Kantorovich-Vlasov Method for the Flexural Analysis of Kirchhoff Plates with Opposite Edges Clamped, and Simply Supported (CSCS Plates)

Transcription

1 ISSN (Prit) : ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) Ktorovih-Vlsov Method for the Flexurl Alysis of Kirhhoff Pltes with Opposite Edges Clmped, d Simply Supported (CSCS Pltes) Oh H.N. #1, Mm B.O. #, Ike, C.C. *3, Nwoji, C.U. #1, #, # ept of Civil Egieerig *3 Uiversity of Nigeri, Nsukk, Eugu Stte, Nigeri. Eugu Stte Uiversity of Siee & Tehology, Eugu Stte, Nigeri. 1 hygius.oh@u.edu.g bejmi.mm@u.edu.g 3 ike7@yhoo.om lifford.woji@u.edu.g Abstrt - The Ktorovih-Vlsov method ws used, i this study, for the flexurl lysis of retgulr Kirhhoff pltes with opposite edges (x =, d x = ) simply supported d the other opposite edges (y =, d y = b) lmped (CSCS pltes). The plte ws subjeted to lier distributio of lod over the etire plte domi. Vlsov method ws used i fidig the oordite futio i the x-diretio, d Ktorovih method ws used to osider the displemet futio for the plte. The totl potetil eergy futiol, d the orrespodig Euler-Lgrge differetil equtios were the obtied for the plte problem. This ws solved subjet to the boudry oditios i the y diretio to obti the displemet futio whih miimized the totl potetil eergy futiol. Bedig momet distributios were obtied usig the bedig momet-displemet equtios. The solutios obtied for defletio d bedig momet distributios were foud to be rpidly overget sigle series. efletios d bedig momet omputed t the eter of the plte were lso rpidly overget series. The solutios obtied for defletios d bedig momets (M xx d M yy ) were extly idetil with solutios preseted by Timosheko d Woiowsky-Krieger who used the method of superpositio. Keywords: Ktorovih-Vlsov method, Kirhhoff plte, totl potetil eergy futiol, Euler-Lgrge differetil equtio. I. INTROUCTION Pltes re struturl members hrterized by trsverse dimesio tht is muh smller th the other iple dimesios. They re used i ivil, mehil, eroutil d mrie egieerig to model roof d floor slbs, ship hulls, bridge dek slbs, sperft pels, retiig wlls, foudtios slbs, d irrft pels [1,, 3]. Pltes my be subjet to trsverse stti or dymi lods or iple ompressive lods resultig i stti flexure, dymi flexure or buklig behviours. They re lssified ordig to shpes s: retgulr, irulr, elliptil, rhombi, skew, or ordig to the rtio of their thikess to the lest iple dimesio s thi pltes, modertely thik pltes d thik pltes [1,, 3]. They re lso lssified bsed o the mteril elsti properties s homogeeous, heterogeeous, isotropi, isotropi, d orthotropi. Severl theories hve bee used to desribe the respose d behviours of pltes uder lods. They re: Kirhhoff plte theory whih is suitble for thi pltes, Reisser [] plte theory, Midli [] plte theory, Leviso s sher deformtio plte theories, Reddy s plte theory, Shimpi s refied plte theory [6], [7]. Theory of elstiity methods hve bee lso used to desribe plte behviour. Kirhhoff plte theory, lso lled the lssil plte theory hs bee foud to be suitble for desribig the behviours of thi pltes uder stti flexure, dymi flexure d buklig; d is dopted i this study. Reisser s stress bsed theory d Midli s first order sher deformtio theory re suitble for modertely thik pltes d tke out of the effet of sher deformtio o the plte respose uder trsverse lods. Other theories tht tke out of the sher deformtio of the plte d thus re suitble for desribig modertely thik pltes ilude sher deformtio plte theories proposed by Leviso; Reddy, d Shimpi s refied plte theories. Thik pltes re desribed usig the theories of elstiity for three dimesiol bodies. OI: /ijet/17/v9i6/ Vol 9 No 6 e 17-J

