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J ournal o f A pp Published Onli n http://dx.doi.or g Open Access ABSTRA C In the analys i internal force s loss of precis i p resent accur a variational pr i b est stress fi e with drilling D two displace m Keywords: F i 1. Introdu c For traditio n from the prin c functions are displacement s b e derived b y of formulatio n curate displa c p rocedures o f most situatio n tant than the high-rise bui l are depende d stresses. For t the accuracy p lacement re s overcoming t formulations Taylor [2]. On the oth e veloped from Washisu, suc h Pian and Su m ment- b ased e l internal force first. Without p lied Mathemat i n e November 2 0 g /10.4236/jamp . Hy D i 2 Department o C T i s of high-rise s of the shear w i on induced b y a cy enough fo r i nciple is use d e ld, three diff e D OF. Numeri c m en t - b ased pla n i nite Element; c tion n al displacem c iple of mini m usually assu m s , and then the y stress-strain n , these kind s c ement result s f these formul n s, the stress r displacemen t l dings. The r e d on the int e t he sake of d i of stresses i s s ults. Many e t his shortage, developed b y e r hand, the h y the modifie d h as the hybrid m ihara [4]. It i l ements. To f field or stres the differenti a i cs and Ph y sics , 0 13 (http://ww w .2013.16004 brid P o i splace m Xiaoming C 1 China S t o f Engineering M E building, tra d w alls by stres s y using differe n r design. In th i d for improvin g e rent forms ar c al results sho w n e elements c a Displacemen t m en t - b ased el e m um potential m ed as a poly n internal force s relationship. B s of elements s , otherwise t h ations are als o r esults are mu c t s such as t h e inforcements e rnal forces i i fferential wit h s usually low e e fforts have b such as the y MacNeal [ y brid/mixed e d variational p method prop o s different fr o f ormulate thes s field shoul d a l process whi c , 2013, 1, 15-1 9 w .scirp.org/jour n o st-Pro c m ent-B a C hen 1 , Song C t ate Constructio n M echanics, Sch o E mail: chenxia o Recei v d itional displa c s integration. L n tial method t o i s paper, the h y g the stress p r e assumed fo r w that by usi n a n be improve d t -Based Plane e ments deriv e e nergy, the tr i n omial of no d s or stresses c a B y this proce can present a h e constructi o o easier. But i c h more imp o h e designing o of shear wa l i ntegrated fro m h displaceme n e r than the d i b een made fo assumed stra i 1], Piltner a n e lements are d p rinciple of H u o sed b y Pian [ 3 om the displac e elements, t h d be assumed c h is needed fo 9 n al / jamp ) c essing a sed P l C en 2 , Jiany u n Technical Ce n o ol of Aerospa c o ming00@tsing h v e d August 201 3 c emen t - b ased L imited by th e o derive strai n y brid pos t - p ro c r ecision of tw o r the displace m n g the propose d d . Element; Hyb e d i al d al a n ss c- o n in or - o f l ls m n t, i s- fo r in n d e- u - 3 ], e- h e at fo r the di s they c a sent o b tion pr o tion of dable, f usuall y eleme n By c displac b rid p o metho d place m interna l Equati o functio n mined p b e sol v calcula t ing th e paper, t ments. 2. H yb For th e tional c Proce d l ane El e u n Sun 1 , Yu n n ter, Beijing, C h c e, Tsinghua U n h ua.org.cn 3 plane elemen t e singular pro b n s, the displac e c essing proce d o quadrilateral m en t - b ased pl d method, the rid Pos t -Proc e s placemen t - b a s a n exhibit bet t b vious disadv a o cedures are m matrix invers i f urthermore, t h y are not as a n ts. c ombining the emen t - b ased e o st procedure d , the nodal d m en t - b ased pla t l force field w o n is assumed n al, by the p p arameters of v ed out, and fi n t ed easily. It i e internal forc e t his method i s b rid Post- P e hybrid plate c an be written d ure fo r e ments n gui Li 1 h ina n iversity, Beijin g t s are often u s b lem produce d e men t - b ased e l d ure based on plane eleme n ane elements accuracy of s t e ssing Proced u s ed elements t er stress resu l a ntages. For e m uch more co m i on and conde n h e accuracy o a ccurate as t h merits of bot h e lements, a n e was propose d d isplacements t e elements at w hich should