Iterative displacement coefficient method : mathematical formulation and numerical analyses

The mathematical formulation and numerical tests of the originally developed Iterative Displacement Coefficient Method (IDCM) are presented in the paper. The IDCM method is a procedure in the field of seismic analysis of structures that is based on the Nonlinear Static Pushover Analysis. The target displacement level is determined by means of a double iterative procedure developed in the IDCM. The first iterative procedure is carried out simultaneously by forces and displacements, while the second iterative procedure is conducted by correction of the solution obtained for the level of target displacement along a pushover curve. The IDCM method is implemented in the computer code Nonlin Quake TD.


Introduction
A new method based on the nonlinear model of structural behaviour due to seismic action, broadly called the Nonlinear Static Pushover Analysis or NSPA, has been developing over the past two decades, and an extensive research aimed at its further improvement is still under way.The NSPA analysis is founded on the modelling of geometrically and materially non-linear behaviour of structures, while treating seismic actions as a static load, explicitly through forces or implicitly through displacements.Therefore, the NSPA idea is based on the principle involving the closest possible approximation in numerical model, and the nonlinear material and structural behaviour, while seismic forces in incremental steps are not determined in time domain, but rather in the capacity domain.In such a way, the time needed for numerical simulations is greatly reduced with respect to computing time needed in the Incremental Nonlinear Dynamic Analysis or INDA.Also, there is a considerable reduction in the quantity of processing results that are necessary in additional treatment and presentation.The NSPA analysis is generally conducted in two phases.The first phase is performed using the multi-degree-offreedom (MDOF) model, while in the second phase the target displacement analysis is done using the singledegree-of-freedom (SDOF) system, or a direct approach is used.From the mathematical aspect, the numerical model used in the NSPA is a discrete model, formulated using the Finite Element Method (FEM).It is particularly emphasized that complex numerical models can efficiently be formulated by the finite element discretization, with realistic presentation of important structural features, including relevant details.The NSPA MDOF analysis may be independent from the target displacement analysis in the sense that by using it one could analyse key structural parameters, such as the load capacity, stiffness and ductility in the linear, non-linear or collapse domain.The diagnosis of these parameters is important in evaluation of the current state of structures, and in assessing structural behaviour in future earthquakes.The target displacement analysis may not be independent, since the procedure to determine target displacements depends on the type and character of the pushover curve (PC).

Summary of existing methods for target displacement analysis
The development of the NSPA concept and analyses of target displacement of buildings due to seismic actions was initiated more then two decades ago, and official implementations are presented in ATC 40 [1], FEMA 273 [2], Eurocode EC 8 [3], FEMA 356 [4] and FEMA 440 [5] standards.There is nowadays a wide range of NSPA analyses and target displacement analyses but, by their efficiency and superiority, the following ones may be selected: -Capacity Spectrum Method (CSM) -Displacement Coefficient Method (DCM) -Equivalent Linearization Method (ELM) -Displacement Modification Method (DMM) -N2 Method -Modal Pushover Analysis (MPA) -Method of Modal Combinations (MMC) The Capacity Spectrum Method is implemented in the code ATC 40 [1], while the research related to the development, testing and improvement of this method is presented in [6,7].According to the Capacity Spectrum Method, the level of target displacements is determined on the basis of intersection of the pushover curve and the demand curve, which is constructed by an iterative procedure.A graphical presentation of the pushover curve, capacity curve, elastic response spectrum, and radial lines representing periods of vibrations, is given in the ADRS format (accelerationdisplacement response spectrum).In addition, an important research related to the Capacity Spectrum Method with additional improvements, presented as the AutoCSM method, is given in [8,9].A further step in the improvement of the capacity spectrum method involves development of the Adaptive Capacity Spectrum Method (ACSM), cf.[10], while other significant improvements are given in the Improved Capacity Spectrum Method (ICSM), cf.[11][12][13][14].On the other hand, in order to enable a quicker assessment of the target displacement level, the Non-Iterative Capacity Spectrum Method (NICSM) was developed, [15].It is based on the principle of equivalent linearization in the assessment of the required nonlinear structural response.Also, the method of equivalent linearization is presented in FEMA 440, [5], in form of a modified acceleration-displacement response spectrum (MADRS).The displacement coefficient method is based on multiplication of certain coefficients according to FEMA 356, [4], and so the level of target displacement is not determined in an iterative manner.The research related to determination of coefficients applicable in this method is presented in papers [16,17].An improvement of this method, represented through the Displacement Modification Method, is given in FEMA 440, [5].The N2 Method is implemented in Eurocode EC8, [3], while a more detailed research related to the N2 Method is given in [18][19][20].The Modal Pushover Method, based on reformulation of the Response Spectrum Method, turned out to be very superior and efficient in the target displacement analysis, [21].With respect to this method, the Multi-Mode Pushover Procedure (MMPP) was developed, cf.[22,23].In addition, the modal combinations method was developed, [24], as well as an alternative with the Adaptive Modal Combination (AMC), [25].
Iterative displacement coefficient method: mathematical formulation and numerical analyses