2 ISSN (Prit) : ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) II. METHOS OF SOLVING PLATE PROBLEMS Plte problems s desribed by vrious theories re boudry vlue problems. Two brod methods of solvig boudry vlue problems of pltes re: mthemtil or lytil methods, d umeril or pproximte methods. Mthemtil or lytil methods re methods used to obti solutios tht stisfy the goverig differetil equtios of the plte problem t ll poits o the plte domi, d o the plte boudries. They result i losed form solutios. They ilude: Nvier [8] method, Levy [9] method, seprtio of vribles method, itegrl trsform methods. Numeril methods re methods tht im to obti pproximte solutios to the goverig boudry vlue problem of pltes. They ilude: fiite differee methods [1], fiite elemet methods, boudry elemet methods, vritiol Ritz methods [11], vritiol Glerki method [1, 13], ollotio methods, Bubov-Glerki method, Ktorovih method [1, 1], Vlsov method [1, 1]. III. RESEARCH AIM AN OBJECTIVES The im of this study is to pply the Ktorovih-Vlsov method to the lysis of Kirhhoff CSCS pltes uder lierly distributed trsverse lods o the etire plte domi. The speifi objetives re: (i) to obti the totl potetil eergy futiol for the CSCS Kirhhoff plte bedig problem bsed o shpe futios obtied by Ktorovih d Vlsov pprohes. (ii) to obti the Euler-Lgrge-Ostrgrdsky differetil equtios for the extremiztio of the totl potetil eergy futiol. (iii) to solve the Euler-Lgrge-Ostrgrdsky differetil equtios subjet to the boudry oditios of the plte. (iv) to obti the bedig momet distributios. (v) to obti the defletios d bedig momets t the eter of the plte for vrious plte spet rtios. IV. THEORETICAL FRAMEWORK AN METHOOLOGY The Ktorovih-Vlsov method is vritiol method bsed o fidig the displemet futio tht miimizes the totl potetil eergy futiol. The displemet futio is ssumed to be i vribleseprble form s the sum of the produt of oordite futios i the x d y diretios, with the oordite futio i oe of the diretios hose s the eigefutios of vibrtig Euler-Beroulli bem with equivlet ed supports s the plte. The totl potetil eergy futiol for retgulr Kirhhoff plte uder pplied lod is give by: w w w w w (, ) R x y x y dxdy pw x y dxdy x y (1) R where R is the plte domi, w(x, y) is the trsverse defletio of the plte, p(x, y) is the trsverse lod distributio, is the Poisso s rtio, x d y re the iple Crtesi oordites d is the flexurl rigidity of the plte. I the Ktorovih-Vlsov method, the defletio futio w(x, y) is ssumed s the ifiite series: wxy (, ) g( y) f( x) () 1 where either g (y) or f (x) is kow to be the eigefutios of vibrtig Euler-Beroulli bem with the sme ed supports s the plte i the orrespodig oordite diretio; d the ukow futio sought i order to miimize the totl potetil eergy futiol. V. APPLICATION OF KANTOROVICH-VLASOV METHO PROBLEM ESCRIPTION Cosider retgulr Kirhhoff plte with iple dimesios b, simply supported o the opposite edges x =, d x =, d lmped o the other opposite edges y = d y = b, s show i Figure 1. OI: /ijet/17/v9i6/ Vol 9 No 6 e 17-J 18 33

3 ISSN (Prit) : ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) Figure 1: Retgulr Kirhhoff plte, with opposite edges lmped d simply supported uder lier distributed trsverse lod The etire plte domi is subjeted to lierly distributed trsverse lod of itesity p(x) = p x/ VI. EFLECTION FUNCTION Followig Ktorovih proedure, the defletio futio w(x, y) is ssumed to be lier ombitio of the sum of produts of lierly idepedet oordite (shpe or bsis) futios s Equtio (). The oordite futios f (x) i the x-diretio re hose by the Vlsov proedure s the vibrtig modl shpe futios of freely vibrtig prismti Euler-Beroulli bem simply supported t x =, d x =. For simply supported Euler-Beroulli bems, the eigefutios re: ( ) si x f x (3) = 1,, 3,, The the defletio futio beomes: x wxy (, ) g( y)si () 1 TOTAL POTENTIAL ENERGY FUNCTIONAL By substitutio of Equtio () ito the totl potetil eergy futiol, we obti: x x g( y)si g( y)si R 1 1 x x g ( y)os g( y)si 1 1 x x g( y)si dxdy p( x, y) g( y)si dxdy 1 () R 1 The lod distributio is expressed i terms of sigle sie series to obti: x pxy (, ) p si (6) 1 Usig Equtio (6) d simplifyig the totl potetil eergy expressio i Equtio (), we obti: b x x x g( y) si g ( y) si 1 g( y) g( y)si g ( y) os b 1 x x g ( y) g ( y)si dxdy x p( y) g( y)si dxdy (7) OI: /ijet/17/v9i6/ Vol 9 No 6 e 17-J 18 33

