Seismic Performance of Knee Braced Frame

World Academy of Science, Engineering and Technology 5 9 Seismic Performance of Knee Braced Frame Mina Naeemi and Majid Bozorg Abstract In order to dissipate inpt earthqake energy in the Moment Resisting Frame (MRF) and Concentrically Braced Frame (CBF), inelastic deformation in main strctral members, reqires high expense to repair or replace the damaged strctral parts. The new proposed knee braced frame in which the diagonal brace provide most of the lateral stiffness and the knee anchor that is a secondary member, provides dctility throgh flexral yielding. In this case, the strctral damages cased by an earthqake will be concentrated on these members, which can be easily replaced by reasonable cost. In this investigation, sing non-linear and linear static analysis of several knee Braced Frames (KBF), the seismic behavior of this system is assessed for controlling the vlnerability of the main and the secondary elements. Seismic parameters and mechanism of plastic hinges formation of both frame types are investigated by sing the non-linear analysis. Keywords knee braced frame, seismic parameters, energy dissipation. I. INTRODUCTION HE seismic design of steel strctres mst satisfy two T main criteria. These strctres mst have adeqate strength and stiffness to control interstory drift in order that prevent damage to the strctral and non-strctral elements dring moderate bt freqent excitations. Under extreme seismic excitations, the strctres mst have sfficient strength and dctility to prevent collapse. MRF and CBF have been sed as lateral load resisting strctral systems in steel bildings, since stiffness and dctility are generally two opposing properties, neither of MRF or CBF, alone can economically flfill these two criteria. Althogh the MRF is good for dctility and the CBF is good for stiffness, by combining the good featres of these two systems into a hybrid system, an economical seismicresistant strctral system can be obtained. One sch system is Eccentric Braced Frame (EBF) proposed by Roeder and Popov []. Recently, Aristizabal Ochoa [3] has proposed an alternative system, the Knee Braced Frame (KBF). In this system, the knee element acts as a `dctile fse` to prevent collapse of the strctre nder extreme seismic excitations by dissipating energy throgh flexral yielding. A diagonal brace with at least one end connected to the knee element provides most of the elastic lateral stiffness. In this system, however, M. Naeemi was with Department of Civil Engineering, Iran University of Science and Technology, Tehran, Iran., (e-mail: m.naeemi@dot-corp.com). M. Bozorg, is with Engineering Department, Tarbiat Modares University, Tehran, Iran. (Phone: 1-8887683; e-mail: majidcivil3@yahoo.com). the brace was not designed for compression and ths allowed to bckle. Conseqently, the hysteretic response of this strctre will be very similar to that of CBF with pinching in the hysteretic loops, which is not a desirable featre for energy dissipation. II. SAMPLE MODELS OF FRAMES In Fig. 1 different types of KBF systems are shown. They are referred to as K-knee braced frame X-knee braced frame (c) knee braced frame with single brace and one knee element (d) knee braced frame with single brace and with two knee elements. Fig. 1 Different knee brace frames: K-KBF, X-KBF, (c) KBF with single brace and 1KE, (d) KBF with single brace and KEs The optimal shape of KBF is selected from the above systems according to the elastic analysis reslts of them. And the optimal angle of the knee element achieved when the frame has the maximm stiffness, which the tangential ratio of (b/h)/(b/h) is nearly one, it means that the knee element shold be parallel to the diagonal direction of the frame, and the diagonal element passes throgh the mid point of the knee element and the beam-colmn intersection, as shown in Fig.. H = 3m B = 4m h =.5 h =.75m H B 4 b = = 1.33, = 1.33 b = 1.m H 3 h Fig. The selected shape and dimension of the sample frames In this stdy the framing systems with two eqal side spans 4m long are braced and length of the middle span is 5m. The 976

