Bending behaviors of simply supported rectangular plates with an internal line sagged and unsagged supports

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1 Songklanakarin J. Sci. Tchnol. 30 (1) Jan. - F Original Articl Bnding haviors of simpl supportd rctangular plats with an intrnal lin saggd and unsaggd supports Yos Sompornjaronsuk 1 * and Kraiwood Kiattikomol 2 1 Dpartmnt of Civil Enginring Facult of Enginring Mahanakorn Univrsit of Tchnolog Nong Chok Bangkok Thailand 2 Dpartmnt of Civil Enginring Facult of Enginring King Mongkut s Univrsit of Tchnolog Thonuri Bangmod Thung Khru Bangkok Thailand Rcivd 22 Novmr 2007; Accptd 14 Fruar 2008 Astract This rsarch work dals with th application of dual sris quations to th prolms of simpl supportd rctangular plat with an intrnal lin support undr uniforml distriutd load. Two diffrnt tps of prolm ar considrd dpnding on th natur of singularitis that allowd in th filds Th first is th plat having an intrnal lin unsaggd support that th momnt singularit is assumd at th tip of intrnal lin support. Th scond involvs th advancing contact prolm twn th plat and th intrnal lin saggd support in which th shar singularit hiits at th tip of contact. Howvr oth tps of singularit ar in th ordr of an invrs squar root. B choosing th propr finit Hankl transform th dual sris can convrtd to th inhomognous Frdholm intgral quation. This quation with a numrical tchniqu is thn rducd to a st of simultanous quations suital for numrical solution. Th phsical quan-titis of th plats and th tnt of contact rlatd to th lvl of loading in th cas of fr contact ar providd in th prsnt work. Kwords: plat nding advancing contact singularit dual sris quations Frdholm intgral quation 1. Introduction Focusing on th prolms of plat with mid oundar conditions thr ar numrous analtical and numrical mthods usd to analz th prolm. Th numrical mthods ar gnrall found to unsatisfactor (Lissa t al. 1969) spciall at th transition point of discontinuous oundar du to th prolm singularit (Williams 1952) and thn th us of intgral transform (Snddon 1972) is on of th appropriatl analtical mthods to solv th prolm which lads to dtrmin th solution of an intgral quation. Much attntion has n said man rsarchrs to invstigat th static nding prolms (Yang 1968; Kr and Sv 1970; Kiattikomol t al. 1974; Kiattikomol and *Corrsponding author. addrss: os@mut.ac.th Porn-anupapkul1985; Kiattikomol and Sriswasdi 1988) viration charactristics and stailit and uckling haviors (Lissa 1969; Kr and Stahl 1972; Stahl and Kr 1972). Howvr th mntiond works ar th prolms of plat whr th supports hav th sam lvl. Dundurs t al. (1974) first invstigatd th contact twn th plats and th saggd supports in which th saggd supports ar locatd at th plat dgs. For th cass of saggd support placd in domain of th plat rcntl Sompornjaronsuk and Kiattikomol (2006) tratd th two prolms of rctangular plat simpl supportd on th two opposit dgs and ithr fr or clampd on th two othr dgs with an intrnal lin saggd support. Th singularit at th tips of contact twn th plat and an intrnal lin saggd support is introducd in th ordr of an invrs squar root in th shar. Thrfor this rsarch considrs th two prolms of simpl supportd rctangular plat with an intrnal lin