4 ISSN (Prit) : ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) Simplifyig further, b g( y) g( y) g( y) g( y) g( y) 1 b g( y) g( y) dy p( y) g( y) dy (8) Altertively, b g( y) g( y) g( y) g( y) 1 g ( y) g( y) g ( y) p( y) g( y) dy Equtio (9) is the totl potetil eergy futiol for the plte whose miimum with respet to g(y) we wish to F y, g( y) g( y) g ( y) is fid, i order to fully solve the Kirhhoff plte flexure problem. The itegrd F y, g( y) g( y), g ( y) ( g ( y)) ( g( y)) g ( y) g( y) g( y) g( y) g( y) p( y) g( y) EULER-LAGRANGE-OSTRAGRASKY IFFERENTIAL EQUATION The Euler-Lgrge-Ostrgrdsky differetil equtio of equilibrium whih represets the oditio for miimum of the totl potetil eergy futiol is give by: dy F d F d F g( y) dyg ( y) g ( y) Applyig Equtio (11) we obti: iv g g( y) g( y) g( y) g( y) g( y) p( y) Simplifitio yields: iv p ( y) g ( y) g( y) g( y) Equtio (13) is fourth order lier ordiry differetil equtio i g (y) s ukow; d is solved subjet to the boudry oditios of the plte support t the edges y = d y = b. LOA ISTRIBUTION SERIES COEFFICIENT P N The sigle Fourier sie series oeffiiet for the hydrostti lod distributio is, from sigle Fourier sie series theory, x p p( x, y)si dx (1) px x dx si (1) p x x si dx (16) (9) (1) (11) (1) (13) OI: /ijet/17/v9i6/ Vol 9 No 6 e 17-J

5 ISSN (Prit) : ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) 1 p os p ( 1) p (17) SOLUTION OF THE EULER-LAGRANGE-OSTRAGRASKY EQUATION Usig equtio (17), the Euler-Lgrge-Ostrgrdsky differetil equtio for the Kirhhoff plte problem beomes: 1 iv ( 1) p g ( y) g( y) g( y) (18) The homogeeous solutio is the solutio to the fourth order homogeeous lier ordiry differetil equtio (OE): iv g ( y) g( y) g( y) (19) Usig the method of tril futios, it is ssumed tht the homogeeous solutio to g (y) is i the form of the expoetil futio: sy g( y) expsy e () where s is udetermied prmeter we seek to determie. The, the homogeeous OE beomes: sy sy sy se se e (1) Simplifyig, sy s s e () For o-trivil solutios, sy e (3) The uxiliry (hrteristi) polyomil is: s s () s () The roots re s1, double root (6) s3, double root (7) The homogeeous solutio the beomes: ( ) 1 exp y exp y 3 exp y exp g y y y y (8) where 1,, 3 d re the itegrtio ostts. From the distributed lod whih is ot futio of y, we osider tht prtiulr solutios g p would ot deped upo y d thus: iv gp gp (9) The, by substitutio ito the ihomogeeous OE, Equtio (18), we obti, fter simplifitio; 1 ( 1) p gp ( ) (3) OI: /ijet/17/v9i6/ Vol 9 No 6 e 17-J

6 ISSN (Prit) : ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) The geerl solutio is obtied by usig the superpositio (lierity) priiple s the sum of the homogeeous d prtiulr solutios, s follows: ( ) 1 exp y exp y 3 exp y 1 y ( 1) p g y y yexp (31) ( ) ENFORCEMENT OF BOUNARY CONITIONS The boudry oditios t the lmped edges y =, d y = b re the four oditios for eh vlue of : g ( y ) (3) g ( y b ) (33) g ( y ) (3) g ( y b ) (3) The eforemet/pplitio of the four boudry oditios o g (y) results i system of four equtios i terms of the four itegrtio ostts for eh vlue. These four equtios re the solved simulteously to obti the four ostts of itegrtio 1,, 3 d t the th term. Eforemet of the boudry oditios yield: ( 1) p 1 ( ) (36) (37) 1 3 ( 1) b b b b 1 3 e e be be p ( ) (38) where b b b b 1 3 e e (1 b) e (1 b) e (39) () Let p ( 1) p ( ) The, the system of four lgebri equtios be expressed i mtrix form s: b b b b p e e be be 3 1 b b b b e e be be This is solved usig Crmer s rule to obti the four ostts s follows: where 1 3 (1) () 1 (3) () 3 () (6) OI: /ijet/17/v9i6/ Vol 9 No 6 e 17-J

