Calculation of Initial Stiffness of Semirigid Connections with Consideration of Rotational Constraint on Angle from Beam Contact Surface

Transcription

1 Calulaion of Iniial Siffness of Semirigid Conneions wih Consideraion of Roaional Consrain on Angle from Beam Cona Surfae X.G. Lin Osaka Insiue of Tehnology, Japan K. Asada Oayashi Corporaion, Japan SUMMARY: The ojeive of his paper is o derive a new formula o improve iniial siffness alulaion of semi-rigid seel eam-o-olumn onneions wih op and sea flange angles. In order o improve prediion of iniial siffness, deformaion of he onneion from FEM model is losely examined and some deformaion haraerisis are oserved suh as how and when angle an e separaed from eam in he viiniy of ol. Based on hose oservaions, some mehanial assumpions are made aou how horizonal arm of angle is en and hen a new iniial siffness orreion faor is proposed. The iniial siffness formula is furher modified o allow a small separaion eween angle and olumn a ol loaions when angle sars o fail. The new formula also inludes we angle onriuion. When iniial siffness alulaed using he formula proposed here is ompared wih la es resuls, mos of daa shows heir differene remains wihin 1%. Keywords: semirigid onneion, high srengh ol, seel frame, iniial siffness 1. INTRODUCTION Afer many weld fraures were oserved in rigid (welded) onneions of uilding sruures during Hyogo- ken-namu Earhquake of Japan (on January 17, 1995), sruural engineers are geing more and more ineres in designing semi-rigid (oled) eam-o-olumn onneions wih angle or spli ee eause semi-rigid onneions allow more deformaion and asor more energy and an preven possile rual weld fraure failure given y rigid onneions during an earhquake. Bu displaemens of frames are no welome in general and have o e alulaed and onrolled. So design of semi-rigidly joined seel frames will e more hallenging han design of rigidly joined seel frames. Therefore aurae alulaions of displaemens or iniial siffness are needed in design of semi-rigid eam-o-olumn onneions. In order o esimae iniial siffness of semi-rigid eam-o-olumn onneions for displaemen alulaion, C.Faella(1999), N.Kishi and Wai-Fan Chen (199) have developed and inrodued several formulas for he siffness alulaion. In heir work, he verial arm of an angle (op or sea) used o e simplified as anilever eams and deformaion in horizonal direion of he verial arms was assumed uniformed or same. Auhors of his paper has pulished an iniial siffness reduion faor in he pas, whih ries o onsider deformaion differene in he horizonal direion of he angles. Alhough he iniial siffness reduion faor is an improvemen in iniial siffness alulaion, u i sill has following limiaions: (1) he siffness orreion faor proposed earlier o onsider roaional onsrain on eam ona surfae is deermined ased on averaged numerial resuls while i ould vary from one sruure o anoher sruure; (2) earlier siffness formula is ased on assumpion ha failure ours in angle insead of ol and angle never separae from olumn a ol loaions; () earlier siffness formula is only verified y angle ension ess; (4) earlier siffness formula doesn' oun onriuion from we angle and an no e used in he onneion wih we angle in addiion o op and sea angles. In his sudy, some -D finie elemen modeling and analysis using ABAQUS are ondued. In ABAQUS nonlinear analysis he separaion eween angle and olumn or eam a ol loaion is

2 onsidered and also plasi deformaion in angle is inluded. Based on numerial simulaion resuls, a new iniial siffness orreion faor is proposed and he iniial siffness formula is furher modified afer allowing a small separaion eween angle and olumn a ol loaions when angle saring o fail. In finie elemen modeling and analysis, we angle onriuion is ompared wih op and sea angles. A new formula is derived o inlude we angle onriuion. Before finie elemen modeling and analysis of differen join onfiguraions, eigh physial es speimens are uil and esed. Among hem, 4 are flange angle onneions and 4 are we angle onneions. Resuls alulaed using he proposed iniial siffness formula is ompared wih ending es resuls and he auray is verified. 2. BENDING TEST OF SEMIRIGID BEAM-TO-COLUMN CONNECTION This paper is o sudy appliaion of op and sea flange angle as well as doule we angle o eam-oolumn onneion. In order o sudy individual onriuion of hem, op and sea flange angle is used in some ess and doule we angle is used in some oher ess. Fig. 1 shows piures for wo ypial semi-rigid eam-o-olumn onneions. The firs onneion onsiss of wo eams, one ener plae and four flange angles. Column is replaed y a ener plae o exlude olumn onriuion and make sudy fous on angles onriuion. The seond onneion onsiss of wo eams, one ener plae and wo op flange angles and four we angles. The use of op flange angle in he seond ype of onneion is o keep eams having he same roaion ener (he op orner of eams) and has lile onriuion in siffness eause i is loaed in ompressive side. (a) Top and sea flange angle onneion () Doule we angle onneion Figure 1. Bending es speimens (a) Flange angle () We angle Figure 2. Conneion deails for speimen of oh ypes

