Application of MS-Excel Solver to Non-linear Beam Analysis

Transcription

1 / Application of S-cl Solr to Non-linar Bam Analysis Toshimi Taki arch 4, 007 April 8, 007, R. A. ntroction Sprasht softwar in crrnt prsonal comptrs is high prformanc an th softwar has nogh fnctions for application to nginring prolms. lop a mtho to sol non-linar strctral prolms y optimization fnction Solr of S-cl.. S-cl Solr Th most poplar sprasht softwar, S-cl has an a-in tool call Solr which prforms nmrical optimization. Solr fins paramtrs to optimiz ojcti al with mltipl constraints. W can s Solr to sol strctral prolms applying principl of total potntial nrgy or principl of total complimntary nrgy.. qations Non-linar Bam lmnt Bam strctr is iscrtiz to lmnts as finit lmnt mtho (F). Thn, total potntial nrgy is calclat. Shap fnction is th sam as F formlation. This mans that th formlation is th sam as F, t w on t n to prform ariational opration. W jst n to calclat strain nrgy of ach am lmnt an sm p th strain nrgy for all th lmnts. Following formlation show strain nrgy of -imntional am lmnt. s () Bam lmnt Assm that proprtis of a am lmnt as follows. niform cross sction in a lmnt. Sction Ara, A, omnt of nrtia,, ngth, Yong s ols, As shown in figr, following symols ar s., y flction an rotation at Gri in lmnt coorinats for flction: (,, ), y flction an rotation at Gri in lmnt coorinats for flction: (,, ) (, ) -φ 0 y -φ y φ (, ) 0 Figr. Bam lmnt () Bning Dflction of lmnt, is prss as follows.

2 / c ) (, c, at 0,, c 0 at, c a, a Soling th qations, ( ) a, ( ) c, Strain nrgy for ning is prss as follows. ( ) ( ) 0 0 a a To calclat shar forc an ning momnt, s following am qations. Shar forc an ning momnt at Gri, (V, ) Shar forc an ning momnt at Gri, (V, ) qation of am: c V, V V, V () tnsion ngth aftr flction is prss as follows. ( ) ( ) ' tnsion is, ( ) ( ) Strain nrgy is, A

3 / To calclat ial forc at Gri, P an Gri, P, s following qations. P A P P, (4) Consiration of Gomtrical Non-linarity in Coorinat Systm aftr Dformation lmnt coorinat systm aftr flction is consir as shown in figr. lmnt fctions in lmnt coorinat systm aftr flction,, com zro an strain nrgy for ning of th lmnt is prss as follows. a, ' ' ' ', c ', 0 whr, ' φ, ' φ φ is angl twn lmnt coorinat aftr an for flction. Thn, strain nrgy for ning (gomtrical non-linarity is consir), is prss in th following qation. ( a a ) (5) Total Potntial nrgy Total potntial nrgy of th lmnt, total is prss as th following qation. total ( ) ( P Py z z) all lmnts all trnal forcs sing flctions an rotations at all gris as paramtrs, minimiz total potntial nrgy, total y Solr of S-cl, thn yo will ha th soltion. 4. ampl Prolm lastica Post-ckling flction of a simply spport am nr ial forc is analyz. This prolm is a typical gomtrically non-linar prolm known as lastica. Analytical soltion is aailal for lastica an th rslt of th prsnt mtho is compar with th analytical soltion. A am with niform cross sction is ii to 0 lmnts as shown in figr. A tmplat of S-cl to calclat total potntial nrgy was lop. Thn, th total potntial nrgy was minimiz y Solr.

4 4/ y, lmnt D oa , Gri D Figr. ol for lastica Figr shows flction shap for arios ial forcs. Figr 4 shows rlationship twn cntr flction an ial forc. Cntr flction an ial forc ar normaliz with am lngth, an ckling loa, P cr, rspctily. act analytical soltion is shown in th figr for comparison (rfrnc []) y/ / Figr. Dflction Shap

5 5/.. act Soltion Propos tho P/Pcr m/ Figr 4. Rlationship twn Cntr Dflction an Aial Forc 5. Othr Applications Th mtho introc in this papr can appli to following prolms in aircraft strctral analysis. lop th mtho 0 yars ago an ha appli to many prolms of actal aircraft strctral analysis. Trss (linar, or gomtrically non-linar) Bam an Ramn (linar, or gomtrically non-linar) Bolt Joint (linar) Bam Colmn (gomtrically non-linar) -imnsional lastic Prolms (linar, or gomtrically non-linar) Shar Fil Prolms (linar)

6 /. Conclsions Th mtho prsnt in this papr has following fatrs an pct that th mtho will wily s in th aircraft instry. Straight forwar mtho. Spcial tchniq an tios iation of qations ar not ncssary to sol non-linar prolms. No spcial programs ar rqir. Gnral prpos sprasht softwar is s. Applical to actal aircraft strctral analysis. sfl to cation of nrgy mtho. 7. Rfrncs [] C.. Dym an. H. Shams, Soli chanics, A Variational Approach