- Article
- Open Access
Finite Element Modeling and Nonlinear Analysis for Seismic Assessment of Off-Diagonal Steel Braced RC Frame
- Keyvan Ramin^{1}Email author and
- Mitra Fereidoonfar^{2}
https://doi.org/10.1007/s40069-014-0089-9
© The Author(s) 2014
- Received: 9 November 2013
- Accepted: 19 August 2014
- Published: 16 September 2014
Abstract
The geometric nonlinearity of off-diagonal bracing system (ODBS) could be a complementary system to covering and extending the nonlinearity of reinforced concrete material. Finite element modeling is performed for flexural frame, x-braced frame and the ODBS braced frame system at the initial phase. Then the different models are investigated along various analyses. According to the experimental results of flexural and x-braced frame, the verification is done. Analytical assessments are performed in according to three dimensional finite element modeling. Nonlinear static analysis is considered to obtain performance level and seismic behaviour, and then the response modification factors calculated from each model’s pushover curve. In the next phase, the evaluation of cracks observed in the finite element models, especially for RC members of all three systems is performed. The finite element assessment is performed on engendered cracks in ODBS braced frame for various time steps. The nonlinear dynamic time history analysis accomplished in different stories models for three records of Elcentro, Naghan and Tabas earthquake accelerograms. Dynamic analysis is performed after scaling accelerogram on each type of flexural frame, x-braced frame and ODBS braced frame one by one. The base-point on RC frame is considered to investigate proportional displacement under each record. Hysteresis curves are assessed along continuing this study. The equivalent viscous damping for ODBS system is estimated in according to references. Results in each section show the ODBS system has an acceptable seismic behaviour and their conclusions have been converged when the ODBS system is utilized in reinforced concrete frame.
Keywords
- FEM
- seismic behaviour
- pushover analysis
- geometric nonlinearity
- time history analysis
- equivalent viscous damping
- passive control
- crack investigation
- hysteresis curve
1 Introduction
Since, the experimental researches are very expensive and time-consuming (Altun and Birdal 2012), the application of computer modeling methods as initial investigation and also in the next step, calibration of present computer models with a similar previous experimental research can be a certainty. In this study, the finite element model is calibrated in the first step for flexural frame and x-braced frame. Then the main model is simulated according to verified characteristics of the last step’s model. The FE model is considered to different analysis and design.
The design of seismic resistant structures in seismic regions should satisfy two criteria. First, under frequent and low to moderate earthquakes, the structure should have sufficient strength and stiffness to control deflection and prevent any structural damage. Second, under rare and severe earthquakes the structures must have sufficient ductility to prevent collapse (Roeder and Popov 1977).
Reinforced concrete structures usually have dual behaviour against lateral loads, first the behaviour before cracking (pre-cracking) by limited resistant and the other behaviour after cracking (post-cracking) by increasing the ductility. Although the reinforced concrete behaviour of post cracking stages is complicated and extended by multi steps (elastic, yielding, elastic-perfectly-plastic (EPP), plastic and collapse), merely whatever is obvious in all steps after cracking is the high amount of ductility proportional to the other stage.
So, many kinds of lateral load bearings have been used in steel and Reinforced Concrete (RC) structures for recent years, which contain useful performances.
The important thing that should be mentioned here is if the additional system to RC frame being the occasion of imperfect energy absorption in reinforced concrete members, then some part of the structure will not contribute in energy dissipation and actually this system’s application has no economic advantages. For example in a reinforced concrete x-braced frame, just upon the imposing lateral loads, the diagonal members of bracing system make a directional component with lateral loads and so they experience a high percent of those lateral load. Now if the lateral loads be increased, the axial plastic hinge will be formed in steel bracing members before the formation of flexural plastic hinge in RC members. By continuing the imposed lateral load on the x-braced RC frame, the limited nonlinear behaviour is started in term of large deformation in braced frame members. Based on the high capacity of energy absorption in diagonal steel bracing members proportional to reinforced concrete members, the RC frame will be collapsing in possible minimum time after occurrence the large displacement and collapse in diagonal steel members and it won’t let reinforced concrete members to make plastic hinge in a short time domain. So this phenomenon will not let the reinforced concrete members being contributed in energy dissipation and on the other hand the plastic limit is abbreviated along this short time for the load transferring.
The off-diagonal bracing system induces new properties of the reinforced concrete frame. The different performances of the RC frame braced by off-diagonal bracing system (ODBS) are about the each member’s opportunity and possibility for the formation of plastic hinges.
The specific geometry of the ODBS, one of the steel bracing members absorb the amount of energy until, cracking extend to RC members. Additional to large deformation in third member of ODBS, the cracking is increased in RC frame, even may be observed the frame’s plastic hinges. While the other steel diagonal members, being oriented in parallel form, it can be a confident fuse to prevent declining the lateral resistance of the RC frame.
The preceding reasons demonstrate the ODBS braced frame has two stages behaviour for the elastic, elasto-plastic and plastic treatments. The extended amount of energy dissipation for ODBS system is not only because of the inherent nonlinear properties of materials but also the particular nonlinear geometry is effective on damping and energy dissipation and anyhow presents a particular system of passive energy dissipative.
The two basic requirements for seismic design are high stiffness at working load level and large ductility at severe over loadings. These requirements are difficult to be satisfied when the above conventional frames are used. On the contrary, Eccentrically Braced Frames offer an economical framing system satisfying both requirements.
In all types of this system, the vertical components of axial forces in the braces are held in equilibrium by shear and bending moments in short beams of lengths, which is the active links. Active links are designed to remain elastic at working loads and deform inelastic on over loading of structure, thereby dissipating large amount of energy. In this system the hazardous brace buckling can be entirely prevented since the link acts as fuse to limit the brace axial force. Also this frame has a much greater lateral resisting capacity than that of an MRF if the beam section used are the same (Mastrandrea and Piluso 2009a, b; Mastrandrea and Piluso 2009a, b). On the other hand in ODBS system the third member of bracing has similar treatment to active link but by another mechanism.
