The Effet Of Panel Zone On The Column-To-Beam Strength Ratio Required To Prevent The Soft Stor Of Steel Frames S.W. Choi, Y.S. Kim & H.S. Park Yonsei Universit, Seoul, South Korea SUMMARY: THE EFFECT OF PANEL ZONE ON THE COLUMN-TO-BEAM STRENGTH RATIO The strong olumn-weak beam ondition has widel used to ensure the dutile aait of steel moment frames. Man researhes showed that soft stor ould be develoed desite satisfing this ondition. A lot of researhes on the minimum olumn-to-beam strength ratio required to revent the soft stor were erformed. Most of them used the analti modeling methods without the anel zone. The urose of this stud is to evaluate the effet of anel zone on the minimum olumn-to-beam strength ratio required to revent the soft stor of steel moment frames. To identif the effet of anel zone, three analti modeling methods (nonlinear enterline model without rigid end offsets, nonlinear enterline model with rigid end offsets, nonlinear model with anel zones) were used. NSGA-II was used to find the minimum olumn-to-beam strength ratio required to revent the soft stor. These ratios were omared and assessed through aling this method to 3-stor examle. Kewords: olumn-to-beam strength ratio, anel zone 1. INTRODUCTION ANSI/AISC 341-05 and ACI 318-05 use the strong olumn-weak beam onet to revent earl flexural failure, and to indue beam-hinge ollase mehanism. As shown in Eqns. 1.1 and 1.2, the sum of the uer and lower olumn bending strength ( M ) have to be greater than the sum of left and right olumn bending strength ( M b ) on the olumn-to-beam joint to indue the ourrene of lasti hinge on beam in rior to the olumn. But sine this is based on double urvature bending moment distribution, even if this ondition is satisfied, lasti hinge ould our in olumn whih reating a soft stor. On atual struture, infletion oint whih was resumed to lae in the enter of the beam an move to end of the olumn or bending moment distribution of the olumn an realize the single urvature due to higher order mode and hange of motion after the ield. (Park and Paula, 1975). M ANSI/AISC 341-05 1.0 M M ACI 318-05 1.2 M b b (1.1) (1.2) There have been studies about olumn-to-beam strength ratio whih is required to revent a soft stor in moment framework. (Lee, 1996; Nakashima and Sawaizumi, 2000; Doole and Brai, 2001; Kuntz and Brouning, 2003; Medina and Krawinkler, 2005). But, existing studies were onduted using examles designed b engineers exerienes. The all used nonlinear enterline model, and no influene assessment aording to analtial model was made.
Thus in this stud, influene of analtial model on minimum bending strength ratio required to lead beam-hinge ollase mode of steel moment frame was analzed. For an objetive and rational assessment, otimal design method was used to obtain the design model whih has beam-hinge ollase mode. This otimal design method uses the objetive funtions whih minimize the strutural weight and olumn-to-beam strength ratio, the onstraints suh as the inter-stor drift onstraint, lasti hinge formation onstraint on olumn in joint of olumn-to-beam, and onstraint uon the ross setion ratio of vertiall ontinuing olumns. Panel zone is onl onsidered as an influene fator in analtial model regarding the minimum bending strength ratio and 3 analtial models (nonlinear enterline model without rigid end offsets, nonlinear enterline model with rigid end offsets, nonlinear enterline model with anel zone) are used. Influene of analtial model was investigated b omaring the strutural weight and minimum bending strength ratio of otimal designs while using the same otimal design method but differentiating the analtial model. This stud emloed 3 stor steel moment frame examle. 2. ANALYTIC MODELING As showed in Fig. 1, the anel zone signifies ross-domain of olumn and beam at the joint. A shear deformation of anel zone has a signifiant effet on strength, stiffness and distribution of inelasti deformation of moment frame whih reeive earthquake load (Krawinkler and Mohasseb, 1987). Yet, existing studies on bending strength ratio (Lee, 1996; Nakashima and Sawaizumi, 2000; Doole and Brai, 2001; Kuntz and Brouning, 2003; Medina and Krawinkler, 2005) did not onsider this fat. Therefore, this stud omared to influene of anel zone of minimum bending strength ratio required to indue beam-hinge ollase mode of moment frame while onsidering nonlinear enterline model without rigid end offsets (Model M1), whih was mostl used in existing stud, nonlinear enterline with rigid end offsets (Model M2), and also nonlinear enterline model with anel zone (Model M3). As shown in Fig. 2, Model M1 exresses olumn and beam using lines without onsidering anel zone. Two lines smbolizing olumn and beam are laed at the enter of eah olumn and beam, and the meet in the enter of the anel zone. Element of the olumn and beam reates lasti behavior to our on the both ends. As shown in Fig. 3, Model M2 is modeling anel zone b utilizing rigid end offsets. The node loated in the enter of anel zone is onneted to nodes loated in sot in distane of half of width of the beam b using rigid link for anel zone to make a rigid bod motion then the line smbolizing the olumn and beam is onneted to the alread onneted node. Element of olumn and beam is for lasti behavior to our onl at both ends. Panel zone d b d Figure 1. Definition of anel zone