3 Tale 2.1. Deails of ending es speimens Speimen Angle hikness Angle widh g 1 g 2 g Speimen Angle hikness Angle Widh g 1 g 2 L WL L WL L WL L WL P (kn) L15-7 δ P (kn) WL15-2 δ Tale 2.2. Resuls of ending ess Iniial siffness Speimen Loading Unloading (kn/mm) (kn/mm) 125 P (kn) δ P (kn) L15-6 WL15- δ P (kn) Figure. Load versus displaemen urves for ending ess Yield srengh (kn) Speimen L2-7 δ P (kn) Loading (kn/mm) WL2-25 δ Iniial siffness Unloading (kn/mm) Yield srengh (kn) L WL L WL L WL L WL All es speimens are made of JIS seel grade SS4. Angles have wo hikness of 15 mm and 2 mm. Angles are u from eiher L-2x2x2 or L-2x2x15. Fig. 2 shows all major dimensions for all differen angles. Beam is H-294x2x8x12. Cener plae is 6 mm hik. High srengh ols are M22 of eiher F1T or S1T. All ol holes have 2 mm learane. More deailed daa of all eigh speimens are lised in Tale 2.1. All loading es are ondued using Amsler ype universal esing mahine. For eah of ess, a verial load is applied o ener plae and load is inreased slowly unil angles deformaion ge ino large plasi range. Beams are unloaded wie during esing. One unloading is performed in elasi range and anoher in plasi range. Only verial displaemen in ener plae is used in siffness alulaion. Sine axial fore in eam is no desired, roller suppors are used o give eams free moving in axial direion. Fig. presens some of ypial load-defleion urves. Tale 2.2 summarizes alulaed iniial siffness and yield srengh. In Tale 2.2, iniial siffness is alulaed wie. One is alulaed ased on elasi

4 loading urve slope and anoher is ased on elasi unloading urve slope. Beause defleion is very small from elasi loading urve and i is no easy o measure i auraely, iniial siffness ased on unloading urve is hosen for laer omparison wih iniial siffness prediion. The yield srengh is deermined using general yielding alulaion. I is easily noied ha iniial siffness of eah speimen depends mainly on angle hikness and numer of ols. All eams remains in elasi range. When iniial siffness is alulaed using es daa, eam deformaion is required and i is alulaed using elasi heoreial alulaion insead of es measuremen.. IMPROVEMENT OF STIFFNESS PREDICTION FOR FLANGE ANGLE CONNECTION Fig. 4 is a piure of semi-rigid eam-o-olumn onneion. In he pas, for ensional siffness sudy auhors of his paper proposed mehanial mehanism illusraed in Fig. 5. Eqn..1 is he ension siffness formula developed ased on his mehanism. The main feaure of his formula is onsideraion of angle deformaion in he viiniy of ols and inrodues an angle siffness reduion oeffiien in widh direion. K a ( B B B ) K r 1 f 1 2 f 2 2 ( 1.4).65B f E (.1), 1, K,.12 (1 ) K is heoreial siffness of uni widh of he rigid segmen, is hikness of he angle and E is elasi modulus of seel and is oeffiien of shear deformaion. Br and C is widh and lengh of he rigid segmen respeively. Bf is he widh of he flexile segmen, is a siffness reduion oeffiien, whih is a siffness raio eween he rigid segmen and he flexile segmen wih same widh. In he righ end of mehanial mehanism illusraed in Fig. 5 if eam flange and angle do no separae, he righ end anno roae and i will e similar o a fixed end. Afer flange and angle separae, he righ end will roae even i ould ge some resisane from ona surfaes. Compared wih fixed end, is siffness will e redued. A orreion faor a is refleing he siffness reduion. Auhors of his paper adoped average value of numerial resuls of he orreion faor in previous sudy. Bu differen eam-o-olumn onneion should have differen orreion faor. Use of average value ould resul in error in he alulaion. In addiion, rigid segmen lengh alulaion up o he fron of ol is anoher one required furher invesigaion eause i is only good for he failure where ols have lile deformaion and angle auses he failure. When ol sreh inreases, inauray will inrease. Figure 4. Semi-rigid onneion Figure 5. Segmenal eam model for verial arm of op angle