2 Aims and Objectives for this Research
The main aim of this research is related to investigation of ODBS system’s behaviour. Corresponding to the title, in this assessment focused on seismic behaviour of ODBS braced RC frame under real registered earthquake records and also the spectral forces parallel to exact modeling by finite element method (FEM). FEM models were developed to simulate various RC frames with and without steel bracing systems of three full size frames for nonlinear response up to collapse, using the ANSYS program (ANSYS 2015). Then models verified for several analysis and investigation. As it is known to us, a considerable impact load induced at difference modes of vibration through earthquake and exerted on bracing components if the direction of forces change and components stretch under the influence of components’ buckling caused by pressure (Ravi Kumar et al. 2007). This study expects that the amount of lateral forces being transmitted from earth to upper levels, subsequently the effect of impact will decrease as a result of using ODBS, the high energy absorption capacity, in lower floors of the structure. Time history analysis is done for high rise models by different properties. Also several comparisons assessed for applicable results and Conclusions from the current research efforts and recommendations for future studies are included.
3 Research Background
Basis on collapse prevention and life safety performance levels, a structure has to experience the large inelastic deformations in term of large capacity of energy dissipation during any excitation. Actually, the reason of structural stability of a system under any inelastic earthquake load is the condition of hysteretic loops, which means the stability of structural system depends on stability of hysteresis curves in each cycles. Such stable loops of a cyclic load or time history acceleration under an earthquake load can be a provision of sufficient ductility and large amount of energy dissipation for structural system’s element (Khatib et al. 1988; Asgarian et al. 2010).
For high and medium rise buildings, structural steel has been used extensively due to its high strength and ductile properties. In general, bracing systems are divided into two general types: concentric and eccentric (Ghobarah and Abou Elfath 2001; Kim and Choi 2005; Moghaddam et al. 2005). Concentric braced systems are more desirable because of a relative high stiffness, along with their easy construction and economy aspects; hence these important criteria make this type more common than eccentrically braced frames (Davaran and Hoveidae 2009).
Eccentric braces need more construction accuracy thereby resulting in a decrease of construction speed and higher cost in spite of better stiffness performance and higher energy dissipation (Özhendekci and Özhendekci 2008; Bosco and Rossi 2009).
This system allows the architects to have more openings in panel areas (Moghaddam and Estekanchi 1995; Moghaddam and Estekanchi 1999). Moreover, because of the cyclic nature of seismic loads, these brace elements are designed symmetrically and so they should perform in two span to work symmetric.
The idea of steel bracing system application in reinforced concrete structures was first suggested for seismic strengthening of concrete buildings. From the viewpoint of both research and application, this idea has been very prevalent during past two decades because of its simplicity, implementation and its lower relative cost compare with shear wall. For example, Sugano and Fujimura performed a series of experiments on a model of one-story frame which had been strengthened through various methods. They examined x-bracing and k-bracing systems and compared them with samples strengthened by concrete and masonry in-filled walls. They aimed to determine the effect of each system on enhancement of in-plane strength and ductility of the samples (Sugano and Fujimura, 1980). Furthermore, Kavamata and Ohnuma demonstrated the possibility of the effective use of steel bracing systems in concrete buildings (Kawamata and Ohnuma 1981).
A model of two spans, two stories reinforced concrete frame in scale of 1:3 was chosen to represent the seismic weakness and behaviour. The strengthened frame was exposed to lateral and gravity loads and its displacements were allowed to increase by one fiftieth of the frame’s Original height (inter-story drift). The strengthened inside frame by a ductile steel brace, demonstrated better behavior considerably than the preliminary reinforced concrete frame (Masri and Goel 1996) or applied from outside the frame (Bush et al. 1991).
In 1999, the direct internal use of steel bracing system in concrete frame was studied in laboratory. Experiments were carried out on five one span, one story frame samples with a scale of 1:2.5. Two of them had no bracing system but the other three samples strengthened by x-bracing system with different connector’s component including bolt and nut, cover of RC column, and gusset plates placed in concrete. Results showed, depending upon various connectors’ component, the bracing system considerably increases the equivalent stiffness of the frame and notably changes its behavior. When the bracing system’s connector is implanted inside concrete, the performance of frame gets even better and further energy is absorbed. Generally, experiments demonstrated that the bracing system tolerates a major part of lateral load in reinforced concrete frame (Tasnimi and Masoumi 1999).
Dynamic behavior of the concrete buildings strengthened with concentric bracing systems has investigated by Abou Elfath and Ghobarah. A three story building was dynamically analyzed with various earthquake records and the effect of steel bracing system on building as well as the effect of bracing system distribution throughout frame’s height was studied. The position of braces investigated the proportional seismic performance, inter-story drift, and damage index to show the effect of this type of bracing systems (Abou Elfath and Ghobarah 2000).
Maheri et al. (2003) first reviewed previous studies on strengthening by steel bracing systems and then investigated three models including a simple frame, a frame strengthened with x-bracing, and a frame strengthened with knee bracing system under lateral load until failure stage. They found that ductility of RC frame is considerably increases when using knee bracing system (Maheri and Akbari 2003).
In 1994, Moghaddam and Estekanchi modeled and tested Off-Centre bracing systems in steel frames for the first time. Later in 1999, they analyzed the seismic behavior of Off-Diagonal bracing system. They confirmed that this system’s behavior resembles with seismic isolators and play a considerable role in reduction of seismic forces (Moghaddam and Estekanchi 1999, Moghaddam and Estekanchi 1994). All previous studies confirm the effectiveness of steel braces in rehabilitating and retrofitting of RC frame.