s to substitute i p rinciple of st the assu m ed i n ally the new s seemed ver y e accuracy of s extended to i P rocessin g P elements, the as follows [5, 6 r g , China s ed to get the d by wall hole s l ements cann o the Hellinge r - n ts. In order to AQ4 and t ress solutions u re to derive st r l ts. But they a e xample, the c m plex, and the n se are usuall y o f displaceme n h e displacem e h hybrid ele m e w method na d by Cen [5] are derived f r first, after thi s atisfy the eq u i nto the hybri d ationary, the i nternal force f internal forc e y effective for plate elemen t i mprove the pl P rocedure discrete ener g 6 ]: JAMP in-plane s and the o t always Reissner find the AQ4 of these r ains, so a lso pre- c onstruc- calcula- y unavoi- n t results e n t -based m ents and med hy- . In this r om dis- i s, a new u ilibrium d energy undete r - f ield can e s can be improv- t . In this l ane ele- g y func- X. M. CHEN ET AL. Open Access JAMP 16 1 1 1{}[ ]{} 2 1{}[ ]{} 2 {}{} {}{} () e e eee eT Rb A T s A TT z AAA nxnxyny s dA dA dAdAf wdA Tw MMds MD M TD T MκTγ (1) where Wexp is the work of external forces: exp () eznxnxyny As WfwdATwMMds (2) Assume the new internal force field as follows: ,, ,, {}[ ]{} {}[ ]{} MM xxxxyy TM yxyxyy TMM TMM MPα TPα (3) where 11 121112 21 222122 21 222122 11 121112 1211 12 {}[ ] 100000000 [] 000010000 000 0000 01 00000 [] 00 000 0 0 {} T xyxy M T T M MMM jjj j jjj j jjj j jjjj M P P α (4) and 11 j, 12 j, 21 j, 22 j are components of the inver- sion matrix of Jacobian . After substituting Equation (3) into Equation (1), the new form of energy functional can be written as: 1 1 exp [][][] 1{} {} 2[][][] {} ([][] [][]) {} e e e T MbM A eT R MM T TsT A TT MMb A Te Ts dA dA dA W PDP αα PD P αPB PB q (5) By the principle of stationary: 0 {} e R M α (6) the parameters of the assumed internal force field can be written as: 1 {} [][]{} e MMMMq αKKq (7) where 1 1 [] ([][][] [][][]) []([] [][][]) e e T MMMb M A T TsT TT MqM bT s A dA dA KPDP PD P KPBPB (8) Substitute Equation (7) into Equation (3), the new in- ternal force can be obtained. Compared with plate elements, it is much easier to ex- tend this method to plane elements. The according hybrid discrete energy functional of plane elements can be writ- ten as [6]: 1 1 {}{}{}[] {}V 2 e eT T RVDd σuσDσ (9) Similar to Equation (4), a new form of stress field of displacement-based plane element can be assumed as: {} []{} σPα (10) Substitute Equation (10) into Equation (9), the para- meters i in [] P can be obtained by using the prin- ciple of stationary. The according matrix named [] K and [] q K for plane elements are as follows: 1 [] [][][] [] [][] e e T A T qb A tdA tdA KPDP KPB (11) With different matrix of [] P, the stress field will be different too. This new method will be used to try to im- prove the stress accuracy of plane element named AQ4 and AQ4 [7]. 3. Introduction of AQ4θ and AQ4θλ AQ4 and AQ4 are two plane elements with drill- ing DOF formulated using quadrilateral area coordinate methods presented by Long et al. [8,9]. They have the merits of high accuracy and robust against mesh distor- tions. After having been programmed into the software for the analysis of high-rise buildings, the results show that better accuracy is still needed for the stress of shear walls with holes. The definition of DOF of these two elements is: 1234 T T eTT T qqqqq (12) where T iiii uv q 1, 2,3, 4i (13) and i is the additional rigid rotation at element node. The displacements of element are as follows: 0 uu u (14) where 0 u is a polynomial about i u and i v, u is X. M. CHEN ET AL. Open Access JAMP 17 the additional displacement field induced only by the rigid rotation at nodes denoted as i . In order to determine the displacement field 0 u, the shape functions in reference [10] is used, they are as fol- lows: 4 00 1 4 00 1 ii i ii i uNu vNv (15) where: 0 212 1 2 1, 2,3, 4 iiiiiii NgLLgP i (16) and 314 2233112422413 132 4 1 3( )()()()()()() 2 1 P LLLLg gLLggLLgg gg gg gg (17) Assume the rotational displacement field as polyno- mials of quadrilateral area coordinates: 1231 342 431 42 513624 1231 342 431 42 513624 ()( ) ()() ()( ) ()() uLLLL LLLL LL LL vLLLL LLLL