure—including only the first two or three modes with nonlinear RHA demonstrated that the approximate procedure provided good estimates of floor displacements and storey drifts, and identified locations of most plastic hinges; however, plastic hinge rotations were less accurate. MPA estimates the response of buildings responding well into the inelastic range to similar degree of accuracy as standard RSA is capable of estimating peak response of elastic systems. Thus, the MPA procedure is accurate enough for practical application in building evaluation and design. That said, however, all pushover analysis procedures considered do not seem to compute accurately local response quantities, such as hinge plastic rotations.
Modified MPA Procedure was introduced by Anil K. chopra (4),