2 102 Sompornjaronsuk & Kiattikomol / Songklanakarin J. Sci. Tchnol. 30 (1) : (4ac) (a) () Unsaggd Support Saggd Support Figur 1. Rctangular plats with intrnal lin: (a) unsaggd support and () saggd support saggd and unsaggd supports. Th solution tchniqu similar to th prvious work is applid to th prsnt work whil oth tps of prolm ar tratd hr within th sam formulation cpt that for th prolm of plat with unsaggd support th ordr of singularit is an invrs squar root in th momnt instad of an invrs squar root in th shar as in th cas of an intrnal lin saggd support for which th momnt is oundd at th tips of contact. 2. Govrning Equation and Boundar Conditions Considring th gomtr of plat as illustratd in Figur 1 th actual dimnsions of th plat ar a and and scald th factor. Th partial diffrntial quation govrning th dflction function of th plat undr th uniforml distriutd load q is prssd (1) whr flural rigidit E = Young s modulus Poisson s ratio and h = plat thicknss. Du to th two-fold smmtr in dflction function th oundar conditions nd onl to writtn on on quadrant and as follows: : (2a) : (5) whilst th initial gap twn th plat and th intrnal lin saggd support. Sinc th prolm considrd coms th plat with an intrnal lin unsaggd support. It is also rmarkal that Eqs (4) and (5) ar th mid oundar conditions and th othrs ar th rgular oundar conditions. 3. Formulation Utilizing th Lv-Nadai approach (Timoshnko and Woinowsk-Krigr 1959) th dflction function satisfing th support conditions of th plat dgs at is takn as th sum of th particular (w p ) and complmntar (w c ) solutions of Eq.(1) whr and (6) ; (7a) (8) in which th constants A m B m C m and D m ar dtrmind from th oundar conditions of Eqs.(2) and (3) that ild th following rlations with ltting β = m (9) (10). (11) It can notd that th prolms ar now rducd to th dtrmination of a singl constant D m. Th application of th mid oundar conditions givn in Eqs.(4c) and (5) lads to th dual sris quations as follows: whr (12) (13) (14) : (3) (15)

Sompornjaronsuk & Kiattikomol / Songklanakarin J. Sci. Tchnol. 30 (1) 101-107 2008 103.

3 Sompornjaronsuk & Kiattikomol / Songklanakarin J. Sci. Tchnol. 30 (1) (16) For th cas of an intrnal lin unsaggd support th dual sris quations can rducd to a singl intgral quation rprsnting th unknown function P m a finit Hankl transform (Kiattikomol t al. 1974; Kiattikomol and Porn-anupapkul 1985) (17) and for an intrnal lin saggd support th function P m is rplacd (Dundurs t al. 1974; Sompornjaronsuk and Kiattikomol 2006) (18) whil is th unknown auiliar function and J l ( ) is th Bssl function of th first kind and ordr 1. B th sam procdur as usd in th prvious works (Kiattikomol t al. 1974; Sompornjaronsuk and Kiattikomol 2006) sustituting P m from Eqs.(17) and (18) that satisf to ach cas of th plats into Eq.(13) and using th idntit givn in Stahl and Kr (1972) (19) whr H( ) I 1 ( ) ar th Havisid unit stp function and th modifid Bssl function of th first kind and ordr 1 rspctivl and thn aftr som manipulations and with th hlp of crtain idntitis found in Gradshtn and Rzhic (1956) th scond dual sris quations Eq.(13) can rducd to th final form of an inhomognous Frdholm intgral quation of th scond kind ν = 0.3 = 0.45 = 0.40 = 0.35 Saggd Support = = 0.49 in which (20) (21) (22) 2 () ρ = Figur 2. Auiliar functions of squar plat with intrnal lin: (a) unsaggd support and () saggd support (23a) (23). (24) An intgral quation prsntd in Eq.(20) can solvd numricall to otain th unknown auiliar function using standard mthods. 4. Numrical Analsis and Rsults Two diffrnt solutions can dtrmind from th sam formulation solving Eq.(20). In ordr to valuat th

104 Sompornjaronsuk & Kiattikomol / Songklanakarin J. Sci. Tchnol. 30 (1) 101-107 2008 Figur 3.

Dflctions surfacs oundd th rgion for squar plat with intrnal lin: (a) unsaggd support and () saggd support algraic quations and using th Gaussian

It should rmarkd that th closd form prssions for th impropr infinit intgral in Eqs. (23a) wr not found.

Th infinit sris in th krnl and in th function wr calculatd to a rlativ rror of 0.00001.

c and th Poisson s ratio was takn as 0.3. Thrfor th dflction function of th plats that dmonstratd in Figurs 3 to 5 can valuatd from Eq.(6).