Iterative Displacement Coefficient Method
The Iterative Displacement Coefficient Method (IDCM), originally formulated and presented in this work, is based on the Displacement Coefficient Method (DCM), according to FEMA 356, [4].The DCM method is operated through multiplication of certain coefficients, while in the IDCM method the level of target displacements is determined by the double iterative procedure.
The first iterative procedure is simultaneously conducted through forces and displacements, by incremental increase from zero to maximum values.The second iterative procedure is conducted by correcting the solution obtained for target displacement along the pushover curve.In the double iterative algorithm, the first iterative procedure is conducted for all iterations, and then the values are iterated, according to the second iteration procedure, for each iteration from the first iterative procedure.The continuity of the algorithm is achieved by successive implementation of the previously described procedure.
The target displacement level D t according to the DCM method, FEMA 356, [4], is determined by multiplication of the coefficients: where C 0 , C 1 , C 2 , C 3 are modification coefficients, S a is the spectral acceleration, and T eff is the effective period of vibrations.C 0 is the modifying coefficient by which the spectral displacement of the equivalent SDOF system is converted into displacement of the top-most node in the MDOF system, which is calculated via the participation factor Γ 1 of the first natural mode and the control node, or according to FEMA 356, [4].C 1 is the modifying coefficient which represents the ratio of the expected maximum non-linear displacement to the displacement of the linearly elastic response.According to FEMA 356, [4], it is determined as: where T S is the characteristic vibration period at the response spectrum changing from the domain of the constant acceleration to the domain of constant velocities, R is the coefficient of the ratio of the elastic bearing capacity and the yield limit bearing capacity.C 2 is the modifying coefficient which represents the effect of the pinched hysteretic shape, stiffness degradation, and strength deterioration at the maximum level of displacement.Values of this coefficient depend on structural performance: immediate occupancy (IO), life safety (LS) or collapse prevention (CP), type of structural system and the value of the initial elastic period of vibrations T e ≤0.1s and T e ≥T S .C 3 is the modifying coefficient that introduces the increase of displacement due to dynamic P-Δ effects.The value C 3 =1 is taken as positive stiffness in the non-linear range of behaviour K n,PC >0, while it is determined according to FEMA 356 for the negative stiffness in the nonlinear range K n,PC <0, [4]: where α BC is the stiffness ratio in non-linear range K n,BC , determined by bilinearization according to the effective stiffness K eff .The effective period of vibrations T eff is determined according to FEMA 356, [4]: where K e is the initial elastic stiffness.The total base shear force for the level of effective displacement V eff is determined from the intersection of the pushover curve and effective displacement D eff , while the effective stiffness K eff is determined according to: In a general case, the domain of possible values of the effective stiffness K eff is given from the initial elastic stiffness K e to the secant stiffness K sec .The level of the target displacement D t , as calculated according to previously given expressions, is expected to be in the range of D IO ≤D t ≤D LS , and then it is possible to provide a favourable ductile behaviour of the system, while in exceptional cases the level of target displacement may be within the bounds of D IO ≤D t ≤D CP .If the previous condition can not be fulfilled, the structure is considered unable to develop the minimum of the necessary ductile behaviour, and so the structural response is unfavourable.The domain of possible values for the K n,BC bilinear curve (BC) is defined by limits in Figure 1, and by the maximum value of the total relative base shear of the structure (V/W) max .In certain situations, the maximum value of the total relative base shear (V/W) max needs to be increased by the value of Δ(V/W) max .The alternative is introduced since problems may arise in the quality of bilinearization result in case of NSPA pushover curves with K n,PC < 0. The knowledge of all modifying coefficients is required in the determination of target displacement according to (1), but, as may be seen from previous expressions, the direct determination Mladen Ćosić, Stanko Brčić of C 0 and C 2 is possible, while in some situations C 1 and C 3 could remain unknown, unless it is assumed that C 1 and C 3 are equal to 1.The unknown values in these coefficients are R and α BC.Also, R contains the unknown V y,BC (force at the yield limit for bilinear system), while α depends upon D y,BC (displacement at the yield limit for bilinear system) and V y,BC .
In general terms, the process of determination of target displacement of the MDOF system is based on bilinearization by the SDOF system, with additional improvements of non-linear behaviour of SDOF as MDOF system (Figure 2).It is recommended in FEMA 356 [4] to perform balancing of the area below the NSPA pushover curve and the bilinear curve.The solution applied in this method is based on the principle of equal energy of elastoplastic deformations achieved by the NSPA analysis, and presented through the ratio of the bearing capacity-deformation E NSPA, and the energy achieved by elastoplastic deformation from bilinearization The energy of elastoplastic deformations achieved by the NSPA analyses, E NSPA, is determined from the area whose contour is the polygonal line (pushover curve), vertical line at the position of the target displacement D t and the abscissa.The term polygon for bilinear system is completely appropriate, since it consists of two lines while, at the first glance, the term seems mathematically incorrectly formulated for the MDOF system.However, since the pushover curve for the MDOF system is generated by connecting discrete values from incremental situations of NSPA analyses, the realistic shape of the pushover curve is also polygonal.The graphical presentation gives the impression of a smooth curve due to a large number of discrete values which are close to one another, and also due to additional interpolations by splines.

