INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 6, No 3, 2016

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1 INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 6, No 3, 016 Copyright by the authors - Licensee IPA- Under Creative Commons license 3.0 Research article ISSN Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling Peyman Shadman Heidari 1, Ali Golara 1- F. A. Department of Civil Engineering, East Tehran Branch, Islamic Azad University, Tehran, IRAN - Responsible with Strategic Planning Studies at National Iranian Gas Company (NIGC). a.golara@gmail.com doi: /ijcser.6018 ABSTRACT The current research presents a quic method for estimating the lateral stiffness and torsional stiffness of 3D MRF (moment-resistant frame) structures, considering irregular moment frames. This study also provides a method for calculating displacement and rotation and natural frequencies with respect to different lateral load patterns. This study proposes a method for calculating lateral and torsional stiffness for each frame in two directions, and then converting the stiffness of all frames to one frame to obtain the deformation and natural frequency for two directions. The basic idea of the proposed innovative method was developed through the force method to obtain the lateral deformation and stiffness of D building structures. Then, the mentioned procedure was expanded into 3D building structures. Some examples have been made to compare the latter method with linear analysis. The results showed that the suggested method can capture 3D dynamic characteristics with accuracy compared with linear analysis. Keywords: Lateral stiffness, torsional stiffness, natural frequency, force method, 3D building structure. 1. Introduction All computer programs need initial values for calculating the cross-section and material properties of a building s structural elements. These initial values are obtained through some preliminarily calculations. Cross-section and material properties will be the values most referred for optimizing in analysis and design procedures. On the other hand, there is usually no certainty of the correctness of the data entry or matching of the data entered by the user with reality, especially in the case of inexperienced engineers woring with complicated software. In this case, a control or final checing tool is very useful or even necessary. Most methods used to analyze building frames, such as the cantilever, portal, factor, Spurr, Bowman, and Witmer methods (Utu S, 1991), are limited to only regular geometric D moment frames. They cannot calculate the lateral displacement or the lateral and torsional stiffness of the 3D frame systems in the building. The Kan and PCA methods have been presented and used for many years for regular moment D frames with shear walls. However, these methods are not matched with 3D frames and they can result in errors of even more than 50% in the calculation of lateral displacements. Moreover, they have employed some complex formulations that are very time-consuming when they are calculated by hand. Grigorian presented a method for calculating the lateral response of regular D frames based on an analogy between the frame Received on December, 015 Published on February

2 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling and discretized Timosheno beam-columns with similar boundaries. Implementation of this method, however, is not easy enough for use in calculating the 3D irregular frame response with respect to lateral load. Moreover, the calculation of lateral load pattern distribution to determine lateral displacement is more necessary and important when there are eccentricities between the mass center and the rigid center of the 3D frame system. The main aim of the proposed method in this paper is to deal with irregular 3D frames. Miranda and Taghavi [3] used the HS73 model to acquire the approximate structural behavior up to 3 modes. As a follow-up study, Miranda and Aar (Miranda E., 006) extended the use of HS73 (Heidebrecht A.C, 1973) to compute generalized drift spectra with higher mode effects. In this study, the continuum model is also used to estimate the fundamental periods of high-rise buildings More recently, Gengshu et al., studied second order and bucling effects on buildings through the closed form solutions of continuous systems. Eroglu and Aar proposed lateral stiffness estimation in frames and its implementation in continuum models for linear and nonlinear static analyses. Mohsen Shahrouzi M 004, suggested a quic method for estimating the eigenvalues of multi buildings. His proposed method was based on the developed mapping between the chain structure and an equivalent beam model; thus, it led to a dimensionless frequency equation. The procedure presented in this study introduces a new definition for plan irregular structure but regular in height, especially for the mass/stiffness of the last with respect to the others. Hosseini and Imagh-e-naiini presented a quic method for estimating the lateral stiffness of building structures, including regular and irregular moments and braced D frames. The present paper extends the method of their study from D to 3D analytical modeling. By using proposed method, the dynamic specification of a 3D model of moment frame, including the lateral stiffness, torsional stiffness, displacement and rotation of subjected lateral load, and natural frequency of a 3D system, can be calculated with good precision. In other words, this paper provides a more accurate estimation of lateral deformation profiles of discrete systems through the simplified continuum model. Finally, the results obtained from this study s proposed method and numerical modeling results were compared to validate the accuracy of the proposed method.. Methodology The main concept of this study is based on the simplification of D modeling. In this method, a D frame with definite mechanical specifications and multi-bays is converted to many one-bay D frames combined by hinges. These D frames can be summarized by a one-bay, one- D frame, hereinafter referred to as a module of the simplified system. Figure 1 shows a schematic view of the proposed method to simplify a D frame system. The simplified equivalent D system was initially proposed by Hosseini and Imagh-e-naiini. The basic ideas of the proposed method are based on the following facts: A) In a ordinary moment frame subjected to a lateral load, beam and columns in all bays deform similarly. B) The lateral stiffness of a multi- frame at each floor is mainly due to columns and beams just below and above that floor. International Journal of Civil and Structural Engineering 196 Volume 6 Issue 3 016