4 104 Sompornjaronsuk & Kiattikomol / Songklanakarin J. Sci. Tchnol. 30 (1) Figur 3. Dflctions at for squar plat with intrnal lin: (a) unsaggd support and () saggd support Figur 5. Dflctions surfacs oundd th rgion for squar plat with intrnal lin: (a) unsaggd support and () saggd support algraic quations and using th Gaussian limination with partial pivoting to solv for th discrtizd valus of as shown in Figur 2. It should rmarkd that th closd form prssions for th impropr infinit intgral in Eqs. (23a) wr not found. Howvr a numrical intgration is asil prformd with using a 16-point Gauss-Lgndr quadratur formula. Th infinit sris in th krnl and in th function wr calculatd to a rlativ rror of Th numrical valuation was carrid out onl for th cas of a squar plat of scald lngth with various cass of th scald half-lngth of intrnal lin support c and th Poisson s ratio was takn as 0.3. Thrfor th dflction function of th plats that dmonstratd in Figurs 3 to 5 can valuatd from Eq.(6). It is sn that th dflctions of oth cass ar incrasd with th incrasing of -ratio. Aftr th dflction is otaind th strss rsultants ar calculatd as follows: (25) (26) Figur 4. Dflctions at for squar plat with intrnal lin: (a) unsaggd support and () saggd support phsical quantitis th unknown auiliar function must otaind. Simpson s rul was chosn for this purpos caus it is a simpl mthod lading to a sstm of linar (27) (28) whr M M ar th nding momnts and V V ar th support ractions.

Sompornjaronsuk & Kiattikomol / Songklanakarin J. Sci. Tchnol. 30 (1) 101-107 2008 105 Figur 6.

Support ractions for squar plat with intrnal lin saggd support: (a) V (0 ) and () and (c) V ( 0 ) and 0 < < Figur 7.

momnts outsid an intrnal lin support ( 0 = 0) for oth cass of th plat with intrnal lin saggd and unsaggd supports.

rsults ar illustratd as in Figur 8 for th plat with an intrnal lin saggd support.

5 Sompornjaronsuk & Kiattikomol / Songklanakarin J. Sci. Tchnol. 30 (1) Figur 6. Bnding momnts ( 0) M and 0 for squar plat with intrnal lin: (a) unsaggd support and () saggd support Figur 8. Support ractions for squar plat with intrnal lin saggd support: (a) V (0 ) and () and (c) V ( 0 ) and 0 < < Figur 7. Bnding momnts M ( 0 ) and 0 < < for squar plat with intrnal lin: (a) unsaggd support and () saggd support Figurs 6 and 7 show th nding momnts outsid an intrnal lin support ( 0 = 0) for oth cass of th plat with intrnal lin saggd and unsaggd supports. Th momnts at = ar singular for plat having an intrnal lin unsaggd support ut oundd for th cas of plat with an intrnal lin saggd support. This is th diffrnc of th two prolms. In cas of fr contact sinc th distriution of support raction is singular of ordr squar root at th tips of contact it is intgral and th rsults ar illustratd as in Figur 8 for th plat with an intrnal lin saggd support. It is sn that at = th support raction V (0) is singular and th ractions ar not proportional to th applid load. Onl for th cas of fr contact twn th plat and th intrnal lin saggd support th contact curv (Dundurs t al. 1974) of plat can dtrmind from th dflction w(0) supposing th contact lngth. If th sag of intrnal lin support is spcifid as th uniform load rquird to produc th dflction for ach spcifid

