About the Finite Element Analysis for Beam-Hinged Frame. Duan Jin1,a, Li Yun-gui1

Transcription

1 Advnces in Engineering Reserch (AER), volume 143 6th Interntionl Conference on Energy nd Environmentl Protection (ICEEP 2017) About the Finite Element Anlysis for Bem-Hinged Frme Dun Jin1,, Li Yun-gui1 1 Chin Stte Construction Technicl Center, Beijing, , PR Chin dunjin78@126.com Key words: structurl simultion, DOFs coupling, DOFs condenstion, finite element method Abstrct: In this pper, the method of DOFs condenstion is dopted to simulte hinge node in finite element nlysis. Other thn dding dditionl overlpped nodes for the DOFs coupling method, the principle ide of DOFs condenstion method is tht ech two elements connected by hinge node do hve common node, which is exctly the hinge node, while the non-constrined DOFs of the hinge node would be condensed into the other constrined DOFs of the element. The procedure of condenstion is much like the condenstion of sub-structures. The detiled condensed stiffness mtrices for bem elements is derived in this pper. A numericl exmple for hinged frme is presented. The vlidity of the present theory would be demonstrted by compring the present result nd tht of ABAQUS softwre. Introduction As connection mode, hinge is widely used in structurl engineering, especilly for frme structures. In commercil softwre, such s ABAQUS, the most usully method to simulte hinge node is the coupling of degrees of freedom, or DOFs coupling for simplicity. Its principle ide is tht ech two elements connected by hinge node don t hve common node while hve two overlpped nodes. The two elements re connected by dding constrint equtions for the two overlpped nodes. In other words, dditionl element nodes, or degree of freedom (DOF), would be introduced into the finite element model to simulte the hinge node. In this pper, the method of DOFs condenstion [1-6] is dopted to simulte hinge node in finite element nlysis. Other thn dding dditionl overlpped nodes for the DOFs coupling method, the principle ide of DOFs condenstion method is tht ech two elements connected by hinge node do hve common node, which is exctly the hinge node, while the non-constrined DOFs of the hinge node would be condensed into the other constrined DOFs of the element. The procedure of condenstion is much like the condenstion of sub-structures. The detiled condensed stiffness mtrices for bem elements is derived in this pper. A numericl exmple for hinged frme is presented. The vlidity of the present theory would be demonstrted by compring the present result nd tht of ABAQUS softwre. The DOFs condenstion for hinged-connection restrint The equivlent eqution of bem element is s following [1-6]: K e e = P e (1) where e represents the nodl displcement vector of the bem element The degree of freedom (DOF) of the element could be split into two prts: non-constrint DOF nd constrint DOF. The former one usully represents the lterl displcement ngle nd the Copyright 2017, the Authors. Published by Atlntis Press. This is n open ccess rticle under the CC BY-NC license ( 231

2 torsionl ngle, denoted s denoted s displcement vector i. The ltter one usully represents the xil nd lterl displcement, b. Decomposing the stiffness mtrix K nd lod vector P ccording to tht of Kbb Kbi b Pb = K K P ib ii i i e, Eqution (1) could be trnsformed into the following one: From the second line of bove eqution, the following could be derived: ( ) = K P K (3) 1 i ii i ib b Substituting Eqution (2) into Eqution (2), the condensed eqution could be obtined s following: Where ( ) K K K K = P K K P (4) 1 1 bb bi ii ib b b bi ii i For simplicity, bove eqution could be simplified s following: K = P (5) * * bb b b K = K K K K (6) * 1 bb bb bi ii ib (2) P = P K K P (7) * 1 b b bi ii i The stiffness mtrix for hinged-bem element Generlly speking, the stiffness mtrix K for 3d bem element could be represented s following [1,2]: (8) 232

3 Where I y nd I z represent the principle moment of inerti, J represents the torsionl moment of inerti Substituting Eqution (8) into Eqution (6), the stiffness mtrix for ll kinds of hinged-bem element could be deduced. For exmple, the stiffness mtrix for one lterl displcement ngle hinge could be represented s Eqution (9), nd the mtrix for two lterl displcement ngle hinge could be represented s Eqution (9), nd the mtrix for three dimension ngle hinge (3D hinge) could be represented s Eqution (10). Here it should be pointed out tht in coding progrm the condenstion from Eqution (2) to Eqution (5) is not ccording to the Eqution (4) while ccording to the Guss-Jordn elimintion method [2,7]. (9) (10) 233

4 (11) Numericl exmple To demonstrte the bove theory, numericl exmple is presented in the following. The originl configurtion nd deformtion of right-ngle-frme with 3D hinge is shown in Figure 1. The cross-section of the frme is circle with 9mm inner rdius nd 11mm outer rdius. Its Young s modulus is 2.0e11 P nd its Poisson rtio is 0.3. The lterl rod is subjected 1KN verticl concentrted lod nd the verticl rod is subjected 3KN lterl concentrted lod, see Figure 1. The deformtion from the present method, i.e. DOFs condenstion method, nd from the ABAQUS softwre, i.e. DOFs coupling, re presented simultneously in Figure 1. It is obvious tht the result of the present method is identicl with tht of ABQUS softwre. hinged node 0 fixed node 1KN xis-y (m) -1-2 the present result result by ANSYS the originl configurtion 3KN -3 fixed node xis-x (m) Figure 1 the deformtion of the right-ngle-frme with 3D hinge 234

5 Summry In this pper, the method of DOFs condenstion is dopted to simulte hinge node in finite element nlysis. Other thn dding dditionl overlpped nodes for the DOFs coupling method, the principle ide of DOFs condenstion method is tht ech two elements connected by hinge node do hve common node, which is exctly the hinge node, while the non-constrined DOFs of the hinge node would be condensed into the other constrined DOFs of the element. The procedure of condenstion is much like the condenstion of sub-structures. The detiled condensed stiffness mtrices for bem elements is derived in this pper. A numericl exmple for hinged frme is presented. The vlidity of the present theory would be demonstrted by compring the present result nd tht of ABAQUS softwre. References [1] Bthe K J. Finite Element Procedures. NJ: Prentice-Hll, 1996 [2] Wng X C. Finite Element Method. Beijing, Chin: Tsinghu University Press, 2003 (in Chinese) [3] Oden J T. Finite Element of Nonlier Continu. New York: McGrw-Hill, 1972 [4] Simo J C, Hughes T J R. Computtionl Inelsticity. New York: Springer, 1998 [5] Belytschko T, Liu W K, Morn B. Nonliner Finite Elements for Continu nd Structures. New York: Wiley, 2000 [6] Yng Y B, Kuo S R. Theory & Anlysis of Nonliner Frmed Structures. New York: Prentice Hll, 1994 [7] Gun Z & Chen J L. Numericl Computtion Method. Beijing, Chin: Tsinghu University Press, 1990 (in Chinese) 235