4 Finite Element Modeling
4.1 Models Characteristics
4.2 Material Nonlinearity
Concrete and steel are the two constituents of RC braced frame. Among them, concrete is much stronger in compression than in tension (tensile strength is of the order of one-tenth of compressive strength). While its tensile stress–strain relationship is almost linear, the stress–strain relationship in compression is nonlinear from the beginning. The Elastic-Perfectly-Plastic (EPP) model for steel, which is used in this work, assumes the stress to vary linearly with strain up to yield point and remain constant beyond that (Anam and Shoma 2011).
In this research the Willam and Warnke (1974), the yield and failure criteria, is considered for concrete model behaviour. Also, since the SAP2000 (2010) assumption applies the Drucker–Prager criteria for concrete material modeling and its behaviour, both of mentioned criteria’s are considered in analysis of models. By this method, the analytical comparison of applied criteria’s is done.
On the other hand, the uniaxial stress–strain relationship for confined concrete, known as the modified Kent and Park model, has been incorporated in the FE model constructed here. This model shows a good agreement with the experimental results (Kent and Park 1971; Scott et al. 1982) and offers a good balance between simplicity and accuracy (Taucer et al. 1991).
In non-linear dynamic analysis, the non-linear properties of the structure are considered as part of a time domain analysis. This approach is the most rigorous, and is required by some building codes for buildings of unusual configuration or of special importance. However, the calculated response can be very sensitive to the characteristics of the individual ground motion used as seismic input; therefore, several analyses are required using different ground motion records to achieve a reliable estimation of the probabilistic distribution of structural response.
Since the properties of the seismic response depend on the intensity, or severity, of the seismic shaking, a comprehensive assessment calls for numerous nonlinear dynamic analyses at various levels of intensity to represent different possible earthquake scenarios. This has led to the emergence of methods like the Incremental Dynamic Analysis (Bozorgnia and Bertero 2004).
4.3 Geometric Nonlinearity
For specification of nonlinear geometry of ODBS, concrete nonlinearity is added to material nonlinearity in this paper. Only steel nonlinearity for third member of ODBS is considered in this article’s analysis. Steel, on the other hand, is linearly elastic up to a certain stress (called the proportional limit) after which it reaches yield point (f_{ y }) where the stress remains almost constant despite changes in strain. Beyond the yield point, the stress increases again with strain (strain hardening) up to the maximum stress (ultimate strength,f_{ ult }) when it decreases until failure at about a stress quite close to the yield strength.
Nonlinear static procedures use equivalent SDOF structural models and represent seismic ground motion with response spectra. Story drifts and component actions are related subsequently to the global demand parameter by the Pushover or capacity curves that are the basis of the non-linear static procedures. Nonlinear dynamic analysis utilizes the combination of ground motion records with a detailed structural model, therefore is capable of producing results with relatively low uncertainty. In nonlinear dynamic analyses, the detailed structural model subjected to a ground-motion record produces estimates of component deformations for each degree of freedom in the model and the modal responses are combined using schemes such as the sum of squares square root.
Complete comparisons of the studied Retrofitted Frames in ANSYS program (version 15) with the Micro modeling structural element indicate that ODBS steel bracing RC frame has two yielding point that were related to main RC flexural frame and third steel member of ODBS. It’s so useful for structures that are under Impact Loads and loads by high velocity specifically according to Fig. 2 of the previous page.
5 Finite Element Models
5.1 Finite Element Modeling Procedure
The main Flexural RC frame is calibrated by results of experimental modeling of the same flexural frame and X-Braced frame that constructed in laboratory (Maheri and Hadjipour 2003).
5.2 Comparison Between FE and Experimental Results
Figure 7 shows a front view of this modeled RC frame. In Fig. 7a and b, the experimental and numerical force–displacement curves are compared for flexural frame and x-braced frame, respectively. Lateral displacements are measured at the base points at the top height of the frames. A comparison between these two figures shows the amount of calibration level in compare with the experimental modeling frames. The specific purpose of 3D finite element modeling is based upon the complex behavior of reinforced concrete nonlinearity and the geometric nonlinearity that applied for this paper.
The response modification factor (R factor) is the one of important factors for introducing the structural behavior specially to define ductility and stiffness of structures. To doing this aim, the separated calculations are performed to obtaining the factor of behaviour. Related force–displacement diagrams in form of bilinear pushover curves and the samples of finite element modeling are shown also in left side of Fig. 7.
After generating the pushover curves, the response modification factors are obtained for each model. The highest ductility and the large amount of energy dissipation are from ODBS system results.
5.3 The Response Modification Factor for ODBS Braced RC Frame
The maximum drift values (regardless of drift limitations) and ductility andR factor for floor.
Model | Δ_{ y }(m) | Δ_{ max } (m) | Δ_{ s }(m) | Δ_{ w } (m) | V_{ e }(kg) | V_{ y } (kg) | V_{ s } (kg) | V_{ w } (kg) | µ | R |
---|---|---|---|---|---|---|---|---|---|---|
(flexural RC frame) F_{1} | 0.0173 | 0.0665 | 0.0112 | 0.067 | 17,272 | 4,500 | 2,875 | 1,740 | 3.84 | 9.92 |
(X-Braced frame) F_{X} | 0.0103 | 0.039 | 0.0058 | 0.0547 | 33,034 | 12,200 | 9,147 | 5,480 | 2.69 | 6.04 |
(ODBS-Braced frame) F_{ODBS} | 0.0125 | 0.1101 | 0.0082 | 0.0781 | 26,177 | 7,800 | 5,032 | 3,012 | 9.58 | 23.2 |
The response modification factor (R) is included of the inherent ductility and ductility and overstrength effects of a structure and the difference in the design methods and limitations about related manual. Also Ductility reduction factor Rμ is a function of both of the characteristics of the structure including ductility, damping and fundamental period of vibration (T), and the characteristics of earthquake ground motion. Figure 8 explains the schematic behaviour and its corresponding parameters to calculating response modification factor.
WhereR_{ s } is the overstrength factor and Y is termed the allowable stress factor (Maheri and Akbari 2003).