3 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling Figure 1: The simple multi-bay frame subjected lateral load, and its simplified equivalent systems: (a) the main D system, (b) the simplified equivalent D system, (c) the basic module of simplified In a moment D frame with regular geometric planes that are connected to each other by hinges as shown in Figure. 1, the value moment of inertia for column I c and the moment of inertia for beam I g are given by: I c m j1 I cj m (1) I g L m j1 I I m gi j () Where L and m are span length value and number of spans, respectively. It is obvious that the frames shown in Figure 1(b) are equivalent to m of the single frame shown in Figure 1(c). A similar idea was used for the n- D frame shown in Figure. The values of I cj and I gi in Figures (b) and (c) are given by: m I 1 I cj cij (3) j1 I gi L m j1 I I gij j (4) In fact, each of the sub-frames in Figure (c) is a simple frame module, lie that shown in Figure 3, which has lateral stiffness of : International Journal of Civil and Structural Engineering 197 Volume 6 Issue 3 016

4 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling Figure : Model of m-bay, n- D moment frame, and its simplified equivalent systems: (a) the main D system, (b) the one-step-simplified equivalent D system, (c) the final simplified equivalent system 1c c ( d u ) 6du fm ( )( ) (5) h ( ) 3 c c d u d u Where: EI h c c (6) EI h EI gd d (7) gu u (8) h Where h, I c, I gd and I gu are the dimension and the cross-sectional properties of the frame module, respectively, as shown in Figure 3, and E is the modulus of elasticity of the frame material. International Journal of Civil and Structural Engineering 198 Volume 6 Issue 3 016

5 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling Figure 3: The main frame module of the simplified D system for regular moment frames.1 Main concept of approximation lateral and torsional stiffness for 3D systems The lateral stiffness and torsional stiffness in a 3D moment frame for X and Y directions were obtained with the approximating method for the D moment frame. Then, the simplified stiffness in the X and Y directions were summed in each direction, and the 3D system was exchanged for two D moment frames in each direction of X and Y. With this method, it can be given by: n fm x i1 m fm y j1 fm i fm j Where n is the number of moment frames in the X direction and m is the number of moment frames in the Y direction. The coordinate of the center of stiffness (CR) for irregular 3D building systems can be given by: fmy x X CR (11) fmy (9) (10) fmx y Y CR (1) fmx The value of torsional stiffness of each can be obtained from the stiffness of each moment frame in the X and Y direction, and then the sum of these values in each direction. n. x. i fmy i xi i1 (13) n. y. i fmx i yi i1 (14) The sum of torsional stiffness of columns for each of a building system can be given by: n GJ. c. i (15) h i1 International Journal of Civil and Structural Engineering 199 Volume 6 Issue 3 016

