Deflection of Beams by Means of Static Green Functions

Transcription

1 Uniersl Jornl o ehnil Engineering : 9- DOI:.89/jme.. Deletion o ems y ens o Stti Green ntions. A-Hill Deprtment o ehnil Engineering Zrq Uniersity Jorn Copyright y thors ll rights resere. Athors gree tht this rtile remins permnently open ess ner the terms o the Cretie Commons Attrition iense. Interntionl iense Astrt A Green s ntion metho or etermining the eletion o sttilly loe ems is presente. This metho my e pplie to single n mlti-spn sttilly eterminte n ineterminte ems sjete to single or seerl los. Also Green s ntions or ierent ems re gien. inlly seerl emples re gien to illstrte the se o Green s ntions. Keyors Green ntion Deletion o ems Stti o. Introtion The eletion o em n e etermine y the integrtion metho or the se o isontinity ntions n. To in the eletion o em y the metho o integrtion e integrte the moment eqtion EEEE tie hih reqires the knolege o s ontinos ntion throghot the em s entire length n the introing o integrtion onstnts. These integrtion onstnts re then elte sing the onry onitions o the onsiere em. The metho o integrtion is onenient or single-spn ems ith single lo. I the loing on the em onsists o seerl istrite n onentrte los ting on ierent positions on the em the metho o integrtion eomes more teios to pply sine seprte moment ntions he to e gien or eh region eteen the los. Also integrtion o these moment ntions reqires the eltion o to integrtion onstnts or eh region sing onry n/or ontinity onitions. The metho o isontinity ntions is prtilrly ntgeos or soling prolems inoling mltiply loe ems. y this metho isontinity ntions re se to gie the internl moment s single epression inepenent o the nmer o los ting on the em. Integrting the moment eqtion EEEE tie yiels n epression or the eletion o the em ith only to integrtion onstnts inle. These onstnts n then e elte y sing only the onry onitions. Note tht y the se o the orementione methos e he to in the spport retions n the internl moments eore e n in the eletion o the em. In this pper metho sing Green s ntions or etermining the eletion o sttilly loe prismti ems is presente. This metho my e se or single n mlti-spn ems single n mlti loe ems n sttilly eterminte n ineterminte ems. The metho o Green ntions is esier to se thn the orementione methos ese this metho oes not reqire etermining the spport retions or the internl moment epressions o the onsiere em. rthermore sine no integrtion onstnts or in the integrte Green s ntions e on t nee to onsier the onry or the ontinity onitions or the sttilly eterminte ems. The metho o Green s ntions is prtilrly onenient or soling prolems inoling mlti-loe or sttilly ineterminte ems. Also Green ntions or ierent ems re etermine. In orer to emonstrte the se o the Green ntions metho seerl pplition emples re gien.. Green s ntions or Sttilly oe ems A Green s ntion o em is its response t point e to nit onentrte ore ting on n ritrry position eteen the to ens o the em. Ths Green s ntions n implse responses re ientil eept tht the ormer re inole in the soltions to onry le prolems heres the ltter re inole in soltions to initil le prolems. In the olloing Green s ntions or elsti homogeneos isotropi niorm ems ith ierent onry onitions ill e erie sing ple trnsorm. The elsti eletion o n initilly stright prismti em e to stti lo n e esrie y the ierentil eqtion EEEE here is the ening rigiity o the em. The eletion is positie pr n the -is etens positie to the right long the initilly stright longitinl is o the em s shon in ig.. A prime enotes eritie ith respet to.