World Academy of Science, Engineering and Technology 5 9 nmber of frame stories is selected so that investigates the rigid, semi-rigid, moderate and dctile strctres. Therefore the frames are chosen in for levels 5-story, 1-story, 15- story, and -story. For instant, the 5-story frame is shown in Fig. 3 Fig. 3 An example of nder-stdy frames III. LOADING AND DESIGN The gravity loads inclde dead and live load of 6kg/m and kg/m respectively. To calclate the eqivalent static lateral seismic loads Refer to (1), assme that the behavior factor R for Knee-bracing system is 7. V = C. W A. B. I C = R Where V is the base shear, A is design base acceleration ratio (for very high seismic zone=.35g), B is response factor of bilding (is depending on the basic period T), and I is the importance factor of bilding (is depending on the bilding performance considered 1. in this paper). All of the frames are designed according to the AISC89, allowable stress design. IV. NONLINEAR STATIC ANALYSIS (PUSHOVER) The most basic inelastic analysis method is the complete nonlinear time history analysis, which at this time is considered overly complex and impractical for general se. Available simplified nonlinear analysis method referred to as nonlinear static analysis procedres. This method ses a series of seqential elastic analysis, sperimposed to approximate a force-displacement capacity diagram of the overall strctre. The capacity crve is generally constrcted to represent the first mode response of the strctre based on the assmption that the fndamental mode of vibration is predominant response of the strctre. This is generally valid for bildings with fndamental periods of vibration p to abot one second, (1) for more flexible bildings with a fndamental period greater than one second; the analysis shold be considered addressing higher mode effects. The higher mode effects maybe determined by loading progressively applied in proportion to a mode shape other than the fndamental mode shape. The step by step procedres are as followed: 1) Create a compter model of the strctre and apply gravity loads. It is necessary to define bilinear model behavior for each member. (The bilinear models which represent the plastic joint behavior of SAP defalts are sed for beams and colmns in this paper and the models which relate to plastic joint behavior of knee and diagonal elements represented in next section.) ) Apply lateral story forces to the strctre in proportion to the prodct of the mass and fndamental mode shape. 3) Increase the lateral force level ntil some element (or a grop of elements) yields and revise the model sing zero (or very small) stiffness for the yielding elements. 4) Apply new increment of lateral load to the revised strctre sch that other elements yields and the strctre reaches an ltimate limit, sch as: reaching the lateral displacement of control point (roof level) a limit state as defined follow for design earthqake: m <.5h T <.7sec m <.h T.7sec Where is inelastic displacement of the control point, h m is the story height, and T is the first mode of strctre. 5) Record the base shear and the roof-displacement so that create the capacity crve which represents the nonlinear behavior of strctre. V. FORCE-DISPLACEMENT RELATION OF COMPONENTS Component behavior generally will be modeled sing nonlinear load-deformation relations defined by a series of straight line segments. Fig. 4 illstrates two kinds of representations which are sed for compter modeling that is created according to modeling parameters and acceptance criteria for nonlinear approach in FEMA73. Q/Q CE 1.4 1. 1.8.6.4. 4 6 8 1 1 /y 977

World Academy of Science, Engineering and Technology 5 9 Stress 1.5 1.5-1 -5 5 1 15 -.5-1 1 1 8 6 4-1.5 Strain Fig. 4 Load- deformation relations for a knee element, BOX18x18x1, a diagonal element, UNP1 VI. NONLINEAR ANALYSIS RESULTS Fig. 5 illstrates the pshover nonlinear reslts for KBF system in the form of force-displacement crve of sample frames. 7 6 5 4 3 1 1 1 4 6 8 1 1 8 6 4 9 8 7 6 5 4 3 1 (c) 4 6 8 1 1 14 5 1 15 5 (d) 5 1 15 5 Fig. 5 Sample frames capacity crves, 5-story, 1-story, (c) 15-story, (d) -story VII. ESTIMATION OF SEISMIC PARAMETERS In order to investigate the seismic performance of sample frames the seismic parameters sch as: dctility, factor of behavior and formation of plastic hinge can be estimated by sing the force-displacement crves and pshover analysis. A. Dctility Effect in redcing strength factor, R µ Different relations are proposed to determine this factor, in each relation have been attempted to se most of the seismic effective components, the most comprehensive relation is proposed by Miranda, whereas his proposed eqation incldes some more effective components sch as period of strctre, soil properties and earthqake acceleration. Based on Miranda s [1] assmption R µ is calclated as in () µ 1 R µ = +1 φ () φ 1 1 3 = 1+ 3/ (lnt ) 1T µt T 5 For rock earth φ 1 1 = 1+ (lnt ) 1T µt 5T 5 For residal soil Tg 3T g T 1 φ = 1+ 3(ln ) For soft soil 3T 4T Tg 4 Where µ is dctility, T is period of strctre, and dominant period of earthqake. T g is B. Over strength factor, Ω In addition to laboratorial method the analytical method sch nonlinear static analysis can be sed to calclate the Ω factor related to overall yielding of strctre as the collapse mechanismv, to the force in which the first plastic hinge is y formed in strctrev s ; therefore the Ω factor can be fond Refer to (3). Vy Ω = (3) V s 978