6 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling Where G is the torsion constant for the section and J is polar inertia of column, that: J (16) I xc I yc The model is composed of torsional stiffness contributions to overall lateral stiffness. n GJ. i ( fmy i xi fmx i yi ) (17) h i1 Or: n GJ. i. x i. y i i 1 h E where G (1 ) (18) Where i, n, x i, y i, j and G are the counters of each, number of stories, distance from each frame to stiffness center of each in two directions, polar moment inertia, and shear modulus of a material computed by adding moment of inertia in two directions, respectively. Therefore, the displacement in each direction of X and Y can be given by: Vxi xi (19) yi, xi V fmx yi yi (0) fmy Where xi, V, V yi, fmy and fmx are the lateral displacement values in X and Y directions, shear force in X and Y directions, and estimated lateral stiffness in two directions X and Y, respectively. Accumulated lateral displacement values of stories versus base shear in each have a relationship with lateral stiffness.considering the computed torsional stiffness values for each, torsional moment of rigidity center, and Hooe's law, the total in-plan rotation of each can be computed as follows: T i i (1) 3. Evaluation of proposed method for 3D frame behavior To show the high efficiency of the proposed methods for calculating the lateral displacement, torsion of each, and main period of structures, some numerical examples are presented. The examples presented here are 4, 6, 1, 18 and 4- steel frames with irregularity inplan. In all models, the plan view is the same (Figure 4) and the height of the stories and the length of beams are 3m and 5m, respectively. The modulus of elasticity is supposed to be gfcm. For calculating natural frequency and base shear, a mass of 448tons has been considered for all floors. Lateral loading of frames is defined as static loads and calculated based on the regulations of Iranian seismic design code (IS ) with the design basis acceleration of 0.35 for South Pars region and soil period of 0.5 second for soil International Journal of Civil and Structural Engineering 00 Volume 6 Issue 3 016

7 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling type II. The IS [1] is derived from UBC 1994 and BOCA 1978 and have undergone major changes over the years. For summarizing the content of present wor and avoiding time-consuming, the analysis of a 6- frame will be described in detail. The specification of 3D modeled 6- frame with irregular structure is shown in Figure 4. The stiffness center in X and Y direction and the values of stiffness based on the proposed method are illustrated in Table 1 and Table to 4, respectively. For this frame and also other models, the error of the proposed method in calculating the lateral displacement, rotation and the main period of structure as the analytical parameter values will be compared with numerical result from an analysing program. Evaluation of the methodology presented here requires structural model to be accurate. To analytically predict the response of structural system under seismic loads, the building structure should be accurately described. Using reliable analytical software and definition of strength of materials and yielding behavior of elements are necessary. The models were analyzed herein by employing OpenSees software. All of the beams end connections within the structure are assumed to be pinned. Therefore, the beams are modeled as elastic elements with steel0 material. These models are built with nonlinear beam column element for columns as well as the P-delta effects are taen into account. Fiber elements were used in all of the nonlinear elements. The masses are lumped at floor levels, whereas the horizontal degrees of freedom are defined. The Rayleigh damping with a specified ratio of ξ = 0.05 was assigned at the first-vibration-mode, and the effect of nonstructural elements was not considered. Figure 4a: D plan view of studied structure with asymmetric distribution of stiffness International Journal of Civil and Structural Engineering 01 Volume 6 Issue 3 016

8 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling Figure 4b: D view of AF-1and AF-6 frames of studied structure Based on the presented methodology in this paper, in order to obtain torsional stiffness for each, the tortional stiffness of each frame should be computed in each direction then total value of computed stiffness for both directions should be considered as a total stiffness of the frame. In other words, the torsional stiffness in each direction is computed by multiplying lateral stiffness of each frame and square distance of each frame to stiffness center which result in Equation 18. The shear modulus of steel material was given as 784 cm Based on Equation 18, torsional stiffness in each can be obtained by adding the lateral GJ stiffness of columns ( ). It is worth stiffness of the columns () and torsional. x. i h mentioning that the torsional stiffness of columns in each should be computed, and its value should be included separately with the torsional stiffness of each frame in the total torsional stiffness of a frame (Table 5). With respect to the calculation of the abovementioned components for torsional stiffness, the torsional stiffness value would be very small compared with the torsional stiffness in each frame; thus, it can be neglected. Based on computed lateral stiffness values, the shear force, and Hooe's law, lateral displacement can be computed by Equations 19 and 0 and are presented in Table 6. The total in-plan rotation of each can be computed by using Equation 1 and Table 7 presents estimated rotation values calculated by stiffness center and in-plan torsional moment provided by determined lateral load distributions in the X and Y direction for considered structure. In order to calculate the main period of structure, lateral and torsional stiffness values should be computed based on the above-mentioned equations 15 to 18. Thus, the ton.. y. i International Journal of Civil and Structural Engineering 0 Volume 6 Issue 3 016