Figure 2. Bilinearization of the NSPA pushover curve
The double iterative algorithm procedure is conducted in several steps.First, the maximum value of the total base shear force is determined from the NSPA analysis: and also the adequate displacement D adeq : where V max is the maximum value of the total base shear force of the structure, W is the total weight, and H is the structural height.
In the first phase of the IDCM method, the iterative procedure for D y,BC and V y,BC is conducted in the interval, (Figure 3): where D max is the maximum displacement value at the top of the structure, and ΔV max is the increase of the maximum value of the total base shear force.In this phase of iterations, the coefficients C 1 and C 3 are assumed to be equal to 1, while in the next phase C 1 and C 3 are determined according to expressions ( 2) and ( 3), respectively.In the second phase, the iterative procedure for D t and V t is conducted using the previously determined D y,BC and V y,BC values (Figure 4).Then the maximum value of the total base shear force is calculated for the bilinear model V max,BC : and also the adequate displacement D adeq,BC :  The increase of the maximum value of the total base shear force is due to the fact that the force at the yield limit for bilinear system V y,BC might be greater than the maximum force obtained by NSPA analysis V y,PC .This is almost mandatory for NSPA pushover curves with negative stiffness in the nonlinear range K n,PC <0.The displacement increment value is determined from the difference of the adequate displacement D adeq,BC for V max,BC and the initial displacement D 0 , according to: Iterative displacement coefficient method: mathematical formulation and numerical analyses while the force increment is determined from: where ΔD is the increment of displacement, ΔV is the force increment, and N it1 is the number of iterations in the first iterative procedure.The initial, the first and iteration number i of displacements at the yield limit of the bilinear system are determined according to: and the forces according to: The first index in D y,BC (i,i) and V y,BC (i,i) corresponds to the iteration number in the first iterative procedure, and the second index to the number of iterations in the second iterative procedure.The following iterations for displacement and force at the yield limit are continued by adding the displacement increment and loading increment to previous values until the condition that the last iteration is i ΔD =i ΔD (n) and i ΔV =i ΔV (n) is fulfilled.In the first iterative procedure, the following values are assumed for the modifying coefficients C 1 and C 3 : The target displacement for the first iterative procedure is determined as: and the value of the total base shear force for the first iterative procedure as: The energy of deformation achieved through bilinear curve E B (i,i) is determined as follows: while the energy of deformation obtained from NSPA analyses E NSPA (i,i) is determined according to: and the difference in energies for iterations from: The energy achieved by the NSPA analysis is constant and it is not calculated in iterations conducted in the first iterative procedure.The applicable target displacement for the first iterative procedure is determined by first defining the minimum value of differences in energy from all iterations: while the other values are taken from the data base of calculated iterations (DBget): In the general case, it could be written that D y,BC and V y,BC are in the domain bounded by the intersection of curves: Once the first iterative procedure is over, the calculation continues with the second iterative procedure in which coefficients C 1 and C 3 are calculated.The coefficient R (i) is determined according to: while the coefficient C 1 (i) is determined according to (2), and the coefficient C 3 (i) according to (3).When the coefficients are determined, the target displacement level is calculated according to the expression: Final values of modifying coefficients and the target displacement level are obtained at the end of all iterations for the first and the second iterative procedure.The first iterative procedure requires more iterations then the second iterative procedure.The achieved ductility for the level of target displacement μ t and the maximum available ductility of the system, μ max , is determined through displacements at the yield limit of the system D y,PC of the NSPA analysis: or using K eff : or by calculating displacements at the yield limit of the system D y,PC according to [26]: where the effective height H eff is