7 ISSN (Prit) : ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) Thus, The, b b b b e e be be b b b b e e (1 be ) (1 be ) p b b b p e be be b b b e (1 b) e (1 b) e 1 p 1 1 b b b e p be be b b b e (1 b) e (1 b) e 1 1 p 1 b b b e e p be b b b e e (1 b) e 1 1 b b b 1 e e be p b b b e e (1 b) e 1 ( ) b b b p b b e b e b e 1 b b ( ) b e e b b b ( ) p e b e b e b b b b ( ) b e e b b b ( 1) 1 p e b e e b 3 b b ( ) b e e b b b (1 ) 1 p e b e e b b b ( ) b e e 1 y y y y 1 3 p (7) (8) (9) () (1) () (3) () () wxy (, ) e e ye ye p si x (6) These ostts of itegrtio re expressed i terms of the spet rtio b r, s follows: r r r p 1r ( r) e (1 r) e (1 r) e 1 r r ( r) e e r r r p e (1 r) e (1 r) e r ( r) r r ( r) e e r r r 3 r r p e (r 1) e e 1r ( r) e e (7) (8) (9) OI: /ijet/17/v9i6/ Vol 9 No 6 e 17-J

8 ISSN (Prit) : ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) r r r r r p e (1 r) e e 1 r ( r) e e (6) where d BENING MOMENT ISTRIBUTIONS The bedig momet distributios re obtied from the bedig momet defletio equtios: M w ( w ) (61) xx xx yy M w ( w ) (6) yy yy xx w y si xx 1 x 1 w x e e y y 3 y ye ye p (63) w y y y y y y 1 3 ( ) ( xx ) si y 1 w e e ye e ye e x (6) The defletio equtio, Equtio (6) is used to determie the defletio of the eter of squre Kirhhoff plte uder hydrostti lods for iresig iteger vlues of, d the results re tbulted i Tble 1. The tble revels the rpidly overget property of the series with stisftory results obtied with = 3. Tble shows the overged results for the defletio d bedig momet t the eter of squre Kirhhoff plte uder hydrostti lod. The overgee study of the series for bedig momets is show i Tble 3. Tble shows the vritio of eter bedig momets M xx, d M yy with the plte spet rtio, while Tble presets the vritio of eter defletio with plte spet rtio. TABLE 1: Covergee study for eter defletios of squre Kirhhoff (CSCS) plte uder hydrostti lod Number of terms, Ceter defletio w p TABLE : Coverged results for defletios d bedig momets t the eter of squre Kirhhoff (CSCS) pltes uder hydrostti lod p b w Mxx p yy M p OI: /ijet/17/v9i6/ Vol 9 No 6 e 17-J 18 3

9 ISSN (Prit) : ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) TABLE 3: Covergee study for bedig momet oeffiiets t the eter of squre Kirhhoff CSCS plte uder hydrostti lod. px px ( ) (.3) Number of terms, M xx xx p M yy TABLE : Coverged solutios for bedig momets t the eter of retgulr Kirhhoff CSCS plte uder hydrostti lods px px ( ) b Preset study Timosheko d Woiowsky-Krieger M xx M yy M xx M yy..7p b.1p b.7p b.1p b.7.11p b.p b.11p b.1p b 1..1p.17p.13p.17p 1..1p.1p.1p.1p 1..3p.3p.3p.3p.3p.p.3p.p.63p.19p.63p.19p TABLE : Coverged solutios for Kirhhoff (CSCS) plte uder hydrostti lod px ( ), = 11 b p w VII. ISCUSSION yy p px The Ktorovih-Vlsov method hs bee suessfully implemeted to solve the boudry vlue problem px of Kirhhoff (CSCS) pltes uder lierly distributed trsverse lod px ( ), over the etire plte domi. The boudry vlue problem ws preseted i vritiol form so tht the problem beme oe of fidig the defletio futio tht miimized the totl potetil eergy futiol. Bsed o Vlsov d Ktorovih pprohes, the defletio futio ws osidered s Equtio (). This resulted i the totl potetil eergy futiol give by Equtio (9). The Euler-Lgrge-Ostrgrdsky differetil equtio for the problem ws obtied s Equtio (13) for y distributio of pplied trsverse lod d Equtio (18) for the speifi se of lierly distributed trsverse lod osidered i this study. The geerl solutio to the Euler-Lgrge-Ostrgrdsky equtio ws obtied s Equtio (31) usig the lierity priiples for solvig o-homogeeous ordiry differetil equtios. Boudry oditios were efored to obti the geerl solutio for the defletio futio s Equtio (6) where the itegrtio ostts were obtied i terms of the plte spet rtios s Equtios (7) (6). Bedig momet urvture reltios were used to obti the OI: /ijet/17/v9i6/ Vol 9 No 6 e 17-J 18 31