9 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling rotation and transition component of mass moment inertia for each is calculated by the following equation: 1 m ma b md () Where a, b, m, m and d are rotational mass moment inertia, total mass of each, length and width of panel, and distance between mass center of panel and total mass center of each. The results are shown in Table Numerical result from analysing program At this stage, the exact values of lateral and torsional displacement were calculated by OpenSees software. These results consist of lateral displacement values for each and the torsion of each in the X and Y directions. Table 9 presents the exact values of displacement in directions and the rotations of each. Figure 5a and 5b show schematic views of the lateral and torsional displacement of the stories. Figure 5a: Schematic views of stories lateral and Torsional displacements for sixth stories of studied building Figure 5b: schematic views of stories lateral and tortional displacements for sixth stories of studied building International Journal of Civil and Structural Engineering 03 Volume 6 Issue 3 016

10 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling 3.. Calculation of the error of proposed method Based on the method presented in this study, the analytical parameter values used in this study were compared with the corresponding exact values obtained by OpenSees software; the calculated errors can also be computed as follows: estimate exact % e displacement 100 (3) exact estimate exact % e rotation 100 exact (4) Testimate Texact % e priod 100 (5) T exact Figure 6: The error in displacement calculation in X and Y direction Figure 7: The absolute error in rotation calculation in X and Y direction Where,, and T are lateral displacement of each, torsion of each, and main period of structures obtained by the presented method and exact salutation, respectively. International Journal of Civil and Structural Engineering 04 Volume 6 Issue 3 016

11 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling Table 10 to 1 show the calculated errors of lateral displacement, torsion and the main period values of 6- structure for each stories computed by the exact solution and the proposed method. Combining all the error, from Figure 6 to 8 it can be seen that the error of proposed method in calculating the lateral displacements and rotation of stories and main period of irregular buildings are less than 1%. Therefore, this shows a good agreement between the proposed method with analysing program. Figure 8: The error in main period of models Table 1: Specifications of selected structure Center of Rigidity Story height Story (metre) (metre) X Y Table : The obtained values of stiffness based on simplified model in presented methodology Story h (cm) 6 5 c EI h c 1 EI gu EI 1 gd c c( d u) 6du u d fm x ( )( ) L L h c c( d u) 3du International Journal of Civil and Structural Engineering 05 Volume 6 Issue 3 016

12 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling E Table 3: Calculation of the stiffness center based on simplified model in presented methodology for the structure in Y direction Kfmy Calculate of X CR Story A B C D E F Kfmy Kfmy * x (m) X CR Table 4: Calculation of the stiffness center based on simplified model in presented methodology for the structure in X direction Kfmx Calculate of Y CR Story Kfmx Kfmx * y (m) Y CR International Journal of Civil and Structural Engineering 06 Volume 6 Issue 3 016

13 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling Table 5: Computed tortional stiffness value for each based on presented methodology Story n. x. i fmy i xi i1 GJ h n. y. i fmx i yi i1 ton cm rad Table 6: Estimation of displacement values of rigidity center for each Story fmx ton cm fmy ton cm (ton) V xi (ton) V yi xi (cm) xi (cm) yi (cm) yi (cm) Table 7: Estimated in-plan rotation values of each provided by torsional moment for X and Y direction ton cm Story x, y T i x y ton cm, x,y rad x,y rad rad International Journal of Civil and Structural Engineering 07 Volume 6 Issue 3 016