The results indicate the highest amount of response modification factor (R) is obtained for ODBS proportional to the other systems. By considering the presented formulation, the obtained results are gathered in Table 1. TheR factor for ODBS is same as the system of displacement and vibration control, particularly same as the systems isolator and friction dampers. The author has other research paper about the ODBS innovation considering a new friction dampers substituted by third member. The system of composed damper-spring can be a suitable subject for further researches.
5.4 Crack Evaluation Through Finite Element Models
After confirming the compared cracks, for the next stage, the crack analysis is performed on ODBS braced RC frame for further assessments. The ODBS braced frame is analyzed through 94 time steps by displacement control method. The cracking analysis is checked for three times in steps of 28, 65 and 92 from imposed displacement of ODBS model. The reasons for selecting the mentioned time steps along main cracking investigation are at first, effective parameters about crack development and performing plastic hinges along different rotations and at second existed deformation in steel bracing members concerned about important stage of behavior such as yielding and plastic behavior for steel material.
Through the stages of consideration, the ODBS braced RC frame is experienced various behaviours as elastic, elastic–plastic, secondary elastic, secondary elastic–plastic and plastic, then it was in threshold of collapse and also bracing members are formed parallel to each other and they were on diagonal axis. In this position, if the diagonal members lost their strength, then the whole frame was being collapsed.
6 Numerical Models
7 Time History Analysis Methodology
7.1 Time History Records
Characteristics of scaled accelerograms used for time history analysis.
Records | Duration | PGA | Time | Country | Date of | Station | Position | Components | |
---|---|---|---|---|---|---|---|---|---|
(s) | m/sec^{2} | Step(s) | of event | event | Latitude | Longitude | |||
Tabas | 50 | 3.42 | 0.01 | Iran | 1978/09/16 | Deyhook | 33.3′N, 57.52′E | 3.27 | 4.1 |
Naghan | 5 | 7.09 | 0.001 | Iran | 1977/04/06 | Central | 31.98′N, 50.68′E | 7.61 | 7.61 |
Elcentro | 53.7 | 3.49 | 0.01 | USA | 1940/05/18 | E06 array | 32.44′N, 115.3′W | 3.35 | 4.03 |
Macro modeling method was used to analyze the nonlinear behaviour of reinforced concrete frame strengthened by steel bracing system (macro element by lowest accuracy related to micro modeling elements). The model was calibrated using existing laboratory work results and then, larger number of floors and openings were analyzed. The SAP2000 (2010) software was employed to modeling the reinforced concrete frame braced with ODBS system. Dynamic time history analysis is done for modeling high rise concrete frames by increased height and width. Time history analysis using earthquake accelerograms is one of the suggested methods by most regulations to investigate the seismic behaviour of structures. In this study is used the three accelerograms of Naghan, Tabas and Elcentro. Their general characteristics are listed in Table 2.
The various maximum ground acceleration after scaling is set to 0.3 g. Three groups of records are selected based on two parameters; the closest distance to a fault rupture surface [greater than 50 km (far field), nearer than 10 km (near fault)] and the moment magnitude in every scales (Berberian 1977and Jamison et al. 2000). The other characteristics of the real accelerograms such as directivity, fault mechanism, and etc. are the same. The peak ground acceleration of all accelerograms is greater than 0.1 g. These accelerograms are selected from strong ground motion records. The specifications and classification of each group before doing matching procedure are tabulated in Table 2.
7.2 Structural Modeling
Design sections characteristics and component properties for two-story frame’s members.
Components | Floor | Type | Dimensions (b ×h) | Bars and Stirrups | Hinges | Acceptance criteria/type |
---|---|---|---|---|---|---|
Beams | First floor | Rectangular 25 × 25 (cm) | 3 Ø 18 Ø8 @10,20 cm (top &bottom) | Flexural (M3) | 0.01 rad plastic rotation | |
Second floor | Rectangular 25 × 20 (cm) | 3 Ø 16 Ø8 @10,20 cm (top &bottom) | Flexural (M3) | 0.01 rad plastic rotation | ||
Columns | First floor | Rectangular 30 × 30 (cm) | 8 Ø 18 Ø8 @10,20 cm | Flexural +Axial (P-M3) | 0.012 rad plastic rotation | |
Second floor | Rectangular 25 × 25 (cm) | 8 Ø 16 Ø8 @10,20 cm | Flexural +Axial (P-M3) | 0.012 rad plastic rotation | ||
Braces | First floor | X | Box 8 × 8 × 0.5 (cm) | – | Axial (P) | 7 Δ_{T} Plastic deformation |
Second floor | Box 8 × 8 × 0.5 (cm) | – | Axial (P) | 7 Δ_{T} Plastic deformation | ||
First floor | ODBS | 8 × 8 × 0.5: 1,2 (Members) 2 × 2 × 0.3: 3 (Members) | – | Axial (P) | 7 Δ_{T} 9 Δ_{T} | |
Second floor | 8 × 8 × 0.5: 1,2 (Members) 2 × 2 × 0.3: 3 (Members) | – | Axial (P) | 7 Δ_{T} 9Δ_{T} |
Design sections characteristics and component properties for six-story frame’s members.