World Academy of Science, Engineering and Technology 5 9 V y is the base shear related to point of the redction stiffness in eqivalent bilinear force-displacement of strctre. Ω is the nominal over strength factor which is adjsted by mltiplying some coefficient to consider the effect of yielding stress increase by the reason of strain rate increase in an earthqake, F 1, the difference between nominal and actal yielding stress of material, F, so the actal over strength factor can be obtained as follows: Ω = Ω F F... 1 C. Factor of behavior, R The factor of behavior is calclated in two states according to the method which is sed by every code to design strctre. R is the factor of behavior based on ltimate limit stresses and R W is the factor of behavior based on allowable limit stresses, that the relation between R W and R is defined by a dimensionless parameter Y = R W R, which is evalated arond 1.4 to 1.7 (the UBC-97 code has proposed 1.4 for this parameter) the seismic parameters for sample frames are calclated in Table 1. TABLE I SEISMIC PARAMETERS OF KNEE BRACING SAMPLE FRAMES Nmber of stories 5-Story 1- Story 15- Story -Story T(sec).7 1.4.3 3. V y (KN) 576. 113.1 88.4 957. V s (KN) 15. 48.4 38. 435. Ω.68.1.18. Ω 3.1.43.5.5 µ 3.4.46.7.1 φ 1.3.81 1. 1.3 R µ 3.33.8.67.7 R 1.3 6.79 6.67 5. R w 14.45 9.5 9.34 7.8 From the above tables it can be fond that for 5-story frame the Ω factor obtained abot 3 and that of 1 to -story frames is abot to.5. The above vales are compatible as mentioned in reference [1], which is evalated 3 for short strctres and for tall strctres. The displacement limitation code limits the maximm displacement of strctre, for this reason the R µ factor for 1 to -story frames is smaller than that of 5-story frame. Also the vale of R verss the height of strctre is plotted in Fig. 6. As it can be fond from this figre, the obtained vales of R for KB system is more than that of systems sch as Eccentric or Centric Braced Frames, so more dctility is achieved as it is desired in this paper. R vale 1 1 8 6 4 5 1 15 5 Nmber of story Fig. 6 Investigation of varying R for KBF VIII. THE COMPARISON OF NONLINEAR PERFORMANCE OF SAMPLE FRAMES Fig. 7 illstrates the plastic hinge formation in one of the nonlinear analysis step for 5 and 15-story frames of KBF. By evalating the reslts of the KBF system, it can be fond that, as the lateral force increases the first plastic hinge forms in a knee element, so that most of the plastic hinge is concentrated in the knee elements, which is a secondary member of the KBF system. Therefore most of the strctral damages cased by an earthqake will be occrred on the knee element and after earthqake the damaged members can be replaced more easily and at reasonable cost. 979

World Academy of Science, Engineering and Technology 5 9 Conference on Earthqake Engineering, Lisbon, Portgal, September 1986. [6] Nassar A.A. and Osteraos J.D and Krawinkler H. seismic design based on strength and dctility demand, Proceeding of the Earthqake Engineering 1 th worth Conference, p.5861-5866, 199. [7] Thambirajah Balendra, Ming-Tck Sam, Chih-Yong Liaw and Seng- Lip Lee, Preliminary stdies into the behavior of knee braced frames sbject to seismic loading, Eng. Strct. 1991, Vol. 13, Janary. [8] Thambirajah Balendra, Ming-Tck Sam, Chih-Yong Liaw, Diagonal brace with dctile knee anchor for a seismic steel frame, Earthqake Engineering and Strctral Dynamics, Vol. 19, p. 847-858 (199). [9] Thambirajah Balendra, Ming-Tck Sam, Chih-Yong Liaw, Earthqake-resistant steel frames with energy dissipating knee element, Engineering Strctres, Vol. 17, No. 5, p.334-343, 1995. [1] Miranda,E. and Bertero, V.V., Evalation of Strength Redction Factors for Earthqake-Resistant Design, Earthqake Spectra, 1994, Vol.1, No., pp.357-379. Fig. 7 Plastic hinge formation in one of nonlinear analysis steps, for 5-story and 1-story frames IX. CONCLUSION 1) In the KBF system the diagonal brace provides most of the elastic lateral stiffness where the beams and colmns are hinge-connected. The knee elements prevent collapse of the strctre nder extreme seismic excitations by dissipating energy throgh flexral yielding. Since the cost of repairing the strctre is limited to replacing the knee members only. ) The area nder the force-displacement diagram of the KBF system shows the energy dissipating capacity. 3) According to the vales of dctility effect redcing strength factor and over strength factor calclated in tables 1 and for KBF system, it is assmed R µ =. 51 and Ω =. 476 so for the ltimate limit stresses. REFERENCES [1] Jinkoo Kim, Yongill Seo, Seismic design of steel strctres with bckling-restrained knee braces, Jornal of Constrctional steel research 59, p.1477-1497, Jly 3. [] Roeder, C.W. and Popov, E. P. Eccentrically braced steel frames for earthqakes, J. Strctral Div., ASCE 1978, 14, 391-411. [3] Aristizabal-Ochoa, J. D., Disposable knee bracing: improvement in seismic design of steel frames, J. Strctre. Engineering, ASCE, 1986, 11, (7), 1544-155. [4] Uang C.M, Establishing R (or Rw) and Cd factors for bilding seismic provision, J. of Strctre. Eng., VOL, 117, No.1, Janary. [5] Cosenza E. and Lco A.D. Fealla C. and Mazzolani F.M On a simple evalation of strctral coefficients in steel strctres, 8 th Eropean 98