14 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling Table 8: Calculated lateral and tortional stiffness values, transformed mass moment inertial and tortional mass moment inertial MassX MassY ton cm Stor fmx ton cm fmy ton cm Mass rad ton ton y ton. cm. sec cm cm sec sec stor y stor y stor y stor y stor y stor y Table 9: Exact values of displacements in directions and rotations of each Story Diaphragm Load UX (cm) UY (cm) RZ x (rad) 6 D1 EX D1 EX D1 EX D1 EX D1 EX D1 EX BASE D1 EX D1 EY D1 EY D1 EY D1 EY D1 EY D1 EY BASE D1 EY International Journal of Civil and Structural Engineering 08 Volume 6 Issue 3 016

15 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling Table 10: Calculated errors of lateral displacement values for each stories computed by the exact solution and proposed method Story (cm) UX (cm) xi % yi (cm) UY (cm) y e x % e Table 11: Calculated errors of torsions values for each stories computed by the exact solution and proposed method Story rad e. x RZ x (rad) x % rad % e. y RZ y (rad) y Table 1: Obtained main period of the structure based on the exact solution and the proposed method Mode T exact (sec) T estimate (sec) %e priod Conclusion In this study, a simplified method for estimating lateral stiffness, torsional stiffness, displacement and main period of structure is proposed. Results from the presented method and from the exact analysis by OpenSees software for the irregular MRF structures have been compared to evaluate the validation of the proposed method. The results showed that there is good agreement with insignificant error between the proposed method presented in this study and the exact solution based on analytical modeling. Thus, the proposed methods can provide a proper alternate for estimating initial lateral stiffness and lateral displacement and rotation of even irregular structures and also for final checing of designs. The results showed that the proposed method can provide the lateral displacement and rotation values with a maximum error of less than 1%. Therefore, the proposed method can estimate numerical values with acceptable accuracy and be applied for 3D structures. International Journal of Civil and Structural Engineering 09 Volume 6 Issue 3 016

16 Presenting a quic method for estimation of MRF dynamic characteristics using 3D modeling 5. References 1. Building and Housing Research Center (BHRC). Iranian Code of Practice for Seismic Resistant Design of Buildings, Standard No , 3rd edition, (005) Building and Housing Research Center, Tehran, Iran.. Dym C.L., Williams H.E., (007), Estimating fundamental frequencies of tall buildings, Journal of Structural Engineering, 133 (10), pp Eroğlu T., Aar S., (011), Lateral stiffness estimation in frames and its implementation to continuum models for linear and nonlinear static analysis, Bulletin of Earthquae Engineering, 9(4), pp Gengshu,T.,Y-L Pi, Bradford, M.A., Tin-Loi, F., (008), Bucling and second-order effects in dual shear-flexural systems, Journal of Structural Engineering, 134(11), pp Georgian, M., (1993), On the lateral response of regular high-rise frame, (1993) Struct. Design Tall Build,, pp Golara A, (014), Probabilistic seismic hazard analysis of interconnected infrastructure: a case of Iranian high-pressure gas supply system, Natural hazards, 73(), pp Heidebrecht A.C., Stafford B.S., (1973), approximate analysis of open-section shear walls subjected to torsional loading, Journal of the Structural Division, pp Hosseini, M. and Imagh-e-Naiini, M. R., A quic method for estimating the lateral stiffness of building systems, (1999) Structural. Design Tall Build, 8, pp Miranda E., Taghavi S., (005), Approximate floor acceleration demands in multi Building, I: formulation, Journal of Structural Engineering, ASCE, 131(), pp Miranda, E., Aar, S.D., (006), Generalized inter drift spectrum, Journal of Structural Engineering, ASCE, 13(6), pp Shahrouzi M.A, (004), quic method for eigenvalue estimation of multi, 13th World Conference on Earthquae Engineering, Canada, Paper No Utu S., Norris, C.H., Wilbur J.B., (1991), Elementary Structural Analysis, McGraw- Hill, New Yor. International Journal of Civil and Structural Engineering 10 Volume 6 Issue 3 016