LL LL (18) with the conforming Equations as follows, the rotational displacement can be obtained. )41,34,23,12(0)( 0)( 0)( 4 1 4 1 ijsduu uu uu ij l i ii ii i ii (19) where 12 1 2 4 44 2 14 42 3 3 41 41 41 4 2 3 33 2 43 31 2 2 34 34 34 3 2 2 22 2 32 42 1 1 23 23 23 2 2 1 11 2 21 31 4 4 12 12 11 11 11 11 L gg L gg LL c b v u L gg L gg LL c b v u L gg L gg LL c b v u L gg L gg LL c b v u (20) and 12ii i by y , 21ii i cx x Then the stiffness matrix of element AQ4 can be solved out easily, and the strain matrix is: e qqBε (21) where ][][4321 BBBBB q (22) and x N y N x N y N y N y N x N x N iviuii ivi iui i 00 0 0 0 0 ][B (23) After substituting Equation (21) into the stress-strain relationship, the stress of AQ4 can be obtained. AQ4 is an improved element based on AQ4 by adding a displacement field which is induced by internal parameters. The additional displacement fields mentioned above are as follows: 1231 342 41312 423142 1231 342 '' 41312 423142 ()( ) ()( ) ()( ) ()( ) uLLLL L LLLLLLL vLLLL L LLL LLLL (24) where '' 11 2 2 ,,, are the internal parameters. In order to solve the undetermined parameters, the conforming Equations are taken as: {} 0 ij lds u (25) Substitute Equation (24) into (25), the shape functions of {} u can be formulated out, they are: 12341324 1 14 23 12 31 141423 42 14 23 14 23 13 24 14 23 232311242 314 2 (6 ) 6(1 ) () 6 ()( ) () 6(1 ) 1 1()()2()() 3 ()( ) gg ggggg g Ngg gg gg LL gggg ggLL gggg ggggLL LL gggg NggLLggLL LLLL (26) For element AQ 4 , 1 N is taken as the final inter- nal shape function, then: X. M. CHEN ET AL. Open Access JAMP 18 λNu }{ (27) where ' 11 1 1 [] 0 0 T N N λ N (28) Through the condense calculation, stiffness matrix of AQ4 can be written as: q T qqq e kkkkk 1 (29) where tdA tdA tdA q T q T q T qqq BDBk BDBk BDBk (30) and B is the strain matrix of additional displacement field. The according stress fields are as follows: λBελ (31) Combined Equation (21) with (31), the stress field of AQ4 can be obtained. 4. Hybrid Post-Processing of AQ4θ and AQ4θλ In the formulating of stress in elements AQ4 and AQ4 , the strain matrix q B and B are the differential results of displacements to coordinates. The differential calculation lowered the accurate order of stress. To avoid the differential, Hybrid Post-processing procedures can be used to improve the stress accuracy. Based on this theory, three forms of stress fields are pre- sented for these two displacement-based elements. 4.1. Stress Field I The first form of stress field is assumed as Equation (32). It is the stress field of a hybrid element developed by Pian and Wu [6]: 22 131 22 13 113 35 1 1 1 aa bb ab ab σ (32) where 444 12 3 11 1 444 123 111 11 1 44 4 111 444 ii iii ii ii i ii iii ii iii axa xax byb yby (33) 4.2. Stress Field II In order to raise the complete order of stress field, the second form is assumed as a polynomial of analytical trial function method presented by Fu and Long [11]: 1 22 9 001020 206 01020 2060 100002233 yx xy xy xy xyxy σ(34) 4.3. Stress Field III Try to make the three stress components to be indepen- dent, the third form of stress field is assumed as follows: 1 12 1 1 1 xyxy xyxy xyxy σ(35) 5. Numerical Examples 5.1. Strict Patch Test The constant strain/stress patch test using irregular mesh is shown in Figure 1. Let Young’s modulus E = 1000, Poisson’s ratio = 0.25, and thickness of the patch t = 1. After modified by three different forms of Hybrid Post- processing, these two elements can produce exact solu- tions without any problem. 5.2. Cook’s Skew Beam This example was proposed by Cook et al. [12]. As shown in Figure 2, a skew cantilever beam subjected to distributed shear load along its free edge. The results of max at point A and min at point B are listed in Tables 1 and 2. 