Components | Floor | Type | Dimensions (b ×h) | Bars and Stirrups | Hinges | Acceptance Criteria/Type |
---|---|---|---|---|---|---|
Beams | First and second floors | Rectangular 35 × 35 (cm) | 4 Ø 18 Ø10@10,20 cm (top & bottom) | Flexural (M_{3}) | 0.01 rad Plastic rotation | |
Third and fourth floors | Rectangular 35 × 30 (cm) | 4 Ø 16 Ø8 @10, 20 cm (top & bottom) | Flexural (M_{3}) | 0.01 rad Plastic rotation | ||
Fifth and sixth floors | Rectangular 35 × 25 (cm) | 3 Ø 16 Ø8 @10, 20 cm (top & bottom) | Flexural (M_{3}) | 0.01 rad Plastic rotation | ||
Columns | First and second Floors | Rectangular 40 × 40 (cm) | 12 Ø 20 Ø8 @ 8,16 cm | Flexural +Axial (P-M_{3}) | 0.012 rad Plastic rotation | |
Third and fourth floors | Rectangular 35 × 35 (cm) | 8 Ø 20 Ø8 @ 8,16 cm | Flexural +Axial (P-M_{3}) | 0.012 rad Plastic rotation | ||
Fifth and sixth floors | Rectangular 30 × 30 (cm) | 8 Ø 18 Ø8 @ 10,20 cm | Flexural +Axial (P-M_{3}) | 0.012 rad Plastic rotation | ||
Braces | Second and third floors | X | Box 10 × 10 × 0.7 (cm) | – | Axial (P) | 7 Δ_{T} Plastic deformation |
Fourth, fifth, & sixth floors | Box 8 × 8 × 0.5 (cm) | – | Axial (P) | 7 Δ_{T} Plastic deformation | ||
First floor | ODBS | Components 1 and 2: Box 10 × 10 × 0.7 (cm) | – | Axial (P) | 7 Δ_{T} Plastic deformation | |
Component 3 Box 3 × 3 × 0.3 (cm) | 9 Δ_{T} Plastic deformation |
Design sections characteristics and component properties for fifteen-story frame’s members.
Components | Floor | Type | Dimensions (b ×h) | Bars and Stirrups | Hinges | Acceptance criteria/type |
---|---|---|---|---|---|---|
Beams | First, second, and third floors | Rectangular 70 × 55 (cm) | 7 Ø 20 Ø16@12,25 cm (top & bottom) | Flexural (M_{3}) | 0.01 rad Plastic rotation | |
Fourth, fifth, sixth, and seventh floors | Rectangular 65 × 45 (cm) | 8 Ø 22 Ø12@10,20 cm (top & bottom) | Flexural (M_{3}) | 0.01 rad Plastic rotation | ||
Eighth, ninth, tenth, and eleventh floors | Rectangular 55 × 40 (cm) | 7 Ø 18 Ø12@10,20 cm (top & bottom) | Flexural (M_{3}) | 0.01 rad Plastic rotation | ||
Floors from twelve to fifteen | Rectangular 40 × 30 (cm) | 6 Ø 18 Ø10@10,20 cm (top & bottom) | Flexural (M_{3}) | 0.01 rad Plastic rotation | ||
Columns | First floor and Second floor | Rectangular 70 × 70 (cm) | 44 Ø 25 Ø16@ 10,20 cm | Flexural +Axial (P-M_{3}) | 0.012 rad Plastic rotation | |
Floors from three to six | Rectangular 65 × 65 (cm) | 36 Ø 25 Ø14@ 10,20 cm | Flexural +Axial (P-M_{3}) | 0.012 rad Plastic rotation | ||
Floors from seven to eleven | Rectangular 55×55 (cm) | 28 Ø 22 Ø12@ 10,20 cm | Flexural +Axial (P-M_{3}) | 0.012 rad Plastic rotation | ||
Floors from twelve to fifteen | Rectangular 40 × 40 (cm) | 20 Ø 22 Ø10@ 10,20 cm | Flexural +Axial (P-M_{3}) | 0.012 rad Plastic rotation | ||
Braces | Floors from One to four | X | Box 20 × 20 × 1.0 (cm) | – | Axial (P) | 7 Δ_{T} Plastic deformation |
Floors from five to nine | Box 15 × 15 × 0.9 (cm) | – | Axial (P) | 7 Δ_{T} Plastic deformation | ||
Floors from ten to fifteen | Box 12 × 12 × 0.75 (cm) | – | Axial (P) | 7 Δ_{T} Plastic deformation | ||
First floor | ODBS | Components 1&2: Box 20 × 20 × 1.0 | – | Axial (P) | 9 Δ_{T} Plastic deformation |
One of the important analyses for investigating the seismic behaviour of a structure is the time history analysis. So for supplementary analysis, the nonlinear dynamic analysis (NDA) is performed for various time histories.
8 Analysis for Results
8.1 Inter-Story Drift Investigation
The obtained inter-story drift results are indicated the amount of dissipated displacement and energy dissipation is increased in a system by more ductility and deformability. The flexural frame is contributed more than x-braced frame in strain energy absorption, because of its ductile characteristics. Construction of ODBS system is divided by two patterns. First, adding off-diagonal steel brace to the first story RC frame, since the x-brace system is used for other stories (composed bracing system by ODBS at first story and x-bracing system for other stories). As second pattern, adding the off-diagonal bracing system to all stories of RC frame (without any composed bracing system). If the system of bracing selected from first pattern, the first story treats as a ductile system and other stories were being treat as a semi rigid body.
The first pattern advantage was a performance same as base isolation system that it is absorbed the vibrations of ground motion and the other stories had a minimum proportional displacement (limits of inter-story drift is checked always). What is too important by this pattern is the limitation of first inter-story’s drift in term of seismic manuals. This system is behaved exclusively, if the regarded drifts being in use. The decision making about second pattern is too important because of its complicated behaviour. When the ODBS is performed in all stories, the energy dissipation potential is created in each story of the structure. By considering the basic lateral resisting system along flexural frame for ODBS system, by adding steel braces, the lateral displacement decreased strongly at each story. In the other word each story equipped by a type of damper and vibrations are controlled and dissipated specially for stories up to fifth or sixth.
One another investigation is about recorded displacements in special base points. The base points are usually selected along the point nearest to center of mass and center of lateral stiffness. Some single base points are chosen at the last floor’s upper level, such as for pushover analysis, on each type of models, these points were monitored along the incremental static loads. Also in dynamic time history analysis, the central base points in each story are defined and monitored to control proportional displacement and/or in the other word, inter-story drift controlling.