6 106 Sompornjaronsuk & Kiattikomol / Songklanakarin J. Sci. Tchnol. 30 (1) δ = qa D Saggd Support q q ν = 0.3 Figur 9. Etnt of contact with intrnal lin saggd support vrsus lvl of loading for squar plat contact lngth can calculatd writing th sag in th form (29) whr q o is th givn uniform load for plat without partial intrnal lin saggd support or th load at which th plat starts to touch th intrnal lin saggd support and is th constant valu. Whn th contact lngth is spcifid as th maimum dflction can computd from Eq.(6) with stting = 0 and has th form (30) in which q is th applid uniform load and is th constant valu. In viw of Eqs.(29) and (30) thus th uniform load q rquird to produc th dflction with contact lngth is givn th following rlation (31) Hnc th rlationship twn th contact lngth and th incrasd load rquird to produc this contact is otaind. Figur 9 shows th tnt of contact vrsus th lvl of loading. It rvald that th lngths of contact dpnd on th lvl of loading. 5. Conclusions Th analtical invstigations prsntd in this work ar concrnd with th nding prolm of plat having an intrnal lin unsaggd support and th advancing contact prolm twn th plat and intrnal lin saggd support. Both prolms ar tratd within th sam formulation writing th mid oundar conditions in th form of dual sris quations. Utilizing th propr finit Hankl transform that satisfid th singularit ordr in ach prolm tp thn th dual sris quations can rducd to th inhomognous Frdholm intgral quation of th scond kind which is suital for numrical valuation. Th dflction and strss rsultants of th plat ar dtrmind numricall and prsntd graphicall. From th otaind rsults it found that th magnitud of dflction for oth cass of plat is incrasd whn th ratio of is also incrasd. Th nding momnt in oth dirctions at is singular for plat with intrnal lin unsaggd support ut finit valu for plat with intrnal lin saggd support. Although th support raction at th tips of contact twn th plat and th intrnal lin saggd support is singular howvr it can dtrmind. It is sn that th raction is not proportional to th applid load. Finall th tnt of contact with intrnal saggd support dpnds on th lvl of loading whr th rquird load at which th plat starts to touch th saggd support is for th Poisson s ratio takn as 0.3. Rfrncs Dundurs J. Kiattikomol K. and Kr L.M Contact twn plats and saggd supports. Journal of th Enginring Mchanics Division 100 : Gradshtn I.N. and Rzhik I.M Tal of Intgrals Sris and Products Acadmic Prss Nw York. Kr L.M. and Stahl B Eignvalu prolms of rctangular plats with mid dg conditions. Journal of Applid Mchanics 39 : Kr L.M. and Sv C On th nding of crackd plats. Intrnational Journal of Solids and Structurs 6 : Kiattikomol K. and Sriswasdi V Bnding of a uniforml loadd annular plat with mid oundar conditions. AIAA Journal 26 : Kiattikomol K. Kr L.M. and Dundurs J Application of Dual Sris to Rctangular Plats. Journal of th Enginring Mchanics Division 100 : Kiattikomol K. Porn-anupapkul S. and Thonchanga N Application of dual sris quations to a squar plat with varing cornr supportd lngths. Procdings of Th Rcnt Advancs in Structural Enginring Bangkok Thailand Mar : Lissa A.W Fr virations of lastic plats. AIAA 7 th Arospac Scincs Mting Nw York Cit Nw York Jan : Lissa A.W. Clausn W.E. Hulrt L.E. and Hoppr A.T A comparison of approimat mthods for th solution of plat nding prolms. AIAA Journal 7 : Snddon I.N Th Us of Intgral Transforms McGraw-Hill Nw York. Sompornjaronsuk Y. and Kiattikomol K Singularit of advancing contact prolms of plat nding. Th Tnth East Asia-Pacific Confrnc on Structural Enginring and Construction Bangkok Thailand Aug : Stahl B. and Kr L.M Viration and uckling of a rctangular plat with an intrnal support. Quartrl Journal of Mchanics and Applid Mathmatics 25 :

7 Sompornjaronsuk & Kiattikomol / Songklanakarin J. Sci. Tchnol. 30 (1) Timoshnko S.P. and Woinowsk-Krigr S Thor of Plats and Shlls 2 nd d. McGraw-Hill Singapor. Williams M.L Surfac strss singularitis rsulting from various oundar conditions in angular cornrs of plats undr nding. Procdings of th First U.S. National Congrss of Applid Mchanics Chicago Illinois Jun : Yang W.H On an intgral quation solution for a plat with intrnal support. Quartrl Journal of Mchanics and Applid Mathmatics 21 :

DIFFERENTIAL EQUATION

MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous