Enhancements in idealized capacity curve generation for reinforced concrete regular framed structures subjected to seismic loading/Seismine apkrova veikiamu gelzbetoniniu reminiu konstrukciju laikomosios galios idealizuotos kreives generavimo tobulinimas.

Seifi, Mehrdad ; Noorzaei, Jamaloddin ; Jaafar, Mohamed Saleh 等

1. Introduction

The seismic analysis and design procedures have evolved significantly through the last decade. Nowadays, based on the structural characteristics, different approaches ranging from simple equivalent static analysis to nonlinear dynamic analysis are available in literature. As a result, equivalent static analysis is a common approach in practice to determine pseudo-capacity to resist prescribed lateral force for regular R/C frames. Owing to this fact, this approach does not provide insight into the actual capacity of structures. Hence, distinguishing the actual capacity of structures through the procedure of "Performance-Based Design Engineering" (PBDE), has been frequently highlighted (Shattarat et al. 2008; Chandler, Lam 2001; Freeman 2005). In response to this a non-linear static pushover (NSP) analysis, as a compromise between simplified linear static and complex non-linear dynamic methods, has been developed. Nowadays, this method found its way to seismic guidelines. One of the fundamental uses of this method relies on the extraction of non-linear force-displacement relationship between base shear and the displacement of control node by applying a graphical iterative bilinearization procedure according to FEMA-356 (2000).

In spite of its many deficiencies, the conventional code-based method is the most well-known method utilized in academic works. It neglects higher modes contribution, stiffness degradation and period elongations (Menjivar 2004). In recent decades, several methods have been proposed to overcome the deficiencies of the conventional method. Some of these methods include: modal pushover analysis "MPA" (Chopra, Goel 2001), Incremental Response Spectrum Analysis, IRSA (Aydinoglu, Celep 2005), Method of Modal Combination, MMC (Kalkam, Kunnath 2004) and Adaptive Pushover Analysis, APA (Antoniou, Pinho 2004; Pinho et al. 2005). A majority of them have cumbersome conceptual background and involve computationally intensive procedures. It has been reviewed that APA is one of the most rational novel approaches that overcame these limitations (Seifi et al. 2008).

Parallel to the studies in favour of increasing the accuracy of NSP approaches, some of the researchers focused on reducing excessive computational cost and making the PBDE domain viable for real-life engineering applications. Hence, the use of Artificial Intelligence (AI) is incorporated into this realm of knowledge. As a result, ANN has been successfully applied by some researchers (Correno et al. 2004; Tsompanakis et al. 2005; Gonzalez, Zapico 2007). However, no attempt has been made to employ ANN as an alternative to NSP procedure in favour of predicting the capacity curve. Hence, based on the above-mentioned shortcomings, the present study aims to:

i. Choose the most suitable NSP method by means of a comparative study among various types of conventional and adaptive procedures.

ii. Propose a suitable alternative to the graphical iterative procedure of FEMA-356.

iii. Study the effects of tangible geometric and material variables of R/C frame parameters on idealized curve parameters.

iv. Test the applicability of ANN as a replacement for pushover procedure in favour of minimizing expertise, time and efforts of extracting the idealized parameters of a structure.

2. Domain of the selected R/C frame population

First, to facilitate differentiation among the adopted models, "[x.sub.1][fx.sub.2][sx.sub.3][lx.sub.f][bx.sub.5]r]" has been implemented as a reference point where f s, l, b and r refer to the storeys, spans, length of spans, distance between frames and reinforcement type respectively. Also, [x.sub.1] to [x.sub.5] indicate their corresponding values.

The scope of the present study is confined to R/C regular frames with 2 to 7 stories, 2 or 3 spans by lengths in range of 3.5 to 5 m by 0.5 m increments. The distance between frames was assumed to be constant and equal to 4 m. Two types of longitudinal reinforcement of yield strengths equaling 294.3 MPa and 392.4 MPa have been considered which will be referred to as 3 and 4 with respect to the reference point. Based on the above selected variables, 96 different structures could be possibly modelled. In order to cover the whole range of possible differences among structures, 30 well-distributed models were selected to be considered as structural samples (Table 1).

3.1. Preliminary modelling, analysis and detailing procedure

At this stage, the concrete nominal 28-day compressive strength and modulus of elasticity were assumed equal to 27.458 MPa and 24.787 GPa, respectively. Furthermore, nominal yield strengths of transverse reinforcement were considered equal to the aforesaid values for the longitudinal one. The 30 models of R/C regular frame were created with the aid of SAP2000 program (CSI 2006). Stories and roof were subjected to 6.278 kN/[m.sup.2] and 5.788 kN/[m.sup.2] as dead loads in addition to 1.962 kN/[m.sup.2] and 1.471 kN/[m.sup.2] as live loads respectively. By assuming the high seismic region (seismic hazard zone indicator is 0.3 g), linear static analysis procedure, based on UBC-97, was employed for preliminary analysis of models. In all the models P- [DELTA] effect as well as the "cracked sections" was taken into consideration. Confining the storey drift into a 0.025 storey height was a controlling criterion during the analysis procedure (Uniform Building Code 1997).

Afterward, the models were all designed based on ACI 318-99 code load combinations and the "weak beam/strong column" strategy (ACI 318-99 2000). Practical aspects, including bending and curtailment criteria, minimum allowable amount of the longitudinal and transverse reinforcement, existing bar size etc. were controlled manually for each and everyone of the designed frames.

3.2. Finite element modelling

After attaining the logical beams and columns sections, finite element modelling has been carried out. The most relevant package, "SeismoStruct 4.0.2" (2007), has been employed. Via fibre element modelling the program is capable of considering geometric non-linearities as well as material inelasticity. Moreover, precise modelling of bars by defining their location in cross-section was performed. At physical modelling level, 3D inelastic beam-column fibre element models were employed. To represent the beams and columns, 6 and 5 elements were used, respectively. The lengths of each element are determined based on the distribution of reinforcement and expectation of larger level of inelasticity in the vicinity of beam-column connections (Table 2). T-sections and rectangular sections were used to model beams and columns, respectively (Fig. 1). Due to the scope of the study and introduction of negligible confined height in slab, the slab effective width was chosen equal to the beam width. Also, based on a trial and error process, it was found that about 200 fibres are optimum for modelling R/C sections (SeismoStruct user Manual 2007).

To account for material non-linearity, the "Uniaxial constant confinement concrete" (con-cc) model was chosen for unconfined and confined concrete (Martinez-Rueda, Elnashai 1997; Powanusorn 2003). In addition, the modified "Menegotto-Pinto" model proposed by Filippou et al. (1983) was adopted for reinforcement (Colson, Boulabiza 1992; Byfield et al. 2005; Monti et al. 1993).

[FIGURE 1 OMITTED]

4. Distinguishing the qualified capacity curve type

4.1. Description of seismic analyses procedures

With the aim of properly identifying the simplest type of NSP method capable of estimating the capacity curve, a comparative study has been performed for one of the relatively large produced models addressed as "6f3s4l4b3r". This model covers all concerning issues including higher mode effects, whiplash effect and so on. Consequently, the outcome of this study was applied as a quantifiable approach for other created models in the next stages of investigation.

After the distribution of gravity loads among the crossing points of beam elements, with respect to the length of their adjacent elements, five types of lateral load distributions were imposed on analogous structures. On the one hand, concerning the conventional prevalent approaches, the "Triangular" method, proportional to the linear static procedure and "Uniform" method consistent with the weight of stories and regardless of their height were utilized (FEMA 2000; SeismoStruct user Manual 2007). On the other hand, adaptive pushover approaches put forward by Antoniou and Pinho (2004), were implemented. In these methods, a variable distribution of lateral loads was utilized. These loads are updated at every predefined step, namely by "Incremental updating procedure" with respect to the modal shapes and participation factors of modes. The participation factors and modal shapes are extracted by eigenvalue analysis.

Depending on whether forces or displacements are applied, two variants of the method exist: force-based adaptive pushover (FAP) and displacement-based adaptive pushover (DAP). DAP by itself is classified into displacement-based scaling and interstorey drift-based scaling methods. The major difference between the DAP options refers to the fact that in displacement-based scaling method the storey displacement patterns are calculated from the eigenvalue vectors directly, while in interstorey drift-based scaling technique, the eigenvalue vectors are utilized to determine the inter-storey drifts of each mode (Menjivar 2004).

In line with applying adaptive methods, initially, the inertia mass of the building was modelled at each beam-column joint. In addition, a nominal uniform load [P.sub.0] was introduced along the height of structure. Thereupon, for a "force-based" method the base shear was distributed uniformly among all of beam-column joints, while for the "Displacement-based" and "Interstorey-drift based' techniques, target displacements were imposed on joints (Fig. 2).

[FIGURE 2 OMITTED]

The magnitude of the load vector P at any given analysis step is given by the product of its nominal counterpart [P.sub.0], and the load factor [lambda] at that step:

P = [lambda][P.sub.0]. (1)

During analysis the load factor [lambda] varies between zero and the target load multiplier value (1.0) (SeismoStruct user Manual 2007).

In order to achieve uniformity among various applied pushover methods, the analysis was performed until the control node displacement on the roof reached 2% drifts of the frame height (FEMA 2005). Furthermore, with the aim of improving the accuracy of adaptive methods, spectral amplification was employed and equivalent viscous damping ([xi]) was assumed to be 5% as inherent viscous damping (Chopra 1998). A detailed description of the adaptive approaches can be found in (Pinho et al. 2006).

Consequently, in order to justify the NSP methods, "Incremental Dynamic Analysis", IDA, was performed for another similar created model. "Stepping algorithm " was employed during the study and the model was subjected to a broad range of scaled normalized excitation with the PGA ranging from 0.1 g to 0.8 g with a distinct incremental scaling factor of 0.05 (Vamvatsikos, Cornell 2001). Only the horizontal component of earthquake in a plane of model was imposed on all restraints.

4.2. Earthquake input

The ground motion used during the study was a horizontal north-south component of El-Centro 1940, which caused considerable human and economic losses. The rationale of using this record was the consistency between the seismic zone indicators (0.3 g) exploited for preliminary design and the PGA of the record. The record utilized for APA and IDA analysis, is illustrated in Table 3 and Fig. 3 (Online Reference Documentation 2007).

[FIGURE 4 OMITTED]

[FIGURE 3 OMITTED]

5. Extracting the idealized parameters of capacity curve

5.1. Idealization criteria

It is worth mentioning that the capacity curve by itself is not meaningful. Therefore, it must be idealized for extraction of important parameters which describe the structural behaviour (FEMA 2000). Although miscellaneous techniques have been proposed for NSP analysis, there is an agreement among researchers that the bilinear idealization method that has been proposed by FEMA-356 is the most acceptable (Chopra, Goel 2001; Akkar, Metin 2007). FEMA-356 idealized capacity curve (Fig. 4) must be computed under these circumstances: (i) it must be bilinear with an initial slope [K.sub.e] and post yield slop [[alpha]K.sub.e]; (ii) the effective lateral stiffness, [K.sub.e], shall be taken as the secant stiffness calculated at a base shear force equal to 60% of the yield strength of the structure; (iii) the postyield slope [[alpha]K.sub.e] shall be determined by a line segment passing through the actual curve at the calculated target displacement; (iv) effective yield strength [V.sub.y] shall not be taken as greater than the maximum base shear at any point along the actual curve; (v) an "approximate" balance between the area, which is confined between the bilinear and actual curve, is compulsory (FEMA 2000).

5.2. Alternative proposed for FEMA-356 graphical procedure

In order to generate an idealized capacity curve, FEMA-356 has suggested a graphical iterative procedure. Tackling such an approach manually could not be exempt of error. Also simultaneous fulfillment of all of the abovementioned 5 criteria seems to be impractical. Furthermore, none of the packages are able to perform the FEMA-356 procedure and predict the precise idealized capacity curve directly. Hence, the vital role of extracting the accurate idealization parameters through the next stage of investigation necessitates the development of a program capable of idealizing the capacity curve precisely. Accordingly, the present work attempts to write a computer code namely 'BLestimator" under MATLAB(R) environment capable of precisely fulfilling the bilinearization criteria. The concise flow chart of generating the bilinear capacity curve has been presented in Fig. 5 (Seifi et al. 2007; MATLAB 2006).

[FIGURE 5 OMITTED]

6. Influence of structural variables on idealized capacity curve

By assessing the different previously mentioned NSP method and revealing the most suitable one, this method together with "BLestimator" generated a well thought-out comparative instrument which is applied to each and every of gravitational-loaded finite element models of Table 1. The course of action required for extracting the idealized parameters is illustrated in Fig. 6. Its use will produce the data required to study the influence of the considered geometric and material variables on idealized parameters of structures including [K.sub.e], [[alpha]K.sub.e], [V.sub.y] and [X.sub.y] within the scope of work.

7. Prediction of idealized capacity curve via A.N.N

In recent years neural network has been applied as an alternative to conventional techniques in the realm of PBDE (Correno et al. 2004; Gonzalez, Zapico 2007). Moreover, extracting the idealized parameters is a cumbersome and time-consuming procedure which includes specialized steps, while, the non-linear parameters of the structure are easily extractable by means of training appropriate networks.

[FIGURE 6 OMITTED]

7.1. Designing

Among different neural network types the one which is widely used is the feed-forward back propagation network. This network consists of input layer, one or several hidden layer and an output layer. All nodes called neuron are connected with weighted links to each neuron of the next layer (Fig. 7). The output of each neuron is computed through a feed-forward procedure by imposing activation function f(x)), on input values with different weights (w). Tangent hyperbolic function was used for hidden layer(s) and linear function for output layer. The output of a single neuron is computed by:

[y.sub.j](p) = f(x)[[m.summation over (i=1)][x.sub.i](p).[w.sub.ij](p)-[[theta].sub.j]], (2)

where m is the number of inputs, [[theta].sub.j]--the threshold on neuron j and [w.sub.ij] the preliminary weight of input i for neuron j. By computation of actual outputs, [y.sub.k](p) for the last layer and comparing them with desired outputs, [y.sub.d,k](p) by means of performance function, error is computed and back propagated through the network. Then, by calculation of error gradient the weights are adjusted and this cyclical procedure is repeated until achieving a prescribed error (MATLAB 2006; Schalkoff 1997). Comprehensive description of the method could be found in (Schalkoff 1997).

[FIGURE 7 OMITTED]

7.2. Phases of developing ANN in the present study

a) Learning process. This stage is started by defining structural variables as network input and respective idealized parameters obtained analytically as output. To represent to the network all input and output vectors are normalized to the range of [0, 1] by:

[a.sub.i,s](p) = [a.sub.i](p)-[a.sub.min]/[a.sub.max] - [a.sub.min], (3)

where [a.sub.i], is the value of the specific variables for [p.sup.th] model, [a.sub.min] and [a.sub.max] are the minimum and maximum values of the specific variables among all models, and [a.sub.i,s] is the standardized variable value for [p.sup.th] model.

b) Training the network. This is the next stage which consists of error minimization. "Mean Square Error", MSE is selected as "performance function" for all networks where:

[f.sub.MSE] = 1/n[n.summation over (i=1)][e.sup.2.sub.k](p) = 1/n[n.summation over (i=1)][[y.sub.d,k](p) - [y.sub.k](p)).sup.2], (4)

where n is the number of datasets.

c) Testing the network. After training a network of 27 prepared datasets, testing was performed on the remaining 3 datasets selected randomly. Testing step reveals the efficiency of trained network by comparing the predicted values with the desired one. Based on the accuracy of testing results, necessity of testing another ANN configuration could be judged.

8. Results and discussion

The outcomes include: (i) Selection of the superior NSP techniques, (ii) Application of the "BLestimator" program, (iii) Effects of structural variables on idealized parameters (iv) Testing the applicability of neural network as an alternative to the NSP method.

8.1. Comparison of different NSP techniques

In line with assessing the static capacity curves extracted by the aforesaid NSP methods, they have been compared to the dynamic capacity curve achieved by IDA as in Fig. 8. Afterwards, by tracing the numerical results of different static capacity curves, corresponding base shear for each step of the IDA analysis was determined until it reached the predefined target displacement. By putting the outcomes alongside each other (Table 4), credibility of different NSP approaches were revealed. It was shown that:

* Except for the FEMA-Triangular method that conspicuously underestimated the capacity curve of the structure, the outcomes of other techniques had acceptable estimation of dynamic capacity curve.

* Scrutinizing Table 4 shows that among all of the methods, "DAP Interstorey based" is the unique method that follows the upward trend of the base shears of IDA analysis (especially in the last states, where the structure experiences post-yield behaviour). This fact is of great importance since it has an unmistakable effect on judgment of post-yield stiffness behaviour of the structure.

Due to these facts, "DAP Interstorey based" was adopted as the qualified method to be applied for the other models and in the next steps of the study.

8.2. Estimation of precise idealized parameters using "BLestimator"

Due to the findings so far, only the application of the developed program for DAP-interstorey based capacity curve of 6f3s4l4b3r is presented here.

a) Actual capacity curve against fitted one. The curve fitting procedure is performed by importing the first part of coordinates with respect to different degrees imported by the user. After each attempt, for all of the incremental displacement steps, the difference between the corresponding actual and computed (fitted) base shear will be presented numerically and graphically (Fig. 9). After few iterations that just took a few seconds, the predicted polynomial curve of order six was concluded as the best "fitted curve" (Fig. 9, gray line vs. Black line). A critical examination of 100 numerical output datasets unveils that, while in a majority of steps, the absolute difference between the actual base shear and its corresponding computed one is less than 1.50 kN; the maximum difference is also confined to 7.94 kN. The polynomial curve equation is:

-19.111 x [10.sup.5][x.sup.6] + 23.900 x [10.sup.5]x5 - 11.454 x [10.sup.5] x 4 + 26.897 x [10.sup.4][x.sup.3] - 36.781 x [10.sup.3] [x.sup.2] + 43.180 x [10.sup.2] x - 7.417. (5)

b) Area under the estimated capacity curve. It is major parameter that must be measured to satisfy the fifth-mentioned criteria of FEMA-356. The confined area beneath the capacity curve of 6f3s4l4b3r is equal to 110.4715 kN.

[FIGURE 8 OMITTED]

c) Idealization process and extraction of significant parameters. At this stage based on the user-defined number, all generated cases of idealization are compared to each other. Evaluation of the final results for 6f3s4l4b3r is summarized in Table 5. They certify the fact that the idealization process FEMA-356 has been performed as precisely as possible, while the computational time has been minimized to less than a minute.

[FIGURE 9 OMITTED]

8.3. Study of the effects of structural variables on an idealized capacity curve

By arranging the contents of Table 6 with respect to different structural variables it could be found that:

* Juxtaposition of similar couples with difference in the number of spans indicates that the more are the spans, the greater is [absolute value of [alpha][K.sub.e]]. This observation is also true for negative post-yield stiffness models (e.g. 7f2s3.5l4b3r vs. 7f3s3.5l4b3r). The number of spans has negligible effect on [K.sub.e] as well as [X.sub.y].

* Grouping the models which are only dissimilar in the length of spans, reveals increments in spans length lower [K.sub.e] while enhancing the amount of [[alpha][K.sub.e] (e.g. 4f3s3.5l4b3r vs. 4f3s4l4b3r).

* In positive post-yield stiffness frames, increase in length of spans results in larger value of [X.sub.y] (e.g. 4f3s3.5l4b3r vs. 4f3s4l4b3r). Negative post-yield stiffness models do not follow this regulation and show unexpectedly higher [X.sub.y] compared to their corresponding models with longer spans (e.g. 7f3s3.5l4b3r vs. 7f3s4l4b3r).

* Use of low strength bars ([f.sub.y] = 294.3MPa) in narrow span structures is not suggested because it has unsatisfactory effect on [[alpha][K.sub.e]. Redesigning these models by employing reinforcements with higher strength (fy = 392.4 MPa) causes improvement in value of aKe(4f3s3.5l4b3r vs. 4f3s3.5l4b4r).

* Increments in number of stories and the subsequent emerging of the whiplash effect deteriorate the unsatisfactory effect on [[alpha][K.sub.e] in narrow span frames with low strength bars. This causes undesirable negative post-yield stiffness (e.g. 7f2s3.5l4b3r or 7f3s3.5l4b3r).

8.4. Application of A.N.N

This study was initially aimed at designing a general model for predicting all idealized parameters. However, owing to the use of the entire structural variables as inputs, the outcomes for some parameters were completely deviated. Thus, it was concluded that for the prediction of each idealized parameter, an especial purpose neural network should be designed. This is achieved by keeping an eye on the most influential parameters made known in the former stage of study.

Consequently, the most influential number of inputs in achieving accurate results was selected. The number of neurons and hidden layers were chosen by trial and error. Eventually, for all of the ANNs, the i+2:Ni:Ni:1 architecture is concluded to be the superior one. Configurations of all networks, influential inputs and corresponding outputs are shown in Table 7.

Finally, by means of the inverse trend of Eq. 3 the predicted values for all of the idealized parameters are converted to the real ones and are compared to the desired values by calculating their absolute relative percentage error, ARPE, as shown in Table 8. Based on that table, it is observed that the difference between the predicted non-linear parameters and the desired values for all of the parameters, excluding the [X.sub.y], are negligible. Nonetheless, it should be noticed that for this specific parameter and even in the worst conditions, these straightaway predicted values by ANN is undoubtedly preferable to the one which is manually extractable, through the time consuming FEMA-356 procedure. This procedure apparently could not be exempt of error without utilizing "BLestimator".

9. Conclusions

In this paper a method for identifying idealized capacity curve parameters including [K.sub.e], [[alpha][K.sub.e], [V.sub.y] and [X.sub.y] has been proposed. The method is tested for finite element models of R/C 2D regular frame structures modelled as closely as possible to those of actual structures. In order to generate different models, geometric variables, including number of stories, number of spans, length of spans, in addition to yield strength of reinforcement as material variables, has been considered. The process includes 4 successive stages. Starting with a comparative study among different conventional and adaptive pushover approaches, all of the methods were applied for a 6-storey frame by comparing them to dynamic capacity curve as an outcome of IDA. It was concluded that "DAP interstorey based" outperforms the other approaches. The second step was codifying the procedure for fulfilling FEMA-356 idealization criteria. The "BLestimator" program overcame the deficiencies of the graphical iterative procedure. By means of applying the program, exact idealized parameters are attainable in a fraction of a minute. In the third step by applying DAP interstorey accompanied with the aforesaid developed program (an efficient comparative tool) the influence of the structural variables on idealized parameters were investigated. As a consequence, the most influential parameters and their effects on each idealized parameter have been made known. Prepared datasets have been utilized as an input of the last stage for training feed forward back propagation ANNs. By training the networks, the idealized capacity curve converts to a handy tool for neophytes to be informed about structural behaviour. Also, professionals will be able to effortlessly achieve it for the next stages of their study.

DOI: 10.3846/1392-3730.2008.14.24

Received 04 Sept 2008; accepted 29 Oct 2008

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Mehrdad Seifi, Jamaloddin Noorzaei, Mohamed Saleh Jaafar, Waleed Abdulmalik Thanoon

Dept of Civil Engineering, University Putra Malaysia, 43400 UPM-Serdang, Malaysia E-mail: jamal@eng.upm.edu.my

Mehrdad SEIFI obtained his Master of Science in structural engineering from University Putra Malaysia (UPM). His research interests include earthquake engineering, particularly those in performance-based design engineering domain.

Dr. Jamaloddin NOORZAEI completed his PhD study at the University of Roorkee, India. His research interests include computational techniques in civil engineering applications especially those related to structural engineering, soil- structure interaction and earthquake engineering. Currently Associate Professor and Head of the Structural Engineering Research group at the University Putra Malaysia (UPM).

Dr. Mohamed Saleh JAAFAR obtained his PhD from the University of Sheffield. Currently Associate Professor and Dean of the Faculty of Engineering UPM, Malaysia; his research interests include concrete and prestressed concrete structures, high performance concrete and structural conditions assessment.

Dr. Waleed Abdulmalik THANOON obtained his PhD from the University of Roorkee, India. Currently Professor in Department of civil and environmental engineering in UPM, Malaysia; his main interests are materials engineering research, structural assessment, repair and dynamic behaviour of structures.

Table 1. Modelled structures for comparative study

2f2s3.5l4b3r   4f2s4l4b3r     5f3s4l4b3r
2f2s5l4b3r     4f3s3.5l4b3r   6f2s3.5l4b4r
2f3s3.5l4b4r   4f3s3.5l4b4r   6f2s4l4b4r
2f3s4.5l4b4r   4f3s4.5l4b4r   6f3s4.5l4b3r
3f2s3.5l4b4r   4f3s4l4b3r     6f3s4l4b3r
3f2s4.5l4b3r   5f2s3.5l4b3r   7f2s3.5l4b3r
3f3s4.5l4b4r   5f2s4.5l4b3r   7f2s4l4b3r
3f3s4l4b4r     5f2s4.5l4b4r   7f3s3.5l4b3r
3f3s5l4b3r     5f2s5l4b4r     7f3s4l4b3r
3f3s5l4b4r     5f3s4.5l4b4r   7f3s5l4b4r

Table 2. Assumed element lengths

Member                   Elements length (m)

Beam     0.125L   0.175L    0.2L     0.2L    0.175L   0.125L
Column   0.15L    0.22L          0.26L       0.22L    0.15L

Table 3. El-Centro earthquake characteristics

                                             PGA    PGD   Normalization
Earthquake  Magnitude    Site    Component   (g)   (cm)      factor

El Centro      7.1     Imperial   North-    0.318  13.32      3.13
18.05.1940              valley     South

Table 4. Numerical comparing the static
capacity curves with IDA results

                        IDA (Reference)

                      Roof
                  Displacement       Base          FEMA
Scaling Factors       (m)         Shear (kN)    Triangular

--                     0              0              0
0.10g                0.053         179.698        124.751
0.15g                0.105         283.082        193.597
0.20g                0.147         319.766        233.003
0.25g                0.178         296.725        251.354
0.30g                0.216         325.242        263.163
0.35g                0.249         352.936        268.834
0.40g                0.280         363.903        272.909
0.45g                0.385         370.697        277.232

          Non-linear Static Pushover (NSP) method

                                     DAP            DAP
     FEMA                        Displacement   Interstorey
    Uniform           FAP           based          based

                       Base Shear (kN)

       0               0              0              0
    169.370         156.907        170.403        155.817
    264.776         244.174        273.909        234.162
    310.602         291.720        342.049        295.365
    327.949         322.962        370.884        331.951
    336.546         339.176        383.228        355.698
    337.997         350.207        385.040        367.590
    337.356         354.085        379.184        375.400
    321.070         335.115        343.542        382.077

Table 5. Final results of applying idealization program for 6f3s4l4b3r

Parameter description                  Sign           Value      Unit

Maximum base shear of the          [V.sub.maxim]     382.3963     kN
  fitted curve
Coordination of the "effective       [X.sub.y]        0.1295      m
  yield strength point"              [V.sub.y]       327. 981     kN
Coordination of the              [X.sub.intersect]    0.0777      m
  intersection point between     [Y.sub.intersect]   196.7886     kN
  idealized and the main curve
Effective lateral stiffness of          Ke            58.207    Degree
  the building
Post-yield stiffness of the           alphaKe         7.8052    Degree
  building
Relative percentage error of            RPE           0.0015      --
  area comparison

Table 6. Structural variable vs. corresponding
extracted idealized parameters

                             Structural variable
Model Name
                No. of       No. of     Length of     [F.sub.y]
                stories      spans      spans (m)   (kN/[m.sup.2])

2f2s3.5l4b3r       2           2           3.5          294300
2f2s5l4b4r         2           2            5           294300
2f3s3.5l4b4r       2           3           3.5          392400
2f3s4.5l4b4r       2           3           4.5          392400
3f2s3.5l4b4r       3           2           3.5          392400
3f2s4.5l4b3r       3           2           4.5          294300
3f3s4l4b4r         3           3            4           392400
3f3s4.5l4b4r       3           3           4.5          392400
3f3s5l4b3r         3           3            5           294300
3f3s5l4b4r         3           3            5           392400
4f2s4l4b3r         4           2            4           294300
4f3s3.5l4b4r       4           3           3.5          392400
4f3s3.5l4b3r       4           3           3.5          294300
4f3s4l4b3r         4           3            4           294300
4f3s4.5l4b4r       4           3           4.5          392400
5f2s3.5l4b3r       5           2           3.5          294300
5f2s4.5l4b4r       5           2           4.5          392400
5f2s4.5l4b3r       5           2           4.5          294300
5f2s5l4b4r         5           2            5           392400
5f3s4l4b3r         5           3            4           294300
5f3s4.5l4b4r       5           3           4.5          392400
6f2s3.5l4b4r       6           2           3.5          392400
6f2s4l4b4r         6           2            4           392400
6f3s4l4b3r         6           3            4           294300
6f3s4.5l4b3r       6           3           4.5          294300
7f2s3.5l4b3r       7           2           3.5          294300
7f2s4l4b3r         7           2            4           294300
7f3s3.5l4b3r       7           3           3.5          294300
7f3s4l4b3r         7           3            4           294300
7f3s5l4b4r         7           3            5           392400

                             Idealized parameter
Model Name
               [K.sub.e]   a[K.sub.e]   [V.sub.y]     [X.sub.y]
               (degree)     (degree)      (kN)           (m)

2f2s3.5l4b3r    64.930       7.280       107.200        0.032
2f2s5l4b4r      56.473       8.564       158.039        0.045
2f3s3.5l4b4r    59.939       10.247      164.806        0.037
2f3s4.5l4b4r    52.450       10.170      202.960        0.052
3f2s3.5l4b4r    59.011       8.690       134.020        0.061
3f2s4.5l4b3r    56.516       9.021       184.075        0.068
3f3s4l4b4r      54.215       11.988      220.114        0.069
3f3s4.5l4b4r    51.061       12.095      247.066        0.078
3f3s5l4b3r      54.916       9.896       317.276        0.072
3f3s5l4b4r      49.033       13.718      283.970        0.082
4f2s4l4b3r      57.279       5.744       187.940        0.093
4f3s3.5l4b4r    58.450       8.440       234.840        0.085
4f3s3.5l4b3r    61.189       5.390       257.678        0.079
4f3s4l4b3r      57.260       6.607       283.890        0.091
4f3s4.5l4b4r    52.257       12.626      295.390        0.098
5f2s3.5l4b3r    60.220       1.927       191.940        0.110
5f2s4.5l4b4r    54.190       11.650      231.010        0.119
5f2s4.5l4b3r    57.460       7.646       249.260        0.114
5f2s5l4b4r      53.980       12.365      272.777        0.129
5f3s4l4b3r      58.240       5.029       328.843        0.112
5f3s4.5l4b4r    54.315       14.465      309.717        0.106
6f2s3.5l4b4r    57.548       9.280       189.650        0.127
6f2s4l4b4r      56.404       10.320      222.281        0.132
6f3s4l4b3r      58.210       7.805       327.980        0.130
6f3s4.5l4b3r    57.908       8.630       376.910        0.136
7f2s3.5l4b3r    60.837       -2.767      240.471        0.158
7f2s4l4b3r       57.96       9.499       249.290        0.144
7f3s3.5l4b3r    60.553       -5.281      367.582        0.160
7f3s4l4b3r      59.520       11.290      361.350        0.131
7f3s5l4b4r      55.070       13.320      411.087        0.151

Table 7. Superior trained networks, influential
input and corresponding output

Considered   Network
parameter    structure   Considered inputs

[K.sub.e]    3-12-12-1   Length of spans
a[K.sub.e]   6-12-12-1   No. of stories, No. of spans,
                           Length of spans, [F.sub.y]
[V.sub.y]    5-10-10-1   No. of stories, No. of spans, Length of spans
[X.sub.y]    4-12-12-1   No. of stories, Length of spans

Table 8. Final results of ANNs application for testing datasets

Model name                       Desired values

               [K.sub.e]   a[K.sub.e]   [V.sub.y]   [X.sub.y]
                (deg.)       (deg.)       (KN)         (m)

4f3s3.5l4b3r     61.19        5.39       257.68       0.08
6f2s4l4b4r       56.40       10.32       222.28       0.13
5f2s4.5l4b4r     54.19       11.65       231.01       0.12
MARPE (%)

Model name                 Predicted values by ANNs

               [K.sub.e]   a[K.sub.e]   [V.sub.y]   [X.sub.y]
                (deg.)       (deg.)       (KN)         (m)

4f3s3.5l4b3r     60.19        6.09       234.83       0.10
6f2s4l4b4r       57.53       11.40       220.66       0.16
5f2s4.5l4b4r     54.71       11.96       249.27       0.13
MARPE (%)

Model name                           ARPE (%)

4f3s3.5l4b3r     1.64        13.06        8.87        25.86
6f2s4l4b4r       1.99        10.48        0.73        18.25
5f2s4.5l4b4r     0.96         2.67        7.90        9.88
MARPE (%)        1.53         8.74        5.83        18.00