Inter-story drift values for each model by second, sixth, and fifteenth stories under the earthquakes Naghan, Tabas, and Elcentro respectively (from left to right) and also in types of flexural, x-braced and off-diagonal bracing systems.
Inter story drift for Naghan, Tabas & Elcentro respectively (cm) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Floor | 2- Stories Model | |||||||||||
Flexural frame | x-braced frame | ODBS braced frame (all stories) | ODBS braced frame (first story) | |||||||||
1 | 0.35 | 0.32 | 0.38 | 0.17 | 0.16 | 0.21 | 0.23 | 0.26 | 0.28 | 0.37 | 0.29 | 0.56 |
2 | 0.28 | 0.21 | 0.24 | 0.11 | 0.09 | 0.12 | 0.21 | 0.18 | 0.25 | 0.17 | 0.12 | 0.21 |
Floor | 6- Stories Model | |||||||||||
1 | 2.95 | 1.68 | 2.93 | 0.62 | 0.6 | 1.4 | 1.08 | 1.67 | 2.63 | 3.28 | 2.96 | 3.95 |
2 | 1.05 | 1.98 | 2.85 | 0.7 | 0.83 | 1.71 | 2.76 | 1.24 | 3.12 | 0.94 | 0.91 | 1.08 |
3 | 2.48 | 1.85 | 2.33 | 0.8 | 1.15 | 1.93 | 2.05 | 2.93 | 3.34 | 0.85 | 0.63 | 0.79 |
4 | 1.49 | 1.08 | 1.45 | 1.45 | 2.41 | 1.68 | 1.46 | 1.25 | 1.68 | 0.72 | 0.85 | 1.13 |
5 | 1.2 | 0.9 | 1.21 | 0.72 | 1.52 | 0.79 | 0.77 | 0.72 | 0.9 | 0.43 | 0.52 | 0.54 |
6 | 0.8 | 0.58 | 0.86 | 0.51 | 0.89 | 0.46 | 0.68 | 0.55 | 0.88 | 0.08 | 0.29 | 0.42 |
Floor | 15- Stories Model | |||||||||||
1 | 1.49 | 0.47 | 0.98 | 0.95 | 0.6 | 0. 8 | 3.11 | 2.12 | 3.43 | 3.43 | 4.05 | 5.93 |
2 | 3.13 | 0.51 | 1.83 | 0.9 | 0.45 | 0.4 | 3.66 | 3.03 | 4.21 | 4.21 | 1.03 | 1.21 |
3 | 2.45 | 1.37 | 2.93 | 0.75 | 0.39 | 1 | 3.71 | 3.47 | 3.63 | 3.63 | 0.96 | 0.99 |
4 | 2.17 | 1.21 | 3.91 | 1.47 | 0.51 | 1.1 | 3.43 | 3.42 | 3.36 | 3.36 | 0.89 | 0.93 |
5 | 2.96 | 1.53 | 4.85 | 1.68 | 1.41 | 2.1 | 2.84 | 2.98 | 3.02 | 3.02 | 0.98 | 0.76 |
6 | 3.85 | 1.92 | 3.72 | 1.93 | 1.5 | 2,6 | 1.98 | 2.18 | 2.45 | 2.45 | 0.85 | 0.85 |
7 | 4.61 | 2.13 | 3.53 | 0.76 | 0.95 | 3.01 | 2.45 | 1.97 | 3.38 | 3.38 | 0.77 | 1.45 |
8 | 3.42 | 1.94 | 2.98 | 0.93 | 2.1 | 1.76 | 3.53 | 2.63 | 3.23 | 3.23 | 0.63 | 1.23 |
9 | 3.23 | 2.73 | 2.45 | 0.81 | 2.05 | 2.51 | 2.81 | 2.51 | 3.87 | 3.87 | 1.51 | 0.82 |
10 | 3.12 | 2.92 | 2.95 | 0.75 | 2.5 | 1.12 | 1.71 | 1.33 | 2.67 | 2.67 | 1.84 | 0.79 |
11 | 2.31 | 3.43 | 1.95 | 0.6 | 1.63 | 0.93 | 1.58 | 1.04 | 2.32 | 2.32 | 1.51 | 0.71 |
12 | 1.86 | 2.95 | 1.43 | 0.72 | 0.95 | 0.81 | 1.62 | 0.73 | 2.13 | 2.13 | 1.06 | 0.62 |
13 | 1.03 | 2.08 | 1.32 | 0.58 | 0.71 | 0.6 | 1.08 | 0.66 | 1.48 | 1.48 | 0.43 | 0.51 |
14 | 0.68 | 1.81 | 1.11 | 0.49 | 0.45 | 0.48 | 0.78 | 0.51 | 0.94 | 0.94 | 0.41 | 0.43 |
15 | 0.34 | 0.71 | 0.95 | 0.33 | 0.2 | 0.23 | 0.54 | 0.36 | 0.67 | 0.67 | 0.32 | 0.36 |
If the results of various types of bracing system have been considered, the maximum inter-story drift is happened in the medium height of flexural frame and x-bracing frame wherever their fractural mode was happened there, at the same levels. Whereas in the ODBS braced frame, the maximum inter-story drift was happened in the first story and the other inter-story drifts were occurred proportional to the first story and almost decreased. This is the optimal and the ideal behaviour in a structure but, in the other systems the pattern of stiffness sorting that, it is from high to low stiffness, have not been respected and this is the reason of increasing the cost of construction, especially for structural skeleton. ODBS system has not only decrease of drift in upper story levels but also decrease of stiffness. In an approximately estimation of cost, the ODBS system has lowest cost for construction compare with the other RC bracing systems.