10 ISSN (Prit) : ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) bedig momet distributios M xx d M yy from Equtios (61), (6), (63) d (6). The defletio d bedig momet expressios were foud to be sigle series expressios. Vlues of the defletio d bedig momets were lulted for vrious spet rtios of the plte d preseted i Tbles, d. The eter defletios d bedig momets for squre Kirhhoff CSCS plte uder the lierly distributed lod osidered were lso obtied for vrious terms of the sigle series d preseted i Tbles 1 d 3 i order to study the overgee properties of the series for defletio d the series for bedig momet. Tble 1 shows tht the series for defletio overges rpidly d resobly urte results for defletio re obtied by usig =. Similrly, the series for bedig momets overge less rpidly, but resobly urte results re obtied for squre pltes usig = 7 i the series for both M xx d M yy. Tbles, 3 d whih preset overged results for defletio d bedig momets for vrious plte spet rtios illustrte the good greemet betwee the results of the preset study d Timosheko d Woiowsky-Krieger s solutio for the sme problem. Timosheko d Woiowsky-Krieger obtied their solutios by lier superpositio of the solutio for simply supported retgulr Kirhhoff plte uder the lierly distributed trsverse lod p(x) = p x/ with the solutio for the sme problem uder pplied distributio of torque log the fixed edges (y = d y = b), where the torque is of suh mgitude tht rottiol displemet is fully restried t the fixed edges. VIII. CONCLUSIONS From the study, the followig olusios re mde: (i) the Ktorovih-Vlsov method be suessfully implemeted for the solutio of the retgulr Kirhhoff CSCS plte uder lierly distributed trsverse lod o the etire plte regio (ii) the solutios obtied for the defletio w(x, y) d bedig momet expressios M xx d M yy re sigle series otiig expoetil futios (iii) the series for defletio d bedig momet expressios re overget but the series for defletio is more rpidly overget th the series for bedig momets (iv) the overged results obtied for the defletios d bedig momets t the eter of the plte for vrious plte spet rtios gree exelletly well with the solutios preseted by Timosheko d Woiowsky-Krieger. REFERENCES [1] K. Chdrshekhr Theory of Pltes, Uiversity Press (Idi) Limited, Hyderbd, 11. [] R. Szilrd Theories d Applitios of Plte Alysis: Clssil, Numeril d Egieerig Methods, Joh Wiley d Sos I.. [3] S. Timosheko d S. Woiowsky-Krieger Theory of Pltes d Shells, 3rd Editio, MGrw Hill Book Co. New York, 199. [] E. Reisser The effet of trsverse sher deformtio o the bedig of elsti pltes, Jourl of Applied Mehis, Vol 1, pp , 19. [] R.. Midli Ifluee of rotry ierti d sher o flexurl motios of isotropi, elsti pltes, J. App. Meh. Vol 18, No 1, pp 31-38, 191. [6] R.P. Shimpi Refied plte theory d its vrits, AIAA Jourl Vol, No 1, pp , Jury. [7] Y. Suetke Plte bedig lysis by usig modified plte theory, CMES Vol 11, No 3, pp 13-11, Teho Siee Press. 6. [8] C.L.M.N. Nvier Bulleti des siee de l soietie philomrtrique de Pris, pp 9-1, Extrit des reherhé sur l flexio des ples elstiques, 183. [9] M. Levy Memoire sur l theorie des plques elstiques ples, Jourl de Mthemtiques Pures et Appliqueés, Vol 3, pp , [1] J.C. Ezeh, O.M. Iberugbulem d C.I. Oyehere Pure bedig lysis of thi retgulr flt pltes usig ordiry fiite differee method, Itertiol Jourl of Emergig Tehology d Adved Egieerig (IJETAE) Volume 3, Issue 3, pp. - 3, Mrh 13. [11] C.H. Agim, C.A. Chidolue d C.A. Ezegu Applitio of iret Vritiol Method i the Alysis of Isotropi, thi, retgulr pltes, ARPN Jourl of Egieerig d Applied Siees, Vol 7 No 9. Asi Reserh Publishig Network pp September 1. [1] N.N. Osdebe, C.C. Ike, H.N. Oh, C.U. Nwoji d F.O. Okfor Applitio of the Glerki-Vlsov method for the lysis of simply supported retgulr Kirhhoff pltes uder uiform lods, Nigeri Jourl of Tehology (NIJOTECH), Vol 3, No, pp , Otober 16. [13] C.U. Nwoji, B.O. Mm, C.C. Ike d H.N. Oh Glerki-Vlsov method for the flexurl lysis of retgulr Kirhhoff pltes with lmped d simply supported edges, IOSR Jourl of Mehil d Civil Egieerig (IOSR JMCE) Volume 1, Issue, Versio 7, pp 61-7, Mrh 17. [1] C.U. Nwoji, B.O. Mm, H.N. Oh d C.C. Ike Ktorovih-Vlsov method for simply supported pltes uder uiformly distributed lods, Itertiol Jourl of Civil, Mehil d Eergy Siee (IJCMES) Volume 3, Issue, pp 69-77, Mrh-April 17. [1] C.C. Ike Ktorovih-Euler Lgrge-Glerki method for bedig lysis of thi pltes, Nigeri Jourl of Tehology, NIJOTECH, Volume 36, No, pp 31-36, April 17. OI: /ijet/17/v9i6/ Vol 9 No 6 e 17-J 18 3