d b Node al sring d Figure 2. Nonlinear enterline model without rigid end offsets (Model M1) d b Node al sring Rigid link (rigid end offset) d Figure 3. Nonlinear enterline model with rigid end offsets (Model M2) Unrestrained rotation d b Rigid link Node Shear sring al sring d Figure 4. Nonlinear enterline model with anel zone (Model M3)
To onsider the shear stiffness and strength of the anel zone, as shown in Fig.4, Model M3 used suggested model in FEMA 355. A shear behavior of anel zone is exressed with shear sring whih motion like shown in Fig. 5, this is deided b ielding oint (γ, V ) and lasti oint (γ, V ). Values of ield oint and lasti oint an be alulated b using Eqns. (2.1)-(2.4). V V V γ γ γ Figure 5. Trilinear shear fore shear distortion relationshi of anel zone V 0.55F d t (2.1) F 3G (2.2) 2 3bt f V 0.55 Fdt (1 ) d d t (2.3) b 4 (2.4) where V is the anel zone shear ield strength, F is the ield strength of the material, d is the deth of the olumn, t is the thikness of the web inluding an doubler lates, is the anel zone shear ield distortion, G is the shear modulus of the olumn material, V is the anel zone shear lasti strength, b is the width of the olumn flange, t f is the thikness of the olumn flange, d b is the deth of the beam, is the anel zone shear lasti distortion. 3. THE OPTIMAL DESIGN METHOD FOR INDUCING BEAM-HINGE COLLAPSE OF STEEL MOMENT FRAMES In this stud, the otimal design method is emloed to get steel moment frame design model whih has beam-hinge ollase mode. This uses two objetive funtions. The urose of this stud is to evaluate minimum olumn-to-beam bending strength ratio required while induing beam-hinge ollase mode aording to the joint. Thus, the first objetive funtion is set to minimize the strutural weight of struture as shown in Eqn. (3.1). The seond objetive funtion is set to minimize the largest bending strength ratio at olumn-to-beam strength ratio of design model, as dislaed in Eqn. (3.2). This is to analze with ratial design model in order to evaluate rationall minimum bending strength ratio. Minimize m i Al i i (3.1) i1 Minimize (3.2) max
where i is the densit of the ith element, A i is the ross-setional area of the ith element, l i is the length of the ith element, m is the number of elements onsisting of the struture, max is the maximum lasti moment strength ratio among joints of the struture. The stud has basiall onsidered three onstraint onditions. The first ondition is to onstraint inter-stor drift as shown in Eqn. (3.3). Generall, earthquake-resistant design of moment frame is determined b the inter-stor drift ondition of struture than strength ondition of elements (Fouth and Yun, 2002). Therefore, this stud has onl onsidered inter-stor drift ondition while exluding strength ondition. The seond ondition is to onsider onstraint ondition where lasti hinge does not our on end of olumn whih forms the joint to indue the beam-hinge ollase mehanism as shown in Eqn. (3.4). The third ondition is that ross-setional area of lower olumn is bigger or equal to ross-setional area of uer olumn in vertiall ontinuing olumn to onsider the onstrutabilit as shown in Eqn. (3.5). In ase of otimizing design using the Model M3, Eqn. (3.6) is onsidered as an additional onstraint ondition. This means lasti deformation in anel zone is not ermitted. This stud has emloed Non-dominated Sorting Geneti Algorithm II (NSGA-II) to solve the roblem omosed of objetive funtion and onstraint ondition. (Deb et al., 2002). 1.0 a (3.3) N 0 (3.4) A i1, j A i, j max where and 1.0 (3.5) 1.0 (3.6) a are the maximum and allowable inter-stor drift ratio, resetivel, N is the number of lasti hinges formed at the olumn art onsisting A i, j is the ross setional area of the ith stor and jth olumn, max is the maximum shear distortion of anel zone obtained from the analsis result. 4. APPLICATION 9 9 9 9 3 6 6 6 3 8 8 8 8 2 5 5 5 2 7 7 7 7 3@3.96m 1 4 4 4 1 4@9.14m : Design variable Figure 6. Examle