8.2 Plastic Hinges in terms of Levels of Performance
Generated plastic hinges is indicated in Fig. 19. As shown in this figure, the off-diagonal system has the highest level of deformation not only in RC frame members but also in steel members, especially in third member of steel ODBS. The rotational capacity is increased in ODBS, the most ductile system. Performance levels of flexural and X-braced frames are limited to Life Safety (LS) level, but for ODBS, level of performance has been extended to higher ductility about related design criteria. Six and fifteen stories flexural frames have plastic hinges more than X-braced frame. On the other hand in ODBS, more members are contributed in absorption of defined existing energy by nonlinear ductile behavior of plastic rotation and deformation proportional to flexural frame. Non-linear static analyses as well as dynamic step by step seismic analyses are performed and special purpose elements are employed for the needs of this study. Results showing the influence of the maximum rotation of the multi-storey frame members in terms of ductility requirements and rotational requirements of the frame members (Karayannis et al.).
Acceptance criteria for flexural frame of LS level is 0.02 for primary components and also the acceptance criteria for ODBS braced frame of CP level is 0.025 and 0.05 for primary and secondary components respectively. A more detailed scrutinizing of the results reveals that the hinges formed in ODBS system endure the maximum deformation and earn the structure a very high performance level along ductile behavior. In addition, the more performance levels of plastic hinges are gathered in the structure, so by this level of ductility, the structure will be absorbed more quantity of energy. These results are deduced based on non-linear dynamic step by step analyses. The time steps for this analysis is considered less than ΔT = 0.02 s. Future research will be dedicated to the full time history analysis and investigate the proportional hysteresis curves. Assessing the stiffness and/or the strength degrading is the most important to diagnosing the exact behavior of this system.
The values of maximum response of ODBS braced frame under various eccentricities.
Elcentro time-history acceleration response | General characteristics | ||||||
---|---|---|---|---|---|---|---|
Models & index | Max. Ecc (e_{1}) | Max. Acc (g) | Max.Vel (cm/s) | Max. Displ (cm) | Initial stiffness (kN/cm) | Period (1st mode) (s) | Δ at first yield point (cm) |
ODBS (p0) | 0.00 | 0.342 | 46.08 | 3.93 | 491.78 | 0.47 | 2.83 |
ODBS (p1) | 0.1 | 0.261 | 40.14 | 4.67 | 315.59 | 0.59 | 2.49 |
ODBS (p2) | 0.2 | 0.175 | 37.72 | 5.10 | 158.51 | 0.83 | 2.31 |
ODBS (p3) | 0.3 | 0.073 | 33.3 | 6.06 | 77.49 | 1.19 | 2.1 |
ODBS (p4) | 0.4 | 0.034 | 34.71 | 6.81 | 37.77 | 1.71 | 1.7 |
ODBS (p5) | 0.5 | 0.028 | 36.95 | 6.22 | 17.36 | 2.52 | 1.6 |
ODBS (p6) | 0.6 | 0.023 | 38.71 | 6.16 | 6.50 | 4.12 | 1.3 |
ODBS (p7) | 0.7 | 0.018 | 40.08 | 6.00 | 1.23 | 9.48 | 1.1 |
X-Bracing | – | 0.581 | 78.35 | 2.04 | 894.39 | 0.39 | 2.96 |
8.3 Effects of Eccentricity on ODBS System
By investigating the recent results, it seems that the energy dissipation is increased by increasing eccentricity and indicates the optimum amount of eccentricity is about 0.1–0.4 for ODBS systems. The eccentricities out of the range of 0.1–0.4 are not suggested to use in structures. Table 7 includes the recorded spectral response as acceleration, velocity and spectral displacement for ODBS system by various eccentricities. By continuing the assessments, the spectral analysis is performed on x-braced frame too. The spectral response of ODBS in compare with x-braced frame indicated the advantage of ODBS system. The stiffness and displacement characteristics of models under Elcentro earthquake are illustrated in Table 7. The intense velocity of x-brace model proportional to ODBS is generated imposing loads by impulsive tendency within the time domain of acceleration. This phenomenon may be the cause of structural concrete deteriorations. The minimum amount of spectral velocity is concerned about ODBS braced frame by eccentricity equal to 0.3. The lowest amount of spectral displacement in concerned about x-braced frame and ODBS braced frame, respectively. Finally the lowest response acceleration is recorded through 0.7 for eccentricity in ODBS braced frame.
The optimum levels for displacement and acceleration is affected from the structural stiffness and the mass inertia respectively. Manual limits should be considered for minimum displacement and allowable rotation. By knowing the optimum eccentricity about 0.2–0.5, the optimum behaviour of ODBS is generated. The relation between maximum acceleration response of ODBS and eccentricities variations under mentioned earthquake is specified as Fig. 21 (left). The results are converged for eccentricity about 0.1–0.3 under every excitation. Pushover curve due to different eccentricities is shown in Fig. 21 (right). Results are explained as how much the ductility levels are increased.
Approximation of the equivalent viscous damping ratio is effective parameters to verify the dynamic behaviour and is considered to predict the structural damping treatment and its energy dissipation capacity. The dynamic characteristic for comparison the various load bearing systems is proportional to equivalent damping and effective stiffness. Possibility to solving a simple linear braced frame instead of nonlinear ODBS braced frame is the reason why the equivalent viscous damping is estimated considering both elastic and inelastic energy dissipation.
The previous methods to approximate the equivalent viscous damping for present structure that this system is the composed structural material has indicated just for some hysteretic models, deformability or ductility range and frequency. Time history analysis carried out correspondent some results by amount of difference in present research from exact equivalent viscous damping. The exact and effective equivalent viscous damping ratio is related about two dependent factors, ductility and frequency (or period), because of this estimated values of equivalent viscous damping ratio has been changed along any hysteresis analysis. Then the average equivalent viscous damping should be considered to investigate the present research subject.
The average value has minimum variation compare with exact value. This means that it is not an essential object to have the exact evaluation for damping ratio.