calculated according to the expression: Mladen Ćosić, Stanko Brčić The design displacement Δ i of the floor number i is: where: while for the regular framed systems: where ε y is the steel dilatation at the yield limit, L b is the beam length (centre to centre of columns), h b is the beam height, m i is the mass of i-th floor, and H i is the height of the floor i.The maximum displacement and the corresponding total base shear force are determined from: so, the achieved ductility μ t for the level of the target displacement and the maximum available ductility of the system μ max are: Other parameters that are important for the assessment of seismic performances of the system are the secant stiffness K t,sec and the secant period of vibrations T t,sec for the target displacement level: the minimum necessary displacement in order to achieve a favourable ductile behaviour of the system: . , = = 1 5 (39) and the (conceptual) damage index for the level of target displacement DI t [27]: 4. Specifics on determination of the target displacement level using the Nonlin Quake TD The Iterative displacement coefficient method (IDCM) is implemented in the original software solution Nonlin Quake TD (TD -target displacement).Program Nonlin Quake TD is a part of the complex software system Nonlin Quake for the non-linear performance-based seismic analysis of framed buildings.Nonlin Quake TD code is written in the programming language VB/VBA (Visual Basic/Visual Basic for Application), [28,29], and the user-software interaction is done through the corresponding graphical user interfaces (GUI).The initiation of the IDCM method for determining the level of target displacements, using Nonlin Quake TD, is performed by correction of discrete values of the pushover curve, obtained by the nonlinear static pushover analysis: The conversion into absolute values is performed, since negative values are obtained for V i in a large number of cases, as the result of numerical solution.The initial elastic structural stiffness K e is determined according to: where V 1 is the total base shear force for the first discrete value, D 1 is the displacement of the top of the structure for the first discrete value.Then the transformation from absolute into relative values is performed: The decision about the sign of non-linear stiffness of the MDOF system K n,PC is made by first calculating the tangent stiffness of the system K t,I for two consecutive discrete values of the pushover curve:  where D P,max and D P,min are the maximum and the minimum values, respectively, for the selected domain according to (45)-(47).This is followed by summation of all coefficients: and finally, the decision about the sign of K n,PC is made according to the positive or negative value of C w (a positive stiffness corresponds to positive value of C w ).In certain situations K t,I may significantly change its sign in the non-linear domain, from positive, zero and to the negative value (saw-tooth forcedeflection behaviour) [31].On the other hand, the difference in displacements of two consecutive discrete values of the pushover curve may be very significant, and so the general assessment of the sign of the non-linear stiffness may be complicated.This is particularly important at the crossover from the linear into the non-linear domain, and also in the domain of the pre-collapse state.In the research [32][33][34], which was done using one-dimensional finite elements for modelling framed structures and the SeismoStruct software [35], pushover curves obtained by NSPA analyses were presented without a frequent change of stiffness in the non-linear domain.Domains (45)-( 47) are selected since it is expected from the structure to develop the non-linear behaviour whose maximum displacement is greater then the LS performance level.If the maximum displacement greater than the LS performance level is not achieved, the lower achieved displacements are considered, and in that case the number of discrete response values is lower.The Nonlin Quake TD can iteratively determine the α BC (ratio of the stiffness in the non-linear domain K n,BC , determined through bilinearization procedure, to the effective stiffness K eff ), while the sign may be different from α PC or through iterations to determine the most favourable case with the same sign as for α PC (Figure 5).The expected (preliminary) global DR t,exp and the interstorey drift IDR t,exp are determined using the modified beta distribution [36]:  All calculated parameters of the IDCM method, and the determined target displacement level, are presented in Figure 6.