5 (a) L15-7 () L2-7 Figure 6. Separaion paern in he viiniy of ol area Ks Figure 7. Segmenal eam model for horizonal arm of op angle In order o improve siffness prediion, his paper ondus FEM modeling and analysis on es speimens and sudies he way how angle deforms in he finie elemen models. Fig. 6 shows angle ona separaion urves on he verial arm. Compared wih 15 mm hik L15-7, 2 mm hik L2-7 shows a lower separaion urve, whih exends ino he ol hole. In he horizonal arms, separaion lines show similar rend, u hey do no pass ol holes a all and all sop in he fron of ols. In order o develop orreion faor formula, roaional siffness Ks is required o alulae firs aording o Fig. 7. Beause ona separaion urves are similar in he verial arm and horizonal arm, afer lengh C is modified in Fig. 5, new mehanial mehanism is oained as in Fig. 7. To use his new mehanial mehanism, roaional siffness an e derived as follow. K s ( B r B 1 f 1 B 2 f 2 ) K s (.52).65B f E (.2), 1, Ks (1 ), In he aove formula,,,, are ona parameers in he horizonal arm and hey have definiion similar o hose in Eqn..1. Ks is roaional siffness of uni widh of rigid segmen. Formula for is a simplified one and he error inrodued y his simplifiaion is igger han same alulaion in he verial arm. To use esalished aove, orreion faor a an e esimaed as follow. a , ( Ba / B ) ( B / B a 1/ ) (.)

6 T1 T2 T T4 B 4 Mode1 Mode2 Mode Mode 4 Figure 8. Failure modes for semi-rigid onneion Figure 9. Model for parameer l Figure 1. Separaion haraerisi in he we B a B B g 2 r f 1 B f 2.5( B ),, B B B r 1 f 1 B 2 f 2 g.5b (.7 ) 1, B B B r 1 f 1 B 2 f 2 (.4) B and B are equivalen widh of horizonal arm and verial arm. Regarding o parameer l, i an e aken as zero if ol does no sreh and angle fails firs. I is oserved from numerial resuls in Fig. 6 ha when angle is hiker, ol may have some small srehing and make ona separaion urve pass ol and reah he area ehind ol. This ype of failure is mode in Fig. 8. In general here are four ypes of onneion failure modes as illusraed in Fig. 8. In order o ahieve enough deformaion ailiy in onneion design, failure mode 1 and 2 should e avoided. Failure mode 4 is he es one. Therefore all sudies ondued y auhors of his paper are aou how o do iniial siffness alulaion on failure mode 4. Failure mode has less apaiy of deformaion han mode 4, u muh eer han mode 1 and 2. Mode is a possile rue failure mode in sruural design and should e inluded in iniial siffness invesigaion. This paper ries o modify exising siffness alulaion formula o over oh 4 and failure modes. The sraegy is o use yield srengh differene eween mode 4 and o move ona separaion urve owards rear edge of ol head. When he differene (T4-T) reahes 2% of T4, move he ona separaion urve o he rear edge of ol head. This will e exreme ase. Based on all earlier reasoning, parameer l alulaion an e expressed as Eqn..5. Fig. 9 shows he relaionship eween l and f. Le When. B is he size of ol head as indiaed in Fig. 5. B 5 (.7), ( T4 T ) / T (.5) 4 4. WEB ANGLE CONTRIBUTION AND STIFFNESS PREDICTION OF SPECIMENS Fig. 1 shows we deformaion and ona separaion urve from finie elemen analysis. Alhough sress disriuion of we is more ompliaed han op or sea angle, ona separaion urve is sill