11 ISSN (Prit) : ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) AUTHOR PROFILE Hygius Nwkwo Oh (B.Eg, M.Eg, Ph) Bor i Nigeri o My th 199, I obtied my B.Eg. (Civil Egieerig) i After brief work i Lgos Nigeri, I trvelled to Fre i 1987 where I obtied M.Eg (Civil Egieerig) t INSA de Lyo d Ph. (Struturl Mehis) t CNAM Pris i The I worked s reserher t Reserh d evelopmet (R&) uit of RENAULT Motors Reuil Mlmiso Fre, eprtmet of Egieerig Uiversity of Mhester Egld d Behis Osiride of FIAT Motors Torio Itly for oe yer eh. From Jury 1996 to dte s leturer i Civil Egieerig eprtmet of Uiversity of Nigeri, Nsukk, I teh d supervise both udergrdute d Postgrdute studets d rry out reserh tivities i mehis d behvior of strutures Egr r Bejmi Okwudili Mm I got my Ph i the yer 9 i the Uiversity of Nigeri, Nsukk. My mster s degree i ivil Struturl Egieerig i 199 i the sme Shool. My first degree i Civil Egieerig i the Ambr Stte Uiversity of Tehology, Eugu. I hve bee leturig i the eprtmet of Civil Egieerig, Uiversity of Nigeri, Nsukk from 199 till dte. Egr r. Chrles C. Ike is leturer t the Eugu Stte Uiversity of Siee d Tehology, Eugu, Nigeri. He holds Ph i struturl egieerig, d is member of the Nigeri Soiety of Egieers. He is registered with the Couil for the Regultio of Egieerig i Nigeri (COREN). He hs umerous publitios i reputble itertiol jourls. Egr r. Clifford U. Nwoji obtied his ivil egieerig edutio from Uiversity of Nigeri Nsukk. He is member of Nigeri Soiety of Egieers d lso registered egieer with Couil for Regultio of Egieerig i Nigeri. He hs substtil idustril experiee. Sie obtiig his higher degrees i egieerig he hs bee tehig struturl egieerig ourses t both udergrdute d postgrdute levels. He hs suessfully supervised higher degree theses. He hs to his redit good umber of published reserh ppers i reputble itertiol jourls. OI: /ijet/17/v9i6/ Vol 9 No 6 e 17-J 18 33