Numerical testing
The developed iterative displacement coefficient method, implemented in Nonlin Quake TD for NSPA analysis of target displacement, is tested in order to estimate performances, to verify and compare solutions with the solutions obtained by the non-linear dynamic analysis (NDA).The testing was conducted for: -15 different NSPA pushover curves generated for standard response models of framed buildings and defining the stiffness, bearing and ductility by accidental choice (using the random function), within empirical boundaries for those parameters, based on the literature and experience, as presented in Figure 7, -2D 8-story 4-bay regular framed system according to [34] for seismic action in the plane of the frame, -SDOF model of the bridge pier 10 m in height, of circular cross-section, 2.5 m in diameter, mass of 1000 t, for Mladen Ćosić, Stanko Brčić bidirectional seismic action, and designed according to the methodology given in the Displacement-Based Seismic Design [26].
First, the values of the coefficient C 3 for 30 NSPA pushover curves with K n <0 were considered, which were also classified into two groups.The first group includes NSPA pushover curves that were generated for different values of the initial elastic stiffness K e and ductility levels μ (Figure 7a): While the second group includes the NSPA pushover curves that were generated using the random function while obeying the limits for the linear, non-linear and the collapse domain (Figure 7b): The influence of dynamic P-Δ effects is introduced through the coefficient C 3 , and since these effects are important for the shape of NSPA pushover curves for K n <0, the research was performed particularly for this type of curves.Using the IDCM method, the values of the coefficient C 3 were determined for the previously defined NSPA pushover curves in the total of 180 IDCM analyses.After that, series of non-linear regression analyses with different types of functions were performed.An optimum solution was achieved with the exponential function, since the correlation coefficient was the largest (Figure 8).The correlation coefficient r 2 is determined according to [38] as: while the standard deviation is determined as: where C 3,IDCM,i is the discrete value calculated according to the IDCM method, C 3,IDCM,m is the average value calculated according to the IDCM method, C 3,Reg,i is the discrete value determined by the regression analysis, while C 3,RegM,m is an average value determined by the regression analysis.Using the non-linear regression analysis with the exponential function, the expression for the coefficient C 3 was derived in function of the spectral acceleration S a , and for the standardized response model of the NSPA pushover curve, as:  Iterative displacement coefficient method: mathematical formulation and numerical analyses Finally, the following expression for the coefficient C 3 was derived for all models of the NSPA pushover curves: With reference to paper [39] where the research was done for the bilinear elastoplastic system, we applied in this research 30 NSPA pushover curves of the real and simulated responses of a building as the MDOF system, and so expressions (60)-(62) may be used directly for practical purposes.A 2D 8-story 4-bay regular framed system according to [34] (Figure 9) was considered in the second part of the research.To use the IDCM method, a parametric analysis (40 analyses) was performed related to number of iterations N it1 , number of iterations N it2, and the additional maximum total base shear force ΔV max : Table 1.Dimensions and reinforcement of the cross sections [34] The preliminary number of iterations in the first iterative procedure N it1 can be determined by considering the total base shear force V.If we take that the minimum increment of force increase is m, then the necessary number of iterations N it1 is equal to V/m.On the other hand, the number of iterations N it1 in the first iterative procedure can also be determined by considering the maximum structural displacement value D max .If one assumes that the minimum increment of displacement increase is n, then the necessary number of iterations N it1 is equal to D max /n.It was established through extensive research that the criterion to determine N it1 from the ratio V/m is more applicable.The analogy used in determining the preliminary number of iterations N it1 in the first iterative procedure can not be applied for the second iterative procedure, N it2 .In this case the parametric analysis and the empirical approach are more reliable.The total of 32 IDCM analyses of target displacements were conducted.Previously, using NDA analysis, the level of target displacement of the frame DR t,NDA =1.39% and (V/W) t,NDA ≈20% was determined for the Loma Prieta earthquake (station Agnews State Hospital, LP89, PGA o =0.17g) according to PEER GMDB [40] (Figure 10).
Another parameter which significantly influences the level of target displacement is the value of the reduction of initial elastic stiffness ΔK, through which K eff can be determined.The parametric IDCM method was conducted for ΔK values (Figure 12): and the target displacement level according to NDA analysis was considered when applicable.
It can generally be stated that some percentage of increase of the maximum total base shear force ΔV max (about 5 to 10 percent) can be recommended for NSPA pushover curves with K n <0.It is also advisable to introduce the difference between the initial and effective stiffness K e ≠K eff when determining the target displacement level.Lower levels of target displacement can be expected in bilinearization procedure if it is assumed in the analysis that K eff =K e .
In the third part of the research, the SDOF model of the bridge pier was considered where, for the IDCM method, the parametric analysis with direct scaling in the peak ground acceleration (PGA) was also conducted.As already stated, the bridge pier is 10m high, of the circular cross section with diametar of 2.5m, and the mass of 1000 t.The pier is designed for the bidirectional seismic action according to the methodology given in the Displacement-Based Seismic Design [26].Analyses were performed for the Loma Prieta (previously presented) two-component accelerogram (fault parallel, FP, and fault normal, FN, accelerograms).Then, the inter-components of the accelerogram were generated for the incremental increase of the angle of Δθ=30º, and so the total number of inter-components considered was 12.The procedure for generating accelerogram inter-components is conducted by rotating components FP and FN in the reference coordinate system, where in the final stage one obtains: where a θ (t) is the accelerogram for the angle of rotation θ, a FP (t) is the accelerogram for direction FP, and a FN (t) accelerogram for FN direction.The angle φ is determined according to: whereα f is the angle of fault direction.By analogy to the number of inter-components of the accelerogram, the same number of pushover curves from NSPA analyses was generated and the target displacements were determined using the IDCM method.
Fragments of NSPA and NDA pushover curves, generated as the levels of target displacements obtained by parametric analyses, are presented in Figures 13÷18.In Iterative displacement coefficient method: mathematical formulation and numerical analyses Non-linear Dynamic Analysis), while in case of NSPA analyses the fragments are parts of the NSPA pushover curve for the scaling level of PGA = 0.3÷0.7g.The total of 60 IDCM and NDA analyses were conducted.A very satisfactory agreement in drift values was obtained by IDCM and NDA analyses, at almost all target displacement (DR t ) levels.Also, disagreements in the relative total base shear force (V/W) t , obtained by IDCM method, compared to the NDA method, are minimum at all target displacement levels.