2 Deletion o ems y ens o Stti Green ntions igre. Deletion o ems The pplie istrite lo is positie in the onr iretion. y the tion o onentrte ore t position the lo is gien s n y the tion o moment lo t position the lo is gien s here is the Dir elt ntion. In orer to etermine the Green s ntion or em e reple the ting ritrry istrite lo ith nit onentrte ore pplie t n ritrry position eteen the to ens o the em s shon in ig.. Hene Eq. eomes. ple trnsorming this eqtion yiels s V s s s. 5 s s e rom this eqtion the ple trnsorm o the eletion is otine s s e V s. s s s s s Inerting this eqtion les to g here - is the nit step ntion eine s 7. 8 > Eqtion 7 is lle the Green s ntion o Eq. n represents the eletion o point on the em hen nit lo hs een pplie t n ritrry point eteen the to ens o the em. One the Green s ntion or the em is otine the eletion o the em e to ny ritrry loing istrition my e gien in the orm g 9 here is the length o the em n g is the inrementl eletion t e to the tl lo intensity t. Depening pon the nmer o the ting los on the em n the nmer o the em s spns the integrl in Eq. 9 hs to e iie into seerl integrls see the olloing emples. The nknon prmeters n in Eq. 7 re to e etermine rom the onry onitions. As n emple or simply spporte em the eletion n internl moment mst nish t n. Ths the onry onition t re n t re. Sstitting Eq. into Eq. 7 yiels. Using Eq. yiels the olloing to eqtions. Soling these eqtions yiels or the prmeters n : 5. Sstitting Eq. 5 n Eq. into Eq. yiels the Green s ntion or simply spporte em in the orm g or in interls orm g 7. 8 Tle in the Appeni inles Green s ntions or ems ith ierent onry onitions erie y sing Eq. 7 n pproprite onry onitions.

3 Uniersl Jornl o ehnil Engineering : 9-. Applition Emples The olloing emples emonstrte the se o Green s ntions or etermining the eletion o ierent ems.. Simply Spporte em ith Conentrte ore igre shos simply spporte em sjete to onentrte ore -. The eletion o this em my e otine iretly rom Eq. 8 s 9 or in simpler orm. igre. Simply spporte em ith onentrte ore.. Simply Spporte em ith Distrite o Using the nit step ntion the ting lo on the em shon in ig. my e gien s. Sstitting this eqtion n Eq. 7 into Eq. 9 gies. igre. Simply spporte em ith onstnt istrite lo The irst integrl in this eqtion is to e integrte only eteen n sine the lo is eql to zero or n > ths } {. igre. Prot o step ntions or emple ese o the three step ntions in the remining integrl in this eqtion this integrl is to e iie into three seprte integrls epening pon hether or. or this reson e se ig. hih gies or the prot o the step ntions ---- or

4 Deletion o ems y ens o Stti Green ntions ll oer the interl : the le s long s the le eept or here the prot is n the le eept or here the prot is. Ths e otin or the seon integrl in Eq. the le o or. or the interl e get. inlly or the interl e otin. 5 Comining the reslts in Eqs. n 5 yiels the inl soltion in the orm: or in the orm 5 7 here 5... Simply Spporte em ith oment o igre 5. Simply spporte em ith moment lo In this emple e onsier simply spporte em sjete to onentrte moment lo s shon in ig. 5. The lo my e esrie orresponing to Eq. s. Sstitting this eqtion n Eq. 7 into Eq. 9 gies. 8 Using orml integrl king se o the reltion n n n 9 gies or the eletion in Eq. 8: or in more milir orm... Sttilly Ineterminte To-spn em We onsier in this emple the sttilly ineterminte em shon in ig.. The spport t s n eternl ore hih n e etermine ter the eletion hs een on rom the onition. Ths the ting lo my e gien s. igre. Sttilly ineterminte ntileere em The Green s ntion or ntileere em is otine rom Tle s g. Sstitting Eqs. n into Eq. 9 yiels

5 Uniersl Jornl o ehnil Engineering : 9-. The irst integrl in this eqtion n e reily elte y sing the reltion or or or. 5 Ths e otin or Eq.. The eltion o the integrl in this eqtion gies three soltions epening pon the prot o the step ntions inle in the integrl. This prot gies or ll oer the interl the le o or - or n or. Ths sing Eq. 5 the integrl in Eq. gies the le or or n or. Comining ll the otine reslts gies or the eletion o the em shon in ig. :. 7 The nknon spport retion n e otine rom the onition. This onition gies or the le:. 8 Aknolegements This Reserh is ne y the Denship o Sientii Reserh in Zrq Uniersity/Jorn. Appeni Tle A. Stti Green ntions or ierent ems em Type g

6 Deletion o ems y ens o Stti Green ntions REERENCES R. C. Hieler. ehnis o mterils 9 th eition Prentie-Hll U.S.A.. A. Higon et l. ehnis o terils th eition John Wiley & sons Ne York A. Pipes n. R. Hrill. Applie mthemtis or Engineers n Physiists r Eition Gr-Hill Tokyo 97.. A-Hill ore irtion o Eler-ernolli ems y mens o ynmi Green ntions Elseier Siene t. Jornl o Son n Virtion Vol. 7 No 9-7.