NUMERICAL RESULTS QUALITY IN DEPENDENCE ON ABAQUS PLANE STRESS ELEMENTS TYPE IN BIG DISPLACEMENTS COMPRESSION TEST

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1 Appled Computer Scence, vol. 13, no. 4, pp do: /acs Submtted: Revsed: Accepted: Abaqus Fnte Elements, Plane Stress, Orthotropc Materal Bartosz KAWECKI *, Jerzy PODGÓRSKI * NUMERICAL RESULTS QUALITY IN DEPENDENCE ON ABAQUS PLANE STRESS ELEMENTS TYPE IN BIG DISPLACEMENTS COMPRESSION TEST Abstract The paper presents a bref descrpton of the Abaqus Smula plane stress quadrlateral elements (CPS4R, CPS4I, CPS4, CPS8R, CPS8). Comparson of the results qualty obtaned usng each of them was done. There was consdered two dmensonal bg dsplacements compresson test for a hghly orthotropc materal. Smulatons were performed for the compresson n two perpendcular drectons. 1. INTRODUCTION Abaqus Smula software makes many fnte elements avalable to ts users. The basc problem s the crteron of choosng an approprate element to the specfc nvestgaton. The paper presents the descrpton of fve plane stress quadrlateral elements avalable n Abaqus. Bref outlne of nodes, degrees of freedom and the Lagrange polynomal shape functons was done (Zenkewcz 2000, Bathe 2014, Lu 2014). The paper provdes a comparson of the results obtaned for a hghly orthotropc materal usng each of elements n a bg dsplacements compresson test. Consttutve law for the materal was defned basng on (Jones 1999, Lekhntsk 1981). The general am of the numercal experment was the need of determnaton element's sutablty for usage n the analyses, where consderable dstorton of the elements s antcpated. The problem of oversze dstorton was rased n several papers (Macneal 1985, Barlow 1989, Lee 1993). * Lubln Unversty of Technology, Faculty of Cvl Engneerng and Archtecture, Department of Structural Mechancs, Nadbystrzycka 40, Lubln, Poland, b.kaweck@pollub.pl, j.podgorsk@pollub.pl 56

2 As a report from numercal analyses there was specfed σ 11 and σ 22 stress dstrbuton n dependence on element type, drecton of compressng and appled mesh. Modellng and mesh study was based on the scentfc papers (Turcke 1974, Cecl 1994). Addtonally, there were prepared dagrams of P-δ relaton to notce any dscontnutes or dsturbances n the model. Basng on the numercal results there were made general conclusons and recommendatons for usng the Abaqus plane stress quadrlateral elements wth hghly orthotropc materals. 2. ELEMENTS DESCRIPTION The smplest elements are 4-node blnear plane stress quadrlateral elements, whch are presented n Fg.1. Fg. 1. Lnear geometrc order (frst order) plane stress elements (author's study basng on Abaqus User's Manual) The dfference between CPS4 and CPS4I s an occurrance of the addtonal nternal degrees of freedom preventng the element from overly stff behavor n bendng, called shear lockng. The phenomenon s more precsely descrbed n (Cook 2002) and Abaqus User's Manual. Fg node element shape functons based on Lagrange polynomals 57

3 General functons descrbng work of the fnte elements are the shape functons (Zenkewcz 2000, Bathe 2014, Lu 2014). In most cases they are based on Lagrange polynomals. Defnng coordnate system usng η and ξ axes, shape functons may be wrtten as t was shown n Fg.2. Denotatons u and v descrbe nodal translaton degrees of freedom. More complex elements are 8-node bquadratc plane stress quadrlateral elements, whch are presented n Fg.3. Fg. 3. Quadratc geometrc order (second order) plane stress elements (author's study basng on Abaqus User's Manual) Defnng coordnate system as before, usng η and ξ axes, shape functons may be wrtten as t was shown n Fg.4. Denotatons u and v descrbe nodal translaton degrees of freedom. Fg node element shape functons based on Lagrange polynomals In both 4-node and 8-node elements the equaton gven below s met: n 1 N 1 (1) 58

4 Dsplacement feld s descrbed by formula: n u N u 1 n v N v In both elements types there are consdered reduced and full ntegraton modes. Reduced ntegraton provdes lower computatonal cost n the numercal analyss, but on the other hand may lead to the lower qualty of the results. These aspects wll be dscussed n the next paragraph. 1 (2) 3. COMPRESSION TEST FEM MODEL There were prepared two numercal models wth dfferent local coordnate systems orentatons. For hghly orthotropc materals, orentaton of the specmen s crucal for obtaned stresses and strans values. Fg. 5. a) presents set of compresson n a longtudnal drecton, and Fg. 5. b) set of compresson n a transversal drecton. Predcted results of the stresses dstrbuton should be completely dfferent from each other. Fg. 5. Compresson test wth dfferent local coordnate systems orentaton: a) longtudnal compresson, b) transversal compresson 59