Conclusion
The mathematical formulation and numerical testing of the originally developed Iterative Displacement Coefficient Method (IDCM), with a step by step algorithm, is presented in the paper.The IDCM method is a procedure for estimating earthquake effects on structures using the Non-Linear Static Pushoevr Analysis.
According to initial idea, the IDCM algorithm is conducted using triple iterations: the first iterative procedure for the effective stiffness, the second iterative procedure generally for forces and displacements, and the third iterative procedure for correcting the solution obtained for the level of target displacements along the pushover curve.By subsequent optimization of the algorithm the number of iterations applied for solution testing was reduced, and so the double iterative algorithm is presented in the final form.The IDCM method is implemented using the originally developed software application Nonlin Quake TD, which is a part of the complex software platform Nonlin Quake for the non-linear performance-based seismic analysis of framed structures.The IDCM method, as implemented in the Nonlin Quake TD, enables a wide spectrum of parameter variations, which directly influences the quality of target displacement obtained.Depending on options implemented in the Nonlin Quake TD for the analysis of target displacements, the resulting solutions may vary to a considerable extent.In some situations even numerically unacceptable quasi-solutions may be obtained, and then the robustness of the software points out, in most cases, to the need to correct the number of iterations N it1 , N it2 and the additional maximum total base shear force ΔV max .
In the first part of the research, the expression for the coefficient C 3 , as a function of the spectral acceleration S a, was derived using the non-linear regression with exponential function.15 NSPA pushover curves generated for the standardized response models of buildings, and 15 NSPA pushover curves generated using the random function, were applied in this research.The coefficient C 3 was determined using the previously presented pushover curves, and also by varying the spectral acceleration in the interval of S a =0.25÷1.5 g, where the total number of target displacement analyses was 180.In the second part of the research, the parametric analysis was conducted in order to determine the necessary number of iterations N it1 and N it2 for a realistic 2D 8-story 4-bay regular framed system.The domain of the optimum number of iterations lies within the interval of 1250 to 200000.The supreme value of 200000 iterations may be required in situations when ΔV max = 0 is selected and when, in exceptional cases, the solution can not be obtained with a fewer number of iterations.Sixty IDCM and NDA analyses were conducted In the third part of the research.The comparison of solutions obtained by IDCM and NDA analyses points to a very good agreement in values obtained for drifts DR t and relative total base shear forces (V/W) t for the target displacement level.