7 showing similar haraerisis. Cona separaion line around ol is urve, no a sraigh line. The lengh of rigid segmen is geing longer when i is away from roaion ener. Considering ha he oom ol has more onriuion, lengh of rigid segmen a oom ol is aken as C in Eqn..1. Tha means his will e he lengh of rigid segmen for all oher rigid segmens. In Fig. 11, he ona area of we angle is divided ino many small areas y ols and resulan reaion fore is assumed o e a ol ener. Then we ension siffness onriuion from eah of areas an e alulaed in a similar way used for op and sea angles using Eqn..1 and.5. Based on all assumpion made aove, for a semi-rigid onneion wih oh flange angles (op and sea) and doule we angles he roaional siffness an e alulaed in Eqn i is he disane from ol o he infleion poin in flange angle K ( H 1.5)( H 1.5 ) K K K K (4.1) i ( B / B ) 2 / 2 a i (4.2) 1/ EBa ( Ba / B ) 6Ks In order o verify he auray of iniial siffness prediion esalished in his sudy, all alulaed resuls are ompared wih la es resuls. In iniial siffness prediion using Eqn. 4.1, he firs porion of Eqn. 4.1 is used for semi-rigid onneions wih op and sea angles. The seond porion of Eqn. 4.1 is used for semi-rigid onneions wih doule we angles. Fig. 12 shows all omparison eween siffness alulaion using Eqn. 4.1 and la es resuls. In he omparison of semi-rigid onneions wih op and sea angles, when ol posiion parameer g 1 is higher, siffness prediion ends o e higher. Siffness prediion is lower when g 1 is smaller. All speimen shows differene less han 1%. For semi-rigid onneions wih doule we angles, he highes siffness onneion is showing highes differene, aou 2% higher. All oher ases have differene wihin 1%. Figure 11. Segmenal eam model for we angle K pre /K es L15-7 L15-6 L2-7 L2-6 (a) Top and sea flange angle onneion K pre /K es WL15-2 WL15- WL2-2 WL2- () Doule we angle onneion Figure 12. Comparison of predied iniial siffness wih es resuls

8 5. CONCLUDING REMARKS 1) In order o improve siffness prediion auray, his paper sudies ona separaion haraerisi of flange angle and we angle and proposes new formula for orreion faor alulaion. 2) This paper modifies earlier siffness alulaion and makes alulaion also valid for he onneion where small deformaion of ol is expeed. ) This paper inrodues a mehod o alulae we angle siffness onriuion. 4) 7 of 8 es resuls prove prediion error wihin 1% and only 1 es resul shows 2% differene eween prediion and es resuls. REFERENCES Lin, X.G. and Hamamoo, N. (28). Prediion of Iniial Siffness of Semirigid Seel Beam-o-Column Conneions wih Bols and Angles. Proeedings of he 1 h World Conferene on Earhquake Engineering, Paper No Kishi, N. and Chen, W.F. (199). Momen-roaion relaions of semirigid onneions wih angles. Journal of Sruural Engineering. 116:7, Fealla, C., Piluso, V. and Rizzano, G. (1999). Sruural Seel Semirigid Conneions: Theory, Design and Sofware. CRC Press LLC, U.S.A. Azizinamini, A., Bordurn, J.H. and Radziminski, J.B. (1987). Iniial Siffness of Semi-rigid Seel Beam-o-Column Conneions. Journal of Consruional Seel Researh. No.8, Lin, X.G. and Sugimoo, H. (24). Experimenal Sudy on Inelasi Behavior and Ulimae Srengh of Seel Beam-o-Column Conneions wih Bols and Angles. Proeedings of he 1 h World Conferene on Earhquake Engineering. Paper No Tanuma, Y. and Hashimoo, K. (1999). Momen-roaion ehaviour of semirigid onneions wih angles. Journal of Sruural and Consruion Engineering. No.515, Tanaka, H. and Tanaka, A. (1975). Design formulas for he srengh of -su onneion. Seel Consruion Engineering. 11:12, 5-1.