This stud used the 2D 3 stor steel moment frame whih Guta and Krawinkler (1999) and Hasan et al. (2002) used like Fig. 6 to evaluate the minimum bending strength ratio required to revent a soft stor. Design variables for strutural otimization were used 6 and 3 as the erformane of ross setion on olumn and beam, resetivel. The list of ross setion whih design variable of olumn and beam an selet arbitraril were set as 16, resetivel and the number of osize used in NSGA-II is 20. The struture is laed is LA region of United States and ground level is D while the imortane fator of struture is resumed to be 1.0. Strutural steels used in beam and olumn are eah A36, A572 Grade 50 steel. The alulation roess of Equivalent lateral fore as roosed in ASCE 7-05 as an earthquake load was used and strutural analsis emloed OenSees. Linear analsis was onduted to evaluate the onstraint ondition of inter-stor drift. In order to review the onstraint ondition for reventing ourrene of lasti hinge on olumn, ushover analsis was erformed to figure out the ollase mehanism. Lateral load attern emloed in non-linear analsis used an inverted triangle attern and target dislaement used value fall under 5% of overall height of struture b referening 5% of maximum inter-stor drift ratio whih is ollase revention level (C.P.) resented in FEMA 356. In order to analze the effet of anel zone on minimum bending strength ratio required to indue the beam-hinge ollase mehanism of examle struture, struture was modeled b emloing Model M1, Model M2, and Model M3 analtial model and otimal design method mentioned in 3 rd hater is alied to eah model. As the result of analsis, total of 19 designs (Model M1 : 6, Model M2 : 9, Model M3 : 4) are obtained as shown in Fig. 7. In all three ases, the minimum bending strength ratio required to indue the beam-hinge ollase of steel moment framework inreased as the strutural weights dereased. The result of Model M1 and Model M2 were onfirmed to show a similar tenden however, in ase of Model M3, whih onsidered the shearing deformation of anel zone and limited its lasti deformation, had relativel bigger value when omared to Model M1 and M2. As shown in Table 4.1, even if beam-hinge ollase mehanism was indued through Eqn. (3.4), onfirmation of onstraint ondition of inter-stor drift like Eqn. (3.) alied as a dominant design element an be made. Minimum olumn-to-beam strength ratio 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 Model M1 Model M2 Model M3 48 50 52 54 56 58 60 62 64 66 Strutural weight (kis*in) Figure 7. The solutions obtained from the otimal design algorithm 5. CONCLUSION This stud analzed influene of analtial model regarding minimum bending strength ratio required to indue beam-hinge ollase mode of a steel moment frame. In order to obtain the design model adating beam-hinge ollase mode, otimal design method using onstraint ondition where lasti
hinge does not inur at end of olumn whih omoses the joint was emloed. Three analtial models where anel zone was not onsidered (Model M1), shear strength of anel zone is onsidered as an infinite stiffness (Model M2), and shear stiffness of anel zone is onsidered while restriting its lasti behavior (Model M3) were emloed to omare the influene of minimum bending strength ratio. The influene of analtial model was evaluated b omaring objetive funtion and value of onstraint ondition in design model obtained from differening the analtial model while aling idential otimal design method to examle of three stor steel moment framework. As the result of analsis, all three ases had minimum bending strength ratio above 1.0 and minimum bending strength ratio required to indue the beam-hinge ollase of steel moment framework is inversel related with the strutural weight. Distributions of Model M1 and M2 had a similar tenden however, in ase of Model M3, whih onsidered the shearing deformation while limiting lasti deformation has a relativel bigger value of bending strength ratio when omared to Model M1 and M2. Table 4.1. The values of objetives funtions and onstraints of Model M1, M2 and M3 Objetive funtions Constraints Model Equation Equation Equation Equation Equation Equation te (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) 53.77 2.17 0.98 0.00 1.0-54.50 1.92 0.98 0.00 1.0 - Model 54.83 1.92 1.00 0.00 1.0 - M1 56.06 1.75 0.96 0.00 1.0-56.18 1.70 0.96 0.00 1.0-57.25 1.69 0.93 0.00 1.0 - Model M2 Model M3 49.12 2.73 0.97 0.00 1.0-50.41 2.67 0.92 0.00 1.0-51.00 2.42 0.88 0.00 1.0-52.17 2.17 0.85 0.00 1.0-53.17 2.14 0.81 0.00 1.0-56.63 1.75 0.72 0.00 1.0-59.38 1.55 0.68 0.00 1.0-64.62 1.49 0.62 0.00 1.0-64.92 1.32 0.66 0.00 1.0-57.74 2.89 1.00 0.00 0.83 0.57 58.34 2.59 1.00 0.00 0.83 0.88 58.42 2.45 1.00 0.00 0.83 0.88 59.54 2.38 0.98 0.00 0.83 0.83 AKCNOWLEDGEMENT This work was suorted b the National Researh Foundation of Korea (NRF) grant funded b the Korea government (MEST) (No. 2012-0001247). REFERENCES Park, R. and Paula, T. (1975) Reinfored onrete struture, John Wile and Sons. AISC. (2005) ANSI/AISC 360-05 Seifiation for strutural steel buildings, Amerian Institute of Steel Constrution. ACI Committee 318. (2005) Building ode requirements for strutural onrete (ACI 318-05) and Commentar (ACI 318R-05), Amerian Conrete Institute. Lee, H. (1996) Revised rule for onets of strong-olumn weak-girder design, Journal of Strutural Engineering 122:4, 359-364. Nakashima, M. and Sawaizumi, S. (2000) Column-to-beam strength ratio requied for ensuring beam-ollase mehanisms in earthquake resonses of steel moment frames, Proeedings of the 12th World Conferene on Earthquake Engineering.
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