Figure 1 .
Figure 1.Domain of possible values of K n,BC for (V/W) max +Δ(V/W) max

Figure 3 .
Figure 3. Determination of D y,BC and V y,BC using the first iterative procedure

Figure 4 .
Figure 4. Determination of D t and V t applying the second iterative procedure

2 ( 47 )
Global drifts DR IO , DR LS and DR CP for reinforced framed systems (i.e.structural performance levels), according to SEAOC[30] and FEMA 356 [4], are given by: Data for walls Data for frames Iterative displacement coefficient method: mathematical formulation and numerical analyses while the corresponding total base shear forces for IO, LS and CP performance levels are determined from the intersection of the pushover curve and drifts DR IO , DR LS and DR CP .After selecting the domain and establishing K t,I the weighting coefficients are determined according to:

Figure 5 .
Figure 5. a) α BC <0, α PC <0; b) α BC >0, α PC <0Minimum and maximum values of the global and interstorey drifts are determined by first determining to which performance level belongs the target displacement.In case of a global drift, the minimum and maximum values are given by (48), while interstorey drifts are presented in[37]:

Figure 6 .
Figure 6.Overview of parameters calculated to determine the target displacement level according to IDCM method random response function of the NSPA pushover curve it was obtained as:

Figure 10 .
Figure 10.Original non-scaled accelerogram, Loma Prieta LP89: a) 1st component; b) 2nd component Figure 11.a presents calculated global drift values for the target displacement level DR t , while Figure 11.b) presents calculated relative values of the total base shear force (V/W) t for the target displacement level, according to IDCM method.