9 Equivalent Viscous Damping
Estimation of the equivalent viscous damping ratio is the one of effective parameters to verifying the dynamic behaviour and is considered to predict the structural damping treatment and its energy dissipation capacity. The dynamic characteristic for comparison the various load bearing systems is proportional to equivalent damping and effective stiffness. Possibility to calculating a simple linear braced frame instead of nonlinear ODBS braced frame is the reason why the equivalent viscous damping is estimated considering both elastic and inelastic energy dissipation.
That ${\mathit{\xi}}_{\text{elst}}$ is linear or primary damping in elastic level and ${\mathit{\xi}}_{\text{hyst}}$ is the nonlinear or secondary damping in inelastic level used for effect of energy dissipated loops. To calculate these quantities, the model should be subjected under harmonic cyclic loading by constant period in each cycle. Elastic damping ratio is assumed 5 % in this study. Because of this part of Eq. (7) is outside of this research’s aim and scope, this quantity is considered in constant form (assumption is based on laboratory conditions in recent researches).
This equation is defined by energy dissipated product by energy stored corresponding to characteristics of loading and time domain for imposed seismic or cyclic load. For example Eq. (8) is the one status of harmonic load substituted by u and $\dot{u}$ from Eq. (13).
As investigated for ODBS spectral response in last pages, the next assessment was about obtaining the equivalent viscous damping for ODBS braced frame. The obtained results according to Fig. 23 is illustrated the viscous damping characteristics for ODBS braced frame under Naghan earthquake. As indicated in recent figures and illustrative equations, the viscous damping ratio and the spectral displacements are related inversely. It means decreasing the spectral displacement is because of increasing the viscous damping for a system. In Fig. 23, the various amount of equivalent viscous damping is compared each model of ODBS and x-braced frame together. The effects of ODBS equivalent viscous damping ratios variations were more than other systems. In Fig. 23a, c and d, the response spectral displacements versus time periods are specified for various accelerograms assessment.
Figure 23 indicates the sequence of inelastic damping ratio for dissipating energy proportional to its exact treatment. This specific means ODBS system may be able to have 11–15 % of equivalent viscous damping ratio by comparing this status to real work. In minimum range of viscous damping, the ODBS gathers 6–10 % of damping ratio in term of hysteresis energy dissipation. On the other hand, according to extended analysis, these equivalent damping quantities are lower in the x-braced and flexural frame. In the other word the ductile capacity of ODBS to absorbing energy is higher than the other systems.
10 Energy Dissipation for Various Models
10.1 Hysteresis Response of Models Under Earthquake Excitations
Connecting the pick points of hysteresis loops obtained the spectral response that, it is the one of important parameters for control the structural response in the range of period domain. Figure 23 indicates the spectral response acceleration and spectral response displacement in some different case studies. The various models by 2, 6 and 15 stories for three types of flexural, x-brace and off-diagonal bracing system are subjected to Elcentro, Naghan and Tabas accelerogram records.
10.2 Time History Analysis Response
For the future, the exact modeling and analysis of ODBS by experimental investigation could be performed and verified by highest level of accuracy. According to obtained results, the influence factor for variation control and energy dissipation of ODBS system is more than other lateral resisting systems, specially compare with flexural and x-braced frame. The ODBS system was very advantages to controlling deteriorations due to loading by short time period just like an isolation damper or viscous damper and/or the composed system of them.
11 Conclusion
- (1)
FE models are treated properly as verified by experimental investigations. Nonlinear analysis of FE models is indicated as converged response along its ductility confirmations. The finite element analyses are performed by comparative models as Flexural, X-braced and ODBS braced frames where their results indicate the highest ductility for ODBS. Pushover diagrams are also performed for investigating the capacity of various mentioned frames.
- (2)
The acceptance criteria for rotations and displacements are compared with related quantities of the ATC manual. The ODBS models had the most rotation at nodal elements, before the collapse level. Each of occurred plastic hinges is deformed elastically and plastically two times and it may be the cause of more energy dissipation.
- (3)
Also the crack analysis is performed according to finite element models in a time domain consisted of several time steps. The time steps were the steps of imposing displacement. The ODBS system was fully cracked pattern in term of flexural and shear crack. The differences between flexural and ODBS systems are indicated in capacity of bearing for both of deformations and loads.
- (4)
High-rise frames in 2, 6 and 15 floors are modeled numerically for considering ODBS effects on a simple flexural frame. The optimum number of stories to use ODBS system in first story is about 4–12 stories but if the ODBS system utilized for all stories, it has no limitation for number of stories. The only disadvantage of using ODBS in all stories is that it’s uneconomically.
- (5)
Application of ODBS system is considered for different regions and their given results are concentrated along its better behaviour under strong ground motion, because the natural period of ODBS system is high and for preventing the occurrence of resonance phenomenon, it is better that, this system being under strong ground excitation by low excitation’s period.
- (6)
The specific property of ODBS system is about its damping and its ability for dissipating received energy. Along analytical investigation of ODBS system, the equivalent viscous damping ratio is calculated about 11–15 % according to the results. The ODBS is absorbed 6–10 % of damping ratio in term of hysteresis energy dissipation. The ODBS equivalent damping is higher than related quantities for x-braced and flexural frame. The ODBS damping ratio is about 1.8–2.5 times of flexural frame damping.
- (7)
Mentioned numerical models are also considered to assessing the ODBS hysteretic behaviour. Many hysteresis diagrams are obtained to various models under several earthquake records. Hysteresis curves indicate the ODBS system had less pinching and the most strain energy. The obtained curves are geometrical comparison to calculating its energy dissipated quantities. The absorbed energy for ODBS model was several times of x-bracing system. The strength degradation for flexural frame is more than the other systems. The ODBS system is a sustainable system versus the large number of cycles.
Declarations
Acknowledgments
I would like to express my deep gratitude to professors of Shiraz University for their life pattern specially thanks to Profs. H. Seyyedian; S. A. Anvar & A. R. Ranjbaran. Finally, I wish to specially thanks to Mehri Zoroufian for her support and encouragement throughout my study.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Authors’ Affiliations
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