5 Compressng steel parts were modeled as rgd bodes to prevent them from deformatons. Lower support was fxed n the reference pont, upper steel plate was constraned n the horzontal drecton and n the plane rotaton. There was added constant vertcal dsplacement of the upper steel plate and frctonal contact between the specmen and steel plates. There were two mesh szes taken nto account: 16 elements and 64 elements, whch s presented n Fg.6. Fg. 6. Number of fnte elements used n the numercal smulaton For both drectons of compresson there was added the same vertcal dsplacement δ = 5 mm. 4. NUMERICAL RESULTS Numercal results contan comparson of the elements behavour under bg dsplacement compresson and descrpton of σ 11 and σ 22 stresses dstrbuton n the specmen. Expected values σ 11 stresses were much hgher than σ 22 because of a great dfference n the elastc modules dependng on the drecton. For better understandng work of the whole specmen there were made dagrams wth P-δ relaton. Results for mesh wth 16 elements are shown n Fg Fg. 7. CPS4R elements dsplacements, σ11, σ22 stresses dstrbuton and P-δ relaton 60

6 Fg. 8. CPS4I elements dsplacements, σ11, σ22 stresses dstrbuton and P-δ relaton Fg. 9. CPS4 elements dsplacements, σ11, σ22 stresses dstrbuton and P-δ relaton Fg. 10. CPS8R elements dsplacements, σ11, σ22 stresses dstrbuton and P-δ relaton Fg. 11. CPS8 elements dsplacements, σ11, σ22 stresses dstrbuton and P-δ relaton CPS8 presented the most precse results. Comparng other elements to the CPS8 t was vsble that CPS8R and CPS4 gave accurate results whle CPS4I and CPS4R gave naccurate results n σ 11 drecton, due to the excessve elements dstorton. 61

7 Addtonally, for CPS4R element, analyss convergence ends n about 80% of the analyss progress and the shape of the elements s unphyscal. P-δ relaton dagrams clearly shows the dsturbances n the whole model caused by CPS4R and CPS4I elements, whle for CPS4, CPS8R and CPS8 relaton s smooth. General prncpal to get the proper results from the numercal model was consderng lower and hgher mesh densty. Only then an nterpretaton of the results mght be correct. Because of that there was prepared the second mesh wth 64 fnte elements n the specmen. Ths approach allowed to fnd out how mesh densty nfluence the soluton. Results of the nvestgaton are presented n Fg Fg. 12. CPS4R elements dsplacements, σ11, σ22 stresses dstrbuton and P-δ relaton Fg. 13. CPS4I elements dsplacements, σ11, σ22 stresses dstrbuton and P-δ relaton The same as n case of 16 elements mesh, there were vsble naccurate stress results for CPS4R and CPS4I. It seemed that mesh densty had a low nfluence on the elements work. There was a vsble progress n proper stress dstrbuton range n case of CPS4R elements, however after reachng some stress lmt, these elements behavour was stll unphyscal. Fg. 14. CPS4 elements dsplacements, σ11, σ22 stresses dstrbuton and P-δ relaton 62

8 Fg. 15. CPS8R elements dsplacements, σ11, σ22 stresses dstrbuton and P-δ relaton Fg. 16. CPS8 elements dsplacements, σ11, σ22 stresses dstrbuton and P-δ relaton 5. CONCLUSIONS Basng on the performed numercal analyses there were made several conclusons about usng Abaqus plane stress quadrlateral elements. CPS4R and CPS4I elements were recommended only to use wth small dsplacements and lnear problems. Nonlnearty would probably cause smlar nadequate effects as an orthotropc materal as presented n the paper. In case of these fnte elements, makng mesh more dense may have no nfluence on gettng better numercal results. CPS4 elements gave the same good results n the compresson as CPS8R and CPS8 elements. However t was worth to remember that shear lockng occurs n CPS4 elements. The best soluton for almost all of the plane stress problems are second order quadrlateral elements CPS8R and CPS8, whch gve proper results n compresson, bg dsplacements, works well wth hghly orthotropc materals and wth bendng. The author's study provdes precse nformaton about how the Abaqus plane stress elements work under compresson. It s a very mportant case for a further research. There are planned several numercal modellng valdatons n delamnaton and damage processes n hghly orthotropc materals. It s possble only when accurate results are provded by the fnte elements. 63

9 REFERENCES Abaqus/CAE V6.14 User's Manual. Armstrong, C. G. (1994), Modellng requrements for fnte-element analyss. In Computer-Aded Desgn, Vol. 26, , Barlow, J. (1989), More on optmal stress ponts reduced ntegraton, element dstortons and error estmaton. Internatonal Journal of Numercal Methods n Engneerng, Vol. 28, , do: /nme Bathe, K. J. (2014). Fnte Element Procedures. Second Edton, Cook R. D., Malkus, D.S., Plesha M. E., Wtt R.J.(2002), Concepts and Applcatons of FEA, 4 th ed., John Wley&Sons, Inc., Jones, R. M. (1999), Mechancs of Composte Materals. Second Edton, Lekhntsk S.G. (1981), Theory of Elastcty of an Ansotropc Elastc Body, Mr Publshers. Lee, N. S., & Bathe, K. J. (1993), Effects of element dstortons on the performance of soparametrc elements. Internatonal Journal of Numercal Methods n Engneerng, Vol. 36, , do: /nme Lu, G. R. & Quek, S. S. (2014), The Fnte Element Method. A Practcal Course. Second Edton, Macneal, R. H., & Harder, R. L. (1985), A proposed standard set of problems to test fnte element accuracy. In Fnte Elements n Analyss and Desgn, Vol. 1, 3 20, Turcke, D. J., & McNece, G. M. (1974), Gudelnes for selectng fnte element grds based on an optmzaton study. In Computers & Structures, Vol. 4, , Zenkewcz, O. C., & Taylor, R. L. (2000). The Fnte Element Method. Ffth Edton. Volume 1: The Bass,