6. Conclusion Using the traditional method to calculate stress of dis- placement-based elements, the accuracy will descend for the reason of differential when strains are derived from the displacement fields. For improving the stress accura- cy, hybrid/mix elements are very effective. However, the formulations of the hybrid elements are more compli- cated than those displacement-based elements. Based on X. M. CHEN ET AL. Open Access JAMP 19 y 4 2 2 2 2.5 1.5 4 E = 1500, μ = 0.25 7 3 5 6 1000 Figure 1. Patch test. 48 P=1 E =1500, μ=1 /3 44 44 A x y B Figure 2. Cook’s skew beam. Table 1. Stress at point A and B of Cook’s Beam of AQ4θ. Method σAmax σBmin 2 × 2 4 × 4 8 × 8 2 × 2 4 × 48 × 8 Field I 0.1791 0.2261 0.2338 −0.1700 −0.1929−0.2002 Field II 0.1951 0.2298 0.2348 −0.1942 −0.1933−0.2010 Field III 0.1914 0.2240 0.2319 −0.1769 −0.1938−0.2009 Source Val. 0.1917 0.2241 0.2377 −0.1877 −0.1939−0.2060 Ref. Val. 0.2362 −0.2023 Table 2. Stress at point A and B of Cook’s Beam of AQ4θλ. Method σAmax σBmin 2 × 2 4 × 4 8 × 8 2 × 2 4 × 48 × 8 Field I 0.1913 0.2271 0.2342 −0.1748 −0.1919 −0.2009 Field II 0.2145 0.2358 0.2364 −0.2084 −0.2032 −0.2027 Field III 0.2147 0.2358 0.2364 −0.2092 −0.2033 −0.2027 Source Val. 0.2498 0.2338 0.2358 −0.1729 −0.1896 −0.2018 Ref. Val. 0.2362 −0.2023 the Hellinger-Reissner variational principle, hybrid post- process procedure can take advantage of the merits of these two kinds of elements to establish the relationship between the displacement and the stress or internal force fields. In this paper, based on this theory, three forms of stress fields are used to improve the stress of plane ele- ments with drilling DOF. Through the numerical results, for element AQ 4 θ, only the second form of stress field is effective, but for AQ 4θ , except for the first form of stress, the other two forms can present better results than the source elements. It is proved that the method of hy- brid post-process procedure is workable. REFERENCES [1] R. H. MacNeal, “Derivation of Element Stiffness Matric- es by Assumed Strain Distributions,” Nuclear Engineer- ing and Design, Vol. 70, No. 1, 1982, pp. 3-12. ttp://dx.doi.org/10.1016/0029-5493(82)90262-X [2] R. Piltner and R. L. Taylor, “A Systematic Constructions of B-Bar Functions for Linear and Nonlinear Mixed-En- hanced Finite Elements for Plane Elasticity Problems,” International Journal for Numerical Methods in Engi- neering, Vol. 44, No. 5, 1997, pp. 615-639. http://dx.doi.org/10.1002/(SICI)1097-0207(19990220)44: 5<615::AID-NME518>3.0.CO;2-U [3] T. H. H. Pian, “Derivation of Element Stiffness Matrices by Assumed Stress Distributions,” AIAA Journal, Vol. 2, No. 7, 1964, pp. 1333-1336. http://dx.doi.org/10.2514/3.2546 [4] T. H. H. Pian and K. Sumihara, “Rational Approach for Assumed Stress Finite Elements,” International Journal for Numerical Methods in Engineering, Vol. 20, No. 9, 1984, pp. 1685-1695. http://dx.doi.org/10.1002/nme.1620200911 [5] S. Cen, Y. Q. Long, et al., “Application of the Quadrila- teral Area Co-Ordinate Method: A New Element for Mindlin-Reissner Plate,” International Journal for Nume- rical Methods in Engineering, Vol. 66, No. 1, 2006, pp. 1-45. http://dx.doi.org/10.1002/nme.1533 [6] T. H. H. Pian and C. C. Wu, “Hybrid and Incompatible Finite Element Methods,” Chapman & Hall/CRC, Boca Raton, 2006. [7] X. M. Chen, Y. Q. Long and Y. Xu, “Construction of Quadrilateral Membrane Elements with Drilling DOF Us- ing Area Coordinate Method,” Engineering Mechanics, Vol. 20, No. 6, 2003, pp. 6-11. [8] Y. Q. Long, J. X. Li, Z. F. Long and S. Cen, “Area Coor- dinates Used in Quadrilateral Elements,” Communica- tions in Numerical Methods in Engineering, Vol. 15, No. 8, 1999, pp. 533-545. http://dx.doi.org/10.1002/(SICI)1099-0887(199908)15:8< 533::AID-CNM265>3.0.CO;2-D [9] Z. F. Long, J. X. Li, S. Cen and Y. Q. Long, “Some Basic Formulae for Area Coordinates Used in Quadrilateral Elements,” Communications in Numerical Methods in Engineering, Vol. 15, No. 12, 1999, pp. 841-852. http://dx.doi.org/10.1002/(SICI)1099-0887(199912)15:12 <841::AID-CNM290>3.0.CO;2-A X. M. CHEN ET AL. Open Access JAMP 20 [10] Z. F.Long, X. M. Chen and Y. Q. Long, “Second-Order Quadrilateral Plane Element Using Area Coordinates,” Engineering Mechanics, Vol. 18, No. 4, 2001, pp. 95-101. [11] X. G. Fu and Y. Q. Long, “Generalized Conforming Qua- drilateral Plane Elements Based on Analytical Trial Func- tions,” Engineering Mechanics, Vol. 19, No. 4, 2002, pp. 12-16. [12] R. D. Cook, D. S. Malkus and M. E. Plesha, “Concepts and Applications of Finite Element Analysis,” 3rd Edition, John Wiley & Sons, Inc., New York, 1989. |